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If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N. For example, 6 is highly composite because d(6) = 4, and for n = 1,2,3,4,5, you get d(n) = 1,2,2,3,2, respectively, which are all less than 4. +A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. +Ramanujan wrote a paper on highly composite numbers in 1915. +The mathematician Jean-Pierre Kahane suggested that Plato must have known about highly composite numbers as he deliberately chose such a number, 5040 (= 7!), as the ideal number of citizens in a city. Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040. + +== Examples == +The first 41 highly composite numbers are listed in the table below (sequence A002182 in the OEIS). The number of divisors is given in the column labeled d(n). Asterisks indicate superior highly composite numbers. + +The divisors of the first 20 highly composite numbers are shown below. + +The table below shows all 72 divisors of 10080 by writing it as a product of two numbers in 36 different ways. + +The 15,000-th highly composite number is the product of 230 primes: + + + + + + a + + 0 + + + 14 + + + + a + + 1 + + + 9 + + + + a + + 2 + + + 6 + + + + a + + 3 + + + 4 + + + + a + + 4 + + + 4 + + + + a + + 5 + + + 3 + + + + a + + 6 + + + 3 + + + + a + + 7 + + + 3 + + + + a + + 8 + + + 2 + + + + a + + 9 + + + 2 + + + + a + + 10 + + + 2 + + + + a + + 11 + + + 2 + + + + a + + 12 + + + 2 + + + + a + + 13 + + + 2 + + + + a + + 14 + + + 2 + + + + a + + 15 + + + 2 + + + + a + + 16 + + + 2 + + + + a + + 17 + + + 2 + + + + a + + 18 + + + 2 + + + + a + + 19 + + + + a + + 20 + + + + a + + 21 + + + ⋯ + + a + + 229 + + + , + + + {\displaystyle a_{0}^{14}a_{1}^{9}a_{2}^{6}a_{3}^{4}a_{4}^{4}a_{5}^{3}a_{6}^{3}a_{7}^{3}a_{8}^{2}a_{9}^{2}a_{10}^{2}a_{11}^{2}a_{12}^{2}a_{13}^{2}a_{14}^{2}a_{15}^{2}a_{16}^{2}a_{17}^{2}a_{18}^{2}a_{19}a_{20}a_{21}\cdots a_{229},} + + +where + + + + + a + + n + + + + + {\displaystyle a_{n}} + + is the + + + + n + + + {\displaystyle n} + +th successive prime number, and all omitted terms (a22 to a228) are factors with exponent equal to one (i.e. the number is + + + + + 2 + + 14 + + + × + + 3 + + 9 + + + × + + 5 + + 6 + + + × + ⋯ + × + 1451 + + + {\displaystyle 2^{14}\times 3^{9}\times 5^{6}\times \cdots \times 1451} + +). More concisely, it is the product of seven distinct primorials: + + + + + + b + + 0 + + + 5 + + + + b + + 1 + + + 3 + + + + b + + 2 + + + 2 + + + + b + + 4 + + + + b + + 7 + + + + b + + 18 + + + + b + + 229 + + + , + + + {\displaystyle b_{0}^{5}b_{1}^{3}b_{2}^{2}b_{4}b_{7}b_{18}b_{229},} + + +where + + + + + b + + n + + + + + {\displaystyle b_{n}} + + is the primorial + + + + + a + + 0 + + + + a + + 1 + + + ⋯ + + a + + n + + + + + {\displaystyle a_{0}a_{1}\cdots a_{n}} + +. + +== Prime factorization == + +Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, but not too many of the same. By the fundamental theorem of arithmetic, every positive integer n has a unique prime factorization: + + + + + n + = + + p + + 1 + + + + c + + 1 + + + + + × + + p + + 2 + + + + c + + 2 + + + + + × + ⋯ + × + + p + + k + + + + c + + k + + + + + + + {\displaystyle n=p_{1}^{c_{1}}\times p_{2}^{c_{2}}\times \cdots \times p_{k}^{c_{k}}} + + +where + + + + + p + + 1 + + + < + + p + + 2 + + + < + ⋯ + < + + p + + k + + + + + {\displaystyle p_{1} 0). A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the region of convergence for causal systems is not the same as that of anticausal systems. +The following functions and variables are used in the table below: + +δ represents the Dirac delta function. +u(t) represents the Heaviside step function. Literature may refer to this by other notation, including + + + + 1 + ( + t + ) + + + {\displaystyle 1(t)} + + or + + + + H + ( + t + ) + + + {\displaystyle H(t)} + +. +Γ(z) represents the Gamma function. +γ is the Euler–Mascheroni constant. +t is a real number. It typically represents time, although it can represent any independent dimension. +s is the complex frequency domain parameter, and Re(s) is its real part. +n is an integer. +α, τ, and ω are real numbers. +q is a complex number. + + +== Table == + + +== See also == +List of Fourier transforms + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_Lie_groups_topics-0.md b/data/en.wikipedia.org/wiki/List_of_Lie_groups_topics-0.md new file mode 100644 index 000000000..91cd7dc38 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_Lie_groups_topics-0.md @@ -0,0 +1,168 @@ +--- +title: "List of Lie groups topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_Lie_groups_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:25.912263+00:00" +instance: "kb-cron" +--- + +This is a list of Lie group topics, by Wikipedia page. + + +== Examples == +See Table of Lie groups for a list + +General linear group, special linear group +SL2(R) +SL2(C) +Unitary group, special unitary group +SU(2) +SU(3) +Orthogonal group, special orthogonal group +Rotation group SO(3) +SO(8) +Generalized orthogonal group, generalized special orthogonal group +The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions. It is the group of hyperbolic motions of the Poincaré disk model of the Hyperbolic plane. +Lorentz group +Spinor group +Symplectic group +Exceptional groups +G2 +F4 +E6 +E7 +E8 +Affine group +Euclidean group +Poincaré group +Heisenberg group + + +== Lie algebras == +Commutator +Jacobi identity +Universal enveloping algebra +Baker–Campbell–Hausdorff formula +Casimir invariant +Killing form +Kac–Moody algebra +Affine Lie algebra +Loop algebra +Graded Lie algebra + + +== Foundational results == +One-parameter group, One-parameter subgroup +Matrix exponential +Infinitesimal transformation +Lie's third theorem +Maurer–Cartan form +Cartan's theorem +Cartan's criterion +Local Lie group +Formal group law +Hilbert's fifth problem +Hilbert–Smith conjecture +Lie group decompositions +Real form (Lie theory) +Complex Lie group +Complexification (Lie group) + + +== Semisimple theory == +Simple Lie group +Compact Lie group, Compact real form +Semisimple Lie algebra +Root system +Simply laced group +ADE classification +Maximal torus +Weyl group +Dynkin diagram +Weyl character formula + + +== Representation theory == + +Representation of a Lie group +Representation of a Lie algebra +Adjoint representation of a Lie group +Adjoint representation of a Lie algebra +Unitary representation +Weight (representation theory) +Peter–Weyl theorem +Borel–Weil theorem +Kirillov character formula +Representation theory of SU(2) +Representation theory of SL2(R) + + +== Applications == + + +=== Physical theories === +Pauli matrices +Gell-Mann matrices +Poisson bracket +Noether's theorem +Wigner's classification +Gauge theory +Grand Unified Theory +Supergroup +Lie superalgebra +Twistor theory +Anyon +Witt algebra +Virasoro algebra + + +=== Geometry === +Erlangen programme +Homogeneous space +Principal homogeneous space +Invariant theory +Lie derivative +Darboux derivative +Lie groupoid +Lie algebroid + + +=== Discrete groups === +Lattice (group) +Lattice (discrete subgroup) +Frieze group +Wallpaper group +Space group +Crystallographic group +Fuchsian group +Modular group +Congruence subgroup +Kleinian group +Discrete Heisenberg group +Clifford–Klein form + + +=== Algebraic groups === +Borel subgroup +Arithmetic group + + +== Special functions == +Dunkl operator + + +=== Automorphic forms === +Modular form +Langlands program + + +== People == +Sophus Lie (1842 – 1899) +Wilhelm Killing (1847 – 1923) +Élie Cartan (1869 – 1951) +Hermann Weyl (1885 – 1955) +Harish-Chandra (1923 – 1983) +Lajos Pukánszky (1928 – 1996) +Bertram Kostant (1928 – 2017) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns-0.md b/data/en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns-0.md new file mode 100644 index 000000000..29b5754ec --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns-0.md @@ -0,0 +1,25 @@ +--- +title: "List of Martin Gardner Mathematical Games columns" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:39.569085+00:00" +instance: "kb-cron" +--- + +Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for Scientific American magazine. During the next 5.5 years, until June 1986, Gardner wrote 9 more columns, bringing his total to 297. During this period other authors wrote most of the columns. In 1981, Gardner's column alternated with a new column by Douglas Hofstadter called "Metamagical Themas" (an anagram of "Mathematical Games"). The table below lists Gardner's columns. +Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by "[cover]" in the table with a hyperlink to the cover. + + +== Other articles by Gardner == +Gardner wrote 5 other articles for Scientific American. His flexagon article in December 1956 was in all but name the first article in the series of Mathematical Games columns and led directly to the series which began the following month. These five articles are listed below. + + +== References == + + +== External links == +A Quarter Century of Recreational Mathematics, by Martin Gardner preserved at the Internet Archive +A subject index for the fifteen books of Martin Gardner's Mathematical Games columns +The Top 10 Martin Gardner Scientific American Articles \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers-0.md b/data/en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers-0.md new file mode 100644 index 000000000..d32246817 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers-0.md @@ -0,0 +1,30 @@ +--- +title: "List of Mersenne primes and perfect numbers" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:25.190446+00:00" +instance: "kb-cron" +--- + +Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1. The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89. +Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. +Euclid proved c. 300 BCE that every prime expressed as Mp = 2p − 1 has a corresponding perfect number Mp × (Mp+1)/2 = 2p − 1 × (2p − 1). For example, the Mersenne prime 22 − 1 = 3 leads to the corresponding perfect number 22 − 1 × (22 − 1) = 2 × 3 = 6. In 1747, Leonhard Euler completed what is now called the Euclid–Euler theorem, showing that these are the only even perfect numbers. As a result, there is a one-to-one correspondence between Mersenne primes and even perfect numbers, so a list of one can be converted into a list of the other. +It is currently an open problem whether there are infinitely many Mersenne primes and even perfect numbers. The density of Mersenne primes is the subject of the Lenstra–Pomerance–Wagstaff conjecture, which states that the expected number of Mersenne primes less than some given x is (eγ / log 2) × log log x, where e is Euler's number, γ is Euler's constant, and log is the natural logarithm. It is widely believed, but not proven, that no odd perfect numbers exist; numerous restrictive conditions have been proven, including a lower bound of 101500. +The following is a list of all 52 currently known (as of November 2025) Mersenne primes and corresponding perfect numbers, along with their exponents p. The largest 18 of these have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS; their discoverers are listed as "GIMPS / name", where the name is the person who supplied the computer that made the discovery. New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers. Due to this efficiency, the largest known prime number has often been a Mersenne prime. +All possible exponents up to the 50th (p = 77,232,917) have been tested and verified by GIMPS as of September 2025. Ranks 51 and up are provisional, and may change in the unlikely event that additional primes are discovered between the currently listed ones. Later entries are extremely long, so only the first and last six digits of each number are shown, along with the number of decimal digits. + + +== Notes == + + +== References == + + +== External links == +OEIS sequence A000043 (Corresponding exponents p) +OEIS sequence A000396 (Perfect numbers) +OEIS sequence A000668 (Mersenne primes) +List on GIMPS, with the full values of large numbers Archived 2020-06-07 at the Wayback Machine +A technical report on the history of Mersenne numbers, by Guy Haworth Archived 2021-10-13 at the Wayback Machine \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_NP-complete_problems-0.md b/data/en.wikipedia.org/wiki/List_of_NP-complete_problems-0.md new file mode 100644 index 000000000..a44cc7d86 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_NP-complete_problems-0.md @@ -0,0 +1,225 @@ +--- +title: "List of NP-complete problems" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/List_of_NP-complete_problems" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:51.446008+00:00" +instance: "kb-cron" +--- + +This is a list of some of the more commonly known problems that are NP-complete when expressed as decision problems. As there are thousands of such problems known, this list is in no way comprehensive. Many problems of this type can be found in Garey & Johnson (1979). + +== Graphs and hypergraphs == +Graphs occur frequently in everyday applications. Examples include biological or social networks, which contain hundreds, thousands and even billions of nodes in some cases (e.g. Facebook or LinkedIn). + +1-planarity +3-dimensional matching +Bandwidth problem +Bipartite dimension +Capacitated minimum spanning tree +Route inspection problem (also called Chinese postman problem) for mixed graphs (having both directed and undirected edges). The program is solvable in polynomial time if the graph has all undirected or all directed edges. Variants include the rural postman problem. +Clique cover problem +Clique problem +Complete coloring, a.k.a. achromatic number +Cycle rank +Degree-constrained spanning tree +Domatic number +Dominating set, a.k.a. domination number +NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. +Feedback vertex set +Feedback arc set +Graph coloring +Graph homomorphism problem +Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms of the number of edges between parts. +Grundy number of a directed graph. +Hamiltonian completion +Hamiltonian path problem, directed and undirected. +Induced subgraph isomorphism problem +Graph intersection number +Longest path problem +Maximum bipartite subgraph or (especially with weighted edges) maximum cut. +Maximum common subgraph isomorphism problem +Maximum independent set +Maximum Induced path +Minimum maximal independent set a.k.a. minimum independent dominating set +NP-complete special cases include the minimum maximal matching problem, which is essentially equal to the edge dominating set problem (see above). +Metric dimension of a graph +Metric k-center +Minimum degree spanning tree +Minimum k-cut +Minimum k-spanning tree +Minor testing (checking whether an input graph + + + + G + + + {\displaystyle G} + + contains an input graph + + + + H + + + {\displaystyle H} + + as a minor); the same holds with topological minors +Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. (The minimum spanning tree for an entire graph is solvable in polynomial time.) +Modularity maximization +Monochromatic triangle +Pathwidth, or, equivalently, interval thickness, and vertex separation number +Rank coloring +k-Chinese postman +Shortest total path length spanning tree +Slope number two testing +Recognizing string graphs +Subgraph isomorphism problem +Treewidth +Testing whether a tree may be represented as Euclidean minimum spanning tree +Vertex cover +Minimum Wiener connector problem + +== Mathematical programming == +3-partition problem +Bin packing problem +Bottleneck traveling salesman +Uncapacitated facility location problem +Flow Shop Scheduling Problem +Generalized assignment problem +Integer programming. The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete +Some problems related to job-shop scheduling +Knapsack problem, quadratic knapsack problem, and several variants +Some problems related to multiprocessor scheduling +Numerical 3-dimensional matching +Open-shop scheduling +Partition problem +Quadratic assignment problem +Quadratic programming (NP-hard in some cases, P if convex) +Subset sum problem +Variations on the traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. + +== Formal languages and string processing == +Closest string +Longest common subsequence problem over multiple sequences +The bounded variant of the Post correspondence problem +Shortest common supersequence over multiple sequences +Extension of the string-to-string correction problem + +== Games and puzzles == +Bag (Corral) +Battleship +Bulls and Cows, marketed as Master Mind: certain optimisation problems but not the game itself. +Edge-matching puzzles +Fillomino +(Generalized) FreeCell +Goishi Hiroi +Hashiwokakero +Heyawake +(Generalized) Instant Insanity +Kakuro (Cross Sums) +Kingdomino +Kuromasu (also known as Kurodoko) +LaserTank +Lemmings (with a polynomial time limit) +Light Up +Mahjong solitaire (with looking below tiles) +Masyu +Minesweeper Consistency Problem (but see Scott, Stege, & van Rooij) +Nonograms +Numberlink +Nurikabe +(Generalized) Pandemic +Peg solitaire +n-Queens completion +Optimal solution for the N×N×N Rubik's Cube +SameGame +Shakashaka +Slither Link on a variety of grids +(Generalized) Sudoku +Tatamibari +Tentai Show +Problems related to Tetris +Verbal arithmetic + +== Other == +Berth allocation problem +Betweenness +Assembling an optimal Bitcoin block. +Boolean satisfiability problem (SAT). There are many variations that are also NP-complete. An important variant is where each clause has exactly three literals (3SAT), since it is used in the proof of many other NP-completeness results. +Circuit satisfiability problem +Conjunctive Boolean query +Cyclic ordering +Exact cover problem. Remains NP-complete for 3-sets. Solvable in polynomial time for 2-sets (this is a matching). +Finding the global minimum solution of a Hartree-Fock problem +Upward planarity testing +Hospitals-and-residents problem with couples +Knot genus +Latin square completion (the problem of determining if a partially filled square can be completed) +Maximum 2-satisfiability +Maximum volume submatrix – Problem of selecting the best conditioned subset of a larger + + + + m + × + n + + + {\displaystyle m\times n} + + matrix. This class of problem is associated with Rank revealing QR factorizations and D optimal experimental design. +Minimal addition chains for sequences. The complexity of minimal addition chains for individual numbers is unknown. +Modal logic S5-Satisfiability +Pancake sorting distance problem for strings +Solubility of two-variable quadratic polynomials over the integers. Given positive integers + + + + + A + , + B + , + C + + + + {\displaystyle \textstyle A,B,C} + +, decide existence of positive integers + + + + x + , + y + + + {\displaystyle x,y} + + such that + + + + A + + x + + 2 + + + + + B + y + − + C + = + 0 + + + {\displaystyle Ax^{2}+By-C=0} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_NP-complete_problems-1.md b/data/en.wikipedia.org/wiki/List_of_NP-complete_problems-1.md new file mode 100644 index 000000000..3745cb94c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_NP-complete_problems-1.md @@ -0,0 +1,117 @@ +--- +title: "List of NP-complete problems" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/List_of_NP-complete_problems" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:51.446008+00:00" +instance: "kb-cron" +--- + +By the same article existence of bounded modular square roots with arbitrarily composite modulus. Given positive integers + + + + + A + , + B + , + C + ≥ + 0 + + + + {\displaystyle \textstyle A,B,C\geq 0} + +, decide existence of an integer + + + + x + ∈ + [ + 0 + , + C + ] + + + {\displaystyle x\in [0,C]} + + such that + + + + + x + + 2 + + + ≡ + A + + mod + + B + + + + + {\displaystyle x^{2}\equiv A{\bmod {B}}} + +. The problem remains NP-complete even if a prime factorization of + + + + B + + + {\displaystyle B} + + is provided. +Serializability of database histories +Set cover (also called "minimum cover" problem). This is equivalent, by transposing the incidence matrix, to the hitting set problem. +Set packing +Set splitting problem +Scheduling to minimize weighted completion time +Block Sorting (Sorting by Block Moves) +Sparse approximation +Variations of the Steiner tree problem. Specifically, with the discretized Euclidean metric, rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. +Three-dimensional Ising model + +== See also == +Existential theory of the reals § Complete problems +Karp's 21 NP-complete problems +List of PSPACE-complete problems +Reduction (complexity) + +== Notes == + +== References == +General + +Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences (1st ed.). New York: W. H. Freeman and Company. ISBN 9780716710455. MR 0519066. OCLC 247570676.. This book is a classic, developing the theory, then cataloguing many NP-Complete problems. +Cook, S.A. (1971). "The complexity of theorem proving procedures". Proceedings, Third Annual ACM Symposium on the Theory of Computing, ACM, New York. pp. 151–158. doi:10.1145/800157.805047. +Karp, Richard M. (1972). "Reducibility among combinatorial problems". In Miller, Raymond E.; Thatcher, James W. (eds.). Complexity of Computer Computations. Plenum. pp. 85–103. +Dunne, P.E. "An annotated list of selected NP-complete problems". COMP202, Dept. of Computer Science, University of Liverpool. Retrieved 21 June 2008. +Crescenzi, P.; Kann, V.; Halldórsson, M.; Karpinski, M.; Woeginger, G. "A compendium of NP optimization problems". KTH NADA, Stockholm. Retrieved 21 June 2008. +Dahlke, K. "NP-complete problems". Math Reference Project. Retrieved 21 June 2008. +Specific problems + +Friedman, E (2002). "Pearl puzzles are NP-complete". Stetson University, DeLand, Florida. Retrieved 9 March 2026. +Grigoriev, A; Bodlaender, H L (2007). "Algorithms for graphs embeddable with few crossings per edge". Algorithmica. 49 (1): 1–11. CiteSeerX 10.1.1.61.3576. doi:10.1007/s00453-007-0010-x. MR 2344391. S2CID 8174422. +Hartung, S; Nichterlein, A (2012). "NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed Acyclic Graphs". How the World Computes. Lecture Notes in Computer Science. Vol. 7318. Springer, Berlin, Heidelberg. pp. 283–292. CiteSeerX 10.1.1.377.2077. doi:10.1007/978-3-642-30870-3_29. ISBN 978-3-642-30869-7. S2CID 6112925. +Holzer, Markus; Ruepp, Oliver (2007). "The Troubles of Interior Design–A Complexity Analysis of the Game Heyawake" (PDF). Proceedings, 4th International Conference on Fun with Algorithms, LNCS 4475. Springer, Berlin/Heidelberg. pp. 198–212. doi:10.1007/978-3-540-72914-3_18. ISBN 978-3-540-72913-6. +Kaye, Richard (2000). "Minesweeper is NP-complete". Mathematical Intelligencer. 22 (2): 9–15. doi:10.1007/BF03025367. S2CID 122435790. Further information available online at Richard Kaye's Minesweeper pages. +Kashiwabara, T.; Fujisawa, T. (1979). "NP-completeness of the problem of finding a minimum-clique-number interval graph containing a given graph as a subgraph". Proceedings. International Symposium on Circuits and Systems. pp. 657–660. +Ohtsuki, Tatsuo; Mori, Hajimu; Kuh, Ernest S.; Kashiwabara, Toshinobu; Fujisawa, Toshio (1979). "One-dimensional logic gate assignment and interval graphs". IEEE Transactions on Circuits and Systems. 26 (9): 675–684. doi:10.1109/TCS.1979.1084695. +Lengauer, Thomas (1981). "Black-white pebbles and graph separation". Acta Informatica. 16 (4): 465–475. doi:10.1007/BF00264496. S2CID 19415148. +Arnborg, Stefan; Corneil, Derek G.; Proskurowski, Andrzej (1987). "Complexity of finding embeddings in a k-tree". SIAM Journal on Algebraic and Discrete Methods. 8 (2): 277–284. doi:10.1137/0608024. +Cormode, Graham (2004). "The hardness of the lemmings game, or Oh no, more NP-completeness proofs". Proceedings of Third International Conference on Fun with Algorithms (FUN 2004). pp. 65–76. + +== External links == +A compendium of NP optimization problems +Graph of NP-complete Problems \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_PPAD-complete_problems-0.md b/data/en.wikipedia.org/wiki/List_of_PPAD-complete_problems-0.md new file mode 100644 index 000000000..977447f51 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_PPAD-complete_problems-0.md @@ -0,0 +1,46 @@ +--- +title: "List of PPAD-complete problems" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_PPAD-complete_problems" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:19.979284+00:00" +instance: "kb-cron" +--- + +This is a list of PPAD-complete problems. + + +== Fixed-point theorems == +Sperner's lemma +Brouwer fixed-point theorem +Kakutani fixed-point theorem + + +== Game theory == +Nash equilibrium +Core of Balanced Games + + +== Equilibria in game theory and economics == +Fisher market equilibria +Arrow-Debreu equilibria +Approximate Competitive Equilibrium from Equal Incomes +Finding clearing payments in financial networks + + +== Graph theory == +Fractional stable paths problems +Fractional hypergraph matching (see also the NP-complete Hypergraph matching) +Fractional strong kernel + + +== Miscellaneous == +Scarf's lemma +Fractional bounded budget connection games + + +== References == +Papadimitriou, Christos (1994). "On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence". Journal of Computer and System Sciences. 48 (3): 498–532. CiteSeerX 10.1.1.321.7008. doi:10.1016/S0022-0000(05)80063-7. Paper available online at Papadimitriou's Homepage. +C. Daskalakis, P. W. Goldberg and C.H. Papadimitriou (2009). "The Complexity of Computing a Nash Equilibrium". SIAM Journal on Computing. 39 (3): 195–259. CiteSeerX 10.1.1.68.6111. doi:10.1137/070699652. +Xi Chen; Xiaotie Deng (2006). "Settling the complexity of two-player Nash equilibrium". Proc. 47th FOCS. pp. 261–272. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_PSPACE-complete_problems-0.md b/data/en.wikipedia.org/wiki/List_of_PSPACE-complete_problems-0.md new file mode 100644 index 000000000..f4f196050 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_PSPACE-complete_problems-0.md @@ -0,0 +1,54 @@ +--- +title: "List of PSPACE-complete problems" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_PSPACE-complete_problems" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:31.189075+00:00" +instance: "kb-cron" +--- + +Here are some of the more commonly known problems that are PSPACE-complete when expressed as decision problems. This list is in no way comprehensive. + + +== Games and puzzles == +Generalized versions of: + + +== Logic == + + +== Lambda calculus == +Type inhabitation problem for simply typed lambda calculus + + +== Automata and language theory == + + +=== Circuit theory === +Integer circuit evaluation + + +=== Automata theory === + + +=== Formal languages === + + +== Graph theory == + + +== Others == + + +== See also == +List of NP-complete problems + + +== Notes == + + +== References == +Garey, M.R.; Johnson, D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W.H. Freeman. ISBN 978-0-7167-1045-5. +Eppstein's page on computational complexity of games +The Complexity of Approximating PSPACE-complete problems for hierarchical specifications \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-0.md b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-0.md new file mode 100644 index 000000000..4528a1faf --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-0.md @@ -0,0 +1,1725 @@ +--- +title: "List of Runge–Kutta methods" +chunk: 1/5 +source: "https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:47.604001+00:00" +instance: "kb-cron" +--- + +Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation + + + + + + + + d + y + + + d + t + + + + = + f + ( + t + , + y + ) + . + + + {\displaystyle {\frac {dy}{dt}}=f(t,y).} + + +Explicit Runge–Kutta methods take the form + + + + + + + + + + y + + n + + + 1 + + + + + + = + + y + + n + + + + + h + + ∑ + + i + = + 1 + + + s + + + + b + + i + + + + k + + i + + + + + + + + k + + 1 + + + + + + = + f + ( + + t + + n + + + , + + y + + n + + + ) + , + + + + + + k + + 2 + + + + + + = + f + ( + + t + + n + + + + + + c + + 2 + + + h + , + + y + + n + + + + + h + ( + + a + + 21 + + + + k + + 1 + + + ) + ) + , + + + + + + k + + 3 + + + + + + = + f + ( + + t + + n + + + + + + c + + 3 + + + h + , + + y + + n + + + + + h + ( + + a + + 31 + + + + k + + 1 + + + + + + a + + 32 + + + + k + + 2 + + + ) + ) + , + + + + + + + + + ⋮ + + + + + + k + + i + + + + + + = + f + + ( + + + t + + n + + + + + + c + + i + + + h + , + + y + + n + + + + + h + + ∑ + + j + = + 1 + + + i + − + 1 + + + + a + + i + j + + + + k + + j + + + + ) + + . + + + + + + + {\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\\k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\\&\;\;\vdots \\k_{i}&=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{i-1}a_{ij}k_{j}\right).\end{aligned}}} + + +Stages for implicit methods of s stages take the more general form, with the solution to be found over all s + + + + + + k + + i + + + = + f + + ( + + + t + + n + + + + + + c + + i + + + h + , + + y + + n + + + + + h + + ∑ + + j + = + 1 + + + s + + + + a + + i + j + + + + k + + j + + + + ) + + . + + + {\displaystyle k_{i}=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{s}a_{ij}k_{j}\right).} + + +Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows: + + + + + + + + + + c + + 1 + + + + + + a + + 11 + + + + + + a + + 12 + + + + + … + + + + a + + 1 + s + + + + + + + + c + + 2 + + + + + + a + + 21 + + + + + + a + + 22 + + + + + … + + + + a + + 2 + s + + + + + + + ⋮ + + + ⋮ + + + ⋮ + + + ⋱ + + + ⋮ + + + + + + c + + s + + + + + + a + + s + 1 + + + + + + a + + s + 2 + + + + + … + + + + a + + s + s + + + + + + + + + b + + 1 + + + + + + b + + 2 + + + + + … + + + + b + + s + + + + + + + + + {\displaystyle {\begin{array}{c|cccc}c_{1}&a_{11}&a_{12}&\dots &a_{1s}\\c_{2}&a_{21}&a_{22}&\dots &a_{2s}\\\vdots &\vdots &\vdots &\ddots &\vdots \\c_{s}&a_{s1}&a_{s2}&\dots &a_{ss}\\\hline &b_{1}&b_{2}&\dots &b_{s}\\\end{array}}} + + +For adaptive and implicit methods, the Butcher tableau is extended to give values of + + + + + b + + i + + + ∗ + + + + + {\displaystyle b_{i}^{*}} + +, and the estimated error is then + + + + + + e + + n + + + 1 + + + = + h + + ∑ + + i + = + 1 + + + s + + + ( + + b + + i + + + − + + b + + i + + + ∗ + + + ) + + k + + i + + + + + {\displaystyle e_{n+1}=h\sum _{i=1}^{s}(b_{i}-b_{i}^{*})k_{i}} + +. + +== Explicit methods == +The explicit methods are those where the matrix + + + + [ + + a + + i + j + + + ] + + + {\displaystyle [a_{ij}]} + + is lower triangular. + +=== First-order methods === + +==== Forward Euler ==== +The Euler method is first order. The lack of stability and accuracy limits its popularity mainly to use as a simple introductory example of a numeric solution method. + + + + + + + + + 0 + + + 0 + + + + + + 1 + + + + + + + {\displaystyle {\begin{array}{c|c}0&0\\\hline &1\\\end{array}}} + + +=== Second-order methods === + +==== Generic second-order method ==== +Second-order methods can be generically written as follows: + + + + + + + + + 0 + + + 0 + + + 0 + + + + + α + + + α + + + 0 + + + + + + 1 + − + + + 1 + + 2 + α + + + + + + + + 1 + + 2 + α + + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0\\\alpha &\alpha &0\\\hline &1-{\frac {1}{2\alpha }}&{\frac {1}{2\alpha }}\\\end{array}}} + + +with α ≠ 0. + +==== Explicit midpoint method ==== +The (explicit) midpoint method is a second-order method with two stages (see also the implicit midpoint method below): + + + + + + + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + 0 + + + + + + 0 + + + 1 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&0&0\\1/2&1/2&0\\\hline &0&1\\\end{array}}} + + +==== Heun's method ==== +Heun's method is a second-order method with two stages. It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method: + + + + + + + + + 0 + + + 0 + + + 0 + + + + + 1 + + + 1 + + + 0 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&0&0\\1&1&0\\\hline &1/2&1/2\\\end{array}}} + + +==== Ralston's method ==== +Ralston's method is a second-order method with two stages and a minimum local error bound: + + + + + + + + + 0 + + + 0 + + + 0 + + + + + 2 + + / + + 3 + + + 2 + + / + + 3 + + + 0 + + + + + + 1 + + / + + 4 + + + 3 + + / + + 4 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&0&0\\2/3&2/3&0\\\hline &1/4&3/4\\\end{array}}} + + +=== Third-order methods === + +==== Generic third-order method ==== +Third-order methods can be generically written as follows: + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + α + + + α + + + 0 + + + 0 + + + + + β + + + + + β + α + + + + + + β + − + 3 + α + ( + 1 + − + α + ) + + + ( + 3 + α + − + 2 + ) + + + + + + − + + + β + α + + + + + + β + − + α + + + ( + 3 + α + − + 2 + ) + + + + + + 0 + + + + + + 1 + − + + + + 3 + α + + + 3 + β + − + 2 + + + 6 + α + β + + + + + + + + + 3 + β + − + 2 + + + 6 + α + ( + β + − + α + ) + + + + + + + + + 2 + − + 3 + α + + + 6 + β + ( + β + − + α + ) + + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0&0\\\alpha &\alpha &0&0\\\beta &{\frac {\beta }{\alpha }}{\frac {\beta -3\alpha (1-\alpha )}{(3\alpha -2)}}&-{\frac {\beta }{\alpha }}{\frac {\beta -\alpha }{(3\alpha -2)}}&0\\\hline &1-{\frac {3\alpha +3\beta -2}{6\alpha \beta }}&{\frac {3\beta -2}{6\alpha (\beta -\alpha )}}&{\frac {2-3\alpha }{6\beta (\beta -\alpha )}}\\\end{array}}} + + +with α ≠ 0, α ≠ 2⁄3, β ≠ 0, and α ≠ β. + +==== Kutta's third-order method ==== + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + 0 + + + 0 + + + + + 1 + + + − + 1 + + + 2 + + + 0 + + + + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0&0\\1/2&1/2&0&0\\1&-1&2&0\\\hline &1/6&2/3&1/6\\\end{array}}} + + +==== Heun's third-order method ==== + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 3 + + + 1 + + / + + 3 + + + 0 + + + 0 + + + + + 2 + + / + + 3 + + + 0 + + + 2 + + / + + 3 + + + 0 + + + + + + 1 + + / + + 4 + + + 0 + + + 3 + + / + + 4 + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0&0\\1/3&1/3&0&0\\2/3&0&2/3&0\\\hline &1/4&0&3/4\\\end{array}}} + + +==== Ralston's third-order method ==== +Ralston's third-order method has a minimum local error bound and is used in the embedded Bogacki–Shampine method. + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + 0 + + + 0 + + + + + 3 + + / + + 4 + + + 0 + + + 3 + + / + + 4 + + + 0 + + + + + + 2 + + / + + 9 + + + 1 + + / + + 3 + + + 4 + + / + + 9 + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0&0\\1/2&1/2&0&0\\3/4&0&3/4&0\\\hline &2/9&1/3&4/9\\\end{array}}} + + +==== Van der Houwen's/Wray's third-order method ==== + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 8 + + / + + 15 + + + 8 + + / + + 15 + + + 0 + + + 0 + + + + + 2 + + / + + 3 + + + 1 + + / + + 4 + + + 5 + + / + + 12 + + + 0 + + + + + + 1 + + / + + 4 + + + 0 + + + 3 + + / + + 4 + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0&0\\8/15&8/15&0&0\\2/3&1/4&5/12&0\\\hline &1/4&0&3/4\\\end{array}}} + + +==== Third-order Strong Stability Preserving Runge-Kutta (SSPRK3) ==== + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + + 1 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 1 + + / + + 4 + + + 1 + + / + + 4 + + + 0 + + + + + + 1 + + / + + 6 + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0&0\\1&1&0&0\\1/2&1/4&1/4&0\\\hline &1/6&1/6&2/3\\\end{array}}} + + +=== Fourth-order methods === + +==== Classic fourth-order method ==== +The "original" Runge–Kutta method. + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 0 + + + 1 + + / + + 2 + + + 0 + + + 0 + + + + + 1 + + + 0 + + + 0 + + + 1 + + + 0 + + + + + + 1 + + / + + 6 + + + 1 + + / + + 3 + + + 1 + + / + + 3 + + + 1 + + / + + 6 + + + + + + + {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\1/2&1/2&0&0&0\\1/2&0&1/2&0&0\\1&0&0&1&0\\\hline &1/6&1/3&1/3&1/6\\\end{array}}} + + +==== 3/8-rule fourth-order method ==== +This method isn't as well known as the "classic" method, but is just as classic because it was proposed in the same paper (Kutta, 1901). \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-1.md b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-1.md new file mode 100644 index 000000000..c40b948ab --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-1.md @@ -0,0 +1,1598 @@ +--- +title: "List of Runge–Kutta methods" +chunk: 2/5 +source: "https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:47.604001+00:00" +instance: "kb-cron" +--- + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 3 + + + 1 + + / + + 3 + + + 0 + + + 0 + + + 0 + + + + + 2 + + / + + 3 + + + − + 1 + + / + + 3 + + + 1 + + + 0 + + + 0 + + + + + 1 + + + 1 + + + − + 1 + + + 1 + + + 0 + + + + + + 1 + + / + + 8 + + + 3 + + / + + 8 + + + 3 + + / + + 8 + + + 1 + + / + + 8 + + + + + + + {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\1/3&1/3&0&0&0\\2/3&-1/3&1&0&0\\1&1&-1&1&0\\\hline &1/8&3/8&3/8&1/8\\\end{array}}} + + +==== Ralston's fourth-order method ==== +This fourth order method has minimum truncation error. + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + 0 + + + + + + + 2 + 5 + + + + + + + 2 + 5 + + + + + 0 + + + 0 + + + 0 + + + + + + + + 14 + − + 3 + + + 5 + + + + 16 + + + + + + + + − + 2 + + 889 + + + 1 + + 428 + + + 5 + + + + + 1 + + 024 + + + + + + + + + 3 + + 785 + − + 1 + + 620 + + + 5 + + + + + 1 + + 024 + + + + + + 0 + + + 0 + + + + + 1 + + + + + + − + 3 + + 365 + + + 2 + + 094 + + + 5 + + + + + 6 + + 040 + + + + + + + + + − + 975 + − + 3 + + 046 + + + 5 + + + + + 2 + + 552 + + + + + + + + + 467 + + 040 + + + 203 + + 968 + + + 5 + + + + + 240 + + 845 + + + + + + 0 + + + + + + + + + 263 + + + 24 + + + 5 + + + + + 1 + + 812 + + + + + + + + + 125 + − + 1000 + + + 5 + + + + + 3 + + 828 + + + + + + + + + 3 + + 426 + + 304 + + + 1 + + 661 + + 952 + + + 5 + + + + + 5 + + 924 + + 787 + + + + + + + + + 30 + − + 4 + + + 5 + + + + 123 + + + + + + + + + {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\{\frac {2}{5}}&{\frac {2}{5}}&0&0&0\\{\frac {14-3{\sqrt {5}}}{16}}&{\frac {-2\,889+1\,428{\sqrt {5}}}{1\,024}}&{\frac {3\,785-1\,620{\sqrt {5}}}{1\,024}}&0&0\\1&{\frac {-3\,365+2\,094{\sqrt {5}}}{6\,040}}&{\frac {-975-3\,046{\sqrt {5}}}{2\,552}}&{\frac {467\,040+203\,968{\sqrt {5}}}{240\,845}}&0\\\hline &{\frac {263+24{\sqrt {5}}}{1\,812}}&{\frac {125-1000{\sqrt {5}}}{3\,828}}&{\frac {3\,426\,304+1\,661\,952{\sqrt {5}}}{5\,924\,787}}&{\frac {30-4{\sqrt {5}}}{123}}\\\end{array}}} + + +=== Fifth-order methods === + +==== Nyström's fifth-order method ==== +This fifth-order method was a correction of the one proposed originally by Kutta's work. + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + 0 + + + 0 + + + 0 + + + + + + + 1 + 3 + + + + + + + 1 + 3 + + + + + 0 + + + 0 + + + 0 + + + 0 + + + 0 + + + + + + + 2 + 5 + + + + + + + 4 + 25 + + + + + + + 6 + 25 + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + + + + 1 + 4 + + + + + − + 3 + + + + + 15 + 4 + + + + + 0 + + + 0 + + + 0 + + + + + + + 2 + 3 + + + + + + + 2 + 27 + + + + + + + 10 + 9 + + + + + − + + + 50 + 81 + + + + + + + 8 + 81 + + + + + 0 + + + 0 + + + + + + + 4 + 5 + + + + + + + 2 + 25 + + + + + + + 12 + 25 + + + + + + + 2 + 15 + + + + + + + 8 + 75 + + + + + 0 + + + 0 + + + + + + + + 23 + 192 + + + + + 0 + + + + + 125 + 192 + + + + + 0 + + + − + + + 27 + 64 + + + + + + + 125 + 192 + + + + + + + + + {\displaystyle {\begin{array}{c|cccccc}0&0&0&0&0&0&0\\{\frac {1}{3}}&{\frac {1}{3}}&0&0&0&0&0\\{\frac {2}{5}}&{\frac {4}{25}}&{\frac {6}{25}}&0&0&0&0\\1&{\frac {1}{4}}&-3&{\frac {15}{4}}&0&0&0\\{\frac {2}{3}}&{\frac {2}{27}}&{\frac {10}{9}}&-{\frac {50}{81}}&{\frac {8}{81}}&0&0\\{\frac {4}{5}}&{\frac {2}{25}}&{\frac {12}{25}}&{\frac {2}{15}}&{\frac {8}{75}}&0&0\\\hline &{\frac {23}{192}}&0&{\frac {125}{192}}&0&-{\frac {27}{64}}&{\frac {125}{192}}\\\end{array}}} + + +== Embedded methods == +The embedded methods are designed to produce an estimate of the local truncation error of a single Runge–Kutta step, and as result, allow to control the error with adaptive stepsize. This is done by having two methods in the tableau, one with order p and one with order p-1. +The lower-order step is given by + + + + + + y + + n + + + 1 + + + ∗ + + + = + + y + + n + + + + + h + + ∑ + + i + = + 1 + + + s + + + + b + + i + + + ∗ + + + + k + + i + + + , + + + {\displaystyle y_{n+1}^{*}=y_{n}+h\sum _{i=1}^{s}b_{i}^{*}k_{i},} + + +where the + + + + + k + + i + + + + + {\displaystyle k_{i}} + + are the same as for the higher order method. Then the error is + + + + + + e + + n + + + 1 + + + = + + y + + n + + + 1 + + + − + + y + + n + + + 1 + + + ∗ + + + = + h + + ∑ + + i + = + 1 + + + s + + + ( + + b + + i + + + − + + b + + i + + + ∗ + + + ) + + k + + i + + + , + + + {\displaystyle e_{n+1}=y_{n+1}-y_{n+1}^{*}=h\sum _{i=1}^{s}(b_{i}-b_{i}^{*})k_{i},} + + +which is + + + + O + ( + + h + + p + + + ) + + + {\displaystyle O(h^{p})} + +. The Butcher Tableau for this kind of method is extended to give the values of + + + + + b + + i + + + ∗ + + + + + {\displaystyle b_{i}^{*}} + + + + + + + + + + + c + + 1 + + + + + + a + + 11 + + + + + + a + + 12 + + + + + … + + + + a + + 1 + s + + + + + + + + c + + 2 + + + + + + a + + 21 + + + + + + a + + 22 + + + + + … + + + + a + + 2 + s + + + + + + + ⋮ + + + ⋮ + + + ⋮ + + + ⋱ + + + ⋮ + + + + + + c + + s + + + + + + a + + s + 1 + + + + + + a + + s + 2 + + + + + … + + + + a + + s + s + + + + + + + + + b + + 1 + + + + + + b + + 2 + + + + + … + + + + b + + s + + + + + + + + + b + + 1 + + + ∗ + + + + + + b + + 2 + + + ∗ + + + + + … + + + + b + + s + + + ∗ + + + + + + + + + {\displaystyle {\begin{array}{c|cccc}c_{1}&a_{11}&a_{12}&\dots &a_{1s}\\c_{2}&a_{21}&a_{22}&\dots &a_{2s}\\\vdots &\vdots &\vdots &\ddots &\vdots \\c_{s}&a_{s1}&a_{s2}&\dots &a_{ss}\\\hline &b_{1}&b_{2}&\dots &b_{s}\\&b_{1}^{*}&b_{2}^{*}&\dots &b_{s}^{*}\\\end{array}}} + + +=== Heun–Euler === +The simplest adaptive Runge–Kutta method involves combining Heun's method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is: + + + + + + + + + 0 + + + + + + 1 + + + 1 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + + 0 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&\\1&1\\\hline &1/2&1/2\\&1&0\end{array}}} + + +The error estimate is used to control the stepsize. + +=== Fehlberg RK1(2) === +The Fehlberg method has two methods of orders 1 and 2. Its extended Butcher Tableau is: + +The first row of b coefficients gives the second-order accurate solution, and the second row has order one. + +=== Bogacki–Shampine === +The Bogacki–Shampine method has two methods of orders 2 and 3. Its extended Butcher Tableau is: + +The first row of b coefficients gives the third-order accurate solution, and the second row has order two. + +=== Fehlberg === +The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4; it is sometimes called RKF45 . Its extended Butcher Tableau is: + + + + + + + + + 0 + + + + + + + + + + 1 + + / + + 4 + + + 1 + + / + + 4 + + + + + + + + 3 + + / + + 8 + + + 3 + + / + + 32 + + + 9 + + / + + 32 + + + + + + + 12 + + / + + 13 + + + 1932 + + / + + 2197 + + + − + 7200 + + / + + 2197 + + + 7296 + + / + + 2197 + + + + + + 1 + + + 439 + + / + + 216 + + + − + 8 + + + 3680 + + / + + 513 + + + − + 845 + + / + + 4104 + + + + + + 1 + + / + + 2 + + + − + 8 + + / + + 27 + + + 2 + + + − + 3544 + + / + + 2565 + + + 1859 + + / + + 4104 + + + − + 11 + + / + + 40 + + + + + + 16 + + / + + 135 + + + 0 + + + 6656 + + / + + 12825 + + + 28561 + + / + + 56430 + + + − + 9 + + / + + 50 + + + 2 + + / + + 55 + + + + + + 25 + + / + + 216 + + + 0 + + + 1408 + + / + + 2565 + + + 2197 + + / + + 4104 + + + − + 1 + + / + + 5 + + + 0 + + + + + + + {\displaystyle {\begin{array}{r|ccccc}0&&&&&\\1/4&1/4&&&\\3/8&3/32&9/32&&\\12/13&1932/2197&-7200/2197&7296/2197&\\1&439/216&-8&3680/513&-845/4104&\\1/2&-8/27&2&-3544/2565&1859/4104&-11/40\\\hline &16/135&0&6656/12825&28561/56430&-9/50&2/55\\&25/216&0&1408/2565&2197/4104&-1/5&0\end{array}}} + + +The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four. +The coefficients here allow for an adaptive stepsize to be determined automatically. + +=== Cash-Karp === +Cash and Karp have modified Fehlberg's original idea. The extended tableau for the Cash–Karp method is + +The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four. + +=== Dormand–Prince === +The extended tableau for the Dormand–Prince method is + +The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order accurate solution. + +== Implicit methods == + +=== Backward Euler === +The backward Euler method is first order. Unconditionally stable and non-oscillatory for linear diffusion problems. + + + + + + + + + 1 + + + 1 + + + + + + 1 + + + + + + + {\displaystyle {\begin{array}{c|c}1&1\\\hline &1\\\end{array}}} + + +=== Implicit midpoint === +The implicit midpoint method is of second order. It is the simplest method in the class of collocation methods known as the Gauss-Legendre methods. It is a symplectic integrator. + + + + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + + + + + + {\displaystyle {\begin{array}{c|c}1/2&1/2\\\hline &1\end{array}}} + + +=== Crank-Nicolson method === +The Crank–Nicolson method corresponds to the implicit trapezoidal rule and is a second-order accurate and A-stable method. + + + + + + + + + 0 + + + 0 + + + 0 + + + + + 1 + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&0&0\\1&1/2&1/2\\\hline &1/2&1/2\\\end{array}}} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-2.md b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-2.md new file mode 100644 index 000000000..dc037016d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-2.md @@ -0,0 +1,1613 @@ +--- +title: "List of Runge–Kutta methods" +chunk: 3/5 +source: "https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:47.604001+00:00" +instance: "kb-cron" +--- + +=== Gauss–Legendre methods === +These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau: + + + + + + + + + + + 1 + 2 + + + − + + + + 3 + + 6 + + + + + + + 1 + 4 + + + + + + + 1 + 4 + + + − + + + + 3 + + 6 + + + + + + + + + 1 + 2 + + + + + + + + 3 + + 6 + + + + + + + 1 + 4 + + + + + + + + 3 + + 6 + + + + + + + 1 + 4 + + + + + + + + + + 1 + 2 + + + + + + + 1 + 2 + + + + + + + + + + 1 + 2 + + + + + + + + 3 + + 2 + + + + + + + 1 + 2 + + + − + + + + 3 + + 2 + + + + + + + + + {\displaystyle {\begin{array}{c|cc}{\frac {1}{2}}-{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}&{\frac {1}{4}}-{\frac {\sqrt {3}}{6}}\\{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\&{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}\\\end{array}}} + + +The Gauss–Legendre method of order six has Butcher tableau: + + + + + + + + + + + 1 + 2 + + + − + + + + 15 + + 10 + + + + + + + 5 + 36 + + + + + + + 2 + 9 + + + − + + + + 15 + + 15 + + + + + + + 5 + 36 + + + − + + + + 15 + + 30 + + + + + + + + + 1 + 2 + + + + + + + 5 + 36 + + + + + + + + 15 + + 24 + + + + + + + 2 + 9 + + + + + + + 5 + 36 + + + − + + + + 15 + + 24 + + + + + + + + + 1 + 2 + + + + + + + + 15 + + 10 + + + + + + + 5 + 36 + + + + + + + + 15 + + 30 + + + + + + + 2 + 9 + + + + + + + + 15 + + 15 + + + + + + + 5 + 36 + + + + + + + + + + 5 + 18 + + + + + + + 4 + 9 + + + + + + + 5 + 18 + + + + + + + + − + + + 5 + 6 + + + + + + + 8 + 3 + + + + + − + + + 5 + 6 + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}{\frac {1}{2}}-{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}&{\frac {2}{9}}-{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{30}}\\{\frac {1}{2}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{24}}&{\frac {2}{9}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{24}}\\{\frac {1}{2}}+{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{30}}&{\frac {2}{9}}+{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}\\\hline &{\frac {5}{18}}&{\frac {4}{9}}&{\frac {5}{18}}\\&-{\frac {5}{6}}&{\frac {8}{3}}&-{\frac {5}{6}}\end{array}}} + + +=== Diagonally Implicit Runge–Kutta methods === +Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems; + +the advantage of this approach is that here the solution may be found sequentially as opposed to simultaneously. +The simplest method from this class is the order 2 implicit midpoint method. +Kraaijevanger and Spijker's two-stage Diagonally Implicit Runge–Kutta method: + + + + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + 0 + + + + + 3 + + / + + 2 + + + − + 1 + + / + + 2 + + + 2 + + + + + + − + 1 + + / + + 2 + + + 3 + + / + + 2 + + + + + + + {\displaystyle {\begin{array}{c|cc}1/2&1/2&0\\3/2&-1/2&2\\\hline &-1/2&3/2\\\end{array}}} + + +Qin and Zhang's two-stage, 2nd order, symplectic Diagonally Implicit Runge–Kutta method: + + + + + + + + + 1 + + / + + 4 + + + 1 + + / + + 4 + + + 0 + + + + + 3 + + / + + 4 + + + 1 + + / + + 2 + + + 1 + + / + + 4 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + + {\displaystyle {\begin{array}{c|cc}1/4&1/4&0\\3/4&1/2&1/4\\\hline &1/2&1/2\\\end{array}}} + + +Pareschi and Russo's two-stage 2nd order Diagonally Implicit Runge–Kutta method: + + + + + + + + + x + + + x + + + 0 + + + + + 1 + − + x + + + 1 + − + 2 + x + + + x + + + + + + + + 1 + 2 + + + + + + + 1 + 2 + + + + + + + + + {\displaystyle {\begin{array}{c|cc}x&x&0\\1-x&1-2x&x\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\\end{array}}} + + +This Diagonally Implicit Runge–Kutta method is A-stable if and only if + + + + x + ≥ + + + 1 + 4 + + + + + {\textstyle x\geq {\frac {1}{4}}} + +. Moreover, this method is L-stable if and only if + + + + x + + + {\displaystyle x} + + equals one of the roots of the polynomial + + + + + x + + 2 + + + − + 2 + x + + + + + 1 + 2 + + + + + {\textstyle x^{2}-2x+{\frac {1}{2}}} + +, i.e. if + + + + x + = + 1 + ± + + + + 2 + + 2 + + + + + {\textstyle x=1\pm {\frac {\sqrt {2}}{2}}} + +. +Qin and Zhang's Diagonally Implicit Runge–Kutta method corresponds to Pareschi and Russo's Diagonally Implicit Runge–Kutta method with + + + + x + = + 1 + + / + + 4 + + + {\displaystyle x=1/4} + +. +Two-stage 2nd order Diagonally Implicit Runge–Kutta method: + + + + + + + + + x + + + x + + + 0 + + + + + 1 + + + 1 + − + x + + + x + + + + + + 1 + − + x + + + x + + + + + + + {\displaystyle {\begin{array}{c|cc}x&x&0\\1&1-x&x\\\hline &1-x&x\\\end{array}}} + + +Again, this Diagonally Implicit Runge–Kutta method is A-stable if and only if + + + + x + ≥ + + + 1 + 4 + + + + + {\textstyle x\geq {\frac {1}{4}}} + +. As the previous method, this method is again L-stable if and only if + + + + x + + + {\displaystyle x} + + equals one of the roots of the polynomial + + + + + x + + 2 + + + − + 2 + x + + + + + 1 + 2 + + + + + {\textstyle x^{2}-2x+{\frac {1}{2}}} + +, i.e. if + + + + x + = + 1 + ± + + + + 2 + + 2 + + + + + {\textstyle x=1\pm {\frac {\sqrt {2}}{2}}} + +. This condition is also necessary for 2nd order accuracy. +Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method: + + + + + + + + + + + 1 + 2 + + + + + + + + 3 + + 6 + + + + + + + 1 + 2 + + + + + + + + 3 + + 6 + + + + + 0 + + + + + + + 1 + 2 + + + − + + + + 3 + + 6 + + + + + − + + + + 3 + + 3 + + + + + + + 1 + 2 + + + + + + + + 3 + + 6 + + + + + + + + + + 1 + 2 + + + + + + + 1 + 2 + + + + + + + + + {\displaystyle {\begin{array}{c|cc}{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&0\\{\frac {1}{2}}-{\frac {\sqrt {3}}{6}}&-{\frac {\sqrt {3}}{3}}&{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\\end{array}}} + + +Crouzeix's three-stage, 4th order Diagonally Implicit Runge–Kutta method: + + + + + + + + + + + + 1 + + + α + + 2 + + + + + + + + 1 + + + α + + 2 + + + + + 0 + + + 0 + + + + + + + 1 + 2 + + + + + − + + + α + 2 + + + + + + + + 1 + + + α + + 2 + + + + + 0 + + + + + + + + 1 + − + α + + 2 + + + + + 1 + + + α + + + − + ( + 1 + + + 2 + + α + ) + + + + + + 1 + + + α + + 2 + + + + + + + + + + 1 + + 6 + + α + + 2 + + + + + + + + 1 + − + + + 1 + + 3 + + α + + 2 + + + + + + + + + + 1 + + 6 + + α + + 2 + + + + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}{\frac {1+\alpha }{2}}&{\frac {1+\alpha }{2}}&0&0\\{\frac {1}{2}}&-{\frac {\alpha }{2}}&{\frac {1+\alpha }{2}}&0\\{\frac {1-\alpha }{2}}&1+\alpha &-(1+2\,\alpha )&{\frac {1+\alpha }{2}}\\\hline &{\frac {1}{6\alpha ^{2}}}&1-{\frac {1}{3\alpha ^{2}}}&{\frac {1}{6\alpha ^{2}}}\\\end{array}}} + + +with + + + + α + = + + + 2 + + 3 + + + + cos + ⁡ + + + π + 18 + + + + + {\textstyle \alpha ={\frac {2}{\sqrt {3}}}\cos {\frac {\pi }{18}}} + +. +Three-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method: + + + + + + + + + x + + + x + + + 0 + + + 0 + + + + + + + + 1 + + + x + + 2 + + + + + + + + 1 + − + x + + 2 + + + + + x + + + 0 + + + + + 1 + + + − + 3 + + x + + 2 + + + + / + + 2 + + + 4 + x + − + 1 + + / + + 4 + + + 3 + + x + + 2 + + + + / + + 2 + − + 5 + x + + + 5 + + / + + 4 + + + x + + + + + + − + 3 + + x + + 2 + + + + / + + 2 + + + 4 + x + − + 1 + + / + + 4 + + + 3 + + x + + 2 + + + + / + + 2 + − + 5 + x + + + 5 + + / + + 4 + + + x + + + + + + + {\displaystyle {\begin{array}{c|ccc}x&x&0&0\\{\frac {1+x}{2}}&{\frac {1-x}{2}}&x&0\\1&-3x^{2}/2+4x-1/4&3x^{2}/2-5x+5/4&x\\\hline &-3x^{2}/2+4x-1/4&3x^{2}/2-5x+5/4&x\\\end{array}}} + + +with + + + + x + = + 0.4358665215 + + + {\displaystyle x=0.4358665215} + + +Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau: + + + + + + + + + x + + + x + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + − + x + + + x + + + 0 + + + + + 1 + − + x + + + 2 + x + + + 1 + − + 4 + x + + + x + + + + + + + + 1 + + 6 + ( + 1 + − + 2 + x + + ) + + 2 + + + + + + + + + + + 3 + ( + 1 + − + 2 + x + + ) + + 2 + + + − + 1 + + + 3 + ( + 1 + − + 2 + x + + ) + + 2 + + + + + + + + + + 1 + + 6 + ( + 1 + − + 2 + x + + ) + + 2 + + + + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}x&x&0&0\\1/2&1/2-x&x&0\\1-x&2x&1-4x&x\\\hline &{\frac {1}{6(1-2x)^{2}}}&{\frac {3(1-2x)^{2}-1}{3(1-2x)^{2}}}&{\frac {1}{6(1-2x)^{2}}}\\\end{array}}} + + +with + + + + x + + + {\displaystyle x} + + one of the three roots of the cubic equation + + + + + x + + 3 + + + − + 3 + + x + + 2 + + + + / + + 2 + + + x + + / + + 2 + − + 1 + + / + + 24 + = + 0 + + + {\displaystyle x^{3}-3x^{2}/2+x/2-1/24=0} + +. The three roots of this cubic equation are approximately + + + + + x + + 1 + + + = + + + 1 + 2 + + + + + + + 1 + + 3 + + + + cos + ⁡ + + + π + 18 + + + = + 1.068579021301629 + + + {\textstyle x_{1}={\frac {1}{2}}+{\frac {1}{\sqrt {3}}}\cos {\frac {\pi }{18}}=1.068579021301629} + +, + + + + + x + + 2 + + + = + 0.1288864005157204 + + + {\textstyle x_{2}=0.1288864005157204} + +, and + + + + + x + + 3 + + + = + 0.3025345781826508 + + + {\textstyle x_{3}=0.3025345781826508} + +. The root + + + + + x + + 1 + + + + + {\displaystyle x_{1}} + + gives the best stability properties for initial value problems. +Four-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-3.md b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-3.md new file mode 100644 index 000000000..a41e91f17 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-3.md @@ -0,0 +1,1520 @@ +--- +title: "List of Runge–Kutta methods" +chunk: 4/5 +source: "https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:47.604001+00:00" +instance: "kb-cron" +--- + + + + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + 0 + + + 0 + + + 0 + + + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + 1 + + / + + 2 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + − + 1 + + / + + 2 + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + 0 + + + + + 1 + + + 3 + + / + + 2 + + + − + 3 + + / + + 2 + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 3 + + / + + 2 + + + − + 3 + + / + + 2 + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + + {\displaystyle {\begin{array}{c|cccc}1/2&1/2&0&0&0\\2/3&1/6&1/2&0&0\\1/2&-1/2&1/2&1/2&0\\1&3/2&-3/2&1/2&1/2\\\hline &3/2&-3/2&1/2&1/2\\\end{array}}} + + +=== Lobatto methods === +There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis. All are implicit methods, have order 2s − 2 and they all have c1 = 0 and cs = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta. + +==== Lobatto IIIA methods ==== +The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule: + + + + + + + + + 0 + + + 0 + + + 0 + + + + + 1 + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + + 0 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&0&0\\1&1/2&1/2\\\hline &1/2&1/2\\&1&0\\\end{array}}} + + +The fourth-order method is given by + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 5 + + / + + 24 + + + 1 + + / + + 3 + + + − + 1 + + / + + 24 + + + + + 1 + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + + + + − + + + 1 + 2 + + + + + 2 + + + − + + + 1 + 2 + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0&0\\1/2&5/24&1/3&-1/24\\1&1/6&2/3&1/6\\\hline &1/6&2/3&1/6\\&-{\frac {1}{2}}&2&-{\frac {1}{2}}\\\end{array}}} + + +These methods are A-stable, but neither L-stable nor B-stable. + +==== Lobatto IIIB methods ==== +The Lobatto IIIB methods are not collocation methods, but they can be viewed as discontinuous collocation methods (Hairer, Lubich & Wanner 2006, §II.1.4). The second-order method is given by + + + + + + + + + 0 + + + 1 + + / + + 2 + + + 0 + + + + + 1 + + + 1 + + / + + 2 + + + 0 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + + 0 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&1/2&0\\1&1/2&0\\\hline &1/2&1/2\\&1&0\\\end{array}}} + + +The fourth-order method is given by + + + + + + + + + 0 + + + 1 + + / + + 6 + + + − + 1 + + / + + 6 + + + 0 + + + + + 1 + + / + + 2 + + + 1 + + / + + 6 + + + 1 + + / + + 3 + + + 0 + + + + + 1 + + + 1 + + / + + 6 + + + 5 + + / + + 6 + + + 0 + + + + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + + + + − + + + 1 + 2 + + + + + 2 + + + − + + + 1 + 2 + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&1/6&-1/6&0\\1/2&1/6&1/3&0\\1&1/6&5/6&0\\\hline &1/6&2/3&1/6\\&-{\frac {1}{2}}&2&-{\frac {1}{2}}\\\end{array}}} + + +Lobatto IIIB methods are A-stable, but neither L-stable nor B-stable. + +==== Lobatto IIIC methods ==== +The Lobatto IIIC methods also are discontinuous collocation methods. The second-order method is given by + + + + + + + + + 0 + + + 1 + + / + + 2 + + + − + 1 + + / + + 2 + + + + + 1 + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + + 0 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&1/2&-1/2\\1&1/2&1/2\\\hline &1/2&1/2\\&1&0\\\end{array}}} + + +The fourth-order method is given by + + + + + + + + + 0 + + + 1 + + / + + 6 + + + − + 1 + + / + + 3 + + + 1 + + / + + 6 + + + + + 1 + + / + + 2 + + + 1 + + / + + 6 + + + 5 + + / + + 12 + + + − + 1 + + / + + 12 + + + + + 1 + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + + + + − + + + 1 + 2 + + + + + 2 + + + − + + + 1 + 2 + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&1/6&-1/3&1/6\\1/2&1/6&5/12&-1/12\\1&1/6&2/3&1/6\\\hline &1/6&2/3&1/6\\&-{\frac {1}{2}}&2&-{\frac {1}{2}}\\\end{array}}} + + +They are L-stable. They are also algebraically stable and thus B-stable, which makes them suitable for stiff problems. + +==== Lobatto IIIC* methods ==== +The Lobatto IIIC* methods are also known as Lobatto III methods (Butcher, 2008), Butcher's Lobatto methods (Hairer et al., 1993), and Lobatto IIIC methods (Sun, 2000) in the literature. The second-order method is given by + + + + + + + + + 0 + + + 0 + + + 0 + + + + + 1 + + + 1 + + + 0 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&0&0\\1&1&0\\\hline &1/2&1/2\\\end{array}}} + + +Butcher's three-stage, fourth-order method is given by + + + + + + + + + 0 + + + 0 + + + 0 + + + 0 + + + + + 1 + + / + + 2 + + + 1 + + / + + 4 + + + 1 + + / + + 4 + + + 0 + + + + + 1 + + + 0 + + + 1 + + + 0 + + + + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&0&0&0\\1/2&1/4&1/4&0\\1&0&1&0\\\hline &1/6&2/3&1/6\\\end{array}}} + + +These methods are not A-stable, B-stable or L-stable. The Lobatto IIIC* method for + + + + s + = + 2 + + + {\displaystyle s=2} + + is sometimes called the explicit trapezoidal rule. + +==== Generalized Lobatto methods ==== +One can consider a very general family of methods with three real parameters + + + + ( + + α + + A + + + , + + α + + B + + + , + + α + + C + + + ) + + + {\displaystyle (\alpha _{A},\alpha _{B},\alpha _{C})} + + by considering Lobatto coefficients of the form + + + + + + a + + i + , + j + + + ( + + α + + A + + + , + + α + + B + + + , + + α + + C + + + ) + = + + α + + A + + + + a + + i + , + j + + + A + + + + + + α + + B + + + + a + + i + , + j + + + B + + + + + + α + + C + + + + a + + i + , + j + + + C + + + + + + α + + C + ∗ + + + + a + + i + , + j + + + C + ∗ + + + + + {\displaystyle a_{i,j}(\alpha _{A},\alpha _{B},\alpha _{C})=\alpha _{A}a_{i,j}^{A}+\alpha _{B}a_{i,j}^{B}+\alpha _{C}a_{i,j}^{C}+\alpha _{C*}a_{i,j}^{C*}} + +, +where + + + + + + α + + C + ∗ + + + = + 1 + − + + α + + A + + + − + + α + + B + + + − + + α + + C + + + + + {\displaystyle \alpha _{C*}=1-\alpha _{A}-\alpha _{B}-\alpha _{C}} + +. +For example, Lobatto IIID family introduced in (Nørsett and Wanner, 1981), also called Lobatto IIINW, are given by + + + + + + + + + 0 + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + 1 + + + − + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + 1 + + / + + 2 + + + 1 + + / + + 2 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&1/2&1/2\\1&-1/2&1/2\\\hline &1/2&1/2\\\end{array}}} + + +and + + + + + + + + + 0 + + + 1 + + / + + 6 + + + 0 + + + − + 1 + + / + + 6 + + + + + 1 + + / + + 2 + + + 1 + + / + + 12 + + + 5 + + / + + 12 + + + 0 + + + + + 1 + + + 1 + + / + + 2 + + + 1 + + / + + 3 + + + 1 + + / + + 6 + + + + + + 1 + + / + + 6 + + + 2 + + / + + 3 + + + 1 + + / + + 6 + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&1/6&0&-1/6\\1/2&1/12&5/12&0\\1&1/2&1/3&1/6\\\hline &1/6&2/3&1/6\\\end{array}}} + + +These methods correspond to + + + + + α + + A + + + = + 2 + + + {\displaystyle \alpha _{A}=2} + +, + + + + + α + + B + + + = + 2 + + + {\displaystyle \alpha _{B}=2} + +, + + + + + α + + C + + + = + − + 1 + + + {\displaystyle \alpha _{C}=-1} + +, and + + + + + α + + C + ∗ + + + = + − + 2 + + + {\displaystyle \alpha _{C*}=-2} + +. The methods are L-stable. They are algebraically stable and thus B-stable. + +=== Radau methods === +Radau methods are fully implicit methods (matrix A of such methods can have any structure). Radau methods attain order 2s − 1 for s stages. Radau methods are A-stable, but expensive to implement. Also they can suffer from order reduction. + +==== Radau IA methods ==== +The first order method is similar to the backward Euler method and given by + + + + + + + + + 0 + + + 1 + + + + + + 1 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&1\\\hline &1\\\end{array}}} + + +The third-order method is given by + + + + + + + + + 0 + + + 1 + + / + + 4 + + + − + 1 + + / + + 4 + + + + + 2 + + / + + 3 + + + 1 + + / + + 4 + + + 5 + + / + + 12 + + + + + + 1 + + / + + 4 + + + 3 + + / + + 4 + + + + + + + {\displaystyle {\begin{array}{c|cc}0&1/4&-1/4\\2/3&1/4&5/12\\\hline &1/4&3/4\\\end{array}}} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-4.md b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-4.md new file mode 100644 index 000000000..35665b4d9 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-4.md @@ -0,0 +1,662 @@ +--- +title: "List of Runge–Kutta methods" +chunk: 5/5 +source: "https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:47.604001+00:00" +instance: "kb-cron" +--- + +The fifth-order method is given by + + + + + + + + + 0 + + + + + 1 + 9 + + + + + + + + − + 1 + − + + + 6 + + + + 18 + + + + + + + + − + 1 + + + + + 6 + + + + 18 + + + + + + + + + 3 + 5 + + + − + + + + 6 + + 10 + + + + + + + 1 + 9 + + + + + + + 11 + 45 + + + + + + + + 7 + + + 6 + + + + 360 + + + + + + + 11 + 45 + + + − + + + + 43 + + + 6 + + + + 360 + + + + + + + + + 3 + 5 + + + + + + + + 6 + + 10 + + + + + + + 1 + 9 + + + + + + + 11 + 45 + + + + + + + + 43 + + + 6 + + + + 360 + + + + + + + 11 + 45 + + + − + + + + 7 + + + 6 + + + + 360 + + + + + + + + + + 1 + 9 + + + + + + + 4 + 9 + + + + + + + + 6 + + 36 + + + + + + + 4 + 9 + + + − + + + + 6 + + 36 + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}0&{\frac {1}{9}}&{\frac {-1-{\sqrt {6}}}{18}}&{\frac {-1+{\sqrt {6}}}{18}}\\{\frac {3}{5}}-{\frac {\sqrt {6}}{10}}&{\frac {1}{9}}&{\frac {11}{45}}+{\frac {7{\sqrt {6}}}{360}}&{\frac {11}{45}}-{\frac {43{\sqrt {6}}}{360}}\\{\frac {3}{5}}+{\frac {\sqrt {6}}{10}}&{\frac {1}{9}}&{\frac {11}{45}}+{\frac {43{\sqrt {6}}}{360}}&{\frac {11}{45}}-{\frac {7{\sqrt {6}}}{360}}\\\hline &{\frac {1}{9}}&{\frac {4}{9}}+{\frac {\sqrt {6}}{36}}&{\frac {4}{9}}-{\frac {\sqrt {6}}{36}}\\\end{array}}} + + +==== Radau IIA methods ==== +The ci of this method are zeros of + + + + + + + + d + + s + − + 1 + + + + d + + x + + s + − + 1 + + + + + + ( + + x + + s + − + 1 + + + ( + x + − + 1 + + ) + + s + + + ) + + + {\displaystyle {\frac {d^{s-1}}{dx^{s-1}}}(x^{s-1}(x-1)^{s})} + +. +The first-order method is equivalent to the backward Euler method. +The third-order method is given by + + + + + + + + + 1 + + / + + 3 + + + 5 + + / + + 12 + + + − + 1 + + / + + 12 + + + + + 1 + + + 3 + + / + + 4 + + + 1 + + / + + 4 + + + + + + 3 + + / + + 4 + + + 1 + + / + + 4 + + + + + + + {\displaystyle {\begin{array}{c|cc}1/3&5/12&-1/12\\1&3/4&1/4\\\hline &3/4&1/4\\\end{array}}} + + +The fifth-order method is given by + + + + + + + + + + + 2 + 5 + + + − + + + + 6 + + 10 + + + + + + + 11 + 45 + + + − + + + + 7 + + + 6 + + + + 360 + + + + + + + 37 + 225 + + + − + + + + 169 + + + 6 + + + + 1800 + + + + + − + + + 2 + 225 + + + + + + + + 6 + + 75 + + + + + + + + + 2 + 5 + + + + + + + + 6 + + 10 + + + + + + + 37 + 225 + + + + + + + + 169 + + + 6 + + + + 1800 + + + + + + + 11 + 45 + + + + + + + + 7 + + + 6 + + + + 360 + + + + + − + + + 2 + 225 + + + − + + + + 6 + + 75 + + + + + + + 1 + + + + + 4 + 9 + + + − + + + + 6 + + 36 + + + + + + + 4 + 9 + + + + + + + + 6 + + 36 + + + + + + + 1 + 9 + + + + + + + + + + 4 + 9 + + + − + + + + 6 + + 36 + + + + + + + 4 + 9 + + + + + + + + 6 + + 36 + + + + + + + 1 + 9 + + + + + + + + + {\displaystyle {\begin{array}{c|ccc}{\frac {2}{5}}-{\frac {\sqrt {6}}{10}}&{\frac {11}{45}}-{\frac {7{\sqrt {6}}}{360}}&{\frac {37}{225}}-{\frac {169{\sqrt {6}}}{1800}}&-{\frac {2}{225}}+{\frac {\sqrt {6}}{75}}\\{\frac {2}{5}}+{\frac {\sqrt {6}}{10}}&{\frac {37}{225}}+{\frac {169{\sqrt {6}}}{1800}}&{\frac {11}{45}}+{\frac {7{\sqrt {6}}}{360}}&-{\frac {2}{225}}-{\frac {\sqrt {6}}{75}}\\1&{\frac {4}{9}}-{\frac {\sqrt {6}}{36}}&{\frac {4}{9}}+{\frac {\sqrt {6}}{36}}&{\frac {1}{9}}\\\hline &{\frac {4}{9}}-{\frac {\sqrt {6}}{36}}&{\frac {4}{9}}+{\frac {\sqrt {6}}{36}}&{\frac {1}{9}}\\\end{array}}} + + +== Notes == + +== References == +Ehle, Byron L. (1969). On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems (PDF) (Thesis). +Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0. +Hairer, Ernst; Wanner, Gerhard (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-60452-5. +Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-30663-4. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_alternative_set_theories-0.md b/data/en.wikipedia.org/wiki/List_of_alternative_set_theories-0.md new file mode 100644 index 000000000..9513aa31c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_alternative_set_theories-0.md @@ -0,0 +1,44 @@ +--- +title: "List of alternative set theories" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_alternative_set_theories" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:54.419564+00:00" +instance: "kb-cron" +--- + +In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory. + + +== Alternative set theories == +Alternative set theories include: + +Vopěnka's alternative set theory +Von Neumann–Bernays–Gödel set theory +Morse–Kelley set theory +Tarski–Grothendieck set theory +Ackermann set theory +Type theory +New Foundations +Positive set theory +Internal set theory +Pocket set theory +Naive set theory +S (set theory) +Double extension set theory +Kripke–Platek set theory +Kripke–Platek set theory with urelements +Scott–Potter set theory +Constructive set theory +Zermelo set theory +General set theory +Mac Lane set theory + + +== See also == +Non-well-founded set theory +List of first-order theories § Set theories + + +== Notes == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems-0.md b/data/en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems-0.md new file mode 100644 index 000000000..7d499c17d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems-0.md @@ -0,0 +1,30 @@ +--- +title: "List of books on history of number systems" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:29.597547+00:00" +instance: "kb-cron" +--- + +This list compiles notable works that explore the history and development of number systems across various civilizations and time periods. These works cover topics ranging from ancient numeral systems and arithmetic methods to the evolution of mathematical notations and the impact of numerals on science, trade, and culture. + + +== Overview == +Number systems have been central to the development of human civilization, enabling record-keeping, commerce, astronomy, and scientific advancement. Early systems such as tally marks and Roman numerals gradually gave way to more abstract and efficient representations like the Babylonian base-60 system and the Hindu–Arabic numerals, now standard worldwide. The invention of zero, positional notation, and symbolic mathematics has had profound philosophical and technological implications. + + +== Notable works on the history of number systems == + + +== Works on the history of zero == + + +== Children's books on the history of numbers == + + +== Historical texts == + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_countries_by_medal_count_at_International_Mathematical_Olympiad-0.md b/data/en.wikipedia.org/wiki/List_of_countries_by_medal_count_at_International_Mathematical_Olympiad-0.md new file mode 100644 index 000000000..505021e66 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_countries_by_medal_count_at_International_Mathematical_Olympiad-0.md @@ -0,0 +1,19 @@ +--- +title: "List of countries by medal count at International Mathematical Olympiad" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_countries_by_medal_count_at_International_Mathematical_Olympiad" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:15.567788+00:00" +instance: "kb-cron" +--- + +The following is the top 100 list of countries by medal count at the International Mathematical Olympiad: + + +== Notes == + + ^ This team is now defunct. + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-0.md b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-0.md new file mode 100644 index 000000000..492f5af15 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-0.md @@ -0,0 +1,1159 @@ +--- +title: "List of formulae involving π" +chunk: 1/8 +source: "https://en.wikipedia.org/wiki/List_of_formulae_involving_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:10.037546+00:00" +instance: "kb-cron" +--- + +The following is a list of significant formulae involving the mathematical constant π. Many of these formulae can be found in the article Pi, or the article Approximations of π. + +== Euclidean geometry == + + + + + π + = + + + C + d + + + = + + + C + + 2 + r + + + + + + {\displaystyle \pi ={\frac {C}{d}}={\frac {C}{2r}}} + + +where C is the circumference of a circle, d is the diameter, and r is the radius. More generally, + + + + + π + = + + + L + w + + + + + {\displaystyle \pi ={\frac {L}{w}}} + + +where L and w are, respectively, the perimeter and the width of any curve of constant width. + + + + + A + = + π + + r + + 2 + + + + + {\displaystyle A=\pi r^{2}} + + +where A is the area of a circle. More generally, + + + + + A + = + π + a + b + + + {\displaystyle A=\pi ab} + + +where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b. + + + + + C + = + + + + 2 + π + + + agm + ⁡ + ( + a + , + b + ) + + + + + ( + + + a + + 1 + + + 2 + + + − + + ∑ + + n + = + 2 + + + ∞ + + + + 2 + + n + − + 1 + + + ( + + a + + n + + + 2 + + + − + + b + + n + + + 2 + + + ) + + ) + + + + {\displaystyle C={\frac {2\pi }{\operatorname {agm} (a,b)}}\left(a_{1}^{2}-\sum _{n=2}^{\infty }2^{n-1}(a_{n}^{2}-b_{n}^{2})\right)} + + +where C is the circumference of an ellipse with semi-major axis a and semi-minor axis b and + + + + + a + + n + + + , + + b + + n + + + + + {\displaystyle a_{n},b_{n}} + + are the arithmetic and geometric iterations of + + + + agm + ⁡ + ( + a + , + b + ) + + + {\displaystyle \operatorname {agm} (a,b)} + +, the arithmetic-geometric mean of a and b with the initial values + + + + + a + + 0 + + + = + a + + + {\displaystyle a_{0}=a} + + and + + + + + b + + 0 + + + = + b + + + {\displaystyle b_{0}=b} + +. + + + + + A + = + 4 + π + + r + + 2 + + + + + {\displaystyle A=4\pi r^{2}} + + +where A is the area between the witch of Agnesi and its asymptotic line; r is the radius of the defining circle. + + + + + A + = + + + + Γ + ( + 1 + + / + + 4 + + ) + + 2 + + + + + 2 + + + π + + + + + + + r + + 2 + + + = + + + + π + + r + + 2 + + + + + agm + ⁡ + ( + 1 + , + 1 + + / + + + + 2 + + + ) + + + + + + {\displaystyle A={\frac {\Gamma (1/4)^{2}}{2{\sqrt {\pi }}}}r^{2}={\frac {\pi r^{2}}{\operatorname {agm} (1,1/{\sqrt {2}})}}} + + +where A is the area of a squircle with minor radius r, + + + + Γ + + + {\displaystyle \Gamma } + + is the gamma function. + + + + + A + = + ( + k + + + 1 + ) + ( + k + + + 2 + ) + π + + r + + 2 + + + + + {\displaystyle A=(k+1)(k+2)\pi r^{2}} + + +where A is the area of an epicycloid with the smaller circle of radius r and the larger circle of radius kr ( + + + + k + ∈ + + N + + + + {\displaystyle k\in \mathbb {N} } + +), assuming the initial point lies on the larger circle. + + + + + A + = + + + + ( + − + 1 + + ) + + k + + + + + 3 + + 8 + + + π + + a + + 2 + + + + + {\displaystyle A={\frac {(-1)^{k}+3}{8}}\pi a^{2}} + + +where A is the area of a rose with angular frequency k ( + + + + k + ∈ + + N + + + + {\displaystyle k\in \mathbb {N} } + +) and amplitude a. + + + + + L + = + + + + Γ + ( + 1 + + / + + 4 + + ) + + 2 + + + + + π + + + + c + = + + + + 2 + π + c + + + agm + ⁡ + ( + 1 + , + 1 + + / + + + + 2 + + + ) + + + + + + {\displaystyle L={\frac {\Gamma (1/4)^{2}}{\sqrt {\pi }}}c={\frac {2\pi c}{\operatorname {agm} (1,1/{\sqrt {2}})}}} + + +where L is the perimeter of the lemniscate of Bernoulli with focal distance c. + + + + + V + = + + + 4 + 3 + + + π + + r + + 3 + + + + + {\displaystyle V={4 \over 3}\pi r^{3}} + + +where V is the volume of a sphere and r is the radius. + + + + + S + A + = + 4 + π + + r + + 2 + + + + + {\displaystyle SA=4\pi r^{2}} + + +where SA is the surface area of a sphere and r is the radius. + + + + + H + = + + + 1 + 2 + + + + π + + 2 + + + + r + + 4 + + + + + {\displaystyle H={1 \over 2}\pi ^{2}r^{4}} + + +where H is the hypervolume of a 3-sphere and r is the radius. + + + + + S + V + = + 2 + + π + + 2 + + + + r + + 3 + + + + + {\displaystyle SV=2\pi ^{2}r^{3}} + + +where SV is the surface volume of a 3-sphere and r is the radius. + +=== Regular convex polygons === +Sum S of internal angles of a regular convex polygon with n sides: + + + + + S + = + ( + n + − + 2 + ) + π + + + {\displaystyle S=(n-2)\pi } + + +Area A of a regular convex polygon with n sides and side length s: + + + + + A + = + + + + n + + s + + 2 + + + + 4 + + + cot + ⁡ + + + π + n + + + + + {\displaystyle A={\frac {ns^{2}}{4}}\cot {\frac {\pi }{n}}} + + +Inradius r of a regular convex polygon with n sides and side length s: + + + + + r + = + + + s + 2 + + + cot + ⁡ + + + π + n + + + + + {\displaystyle r={\frac {s}{2}}\cot {\frac {\pi }{n}}} + + +Circumradius R of a regular convex polygon with n sides and side length s: + + + + + R + = + + + s + 2 + + + csc + ⁡ + + + π + n + + + + + {\displaystyle R={\frac {s}{2}}\csc {\frac {\pi }{n}}} + + +== Physics == +The cosmological constant: + + + + + Λ + = + + + + 8 + π + G + + + 3 + + c + + 2 + + + + + + ρ + + + {\displaystyle \Lambda ={{8\pi G} \over {3c^{2}}}\rho } + + +Heisenberg's uncertainty principle: + + + + + Δ + x + + Δ + p + ≥ + + + h + + 4 + π + + + + + + {\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}} + + +Einstein's field equation of general relativity: + + + + + + R + + μ + ν + + + − + + + 1 + 2 + + + + g + + μ + ν + + + R + + + Λ + + g + + μ + ν + + + = + + + + 8 + π + G + + + c + + 4 + + + + + + T + + μ + ν + + + + + {\displaystyle R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }} + + +Coulomb's law for the electric force in vacuum: + + + + + F + = + + + + + | + + + q + + 1 + + + + q + + 2 + + + + | + + + + 4 + π + + ε + + 0 + + + + r + + 2 + + + + + + + + {\displaystyle F={\frac {|q_{1}q_{2}|}{4\pi \varepsilon _{0}r^{2}}}} + + +Magnetic permeability of free space: + + + + + + μ + + 0 + + + ≈ + 4 + π + ⋅ + + 10 + + − + 7 + + + + + N + + + / + + + + A + + + 2 + + + + + {\displaystyle \mu _{0}\approx 4\pi \cdot 10^{-7}\,\mathrm {N} /\mathrm {A} ^{2}} + + +Approximate period of a simple pendulum with small amplitude: + + + + + T + ≈ + 2 + π + + + + L + g + + + + + + {\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}} + + +Exact period of a simple pendulum with amplitude + + + + + θ + + 0 + + + + + {\displaystyle \theta _{0}} + + ( + + + + agm + + + {\displaystyle \operatorname {agm} } + + is the arithmetic–geometric mean): + + + + + T + = + + + + 2 + π + + + agm + ⁡ + ( + 1 + , + cos + ⁡ + ( + + θ + + 0 + + + + / + + 2 + ) + ) + + + + + + + L + g + + + + + + {\displaystyle T={\frac {2\pi }{\operatorname {agm} (1,\cos(\theta _{0}/2))}}{\sqrt {\frac {L}{g}}}} + + +Period of a spring-mass system with spring constant + + + + k + + + {\displaystyle k} + + and mass + + + + m + + + {\displaystyle m} + +: + + + + + T + = + 2 + π + + + + m + k + + + + + + {\displaystyle T=2\pi {\sqrt {\frac {m}{k}}}} + + +Kepler's third law of planetary motion: + + + + + + + + R + + 3 + + + + T + + 2 + + + + + = + + + + G + M + + + 4 + + π + + 2 + + + + + + + + {\displaystyle {\frac {R^{3}}{T^{2}}}={\frac {GM}{4\pi ^{2}}}} + + +The buckling formula: + + + + + F + = + + + + + π + + 2 + + + E + I + + + L + + 2 + + + + + + + {\displaystyle F={\frac {\pi ^{2}EI}{L^{2}}}} + + +A puzzle involving "colliding billiard balls": + + + + + ⌊ + + + b + + N + + + π + + ⌋ + + + {\displaystyle \lfloor {b^{N}\pi }\rfloor } + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-1.md b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-1.md new file mode 100644 index 000000000..20547986c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-1.md @@ -0,0 +1,1445 @@ +--- +title: "List of formulae involving π" +chunk: 2/8 +source: "https://en.wikipedia.org/wiki/List_of_formulae_involving_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:10.037546+00:00" +instance: "kb-cron" +--- + +is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b2Nm, when struck by the other object. (This gives the digits of π in base b up to N digits past the radix point.) + +== Formulae yielding π == + +=== Integrals === + + + + + 2 + + ∫ + + − + 1 + + + 1 + + + + + 1 + − + + x + + 2 + + + + + + d + x + = + π + + + {\displaystyle 2\int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx=\pi } + + (integrating two halves + + + + y + ( + x + ) + = + + + 1 + − + + x + + 2 + + + + + + + {\displaystyle y(x)={\sqrt {1-x^{2}}}} + + to obtain the area of the unit circle) + + + + + + ∫ + + 0 + + + 2 + + + + + 4 + − + + x + + 2 + + + + + + d + x + = + π + + + {\displaystyle \int _{0}^{2}{\sqrt {4-x^{2}}}\,dx=\pi } + + (integrating a quarter of a circle with a radius of two + + + + + x + + 2 + + + + + + y + + 2 + + + = + 4 + + + {\displaystyle x^{2}+y^{2}=4} + + to obtain + + + + + 4 + π + + + / + + 4 + + + {\displaystyle {4\pi }/4} + +) + + + + + + ∫ + + − + ∞ + + + ∞ + + + sech + ⁡ + x + + d + x + = + π + + + {\displaystyle \int _{-\infty }^{\infty }\operatorname {sech} x\,dx=\pi } + + + + + + + ∫ + + − + ∞ + + + ∞ + + + + ∫ + + t + + + ∞ + + + + e + + − + 1 + + / + + 2 + + t + + 2 + + + − + + x + + 2 + + + + + x + t + + + + d + x + + d + t + = + + ∫ + + − + ∞ + + + ∞ + + + + ∫ + + t + + + ∞ + + + + e + + − + + t + + 2 + + + − + 1 + + / + + 2 + + x + + 2 + + + + + x + t + + + + d + x + + d + t + = + π + + + {\displaystyle \int _{-\infty }^{\infty }\int _{t}^{\infty }e^{-1/2t^{2}-x^{2}+xt}\,dx\,dt=\int _{-\infty }^{\infty }\int _{t}^{\infty }e^{-t^{2}-1/2x^{2}+xt}\,dx\,dt=\pi } + + + + + + + ∫ + + − + 1 + + + 1 + + + + + + d + x + + + 1 + − + + x + + 2 + + + + + + = + π + + + {\displaystyle \int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}=\pi } + + + + + + + ∫ + + − + ∞ + + + ∞ + + + + + + d + x + + + 1 + + + + x + + 2 + + + + + + = + π + + + {\displaystyle \int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}=\pi } + + (see also Cauchy distribution) + + + + + + ∫ + + − + ∞ + + + ∞ + + + + + + sin + ⁡ + x + + x + + + + d + x + = + π + + + {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin x}{x}}\,dx=\pi } + + (see Dirichlet integral) + + + + + + ∫ + + − + ∞ + + + ∞ + + + + e + + − + + x + + 2 + + + + + + d + x + = + + + π + + + + + {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}} + + (see Gaussian integral). + + + + + ∮ + + + + d + z + + z + + + = + 2 + π + i + + + {\displaystyle \oint {\frac {dz}{z}}=2\pi i} + + (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula). + + + + + + ∫ + + 0 + + + ∞ + + + ln + ⁡ + + ( + + 1 + + + + + 1 + + x + + 2 + + + + + + ) + + + d + x + = + π + + + {\displaystyle \int _{0}^{\infty }\ln \left(1+{\frac {1}{x^{2}}}\right)\,dx=\pi } + + + + + + + ∫ + + 0 + + + 1 + + + + + + + x + + 4 + + + ( + 1 + − + x + + ) + + 4 + + + + + 1 + + + + x + + 2 + + + + + + + d + x + = + + + 22 + 7 + + + − + π + + + {\displaystyle \int _{0}^{1}{x^{4}(1-x)^{4} \over 1+x^{2}}\,dx={22 \over 7}-\pi } + + (see also Proof that 22/7 exceeds π). + + + + + + ∫ + + 0 + + + 1 + + + + + + + x + + 2 + + + ( + 1 + + + x + + ) + + 4 + + + + + 1 + + + + x + + 2 + + + + + + + d + x + = + π + − + + + 17 + 15 + + + + + {\displaystyle \int _{0}^{1}{x^{2}(1+x)^{4} \over 1+x^{2}}\,dx=\pi -{17 \over 15}} + + + + + + + ∫ + + 0 + + + ∞ + + + + + + x + + α + − + 1 + + + + x + + + 1 + + + + + d + x + = + + + π + + sin + ⁡ + π + α + + + + , + + 0 + < + α + < + 1 + + + {\displaystyle \int _{0}^{\infty }{\frac {x^{\alpha -1}}{x+1}}\,dx={\frac {\pi }{\sin \pi \alpha }},\quad 0<\alpha <1} + + + + + + + ∫ + + 0 + + + ∞ + + + + + + d + x + + + x + ( + x + + + a + ) + ( + x + + + b + ) + + + + = + + + π + + agm + ⁡ + ( + + + a + + + , + + + b + + + ) + + + + + + {\displaystyle \int _{0}^{\infty }{\frac {dx}{\sqrt {x(x+a)(x+b)}}}={\frac {\pi }{\operatorname {agm} ({\sqrt {a}},{\sqrt {b}})}}} + + (where + + + + agm + + + {\displaystyle \operatorname {agm} } + + is the arithmetic–geometric mean; see also elliptic integral) +Note that with symmetric integrands + + + + f + ( + − + x + ) + = + f + ( + x + ) + + + {\displaystyle f(-x)=f(x)} + +, formulas of the form + + + + + ∫ + + − + a + + + a + + + f + ( + x + ) + + d + x + + + {\textstyle \int _{-a}^{a}f(x)\,dx} + + can also be translated to formulas + + + + 2 + + ∫ + + 0 + + + a + + + f + ( + x + ) + + d + x + + + {\textstyle 2\int _{0}^{a}f(x)\,dx} + +. + +=== Efficient infinite series === + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + k + ! + + + ( + 2 + k + + + 1 + ) + ! + ! + + + + = + + ∑ + + k + = + 0 + + + ∞ + + + + + + + 2 + + k + + + k + + ! + + 2 + + + + + ( + 2 + k + + + 1 + ) + ! + + + + = + + + π + 2 + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}={\frac {\pi }{2}}} + + (see also Double factorial) + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + k + ! + + + + 2 + + k + + + ( + 2 + k + + + 1 + ) + ! + ! + + + + = + + + + 2 + π + + + 3 + + + 3 + + + + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {k!}{2^{k}(2k+1)!!}}={\frac {2\pi }{3{\sqrt {3}}}}} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + k + ! + + ( + 2 + k + ) + ! + + ( + 25 + k + − + 3 + ) + + + ( + 3 + k + ) + ! + + + 2 + + k + + + + + + = + + + π + 2 + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {k!\,(2k)!\,(25k-3)}{(3k)!\,2^{k}}}={\frac {\pi }{2}}} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + ( + 6 + k + ) + ! + ( + 13591409 + + + 545140134 + k + ) + + + ( + 3 + k + ) + ! + ( + k + ! + + ) + + 3 + + + + 640320 + + 3 + k + + + + + + = + + + 4270934400 + + + + 10005 + + + π + + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k}}}={\frac {4270934400}{{\sqrt {10005}}\pi }}} + + (see Chudnovsky algorithm) + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + 4 + k + ) + ! + ( + 1103 + + + 26390 + k + ) + + + ( + k + ! + + ) + + 4 + + + + 396 + + 4 + k + + + + + + = + + + 9801 + + 2 + + + 2 + + + π + + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}={\frac {9801}{2{\sqrt {2}}\pi }}} + + (see Srinivasa Ramanujan, Ramanujan–Sato series) +The following are efficient for calculating arbitrary binary digits of π: + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + 4 + + k + + + + + + ( + + + + 2 + + 4 + k + + + 1 + + + + + + + + 2 + + 4 + k + + + 2 + + + + + + + + 1 + + 4 + k + + + 3 + + + + + ) + + = + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4^{k}}}\left({\frac {2}{4k+1}}+{\frac {2}{4k+2}}+{\frac {1}{4k+3}}\right)=\pi } + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + 16 + + k + + + + + + ( + + + + 4 + + 8 + k + + + 1 + + + + − + + + 2 + + 8 + k + + + 4 + + + + − + + + 1 + + 8 + k + + + 5 + + + + − + + + 1 + + 8 + k + + + 6 + + + + + ) + + = + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)=\pi } + + (see Bailey–Borwein–Plouffe formula) + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + 16 + + k + + + + + + ( + + + + 8 + + 8 + k + + + 2 + + + + + + + + 4 + + 8 + k + + + 3 + + + + + + + + 4 + + 8 + k + + + 4 + + + + − + + + 1 + + 8 + k + + + 7 + + + + + ) + + = + 2 + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {8}{8k+2}}+{\frac {4}{8k+3}}+{\frac {4}{8k+4}}-{\frac {1}{8k+7}}\right)=2\pi } + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-2.md b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-2.md new file mode 100644 index 000000000..e417a6ed4 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-2.md @@ -0,0 +1,1894 @@ +--- +title: "List of formulae involving π" +chunk: 3/8 +source: "https://en.wikipedia.org/wiki/List_of_formulae_involving_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:10.037546+00:00" +instance: "kb-cron" +--- + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + + ( + − + 1 + ) + + + k + + + + 2 + + 10 + k + + + + + + ( + + − + + + + 2 + + 5 + + + + 4 + k + + + 1 + + + + − + + + 1 + + 4 + k + + + 3 + + + + + + + + + 2 + + 8 + + + + 10 + k + + + 1 + + + + − + + + + 2 + + 6 + + + + 10 + k + + + 3 + + + + − + + + + 2 + + 2 + + + + 10 + k + + + 5 + + + + − + + + + 2 + + 2 + + + + 10 + k + + + 7 + + + + + + + + 1 + + 10 + k + + + 9 + + + + + ) + + = + + 2 + + 6 + + + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {{(-1)}^{k}}{2^{10k}}}\left(-{\frac {2^{5}}{4k+1}}-{\frac {1}{4k+3}}+{\frac {2^{8}}{10k+1}}-{\frac {2^{6}}{10k+3}}-{\frac {2^{2}}{10k+5}}-{\frac {2^{2}}{10k+7}}+{\frac {1}{10k+9}}\right)=2^{6}\pi } + + +Plouffe's series for calculating arbitrary decimal digits of π: + + + + + + ∑ + + k + = + 1 + + + ∞ + + + k + + + + + 2 + + k + + + k + + ! + + 2 + + + + + ( + 2 + k + ) + ! + + + + = + π + + + 3 + + + {\displaystyle \sum _{k=1}^{\infty }k{\frac {2^{k}k!^{2}}{(2k)!}}=\pi +3} + + +=== Other infinite series === + + + + + ζ + ( + 2 + ) + = + + + 1 + + 1 + + 2 + + + + + + + + + 1 + + 2 + + 2 + + + + + + + + + 1 + + 3 + + 2 + + + + + + + + + 1 + + 4 + + 2 + + + + + + + ⋯ + = + + + + π + + 2 + + + 6 + + + + + {\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}} + + (see also Basel problem and Riemann zeta function) + + + + + ζ + ( + 4 + ) + = + + + 1 + + 1 + + 4 + + + + + + + + + 1 + + 2 + + 4 + + + + + + + + + 1 + + 3 + + 4 + + + + + + + + + 1 + + 4 + + 4 + + + + + + + ⋯ + = + + + + π + + 4 + + + 90 + + + + + {\displaystyle \zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}} + + + + + + ζ + ( + 2 + n + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + k + + 2 + n + + + + + + = + + + 1 + + 1 + + 2 + n + + + + + + + + + 1 + + 2 + + 2 + n + + + + + + + + + 1 + + 3 + + 2 + n + + + + + + + + + 1 + + 4 + + 2 + n + + + + + + + ⋯ + = + ( + − + 1 + + ) + + n + + + 1 + + + + + + + B + + 2 + n + + + ( + 2 + π + + ) + + 2 + n + + + + + 2 + ( + 2 + n + ) + ! + + + + + + {\displaystyle \zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}\,={\frac {1}{1^{2n}}}+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+{\frac {1}{4^{2n}}}+\cdots =(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}} + + , where B2n is a Bernoulli number. + + + + + + ∑ + + n + = + 1 + + + ∞ + + + + + + + 3 + + n + + + − + 1 + + + 4 + + n + + + + + + ζ + ( + n + + + 1 + ) + = + π + + + {\displaystyle \sum _{n=1}^{\infty }{\frac {3^{n}-1}{4^{n}}}\,\zeta (n+1)=\pi } + + + + + + + ∑ + + n + = + 1 + + + ∞ + + + + + + + 7 + + n + + + − + 1 + + + 8 + + n + + + + + + ζ + ( + n + + + 1 + ) + = + ( + 1 + + + + + 2 + + + ) + π + + + {\displaystyle \sum _{n=1}^{\infty }{\frac {7^{n}-1}{8^{n}}}\,\zeta (n+1)=(1+{\sqrt {2}})\pi } + + + + + + + ∑ + + n + = + 2 + + + ∞ + + + + + + 2 + ( + 3 + + / + + 2 + + ) + + n + + + − + 3 + + n + + + ( + ζ + ( + n + ) + − + 1 + ) + = + ln + ⁡ + π + + + {\displaystyle \sum _{n=2}^{\infty }{\frac {2(3/2)^{n}-3}{n}}(\zeta (n)-1)=\ln \pi } + + + + + + + ∑ + + n + = + 1 + + + ∞ + + + ζ + ( + 2 + n + ) + + + + x + + 2 + n + + + n + + + = + ln + ⁡ + + + + π + x + + + sin + ⁡ + π + x + + + + , + + 0 + < + + | + + x + + | + + < + 1 + + + {\displaystyle \sum _{n=1}^{\infty }\zeta (2n){\frac {x^{2n}}{n}}=\ln {\frac {\pi x}{\sin \pi x}},\quad 0<|x|<1} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + n + + + + + 2 + n + + + 1 + + + + = + 1 + − + + + 1 + 3 + + + + + + + 1 + 5 + + + − + + + 1 + 7 + + + + + + + 1 + 9 + + + − + ⋯ + = + arctan + ⁡ + + 1 + + = + + + π + 4 + + + + + {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}={\frac {\pi }{4}}} + + (see Leibniz formula for pi) + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + ( + + + + 1 + + 3 + n + + + 1 + + + + − + + + 1 + + 3 + n + + + 2 + + + + + ) + + = + 1 + − + + + 1 + 2 + + + + + + + 1 + 4 + + + − + + + 1 + 5 + + + + + ⋯ + = + + + π + + 3 + + + 3 + + + + + + + + {\displaystyle \sum _{n=0}^{\infty }\left({\frac {1}{3n+1}}-{\frac {1}{3n+2}}\right)=1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{5}}+\cdots ={\frac {\pi }{3{\sqrt {3}}}}} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + ( + + n + + 2 + + + − + n + ) + + / + + 2 + + + + + 2 + n + + + 1 + + + + = + 1 + + + + + 1 + 3 + + + − + + + 1 + 5 + + + − + + + 1 + 7 + + + + + + + 1 + 9 + + + + + + + 1 + 11 + + + − + ⋯ + = + + + π + + 2 + + + 2 + + + + + + + + {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{(n^{2}-n)/2}}{2n+1}}=1+{\frac {1}{3}}-{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}-\cdots ={\frac {\pi }{2{\sqrt {2}}}}} + + (Newton, Second Letter to Oldenburg, 1676) + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + n + + + + + + 3 + + n + + + ( + 2 + n + + + 1 + ) + + + + = + 1 + − + + + 1 + + + 3 + + 1 + + + ⋅ + 3 + + + + + + + + 1 + + + 3 + + 2 + + + ⋅ + 5 + + + + − + + + 1 + + + 3 + + 3 + + + ⋅ + 7 + + + + + + + + 1 + + + 3 + + 4 + + + ⋅ + 9 + + + + − + ⋯ + = + + + 3 + + + arctan + ⁡ + + + 1 + + 3 + + + + = + + + π + + 2 + + + 3 + + + + + + + + {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{3^{n}(2n+1)}}=1-{\frac {1}{3^{1}\cdot 3}}+{\frac {1}{3^{2}\cdot 5}}-{\frac {1}{3^{3}\cdot 7}}+{\frac {1}{3^{4}\cdot 9}}-\cdots ={\sqrt {3}}\arctan {\frac {1}{\sqrt {3}}}={\frac {\pi }{2{\sqrt {3}}}}} + + (Madhava series) + + + + + + ∑ + + n + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + n + + + 1 + + + + + n + + 2 + + + + + = + + + 1 + + 1 + + 2 + + + + + − + + + 1 + + 2 + + 2 + + + + + + + + + 1 + + 3 + + 2 + + + + + − + + + 1 + + 4 + + 2 + + + + + + + ⋯ + = + + + + π + + 2 + + + 12 + + + + + {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}-{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{12}}} + + + + + + + ∑ + + n + = + 1 + + + ∞ + + + + + 1 + + ( + 2 + n + + ) + + 2 + + + + + + = + + + 1 + + 2 + + 2 + + + + + + + + + 1 + + 4 + + 2 + + + + + + + + + 1 + + 6 + + 2 + + + + + + + + + 1 + + 8 + + 2 + + + + + + + ⋯ + = + + + + π + + 2 + + + 24 + + + + + {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(2n)^{2}}}={\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}+{\frac {1}{6^{2}}}+{\frac {1}{8^{2}}}+\cdots ={\frac {\pi ^{2}}{24}}} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + ( + + + 1 + + 2 + n + + + 1 + + + + ) + + + 2 + + + = + + + 1 + + 1 + + 2 + + + + + + + + + 1 + + 3 + + 2 + + + + + + + + + 1 + + 5 + + 2 + + + + + + + + + 1 + + 7 + + 2 + + + + + + + ⋯ + = + + + + π + + 2 + + + 8 + + + + + {\displaystyle \sum _{n=0}^{\infty }\left({\frac {1}{2n+1}}\right)^{2}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots ={\frac {\pi ^{2}}{8}}} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + ( + + + + ( + − + 1 + + ) + + n + + + + + 2 + n + + + 1 + + + + ) + + + 3 + + + = + + + 1 + + 1 + + 3 + + + + + − + + + 1 + + 3 + + 3 + + + + + + + + + 1 + + 5 + + 3 + + + + + − + + + 1 + + 7 + + 3 + + + + + + + ⋯ + = + + + + π + + 3 + + + 32 + + + + + {\displaystyle \sum _{n=0}^{\infty }\left({\frac {(-1)^{n}}{2n+1}}\right)^{3}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+\cdots ={\frac {\pi ^{3}}{32}}} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + ( + + + 1 + + 2 + n + + + 1 + + + + ) + + + 4 + + + = + + + 1 + + 1 + + 4 + + + + + + + + + 1 + + 3 + + 4 + + + + + + + + + 1 + + 5 + + 4 + + + + + + + + + 1 + + 7 + + 4 + + + + + + + ⋯ + = + + + + π + + 4 + + + 96 + + + + + {\displaystyle \sum _{n=0}^{\infty }\left({\frac {1}{2n+1}}\right)^{4}={\frac {1}{1^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{5^{4}}}+{\frac {1}{7^{4}}}+\cdots ={\frac {\pi ^{4}}{96}}} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + ( + + + + ( + − + 1 + + ) + + n + + + + + 2 + n + + + 1 + + + + ) + + + 5 + + + = + + + 1 + + 1 + + 5 + + + + + − + + + 1 + + 3 + + 5 + + + + + + + + + 1 + + 5 + + 5 + + + + + − + + + 1 + + 7 + + 5 + + + + + + + ⋯ + = + + + + 5 + + π + + 5 + + + + 1536 + + + + + {\displaystyle \sum _{n=0}^{\infty }\left({\frac {(-1)^{n}}{2n+1}}\right)^{5}={\frac {1}{1^{5}}}-{\frac {1}{3^{5}}}+{\frac {1}{5^{5}}}-{\frac {1}{7^{5}}}+\cdots ={\frac {5\pi ^{5}}{1536}}} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + ( + + + 1 + + 2 + n + + + 1 + + + + ) + + + 6 + + + = + + + 1 + + 1 + + 6 + + + + + + + + + 1 + + 3 + + 6 + + + + + + + + + 1 + + 5 + + 6 + + + + + + + + + 1 + + 7 + + 6 + + + + + + + ⋯ + = + + + + π + + 6 + + + 960 + + + + + {\displaystyle \sum _{n=0}^{\infty }\left({\frac {1}{2n+1}}\right)^{6}={\frac {1}{1^{6}}}+{\frac {1}{3^{6}}}+{\frac {1}{5^{6}}}+{\frac {1}{7^{6}}}+\cdots ={\frac {\pi ^{6}}{960}}} + + +In general, \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-3.md b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-3.md new file mode 100644 index 000000000..13b076490 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-3.md @@ -0,0 +1,1707 @@ +--- +title: "List of formulae involving π" +chunk: 4/8 +source: "https://en.wikipedia.org/wiki/List_of_formulae_involving_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:10.037546+00:00" +instance: "kb-cron" +--- + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + n + + + + + ( + 2 + n + + + 1 + + ) + + 2 + k + + + 1 + + + + + + = + ( + − + 1 + + ) + + k + + + + + + E + + 2 + k + + + + 2 + ( + 2 + k + ) + ! + + + + + + ( + + + π + 2 + + + ) + + + 2 + k + + + 1 + + + , + + k + ∈ + + + N + + + 0 + + + + + {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2k+1}}}=(-1)^{k}{\frac {E_{2k}}{2(2k)!}}\left({\frac {\pi }{2}}\right)^{2k+1},\quad k\in \mathbb {N} _{0}} + + +where + + + + + E + + 2 + k + + + + + {\displaystyle E_{2k}} + + is the + + + + 2 + k + + + {\displaystyle 2k} + +th Euler number. + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + ( + + + + 1 + 2 + + n + + + ) + + + + + + + ( + − + 1 + + ) + + n + + + + + 2 + n + + + 1 + + + + = + 1 + − + + + 1 + 6 + + + − + + + 1 + 40 + + + − + ⋯ + = + + + π + 4 + + + + + {\displaystyle \sum _{n=0}^{\infty }{\binom {\frac {1}{2}}{n}}{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{6}}-{\frac {1}{40}}-\cdots ={\frac {\pi }{4}}} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + 1 + + ( + 4 + n + + + 1 + ) + ( + 4 + n + + + 3 + ) + + + + = + + + 1 + + 1 + ⋅ + 3 + + + + + + + + 1 + + 5 + ⋅ + 7 + + + + + + + + 1 + + 9 + ⋅ + 11 + + + + + + ⋯ + = + + + π + 8 + + + + + {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{(4n+1)(4n+3)}}={\frac {1}{1\cdot 3}}+{\frac {1}{5\cdot 7}}+{\frac {1}{9\cdot 11}}+\cdots ={\frac {\pi }{8}}} + + + + + + + ∑ + + n + = + 1 + + + ∞ + + + ( + − + 1 + + ) + + ( + + n + + 2 + + + + + n + ) + + / + + 2 + + + 1 + + + + | + + G + + + ( + + ( + − + 1 + + ) + + n + + + 1 + + + + + 6 + n + − + 3 + + ) + + + / + + 4 + + + | + + = + + | + + + G + + 1 + + + + | + + + + + | + + + G + + 2 + + + + | + + − + + | + + + G + + 4 + + + + | + + − + + | + + + G + + 5 + + + + | + + + + + | + + + G + + 7 + + + + | + + + + + | + + + G + + 8 + + + + | + + − + + | + + + G + + 10 + + + + | + + − + + | + + + G + + 11 + + + + | + + + + ⋯ + = + + + + 3 + + π + + + + + {\displaystyle \sum _{n=1}^{\infty }(-1)^{(n^{2}+n)/2+1}\left|G_{\left((-1)^{n+1}+6n-3\right)/4}\right|=|G_{1}|+|G_{2}|-|G_{4}|-|G_{5}|+|G_{7}|+|G_{8}|-|G_{10}|-|G_{11}|+\cdots ={\frac {\sqrt {3}}{\pi }}} + + (see Gregory coefficients) + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + ( + 1 + + / + + 2 + + ) + + n + + + 2 + + + + + + 2 + + n + + + n + + ! + + 2 + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + n + ( + 1 + + / + + 2 + + ) + + n + + + 2 + + + + + + 2 + + n + + + n + + ! + + 2 + + + + + + = + + + 1 + π + + + + + {\displaystyle \sum _{n=0}^{\infty }{\frac {(1/2)_{n}^{2}}{2^{n}n!^{2}}}\sum _{n=0}^{\infty }{\frac {n(1/2)_{n}^{2}}{2^{n}n!^{2}}}={\frac {1}{\pi }}} + + (where + + + + ( + x + + ) + + n + + + + + {\displaystyle (x)_{n}} + + is the rising factorial) + + + + + + ∑ + + n + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + n + + + 1 + + + + + n + ( + n + + + 1 + ) + ( + 2 + n + + + 1 + ) + + + + = + π + − + 3 + + + {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n(n+1)(2n+1)}}=\pi -3} + + (Nilakantha series) + + + + + + ∑ + + n + = + 1 + + + ∞ + + + + + + F + + 2 + n + + + + + n + + 2 + + + + + + ( + + + + 2 + n + + n + + + ) + + + + + + + = + + + + 4 + + π + + 2 + + + + + 25 + + + 5 + + + + + + + + {\displaystyle \sum _{n=1}^{\infty }{\frac {F_{2n}}{n^{2}{\binom {2n}{n}}}}={\frac {4\pi ^{2}}{25{\sqrt {5}}}}} + + (where + + + + + F + + 2 + n + + + + + {\displaystyle F_{2n}} + + is the + + + + 2 + n + + + {\displaystyle 2n} + +th Fibonacci number) + + + + + + ∑ + + n + = + 1 + + + ∞ + + + + + + L + + 2 + n + + + + + n + + 2 + + + + + + ( + + + + 2 + n + + n + + + ) + + + + + + + = + + + + π + + 2 + + + 5 + + + + + {\displaystyle \sum _{n=1}^{\infty }{\frac {L_{2n}}{n^{2}{\binom {2n}{n}}}}={\frac {\pi ^{2}}{5}}} + + (where + + + + + L + + n + + + + + {\displaystyle L_{n}} + + is the + + + + n + + + {\displaystyle n} + +th Lucas number) + + + + + + ∑ + + n + = + 1 + + + ∞ + + + σ + ( + n + ) + + e + + − + 2 + π + n + + + = + + + 1 + 24 + + + − + + + 1 + + 8 + π + + + + + + {\displaystyle \sum _{n=1}^{\infty }\sigma (n)e^{-2\pi n}={\frac {1}{24}}-{\frac {1}{8\pi }}} + + (where + + + + σ + + + {\displaystyle \sigma } + + is the sum-of-divisors function) + + + + + π + = + + ∑ + + n + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + ε + ( + n + ) + + + + n + + + = + 1 + + + + + 1 + 2 + + + + + + + 1 + 3 + + + + + + + 1 + 4 + + + − + + + 1 + 5 + + + + + + + 1 + 6 + + + + + + + 1 + 7 + + + + + + + 1 + 8 + + + + + + + 1 + 9 + + + − + + + 1 + 10 + + + + + + + 1 + 11 + + + + + + + 1 + 12 + + + − + + + 1 + 13 + + + + + ⋯ + + + {\displaystyle \pi =\sum _{n=1}^{\infty }{\frac {(-1)^{\varepsilon (n)}}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}-{\frac {1}{10}}+{\frac {1}{11}}+{\frac {1}{12}}-{\frac {1}{13}}+\cdots } + + (where + + + + ε + ( + n + ) + + + {\displaystyle \varepsilon (n)} + + is the number of prime factors of the form + + + + p + ≡ + 1 + + ( + + m + o + d + + + 4 + ) + + + {\displaystyle p\equiv 1\,(\mathrm {mod} \,4)} + + of + + + + n + + + {\displaystyle n} + +) + + + + + + + π + 2 + + + = + + ∑ + + n + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + ε + ( + n + ) + + + + n + + + = + 1 + + + + + 1 + 2 + + + − + + + 1 + 3 + + + + + + + 1 + 4 + + + + + + + 1 + 5 + + + − + + + 1 + 6 + + + − + + + 1 + 7 + + + + + + + 1 + 8 + + + + + + + 1 + 9 + + + + + ⋯ + + + {\displaystyle {\frac {\pi }{2}}=\sum _{n=1}^{\infty }{\frac {(-1)^{\varepsilon (n)}}{n}}=1+{\frac {1}{2}}-{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}-{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}+\cdots } + + (where + + + + ε + ( + n + ) + + + {\displaystyle \varepsilon (n)} + + is the number of prime factors of the form + + + + p + ≡ + 3 + + ( + + m + o + d + + + 4 + ) + + + {\displaystyle p\equiv 3\,(\mathrm {mod} \,4)} + + of + + + + n + + + {\displaystyle n} + +) + + + + + π + = + + ∑ + + n + = + − + ∞ + + + ∞ + + + + + + ( + − + 1 + + ) + + n + + + + + n + + + 1 + + / + + 2 + + + + + + {\displaystyle \pi =\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{n+1/2}}} + + + + + + + π + + 2 + + + = + + ∑ + + n + = + − + ∞ + + + ∞ + + + + + 1 + + ( + n + + + 1 + + / + + 2 + + ) + + 2 + + + + + + + + {\displaystyle \pi ^{2}=\sum _{n=-\infty }^{\infty }{\frac {1}{(n+1/2)^{2}}}} + + +The last two formulas are special cases of + + + + + + + + + + + π + + sin + ⁡ + π + x + + + + + + + = + + ∑ + + n + = + − + ∞ + + + ∞ + + + + + + ( + − + 1 + + ) + + n + + + + + n + + + x + + + + + + + + + + ( + + + π + + sin + ⁡ + π + x + + + + ) + + + 2 + + + + + + = + + ∑ + + n + = + − + ∞ + + + ∞ + + + + + 1 + + ( + n + + + x + + ) + + 2 + + + + + + + + + + + + {\displaystyle {\begin{aligned}{\frac {\pi }{\sin \pi x}}&=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{n+x}}\\\left({\frac {\pi }{\sin \pi x}}\right)^{2}&=\sum _{n=-\infty }^{\infty }{\frac {1}{(n+x)^{2}}}\end{aligned}}} + + +which generate infinitely many analogous formulas for + + + + π + + + {\displaystyle \pi } + + when + + + + x + ∈ + + Q + + ∖ + + Z + + . + + + {\displaystyle x\in \mathbb {Q} \setminus \mathbb {Z} .} + + + + + + π + = + + + + ∑ + + n + = + 1 + + + ∞ + + + + + 6 + + n + + 2 + + + + + + + + + {\displaystyle \pi ={\sqrt {\sum _{n=1}^{\infty }{\frac {6}{n^{2}}}}}} + + (derived from Euler's solution to the Basel problem) +Some formulas relating π and harmonic numbers are given here. Further infinite series involving π are: + +where + + + + ( + x + + ) + + n + + + + + {\displaystyle (x)_{n}} + + is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series. + +=== Machin-like formulae === + + + + + + + π + 4 + + + = + arctan + ⁡ + 1 + + + {\displaystyle {\frac {\pi }{4}}=\arctan 1} + + + + + + + + π + 4 + + + = + arctan + ⁡ + + + 1 + 2 + + + + + arctan + ⁡ + + + 1 + 3 + + + + + {\displaystyle {\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}}} + + + + + + + + π + 4 + + + = + 2 + arctan + ⁡ + + + 1 + 2 + + + − + arctan + ⁡ + + + 1 + 7 + + + + + {\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{2}}-\arctan {\frac {1}{7}}} + + + + + + + + π + 4 + + + = + 2 + arctan + ⁡ + + + 1 + 3 + + + + + arctan + ⁡ + + + 1 + 7 + + + + + {\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}} + + + + + + + + π + 4 + + + = + 4 + arctan + ⁡ + + + 1 + 5 + + + − + arctan + ⁡ + + + 1 + 239 + + + + + {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}} + + (the original Machin's formula) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-4.md b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-4.md new file mode 100644 index 000000000..9a64597c1 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-4.md @@ -0,0 +1,1546 @@ +--- +title: "List of formulae involving π" +chunk: 5/8 +source: "https://en.wikipedia.org/wiki/List_of_formulae_involving_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:10.037546+00:00" +instance: "kb-cron" +--- + + + + + + + π + 4 + + + = + 5 + arctan + ⁡ + + + 1 + 7 + + + + + 2 + arctan + ⁡ + + + 3 + 79 + + + + + {\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}} + + + + + + + + π + 4 + + + = + 6 + arctan + ⁡ + + + 1 + 8 + + + + + 2 + arctan + ⁡ + + + 1 + 57 + + + + + arctan + ⁡ + + + 1 + 239 + + + + + {\displaystyle {\frac {\pi }{4}}=6\arctan {\frac {1}{8}}+2\arctan {\frac {1}{57}}+\arctan {\frac {1}{239}}} + + + + + + + + π + 4 + + + = + 12 + arctan + ⁡ + + + 1 + 49 + + + + + 32 + arctan + ⁡ + + + 1 + 57 + + + − + 5 + arctan + ⁡ + + + 1 + 239 + + + + + 12 + arctan + ⁡ + + + 1 + 110443 + + + + + {\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}} + + + + + + + + π + 4 + + + = + 44 + arctan + ⁡ + + + 1 + 57 + + + + + 7 + arctan + ⁡ + + + 1 + 239 + + + − + 12 + arctan + ⁡ + + + 1 + 682 + + + + + 24 + arctan + ⁡ + + + 1 + 12943 + + + + + {\displaystyle {\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}} + + +=== Infinite products === + + + + + + + π + 4 + + + = + + ( + + + ∏ + + p + ≡ + 1 + + + ( + mod + + 4 + ) + + + + + + p + + p + − + 1 + + + + + ) + + ⋅ + + ( + + + ∏ + + p + ≡ + 3 + + + ( + mod + + 4 + ) + + + + + + p + + p + + + 1 + + + + + ) + + = + + + 3 + 4 + + + ⋅ + + + 5 + 4 + + + ⋅ + + + 7 + 8 + + + ⋅ + + + 11 + 12 + + + ⋅ + + + 13 + 12 + + + ⋯ + , + + + {\displaystyle {\frac {\pi }{4}}=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p-1}}\right)\cdot \left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p+1}}\right)={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdots ,} + + (Euler) +where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator. + + + + + + + + + + 3 + + + π + + 6 + + + = + + ( + + + ∏ + + + + p + ≡ + 1 + + + ( + mod + + 6 + ) + + + + p + ∈ + + P + + + + + + + + p + + p + − + 1 + + + + + ) + + ⋅ + + ( + + + ∏ + + + + p + ≡ + 5 + + + ( + mod + + 6 + ) + + + + p + ∈ + + P + + + + + + + + p + + p + + + 1 + + + + + ) + + = + + + 5 + 6 + + + ⋅ + + + 7 + 6 + + + ⋅ + + + 11 + 12 + + + ⋅ + + + 13 + 12 + + + ⋅ + + + 17 + 18 + + + ⋯ + + + {\displaystyle {\frac {{\sqrt {3}}\pi }{6}}=\left(\displaystyle \prod _{p\equiv 1{\pmod {6}} \atop p\in \mathbb {P} }{\frac {p}{p-1}}\right)\cdot \left(\displaystyle \prod _{p\equiv 5{\pmod {6}} \atop p\in \mathbb {P} }{\frac {p}{p+1}}\right)={\frac {5}{6}}\cdot {\frac {7}{6}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdot {\frac {17}{18}}\cdots } + + + + + + + + + 2 + + π + + 2 + + + + 25 + + + = + + ( + + + ∏ + + + + p + ≡ + 1 + + + ( + mod + + 5 + ) + + + + p + ∈ + + P + + + + + + + + + p + + 2 + + + + ( + p + − + 1 + + ) + + 2 + + + + + + + ) + + + ( + + + ∏ + + + + p + ≡ + 2 + , + 3 + + + ( + mod + + 5 + ) + + + + p + ∈ + + P + + + + + + + + + p + + 2 + + + + + p + + 2 + + + + + 1 + + + + + ) + + + ( + + + ∏ + + + + p + ≡ + 4 + + + ( + mod + + 5 + ) + + + + p + ∈ + + P + + + + + + + + + p + + 2 + + + + ( + p + + + 1 + + ) + + 2 + + + + + + + ) + + = + + + 4 + 5 + + + ⋅ + + + 9 + 10 + + + ⋅ + + + 49 + 50 + + + ⋅ + + + 121 + 100 + + + ⋅ + + + 169 + 170 + + + ⋯ + + + {\displaystyle {\frac {2\pi ^{2}}{25}}=\left(\displaystyle \prod _{p\equiv 1{\pmod {5}} \atop p\in \mathbb {P} }{\frac {p^{2}}{(p-1)^{2}}}\right)\left(\displaystyle \prod _{p\equiv 2,3{\pmod {5}} \atop p\in \mathbb {P} }{\frac {p^{2}}{p^{2}+1}}\right)\left(\displaystyle \prod _{p\equiv 4{\pmod {5}} \atop p\in \mathbb {P} }{\frac {p^{2}}{(p+1)^{2}}}\right)={\frac {4}{5}}\cdot {\frac {9}{10}}\cdot {\frac {49}{50}}\cdot {\frac {121}{100}}\cdot {\frac {169}{170}}\cdots } + +, + + + + + + + π + 2 + + + = + + ∏ + + n + = + 1 + + + ∞ + + + + + + ( + 2 + n + ) + ( + 2 + n + ) + + + ( + 2 + n + − + 1 + ) + ( + 2 + n + + + 1 + ) + + + + = + + + 2 + 1 + + + ⋅ + + + 2 + 3 + + + ⋅ + + + 4 + 3 + + + ⋅ + + + 4 + 5 + + + ⋅ + + + 6 + 5 + + + ⋅ + + + 6 + 7 + + + ⋅ + + + 8 + 7 + + + ⋅ + + + 8 + 9 + + + ⋯ + + + {\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }{\frac {(2n)(2n)}{(2n-1)(2n+1)}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots } + + (see also Wallis product) + + + + + + + π + 2 + + + = + + ∏ + + n + = + 1 + + + ∞ + + + + + ( + + 1 + + + + + 1 + n + + + + ) + + + ( + − + 1 + + ) + + n + + + 1 + + + + + = + + + ( + + 1 + + + + + 1 + 1 + + + + ) + + + + + 1 + + + + + ( + + 1 + + + + + 1 + 2 + + + + ) + + + − + 1 + + + + + ( + + 1 + + + + + 1 + 3 + + + + ) + + + + + 1 + + + ⋯ + + + {\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{n}}\right)^{(-1)^{n+1}}=\left(1+{\frac {1}{1}}\right)^{+1}\left(1+{\frac {1}{2}}\right)^{-1}\left(1+{\frac {1}{3}}\right)^{+1}\cdots } + + (another form of Wallis product) +Viète's formula: + + + + + + + 2 + π + + + = + + + + 2 + + 2 + + + ⋅ + + + + 2 + + + + + 2 + + + + 2 + + + ⋅ + + + + 2 + + + + + 2 + + + + + 2 + + + + + + 2 + + + ⋅ + ⋯ + + + {\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots } + + +A double infinite product formula involving the Thue–Morse sequence: + + + + + + + π + 2 + + + = + + ∏ + + m + ≥ + 1 + + + + ∏ + + n + ≥ + 1 + + + + + ( + + + + ( + 4 + + m + + 2 + + + + + n + − + 2 + ) + ( + 4 + + m + + 2 + + + + + 2 + n + − + 1 + + ) + + 2 + + + + + 4 + ( + 2 + + m + + 2 + + + + + n + − + 1 + ) + ( + 4 + + m + + 2 + + + + + n + − + 1 + ) + ( + 2 + + m + + 2 + + + + + n + ) + + + + ) + + + + ϵ + + n + + + + + , + + + {\displaystyle {\frac {\pi }{2}}=\prod _{m\geq 1}\prod _{n\geq 1}\left({\frac {(4m^{2}+n-2)(4m^{2}+2n-1)^{2}}{4(2m^{2}+n-1)(4m^{2}+n-1)(2m^{2}+n)}}\right)^{\epsilon _{n}},} + + +where + + + + + ϵ + + n + + + = + ( + − + 1 + + ) + + + t + + n + + + + + + + {\displaystyle \epsilon _{n}=(-1)^{t_{n}}} + + and + + + + + t + + n + + + + + {\displaystyle t_{n}} + + is the Thue–Morse sequence (Tóth 2020). +One elegant infinite product formula: + + + + + π + = + + ∏ + + n + = + 1 + + + ∞ + + + + ( + + 1 + + + + + + ( + n + + + 2 + ) + + + ( + 2 + n + + + 1 + ) + + + ( + 2 + n + − + 1 + ) + + + ( + n + + + 2 + ) + × + ( + 2 + n + + + 1 + ) + × + ( + 2 + n + − + 1 + ) + + + + + ) + + + + {\displaystyle \pi =\prod _{n=1}^{\infty }\left(1+{\frac {(n+2)+(2n+1)+(2n-1)}{(n+2)\times (2n+1)\times (2n-1)}}\right)} + + +=== Arctangent formulas === + + + + + + + π + + 2 + + k + + + 1 + + + + + = + arctan + ⁡ + + + + 2 + − + + a + + k + − + 1 + + + + + a + + k + + + + + , + + + k + ≥ + 2 + + + {\displaystyle {\frac {\pi }{2^{k+1}}}=\arctan {\frac {\sqrt {2-a_{k-1}}}{a_{k}}},\qquad \qquad k\geq 2} + + + + + + + + π + 4 + + + = + + ∑ + + k + ≥ + 2 + + + arctan + ⁡ + + + + 2 + − + + a + + k + − + 1 + + + + + a + + k + + + + + , + + + {\displaystyle {\frac {\pi }{4}}=\sum _{k\geq 2}\arctan {\frac {\sqrt {2-a_{k-1}}}{a_{k}}},} + + +where + + + + + a + + k + + + = + + + 2 + + + + a + + k + − + 1 + + + + + + + {\displaystyle a_{k}={\sqrt {2+a_{k-1}}}} + + such that + + + + + a + + 1 + + + = + + + 2 + + + + + {\displaystyle a_{1}={\sqrt {2}}} + +. + + + + + + + π + 2 + + + = + + ∑ + + k + = + 0 + + + ∞ + + + arctan + ⁡ + + + 1 + + F + + 2 + k + + + 1 + + + + + = + arctan + ⁡ + + + 1 + 1 + + + + + arctan + ⁡ + + + 1 + 2 + + + + + arctan + ⁡ + + + 1 + 5 + + + + + arctan + ⁡ + + + 1 + 13 + + + + + ⋯ + + + {\displaystyle {\frac {\pi }{2}}=\sum _{k=0}^{\infty }\arctan {\frac {1}{F_{2k+1}}}=\arctan {\frac {1}{1}}+\arctan {\frac {1}{2}}+\arctan {\frac {1}{5}}+\arctan {\frac {1}{13}}+\cdots } + + +where + + + + + F + + k + + + + + {\displaystyle F_{k}} + + is the + + + + k + + + {\displaystyle k} + +th Fibonacci number. + + + + + + + π + 4 + + + = + arctan + ⁡ + + + a + + b + + + c + + + + + + arctan + ⁡ + + + b + + a + + + c + + + + , + + + {\displaystyle {\frac {\pi }{4}}=\arctan {\frac {a}{b+c}}+\arctan {\frac {b}{a+c}},} + + +For Pythagorean triple (a,b,c). + + + + + π + = + arctan + ⁡ + a + + + arctan + ⁡ + b + + + arctan + ⁡ + c + + + {\displaystyle \pi =\arctan a+\arctan b+\arctan c} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-5.md b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-5.md new file mode 100644 index 000000000..33658ccd0 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-5.md @@ -0,0 +1,2098 @@ +--- +title: "List of formulae involving π" +chunk: 6/8 +source: "https://en.wikipedia.org/wiki/List_of_formulae_involving_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:10.037546+00:00" +instance: "kb-cron" +--- + +whenever + + + + a + + + b + + + c + = + a + b + c + + + {\displaystyle a+b+c=abc} + + and + + + + a + + + {\displaystyle a} + +, + + + + b + + + {\displaystyle b} + +, + + + + c + + + {\displaystyle c} + + are positive real numbers (see List of trigonometric identities). A special case is + + + + + π + = + arctan + ⁡ + 1 + + + arctan + ⁡ + 2 + + + arctan + ⁡ + 3. + + + {\displaystyle \pi =\arctan 1+\arctan 2+\arctan 3.} + + +=== Complex functions === + + + + + + e + + i + π + + + + + 1 + = + 0 + + + {\displaystyle e^{i\pi }+1=0} + + (Euler's identity) +The following equivalences are true for any complex + + + + z + + + {\displaystyle z} + +: + + + + + + e + + z + + + ∈ + + R + + ↔ + ℑ + z + ∈ + π + + Z + + + + {\displaystyle e^{z}\in \mathbb {R} \leftrightarrow \Im z\in \pi \mathbb {Z} } + + + + + + + e + + z + + + = + 1 + ↔ + z + ∈ + 2 + π + i + + Z + + + + {\displaystyle e^{z}=1\leftrightarrow z\in 2\pi i\mathbb {Z} } + + +Also + + + + + + + 1 + + + e + + z + + + − + 1 + + + + = + + lim + + N + → + ∞ + + + + ∑ + + n + = + − + N + + + N + + + + + 1 + + z + − + 2 + π + i + n + + + + − + + + 1 + 2 + + + , + + z + ∈ + + C + + . + + + {\displaystyle {\frac {1}{e^{z}-1}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z-2\pi in}}-{\frac {1}{2}},\quad z\in \mathbb {C} .} + + +Suppose a lattice + + + + Ω + + + {\displaystyle \Omega } + + is generated by two periods + + + + + ω + + 1 + + + , + + ω + + 2 + + + + + {\displaystyle \omega _{1},\omega _{2}} + +. We define the quasi-periods of this lattice by + + + + + η + + 1 + + + = + ζ + ( + z + + + + ω + + 1 + + + ; + Ω + ) + − + ζ + ( + z + ; + Ω + ) + + + {\displaystyle \eta _{1}=\zeta (z+\omega _{1};\Omega )-\zeta (z;\Omega )} + + and + + + + + η + + 2 + + + = + ζ + ( + z + + + + ω + + 2 + + + ; + Ω + ) + − + ζ + ( + z + ; + Ω + ) + + + {\displaystyle \eta _{2}=\zeta (z+\omega _{2};\Omega )-\zeta (z;\Omega )} + + where + + + + ζ + + + {\displaystyle \zeta } + + is the Weierstrass zeta function ( + + + + + η + + 1 + + + + + {\displaystyle \eta _{1}} + + and + + + + + η + + 2 + + + + + {\displaystyle \eta _{2}} + + are in fact independent of + + + + z + + + {\displaystyle z} + +). Then the periods and quasi-periods are related by the Legendre identity: + + + + + + η + + 1 + + + + ω + + 2 + + + − + + η + + 2 + + + + ω + + 1 + + + = + 2 + π + i + . + + + {\displaystyle \eta _{1}\omega _{2}-\eta _{2}\omega _{1}=2\pi i.} + + +=== Continued fractions === + + + + + + + 4 + π + + + = + 1 + + + + + + + + + + + + 1 + + 2 + + + + + + + + + + + + 2 + + + + + + + + + + + + 3 + + 2 + + + + + + + + + + + + 2 + + + + + + + + + + + + 5 + + 2 + + + + + + + + + + + + 2 + + + + + + + + + + + + 7 + + 2 + + + + + + + + + + + + 2 + + + ⋱ + + + + + + + + + + + + + + + + + + + + + + + {\displaystyle {\frac {4}{\pi }}=1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+\ddots }}}}}}}}} + + + + + + + + + ϖ + + 2 + + + π + + + = + + 2 + + + + + + + + + + + + 1 + + 2 + + + + + + + + + + + + 4 + + + + + + + + + + + + 3 + + 2 + + + + + + + + + + + + 4 + + + + + + + + + + + + 5 + + 2 + + + + + + + + + + + + 4 + + + + + + + + + + + + 7 + + 2 + + + + + + + + + + + + 4 + + + ⋱ + + + + + + + + + + + + + + + + + + + + + + + + + + {\displaystyle {\frac {\varpi ^{2}}{\pi }}={2+{\cfrac {1^{2}}{4+{\cfrac {3^{2}}{4+{\cfrac {5^{2}}{4+{\cfrac {7^{2}}{4+\ddots \,}}}}}}}}}\quad } + + (Ramanujan, + + + + ϖ + + + {\displaystyle \varpi } + + is the lemniscate constant) + + + + + π + = + + 3 + + + + + + + + + + + + 1 + + 2 + + + + + + + + + + + + 6 + + + + + + + + + + + + 3 + + 2 + + + + + + + + + + + + 6 + + + + + + + + + + + + 5 + + 2 + + + + + + + + + + + + 6 + + + + + + + + + + + + 7 + + 2 + + + + + + + + + + + + 6 + + + ⋱ + + + + + + + + + + + + + + + + + + + + + + + + + {\displaystyle \pi ={3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots \,}}}}}}}}}} + + + + + + π + = + + + + + + + + + 4 + + + + + + + + + + 1 + + + + + + + + + + + + 1 + + 2 + + + + + + + + + + + + 3 + + + + + + + + + + + + 2 + + 2 + + + + + + + + + + + + 5 + + + + + + + + + + + + 3 + + 2 + + + + + + + + + + + + 7 + + + + + + + + + + + + 4 + + 2 + + + + + + + + + + + + 9 + + + ⋱ + + + + + + + + + + + + + + + + + + + + + + + + + + + + {\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+{\cfrac {4^{2}}{9+\ddots }}}}}}}}}}} + + + + + + 2 + π + = + + 6 + + + + + + + + + + + + 2 + + 2 + + + + + + + + + + + + 12 + + + + + + + + + + + + 6 + + 2 + + + + + + + + + + + + 12 + + + + + + + + + + + + 10 + + 2 + + + + + + + + + + + + 12 + + + + + + + + + + + + 14 + + 2 + + + + + + + + + + + + 12 + + + + + + + + + + + + 18 + + 2 + + + + + + + + + + + + 12 + + + ⋱ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + {\displaystyle 2\pi ={6+{\cfrac {2^{2}}{12+{\cfrac {6^{2}}{12+{\cfrac {10^{2}}{12+{\cfrac {14^{2}}{12+{\cfrac {18^{2}}{12+\ddots }}}}}}}}}}}} + + + + + + π + = + 4 + − + + + + + + + + + 2 + + + + + + + + + + 1 + + + + + + + + + + + 1 + + + + + + + + + + 1 + − + + + + + + + + + 1 + + + + + + + + + + 1 + + + + + + + + + + + 2 + + + + + + + + + + 1 + − + + + + + + + + + 2 + + + + + + + + + + 1 + + + + + + + + + + + 3 + + + + + + + + + + 1 + − + + + + + + + + + 3 + + + + + + + + + + ⋱ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + {\displaystyle \pi =4-{\cfrac {2}{1+{\cfrac {1}{1-{\cfrac {1}{1+{\cfrac {2}{1-{\cfrac {2}{1+{\cfrac {3}{1-{\cfrac {3}{\ddots }}}}}}}}}}}}}}} + + +For more on the fourth identity, see Euler's continued fraction formula. + +=== Iterative algorithms === + + + + + + a + + 0 + + + = + 1 + , + + + a + + n + + + 1 + + + = + + ( + + 1 + + + + + 1 + + 2 + n + + + 1 + + + + + ) + + + a + + n + + + , + + π + = + + lim + + n + → + ∞ + + + + + + a + + n + + + 2 + + + n + + + + + {\displaystyle a_{0}=1,\,a_{n+1}=\left(1+{\frac {1}{2n+1}}\right)a_{n},\,\pi =\lim _{n\to \infty }{\frac {a_{n}^{2}}{n}}} + + + + + + + a + + 1 + + + = + 0 + , + + + a + + n + + + 1 + + + = + + + 2 + + + + a + + n + + + + + , + + π + = + + lim + + n + → + ∞ + + + + 2 + + n + + + + + 2 + − + + a + + n + + + + + + + {\displaystyle a_{1}=0,\,a_{n+1}={\sqrt {2+a_{n}}},\,\pi =\lim _{n\to \infty }2^{n}{\sqrt {2-a_{n}}}} + + (closely related to Viète's formula) + + + + + ω + ( + + i + + n + + + , + + i + + n + − + 1 + + + , + … + , + + i + + 1 + + + ) + = + 2 + + + + i + + n + + + + + 2 + + + + i + + n + − + 1 + + + + + 2 + + + ⋯ + + + + i + + 1 + + + + + 2 + + + + + + + = + ω + ( + + b + + n + + + , + + b + + n + − + 1 + + + , + … + , + + b + + 1 + + + ) + , + + + i + + k + + + ∈ + { + − + 1 + , + 1 + } + , + + + b + + k + + + = + + + { + + + + 0 + + + + if + + + i + + k + + + = + 1 + + + + + 1 + + + + if + + + i + + k + + + = + − + 1 + + + + + + + , + + π + = + + + + lim + + n + → + ∞ + + + + + + 2 + + n + + + 1 + + + + 2 + h + + + 1 + + + + + + ω + + ( + + + + + + 10 + … + 0 + + ⏟ + + + + n + − + m + + + + g + + m + , + h + + + 1 + + + + ) + + + + + + + + {\displaystyle \omega (i_{n},i_{n-1},\dots ,i_{1})=2+i_{n}{\sqrt {2+i_{n-1}{\sqrt {2+\cdots +i_{1}{\sqrt {2}}}}}}=\omega (b_{n},b_{n-1},\dots ,b_{1}),\,i_{k}\in \{-1,1\},\,b_{k}={\begin{cases}0&{\text{if }}i_{k}=1\\1&{\text{if }}i_{k}=-1\end{cases}},\,\pi ={\displaystyle \lim _{n\rightarrow \infty }{\frac {2^{n+1}}{2h+1}}{\sqrt {\omega \left(\underbrace {10\ldots 0} _{n-m}g_{m,h+1}\right)}}}} + + (where + + + + + g + + m + , + h + + + 1 + + + + + {\displaystyle g_{m,h+1}} + + is the h+1-th entry of m-bit Gray code, + + + + h + ∈ + + { + + 0 + , + 1 + , + … + , + + 2 + + m + + + − + 1 + + } + + + + {\displaystyle h\in \left\{0,1,\ldots ,2^{m}-1\right\}} + +) + + + + + ∀ + k + ∈ + + N + + , + + + a + + 1 + + + = + + 2 + + − + k + + + , + + + a + + n + + + 1 + + + = + + a + + n + + + + + + 2 + + − + k + + + ( + 1 + − + tan + ⁡ + ( + + 2 + + k + − + 1 + + + + a + + n + + + ) + ) + , + + π + = + + 2 + + k + + + 1 + + + + lim + + n + → + ∞ + + + + a + + n + + + + + {\displaystyle \forall k\in \mathbb {N} ,\,a_{1}=2^{-k},\,a_{n+1}=a_{n}+2^{-k}(1-\tan(2^{k-1}a_{n})),\,\pi =2^{k+1}\lim _{n\to \infty }a_{n}} + + (quadratic convergence) + + + + + + a + + 1 + + + = + 1 + , + + + a + + n + + + 1 + + + = + + a + + n + + + + + sin + ⁡ + + a + + n + + + , + + π + = + + lim + + n + → + ∞ + + + + a + + n + + + + + {\displaystyle a_{1}=1,\,a_{n+1}=a_{n}+\sin a_{n},\,\pi =\lim _{n\to \infty }a_{n}} + + (cubic convergence) + + + + + + a + + 0 + + + = + 2 + + + 3 + + + , + + + b + + 0 + + + = + 3 + , + + + a + + n + + + 1 + + + = + hm + ⁡ + ( + + a + + n + + + , + + b + + n + + + ) + , + + + b + + n + + + 1 + + + = + gm + ⁡ + ( + + a + + n + + + 1 + + + , + + b + + n + + + ) + , + + π + = + + lim + + n + → + ∞ + + + + a + + n + + + = + + lim + + n + → + ∞ + + + + b + + n + + + + + {\displaystyle a_{0}=2{\sqrt {3}},\,b_{0}=3,\,a_{n+1}=\operatorname {hm} (a_{n},b_{n}),\,b_{n+1}=\operatorname {gm} (a_{n+1},b_{n}),\,\pi =\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}} + + (Archimedes' algorithm, see also harmonic mean and geometric mean) +For more iterative algorithms, see the Gauss–Legendre algorithm and Borwein's algorithm. + +=== Asymptotics === \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-6.md b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-6.md new file mode 100644 index 000000000..73ba131b0 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-6.md @@ -0,0 +1,1485 @@ +--- +title: "List of formulae involving π" +chunk: 7/8 +source: "https://en.wikipedia.org/wiki/List_of_formulae_involving_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:10.037546+00:00" +instance: "kb-cron" +--- + + + + + + + + ( + + + + 2 + n + + n + + + ) + + + + ∼ + + + + 4 + + n + + + + π + n + + + + + + {\displaystyle {\binom {2n}{n}}\sim {\frac {4^{n}}{\sqrt {\pi n}}}} + + (asymptotic growth rate of the central binomial coefficients) + + + + + + C + + n + + + ∼ + + + + 4 + + n + + + + π + + n + + 3 + + + + + + + + {\displaystyle C_{n}\sim {\frac {4^{n}}{\sqrt {\pi n^{3}}}}} + + (asymptotic growth rate of the Catalan numbers) + + + + + n + ! + ∼ + + + 2 + π + n + + + + + ( + + + n + e + + + ) + + + n + + + + + {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} + + (Stirling's approximation) + + + + + log + ⁡ + n + ! + ≃ + + ( + + n + + + + + 1 + 2 + + + + ) + + log + ⁡ + n + − + n + + + + + + log + ⁡ + 2 + π + + 2 + + + + + {\displaystyle \log n!\simeq \left(n+{\frac {1}{2}}\right)\log n-n+{\frac {\log 2\pi }{2}}} + + + + + + + ∑ + + k + = + 1 + + + n + + + φ + ( + k + ) + ∼ + + + + 3 + + n + + 2 + + + + + π + + 2 + + + + + + + {\displaystyle \sum _{k=1}^{n}\varphi (k)\sim {\frac {3n^{2}}{\pi ^{2}}}} + + (where + + + + φ + + + {\displaystyle \varphi } + + is Euler's totient function) + + + + + + ∑ + + k + = + 1 + + + n + + + + + + φ + ( + k + ) + + k + + + ∼ + + + + 6 + n + + + π + + 2 + + + + + + + {\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}\sim {\frac {6n}{\pi ^{2}}}} + + +The symbol + + + + ∼ + + + {\displaystyle \sim } + + means that the ratio of the left-hand side and the right-hand side tends to one as + + + + n + → + ∞ + + + {\displaystyle n\to \infty } + +. +The symbol + + + + ≃ + + + {\displaystyle \simeq } + + means that the difference between the left-hand side and the right-hand side tends to zero as + + + + n + → + ∞ + + + {\displaystyle n\to \infty } + +. + +=== Hypergeometric inversions === +With + + + + + + + + + 2 + + + + F + + 1 + + + + + {\displaystyle {}_{2}F_{1}} + + being the hypergeometric function: + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + r + + 2 + + + ( + n + ) + + q + + n + + + = + + + + + + 2 + + + + F + + 1 + + + + ( + + + + 1 + 2 + + + , + + + 1 + 2 + + + , + 1 + , + z + + ) + + + + {\displaystyle \sum _{n=0}^{\infty }r_{2}(n)q^{n}={}_{2}F_{1}\left({\frac {1}{2}},{\frac {1}{2}},1,z\right)} + + +where + + + + + q + = + exp + ⁡ + + ( + + − + π + + + + + + + + + 2 + + + + F + + 1 + + + ( + 1 + + / + + 2 + , + 1 + + / + + 2 + , + 1 + , + 1 + − + z + ) + + + + + + + + 2 + + + + F + + 1 + + + ( + 1 + + / + + 2 + , + 1 + + / + + 2 + , + 1 + , + z + ) + + + + + ) + + , + + z + ∈ + + C + + ∖ + { + 0 + , + 1 + } + + + {\displaystyle q=\exp \left(-\pi {\frac {{}_{2}F_{1}(1/2,1/2,1,1-z)}{{}_{2}F_{1}(1/2,1/2,1,z)}}\right),\quad z\in \mathbb {C} \setminus \{0,1\}} + + +and + + + + + r + + 2 + + + + + {\displaystyle r_{2}} + + is the sum of two squares function. +Similarly, + + + + + 1 + + + 240 + + ∑ + + n + = + 1 + + + ∞ + + + + σ + + 3 + + + ( + n + ) + + q + + n + + + = + + + + + + 2 + + + + F + + 1 + + + + + ( + + + + 1 + 6 + + + , + + + 5 + 6 + + + , + 1 + , + z + + ) + + + 4 + + + + + {\displaystyle 1+240\sum _{n=1}^{\infty }\sigma _{3}(n)q^{n}={}_{2}F_{1}\left({\frac {1}{6}},{\frac {5}{6}},1,z\right)^{4}} + + +where + + + + + q + = + exp + ⁡ + + ( + + − + 2 + π + + + + + + + + + 2 + + + + F + + 1 + + + ( + 1 + + / + + 6 + , + 5 + + / + + 6 + , + 1 + , + 1 + − + z + ) + + + + + + + + 2 + + + + F + + 1 + + + ( + 1 + + / + + 6 + , + 5 + + / + + 6 + , + 1 + , + z + ) + + + + + ) + + , + + z + ∈ + + C + + ∖ + { + 0 + , + 1 + } + + + {\displaystyle q=\exp \left(-2\pi {\frac {{}_{2}F_{1}(1/6,5/6,1,1-z)}{{}_{2}F_{1}(1/6,5/6,1,z)}}\right),\quad z\in \mathbb {C} \setminus \{0,1\}} + + +and + + + + + σ + + 3 + + + + + {\displaystyle \sigma _{3}} + + is a divisor function. +More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. +Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function + + + + τ + + + {\displaystyle \tau } + + and the Fourier coefficients + + + + + j + + + + {\displaystyle \mathrm {j} } + + of the J-invariant ((sequence A000521 in the OEIS)): + + + + + + ∑ + + n + = + − + 1 + + + ∞ + + + + + j + + + n + + + + q + + n + + + = + 256 + + + + + ( + 1 + − + z + + + + z + + 2 + + + + ) + + 3 + + + + + + z + + 2 + + + ( + 1 + − + z + + ) + + 2 + + + + + + + , + + + {\displaystyle \sum _{n=-1}^{\infty }\mathrm {j} _{n}q^{n}=256{\dfrac {(1-z+z^{2})^{3}}{z^{2}(1-z)^{2}}},} + + + + + + + ∑ + + n + = + 1 + + + ∞ + + + τ + ( + n + ) + + q + + n + + + = + + + + + + z + + 2 + + + ( + 1 + − + z + + ) + + 2 + + + + 256 + + + + + + + + + 2 + + + + F + + 1 + + + + + ( + + + + 1 + 2 + + + , + + + 1 + 2 + + + , + 1 + , + z + + ) + + + 12 + + + + + {\displaystyle \sum _{n=1}^{\infty }\tau (n)q^{n}={\dfrac {z^{2}(1-z)^{2}}{256}}{}_{2}F_{1}\left({\frac {1}{2}},{\frac {1}{2}},1,z\right)^{12}} + + +where in both cases + + + + + q + = + exp + ⁡ + + ( + + − + 2 + π + + + + + + + + + 2 + + + + F + + 1 + + + ( + 1 + + / + + 2 + , + 1 + + / + + 2 + , + 1 + , + 1 + − + z + ) + + + + + + + + 2 + + + + F + + 1 + + + ( + 1 + + / + + 2 + , + 1 + + / + + 2 + , + 1 + , + z + ) + + + + + ) + + , + + z + ∈ + + C + + ∖ + { + 0 + , + 1 + } + . + + + {\displaystyle q=\exp \left(-2\pi {\frac {{}_{2}F_{1}(1/2,1/2,1,1-z)}{{}_{2}F_{1}(1/2,1/2,1,z)}}\right),\quad z\in \mathbb {C} \setminus \{0,1\}.} + + +Furthermore, by expanding the last expression as a power series in + + + + + + + + 1 + 2 + + + + + + + + 1 + − + ( + 1 + − + z + + ) + + 1 + + / + + 4 + + + + + 1 + + + ( + 1 + − + z + + ) + + 1 + + / + + 4 + + + + + + + + + {\displaystyle {\dfrac {1}{2}}{\dfrac {1-(1-z)^{1/4}}{1+(1-z)^{1/4}}}} + + +and setting + + + + z + = + 1 + + / + + 2 + + + {\displaystyle z=1/2} + +, we obtain a rapidly convergent series for + + + + + e + + − + 2 + π + + + + + {\displaystyle e^{-2\pi }} + +: + + + + + + e + + − + 2 + π + + + = + + w + + 2 + + + + + 4 + + w + + 6 + + + + + 34 + + w + + 10 + + + + + 360 + + w + + 14 + + + + + 4239 + + w + + 18 + + + + + ⋯ + , + + w + = + + + + 1 + 2 + + + + + + + + + 2 + + 1 + + / + + 4 + + + − + 1 + + + + 2 + + 1 + + / + + 4 + + + + + 1 + + + + + . + + + {\displaystyle e^{-2\pi }=w^{2}+4w^{6}+34w^{10}+360w^{14}+4239w^{18}+\cdots ,\quad w={\dfrac {1}{2}}{\dfrac {2^{1/4}-1}{2^{1/4}+1}}.} + + +=== Miscellaneous === + + + + + Γ + ( + s + ) + Γ + ( + 1 + − + s + ) + = + + + π + + sin + ⁡ + π + s + + + + + + {\displaystyle \Gamma (s)\Gamma (1-s)={\frac {\pi }{\sin \pi s}}} + + (Euler's reflection formula, see Gamma function) + + + + + π + = + + + 6 + ζ + ( + 2 + ) + + + + + {\displaystyle \pi ={\sqrt {6\zeta (2)}}} + + (derived from Euler's solution to Basel problem, see Riemann zeta function) + + + + + + π + + − + s + + / + + 2 + + + Γ + + ( + + + s + 2 + + + ) + + ζ + ( + s + ) + = + + π + + − + ( + 1 + − + s + ) + + / + + 2 + + + Γ + + ( + + + + 1 + − + s + + 2 + + + ) + + ζ + ( + 1 + − + s + ) + + + {\displaystyle \pi ^{-s/2}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\pi ^{-(1-s)/2}\Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)} + + (the functional equation of the Riemann zeta function) + + + + + + e + + − + + ζ + ′ + + ( + 0 + ) + + + = + + + 2 + π + + + + + {\displaystyle e^{-\zeta '(0)}={\sqrt {2\pi }}} + + + + + + + e + + + ζ + ′ + + ( + 0 + , + 1 + + / + + 2 + ) + − + + ζ + ′ + + ( + 0 + , + 1 + ) + + + = + + + π + + + + + {\displaystyle e^{\zeta '(0,1/2)-\zeta '(0,1)}={\sqrt {\pi }}} + + (where + + + + ζ + ( + s + , + a + ) + + + {\displaystyle \zeta (s,a)} + + is the Hurwitz zeta function and the derivative is taken with respect to the first variable) + + + + + π + = + + B + + ( + 1 + + / + + 2 + , + 1 + + / + + 2 + ) + = + Γ + ( + 1 + + / + + 2 + + ) + + 2 + + + + + {\displaystyle \pi =\mathrm {B} (1/2,1/2)=\Gamma (1/2)^{2}} + + (see also Beta function) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-7.md b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-7.md new file mode 100644 index 000000000..104cbeec0 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulae_involving_π-7.md @@ -0,0 +1,920 @@ +--- +title: "List of formulae involving π" +chunk: 8/8 +source: "https://en.wikipedia.org/wiki/List_of_formulae_involving_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:10.037546+00:00" +instance: "kb-cron" +--- + + + + + π + = + + + + Γ + ( + 3 + + / + + 4 + + ) + + 4 + + + + + agm + ⁡ + ( + 1 + , + 1 + + / + + + + 2 + + + + ) + + 2 + + + + + + = + + + + Γ + + + ( + + 1 + + / + + 4 + + ) + + + 4 + + / + + 3 + + + agm + ⁡ + ( + 1 + , + + + 2 + + + + ) + + 2 + + / + + 3 + + + + 2 + + + + + {\displaystyle \pi ={\frac {\Gamma (3/4)^{4}}{\operatorname {agm} (1,1/{\sqrt {2}})^{2}}}={\frac {\Gamma \left({1/4}\right)^{4/3}\operatorname {agm} (1,{\sqrt {2}})^{2/3}}{2}}} + + (where agm is the arithmetic–geometric mean) + + + + + π + = + agm + ⁡ + + ( + + + θ + + 2 + + + 2 + + + ( + 1 + + / + + e + ) + , + + θ + + 3 + + + 2 + + + ( + 1 + + / + + e + ) + + ) + + + + {\displaystyle \pi =\operatorname {agm} \left(\theta _{2}^{2}(1/e),\theta _{3}^{2}(1/e)\right)} + + (where + + + + + θ + + 2 + + + + + {\displaystyle \theta _{2}} + + and + + + + + θ + + 3 + + + + + {\displaystyle \theta _{3}} + + are the Jacobi theta functions) + + + + + agm + ⁡ + ( + 1 + , + + + 2 + + + ) + = + + + π + ϖ + + + + + {\displaystyle \operatorname {agm} (1,{\sqrt {2}})={\frac {\pi }{\varpi }}} + + (due to Gauss, + + + + ϖ + + + {\displaystyle \varpi } + + is the lemniscate constant) + + + + + N + ⁡ + ( + 2 + ϖ + ) + = + + e + + 2 + π + + + , + + N + ⁡ + ( + ϖ + ) + = + + e + + π + + / + + 2 + + + + + {\displaystyle \operatorname {N} (2\varpi )=e^{2\pi },\quad \operatorname {N} (\varpi )=e^{\pi /2}} + + (where + + + + N + + + {\displaystyle \operatorname {N} } + + is the Gauss N-function) + + + + + i + π + = + Log + ⁡ + ( + − + 1 + ) + = + + lim + + n + → + ∞ + + + n + + ( + + ( + − + 1 + + ) + + 1 + + / + + n + + + − + 1 + + ) + + + + {\displaystyle i\pi =\operatorname {Log} (-1)=\lim _{n\to \infty }n\left((-1)^{1/n}-1\right)} + + (where + + + + Log + + + {\displaystyle \operatorname {Log} } + + is the principal value of the complex logarithm) + + + + + 1 + − + + + + π + + 2 + + + 12 + + + = + + lim + + n + → + ∞ + + + + + 1 + + n + + 2 + + + + + + ∑ + + k + = + 1 + + + n + + + ( + n + + mod + + k + + + ) + + + {\displaystyle 1-{\frac {\pi ^{2}}{12}}=\lim _{n\rightarrow \infty }{\frac {1}{n^{2}}}\sum _{k=1}^{n}(n{\bmod {k}})} + + (where + + + + n + + mod + + k + + + + + {\textstyle n{\bmod {k}}} + + is the remainder upon division of n by k) + + + + + π + = + + lim + + r + → + ∞ + + + + + 1 + + r + + 2 + + + + + + ∑ + + x + = + − + r + + + r + + + + + ∑ + + y + = + − + r + + + r + + + + + { + + + + 1 + + + + if + + + + + x + + 2 + + + + + + y + + 2 + + + + + ≤ + r + + + + + 0 + + + + if + + + + + x + + 2 + + + + + + y + + 2 + + + + + > + r + + + + + + + + + {\displaystyle \pi =\lim _{r\to \infty }{\frac {1}{r^{2}}}\sum _{x=-r}^{r}\;\sum _{y=-r}^{r}{\begin{cases}1&{\text{if }}{\sqrt {x^{2}+y^{2}}}\leq r\\0&{\text{if }}{\sqrt {x^{2}+y^{2}}}>r\end{cases}}} + + (summing a circle's area) + + + + + π + = + + lim + + n + → + ∞ + + + + + 4 + + n + + 2 + + + + + + ∑ + + k + = + 1 + + + n + + + + + + n + + 2 + + + − + + k + + 2 + + + + + + + {\displaystyle \pi =\lim _{n\rightarrow \infty }{\frac {4}{n^{2}}}\sum _{k=1}^{n}{\sqrt {n^{2}-k^{2}}}} + + (Riemann sum to evaluate the area of the unit circle) + + + + + π + = + + lim + + n + → + ∞ + + + + + + + 2 + + 4 + n + + + n + + ! + + 4 + + + + + n + ( + 2 + n + ) + + ! + + 2 + + + + + + = + + lim + + n + → + ∞ + + + + + + 2 + + 4 + n + + + + n + + + + + ( + + + + 2 + n + + n + + + ) + + + + + 2 + + + + + + = + + lim + + n + → + ∞ + + + + + 1 + n + + + + + ( + + + + ( + 2 + n + ) + ! + ! + + + ( + 2 + n + − + 1 + ) + ! + ! + + + + ) + + + 2 + + + + + {\displaystyle \pi =\lim _{n\to \infty }{\frac {2^{4n}n!^{4}}{n(2n)!^{2}}}=\lim _{n\rightarrow \infty }{\frac {2^{4n}}{n{2n \choose n}^{2}}}=\lim _{n\rightarrow \infty }{\frac {1}{n}}\left({\frac {(2n)!!}{(2n-1)!!}}\right)^{2}} + + (by combining Stirling's approximation with Wallis product) + + + + + π + = + + lim + + n + → + ∞ + + + + + 1 + n + + + ln + ⁡ + + + 16 + + λ + ( + n + i + ) + + + + + + {\displaystyle \pi =\lim _{n\to \infty }{\frac {1}{n}}\ln {\frac {16}{\lambda (ni)}}} + + (where + + + + λ + + + {\displaystyle \lambda } + + is the modular lambda function) + + + + + π + = + + lim + + n + → + ∞ + + + + + 24 + + n + + + + ln + ⁡ + + ( + + + 2 + + 1 + + / + + 4 + + + + G + + n + + + + ) + + = + + lim + + n + → + ∞ + + + + + 24 + + n + + + + ln + ⁡ + + ( + + + 2 + + 1 + + / + + 4 + + + + g + + n + + + + ) + + + + {\displaystyle \pi =\lim _{n\to \infty }{\frac {24}{\sqrt {n}}}\ln \left(2^{1/4}G_{n}\right)=\lim _{n\to \infty }{\frac {24}{\sqrt {n}}}\ln \left(2^{1/4}g_{n}\right)} + + (where + + + + + G + + n + + + + + {\displaystyle G_{n}} + + and + + + + + g + + n + + + + + {\displaystyle g_{n}} + + are Ramanujan's class invariants) + +== See also == +List of mathematical identities +Lists of mathematics topics +List of trigonometric identities +List of topics related to π +List of representations of e + +== References == + +=== Notes === + +=== Other === + +Tóth, László (2020), "Transcendental Infinite Products Associated with the +-1 Thue-Morse Sequence" (PDF), Journal of Integer Sequences, 23: 20.8.2, arXiv:2009.02025. + +== Further reading == +Borwein, Peter (2000). "The amazing number π" (PDF). Nieuw Archief voor Wiskunde. 5th series. 1 (3): 254–258. Zbl 1173.01300. +Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Number Theory 1: Fermat's Dream. American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-0.md b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-0.md new file mode 100644 index 000000000..24d02304d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-0.md @@ -0,0 +1,1549 @@ +--- +title: "List of formulas in Riemannian geometry" +chunk: 1/6 +source: "https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:45.030637+00:00" +instance: "kb-cron" +--- + +This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise. + +== Christoffel symbols, covariant derivative == +In a smooth coordinate chart, the Christoffel symbols of the first kind are given by + + + + + + Γ + + k + i + j + + + = + + + 1 + 2 + + + + ( + + + + ∂ + + ∂ + + x + + j + + + + + + + g + + k + i + + + + + + + ∂ + + ∂ + + x + + i + + + + + + + g + + k + j + + + − + + + ∂ + + ∂ + + x + + k + + + + + + + g + + i + j + + + + ) + + = + + + 1 + 2 + + + + ( + + + g + + k + i + , + j + + + + + + g + + k + j + , + i + + + − + + g + + i + j + , + k + + + + ) + + + , + + + {\displaystyle \Gamma _{kij}={\frac {1}{2}}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,,} + + +and the Christoffel symbols of the second kind by + + + + + + + + + + Γ + + m + + + + + + + + i + j + + + + + + = + + g + + m + k + + + + Γ + + k + i + j + + + + + + + + + = + + + 1 + 2 + + + + + g + + m + k + + + + ( + + + + ∂ + + ∂ + + x + + j + + + + + + + g + + k + i + + + + + + + ∂ + + ∂ + + x + + i + + + + + + + g + + k + j + + + − + + + ∂ + + ∂ + + x + + k + + + + + + + g + + i + j + + + + ) + + = + + + 1 + 2 + + + + + g + + m + k + + + + ( + + + g + + k + i + , + j + + + + + + g + + k + j + , + i + + + − + + g + + i + j + , + k + + + + ) + + + . + + + + + + + {\displaystyle {\begin{aligned}\Gamma ^{m}{}_{ij}&=g^{mk}\Gamma _{kij}\\&={\frac {1}{2}}\,g^{mk}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\,g^{mk}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,.\end{aligned}}} + + +Here + + + + + g + + i + j + + + + + {\displaystyle g^{ij}} + + is the inverse matrix to the metric tensor + + + + + g + + i + j + + + + + {\displaystyle g_{ij}} + +. In other words, + + + + + + δ + + i + + + + + + + + j + + + = + + g + + i + k + + + + g + + k + j + + + + + {\displaystyle \delta ^{i}{}_{j}=g^{ik}g_{kj}} + + +and thus + + + + + n + = + + δ + + i + + + + + + + + i + + + = + + g + + i + + + + + + + + i + + + = + + g + + i + j + + + + g + + i + j + + + + + {\displaystyle n=\delta ^{i}{}_{i}=g^{i}{}_{i}=g^{ij}g_{ij}} + + +is the dimension of the manifold. Additionally, we can take the trace of contravariant tensors with respect to + + + + g + + + {\displaystyle g} + + as follows (and similarly for covariant ones): let + + + + T + + + {\displaystyle T} + + be a + + + + ( + 2 + , + 0 + ) + + + {\displaystyle (2,0)} + + tensor, then its trace with respect to + + + + g + + + {\displaystyle g} + + is + + + + + + tr + + g + + + ⁡ + T + = + tr + ⁡ + ( + v + ↦ + + g + + − + 1 + + + T + ( + v + , + − + ) + ) + + + {\displaystyle \operatorname {tr} _{g}T=\operatorname {tr} (v\mapsto g^{-1}T(v,-))} + + +where + + + + + g + + − + 1 + + + + + {\displaystyle g^{-1}} + + is the isomorphism between the cotangent space and the tangent space. +Christoffel symbols satisfy the symmetry relations + + + + + + Γ + + k + i + j + + + = + + Γ + + k + j + i + + + + + {\displaystyle \Gamma _{kij}=\Gamma _{kji}} + + or, respectively, + + + + + Γ + + i + + + + + + + + j + k + + + = + + Γ + + i + + + + + + + + k + j + + + , + + + {\displaystyle \Gamma ^{i}{}_{jk}=\Gamma ^{i}{}_{kj},} + + +the second of which is equivalent to the torsion-freeness of the Levi-Civita connection. +The contracting relations on the Christoffel symbols are given by + + + + + + Γ + + i + + + + + + + + k + i + + + = + + + 1 + 2 + + + + g + + i + m + + + + + + ∂ + + g + + i + m + + + + + ∂ + + x + + k + + + + + + = + + + 1 + + 2 + g + + + + + + + ∂ + g + + + ∂ + + x + + k + + + + + + = + + + + ∂ + log + ⁡ + + + + | + + g + + | + + + + + + ∂ + + x + + k + + + + + + + + {\displaystyle \Gamma ^{i}{}_{ki}={\frac {1}{2}}g^{im}{\frac {\partial g_{im}}{\partial x^{k}}}={\frac {1}{2g}}{\frac {\partial g}{\partial x^{k}}}={\frac {\partial \log {\sqrt {|g|}}}{\partial x^{k}}}} + + +and + + + + + + g + + k + ℓ + + + + Γ + + i + + + + + + + + k + ℓ + + + = + + + + − + 1 + + + + | + + g + + | + + + + + + + + + ∂ + + ( + + + + + | + + g + + | + + + + + + g + + i + k + + + + ) + + + + ∂ + + x + + k + + + + + + + + {\displaystyle g^{k\ell }\Gamma ^{i}{}_{k\ell }={\frac {-1}{\sqrt {|g|}}}\;{\frac {\partial \left({\sqrt {|g|}}\,g^{ik}\right)}{\partial x^{k}}}} + + +where |g| is the absolute value of the determinant of the matrix of scalar coefficients of the metric tensor + + + + + g + + i + k + + + + + {\displaystyle g_{ik}} + +. These are useful when dealing with divergences and Laplacians (see below). +The covariant derivative of a vector field with components + + + + + v + + i + + + + + {\displaystyle v^{i}} + + is given by: + + + + + + v + + i + + + + + + + + ; + j + + + = + ( + + ∇ + + j + + + v + + ) + + i + + + = + + + + ∂ + + v + + i + + + + + ∂ + + x + + j + + + + + + + + + Γ + + i + + + + + + + + j + k + + + + v + + k + + + + + {\displaystyle v^{i}{}_{;j}=(\nabla _{j}v)^{i}={\frac {\partial v^{i}}{\partial x^{j}}}+\Gamma ^{i}{}_{jk}v^{k}} + + +and similarly, the covariant derivative of a + + + + ( + 0 + , + 1 + ) + + + {\displaystyle (0,1)} + +-tensor field with components + + + + + v + + i + + + + + {\displaystyle v_{i}} + + is given by: + + + + + + v + + i + ; + j + + + = + ( + + ∇ + + j + + + v + + ) + + i + + + = + + + + ∂ + + v + + i + + + + + ∂ + + x + + j + + + + + + − + + Γ + + k + + + + + + + + i + j + + + + v + + k + + + + + {\displaystyle v_{i;j}=(\nabla _{j}v)_{i}={\frac {\partial v_{i}}{\partial x^{j}}}-\Gamma ^{k}{}_{ij}v_{k}} + + +For a + + + + ( + 2 + , + 0 + ) + + + {\displaystyle (2,0)} + +-tensor field with components + + + + + v + + i + j + + + + + {\displaystyle v^{ij}} + + this becomes + + + + + + v + + i + j + + + + + + + + ; + k + + + = + + ∇ + + k + + + + v + + i + j + + + = + + + + ∂ + + v + + i + j + + + + + ∂ + + x + + k + + + + + + + + + Γ + + i + + + + + + + + k + ℓ + + + + v + + ℓ + j + + + + + + Γ + + j + + + + + + + + k + ℓ + + + + v + + i + ℓ + + + + + {\displaystyle v^{ij}{}_{;k}=\nabla _{k}v^{ij}={\frac {\partial v^{ij}}{\partial x^{k}}}+\Gamma ^{i}{}_{k\ell }v^{\ell j}+\Gamma ^{j}{}_{k\ell }v^{i\ell }} + + +and likewise for tensors with more indices. +The covariant derivative of a function (scalar) + + + + ϕ + + + {\displaystyle \phi } + + is just its usual differential: + + + + + + ∇ + + i + + + ϕ + = + + ϕ + + ; + i + + + = + + ϕ + + , + i + + + = + + + + ∂ + ϕ + + + ∂ + + x + + i + + + + + + + + {\displaystyle \nabla _{i}\phi =\phi _{;i}=\phi _{,i}={\frac {\partial \phi }{\partial x^{i}}}} + + +Because the Levi-Civita connection is metric-compatible, the covariant derivative of the metric vanishes, + + + + + ( + + ∇ + + k + + + g + + ) + + i + j + + + = + 0 + , + + ( + + ∇ + + k + + + g + + ) + + i + j + + + = + 0 + + + {\displaystyle (\nabla _{k}g)_{ij}=0,\quad (\nabla _{k}g)^{ij}=0} + + +as well as the covariant derivatives of the metric's determinant (and volume element) + + + + + + ∇ + + k + + + + + + | + + g + + | + + + + = + 0 + + + {\displaystyle \nabla _{k}{\sqrt {|g|}}=0} + + +The geodesic + + + + X + ( + t + ) + + + {\displaystyle X(t)} + + starting at the origin with initial speed + + + + + v + + i + + + + + {\displaystyle v^{i}} + + has Taylor expansion in the chart: + + + + + X + ( + t + + ) + + i + + + = + t + + v + + i + + + − + + + + t + + 2 + + + 2 + + + + Γ + + i + + + + + + + + j + k + + + + v + + j + + + + v + + k + + + + + O + ( + + t + + 3 + + + ) + + + {\displaystyle X(t)^{i}=tv^{i}-{\frac {t^{2}}{2}}\Gamma ^{i}{}_{jk}v^{j}v^{k}+O(t^{3})} + + +A coordinates-free formula for the Levi-Civita connection (albeit being implicit) is the following: + + + + + g + ( + Z + , + + ∇ + + Y + + + X + ) + = + + + 1 + 2 + + + + + ( + + + X + ( + g + ( + Y + , + Z + ) + ) + + + Y + ( + g + ( + X + , + Z + ) + ) + − + Z + ( + g + ( + X + , + Y + ) + ) + + + g + ( + Y + , + [ + Z + , + X + ] + ) + + + g + ( + X + , + [ + Z + , + Y + ] + ) + − + g + ( + Z + , + [ + X + , + Y + ] + ) + + + ) + + + + + {\displaystyle g(Z,\nabla _{Y}X)={\frac {1}{2}}{\Big (}X(g(Y,Z))+Y(g(X,Z))-Z(g(X,Y))+g(Y,[Z,X])+g(X,[Z,Y])-g(Z,[X,Y]){\Big )}} + + +== Curvature tensors == + +=== Definitions === + +==== (3,1) Riemann curvature tensor ==== \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-1.md b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-1.md new file mode 100644 index 000000000..f1850ad3d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-1.md @@ -0,0 +1,1600 @@ +--- +title: "List of formulas in Riemannian geometry" +chunk: 2/6 +source: "https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:45.030637+00:00" +instance: "kb-cron" +--- + + + + + + + + R + + i + j + k + + + + + l + + + = + + + + ∂ + + Γ + + j + k + + + l + + + + + ∂ + + x + + i + + + + + + − + + + + ∂ + + Γ + + i + k + + + l + + + + + ∂ + + x + + j + + + + + + + + + + ( + + + + Γ + + j + k + + + p + + + + Γ + + i + p + + + l + + + − + + Γ + + i + k + + + p + + + + Γ + + j + p + + + l + + + + + ) + + + + + {\displaystyle {R_{ijk}}^{l}={\frac {\partial \Gamma _{jk}^{l}}{\partial x^{i}}}-{\frac {\partial \Gamma _{ik}^{l}}{\partial x^{j}}}+{\big (}\Gamma _{jk}^{p}\Gamma _{ip}^{l}-\Gamma _{ik}^{p}\Gamma _{jp}^{l}{\big )}} + + + + + + R + ( + u + , + v + ) + w + = + + ∇ + + u + + + + ∇ + + v + + + w + − + + ∇ + + v + + + + ∇ + + u + + + w + − + + ∇ + + [ + u + , + v + ] + + + w + + + {\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w} + + +==== (3,1) Riemann curvature tensor ==== + + + + + + + R + + j + k + l + + + i + + + + = + + + + ∂ + + Γ + + l + j + + + i + + + + + ∂ + + x + + k + + + + + + − + + + + ∂ + + Γ + + k + j + + + i + + + + + ∂ + + x + + l + + + + + + + + + + ( + + + + Γ + + k + p + + + i + + + + Γ + + l + j + + + p + + + − + + Γ + + l + p + + + i + + + + Γ + + k + j + + + p + + + + + ) + + + + + {\displaystyle {R_{jkl}^{i}}={\frac {\partial \Gamma _{lj}^{i}}{\partial x^{k}}}-{\frac {\partial \Gamma _{kj}^{i}}{\partial x^{l}}}+{\big (}\Gamma _{kp}^{i}\Gamma _{lj}^{p}-\Gamma _{lp}^{i}\Gamma _{kj}^{p}{\big )}} + + +==== Ricci curvature ==== + + + + + + R + + i + k + + + = + + + + R + + i + j + k + + + + + j + + + + + {\displaystyle R_{ik}={R_{ijk}}^{j}} + + + + + + Ric + ⁡ + ( + v + , + w + ) + = + tr + ⁡ + ( + u + ↦ + R + ( + u + , + v + ) + w + ) + + + {\displaystyle \operatorname {Ric} (v,w)=\operatorname {tr} (u\mapsto R(u,v)w)} + + +==== Scalar curvature ==== + + + + + R + = + + g + + i + k + + + + R + + i + k + + + + + {\displaystyle R=g^{ik}R_{ik}} + + + + + + R + = + + tr + + g + + + ⁡ + Ric + + + {\displaystyle R=\operatorname {tr} _{g}\operatorname {Ric} } + + +==== Traceless Ricci tensor ==== + + + + + + Q + + i + k + + + = + + R + + i + k + + + − + + + 1 + n + + + R + + g + + i + k + + + + + {\displaystyle Q_{ik}=R_{ik}-{\frac {1}{n}}Rg_{ik}} + + + + + + Q + ( + u + , + v + ) + = + Ric + ⁡ + ( + u + , + v + ) + − + + + 1 + n + + + R + g + ( + u + , + v + ) + + + {\displaystyle Q(u,v)=\operatorname {Ric} (u,v)-{\frac {1}{n}}Rg(u,v)} + + +==== (4,0) Riemann curvature tensor ==== + + + + + + R + + i + j + k + l + + + = + + + + R + + i + j + k + + + + + p + + + + g + + p + l + + + + + {\displaystyle R_{ijkl}={R_{ijk}}^{p}g_{pl}} + + + + + + Rm + ⁡ + ( + u + , + v + , + w + , + x + ) + = + g + + + ( + + + R + ( + u + , + v + ) + w + , + x + + + ) + + + + + {\displaystyle \operatorname {Rm} (u,v,w,x)=g{\big (}R(u,v)w,x{\big )}} + + +==== (4,0) Weyl tensor ==== + + + + + + W + + i + j + k + l + + + = + + R + + i + j + k + l + + + − + + + 1 + + n + ( + n + − + 1 + ) + + + + R + + + ( + + + + g + + i + k + + + + g + + j + l + + + − + + g + + i + l + + + + g + + j + k + + + + + ) + + + − + + + 1 + + n + − + 2 + + + + + + ( + + + + Q + + i + k + + + + g + + j + l + + + − + + Q + + j + k + + + + g + + i + l + + + − + + Q + + i + l + + + + g + + j + k + + + + + + Q + + j + l + + + + g + + i + k + + + + + ) + + + + + {\displaystyle W_{ijkl}=R_{ijkl}-{\frac {1}{n(n-1)}}R{\big (}g_{ik}g_{jl}-g_{il}g_{jk}{\big )}-{\frac {1}{n-2}}{\big (}Q_{ik}g_{jl}-Q_{jk}g_{il}-Q_{il}g_{jk}+Q_{jl}g_{ik}{\big )}} + + + + + + W + ( + u + , + v + , + w + , + x + ) + = + Rm + ⁡ + ( + u + , + v + , + w + , + x + ) + − + + + 1 + + n + ( + n + − + 1 + ) + + + + R + + + ( + + + g + ( + u + , + w + ) + g + ( + v + , + x + ) + − + g + ( + u + , + x + ) + g + ( + v + , + w + ) + + + ) + + + − + + + 1 + + n + − + 2 + + + + + + ( + + + Q + ( + u + , + w + ) + g + ( + v + , + x + ) + − + Q + ( + v + , + w + ) + g + ( + u + , + x + ) + − + Q + ( + u + , + x + ) + g + ( + v + , + w + ) + + + Q + ( + v + , + x + ) + g + ( + u + , + w + ) + + + ) + + + + + {\displaystyle W(u,v,w,x)=\operatorname {Rm} (u,v,w,x)-{\frac {1}{n(n-1)}}R{\big (}g(u,w)g(v,x)-g(u,x)g(v,w){\big )}-{\frac {1}{n-2}}{\big (}Q(u,w)g(v,x)-Q(v,w)g(u,x)-Q(u,x)g(v,w)+Q(v,x)g(u,w){\big )}} + + +==== Einstein tensor ==== + + + + + + G + + i + k + + + = + + R + + i + k + + + − + + + 1 + 2 + + + R + + g + + i + k + + + + + {\displaystyle G_{ik}=R_{ik}-{\frac {1}{2}}Rg_{ik}} + + + + + + G + ( + u + , + v + ) + = + Ric + ⁡ + ( + u + , + v + ) + − + + + 1 + 2 + + + R + g + ( + u + , + v + ) + + + {\displaystyle G(u,v)=\operatorname {Ric} (u,v)-{\frac {1}{2}}Rg(u,v)} + + +=== Identities === + +==== Basic symmetries ==== + + + + + + + + R + + i + j + k + + + + + l + + + = + − + + + + R + + j + i + k + + + + + l + + + + + {\displaystyle {R_{ijk}}^{l}=-{R_{jik}}^{l}} + + + + + + + R + + i + j + k + l + + + = + − + + R + + j + i + k + l + + + = + − + + R + + i + j + l + k + + + = + + R + + k + l + i + j + + + + + {\displaystyle R_{ijkl}=-R_{jikl}=-R_{ijlk}=R_{klij}} + + +The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: + + + + + + W + + i + j + k + l + + + = + − + + W + + j + i + k + l + + + = + − + + W + + i + j + l + k + + + = + + W + + k + l + i + j + + + + + {\displaystyle W_{ijkl}=-W_{jikl}=-W_{ijlk}=W_{klij}} + + + + + + + g + + i + l + + + + W + + i + j + k + l + + + = + 0 + + + {\displaystyle g^{il}W_{ijkl}=0} + + +The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors: + + + + + + R + + j + k + + + = + + R + + k + j + + + + + {\displaystyle R_{jk}=R_{kj}} + + + + + + + G + + j + k + + + = + + G + + k + j + + + + + {\displaystyle G_{jk}=G_{kj}} + + + + + + + Q + + j + k + + + = + + Q + + k + j + + + + + {\displaystyle Q_{jk}=Q_{kj}} + + +==== First Bianchi identity ==== + + + + + + R + + i + j + k + l + + + + + + R + + j + k + i + l + + + + + + R + + k + i + j + l + + + = + 0 + + + {\displaystyle R_{ijkl}+R_{jkil}+R_{kijl}=0} + + + + + + + W + + i + j + k + l + + + + + + W + + j + k + i + l + + + + + + W + + k + i + j + l + + + = + 0 + + + {\displaystyle W_{ijkl}+W_{jkil}+W_{kijl}=0} + + +==== Second Bianchi identity ==== + + + + + + ∇ + + p + + + + R + + i + j + k + l + + + + + + ∇ + + i + + + + R + + j + p + k + l + + + + + + ∇ + + j + + + + R + + p + i + k + l + + + = + 0 + + + {\displaystyle \nabla _{p}R_{ijkl}+\nabla _{i}R_{jpkl}+\nabla _{j}R_{pikl}=0} + + + + + + ( + + ∇ + + u + + + Rm + ) + ( + v + , + w + , + x + , + y + ) + + + ( + + ∇ + + v + + + Rm + ) + ( + w + , + u + , + x + , + y + ) + + + ( + + ∇ + + w + + + Rm + ) + ( + u + , + v + , + x + , + y + ) + = + 0 + + + {\displaystyle (\nabla _{u}\operatorname {Rm} )(v,w,x,y)+(\nabla _{v}\operatorname {Rm} )(w,u,x,y)+(\nabla _{w}\operatorname {Rm} )(u,v,x,y)=0} + + +==== Contracted second Bianchi identity ==== + + + + + + ∇ + + j + + + + R + + p + k + + + − + + ∇ + + p + + + + R + + j + k + + + = + + ∇ + + l + + + + R + + j + p + k + l + + + + + {\displaystyle \nabla _{j}R_{pk}-\nabla _{p}R_{jk}=\nabla ^{l}R_{jpkl}} + + + + + + ( + + ∇ + + u + + + Ric + ) + ( + v + , + w + ) + − + ( + + ∇ + + v + + + Ric + ) + ( + u + , + w + ) + = + − + + tr + + g + + + ⁡ + + + ( + + + ( + x + , + y + ) + ↦ + ( + + ∇ + + x + + + Rm + ) + ( + u + , + v + , + w + , + y + ) + + + ) + + + + + {\displaystyle (\nabla _{u}\operatorname {Ric} )(v,w)-(\nabla _{v}\operatorname {Ric} )(u,w)=-\operatorname {tr} _{g}{\big (}(x,y)\mapsto (\nabla _{x}\operatorname {Rm} )(u,v,w,y){\big )}} + + +==== Twice-contracted second Bianchi identity ==== + + + + + + g + + p + q + + + + ∇ + + p + + + + R + + q + k + + + = + + + 1 + 2 + + + + ∇ + + k + + + R + + + {\displaystyle g^{pq}\nabla _{p}R_{qk}={\frac {1}{2}}\nabla _{k}R} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-2.md b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-2.md new file mode 100644 index 000000000..afc80c532 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-2.md @@ -0,0 +1,1657 @@ +--- +title: "List of formulas in Riemannian geometry" +chunk: 3/6 +source: "https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:45.030637+00:00" +instance: "kb-cron" +--- + + + + + + div + + g + + + ⁡ + Ric + = + + + 1 + 2 + + + d + R + + + {\displaystyle \operatorname {div} _{g}\operatorname {Ric} ={\frac {1}{2}}dR} + + +Equivalently: + + + + + + g + + p + q + + + + ∇ + + p + + + + G + + q + k + + + = + 0 + + + {\displaystyle g^{pq}\nabla _{p}G_{qk}=0} + + + + + + + div + + g + + + ⁡ + G + = + 0 + + + {\displaystyle \operatorname {div} _{g}G=0} + + +==== Ricci identity ==== +If + + + + X + + + {\displaystyle X} + + is a vector field then + + + + + + ∇ + + i + + + + ∇ + + j + + + + X + + k + + + − + + ∇ + + j + + + + ∇ + + i + + + + X + + k + + + = + − + + + + R + + i + j + p + + + + + k + + + + X + + p + + + , + + + {\displaystyle \nabla _{i}\nabla _{j}X^{k}-\nabla _{j}\nabla _{i}X^{k}=-{R_{ijp}}^{k}X^{p},} + + +which is just the definition of the Riemann tensor. If + + + + ω + + + {\displaystyle \omega } + + is a one-form then + + + + + + ∇ + + i + + + + ∇ + + j + + + + ω + + k + + + − + + ∇ + + j + + + + ∇ + + i + + + + ω + + k + + + = + + + + R + + i + j + k + + + + + p + + + + ω + + p + + + . + + + {\displaystyle \nabla _{i}\nabla _{j}\omega _{k}-\nabla _{j}\nabla _{i}\omega _{k}={R_{ijk}}^{p}\omega _{p}.} + + +More generally, if + + + + T + + + {\displaystyle T} + + is a (0,k)-tensor field then + + + + + + ∇ + + i + + + + ∇ + + j + + + + T + + + l + + 1 + + + ⋯ + + l + + k + + + + + − + + ∇ + + j + + + + ∇ + + i + + + + T + + + l + + 1 + + + ⋯ + + l + + k + + + + + = + + + + R + + i + j + + l + + 1 + + + + + + + p + + + + T + + p + + l + + 2 + + + ⋯ + + l + + k + + + + + + + ⋯ + + + + + + R + + i + j + + l + + k + + + + + + + p + + + + T + + + l + + 1 + + + ⋯ + + l + + k + − + 1 + + + p + + + . + + + {\displaystyle \nabla _{i}\nabla _{j}T_{l_{1}\cdots l_{k}}-\nabla _{j}\nabla _{i}T_{l_{1}\cdots l_{k}}={R_{ijl_{1}}}^{p}T_{pl_{2}\cdots l_{k}}+\cdots +{R_{ijl_{k}}}^{p}T_{l_{1}\cdots l_{k-1}p}.} + + +==== Remarks ==== +A classical result says that + + + + W + = + 0 + + + {\displaystyle W=0} + + if and only if + + + + ( + M + , + g + ) + + + {\displaystyle (M,g)} + + is locally conformally flat, i.e. if and only if + + + + M + + + {\displaystyle M} + + can be covered by smooth coordinate charts relative to which the metric tensor is of the form + + + + + g + + i + j + + + = + + e + + φ + + + + δ + + i + j + + + + + {\displaystyle g_{ij}=e^{\varphi }\delta _{ij}} + + for some function + + + + φ + + + {\displaystyle \varphi } + + on the chart. + +== Gradient, divergence, Laplace–Beltrami operator == +The gradient of a function + + + + ϕ + + + {\displaystyle \phi } + + is obtained by raising the index of the differential + + + + + ∂ + + i + + + ϕ + d + + x + + i + + + + + {\displaystyle \partial _{i}\phi dx^{i}} + +, whose components are given by: + + + + + + ∇ + + i + + + ϕ + = + + ϕ + + ; + i + + + = + + g + + i + k + + + + ϕ + + ; + k + + + = + + g + + i + k + + + + ϕ + + , + k + + + = + + g + + i + k + + + + ∂ + + k + + + ϕ + = + + g + + i + k + + + + + + ∂ + ϕ + + + ∂ + + x + + k + + + + + + + + {\displaystyle \nabla ^{i}\phi =\phi ^{;i}=g^{ik}\phi _{;k}=g^{ik}\phi _{,k}=g^{ik}\partial _{k}\phi =g^{ik}{\frac {\partial \phi }{\partial x^{k}}}} + + +The divergence of a vector field with components + + + + + V + + m + + + + + {\displaystyle V^{m}} + + is + + + + + + ∇ + + m + + + + V + + m + + + = + + + + ∂ + + V + + m + + + + + ∂ + + x + + m + + + + + + + + + V + + k + + + + + + ∂ + log + ⁡ + + + + | + + g + + | + + + + + + ∂ + + x + + k + + + + + + = + + + 1 + + + | + + g + + | + + + + + + + + ∂ + ( + + V + + m + + + + + + | + + g + + | + + + + ) + + + ∂ + + x + + m + + + + + + . + + + {\displaystyle \nabla _{m}V^{m}={\frac {\partial V^{m}}{\partial x^{m}}}+V^{k}{\frac {\partial \log {\sqrt {|g|}}}{\partial x^{k}}}={\frac {1}{\sqrt {|g|}}}{\frac {\partial (V^{m}{\sqrt {|g|}})}{\partial x^{m}}}.} + + +The Laplace–Beltrami operator acting on a function + + + + f + + + {\displaystyle f} + + is given by the divergence of the gradient: + + + + + + + + + Δ + f + + + + = + + ∇ + + i + + + + ∇ + + i + + + f + = + + + 1 + + + | + + g + + | + + + + + + + ∂ + + ∂ + + x + + j + + + + + + + ( + + + g + + j + k + + + + + + | + + g + + | + + + + + + + ∂ + f + + + ∂ + + x + + k + + + + + + + ) + + + + + + + + = + + g + + j + k + + + + + + + ∂ + + 2 + + + f + + + ∂ + + x + + j + + + ∂ + + x + + k + + + + + + + + + + + ∂ + + g + + j + k + + + + + ∂ + + x + + j + + + + + + + + + ∂ + f + + + ∂ + + x + + k + + + + + + + + + + 1 + 2 + + + + g + + j + k + + + + g + + i + l + + + + + + ∂ + + g + + i + l + + + + + ∂ + + x + + j + + + + + + + + + ∂ + f + + + ∂ + + x + + k + + + + + + = + + g + + j + k + + + + + + + ∂ + + 2 + + + f + + + ∂ + + x + + j + + + ∂ + + x + + k + + + + + + − + + g + + j + k + + + + Γ + + l + + + + + + + + j + k + + + + + + ∂ + f + + + ∂ + + x + + l + + + + + + + + + + + + {\displaystyle {\begin{aligned}\Delta f&=\nabla _{i}\nabla ^{i}f={\frac {1}{\sqrt {|g|}}}{\frac {\partial }{\partial x^{j}}}\left(g^{jk}{\sqrt {|g|}}{\frac {\partial f}{\partial x^{k}}}\right)\\&=g^{jk}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}+{\frac {\partial g^{jk}}{\partial x^{j}}}{\frac {\partial f}{\partial x^{k}}}+{\frac {1}{2}}g^{jk}g^{il}{\frac {\partial g_{il}}{\partial x^{j}}}{\frac {\partial f}{\partial x^{k}}}=g^{jk}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}-g^{jk}\Gamma ^{l}{}_{jk}{\frac {\partial f}{\partial x^{l}}}\end{aligned}}} + + +The divergence of an antisymmetric tensor field of type + + + + ( + 2 + , + 0 + ) + + + {\displaystyle (2,0)} + + simplifies to + + + + + + ∇ + + k + + + + A + + i + k + + + = + + + 1 + + + | + + g + + | + + + + + + + + ∂ + ( + + A + + i + k + + + + + + | + + g + + | + + + + ) + + + ∂ + + x + + k + + + + + + . + + + {\displaystyle \nabla _{k}A^{ik}={\frac {1}{\sqrt {|g|}}}{\frac {\partial (A^{ik}{\sqrt {|g|}})}{\partial x^{k}}}.} + + +The Hessian of a map + + + + ϕ + : + M + → + N + + + {\displaystyle \phi :M\rightarrow N} + + is given by + + + + + + + ( + + ∇ + + ( + + d + ϕ + + ) + + + ) + + + i + j + + + γ + + + = + + + + + ∂ + + 2 + + + + ϕ + + γ + + + + + ∂ + + x + + i + + + ∂ + + x + + j + + + + + + + − + + M + + + + Γ + + k + + + + + + + + i + j + + + + + + ∂ + + ϕ + + γ + + + + + ∂ + + x + + k + + + + + + + + + + N + + + + Γ + + γ + + + + + + + + α + β + + + + + + ∂ + + ϕ + + α + + + + + ∂ + + x + + i + + + + + + + + + ∂ + + ϕ + + β + + + + + ∂ + + x + + j + + + + + + . + + + {\displaystyle \left(\nabla \left(d\phi \right)\right)_{ij}^{\gamma }={\frac {\partial ^{2}\phi ^{\gamma }}{\partial x^{i}\partial x^{j}}}-^{M}\Gamma ^{k}{}_{ij}{\frac {\partial \phi ^{\gamma }}{\partial x^{k}}}+^{N}\Gamma ^{\gamma }{}_{\alpha \beta }{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}{\frac {\partial \phi ^{\beta }}{\partial x^{j}}}.} + + +== Kulkarni–Nomizu product == +The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let + + + + A + + + {\displaystyle A} + + and + + + + B + + + {\displaystyle B} + + be symmetric covariant 2-tensors. In coordinates, + + + + + + A + + i + j + + + = + + A + + j + i + + + + + + B + + i + j + + + = + + B + + j + i + + + + + {\displaystyle A_{ij}=A_{ji}\qquad \qquad B_{ij}=B_{ji}} + + +Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted + + + + A + + + ∧ + + + + + + + + + + ◯ + + + B + + + {\displaystyle A{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}B} + +. The defining formula is + + + + + + + ( + + A + + + ∧ + + + + + + + + + + ◯ + + + B + + ) + + + i + j + k + l + + + = + + A + + i + k + + + + B + + j + l + + + + + + A + + j + l + + + + B + + i + k + + + − + + A + + i + l + + + + B + + j + k + + + − + + A + + j + k + + + + B + + i + l + + + + + {\displaystyle \left(A{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}B\right)_{ijkl}=A_{ik}B_{jl}+A_{jl}B_{ik}-A_{il}B_{jk}-A_{jk}B_{il}} + + +Clearly, the product satisfies + + + + + A + + + ∧ + + + + + + + + + + ◯ + + + B + = + B + + + ∧ + + + + + + + + + + ◯ + + + A + + + {\displaystyle A{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}B=B{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}A} + + +== In an inertial frame == +An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations + + + + + g + + i + j + + + = + + δ + + i + j + + + + + {\displaystyle g_{ij}=\delta _{ij}} + + and + + + + + Γ + + i + + + + + + + + j + k + + + = + 0 + + + {\displaystyle \Gamma ^{i}{}_{jk}=0} + + (but these may not hold at other points in the frame). These coordinates are also called normal coordinates. +In such a frame, the expression for several operators is simpler. Note that the formulae given below are valid at the origin of the frame only. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-3.md b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-3.md new file mode 100644 index 000000000..ba902daf6 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-3.md @@ -0,0 +1,1661 @@ +--- +title: "List of formulas in Riemannian geometry" +chunk: 4/6 +source: "https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:45.030637+00:00" +instance: "kb-cron" +--- + + + + + + R + + i + k + ℓ + m + + + = + + + 1 + 2 + + + + ( + + + + + + ∂ + + 2 + + + + g + + i + m + + + + + ∂ + + x + + k + + + ∂ + + x + + ℓ + + + + + + + + + + + + ∂ + + 2 + + + + g + + k + ℓ + + + + + ∂ + + x + + i + + + ∂ + + x + + m + + + + + + − + + + + + ∂ + + 2 + + + + g + + i + ℓ + + + + + ∂ + + x + + k + + + ∂ + + x + + m + + + + + + − + + + + + ∂ + + 2 + + + + g + + k + m + + + + + ∂ + + x + + i + + + ∂ + + x + + ℓ + + + + + + + ) + + + + {\displaystyle R_{ik\ell m}={\frac {1}{2}}\left({\frac {\partial ^{2}g_{im}}{\partial x^{k}\partial x^{\ell }}}+{\frac {\partial ^{2}g_{k\ell }}{\partial x^{i}\partial x^{m}}}-{\frac {\partial ^{2}g_{i\ell }}{\partial x^{k}\partial x^{m}}}-{\frac {\partial ^{2}g_{km}}{\partial x^{i}\partial x^{\ell }}}\right)} + + + + + + + R + + ℓ + + + + + + + + i + j + k + + + = + + + ∂ + + ∂ + + x + + j + + + + + + + Γ + + ℓ + + + + + + + + i + k + + + − + + + ∂ + + ∂ + + x + + k + + + + + + + Γ + + ℓ + + + + + + + + i + j + + + + + {\displaystyle R^{\ell }{}_{ijk}={\frac {\partial }{\partial x^{j}}}\Gamma ^{\ell }{}_{ik}-{\frac {\partial }{\partial x^{k}}}\Gamma ^{\ell }{}_{ij}} + + +== Conformal change == +Let + + + + g + + + {\displaystyle g} + + be a Riemannian or pseudo-Riemannian metric on a smooth manifold + + + + M + + + {\displaystyle M} + +, and + + + + φ + + + {\displaystyle \varphi } + + a smooth real-valued function on + + + + M + + + {\displaystyle M} + +. Then + + + + + + + + g + ~ + + + + = + + e + + 2 + φ + + + g + + + {\displaystyle {\tilde {g}}=e^{2\varphi }g} + + +is also a Riemannian metric on + + + + M + + + {\displaystyle M} + +. We say that + + + + + + + g + ~ + + + + + + {\displaystyle {\tilde {g}}} + + is (pointwise) conformal to + + + + g + + + {\displaystyle g} + +. Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with + + + + + + + g + ~ + + + + + + {\displaystyle {\tilde {g}}} + +, while those unmarked with such will be associated with + + + + g + + + {\displaystyle g} + +.) + +=== Levi-Civita connection === + + + + + + + + + Γ + ~ + + + + + i + j + + + k + + + = + + Γ + + i + j + + + k + + + + + + + + ∂ + φ + + + ∂ + + x + + i + + + + + + + δ + + j + + + k + + + + + + + + ∂ + φ + + + ∂ + + x + + j + + + + + + + δ + + i + + + k + + + − + + + + ∂ + φ + + + ∂ + + x + + l + + + + + + + g + + l + k + + + + g + + i + j + + + + + {\displaystyle {\widetilde {\Gamma }}_{ij}^{k}=\Gamma _{ij}^{k}+{\frac {\partial \varphi }{\partial x^{i}}}\delta _{j}^{k}+{\frac {\partial \varphi }{\partial x^{j}}}\delta _{i}^{k}-{\frac {\partial \varphi }{\partial x^{l}}}g^{lk}g_{ij}} + + + + + + + + + + ∇ + ~ + + + + + X + + + Y + = + + ∇ + + X + + + Y + + + d + φ + ( + X + ) + Y + + + d + φ + ( + Y + ) + X + − + g + ( + X + , + Y + ) + ∇ + φ + + + {\displaystyle {\widetilde {\nabla }}_{X}Y=\nabla _{X}Y+d\varphi (X)Y+d\varphi (Y)X-g(X,Y)\nabla \varphi } + + +=== (4,0) Riemann curvature tensor === + + + + + + + + + R + ~ + + + + + i + j + k + l + + + = + + e + + 2 + φ + + + + R + + i + j + k + l + + + + + + e + + 2 + φ + + + + + ( + + + + g + + i + k + + + + T + + j + l + + + + + + g + + j + l + + + + T + + i + k + + + − + + g + + i + l + + + + T + + j + k + + + − + + g + + j + k + + + + T + + i + l + + + + + ) + + + + + {\displaystyle {\widetilde {R}}_{ijkl}=e^{2\varphi }R_{ijkl}+e^{2\varphi }{\big (}g_{ik}T_{jl}+g_{jl}T_{ik}-g_{il}T_{jk}-g_{jk}T_{il}{\big )}} + + where + + + + + T + + i + j + + + = + + ∇ + + i + + + + ∇ + + j + + + φ + − + + ∇ + + i + + + φ + + ∇ + + j + + + φ + + + + + 1 + 2 + + + + | + + d + φ + + + | + + + 2 + + + + g + + i + j + + + + + {\displaystyle T_{ij}=\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi +{\frac {1}{2}}|d\varphi |^{2}g_{ij}} + + +Using the Kulkarni–Nomizu product: + + + + + + + + Rm + ~ + + + + = + + e + + 2 + φ + + + Rm + + + + e + + 2 + φ + + + g + + + ∧ + + + + + + + + + + ◯ + + + + ( + + Hess + ⁡ + φ + − + d + φ + ⊗ + d + φ + + + + + 1 + 2 + + + + | + + d + φ + + + | + + + 2 + + + g + + ) + + + + {\displaystyle {\widetilde {\operatorname {Rm} }}=e^{2\varphi }\operatorname {Rm} +e^{2\varphi }g{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}\left(\operatorname {Hess} \varphi -d\varphi \otimes d\varphi +{\frac {1}{2}}|d\varphi |^{2}g\right)} + + +=== Ricci tensor === + + + + + + + + + R + ~ + + + + + i + j + + + = + + R + + i + j + + + − + ( + n + − + 2 + ) + + + ( + + + + ∇ + + i + + + + ∇ + + j + + + φ + − + + ∇ + + i + + + φ + + ∇ + + j + + + φ + + + ) + + + − + + + ( + + + Δ + φ + + + ( + n + − + 2 + ) + + | + + d + φ + + + | + + + 2 + + + + + ) + + + + g + + i + j + + + + + {\displaystyle {\widetilde {R}}_{ij}=R_{ij}-(n-2){\big (}\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi {\big )}-{\big (}\Delta \varphi +(n-2)|d\varphi |^{2}{\big )}g_{ij}} + + + + + + + + + Ric + ~ + + + + = + Ric + − + ( + n + − + 2 + ) + + + ( + + + Hess + ⁡ + φ + − + d + φ + ⊗ + d + φ + + + ) + + + − + + + ( + + + Δ + φ + + + ( + n + − + 2 + ) + + | + + d + φ + + + | + + + 2 + + + + + ) + + + g + + + {\displaystyle {\widetilde {\operatorname {Ric} }}=\operatorname {Ric} -(n-2){\big (}\operatorname {Hess} \varphi -d\varphi \otimes d\varphi {\big )}-{\big (}\Delta \varphi +(n-2)|d\varphi |^{2}{\big )}g} + + +=== Scalar curvature === + + + + + + + + R + ~ + + + + = + + e + + − + 2 + φ + + + R + − + 2 + ( + n + − + 1 + ) + + e + + − + 2 + φ + + + Δ + φ + − + ( + n + − + 2 + ) + ( + n + − + 1 + ) + + e + + − + 2 + φ + + + + | + + d + φ + + + | + + + 2 + + + + + {\displaystyle {\widetilde {R}}=e^{-2\varphi }R-2(n-1)e^{-2\varphi }\Delta \varphi -(n-2)(n-1)e^{-2\varphi }|d\varphi |^{2}} + + +if + + + + n + ≠ + 2 + + + {\displaystyle n\neq 2} + + this can be written + + + + + + + R + ~ + + + + = + + e + + − + 2 + φ + + + + [ + + R + − + + + + 4 + ( + n + − + 1 + ) + + + ( + n + − + 2 + ) + + + + + e + + − + ( + n + − + 2 + ) + φ + + / + + 2 + + + Δ + + ( + + e + + ( + n + − + 2 + ) + φ + + / + + 2 + + + ) + + + ] + + + + {\displaystyle {\tilde {R}}=e^{-2\varphi }\left[R-{\frac {4(n-1)}{(n-2)}}e^{-(n-2)\varphi /2}\Delta \left(e^{(n-2)\varphi /2}\right)\right]} + + +=== Traceless Ricci tensor === + + + + + + + + + R + ~ + + + + + i + j + + + − + + + 1 + n + + + + + + R + ~ + + + + + + + + g + ~ + + + + + i + j + + + = + + R + + i + j + + + − + + + 1 + n + + + R + + g + + i + j + + + − + ( + n + − + 2 + ) + + + ( + + + + ∇ + + i + + + + ∇ + + j + + + φ + − + + ∇ + + i + + + φ + + ∇ + + j + + + φ + + + ) + + + + + + + + ( + n + − + 2 + ) + + n + + + + + ( + + + Δ + φ + − + + | + + d + φ + + + | + + + 2 + + + + + ) + + + + g + + i + j + + + + + {\displaystyle {\widetilde {R}}_{ij}-{\frac {1}{n}}{\widetilde {R}}{\widetilde {g}}_{ij}=R_{ij}-{\frac {1}{n}}Rg_{ij}-(n-2){\big (}\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi {\big )}+{\frac {(n-2)}{n}}{\big (}\Delta \varphi -|d\varphi |^{2}{\big )}g_{ij}} + + + + + + + + + Ric + ~ + + + + − + + + 1 + n + + + + + + R + ~ + + + + + + + g + ~ + + + + = + Ric + − + + + 1 + n + + + R + g + − + ( + n + − + 2 + ) + + + ( + + + Hess + ⁡ + φ + − + d + φ + ⊗ + d + φ + + + ) + + + + + + + + ( + n + − + 2 + ) + + n + + + + + ( + + + Δ + φ + − + + | + + d + φ + + + | + + + 2 + + + + + ) + + + g + + + {\displaystyle {\widetilde {\operatorname {Ric} }}-{\frac {1}{n}}{\widetilde {R}}{\widetilde {g}}=\operatorname {Ric} -{\frac {1}{n}}Rg-(n-2){\big (}\operatorname {Hess} \varphi -d\varphi \otimes d\varphi {\big )}+{\frac {(n-2)}{n}}{\big (}\Delta \varphi -|d\varphi |^{2}{\big )}g} + + +=== (3,1) Weyl curvature === + + + + + + + + + + + W + ~ + + + + + i + j + k + + + + + l + + + = + + + + W + + i + j + k + + + + + l + + + + + {\displaystyle {{\widetilde {W}}_{ijk}}^{l}={W_{ijk}}^{l}} + + + + + + + + + W + ~ + + + + ( + X + , + Y + , + Z + ) + = + W + ( + X + , + Y + , + Z + ) + + + {\displaystyle {\widetilde {W}}(X,Y,Z)=W(X,Y,Z)} + + for any vector fields + + + + X + , + Y + , + Z + + + {\displaystyle X,Y,Z} + + +=== Volume form === + + + + + + + det + + + + g + ~ + + + + + + = + + e + + n + φ + + + + + det + g + + + + + {\displaystyle {\sqrt {\det {\widetilde {g}}}}=e^{n\varphi }{\sqrt {\det g}}} + + + + + + d + + μ + + + + g + ~ + + + + + = + + e + + n + φ + + + + d + + μ + + g + + + + + {\displaystyle d\mu _{\widetilde {g}}=e^{n\varphi }\,d\mu _{g}} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-4.md b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-4.md new file mode 100644 index 000000000..f9c67d456 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-4.md @@ -0,0 +1,1401 @@ +--- +title: "List of formulas in Riemannian geometry" +chunk: 5/6 +source: "https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:45.030637+00:00" +instance: "kb-cron" +--- + +=== Hodge operator on p-forms === + + + + + + + + + ∗ + ~ + + + + + + i + + 1 + + + ⋯ + + i + + n + − + p + + + + + + j + + 1 + + + ⋯ + + j + + p + + + + + = + + e + + ( + n + − + 2 + p + ) + φ + + + + ∗ + + + i + + 1 + + + ⋯ + + i + + n + − + p + + + + + + j + + 1 + + + ⋯ + + j + + p + + + + + + + {\displaystyle {\widetilde {\ast }}_{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}=e^{(n-2p)\varphi }\ast _{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}} + + + + + + + + + ∗ + ~ + + + + = + + e + + ( + n + − + 2 + p + ) + φ + + + ∗ + + + {\displaystyle {\widetilde {\ast }}=e^{(n-2p)\varphi }\ast } + + +=== Codifferential on p-forms === + + + + + + + + + + d + + ∗ + + + ~ + + + + + + j + + 1 + + + ⋯ + + j + + p + − + 1 + + + + + + i + + 1 + + + ⋯ + + i + + p + + + + + = + + e + + − + 2 + φ + + + ( + + d + + ∗ + + + + ) + + + j + + 1 + + + ⋯ + + j + + p + − + 1 + + + + + + i + + 1 + + + ⋯ + + i + + p + + + + + − + ( + n + − + 2 + p + ) + + e + + − + 2 + φ + + + + ∇ + + + i + + 1 + + + + + φ + + δ + + + j + + 1 + + + + + + i + + 2 + + + + + ⋯ + + δ + + + j + + p + − + 1 + + + + + + i + + p + + + + + + + {\displaystyle {\widetilde {d^{\ast }}}_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}=e^{-2\varphi }(d^{\ast })_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}-(n-2p)e^{-2\varphi }\nabla ^{i_{1}}\varphi \delta _{j_{1}}^{i_{2}}\cdots \delta _{j_{p-1}}^{i_{p}}} + + + + + + + + + + d + + ∗ + + + ~ + + + + = + + e + + − + 2 + φ + + + + d + + ∗ + + + − + ( + n + − + 2 + p + ) + + e + + − + 2 + φ + + + + ι + + ∇ + φ + + + + + {\displaystyle {\widetilde {d^{\ast }}}=e^{-2\varphi }d^{\ast }-(n-2p)e^{-2\varphi }\iota _{\nabla \varphi }} + + +=== Laplacian on functions === + + + + + + + + Δ + ~ + + + + Φ + = + + e + + − + 2 + φ + + + + + ( + + + Δ + Φ + + + ( + n + − + 2 + ) + g + ( + d + φ + , + d + Φ + ) + + + ) + + + + + {\displaystyle {\widetilde {\Delta }}\Phi =e^{-2\varphi }{\Big (}\Delta \Phi +(n-2)g(d\varphi ,d\Phi ){\Big )}} + + +=== Hodge Laplacian on p-forms === + + + + + + + + + Δ + + d + + + ~ + + + + ω + = + + e + + − + 2 + φ + + + + + ( + + + + Δ + + d + + + ω + − + ( + n + − + 2 + p + ) + d + ∘ + + ι + + ∇ + φ + + + ω + − + ( + n + − + 2 + p + − + 2 + ) + + ι + + ∇ + φ + + + ∘ + d + ω + + + 2 + ( + n + − + 2 + p + ) + d + φ + ∧ + + ι + + ∇ + φ + + + ω + − + 2 + d + φ + ∧ + + d + + ∗ + + + ω + + + ) + + + + + {\displaystyle {\widetilde {\Delta ^{d}}}\omega =e^{-2\varphi }{\Big (}\Delta ^{d}\omega -(n-2p)d\circ \iota _{\nabla \varphi }\omega -(n-2p-2)\iota _{\nabla \varphi }\circ d\omega +2(n-2p)d\varphi \wedge \iota _{\nabla \varphi }\omega -2d\varphi \wedge d^{\ast }\omega {\Big )}} + + +The "geometer's" sign convention is used for the Hodge Laplacian here. In particular it has the opposite sign on functions as the usual Laplacian. + +=== Second fundamental form of an immersion === +Suppose + + + + ( + M + , + g + ) + + + {\displaystyle (M,g)} + + is Riemannian and + + + + F + : + Σ + → + ( + M + , + g + ) + + + {\displaystyle F:\Sigma \to (M,g)} + + is a twice-differentiable immersion. Recall that the second fundamental form is, for each + + + + p + ∈ + M + , + + + {\displaystyle p\in M,} + + a symmetric bilinear map + + + + + h + + p + + + : + + T + + p + + + Σ + × + + T + + p + + + Σ + → + + T + + F + ( + p + ) + + + M + , + + + {\displaystyle h_{p}:T_{p}\Sigma \times T_{p}\Sigma \to T_{F(p)}M,} + + which is valued in the + + + + + g + + F + ( + p + ) + + + + + {\displaystyle g_{F(p)}} + +-orthogonal linear subspace to + + + + d + + F + + p + + + ( + + T + + p + + + Σ + ) + ⊂ + + T + + F + ( + p + ) + + + M + . + + + {\displaystyle dF_{p}(T_{p}\Sigma )\subset T_{F(p)}M.} + + Then + + + + + + + + h + ~ + + + + ( + u + , + v + ) + = + h + ( + u + , + v + ) + − + ( + ∇ + φ + + ) + + ⊥ + + + g + ( + u + , + v + ) + + + {\displaystyle {\widetilde {h}}(u,v)=h(u,v)-(\nabla \varphi )^{\perp }g(u,v)} + + for all + + + + u + , + v + ∈ + + T + + p + + + M + + + {\displaystyle u,v\in T_{p}M} + + +Here + + + + ( + ∇ + φ + + ) + + ⊥ + + + + + {\displaystyle (\nabla \varphi )^{\perp }} + + denotes the + + + + + g + + F + ( + p + ) + + + + + {\displaystyle g_{F(p)}} + +-orthogonal projection of + + + + ∇ + φ + ∈ + + T + + F + ( + p + ) + + + M + + + {\displaystyle \nabla \varphi \in T_{F(p)}M} + + onto the + + + + + g + + F + ( + p + ) + + + + + {\displaystyle g_{F(p)}} + +-orthogonal linear subspace to + + + + d + + F + + p + + + ( + + T + + p + + + Σ + ) + ⊂ + + T + + F + ( + p + ) + + + M + . + + + {\displaystyle dF_{p}(T_{p}\Sigma )\subset T_{F(p)}M.} + + +=== Mean curvature of an immersion === +In the same setting as above (and suppose + + + + Σ + + + {\displaystyle \Sigma } + + has dimension + + + + n + + + {\displaystyle n} + +), recall that the mean curvature vector is for each + + + + p + ∈ + Σ + + + {\displaystyle p\in \Sigma } + + an element + + + + + + + H + + + + p + + + ∈ + + T + + F + ( + p + ) + + + M + + + {\displaystyle {\textbf {H}}_{p}\in T_{F(p)}M} + + defined as the + + + + g + + + {\displaystyle g} + +-trace of the second fundamental form. Then + + + + + + + + + H + + ~ + + + + = + + e + + − + 2 + φ + + + ( + + + H + + + − + n + ( + ∇ + φ + + ) + + ⊥ + + + ) + . + + + {\displaystyle {\widetilde {\textbf {H}}}=e^{-2\varphi }({\textbf {H}}-n(\nabla \varphi )^{\perp }).} + + +Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature + + + + H + + + {\displaystyle H} + + in the hypersurface case is + + + + + + + + H + ~ + + + + = + + e + + − + φ + + + ( + H + − + n + ⟨ + ∇ + φ + , + η + ⟩ + ) + + + {\displaystyle {\widetilde {H}}=e^{-\varphi }(H-n\langle \nabla \varphi ,\eta \rangle )} + + +where + + + + η + + + {\displaystyle \eta } + + is a (local) normal vector field. + +== Variation formulas == +Let + + + + M + + + {\displaystyle M} + + be a smooth manifold and let + + + + + g + + t + + + + + {\displaystyle g_{t}} + + be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives + + + + + v + + i + j + + + = + + + ∂ + + ∂ + t + + + + + + ( + + + ( + + g + + t + + + + ) + + i + j + + + + + ) + + + + + {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} + + exist and are themselves as differentiable as necessary for the following expressions to make sense. + + + + v + = + + + + ∂ + g + + + ∂ + t + + + + + + {\displaystyle v={\frac {\partial g}{\partial t}}} + + is a one-parameter family of symmetric 2-tensor fields. + + + + + + + ∂ + + ∂ + t + + + + + Γ + + i + j + + + k + + + = + + + 1 + 2 + + + + g + + k + p + + + + + ( + + + + ∇ + + i + + + + v + + j + p + + + + + + ∇ + + j + + + + v + + i + p + + + − + + ∇ + + p + + + + v + + i + j + + + + + ) + + + . + + + {\displaystyle {\frac {\partial }{\partial t}}\Gamma _{ij}^{k}={\frac {1}{2}}g^{kp}{\Big (}\nabla _{i}v_{jp}+\nabla _{j}v_{ip}-\nabla _{p}v_{ij}{\Big )}.} + + + + + + + + ∂ + + ∂ + t + + + + + R + + i + j + k + l + + + = + + + 1 + 2 + + + + + ( + + + + ∇ + + j + + + + ∇ + + k + + + + v + + i + l + + + + + + ∇ + + i + + + + ∇ + + l + + + + v + + j + k + + + − + + ∇ + + i + + + + ∇ + + k + + + + v + + j + l + + + − + + ∇ + + j + + + + ∇ + + l + + + + v + + i + k + + + + + ) + + + + + + + 1 + 2 + + + + + + R + + i + j + k + + + + + p + + + + v + + p + l + + + − + + + 1 + 2 + + + + + + R + + i + j + l + + + + + p + + + + v + + p + k + + + + + {\displaystyle {\frac {\partial }{\partial t}}R_{ijkl}={\frac {1}{2}}{\Big (}\nabla _{j}\nabla _{k}v_{il}+\nabla _{i}\nabla _{l}v_{jk}-\nabla _{i}\nabla _{k}v_{jl}-\nabla _{j}\nabla _{l}v_{ik}{\Big )}+{\frac {1}{2}}{R_{ijk}}^{p}v_{pl}-{\frac {1}{2}}{R_{ijl}}^{p}v_{pk}} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-5.md b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-5.md new file mode 100644 index 000000000..8b58e9593 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-5.md @@ -0,0 +1,938 @@ +--- +title: "List of formulas in Riemannian geometry" +chunk: 6/6 +source: "https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:45.030637+00:00" +instance: "kb-cron" +--- + + + + + + + ∂ + + ∂ + t + + + + + R + + i + k + + + = + + + 1 + 2 + + + + + ( + + + + ∇ + + p + + + + ∇ + + k + + + + v + + i + p + + + + + + ∇ + + i + + + ( + div + ⁡ + v + + ) + + k + + + − + + ∇ + + i + + + + ∇ + + k + + + ( + + tr + + g + + + ⁡ + v + ) + − + Δ + + v + + i + k + + + + + ) + + + + + + + 1 + 2 + + + + R + + i + + + p + + + + v + + p + k + + + − + + + 1 + 2 + + + + R + + i + + + + + + + + p + + + + + + + + k + + + + + + + + q + + + + v + + p + q + + + + + {\displaystyle {\frac {\partial }{\partial t}}R_{ik}={\frac {1}{2}}{\Big (}\nabla ^{p}\nabla _{k}v_{ip}+\nabla _{i}(\operatorname {div} v)_{k}-\nabla _{i}\nabla _{k}(\operatorname {tr} _{g}v)-\Delta v_{ik}{\Big )}+{\frac {1}{2}}R_{i}^{p}v_{pk}-{\frac {1}{2}}R_{i}{}^{p}{}_{k}{}^{q}v_{pq}} + + + + + + + + ∂ + + ∂ + t + + + + R + = + + div + + g + + + ⁡ + + div + + g + + + ⁡ + v + − + Δ + ( + + tr + + g + + + ⁡ + v + ) + − + ⟨ + v + , + Ric + + ⟩ + + g + + + + + {\displaystyle {\frac {\partial }{\partial t}}R=\operatorname {div} _{g}\operatorname {div} _{g}v-\Delta (\operatorname {tr} _{g}v)-\langle v,\operatorname {Ric} \rangle _{g}} + + + + + + + + ∂ + + ∂ + t + + + + d + + μ + + g + + + = + + + 1 + 2 + + + + g + + p + q + + + + v + + p + q + + + + d + + μ + + g + + + + + {\displaystyle {\frac {\partial }{\partial t}}d\mu _{g}={\frac {1}{2}}g^{pq}v_{pq}\,d\mu _{g}} + + + + + + + + ∂ + + ∂ + t + + + + + ∇ + + i + + + + ∇ + + j + + + Φ + = + + ∇ + + i + + + + ∇ + + j + + + + + + ∂ + Φ + + + ∂ + t + + + + − + + + 1 + 2 + + + + g + + k + p + + + + + ( + + + + ∇ + + i + + + + v + + j + p + + + + + + ∇ + + j + + + + v + + i + p + + + − + + ∇ + + p + + + + v + + i + j + + + + + ) + + + + + + ∂ + Φ + + + ∂ + + x + + k + + + + + + + + {\displaystyle {\frac {\partial }{\partial t}}\nabla _{i}\nabla _{j}\Phi =\nabla _{i}\nabla _{j}{\frac {\partial \Phi }{\partial t}}-{\frac {1}{2}}g^{kp}{\Big (}\nabla _{i}v_{jp}+\nabla _{j}v_{ip}-\nabla _{p}v_{ij}{\Big )}{\frac {\partial \Phi }{\partial x^{k}}}} + + + + + + + + ∂ + + ∂ + t + + + + Δ + Φ + = + − + ⟨ + v + , + Hess + ⁡ + Φ + + ⟩ + + g + + + − + g + + + ( + + + div + ⁡ + v + − + + + 1 + 2 + + + d + ( + + tr + + g + + + ⁡ + v + ) + , + d + Φ + + + ) + + + + + {\displaystyle {\frac {\partial }{\partial t}}\Delta \Phi =-\langle v,\operatorname {Hess} \Phi \rangle _{g}-g{\Big (}\operatorname {div} v-{\frac {1}{2}}d(\operatorname {tr} _{g}v),d\Phi {\Big )}} + + +== Principal symbol == +The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature. + +The principal symbol of the map + + + + g + ↦ + + Rm + + g + + + + + {\displaystyle g\mapsto \operatorname {Rm} ^{g}} + + assigns to each + + + + ξ + ∈ + + T + + p + + + ∗ + + + M + + + {\displaystyle \xi \in T_{p}^{\ast }M} + + a map from the space of symmetric (0,2)-tensors on + + + + + T + + p + + + M + + + {\displaystyle T_{p}M} + + to the space of (0,4)-tensors on + + + + + T + + p + + + M + , + + + {\displaystyle T_{p}M,} + + given by + + + + + v + ↦ + + + + + ξ + + j + + + + ξ + + k + + + + v + + i + l + + + + + + ξ + + i + + + + ξ + + l + + + + v + + j + k + + + − + + ξ + + i + + + + ξ + + k + + + + v + + j + l + + + − + + ξ + + j + + + + ξ + + l + + + + v + + i + k + + + + 2 + + + = + − + + + 1 + 2 + + + ( + ξ + ⊗ + ξ + ) + + + ∧ + + + + + + + + + + ◯ + + + v + . + + + {\displaystyle v\mapsto {\frac {\xi _{j}\xi _{k}v_{il}+\xi _{i}\xi _{l}v_{jk}-\xi _{i}\xi _{k}v_{jl}-\xi _{j}\xi _{l}v_{ik}}{2}}=-{\frac {1}{2}}(\xi \otimes \xi ){~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}v.} + + +The principal symbol of the map + + + + g + ↦ + + Ric + + g + + + + + {\displaystyle g\mapsto \operatorname {Ric} ^{g}} + + assigns to each + + + + ξ + ∈ + + T + + p + + + ∗ + + + M + + + {\displaystyle \xi \in T_{p}^{\ast }M} + + an endomorphism of the space of symmetric 2-tensors on + + + + + T + + p + + + M + + + {\displaystyle T_{p}M} + + given by + + + + + v + ↦ + v + ( + + ξ + + ♯ + + + , + ⋅ + ) + ⊗ + ξ + + + ξ + ⊗ + v + ( + + ξ + + ♯ + + + , + ⋅ + ) + − + ( + + tr + + + g + + p + + + + + ⁡ + v + ) + ξ + ⊗ + ξ + − + + | + + ξ + + + | + + + g + + + 2 + + + v + . + + + {\displaystyle v\mapsto v(\xi ^{\sharp },\cdot )\otimes \xi +\xi \otimes v(\xi ^{\sharp },\cdot )-(\operatorname {tr} _{g_{p}}v)\xi \otimes \xi -|\xi |_{g}^{2}v.} + + +The principal symbol of the map + + + + g + ↦ + + R + + g + + + + + {\displaystyle g\mapsto R^{g}} + + assigns to each + + + + ξ + ∈ + + T + + p + + + ∗ + + + M + + + {\displaystyle \xi \in T_{p}^{\ast }M} + + an element of the dual space to the vector space of symmetric 2-tensors on + + + + + T + + p + + + M + + + {\displaystyle T_{p}M} + + by + + + + + v + ↦ + + | + + ξ + + + | + + + g + + + 2 + + + + tr + + g + + + ⁡ + v + + + v + ( + + ξ + + ♯ + + + , + + ξ + + ♯ + + + ) + . + + + {\displaystyle v\mapsto |\xi |_{g}^{2}\operatorname {tr} _{g}v+v(\xi ^{\sharp },\xi ^{\sharp }).} + + +== See also == +Liouville equations + +== Notes == + +== References == +Arthur L. Besse. "Einstein manifolds." Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10. Springer-Verlag, Berlin, 1987. xii+510 pp. ISBN 3-540-15279-2 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_homological_algebra_topics-0.md b/data/en.wikipedia.org/wiki/List_of_homological_algebra_topics-0.md new file mode 100644 index 000000000..4e0494eaa --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_homological_algebra_topics-0.md @@ -0,0 +1,50 @@ +--- +title: "List of homological algebra topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_homological_algebra_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:03.084460+00:00" +instance: "kb-cron" +--- + +This is a list of homological algebra topics, by Wikipedia page. +Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. +Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology. + + +== Basic techniques == +Cokernel +Exact sequence +Chain complex +Differential module +Five lemma +Short five lemma +Snake lemma +Nine lemma +Extension (algebra) +Central extension +Splitting lemma +Projective module +Injective module +Projective resolution +Injective resolution +Koszul complex +Exact functor +Derived functor +Ext functor +Tor functor +Filtration (abstract algebra) +Spectral sequence +Abelian category +Triangulated category +Derived category + + +== Applications == +Group cohomology +Galois cohomology +Lie algebra cohomology +Sheaf cohomology +Whitehead problem +Homological conjectures in commutative algebra \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_impossible_puzzles-0.md b/data/en.wikipedia.org/wiki/List_of_impossible_puzzles-0.md new file mode 100644 index 000000000..7a7b170bf --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_impossible_puzzles-0.md @@ -0,0 +1,31 @@ +--- +title: "List of impossible puzzles" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_impossible_puzzles" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:05.485545+00:00" +instance: "kb-cron" +--- + +This is a list of puzzles that cannot be solved. An impossible puzzle is a puzzle that cannot be resolved, either due to lack of sufficient information, or any number of logical impossibilities. + +15 Puzzle – Slide fifteen numbered tiles into numerical order. It is impossible to solve in half of the starting positions. +Five room puzzle – Cross each wall of a diagram exactly once with a continuous line. +MU puzzle – Transform the string MI to MU according to a set of rules. +Mutilated chessboard problem – Place 31 dominoes of size 2×1 on a chessboard with two opposite corners removed. +Coloring the edges of the Petersen graph with three colors. +Seven Bridges of Königsberg – Walk through a city while crossing each of seven bridges exactly once. +Squaring the circle, the impossible problem of constructing a square with the same area as a given circle, using only a compass and straightedge. +Three cups problem – Turn three cups right-side up after starting with one wrong and turning two at a time. +Three utilities problem – Connect three cottages to gas, water, and electricity without crossing lines. +Thirty-six officers problem – Arrange six regiments consisting of six officers each of different ranks in a 6 × 6 square so that no rank or regiment is repeated in any row or column. + + +== See also == +Impossible Puzzle, or "Sum and Product Puzzle", which is not impossible +-gry, a word puzzle +List of undecidable problems, no algorithm can exist to answer a yes–no question about the input + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-0.md b/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-0.md new file mode 100644 index 000000000..3eaec8e41 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-0.md @@ -0,0 +1,37 @@ +--- +title: "List of incomplete proofs" +chunk: 1/4 +source: "https://en.wikipedia.org/wiki/List_of_incomplete_proofs" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:06.686663+00:00" +instance: "kb-cron" +--- + +This page lists notable examples of incomplete or incorrect published mathematical proofs. Most of these were accepted as complete or correct for several years but later discovered to contain gaps or errors. There are both examples where a complete proof was later found, or where the alleged result turned out to be false. + +== Results later proved rigorously == +Euclid's Elements. Euclid's proofs are essentially correct, but strictly speaking sometimes contain gaps because he tacitly uses some unstated assumptions, such as the existence of intersection points. In 1899 David Hilbert gave a complete set of (second order) axioms for Euclidean geometry, called Hilbert's axioms, and between 1926 and 1959 Tarski gave some complete sets of first order axioms, called Tarski's axioms. +Isoperimetric inequality. For three dimensions it states that the shape enclosing the maximum volume for its surface area is the sphere. It was formulated by Archimedes but not proved rigorously until the 19th century, by Hermann Schwarz. +Infinitesimals. In the 18th century there was widespread use of infinitesimals in calculus, though these were not really well defined. Calculus was put on firm foundations in the 19th century, and Robinson put infinitesimals in a rigorous basis with the introduction of nonstandard analysis in the 20th century. +Fundamental theorem of algebra (see History). Many incomplete or incorrect attempts were made at proving this theorem in the 18th century, including by d'Alembert (1746), Euler (1749), de Foncenex (1759), Lagrange (1772), Laplace (1795), Wood (1798), and Gauss (1799). The first rigorous proof was published by Argand in 1806. +Dirichlet's theorem on arithmetic progressions. In 1808 Legendre published an attempt at a proof of Dirichlet's theorem, but as Dupré pointed out in 1859 one of the lemmas used by Legendre is false. Dirichlet gave a complete proof in 1837. +The proofs of the Kronecker–Weber theorem by Kronecker (1853) and Weber (1886) both had gaps. The first complete proof was given by Hilbert in 1896. +In 1879, Alfred Kempe published a purported proof of the four color theorem, whose validity as a proof was accepted for eleven years before it was refuted by Percy Heawood. Peter Guthrie Tait gave another incorrect proof in 1880 which was shown to be incorrect by Julius Petersen in 1891. Kempe's proof did, however, suffice to show the weaker five color theorem. The four-color theorem was eventually proved by Kenneth Appel and Wolfgang Haken in 1976. +Schröder–Bernstein theorem. In 1896 Schröder published a proof sketch which, however, was shown to be faulty by Alwin Reinhold Korselt in 1911 (confirmed by Schröder). +Fermat's Last Theorem. An initial proof was released by Andrew Wiles in June 1993 but was found to contain an error in September of that year. Wiles would go on to publish a corrected proof in 1995. +Jordan curve theorem. There has been some controversy about whether Jordan's original proof of this in 1887 contains gaps. Oswald Veblen in 1905 claimed that Jordan's proof is incomplete, but in 2007 Hales said that the gaps are minor and that Jordan's proof is essentially complete. +In 1905 Lebesgue tried to prove the (correct) result that a function implicitly defined by a Baire function is Baire, but his proof incorrectly assumed that the projection of a Borel set is Borel. Suslin pointed out the error and was inspired by it to define analytic sets as continuous images of Borel sets. +Dehn's lemma. Dehn published an attempted proof in 1910, but Kneser found a gap in 1929. It was finally proved in 1956 by Christos Papakyriakopoulos. +Hilbert's sixteenth problem about the finiteness of the number of limit cycles of a plane polynomial vector field. Henri Dulac published a partial solution to this problem in 1923, but in about 1980 Écalle and Ilyashenko independently found a serious gap, and fixed it in about 1991. +In 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the theorem of the three geodesics, which was later found to be flawed. The proof was completed by Werner Ballmann about 50 years later. +Littlewood–Richardson rule. Robinson published an incomplete proof in 1938, though the gaps were not noticed for many years. The first complete proofs were given by Marcel-Paul Schützenberger in 1977 and Thomas in 1974. +Class numbers of imaginary quadratic fields. In 1952 Heegner published a solution to this problem. His paper was not accepted as a complete proof as it contained a gap, and the first complete proofs were given in about 1967 by Baker and Stark. In 1969 Stark showed how to fill the gap in Heegner's paper. +In 1954 Igor Shafarevich published a proof that every finite solvable group is a Galois group over the rationals. However Schmidt pointed out a gap in the argument at the prime 2, which Shafarevich fixed in 1989. +Nielsen realization problem. Kravetz claimed to solve this in 1959 by first showing that Teichmüller space is negatively curved, but in 1974 Masur showed that it is not negatively curved. The Nielsen realization problem was finally solved in 1980 by Kerckhoff. +Yamabe problem. Yamabe claimed a solution in 1960, but Trudinger discovered a gap in 1968, and a complete proof was not given until 1984. +Mordell conjecture over function fields. Manin published a proof in 1963, but Coleman (1990) found and corrected a gap in the proof. +In 1973 Britton published a 282-page attempted solution of Burnside's problem. In his proof he assumed the existence of a set of parameters satisfying some inequalities, but Adian pointed out that these inequalities were inconsistent. Novikov and Adian had previously found a correct solution around 1968. +Classification of finite simple groups. In 1983, Gorenstein announced that the proof of the classification had been completed, but he had been misinformed about the status of the proof of classification of quasithin groups, which had a serious gap in it. A complete proof for this case was published by Aschbacher and Smith in 2004. +In 1986, Spencer Bloch published the paper "Algebraic Cycles and Higher K-theory" which introduced a higher Chow group, a precursor to motivic cohomology. The paper used an incorrect moving lemma; the lemma was later replaced by 30 pages of complex arguments that "took many years to be accepted as correct." +Kepler conjecture. Hsiang published an incomplete proof of this in 1993. In 1998 Hales published a proof depending on long computer calculations. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-1.md b/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-1.md new file mode 100644 index 000000000..29e3829a6 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-1.md @@ -0,0 +1,12 @@ +--- +title: "List of incomplete proofs" +chunk: 2/4 +source: "https://en.wikipedia.org/wiki/List_of_incomplete_proofs" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:06.686663+00:00" +instance: "kb-cron" +--- + +== Incorrect results == +In 1759 Euler claimed that there were no closed knight tours on a chess board with 3 rows, but in 1917 Ernest Bergholt found tours on 3 × 10 and 3 × 12 boards. Euler's conjecture on Graeco-Latin squares. In the 1780s Euler conjectured that no such squares exist for any oddly even number n ≡ 2 (mod 4). In 1959, R. C. Bose and S. S. Shrikhande constructed counterexamples of order 22. Then E. T. Parker found a counterexample of order 10 using a one-hour computer search. Finally Parker, Bose, and Shrikhande showed this conjecture to be false for all n ≥ 10. In 1798 A. M. Legendre claimed that 6 is not the sum of 2 rational cubes, which as Lamé pointed out in 1865 is false as 6 = (37/21)3 + (17/21)3. In 1803, Gian Francesco Malfatti claimed to prove that a certain arrangement of three circles would cover the maximum possible area inside a right triangle. However, to do so he made certain unwarranted assumptions about the configuration of the circles. It was shown in 1930 that circles in a different configuration could cover a greater area, and in 1967 that Malfatti's configuration was never optimal. See Malfatti circles. In 1806 André-Marie Ampère claimed to prove that a continuous function is differentiable at most points (though it is not entirely clear what he was claiming as he did not give a precise definition of a function). However, in 1872 Weierstrass gave an example of a continuous function that was not differentiable anywhere: The Weierstrass function. Intersection theory. In 1848 Steiner claimed that the number of conics tangent to 5 given conics is 7776 = 65, but later realized this was wrong. The correct number 3264 was found by Berner in 1865 and by Ernest de Jonquieres around 1859 and by Chasles in 1864 using his theory of characteristics. However these results, like many others in classical intersection theory, do not seem to have been given complete proofs until the work of Fulton and Macpherson in about 1978. Dirichlet's principle. This was used by Riemann in 1851, but Weierstrass found a counterexample to one version of this principle in 1870, and Hilbert stated and proved a correct version in 1900. Cayley (1878) incorrectly claimed that there are three different groups of order 6. This mistake is strange because in an earlier 1854 paper he correctly stated that there are just two such groups. In 1885, Evgraf Fedorov classified the convex polyhedra with congruent rhombic faces, but missed a case. Stanko Bilinski in 1960 rediscovered the Bilinski dodecahedron (forgotten after its previous 1752 publication) and proved that, with the addition of this shape, the classification was complete. Wronskians. In 1887 Mansion claimed in his textbook that if a Wronskian of some functions vanishes everywhere then the functions are linearly dependent. In 1889 Peano pointed out the counterexample x2 and x|x|. The result is correct if the functions are analytic. Vahlen (1891) published a purported example of an algebraic curve in 3-dimensional projective space that could not be defined as the zeros of 3 polynomials, but in 1941 Perron found 3 equations defining Vahlen's curve. In 1961 Kneser showed that any algebraic curve in projective 3-space can be given as the zeros of 3 polynomials. In 1898 Miller published a paper incorrectly claiming to prove that the Mathieu group M24 does not exist, though in 1900 he pointed out that his proof was wrong. Little claimed in 1900 that the writhe of a reduced knot diagram is an invariant. However, in 1974 Perko discovered a counterexample called the Perko pair, a pair of knots listed as distinct in tables for many years that are in fact the same. Hilbert's twenty-first problem. In 1908 Plemelj claimed to have shown the existence of Fuchsian differential equations with any given monodromy group, but in 1989 Bolibruch discovered a counterexample. In 1925 Ackermann published a proof that a weak system can prove the consistency of a version of analysis, but von Neumann found an explicit mistake in it a few years later. Gödel's incompleteness theorems showed that it is not possible to prove the consistency of analysis using weaker systems. Groups of order 64. In 1930 Miller published a paper claiming that there are 294 groups of order 64. Hall and Senior showed in 1964 that the correct number is 267. Kurt Gödel proved in 1933 that the truth of a certain class of sentences of first-order arithmetic, known in the literature as [∃*∀2∃*, all, (0)], was decidable. That is, there was a method for deciding correctly whether any statement of that form was true. In the final sentence of that paper, he asserted that the same proof would work for the decidability of the larger class [∃*∀2∃*, all, (0)]=, which also includes formulas that contain an equality predicate. However, in the mid-1960s, Stål Aanderaa showed that Gödel's proof would not go through for the larger class, and in 1982 Warren Goldfarb showed that validity of formulas from the larger class was in fact undecidable. Grunwald–Wang theorem. Wilhelm Grunwald published an incorrect proof in 1933 of an incorrect theorem, and George Whaples later published another incorrect proof. Shianghao Wang found a counterexample in 1948 and published a corrected version of the theorem in 1950. In 1934 Severi claimed that the space of rational equivalence classes of cycles on an algebraic surface is finite-dimensional, but Mumford (1968) showed that this is false for surfaces of positive geometric genus. One of many examples from algebraic geometry in the first half of the 20th century: Severi (1946) claimed that a degree-n surface in 3-dimensional projective space has at most (n+23)−4 nodes, B. Segre pointed out that this was wrong; for example, for degree 6 the maximum number of nodes is 65, achieved by the Barth sextic, which is more than the maximum of 52 claimed by Severi. Rokhlin invariant. Rokhlin incorrectly claimed in 1951 that the third stable stem of the homotopy groups of spheres is of order 12. In 1952 he discovered his error: it is in fact cyclic of order 24. The difference is crucial as it results in the existence of the Rokhlin invariant, a fundamental tool in the theory of 3- and 4-dimensional manifolds. In 1961, Jan-Erik Roos published an incorrect theorem about the vanishing of the first derived functor of the inverse limit functor under certain general conditions. However, in 2002, Amnon Neeman constructed a counterexample. Roos showed in 2006 that the theorem holds if one adds the assumption that the category has a set of generators. The Schur multiplier of the Mathieu group M22 is particularly notorious as it was miscalculated more than once: Burgoyne & Fong (1966) first claimed it had order 3, then in a 1968 correction claimed it had order 6; its order is in fact (currently believed to be) 12. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-2.md b/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-2.md new file mode 100644 index 000000000..c71bb82b3 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-2.md @@ -0,0 +1,11 @@ +--- +title: "List of incomplete proofs" +chunk: 3/4 +source: "https://en.wikipedia.org/wiki/List_of_incomplete_proofs" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:06.686663+00:00" +instance: "kb-cron" +--- + +This caused an error in the title of Janko's paper A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroup on J4: it does not have the full covering group as a subgroup, as the full covering group is larger than was realized at the time. The original statement of the classification of N-groups by Thompson in 1968 accidentally omitted the Tits group, though he soon fixed this. In 1975, Leitzel, Madan, and Queen incorrectly claimed that there are only 7 function fields over finite fields with genus > 0 and class number 1, but in 2013 Stirpe found another; there are in fact exactly 8. Busemann–Petty problem. Zhang published two papers in the Annals of Mathematics in 1994 and 1999, in the first of which he proved that the Busemann–Petty problem in R4 has a negative solution, and in the second of which he proved that it has a positive solution. Algebraic stacks. The book Laumon & Moret-Bailly (2000) on algebraic stacks mistakenly claimed that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. The results depending on this were repaired by Olsson (2007). \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-3.md b/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-3.md new file mode 100644 index 000000000..463eecbd3 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_incomplete_proofs-3.md @@ -0,0 +1,58 @@ +--- +title: "List of incomplete proofs" +chunk: 4/4 +source: "https://en.wikipedia.org/wiki/List_of_incomplete_proofs" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:06.686663+00:00" +instance: "kb-cron" +--- + +== Theories proven inconsistent == +Frege's foundations of mathematics in his 1879 book Begriffsschrift turned out to be inconsistent because of Russell's paradox, found in 1901. +Church's original published attempt in 1932 to define a formal system was inconsistent, as was his correction in 1933. The consistent part of his system later became the lambda calculus. +Quine published his original description of the system Mathematical Logic in 1940, but in 1942 Rosser showed it was inconsistent. Wang found a correction in 1950; the consistency of this revised system is still unclear. +In 1967 Reinhardt proposed Reinhardt cardinals, which Kunen showed to be inconsistent with ZFC in 1971, though they are not known to be inconsistent with ZF. +Per Martin-Löf's original version of intuitionistic type theory proposed in 1971 was shown to be inconsistent by Jean-Yves Girard in 1972, and was replaced by a corrected version. + +== Status unclear == +Uniform convergence. In his Cours d'Analyse of 1821, Cauchy "proved" that if a sum of continuous functions converges pointwise, then its limit is also continuous. However, Abel observed in 1826 that this is not the case. For the conclusion to hold, "pointwise convergence" must be replaced with "uniform convergence". It is not entirely clear that Cauchy's original result was wrong, because his definition of pointwise convergence was a little vague and may have been stronger than the one currently in use, and there are ways to interpret his result so that it is correct. There are many counterexamples using the standard definition of pointwise convergence. For example, a Fourier series of sine and cosine functions, all continuous, may converge pointwise to a discontinuous function such as a step function. +Carmichael's totient function conjecture was stated as a theorem by Robert Daniel Carmichael in 1907, but in 1922 he pointed out that his proof was incomplete. As of 2016 the problem is still open. +Italian school of algebraic geometry. Most gaps in proofs are caused either by a subtle technical oversight, or before the 20th century by a lack of precise definitions. A major exception to this is the Italian school of algebraic geometry in the first half of the 20th century, where lower standards of rigor gradually became acceptable. The result was that there are many papers in this area where the proofs are incomplete, or the theorems are not stated precisely. This list contains a few representative examples, where the result was not just incompletely proved but also hopelessly wrong. +In 1933 George David Birkhoff and Waldemar Joseph Trjitzinsky published a very general theorem on the asymptotics of sequences satisfying linear recurrences. The theorem was popularized by Jet Wimp and Doron Zeilberger in 1985. However, while the result is probably true, as of now (2021) Birkhoff and Trjitzinsky's proof is not generally accepted by experts, and the theorem is (acceptedly) proved only in special cases. +Jacobian conjecture. Keller asked this as a question in 1939, and in the next few years there were several published incomplete proofs, including 3 by B. Segre, but Vitushkin found gaps in many of them. The Jacobian conjecture is (as of 2016) an open problem, and more incomplete proofs are regularly announced. Hyman Bass, Edwin H. Connell, and David Wright (1982) discuss the errors in some of these incomplete proofs. +A strengthening of Hilbert's sixteenth problem asking whether there exists a uniform finite upper bound for the number of limit cycles of planar polynomial vector fields of given degree n. In the 1950s, Evgenii Landis and Ivan Petrovsky published a purported solution, but it was shown wrong in the early 1960s. +In 1954 Zarankiewicz claimed to have solved Turán's brick factory problem about the crossing number of complete bipartite graphs, but Kainen and Ringel later noticed a gap in his proof. +Complex structures on the 6-sphere. In 1969 Alfred Adler published a paper in the American Journal of Mathematics claiming that the 6-sphere has no complex structure. His argument was incomplete, and this is (as of 2016) still a major open problem. +Closed geodesics. In 1978 Wilhelm Klingenberg published a proof that smooth compact manifolds without boundary have infinitely many closed geodesics. His proof was controversial, and there is currently (as of 2016) no consensus on whether his proof is complete. +In 1991, Kapranov and Voevodsky published a paper claiming to prove a version of the homotopy hypothesis. Later, Simpson showed the result of the paper is not true but conjectured that a variant of the result might be true, the variant now known as the Simpson conjecture. +Telescope conjecture. Ravenel announced a refutation of this in 1992, but later withdrew it, and the conjecture is still open. +Matroid bundles. In 2003 Daniel Biss published a paper in the Annals of Mathematics claiming to show that matroid bundles are equivalent to real vector bundles, but in 2009 published a correction pointing out a serious gap in the proof. His correction was based on a 2007 paper by Mnëv. +In 2012, the Japanese mathematician Shinichi Mochizuki released online a series of papers in which he claims to prove the abc conjecture. Despite the publication in a peer-reviewed journal later, his proof has not been accepted as correct in the mainstream mathematical community. + +== See also == +List of long mathematical proofs +List of disproved mathematical ideas +Superseded theories in science + +== Notes == + +== References == + +== Further reading == +Lecat, Maurice (1935), Erreurs de mathématiciens des origines à nos jours, Bruxelles - Louvain: Librairie Castaigne - Ém. Desbarax — Lists over a hundred pages of (mostly trivial) published errors made by mathematicians. + +== External links == +David Mumford email about the errors of the Italian algebraic geometry school under Severi +The first 9 pages of [1] mention some examples of incorrect results in homotopy theory. + +=== MathOverflow questions === +Ilya Nikokoshev, Most interesting mathematics mistake? +Kevin Buzzard what mistakes did the Italian algebraic geometers actually make? +Will Jagy, Widely accepted mathematical results that were later shown wrong? +John Stillwell, What are some correct results discovered with incorrect (or no) proofs? +Moritz. Theorems demoted back to conjectures +Mei Zhang, Proofs shown to be wrong after formalization with proof assistant + +=== StackExchange questions === +Steven-Owen, In the history of mathematics, has there ever been a mistake? \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_inequalities-0.md b/data/en.wikipedia.org/wiki/List_of_inequalities-0.md new file mode 100644 index 000000000..d67d2548f --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_inequalities-0.md @@ -0,0 +1,286 @@ +--- +title: "List of inequalities" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_inequalities" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:09.363624+00:00" +instance: "kb-cron" +--- + +This article lists Wikipedia articles about named mathematical inequalities. + + +== Inequalities in pure mathematics == + + +=== Analysis === +Agmon's inequality +Askey–Gasper inequality +Babenko–Beckner inequality +Bernoulli's inequality +Bernstein's inequality (mathematical analysis) +Bessel's inequality +Bihari–LaSalle inequality +Bohnenblust–Hille inequality +Borell–Brascamp–Lieb inequality +Brezis–Gallouet inequality +Carleman's inequality +Carlson's inequality +Chebyshev–Markov–Stieltjes inequalities +Chebyshev's sum inequality +Clarkson's inequalities +Eilenberg's inequality +Fekete–Szegő inequality +Fenchel's inequality +Friedrichs' inequality +Gagliardo–Nirenberg interpolation inequality +Gårding's inequality +Grothendieck inequality +Grunsky's inequalities +Hanner's inequalities +Hardy's inequality +Hardy–Littlewood inequality +Hardy–Littlewood–Sobolev inequality +Harnack's inequality +Hausdorff–Young inequality +Hermite–Hadamard inequality +Hilbert's inequality +Hölder's inequality +Jackson's inequality +Jensen's inequality +Khabibullin's conjecture on integral inequalities +Kantorovich inequality +Karamata's inequality +Korn's inequality +Ladyzhenskaya's inequality +Landau–Kolmogorov inequality +Lebedev–Milin inequality +Lieb–Thirring inequality +Littlewood's 4/3 inequality +Markov brothers' inequality +Mashreghi–Ransford inequality +Max–min inequality +Minkowski's inequality +Poincaré inequality +Popoviciu's inequality +Prékopa–Leindler inequality +Rayleigh–Faber–Krahn inequality +Remez inequality +Riesz rearrangement inequality +Schur test +Shapiro inequality +Sobolev inequality +Steffensen's inequality +Szegő inequality +Three spheres inequality +Trace inequalities +Trudinger's theorem +Turán's inequalities +Von Neumann's inequality +Wirtinger's inequality for functions +Young's convolution inequality +Young's inequality for products + + +==== Inequalities relating to means ==== +Hardy–Littlewood maximal inequality +Inequality of arithmetic and geometric means +Ky Fan inequality +Levinson's inequality +Maclaurin's inequality +Mahler's inequality +Muirhead's inequality +Newton's inequalities +Stein–Strömberg theorem + + +=== Combinatorics === +Binomial coefficient bounds +Factorial bounds +XYZ inequality +Fisher's inequality +Ingleton's inequality +Lubell–Yamamoto–Meshalkin inequality +Nesbitt's inequality +Rearrangement inequality +Schur's inequality +Shapiro inequality +Stirling's formula (bounds) + + +=== Differential equations === +Grönwall's inequality + + +=== Geometry === + +Alexandrov–Fenchel inequality +Aristarchus's inequality +Barrow's inequality +Berger–Kazdan comparison theorem +Blaschke–Lebesgue inequality +Blaschke–Santaló inequality +Bishop–Gromov inequality +Bogomolov–Miyaoka–Yau inequality +Bonnesen's inequality +Brascamp–Lieb inequality +Brunn–Minkowski inequality +Castelnuovo–Severi inequality +Cheng's eigenvalue comparison theorem +Clifford's theorem on special divisors +Cohn-Vossen's inequality +Erdős–Mordell inequality +Euler's theorem in geometry +Gromov's inequality for complex projective space +Gromov's systolic inequality for essential manifolds +Hadamard's inequality +Hadwiger–Finsler inequality +Hinge theorem +Hitchin–Thorpe inequality +Isoperimetric inequality +Jordan's inequality +Jung's theorem +Loewner's torus inequality +Łojasiewicz inequality +Loomis–Whitney inequality +Melchior's inequality +Milman's reverse Brunn–Minkowski inequality +Milnor–Wood inequality +Minkowski's first inequality for convex bodies +Myers's theorem +Noether inequality +Ono's inequality +Pedoe's inequality +Ptolemy's inequality +Pu's inequality +Riemannian Penrose inequality +Toponogov's theorem +Triangle inequality +Weitzenböck's inequality +Wirtinger inequality (2-forms) + + +=== Information theory === +Inequalities in information theory +Kraft's inequality +Log sum inequality +Welch bounds + + +=== Algebra === +Abhyankar's inequality +Pisier–Ringrose inequality + + +==== Linear algebra ==== +Abel's inequality +Bregman–Minc inequality +Cauchy–Schwarz inequality +Golden–Thompson inequality +Hadamard's inequality +Hoffman-Wielandt inequality +Peetre's inequality +Sylvester's rank inequality +Triangle inequality +Trace inequalities + + +===== Eigenvalue inequalities ===== +Bendixson's inequality +Weyl's inequality in matrix theory +Cauchy interlacing theorem +Poincaré separation theorem + + +=== Number theory === +Bonse's inequality +Large sieve inequality +Pólya–Vinogradov inequality +Turán–Kubilius inequality +Weyl's inequality + + +=== Probability theory and statistics === +Azuma's inequality +Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount +Bhatia–Davis inequality, an upper bound on the variance of any bounded probability distribution +Bernstein inequalities (probability theory) +Boole's inequality +Borell–TIS inequality +BRS-inequality +Burkholder's inequality +Burkholder–Davis–Gundy inequalities +Cantelli's inequality +Chebyshev's inequality +Chernoff's inequality +Chung–Erdős inequality +Concentration inequality +Cramér–Rao inequality +Doob's martingale inequality +Dvoretzky–Kiefer–Wolfowitz inequality +Eaton's inequality, a bound on the largest absolute value of a linear combination of bounded random variables +Emery's inequality +Entropy power inequality +Etemadi's inequality +Fannes–Audenaert inequality +Fano's inequality +Fefferman's inequality +Fréchet inequalities +Gauss's inequality +Gauss–Markov theorem, the statement that the least-squares estimators in certain linear models are the best linear unbiased estimators +Gaussian correlation inequality +Gaussian isoperimetric inequality +Gibbs's inequality +Hoeffding's inequality +Hoeffding's lemma +Jensen's inequality +Khintchine inequality +Kolmogorov's inequality +Kunita–Watanabe inequality +Le Cam's theorem +Lenglart's inequality +Marcinkiewicz–Zygmund inequality +Markov's inequality +McDiarmid's inequality +Paley–Zygmund inequality +Pinsker's inequality +Popoviciu's inequality on variances +Prophet inequality +Rao–Blackwell theorem +Ross's conjecture, a lower bound on the average waiting time in certain queues +Samuelson's inequality +Shearer's inequality +Stochastic Gronwall inequality +Talagrand's concentration inequality +Vitale's random Brunn–Minkowski inequality +Vysochanskiï–Petunin inequality + + +=== Topology === +Berger's inequality for Einstein manifolds + + +== Inequalities particular to physics == +Ahlswede–Daykin inequality +Bell's inequality – see Bell's theorem +Bell's original inequality +CHSH inequality +Clausius–Duhem inequality +Correlation inequality – any of several inequalities +FKG inequality +Ginibre inequality +Griffiths inequality +Heisenberg's inequality +Holley inequality +Leggett–Garg inequality +Riemannian Penrose inequality +Rushbrooke inequality +Tsirelson's inequality + + +== See also == +Comparison theorem +List of mathematical identities +Lists of mathematics topics +List of set identities and relations \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics-0.md b/data/en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics-0.md new file mode 100644 index 000000000..677571fdd --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics-0.md @@ -0,0 +1,112 @@ +--- +title: "List of integration and measure theory topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:12.959879+00:00" +instance: "kb-cron" +--- + +This is a list of integration and measure theory topics, by Wikipedia page. + + +== Intuitive foundations == +Length +Area +Volume +Probability +Moving average + + +== Riemann integral == +Riemann sum +Riemann–Stieltjes integral +Bounded variation +Jordan content + + +== Improper integrals == +Cauchy principal value + + +== Measure theory and the Lebesgue integral == +Measure (mathematics) +Sigma algebra +Separable sigma algebra +Filtration (abstract algebra) +Borel algebra +Borel measure +Indicator function +Lebesgue measure +Lebesgue integration +Lebesgue's density theorem +Counting measure +Complete measure +Haar measure +Outer measure +Borel regular measure +Radon measure +Measurable function +Null set, negligible set +Almost everywhere, conull set +Lp space +Borel–Cantelli lemma +Lebesgue's monotone convergence theorem +Fatou's lemma +Absolutely continuous +Uniform absolute continuity +Total variation +Radon–Nikodym theorem +Fubini's theorem +Double integral +Vitali set, non-measurable set + + +== Extensions == +Henstock–Kurzweil integral +Amenable group +Banach–Tarski paradox +Hausdorff paradox + + +== Integral equations == +Fredholm equation +Fredholm operator +Liouville–Neumann series + + +== Integral transforms == +See also list of transforms, list of Fourier-related transforms + +Kernel (integral operator) +Convolution +Radon transform + + +== Integral geometry == +Buffon's needle +Hadwiger's theorem +mean width +intrinsic volumes + + +== Other == +Stokes theorem +Differentiation under the integral sign +Contour integration +Examples of contour integration + + +== See also == +List of calculus topics +List of multivariable calculus topics +List of real analysis topics +List of integrals +List of integrals of exponential functions +List of integrals of hyperbolic functions +List of integrals of irrational functions +List of integrals of logarithmic functions +List of integrals of rational functions +List of integrals of trigonometric functions +List of integrals of inverse trigonometric functions \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-0.md b/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-0.md new file mode 100644 index 000000000..67c2f5eff --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-0.md @@ -0,0 +1,133 @@ +--- +title: "List of irreducible Tits indices" +chunk: 1/3 +source: "https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:16.864015+00:00" +instance: "kb-cron" +--- + +In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field k, not assumed to be algebraically closed. The possible irreducible indices were classified by Jacques Tits, and this classification is reproduced below. (Because every index is a direct sum of irreducible indices, classifying all indices amounts to classifying irreducible indices.) + +== Organization of the list == +An index can be represented as a Dynkin diagram with certain vertices drawn close to each other (the orbit of the vertices under the *-action of the Galois group of k) and with certain sets of vertices circled (the orbits of the non-distinguished vertices under the *-action). This representation captures the full information of the index except when the underlying Dynkin diagram is D4, in which case one must distinguish between an action by the cyclic group C3 or the permutation group S3. +Alternatively, an index can be represented using the name of the underlying Dynkin diagram together with additional superscripts and subscripts, to be explained momentarily. This representation, together with the labeled Dynkin diagram described in the previous paragraph, captures the full information of the index. +The notation for an index is of the form gXtn,r, where + +X is the letter of the underlying Dynkin diagram (A, B, C, D, E, F, or G), +n is the number of vertices of the Dynkin diagram, +r is the relative rank of the corresponding algebraic group, +g is the order of the quotient of the absolute Galois group that acts faithfully on the Dynkin diagram (so g = 1, 2, 3, or 6), and +t is either +the degree of a certain division algebra (that is, the square root of its dimension) arising in the construction of the algebraic group when the group is of classical type (A, B, C, or D), in which case t is written in parentheses, or +the dimension of the anisotropic kernel of the algebraic group when the group is of exceptional type (E, F, or G), in which case t is written without parentheses. +In the description, there are given (only for classical groups), a representative of the isogeny class of the group +of the given Tits index. +The following complete list of all possible Tits indices over those special fields, which are the finite fields, the local and global fields (in any characteristic) is given (see and +(with full proof)). +The related sources are, and. + +== An == + +=== 1An === +Full name +1A(d)n,r +Image + +Conditions +d · (r + 1) = n + 1, d ≥ 1. +Distinguished vertices +d, 2d, ... , rd. +Description +Algebraic group : The special linear group SLr+1(D) where D is a central division algebra over k. +Special fields +Over a finite field, d = 1; over the reals, d = 1 or 2; over a p-adic field or a number field, or any local or global function field, d is arbitrary. + +=== 2An === +Full name +2A(d)n,r +Image + +Conditions +d | n + 1, d ≥ 1, 2rd ≤ n + 1. +Distinguished vertices +⁠ + + + + ( + d + , + n + + + 1 + − + d + ) + , + ( + 2 + d + , + n + + + 1 + − + 2 + d + ) + , + . + . + . + , + ( + r + d + , + n + + + 1 + − + r + d + ) + + + {\displaystyle (d,n+1-d),(2d,n+1-2d),...,(rd,n+1-rd)} + +⁠. +Description +Algebraic group : The special unitary group SU(n+1)/d(D,h), where D is a central division algebra of degree d over a separable quadratic extension k' of k, and where h is a nondegenerate hermitian form of index r relative to the unique non-trivial k-automorphism of k' . +Special fields +Over a finite field, d = 1 and r = ⌊(n+1)/2⌋; over the reals, d = 1; over a p-adic field or local function field, d = 1 and n = 2r − 1, 2r, 2r+1. Over a real number field, d and r are arbitrary; over a totally imaginary number field, d=1 and n = 2r − 1, 2r, 2r+1, or d>1 and (n+1)/d−2r =0,1; over a global function field, d=1 and n = 2r, 2r+1, 2r+2, or d>1 and (n+1)/d−2r =0,1. + +== Bn == +Full name +Bn,r +Image + +Conditions +None. +Distinguished vertices +1, 2, ... , r. +Description +Algebraic group : The special orthogonal group SO2n+1(k,q), where q is a quadratic form of index r, and defect 1 if k has characteristic 2. +Special fields +Over a finite field, r = n; over a p-adic field or local function field, r = n or n − 1 (and if char.k=2, defect 1); over the reals or a real number field, r is arbitrary; over a totally imaginary number field or a global function field, r = n or n − 1 (and if char.k=2, defect 1). + +== Cn == +Full name: C(d)n,r +Image: +Conditions: d = 2a | 2n, d ≥ 1; n = r if d = 1. +Distinguished vertices: d, 2d,...,rd. +Description: Algebraic group: The special unitary group SU2n/d(D,h), where D is a division algebra of degree d over k and h is a nondegenerate antihermitian form relative to a k-linear involution σ of D (also called an "involution of the first kind") such that the fixed-point subring Dσ has dimension d(d + 1)/2; or equivalently, when d > 1 and char k ≠ 2, the group SU2n/d where D and h are as above except that h is hermitian and Dσ has dimension d(d − 1)/2. When d = 1, this group is the symplectic group Sp2n(k). +Special fields: Over a finite field, d = 1; over the reals or a real number field, d = 1 (and r = n) or d = 2; over a p-adic field, local function field, totally imaginary number field or global function field, d = 1 (and r = n) or d = 2, and n = 2r or 2r − 1. + +== Dn == + +=== 1Dn === +Full name +1D(d)n,r +Image \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-1.md b/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-1.md new file mode 100644 index 000000000..752068485 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-1.md @@ -0,0 +1,384 @@ +--- +title: "List of irreducible Tits indices" +chunk: 2/3 +source: "https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:16.864015+00:00" +instance: "kb-cron" +--- + +Conditions +d = 2a | 2n, d ≥ 1, rd ≤ n, n ≠ rd + 1. +Distinguished vertices +d, 2d, ..., rd. +Description +Algebraic group : If k has characteristic ≠ 2, the special unitary group SU2n/d(D,h), where D is a division algebra of degree d over k and h is a hermitian form of discriminant 1 and index r, relative to a k-linear involution σ of D, an "involution of the first kind such that the subring Dσ has dimension d(d + 1)/2; or equivalently, when d > 1 and char k ≠ 2, the group SU2n/d where D and h are as above except that h is anti-hermitian form of discriminant 1 and index r, and Dσ has dimension d(d − 1)/2. +Special fields +Over a finite field, d = 1 and n = r; over the reals, d = 1 and n − r = 2m, or d = 2 and n = 2r; over a p-adic field or local function field, d = 1 and r = n or n − 2, or d = 2 and n = 2r or 2r + 3; over a number field, d = 1 and n − r = 2m, or d = 2 and n − 2r = 2m or 3; over a totally imaginary number field or a global function field, d=1 and ⁠ + + + + n + − + r + = + 0 + , + 2 + , + + + {\displaystyle n-r=0,2,} + +⁠ or ⁠ + + + + d + = + 2 + + + {\displaystyle d=2} + +⁠ and ⁠ + + + + n + − + 2 + r + = + 0 + , + 3. + + + {\displaystyle n-2r=0,3.} + +⁠ + +=== 2Dn === +Full name +2D(d)n,r +Image + +Conditions +d = 2a | 2n, d ≥ 1, rd ≤ n-1. +Distinguished vertices +d, 2d, ... , rd. The last one is replaced by ⁠ + + + + ( + n + − + 1 + , + n + ) + + + {\displaystyle (n-1,n)} + +⁠ when ⁠ + + + + n + = + r + d + + + 1. + + + {\displaystyle n=rd+1.} + +⁠ +Description +Algebraic group : The same as for 1D(d)n,r, except that all forms +in question have now discriminant ≠ 1. + +Special fields +Over a finite field, ⁠ + + + + d + = + 1 + , + n + = + r + + + 1 + + + {\displaystyle d=1,n=r+1} + +⁠; over the reals, ⁠ + + + + d + = + 1 + , + n + − + r + = + 2 + m + + + 1 + + + {\displaystyle d=1,n-r=2m+1} + +⁠ or ⁠ + + + + d + = + 2 + , + n + = + 2 + r + + + 1. + + + {\displaystyle d=2,n=2r+1.} + +⁠ Over +a p-adic or local function field, ⁠ + + + + d + = + 1 + , + n + = + r + + + 1 + + + {\displaystyle d=1,n=r+1} + +⁠, or ⁠ + + + + d + = + 2 + , + n + − + 2 + r + = + 1 + , + 2 + , + 3 + + + {\displaystyle d=2,n-2r=1,2,3} + +⁠; over a real number field, ⁠ + + + + d + = + 1 + , + r + + + {\displaystyle d=1,r} + +⁠ is arbitrary, +⁠ + + + + d + = + 2 + , + n + − + 2 + r + = + 1 + , + 2 + , + 3 + , + + + {\displaystyle d=2,n-2r=1,2,3,} + +⁠ if D is non-split over the reals, and ⁠ + + + + d + = + 2 + , + r + + + {\displaystyle d=2,r} + +⁠ arbitrary, if D is split over the reals. +Over a totally imaginary number field or global function field, ⁠ + + + + d + = + 1 + , + n + − + r + = + 2 + , + 4 + + + {\displaystyle d=1,n-r=2,4} + +⁠ or ⁠ + + + + d + = + 2 + , + n + − + 2 + r + = + 1 + , + 2 + , + 3 + + + {\displaystyle d=2,n-2r=1,2,3} + +⁠. + +=== 3D284,0 === +Image + +Special fields +This type exists only over some number fields; does not exist over the finite fields, local fields nor global +function fields. + +=== 6D284,0 === +Image + +Special fields +This type exists only over some number fields; does not exist over the finite fields, local fields nor global +function fields. + +=== 3D94,1 === +Image + +Special fields +This type exists only over some number fields; does not exist over the finite fields, local fields nor global +function fields. + +=== 6D94,1 === +Image + +Special fields +This type exists only over some number fields; does not exist over the finite fields, local fields nor global +function fields. + +=== 3D24,2 === +Image + +Special fields +This type exists over any finite field, any local non-archimedean and global field; does not exist over the reals. + +=== 6D24,2 === +Image + +Special fields +This type exists over any local non-archimedean and global field; does not exist over the finite fields nor the reals. + +== E6 == + +=== 1E786,0 === +Image: +Special fields: This type exists only over some number fields; does not exist over the finite fields, +local fields nor global function fields. + +=== 1E286,2 === +Image: +Special fields: This type exists only over the reals and over some number fields; does not exist over any +finite field nor over any local non-archimedean field nor global function field. + +=== 1E166,2 === +Image: +Special fields: This type exists only over some local non-archimedean and global fields; does not exists over the finite fields nor the reals. + +=== 1E06,6 === +Image: +Special fields: This type exists over any field. + +=== 2E786,0 === +Image: +Special fields: This type exists only over the reals and over some number fields; does not exist over any +finite field nor over any local non-archimedean field nor global function field. + +=== 2E356,1 === +Image: +Special fields: This type exists only over some number fields; does not exist over the reals, +any finite field nor over any local field nor global function field. + +=== 2E296,1 === +Image: +Special fields: This type exists only over some number fields; does not exist over the reals, +any finite field nor over any local field nor global function field. + +=== 2E16'6,2 === +Image: +Special fields: This type exists only over the reals and over some number fields; does not exist over any +finite field nor over any local non-archimedean field nor global function field. + +=== 2E16"6,2 === +Image: +Special fields: This type exists only over some number fields; does not exist over any +finite field nor over any local field nor global function field. + +=== 2E26,4 === +Image: +Special fields: This type exists over any finite field, any local and global field. + +== E7 == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-2.md b/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-2.md new file mode 100644 index 000000000..8d4b1d6f8 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-2.md @@ -0,0 +1,118 @@ +--- +title: "List of irreducible Tits indices" +chunk: 3/3 +source: "https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:16.864015+00:00" +instance: "kb-cron" +--- + +=== E1337,0 === +Image: +Special fields: This type exists only over the reals and over some number fields; does not exist over any +finite field nor over any local non-archimedean nor global function field. + +=== E787,1 === +Image: +Special fields: This type does not exist over any finite field nor any local nor global field. + +=== E667,1 === +Image: +Special fields: This type does not exist over any finite field nor over any local nor global field. + +=== E487,1 === +Image: +Special fields: This type does not exist over any finite field nor any local nor global fields. + +=== E317,2 === +Image: +Special fields: This type exists only over some number fields; does not exists over any finite field, +nor any local nor global field. + +=== E287,3 === +Image: +Special fields: This type exists only over the reals and over some number fields; does not exists over any finite field, nor local non-archimedean nor global function fields. + +=== E97,4 === +Image: +Special fields: This type does not exist over any finite field; it exists over any local and global field. + +=== E07,7 === +Image: +Special fields: This type exists over any field. + +== E8 == + +=== E2488,0 === +Image: +Special fields: This type exists only over the reals and over some number fields; does not exists over any finite field, +nor local non-archimedean nor global function fields. + +=== E1338,1 === +Image: +Special fields: This type does not exist over any finite field nor over any local nor global field. + +=== E918,1 === +Image: +Special fields: This type does not exist over any finite field nor over any local nor global field. + +=== E788,2 === +Image: +Special fields: This type does not exist over any finite field nor over any local nor global field. + +=== E668,2 === +Image: +Special fields: This type does not exist over any finite field nor over any local nor global field. + +=== E288,4 === +Image: +Special fields: This type exists only over the reals and over some number fields; does not exists over finite fields, +local non-archimedean nor global function fields. + +=== E08,8 === +Image: +Special fields: This type exists over any field. + +== F4 == + +=== F524,0 === +Image: +Description: Algebraic Group: The automorphism group of an exceptional simple Jordan algebra J that does not contain nonzero nilpotent elements. +Special fields: This type exists only over the reals and over some number fields; does not exist over finite fields, local non-archimedean nor global function fields. + +=== F214,1 === +Image: +Description: Algebraic Group: The automorphism group of an exceptional simple Jordan algebra J containing nonzero nilpotent elements, no two of which are nonproportional and orthogonal. +Special fields: This type exists only over the reals and over some number fields; does not exist over any finite field, nor local non-archimedean nor global function field. + +=== F04,4 === +Image: +Description: Algebraic Group: The automorphism group of an exceptional simple Jordan algebra J containing nonproportional orthogonal nilpotent elements. +Special fields: This type exists over any field. + +== G2 == +A group of type G2 is always the automorphism group of an octonion algebra. + +=== G142,0 === +Image: +Description: Algebraic group: the automorphism group of a division octonion algebra. +Special fields: This type exists over the reals and some number fields; does not exist over any finite field, nor +local non-archimedean nor global function field. + +=== G02,2 === +Image: +Description: Algebraic group: the automorphism group of a split octonion algebra. +Special fields: This type exists over any field. + +== Notes == + +== References == +Jacobson, Nathan (1939), "Cayley numbers and simple Lie algebras of type G", Duke Mathematical Journal, 5: 775–783, doi:10.1215/s0012-7094-39-00562-4 +Satake, I. (1971), Classification theory of semisimple algebraic groups (with an appendix by M. Sugiura), New York: Marcel--Dekker, pp. viii+149, MR 0316588 +Satake, I. (2001), "On classification of semisimple algebraic groups", Class Field Theory - Its centenary and prospect (Tokyo, 1998) (Advances Studies in Pure Math. vol. 30), Tokyo: Math. Soc. Japan, pp. 197–216, MR 1846459 +Selbach, M. (1976), Klassifikationstheorie der halbeinfacher algebraischer Gruppen, Bonner Math. Schriften, No. 83, Bonn: Universitat Bonn, MR 0432776 +Springer, Tonny A. (1998) [1981], Linear Algebraic Groups (2nd ed.), New York: Birkhäuser, ISBN 0-8176-4021-5, MR 1642713 +Sury, B. (2012), What is the Tits index and how to work with it (www.isibang.ac.in/~sury/titsclassbeam.pdf) +Thắng, N. Q. (2022), "On the Tits indices of absolutely almost simple algebraic groups over local and global fields", Journal of Pure and Applied Algebra, 226 (9), doi:10.1016/j.jpaa.2022.107031, MR 4379334 +Tits, Jacques (1966), "Classification of algebraic semisimple groups", Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 33–62, MR 0224710 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_knot_theory_topics-0.md b/data/en.wikipedia.org/wiki/List_of_knot_theory_topics-0.md new file mode 100644 index 000000000..925cc9d84 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_knot_theory_topics-0.md @@ -0,0 +1,195 @@ +--- +title: "List of knot theory topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_knot_theory_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:19.329939+00:00" +instance: "kb-cron" +--- + +Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. + + +== History == + + +== Knots, links, braids == +Knot (mathematics) gives a general introduction to the concept of a knot. +Two classes of knots: torus knots and pretzel knots +Cinquefoil knot also known as a (5, 2) torus knot. +Figure-eight knot (mathematics) the only 4-crossing knot +Granny knot (mathematics) and Square knot (mathematics) are a connected sum of two Trefoil knots +Perko pair, two entries in a knot table that were later shown to be identical. +Stevedore knot (mathematics), a prime knot with crossing number 6 +Three-twist knot is the twist knot with three-half twists, also known as the 52 knot. +Trefoil knot A knot with crossing number 3 +Unknot +Knot complement, a compact 3 manifold obtained by removing an open neighborhood of a proper embedding of a tame knot from the 3-sphere. +Notation used in knot theory: + +Conway notation +Dowker–Thistlethwaite notation (DT notation) +Gauss code (see also Gauss diagrams) +continued fraction + + +=== General knot types === +2-bridge knot +Alternating knot; a knot that can be represented by an alternating diagram (i.e. the crossing alternate over and under as one traverses the knot). +Berge knot a class of knots related to Lens space surgeries and defined in terms of their properties with respect to a genus 2 Heegaard surface. +Cable knot, see Satellite knot +Chiral knot is knot which is not equivalent to its mirror image. +Double torus knot, a knot that can be embedded in a double torus (a genus 2 surface). +Fibered knot +Framed knot +Invertible knot +Prime knot +Legendrian knot are knots embedded in + + + + + + R + + + 3 + + + + + {\displaystyle \mathbb {R} ^{3}} + + tangent to the standard contact structure. +Lissajous knot +Ribbon knot +Satellite knot +Slice knot +Torus knot +Transverse knot +Twist knot +Virtual knot +Wild knot + + +=== Links === +Borromean rings, the simplest Brunnian link +Brunnian link, a set of links which become trivial if one loop is removed +Hopf link, the simplest non-trivial link +Solomon's knot, a two-ring link with four crossings. +Whitehead link, a twisted loop linked with an untwisted loop. +Unlink +General types of links: + +Algebraic link +Hyperbolic link +Pretzel link +Split link +String link + + +=== Tangles === +Tangle (mathematics) +Algebraic tangle +Tangle diagram +Tangle product +Tangle rotation +Tangle sum +Inverse of a tangle +Rational tangle +Tangle denominator closure +Tangle numerator closure +Reciprocal tangle + + +=== Braids === +Braid theory +Braid group + + +== Operations == +Band sum +Flype +Fox n-coloring +Tricolorability +Knot sum +Reidemeister move + + +=== Elementary treatment using polygonal curves === +elementary move (R1 move, R2 move, R3 move) +R-equivalent +delta-equivalent + + +== Invariants and properties == +Knot invariant is an invariant defined on knots which is invariant under ambient isotopies of the knot. +Finite type invariant is a knot invariant that can be extended to an invariant of certain singular knots +Knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. +Alexander polynomial and the associated Alexander matrix; The first knot polynomial (1923). Sometimes called the Alexander–Conway polynomial +Bracket polynomial is a polynomial invariant of framed links. Related to the Jones polynomial. Also known as the Kauffman bracket. +Conway polynomial uses Skein relations. +Homfly polynomial or HOMFLYPT polynomial. +Jones polynomial assigns a Laurent polynomial in the variable t1/2 to the knot or link. +Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. +Arf invariant of a knot +Average crossing number +Bridge number +Crosscap number +Crossing number +Hyperbolic volume +Kontsevich invariant +Linking number +Milnor invariants +Racks and quandles and Biquandle +Ropelength +Seifert surface +Self-linking number +Signature of a knot +Skein relation +Slice genus +Tunnel number, the number of arcs that must be added to make the knot complement a handlebody +Writhe + + +== Mathematical problems == +Berge conjecture +Birman–Wenzl algebra +Clasper (mathematics) +Eilenberg–Mazur swindle +Fáry–Milnor theorem +Gordon–Luecke theorem +Khovanov homology +Knot group +Knot tabulation +Knotless embedding +Linkless embedding +Link concordance +Link group +Link (knot theory) +Milnor conjecture (topology) +Milnor map +Möbius energy +Mutation (knot theory) +Physical knot theory +Planar algebra +Smith conjecture +Tait conjectures +Temperley–Lieb algebra +Thurston–Bennequin number +Tricolorability +Unknotting number +Unknotting problem +Volume conjecture + + +=== Theorems === +Schubert's theorem +Conway's theorem +Alexander's theorem + + +== Lists == +List of mathematical knots and links +List of prime knots \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_large_cardinal_properties-0.md b/data/en.wikipedia.org/wiki/List_of_large_cardinal_properties-0.md new file mode 100644 index 000000000..6172c063d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_large_cardinal_properties-0.md @@ -0,0 +1,92 @@ +--- +title: "List of large cardinal properties" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_large_cardinal_properties" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:21.962706+00:00" +instance: "kb-cron" +--- + +This page includes a list of large cardinal properties in the mathematical field of set theory. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, Vκ satisfies "there is an unbounded class of cardinals satisfying φ". +The following table usually arranges cardinals in order of consistency strength, with size of the cardinal used as a tiebreaker. In a few cases (such as strongly compact cardinals) the exact consistency strength is not known and the table uses the current best guess. + +"Small" cardinals: 0, 1, 2, ..., + + + + + ℵ + + 0 + + + , + + ℵ + + 1 + + + + + {\displaystyle \aleph _{0},\aleph _{1}} + +,..., + + + + κ + = + + ℵ + + κ + + + + + {\displaystyle \kappa =\aleph _{\kappa }} + +, ... (see Aleph number) +the height of the minimal transitive model of ZFC +worldly cardinals +weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals +weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals. +reflecting cardinals +pseudo uplifting cardinals, uplifting cardinals +weakly compact (= Π11-indescribable), Πmn-indescribable, totally indescribable cardinals, ν-indescribable cardinals +λ-unfoldable, unfoldable cardinals, λ-shrewd, shrewd cardinals, strongly uplifting cardinals (not clear how these relate to each other). +ethereal cardinals, subtle cardinals +almost ineffable, ineffable, n-ineffable, totally ineffable cardinals +remarkable cardinals +α-Erdős cardinals (for countable α), 0# (not a cardinal), γ-iterable, γ-Erdős cardinals (for uncountable γ) +almost Ramsey, Jónsson, Rowbottom, Ramsey, ineffably Ramsey, completely Ramsey, strongly Ramsey, super Ramsey cardinals +measurable cardinals, 0† +λ-strong, strong cardinals, tall cardinals +Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin cardinals +superstrong cardinals (=1-superstrong; for n-superstrong for n≥2 see further down.) +subcompact, strongly compact (Woodin< strongly compact≤supercompact), supercompact, hypercompact cardinals +η-extendible, extendible cardinals +almost high jump cardinals +Vopěnka cardinals, Shelah for supercompactness, high jump cardinals, super high jump cardinals +n-superstrong (n≥2), n-almost huge, n-super almost huge, n-huge, n-superhuge cardinals (1-huge=huge, etc.) +exacting cardinals, ultraexacting cardinals +Wholeness axiom, rank-into-rank (Axioms I3, I2, I1, and I0) +The following even stronger large cardinal properties are not consistent with the axiom of choice, but their existence has not yet been refuted in ZF alone (that is, without use of the axiom of choice). + +weakly Reinhardt cardinal, Reinhardt cardinal, proto-Berkeley cardinal, Berkeley cardinal, super Reinhardt cardinal, totally Reinhardt cardinal, club Berkeley cardinal, limit club Berkeley cardinal +Many of these large cardinals axioms also have virtual versions. + + +== References == + +Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2. +Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3. +Kanamori, Akihiro; Magidor, M. (1978). "The evolution of large cardinal axioms in set theory". Higher Set Theory (PDF). Lecture Notes in Mathematics. Vol. 669. Springer Berlin / Heidelberg. pp. 99–275. doi:10.1007/BFb0103104. ISBN 978-3-540-08926-1. +Solovay, Robert M.; Reinhardt, William N.; Kanamori, Akihiro (1978). "Strong axioms of infinity and elementary embeddings" (PDF). Annals of Mathematical Logic. 13 (1): 73–116. doi:10.1016/0003-4843(78)90031-1. + + +== External links == +Cantor's attic +some diagrams of large cardinal properties \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_lemmas-0.md b/data/en.wikipedia.org/wiki/List_of_lemmas-0.md new file mode 100644 index 000000000..7c654c80c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_lemmas-0.md @@ -0,0 +1,311 @@ +--- +title: "List of lemmas" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_lemmas" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:23.303385+00:00" +instance: "kb-cron" +--- + +This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. + + +== Algebra == + +Abhyankar's lemma +Aubin–Lions lemma +Bergman's diamond lemma +Fitting lemma +Injective test lemma +Hua's lemma (exponential sums) +Krull's separation lemma +Schanuel's lemma (projective modules) +Schwartz–Zippel lemma +Shapiro's lemma +Stewart–Walker lemma (tensors) +Whitehead's lemma (Lie algebras) +Zariski's lemma + + +=== Algebraic geometry === +Abhyankar's lemma +Fundamental lemma (Langlands program) + + +=== Category theory === + +Five lemma +Horseshoe lemma +Nine lemma +Short five lemma +Snake lemma +Splitting lemma +Yoneda lemma + + +=== Linear algebra === +Matrix determinant lemma +Matrix inversion lemma + + +=== Group theory === +Burnside's lemma also known as the Cauchy–Frobenius lemma +Frattini's lemma (finite groups) +Goursat's lemma +Mautner's lemma (representation theory) +Ping-pong lemma (geometric group theory) +Schreier's subgroup lemma +Schur's lemma (representation theory) +Zassenhaus lemma + + +=== Polynomials === +Gauss's lemma (polynomials) +Schwartz–Zippel lemma + + +=== Ring theory and commutative algebra === +Artin–Rees lemma +Hensel's lemma (commutative rings) +Nakayama lemma +Noether's normalization lemma +Prime avoidance lemma + + +=== Universal algebra === +Jónsson's lemma + + +== Analysis == + +Fekete's lemma +Fundamental lemma of the calculus of variations +Hopf lemma +Sard's lemma (singularity theory) +Stechkin's lemma (functional and numerical analysis) +Vitali covering lemma (real analysis) +Watson's lemma + + +=== Complex analysis === +Estimation lemma (contour integrals) +Hartogs's lemma (several complex variables) +Jordan's lemma +Lemma on the Logarithmic derivative +Schwarz lemma + + +=== Fourier analysis === +Riemann–Lebesgue lemma + + +=== Differential equations === +Borel's lemma (partial differential equations) +Grönwall's lemma +Lax–Milgram lemma +Pugh's closing lemma +Weyl's lemma (Laplace equation) (partial differential equations) + + +=== Differential forms === +Poincaré lemma of closed and exact differential forms + + +=== Functional analysis === +Cotlar–Stein lemma +Ehrling's lemma +Riesz's lemma + + +=== Mathematical series === +Abel's lemma +Kronecker's lemma + + +=== Numerical analysis === +Bramble–Hilbert lemma +Céa's lemma + + +== Applied mathematics == + +Danielson–Lanczos lemma (Fourier transforms) +Farkas's lemma (linear programming) +Feld–Tai lemma (electromagnetism) +Little's lemma (queuing theory) +Finsler's lemma + + +=== Control theory === +Finsler's lemma +Hautus lemma +Kalman–Yakubovich–Popov lemma + + +=== Computational complexity theory === +Isolation lemma +Switching lemma + + +==== Cryptography ==== +Forking lemma +Leftover hash lemma +Piling-up lemma (linear cryptanalysis) +Yao's XOR lemma + + +==== Formal languages ==== +Interchange lemma +Newman's lemma (term rewriting) +Ogden's lemma +Pumping lemma sometimes called the Bar-Hillel lemma + + +=== Microeconomics === +Hotelling's lemma +Shephard's lemma + + +== Combinatorics == +Cousin's lemma (integrals) +Dickson's lemma +Littlewood–Offord lemma +Pólya–Burnside lemma +Sperner's lemma +Ky Fan lemma (combinatorial geometry) + + +=== Graph theory === + +Berge's lemma +Counting lemma +Crossing lemma +Expander mixing lemma +Handshaking lemma +Kelly's lemma +Kőnig's lemma +Szemerédi regularity lemma + + +=== Order theory === +Higman's lemma +Ultrafilter lemma + + +== Dynamical systems == +Barbalat's lemma +Kac's lemma (ergodic theory) + + +== Geometry == +Shadowing lemma +Big-little-big lemma (mathematics of paper folding) +Gordan's lemma +Hilbert's lemma + + +=== Euclidean geometry === +Archimedes's lemmas +Johnson–Lindenstrauss lemma (Euclidean geometry) + + +=== Hyperbolic geometry === +Margulis lemma + + +=== Metric spaces === +Lebesgue's number lemma (dimension theory) + + +=== Riemannian geometry === +Gauss's lemma (Riemannian geometry) + + +== Mathematical logic == +Craig interpolation lemma +Diagonal lemma +Lindenbaum's lemma +Mostowski collapse lemma +Teichmüller–Tukey lemma also known as Tukey's lemma +Zorn's lemma; equivalent to the axiom of choice + + +=== Set theory === + +Covering lemma +Delta lemma +Dynkin lemma +Fodor's lemma +Fixed-point lemma for normal functions (axiomatic set theory) +Moschovakis coding lemma +Rasiowa–Sikorski lemma + + +== Number theory == + +Bézout's lemma +Dwork's lemma +Euclid's lemma +Gauss's lemma +Hensel's lemma +Zolotarev's lemma +Siegel's lemma (Diophantine approximation) + + +=== Analytic number theory === +Hua's lemma +Vaughan's lemma + + +=== Diophantine equations === +Bhaskara's lemma + + +=== Sieve theory === +Fundamental lemma of sieve theory + + +== Probability theory == +Borel–Cantelli lemma +Doob–Dynkin lemma +Itô's lemma (stochastic calculus) +Lovász local lemma +Stein's lemma +Wald's lemma + + +=== Statistics === +Glivenko–Cantelli lemma +Neyman–Pearson lemma +Robbins lemma + + +=== Measure theory === +Factorization lemma +Fatou's lemma +Frostman's lemma (geometric measure theory) +Malliavin's absolute continuity lemma + + +== Topology == +Lindelöf's lemma +Urysohn's lemma +Tube lemma + + +=== Differential topology === +Morse lemma + + +=== Fixed-point theory === + +Knaster–Kuratowski–Mazurkiewicz lemma + + +=== Geometric topology === +Dehn's lemma + + +=== Topological groups and semigroups === +Ellis–Numakura lemma (topological semigroups) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering-0.md b/data/en.wikipedia.org/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering-0.md new file mode 100644 index 000000000..1eb4a0da4 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering-0.md @@ -0,0 +1,41 @@ +--- +title: "List of letters used in mathematics, science, and engineering" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:24.614221+00:00" +instance: "kb-cron" +--- + +Latin and Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. + + +== Hindu-Arabic numerals == + + +== Latin == + + +== Greek == + + +== Other scripts == + + +=== Hebrew === + + +=== Cyrillic === + + +=== Japanese === + + +=== Modified Latin === + + +=== Modified Greek === + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_limits-0.md b/data/en.wikipedia.org/wiki/List_of_limits-0.md new file mode 100644 index 000000000..e83301821 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_limits-0.md @@ -0,0 +1,1252 @@ +--- +title: "List of limits" +chunk: 1/3 +source: "https://en.wikipedia.org/wiki/List_of_limits" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:28.347754+00:00" +instance: "kb-cron" +--- + +This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x. + +== Limits for general functions == + +=== Definitions of limits and related concepts === + + + + + + lim + + x + → + c + + + f + ( + x + ) + = + L + + + {\displaystyle \lim _{x\to c}f(x)=L} + + if and only if + + + + ∀ + ε + > + 0 + + ∃ + δ + > + 0 + : + 0 + < + + | + + x + − + c + + | + + < + δ + + ⟹ + + + | + + f + ( + x + ) + − + L + + | + + < + ε + + + {\displaystyle \forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon } + +. This is the (ε, δ)-definition of limit. +The limit superior and limit inferior of a sequence are defined as + + + + + lim sup + + n + → + ∞ + + + + x + + n + + + = + + lim + + n + → + ∞ + + + + ( + + + sup + + m + ≥ + n + + + + x + + m + + + + ) + + + + {\displaystyle \limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)} + + and + + + + + lim inf + + n + → + ∞ + + + + x + + n + + + = + + lim + + n + → + ∞ + + + + ( + + + inf + + m + ≥ + n + + + + x + + m + + + + ) + + + + {\displaystyle \liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)} + +. +A function, + + + + f + ( + x + ) + + + {\displaystyle f(x)} + +, is said to be continuous at a point, c, if + + + + + lim + + x + → + c + + + f + ( + x + ) + = + f + ( + c + ) + . + + + {\displaystyle \lim _{x\to c}f(x)=f(c).} + + +=== Operations on a single known limit === +If + + + + + lim + + x + → + c + + + f + ( + x + ) + = + L + + + {\displaystyle \lim _{x\to c}f(x)=L} + + then: + + + + + + lim + + x + → + c + + + + [ + f + ( + x + ) + ± + a + ] + = + L + ± + a + + + {\displaystyle \lim _{x\to c}\,[f(x)\pm a]=L\pm a} + + + + + + + lim + + x + → + c + + + + a + f + ( + x + ) + = + a + L + + + {\displaystyle \lim _{x\to c}\,af(x)=aL} + + + + + + + lim + + x + → + c + + + + + 1 + + f + ( + x + ) + + + + = + + + 1 + L + + + + + {\displaystyle \lim _{x\to c}{\frac {1}{f(x)}}={\frac {1}{L}}} + + if L is not equal to 0. + + + + + + lim + + x + → + c + + + + f + ( + x + + ) + + n + + + = + + L + + n + + + + + {\displaystyle \lim _{x\to c}\,f(x)^{n}=L^{n}} + + if n is a positive integer + + + + + + lim + + x + → + c + + + + f + ( + x + + ) + + + 1 + n + + + + = + + L + + + 1 + n + + + + + + {\displaystyle \lim _{x\to c}\,f(x)^{1 \over n}=L^{1 \over n}} + + if n is a positive integer, and if n is even, then L > 0. +In general, if g(x) is continuous at L and + + + + + lim + + x + → + c + + + f + ( + x + ) + = + L + + + {\displaystyle \lim _{x\to c}f(x)=L} + + then + + + + + + lim + + x + → + c + + + g + + ( + + f + ( + x + ) + + ) + + = + g + ( + L + ) + + + {\displaystyle \lim _{x\to c}g\left(f(x)\right)=g(L)} + + +=== Operations on two known limits === +If + + + + + lim + + x + → + c + + + f + ( + x + ) + = + + L + + 1 + + + + + {\displaystyle \lim _{x\to c}f(x)=L_{1}} + + and + + + + + lim + + x + → + c + + + g + ( + x + ) + = + + L + + 2 + + + + + {\displaystyle \lim _{x\to c}g(x)=L_{2}} + + then: + + + + + + lim + + x + → + c + + + + [ + f + ( + x + ) + ± + g + ( + x + ) + ] + = + + L + + 1 + + + ± + + L + + 2 + + + + + {\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}} + + + + + + + lim + + x + → + c + + + + [ + f + ( + x + ) + g + ( + x + ) + ] + = + + L + + 1 + + + ⋅ + + L + + 2 + + + + + {\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\cdot L_{2}} + + + + + + + lim + + x + → + c + + + + + + f + ( + x + ) + + + g + ( + x + ) + + + + = + + + + L + + 1 + + + + L + + 2 + + + + + + + if + + + L + + 2 + + + ≠ + 0 + + + {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\text{ if }}L_{2}\neq 0} + + +=== Limits involving derivatives or infinitesimal changes === +In these limits, the infinitesimal change + + + + h + + + {\displaystyle h} + + is often denoted + + + + Δ + x + + + {\displaystyle \Delta x} + + or + + + + δ + x + + + {\displaystyle \delta x} + +. If + + + + f + ( + x + ) + + + {\displaystyle f(x)} + + is differentiable at + + + + x + + + {\displaystyle x} + +, + + + + + + lim + + h + → + 0 + + + + + + f + ( + x + + + h + ) + − + f + ( + x + ) + + h + + + = + + f + ′ + + ( + x + ) + + + {\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)} + +. This is the definition of the derivative. All differentiation rules can also be reframed as rules involving limits. For example, if g(x) is differentiable at x, + + + + + + lim + + h + → + 0 + + + + + + f + ∘ + g + ( + x + + + h + ) + − + f + ∘ + g + ( + x + ) + + h + + + = + + f + ′ + + [ + g + ( + x + ) + ] + + g + ′ + + ( + x + ) + + + {\displaystyle \lim _{h\to 0}{f\circ g(x+h)-f\circ g(x) \over h}=f'[g(x)]g'(x)} + +. This is the chain rule. + + + + + + lim + + h + → + 0 + + + + + + f + ( + x + + + h + ) + g + ( + x + + + h + ) + − + f + ( + x + ) + g + ( + x + ) + + h + + + = + + f + ′ + + ( + x + ) + g + ( + x + ) + + + f + ( + x + ) + + g + ′ + + ( + x + ) + + + {\displaystyle \lim _{h\to 0}{f(x+h)g(x+h)-f(x)g(x) \over h}=f'(x)g(x)+f(x)g'(x)} + +. This is the product rule. + + + + + + lim + + h + → + 0 + + + + + ( + + + + f + ( + x + + + h + ) + + + f + ( + x + ) + + + + ) + + + 1 + + / + + h + + + = + exp + ⁡ + + ( + + + + + f + ′ + + ( + x + ) + + + f + ( + x + ) + + + + ) + + + + {\displaystyle \lim _{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{1/h}=\exp \left({\frac {f'(x)}{f(x)}}\right)} + + + + + + + lim + + h + → + 0 + + + + + + ( + + + + f + ( + + e + + h + + + x + ) + + + f + ( + x + ) + + + + ) + + + 1 + + / + + h + + + + = + exp + ⁡ + + ( + + + + x + + f + ′ + + ( + x + ) + + + f + ( + x + ) + + + + ) + + + + {\displaystyle \lim _{h\to 0}{\left({f(e^{h}x) \over {f(x)}}\right)^{1/h}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)} + + +If + + + + f + ( + x + ) + + + {\displaystyle f(x)} + + and + + + + g + ( + x + ) + + + {\displaystyle g(x)} + + are differentiable on an open interval containing c, except possibly c itself, and + + + + + lim + + x + → + c + + + f + ( + x + ) + = + + lim + + x + → + c + + + g + ( + x + ) + = + 0 + + or + + ± + ∞ + + + {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty } + +, L'Hôpital's rule can be used: + + + + + + lim + + x + → + c + + + + + + f + ( + x + ) + + + g + ( + x + ) + + + + = + + lim + + x + → + c + + + + + + + f + ′ + + ( + x + ) + + + + g + ′ + + ( + x + ) + + + + + + {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}} + + +=== Inequalities === +If + + + + f + ( + x + ) + ≤ + g + ( + x + ) + + + {\displaystyle f(x)\leq g(x)} + + for all x in an interval that contains c, except possibly c itself, and the limit of + + + + f + ( + x + ) + + + {\displaystyle f(x)} + + and + + + + g + ( + x + ) + + + {\displaystyle g(x)} + + both exist at c, then + + + + + + lim + + x + → + c + + + f + ( + x + ) + ≤ + + lim + + x + → + c + + + g + ( + x + ) + + + {\displaystyle \lim _{x\to c}f(x)\leq \lim _{x\to c}g(x)} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_limits-1.md b/data/en.wikipedia.org/wiki/List_of_limits-1.md new file mode 100644 index 000000000..997666bec --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_limits-1.md @@ -0,0 +1,1565 @@ +--- +title: "List of limits" +chunk: 2/3 +source: "https://en.wikipedia.org/wiki/List_of_limits" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:28.347754+00:00" +instance: "kb-cron" +--- + +If + + + + + lim + + x + → + c + + + f + ( + x + ) + = + + lim + + x + → + c + + + h + ( + x + ) + = + L + + + {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L} + + and + + + + f + ( + x + ) + ≤ + g + ( + x + ) + ≤ + h + ( + x + ) + + + {\displaystyle f(x)\leq g(x)\leq h(x)} + + for all x in an open interval that contains c, except possibly c itself, + + + + + + lim + + x + → + c + + + g + ( + x + ) + = + L + . + + + {\displaystyle \lim _{x\to c}g(x)=L.} + + This is known as the squeeze theorem. This applies even in the cases that f(x) and g(x) take on different values at c, or are discontinuous at c. + +== Polynomials and functions of the form xa == + + + + + + lim + + x + → + c + + + a + = + a + + + {\displaystyle \lim _{x\to c}a=a} + + +=== Polynomials in x === + + + + + + lim + + x + → + c + + + x + = + c + + + {\displaystyle \lim _{x\to c}x=c} + + + + + + + lim + + x + → + c + + + ( + a + x + + + b + ) + = + a + c + + + b + + + {\displaystyle \lim _{x\to c}(ax+b)=ac+b} + + + + + + + lim + + x + → + c + + + + x + + n + + + = + + c + + n + + + + + {\displaystyle \lim _{x\to c}x^{n}=c^{n}} + + if n is a positive integer + + + + + + lim + + x + → + ∞ + + + x + + / + + a + = + + + { + + + + ∞ + , + + + a + > + 0 + + + + + + does not exist + + , + + + a + = + 0 + + + + + − + ∞ + , + + + a + < + 0 + + + + + + + + + {\displaystyle \lim _{x\to \infty }x/a={\begin{cases}\infty ,&a>0\\{\text{does not exist}},&a=0\\-\infty ,&a<0\end{cases}}} + + +In general, if + + + + p + ( + x + ) + + + {\displaystyle p(x)} + + is a polynomial then, by the continuity of polynomials, + + + + + lim + + x + → + c + + + p + ( + x + ) + = + p + ( + c + ) + + + {\displaystyle \lim _{x\to c}p(x)=p(c)} + + This is also true for rational functions, as they are continuous on their domains. + +=== Functions of the form xa === + + + + + + lim + + x + → + c + + + + x + + a + + + = + + c + + a + + + . + + + {\displaystyle \lim _{x\to c}x^{a}=c^{a}.} + + In particular, + + + + + + lim + + x + → + ∞ + + + + x + + a + + + = + + + { + + + + ∞ + , + + + a + > + 0 + + + + + 1 + , + + + a + = + 0 + + + + + 0 + , + + + a + < + 0 + + + + + + + + + {\displaystyle \lim _{x\to \infty }x^{a}={\begin{cases}\infty ,&a>0\\1,&a=0\\0,&a<0\end{cases}}} + + + + + + + lim + + x + → + c + + + + x + + 1 + + / + + a + + + = + + c + + 1 + + / + + a + + + + + {\displaystyle \lim _{x\to c}x^{1/a}=c^{1/a}} + +. In particular, + + + + + + lim + + x + → + ∞ + + + + x + + 1 + + / + + a + + + = + + lim + + x + → + ∞ + + + + + x + + a + + + + = + ∞ + + for any + + a + > + 0 + + + {\displaystyle \lim _{x\to \infty }x^{1/a}=\lim _{x\to \infty }{\sqrt[{a}]{x}}=\infty {\text{ for any }}a>0} + + + + + + + lim + + x + → + + 0 + + + + + + + + + x + + − + n + + + = + + lim + + x + → + + 0 + + + + + + + + + + 1 + + x + + n + + + + + = + + + ∞ + + + {\displaystyle \lim _{x\to 0^{+}}x^{-n}=\lim _{x\to 0^{+}}{\frac {1}{x^{n}}}=+\infty } + + + + + + + lim + + x + → + + 0 + + − + + + + + + x + + − + n + + + = + + lim + + x + → + + 0 + + − + + + + + + + 1 + + x + + n + + + + + = + + + { + + + + − + ∞ + , + + + + if + + n + + is odd + + + + + + + + ∞ + , + + + + if + + n + + is even + + + + + + + + + + {\displaystyle \lim _{x\to 0^{-}}x^{-n}=\lim _{x\to 0^{-}}{\frac {1}{x^{n}}}={\begin{cases}-\infty ,&{\text{if }}n{\text{ is odd}}\\+\infty ,&{\text{if }}n{\text{ is even}}\end{cases}}} + + + + + + + lim + + x + → + ∞ + + + a + + x + + − + 1 + + + = + + lim + + x + → + ∞ + + + a + + / + + x + = + 0 + + for any real + + a + + + {\displaystyle \lim _{x\to \infty }ax^{-1}=\lim _{x\to \infty }a/x=0{\text{ for any real }}a} + + +== Exponential functions == + +=== Functions of the form ag(x) === + + + + + + lim + + x + → + c + + + + e + + x + + + = + + e + + c + + + + + {\displaystyle \lim _{x\to c}e^{x}=e^{c}} + +, due to the continuity of + + + + + e + + x + + + + + {\displaystyle e^{x}} + + + + + + + lim + + x + → + ∞ + + + + a + + x + + + = + + + { + + + + ∞ + , + + + a + > + 1 + + + + + 1 + , + + + a + = + 1 + + + + + 0 + , + + + 0 + < + a + < + 1 + + + + + + + + + {\displaystyle \lim _{x\to \infty }a^{x}={\begin{cases}\infty ,&a>1\\1,&a=1\\0,&0 + 1 + + + + + 1 + , + + + a + = + 1 + + + + + ∞ + , + + + 0 + < + a + < + 1 + + + + + + + + + {\displaystyle \lim _{x\to \infty }a^{-x}={\begin{cases}0,&a>1\\1,&a=1\\\infty ,&0 + 0 + + + + + 0 + , + + + a + = + 0 + + + + + + does not exist + + , + + + a + < + 0 + + + + + + + + + {\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{a}}=\lim _{x\to \infty }{a}^{1/x}={\begin{cases}1,&a>0\\0,&a=0\\{\text{does not exist}},&a<0\end{cases}}} + + +=== Functions of the form xg(x) === + + + + + + lim + + x + → + ∞ + + + + + x + + x + + + + = + + lim + + x + → + ∞ + + + + + x + + + 1 + + / + + x + + + = + 1 + + + {\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{x}}=\lim _{x\to \infty }{x}^{1/x}=1} + + +=== Functions of the form f(x)g(x) === + + + + + + lim + + x + → + + + ∞ + + + + + ( + + + x + + x + + + k + + + + ) + + + x + + + = + + e + + − + k + + + + + {\displaystyle \lim _{x\to +\infty }\left({\frac {x}{x+k}}\right)^{x}=e^{-k}} + + + + + + + lim + + x + → + 0 + + + + + ( + + 1 + + + x + + ) + + + + 1 + x + + + + = + e + + + {\displaystyle \lim _{x\to 0}\left(1+x\right)^{\frac {1}{x}}=e} + + + + + + + lim + + x + → + 0 + + + + + ( + + 1 + + + k + x + + ) + + + + m + x + + + + = + + e + + m + k + + + + + {\displaystyle \lim _{x\to 0}\left(1+kx\right)^{\frac {m}{x}}=e^{mk}} + + + + + + + lim + + x + → + + + ∞ + + + + + ( + + 1 + + + + + 1 + x + + + + ) + + + x + + + = + e + + + {\displaystyle \lim _{x\to +\infty }\left(1+{\frac {1}{x}}\right)^{x}=e} + + + + + + + lim + + x + → + + + ∞ + + + + + ( + + 1 + − + + + 1 + x + + + + ) + + + x + + + = + + + 1 + e + + + + + {\displaystyle \lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}} + + + + + + + lim + + x + → + + + ∞ + + + + + ( + + 1 + + + + + k + x + + + + ) + + + m + x + + + = + + e + + m + k + + + + + {\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}} + + + + + + + lim + + x + → + 0 + + + + + ( + + 1 + + + a + + ( + + + e + + − + x + + + − + 1 + + ) + + + ) + + + − + + + 1 + x + + + + + = + + e + + a + + + + + {\displaystyle \lim _{x\to 0}\left(1+a\left({e^{-x}-1}\right)\right)^{-{\frac {1}{x}}}=e^{a}} + +. This limit can be derived from this limit. + +=== Sums, products and composites === + + + + + + lim + + x + → + 0 + + + x + + e + + − + x + + + = + 0 + + + {\displaystyle \lim _{x\to 0}xe^{-x}=0} + + + + + + + lim + + x + → + ∞ + + + x + + e + + − + x + + + = + 0 + + + {\displaystyle \lim _{x\to \infty }xe^{-x}=0} + + + + + + + lim + + x + → + 0 + + + + ( + + + + + a + + x + + + − + 1 + + x + + + ) + + = + ln + ⁡ + + a + + , + + + {\displaystyle \lim _{x\to 0}\left({\frac {a^{x}-1}{x}}\right)=\ln {a},} + + for all positive a. + + + + + + lim + + x + → + 0 + + + + ( + + + + + e + + x + + + − + 1 + + x + + + ) + + = + 1 + + + {\displaystyle \lim _{x\to 0}\left({\frac {e^{x}-1}{x}}\right)=1} + + + + + + + lim + + x + → + 0 + + + + ( + + + + + e + + a + x + + + − + 1 + + x + + + ) + + = + a + + + {\displaystyle \lim _{x\to 0}\left({\frac {e^{ax}-1}{x}}\right)=a} + + +== Logarithmic functions == + +=== Natural logarithms === + + + + + + lim + + x + → + c + + + ln + ⁡ + + x + + = + ln + ⁡ + c + + + {\displaystyle \lim _{x\to c}\ln {x}=\ln c} + +, due to the continuity of + + + + ln + ⁡ + + x + + + + {\displaystyle \ln {x}} + +. In particular, + + + + + + lim + + x + → + + 0 + + + + + + + + log + ⁡ + x + = + − + ∞ + + + {\displaystyle \lim _{x\to 0^{+}}\log x=-\infty } + + + + + + + lim + + x + → + ∞ + + + log + ⁡ + x + = + ∞ + + + {\displaystyle \lim _{x\to \infty }\log x=\infty } + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_limits-2.md b/data/en.wikipedia.org/wiki/List_of_limits-2.md new file mode 100644 index 000000000..6360c4457 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_limits-2.md @@ -0,0 +1,1498 @@ +--- +title: "List of limits" +chunk: 3/3 +source: "https://en.wikipedia.org/wiki/List_of_limits" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:28.347754+00:00" +instance: "kb-cron" +--- + + + + + + lim + + x + → + 1 + + + + + + ln + ⁡ + ( + x + ) + + + x + − + 1 + + + + = + 1 + + + {\displaystyle \lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1} + + + + + + + lim + + x + → + 0 + + + + + + ln + ⁡ + ( + x + + + 1 + ) + + x + + + = + 1 + + + {\displaystyle \lim _{x\to 0}{\frac {\ln(x+1)}{x}}=1} + + + + + + + lim + + x + → + 0 + + + + + + − + ln + ⁡ + + ( + + 1 + + + a + + ( + + + e + + − + x + + + − + 1 + + ) + + + ) + + + x + + + = + a + + + {\displaystyle \lim _{x\to 0}{\frac {-\ln \left(1+a\left({e^{-x}-1}\right)\right)}{x}}=a} + +. This limit follows from L'Hôpital's rule. + + + + + + lim + + x + → + 0 + + + x + ln + ⁡ + x + = + 0 + + + {\displaystyle \lim _{x\to 0}x\ln x=0} + +, hence + + + + + lim + + x + → + 0 + + + + x + + x + + + = + 1 + + + {\displaystyle \lim _{x\to 0}x^{x}=1} + + + + + + + lim + + x + → + ∞ + + + + + + ln + ⁡ + x + + x + + + = + 0 + + + {\displaystyle \lim _{x\to \infty }{\frac {\ln x}{x}}=0} + + +=== Logarithms to arbitrary bases === +For b > 1, + + + + + + lim + + x + → + + 0 + + + + + + + + + log + + b + + + ⁡ + x + = + − + ∞ + + + {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-\infty } + + + + + + + lim + + x + → + ∞ + + + + log + + b + + + ⁡ + x + = + ∞ + + + {\displaystyle \lim _{x\to \infty }\log _{b}x=\infty } + + +For b < 1, + + + + + + lim + + x + → + + 0 + + + + + + + + + log + + b + + + ⁡ + x + = + ∞ + + + {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=\infty } + + + + + + + lim + + x + → + ∞ + + + + log + + b + + + ⁡ + x + = + − + ∞ + + + {\displaystyle \lim _{x\to \infty }\log _{b}x=-\infty } + + +Both cases can be generalized to: + + + + + + lim + + x + → + + 0 + + + + + + + + + log + + b + + + ⁡ + x + = + − + F + ( + b + ) + ∞ + + + {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-F(b)\infty } + + + + + + + lim + + x + → + ∞ + + + + log + + b + + + ⁡ + x + = + F + ( + b + ) + ∞ + + + {\displaystyle \lim _{x\to \infty }\log _{b}x=F(b)\infty } + + +where + + + + F + ( + x + ) + = + 2 + H + ( + x + − + 1 + ) + − + 1 + + + {\displaystyle F(x)=2H(x-1)-1} + + and + + + + H + ( + x + ) + + + {\displaystyle H(x)} + + is the Heaviside step function + +== Trigonometric functions == +If + + + + x + + + {\displaystyle x} + + is expressed in radians: + + + + + + lim + + x + → + a + + + sin + ⁡ + x + = + sin + ⁡ + a + + + {\displaystyle \lim _{x\to a}\sin x=\sin a} + + + + + + + lim + + x + → + a + + + cos + ⁡ + x + = + cos + ⁡ + a + + + {\displaystyle \lim _{x\to a}\cos x=\cos a} + + +These limits both follow from the continuity of sin and cos. + + + + + + lim + + x + → + 0 + + + + + + sin + ⁡ + x + + x + + + = + 1 + + + {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1} + +. Or, in general, + + + + + + lim + + x + → + 0 + + + + + + sin + ⁡ + a + x + + + a + x + + + + = + 1 + + + {\displaystyle \lim _{x\to 0}{\frac {\sin ax}{ax}}=1} + +, for a not equal to 0. + + + + + + lim + + x + → + 0 + + + + + + sin + ⁡ + a + x + + x + + + = + a + + + {\displaystyle \lim _{x\to 0}{\frac {\sin ax}{x}}=a} + + + + + + + lim + + x + → + 0 + + + + + + sin + ⁡ + a + x + + + b + x + + + + = + + + a + b + + + + + {\displaystyle \lim _{x\to 0}{\frac {\sin ax}{bx}}={\frac {a}{b}}} + +, for b not equal to 0. + + + + + + lim + + x + → + ∞ + + + x + sin + ⁡ + + ( + + + 1 + x + + + ) + + = + 1 + + + {\displaystyle \lim _{x\to \infty }x\sin \left({\frac {1}{x}}\right)=1} + + + + + + + lim + + x + → + 0 + + + + + + 1 + − + cos + ⁡ + x + + x + + + = + + lim + + x + → + 0 + + + + + + cos + ⁡ + x + − + 1 + + x + + + = + 0 + + + {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=\lim _{x\to 0}{\frac {\cos x-1}{x}}=0} + + + + + + + lim + + x + → + 0 + + + + + + 1 + − + cos + ⁡ + x + + + x + + 2 + + + + + = + + + 1 + 2 + + + + + {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}} + + + + + + + lim + + x + → + + n + + ± + + + + + tan + ⁡ + + ( + + π + x + + + + + π + 2 + + + + ) + + = + ∓ + ∞ + + + {\displaystyle \lim _{x\to n^{\pm }}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty } + +, for integer n. + + + + + + lim + + x + → + 0 + + + + + + tan + ⁡ + x + + x + + + = + 1 + + + {\displaystyle \lim _{x\to 0}{\frac {\tan x}{x}}=1} + +. Or, in general, + + + + + + lim + + x + → + 0 + + + + + + tan + ⁡ + a + x + + + a + x + + + + = + 1 + + + {\displaystyle \lim _{x\to 0}{\frac {\tan ax}{ax}}=1} + +, for a not equal to 0. + + + + + + lim + + x + → + 0 + + + + + + tan + ⁡ + a + x + + + b + x + + + + = + + + a + b + + + + + {\displaystyle \lim _{x\to 0}{\frac {\tan ax}{bx}}={\frac {a}{b}}} + +, for b not equal to 0. + + + + + + lim + + n + → + ∞ + + + + + + + + sin + ⁡ + sin + ⁡ + ⋯ + sin + ⁡ + ( + + x + + 0 + + + ) + + ⏟ + + + + n + + + = + 0 + + + {\displaystyle \lim _{n\to \infty }\ \underbrace {\sin \sin \cdots \sin(x_{0})} _{n}=0} + +, where x0 is an arbitrary real number. + + + + + + lim + + n + → + ∞ + + + + + + + + cos + ⁡ + cos + ⁡ + ⋯ + cos + ⁡ + ( + + x + + 0 + + + ) + + ⏟ + + + + n + + + = + d + + + {\displaystyle \lim _{n\to \infty }\ \underbrace {\cos \cos \cdots \cos(x_{0})} _{n}=d} + +, where d is the Dottie number. x0 can be any arbitrary real number. + +== Sums == +In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. + + + + + + lim + + n + → + ∞ + + + + ∑ + + k + = + 1 + + + n + + + + + 1 + k + + + = + ∞ + + + {\displaystyle \lim _{n\to \infty }\sum _{k=1}^{n}{\frac {1}{k}}=\infty } + +. This is known as the harmonic series. + + + + + + lim + + n + → + ∞ + + + + ( + + + ∑ + + k + = + 1 + + + n + + + + + 1 + k + + + − + log + ⁡ + n + + ) + + = + γ + + + {\displaystyle \lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n\right)=\gamma } + +. This is the Euler Mascheroni constant. + +== Notable special limits == + + + + + + lim + + n + → + ∞ + + + + + n + + + n + ! + + + n + + + + + = + e + + + {\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e} + + + + + + + lim + + n + → + ∞ + + + + + ( + + n + ! + + ) + + + 1 + + / + + n + + + = + ∞ + + + {\displaystyle \lim _{n\to \infty }\left(n!\right)^{1/n}=\infty } + +. This can be proven by considering the inequality + + + + + e + + x + + + ≥ + + + + x + + n + + + + n + ! + + + + + + {\displaystyle e^{x}\geq {\frac {x^{n}}{n!}}} + + at + + + + x + = + n + + + {\displaystyle x=n} + +. + + + + + + lim + + n + → + ∞ + + + + + 2 + + n + + + + + + + 2 + − + + + 2 + + + + + 2 + + + ⋯ + + + + + 2 + + + + + + + + ⏟ + + + + n + + + = + π + + + {\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\dots +{\sqrt {2}}}}}}}} _{n}=\pi } + +. This can be derived from Viète's formula for π. + +== Limiting behavior == + +=== Asymptotic equivalences === +Asymptotic equivalences, + + + + f + ( + x + ) + ∼ + g + ( + x + ) + + + {\displaystyle f(x)\sim g(x)} + +, are true if + + + + + lim + + x + → + ∞ + + + + + + f + ( + x + ) + + + g + ( + x + ) + + + + = + 1 + + + {\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1} + +. Therefore, they can also be reframed as limits. Some notable asymptotic equivalences include + + + + + + lim + + x + → + ∞ + + + + + + x + + / + + ln + ⁡ + x + + + π + ( + x + ) + + + + = + 1 + + + {\displaystyle \lim _{x\to \infty }{\frac {x/\ln x}{\pi (x)}}=1} + +, due to the prime number theorem, + + + + π + ( + x + ) + ∼ + + + x + + ln + ⁡ + x + + + + + + {\displaystyle \pi (x)\sim {\frac {x}{\ln x}}} + +, where π(x) is the prime counting function. + + + + + + lim + + n + → + ∞ + + + + + + + + 2 + π + n + + + + + ( + + + n + e + + + ) + + + n + + + + + n + ! + + + + = + 1 + + + {\displaystyle \lim _{n\to \infty }{\frac {{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}{n!}}=1} + +, due to Stirling's approximation, + + + + n + ! + ∼ + + + 2 + π + n + + + + + ( + + + n + e + + + ) + + + n + + + + + {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} + +. + +=== Big O notation === +The behaviour of functions described by Big O notation can also be described by limits. For example + + + + + f + ( + x + ) + ∈ + + + O + + + ( + g + ( + x + ) + ) + + + {\displaystyle f(x)\in {\mathcal {O}}(g(x))} + + if + + + + + lim sup + + x + → + ∞ + + + + + + + | + + f + ( + x + ) + + | + + + + g + ( + x + ) + + + + < + ∞ + + + {\displaystyle \limsup _{x\to \infty }{\frac {|f(x)|}{g(x)}}<\infty } + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-0.md b/data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-0.md new file mode 100644 index 000000000..0ed8b6192 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-0.md @@ -0,0 +1,45 @@ +--- +title: "List of long mathematical proofs" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:29.984359+00:00" +instance: "kb-cron" +--- + +This is a list of unusually long mathematical proofs. Such proofs often use computational proof methods and may be considered non-surveyable. +As of 2011, the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10,000 pages. There are several proofs that would be far longer than this if the details of the computer calculations they depend on were published in full. + +== Long proofs == +The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. + +1799 – The Abel–Ruffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages. +1890 – Killing's classification of simple complex Lie algebras, including his discovery of the exceptional Lie algebras, took 180 pages in 4 papers. +1894 – The ruler-and-compass construction of a polygon of 65537 sides by Johann Gustav Hermes took over 200 pages. +1905 – Emanuel Lasker's original proof of the Lasker–Noether theorem took 98 pages, but has since been simplified: modern proofs are less than a page long. +1963 – Odd order theorem by Feit and Thompson was 255 pages long, which at the time was over 10 times as long as what had previously been considered a long paper in group theory. +1964 – Resolution of singularities. Hironaka's original proof was 216 pages long; it has since been simplified considerably down to about 10 or 20 pages. +1966 – Abyhankar's proof of resolution of singularities for 3-folds in characteristic greater than 6 covered about 500 pages in several papers. In 2009, Cutkosky simplified this to about 40 pages. +1966 – Discrete series representations of Lie groups. Harish-Chandra's construction of these involved a long series of papers totaling around 500 pages. His later work on the Plancherel theorem for semisimple groups added another 150 pages to these. +1968 – the Novikov–Adian proof solving Burnside's problem on finitely generated infinite groups with finite exponents negatively. The three-part original paper is more than 300 pages long. (Britton later published a 282-page paper attempting to solve the problem, but his paper contained a serious gap.) +1960-1970 – Fondements de la Géometrie Algébrique, Éléments de géométrie algébrique and Séminaire de géométrie algébrique. Grothendieck's work on the foundations of algebraic geometry covers many thousands of pages. Although this is not a proof of a single theorem, there are several theorems in it whose proofs depend on hundreds of earlier pages. +1974 – N-group theorem. Thompson's classification of N-groups used 6 papers totaling about 400 pages, but also used earlier results of his such as the odd order theorem, which bring to total length up to more than 700 pages. +1974 – Ramanujan conjecture and the Weil conjectures. While Deligne's final paper proving these conjectures were "only" about 30 pages long, it depended on background results in algebraic geometry and étale cohomology that Deligne estimated to be about 2000 pages long. +1974 – 4-color theorem. Appel and Haken's proof of this took 139 pages, and also depended on long computer calculations. +1974 – The Gorenstein–Harada theorem classifying finite groups of sectional 2-rank at most 4 was 464 pages long. +1976 – Eisenstein series. Langlands's proof of the functional equation for Eisenstein series was 337 pages long. +1983 – Trichotomy theorem. Gorenstein and Lyons's proof for the case of rank at least 4 was 731 pages long, and Aschbacher's proof of the rank 3 case adds another 159 pages, for a total of 890 pages. +1983 – Selberg trace formula. Hejhal's proof of a general form of the Selberg trace formula consisted of 2 volumes with a total length of 1322 pages. +Arthur–Selberg trace formula. Arthur's proofs of the various versions of this cover several hundred pages spread over many papers. +2000 – Almgren's regularity theorem. Almgren's proof was 955 pages long. +2000 – Lafforgue's theorem on the Langlands conjecture for the general linear group over function fields. Laurent Lafforgue's proof of this was about 600 pages long, not counting many pages of background results. +2003 – Poincaré conjecture, Geometrization theorem, Geometrization conjecture. Perelman's original proofs of the Poincaré conjecture and the Geometrization conjecture were not lengthy, but were rather sketchy. Several other mathematicians have published proofs with the details filled in, which come to several hundred pages. +2004 – Quasithin groups. The classification of the simple quasithin groups by Aschbacher and Smith was 1221 pages long, one of the longest single papers ever written. +2004 – Classification of finite simple groups. The proof of this is spread out over hundreds of journal articles which makes it hard to estimate its total length, which is probably around 10,000 to 20,000 pages. +2004 – Robertson–Seymour theorem. The proof takes about 500 pages spread over about 20 papers. +2005 – Kepler conjecture. Hales's proof of this involves several hundred pages of published arguments, together with several gigabytes of computer calculations. +2006 – the strong perfect graph theorem, by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas. The paper comprised 180 pages in the Annals of Mathematics. + +== Long computer calculations == +There are many mathematical theorems that have been checked by long computer calculations. If these were written out as proofs, many would be far longer than most of the proofs above. There is not really a clear distinction between computer calculations and proofs, as several of the proofs above, such as the 4-color theorem and the Kepler conjecture, use long computer calculations as well as many pages of mathematical argument. For the computer calculations in this section, the mathematical arguments are only a few pages long, and the length is due to long but routine calculations. Some typical examples of such theorems include: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-1.md b/data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-1.md new file mode 100644 index 000000000..59e9c3d44 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-1.md @@ -0,0 +1,42 @@ +--- +title: "List of long mathematical proofs" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:29.984359+00:00" +instance: "kb-cron" +--- + +Several proofs of the existence of sporadic simple groups, such as the Lyons group, originally used computer calculations with large matrices or with permutations on billions of symbols. In most cases, such as the baby monster group, the computer proofs were later replaced by shorter proofs avoiding computer calculations. Similarly, the calculation of the maximal subgroups of the larger sporadic groups uses a lot of computer calculations. +Proving that a particular number is prime +2004 – Verification of the Riemann hypothesis for the first 1013 zeros of the Riemann zeta function. +2007 – Verification that checkers is a draw. +Calculations of large numbers of digits of π. +2010 – Showing that the Rubik's Cube can be solved in 20 moves. +2012 – Showing that Sudoku needs at least 17 clues. +2013 – Ternary Goldbach conjecture: Every odd number greater than 5 can be expressed as the sum of three primes. +2014 – Proof of Erdős discrepancy conjecture for the particular case C=2: every ±1-sequence of the length 1161 has a discrepancy at least 3; the original proof, generated by a SAT solver, had a size of 13 gigabytes and was later reduced to 850 megabytes. +2016 – Solving the Boolean Pythagorean triples problem required the generation of 200 terabytes of proof. +2017 – Marijn Heule, who coauthored solution to the Boolean Pythagorean triples problem, announced a 2 petabytes long proof that the 5th Schur's number is 160. + +== Long proofs in mathematical logic == + +Kurt Gödel showed how to find explicit examples of statements in formal systems that are provable in that system but whose shortest proof is absurdly long. For example, the statement: + +"This statement cannot be proved in Peano arithmetic in less than a googolplex symbols" +is provable in Peano arithmetic but the shortest proof has at least a googolplex symbols. It has a short proof in a more powerful system: in fact, it is easily provable in Peano arithmetic together with the statement that Peano arithmetic is consistent (which cannot be proved in Peano arithmetic by Gödel's incompleteness theorem). +In this argument, Peano arithmetic can be replaced by any more powerful consistent system, and a googolplex can be replaced by any number that can be described concisely in the system. +Harvey Friedman found some explicit natural examples of this phenomenon, giving some explicit statements in Peano arithmetic and other formal systems whose shortest proofs are ridiculously long (Smoryński 1982). For example, the statement + +"there is an integer n such that if there is a sequence of rooted trees T1, T2, ..., Tn such that Tk has at most k+10 vertices, then some tree can be homeomorphically embedded in a later one" +is provable in Peano arithmetic, but the shortest proof has length at least 10002, where 02 = 1 and n+12 = 2(n2) (tetrational growth). The statement is a special case of Kruskal's theorem and has a short proof in second order arithmetic. + +== See also == +List of incomplete proofs +Proof by intimidation + +== References == + +Krantz, Steven G. (2011), The proof is in the pudding. The changing nature of mathematical proof (PDF), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-48744-1, ISBN 978-0-387-48908-7, MR 2789493 +Smoryński, C. (1982), "The varieties of arboreal experience", Math. Intelligencer, 4 (4): 182–189, doi:10.1007/bf03023553, MR 0685558, S2CID 125748405 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_manifolds-0.md b/data/en.wikipedia.org/wiki/List_of_manifolds-0.md new file mode 100644 index 000000000..0a57ea28f --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_manifolds-0.md @@ -0,0 +1,139 @@ +--- +title: "List of manifolds" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_manifolds" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:38.253557+00:00" +instance: "kb-cron" +--- + +This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics. For categorical listings see Category:Manifolds and its subcategories. + + +== Generic families of manifolds == +Euclidean space, Rn +n-sphere, Sn +n-torus, Tn +Real projective space, RPn +Complex projective space, CPn +Quaternionic projective space, HPn +Flag manifold +Grassmann manifold +Stiefel manifold +Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics. + + +== Manifolds of a specific dimension == + + +=== 1-manifolds === +Circle, S1 +Long line +Real line, R +Real projective line, RP1 ≅ S1 + + +=== 2-manifolds === +Cylinder, S1 × R +Klein bottle, RP2 # RP2 +Klein quartic (a genus 3 surface) +Möbius strip +Real projective plane, RP2 +Sphere, S2 +Surface of genus g +Torus +Double torus + + +=== 3-manifolds === +3-sphere, S3 +3-torus, T3 +Poincaré homology sphere +SO(3) ≅ RP3 +Solid Klein bottle +Solid torus +Whitehead manifold +Meyerhoff manifold +Weeks manifold +For more examples see 3-manifold. + + +=== 4-manifolds === +Complex projective plane +Del Pezzo surface +E8 manifold +Enriques surface +Exotic R4 +Hirzebruch surface +K3 surface +For more examples see 4-manifold. + + +== Special types of manifolds == + + +=== Manifolds related to spheres === +Brieskorn manifold +Exotic sphere +Homology sphere +Homotopy sphere +Lens space +Spherical 3-manifold + + +=== Special classes of Riemannian manifolds === +Einstein manifold +Ricci-flat manifold +G2 manifold +Kähler manifold +Calabi–Yau manifold +Hyperkähler manifold +Quaternionic Kähler manifold +Riemannian symmetric space +Spin(7) manifold + + +== Categories of manifolds == + + +=== Manifolds definable by a particular choice of atlas === +Affine manifold +Analytic manifold +Complex manifold +Differentiable (smooth) manifold +Piecewise linear manifold +Lipschitz manifold +Topological manifold + + +=== Manifolds with additional structure === +Almost complex manifold +Almost symplectic manifold +Calibrated manifold +Complex manifold +Contact manifold +CR manifold +Finsler manifold +Hermitian manifold +Hyperkähler manifold +Kähler manifold +Lie group +Pseudo-Riemannian manifold +Riemannian manifold +Sasakian manifold +Spin manifold +Symplectic manifold + + +=== Infinite-dimensional manifolds === +Banach manifold +Fréchet manifold +Hilbert manifold + + +== See also == +List of topological spaces – List of concrete topologies and topological spaces + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematic_operators-0.md b/data/en.wikipedia.org/wiki/List_of_mathematic_operators-0.md new file mode 100644 index 000000000..f27ae54ee --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematic_operators-0.md @@ -0,0 +1,103 @@ +--- +title: "List of mathematic operators" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematic_operators" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:01.392792+00:00" +instance: "kb-cron" +--- + +In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differential operators. +In the following L is an operator + + + + + L + : + + + F + + + → + + + G + + + + + {\displaystyle L:{\mathcal {F}}\to {\mathcal {G}}} + + +which takes a function + + + + y + ∈ + + + F + + + + + {\displaystyle y\in {\mathcal {F}}} + + to another function + + + + L + [ + y + ] + ∈ + + + G + + + + + {\displaystyle L[y]\in {\mathcal {G}}} + +. Here, + + + + + + F + + + + + {\displaystyle {\mathcal {F}}} + + and + + + + + + G + + + + + {\displaystyle {\mathcal {G}}} + + are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions. + + +== See also == +List of transforms +List of Fourier-related transforms +Transfer operator +Fredholm operator +Borel transform +Glossary of mathematical symbols \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_constants-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_constants-0.md new file mode 100644 index 000000000..8ea3b94e8 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_constants-0.md @@ -0,0 +1,62 @@ +--- +title: "List of mathematical constants" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_constants" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:41.282048+00:00" +instance: "kb-cron" +--- + +A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery. +The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them. + + +== List == + + +== Mathematical constants sorted by their representations as continued fractions == +The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one. Decimal representations are rounded or padded to 10 places if the values are known. + + +== Sequences of constants == + + +== See also == +Invariant (mathematics) +Glossary of mathematical symbols +List of mathematical symbols by subject +List of numbers +List of physical constants +Particular values of the Riemann zeta function +Physical constant + + +== Notes == + + +== References == + + +=== Site MathWorld Wolfram.com === + + +=== Site OEIS.org === + + +=== Site OEIS Wiki === + + +== Bibliography == + + +== Further reading == +Wolfram, Stephen. "4: Systems Based on Numbers". A New Kind of Science. Section 5: Mathematical Constants — Continued fractions. + + +== External links == +Inverse Symbolic Calculator, Plouffe's Inverter +Constants – from Wolfram MathWorld +On-Line Encyclopedia of Integer Sequences (OEIS) +Steven Finch's page of mathematical constants +Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_identities-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_identities-0.md new file mode 100644 index 000000000..a295c1ba3 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_identities-0.md @@ -0,0 +1,62 @@ +--- +title: "List of mathematical identities" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_identities" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:04.285513+00:00" +instance: "kb-cron" +--- + +This article lists mathematical identities, that is, identically true relations holding in mathematics. + +Binet-cauchy identity +Binomial inverse theorem +Binomial identity +Brahmagupta–Fibonacci two-square identity +Candido's identity +Cassini and Catalan identities +Degen's eight-square identity +Difference of two squares +Euler's four-square identity +Euler's identity +Fibonacci's identity see Brahmagupta–Fibonacci identity or Cassini and Catalan identities +Heine's identity +Hermite's identity +Lagrange's identity +Lagrange's trigonometric identities +List of logarithmic identities +MacWilliams identity +Matrix determinant lemma +Newton's identity +Parseval's identity +Pfister's sixteen-square identity +Sherman–Morrison formula +Sophie Germain identity +Sun's curious identity +Sylvester's determinant identity +Vandermonde's identity +Woodbury matrix identity + + +== Identities for classes of functions == +Exterior calculus identities +Fibonacci identities: Combinatorial Fibonacci identities and Other Fibonacci identities +Hypergeometric function identities +List of integrals of logarithmic functions +List of topics related to π +List of trigonometric identities +Inverse trigonometric functions +Logarithmic identities +Summation identities +Vector calculus identities + + +== See also == +List of inequalities +List of set identities and relations – Equalities for combinations of sets + + +== External links == +A Collection of Algebraic Identities +Matrix Identities \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_logic_topics-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_logic_topics-0.md new file mode 100644 index 000000000..acff4e7a8 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_logic_topics-0.md @@ -0,0 +1,446 @@ +--- +title: "List of mathematical logic topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_logic_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:35.913586+00:00" +instance: "kb-cron" +--- + +This is a list of mathematical logic topics. +For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and complexity topics for more theory of algorithms. + + +== Working foundations == +Peano axioms +Giuseppe Peano +Mathematical induction +Structural induction +Recursive definition +Naive set theory +Element (mathematics) +Ur-element +Singleton (mathematics) +Simple theorems in the algebra of sets +Algebra of sets +Power set +Empty set +Non-empty set +Empty function +Universe (mathematics) +Axiomatization +Axiomatic system +Axiom schema +Axiomatic method +Formal system +Mathematical proof +Direct proof +Reductio ad absurdum +Proof by exhaustion +Constructive proof +Nonconstructive proof +Tautology +Consistency proof +Arithmetization of analysis +Foundations of mathematics +Formal language +Principia Mathematica +Hilbert's program +Impredicative +Definable real number +Algebraic logic +Boolean algebra (logic) +Dialectica space +categorical logic + + +== Model theory == +Finite model theory +Descriptive complexity theory +Model checking +Trakhtenbrot's theorem +Computable model theory +Tarski's exponential function problem +Undecidable problem +Institutional model theory +Institution (computer science) +Non-standard analysis +Non-standard calculus +Hyperinteger +Hyperreal number +Transfer principle +Overspill +Elementary Calculus: An Infinitesimal Approach +Criticism of non-standard analysis +Standard part function +Set theory +Forcing (mathematics) +Boolean-valued model +Kripke semantics +General frame +Predicate logic +First-order logic +Infinitary logic +Many-sorted logic +Higher-order logic +Lindström quantifier +Second-order logic +Soundness theorem +Gödel's completeness theorem +Original proof of Gödel's completeness theorem +Compactness theorem +Löwenheim–Skolem theorem +Skolem's paradox +Gödel's incompleteness theorems +Structure (mathematical logic) +Interpretation (logic) +Substructure (mathematics) +Elementary substructure +Skolem hull +Non-standard model +Atomic model (mathematical logic) +Prime model +Saturated model +Existentially closed model +Ultraproduct +Age (model theory) +Amalgamation property +Hrushovski construction +Potential isomorphism +Theory (mathematical logic) +Complete theory +Vaught's test +Morley's categoricity theorem +Stability spectrum +Morley rank +Stable theory +Forking extension +Strongly minimal theory +Stable group +Tame group +o-minimal theory +Weakly o-minimal structure +C-minimal theory +Spectrum of a theory +Vaught conjecture +Model complete theory +List of first-order theories +Conservative extension +Elementary class +Pseudoelementary class +Strength (mathematical logic) +Differentially closed field +Exponential field +Ax–Grothendieck theorem +Ax–Kochen theorem +Peano axioms +Non-standard model of arithmetic +First-order arithmetic +Second-order arithmetic +Presburger arithmetic +Wilkie's theorem +Functional predicate +T-schema +Back-and-forth method +Barwise compactness theorem +Skolemization +Lindenbaum–Tarski algebra +Löb's theorem +Arithmetical set +Definable set +Ehrenfeucht–Fraïssé game +Herbrand interpretation / Herbrand structure +Imaginary element +Indiscernibles +Interpretation (model theory) / Interpretable structure +Pregeometry (model theory) +Quantifier elimination +Reduct +Signature (logic) +Skolem normal form +Type (model theory) +Zariski geometry + + +== Set theory == +Algebra of sets +Axiom of choice +Axiom of countable choice +Axiom of dependent choice +Zorn's lemma +Boolean algebra (structure) +Boolean-valued model +Burali-Forti paradox +Cantor's back-and-forth method +Cantor's diagonal argument +Cantor's first uncountability proof +Cantor's theorem +Cantor–Bernstein–Schroeder theorem +Cardinality +Aleph number +Aleph-null +Aleph-one +Beth number +Cardinal number +Hartogs number +Cartesian product +Class (set theory) +Complement (set theory) +Complete Boolean algebra +Continuum (set theory) +Suslin's problem +Continuum hypothesis +Countable set +Descriptive set theory +Analytic set +Analytical hierarchy +Borel equivalence relation +Infinity-Borel set +Lightface analytic game +Perfect set property +Polish space +Prewellordering +Projective set +Property of Baire +Uniformization (set theory) +Universally measurable set +Determinacy +AD+ +Axiom of determinacy +Axiom of projective determinacy +Axiom of real determinacy +Empty set +Forcing (mathematics) +Fuzzy set +Internal set theory +Intersection (set theory) +L +L(R) +Large cardinal property +Musical set theory +Ordinal number +Infinite descending chain +Limit ordinal +Successor ordinal +Transfinite induction +∈-induction +Well-founded set +Well-order +Power set +Russell's paradox +Set theory +Alternative set theory +Axiomatic set theory +Kripke–Platek set theory with urelements +Morse–Kelley set theory +Naive set theory +New Foundations +Positive set theory +Zermelo–Fraenkel set theory +Zermelo set theory +Set (mathematics) +Simple theorems in the algebra of sets +Subset +Θ (set theory) +Tree (descriptive set theory) +Tree (set theory) +Union (set theory) +Von Neumann universe +Zero sharp + + +== Descriptive set theory == +Analytical hierarchy + + +== Large cardinals == +Almost Ramsey cardinal +Erdős cardinal +Extendible cardinal +Huge cardinal +Hyper-Woodin cardinal +Inaccessible cardinal +Ineffable cardinal +Mahlo cardinal +Measurable cardinal +N-huge cardinal +Ramsey cardinal +Rank-into-rank +Remarkable cardinal +Shelah cardinal +Strong cardinal +Strongly inaccessible cardinal +Subtle cardinal +Supercompact cardinal +Superstrong cardinal +Totally indescribable cardinal +Weakly compact cardinal +Weakly hyper-Woodin cardinal +Weakly inaccessible cardinal +Woodin cardinal +Unfoldable cardinal + + +== Recursion theory == +Entscheidungsproblem +Decision problem +Decidability (logic) +Church–Turing thesis +Computable function +Algorithm +Recursion +Primitive recursive function +Mu operator +Ackermann function +Turing machine +Halting problem +Computability theory, computation +Herbrand Universe +Markov algorithm +Lambda calculus +Church–Rosser theorem +Calculus of constructions +Combinatory logic +Post correspondence problem +Kleene's recursion theorem +Recursively enumerable set +Recursively enumerable language +Decidable language +Undecidable language +Rice's theorem +Post's theorem +Turing degree +Effective results in number theory +Diophantine set +Matiyasevich's theorem +Word problem for groups +Arithmetical hierarchy +Subrecursion theory +Presburger arithmetic +Computational complexity theory +Polynomial time +Exponential time +Complexity class +Complexity classes P and NP +Cook's theorem +List of complexity classes +Polynomial hierarchy +Exponential hierarchy +NP-complete +Time hierarchy theorem +Space hierarchy theorem +Natural proof +Hypercomputation +Oracle machine +Rózsa Péter +Alonzo Church +Emil Post +Alan Turing +Jacques Herbrand +Haskell Curry +Stephen Cole Kleene +Definable real number + + +== Proof theory == +Metamathematics +Cut-elimination +Tarski's undefinability theorem +Diagonal lemma +Provability logic +Interpretability logic +Sequent +Sequent calculus +Analytic proof +Structural proof theory +Self-verifying theories +Substructural logics +Structural rule +Weakening +Contraction +Linear logic +Intuitionistic linear logic +Proof net +Affine logic +Strict logic +Relevant logic +Proof-theoretic semantics +Ludics +System F +Gerhard Gentzen +Gentzen's consistency proof +Reverse mathematics +Nonfirstorderizability +Interpretability +Weak interpretability +Cointerpretability +Tolerant sequence +Cotolerant sequence +Deduction theorem +Cirquent calculus + + +== Mathematical constructivism == +Nonconstructive proof +Existence theorem +Intuitionistic logic +Intuitionistic type theory +Type theory +Lambda calculus +Church–Rosser theorem +Simply typed lambda calculus +Typed lambda calculus +Curry–Howard isomorphism +Calculus of constructions +Constructivist analysis +Lambda cube +System F +Introduction to topos theory +LF (logical framework) +Computability logic +Computable measure theory +Finitism +Ultraintuitionism +Luitzen Egbertus Jan Brouwer + + +== Modal logic == +Kripke semantics +Sahlqvist formula +Interior algebra + + +== Theorem provers == +First-order resolution +Automated theorem proving +ACL2 theorem prover +E equational theorem prover +Gandalf theorem prover +HOL theorem prover +Isabelle theorem prover +LCF theorem prover +Otter theorem prover +Paradox theorem prover +Vampire theorem prover +Interactive proof system +Mizar system +QED project +Rocq, formerly Coq + + +== Discovery systems == +Automated Mathematician +Eurisko + + +== Historical == +Begriffsschrift +Systems of Logic Based on Ordinals – Alan Turing's Ph.D. thesis + + +== See also == + +Kurt Gödel +Alfred Tarski +Saharon Shelah \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_proofs-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_proofs-0.md new file mode 100644 index 000000000..7a3bb8e10 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_proofs-0.md @@ -0,0 +1,208 @@ +--- +title: "List of mathematical proofs" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_proofs" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:28.793352+00:00" +instance: "kb-cron" +--- + +A list of articles with mathematical proofs: + + +== Theorems of which articles are primarily devoted to proving them == + +Bertrand's postulate and a proof +Estimation of covariance matrices +Fermat's little theorem and some proofs +Gödel's completeness theorem and its original proof +Mathematical induction and a proof +Proof that 0.999... equals 1 +Proof that 22/7 exceeds π +Proof that e is irrational +Proof that π is irrational +Proof that the sum of the reciprocals of the primes diverges + + +== Articles devoted to theorems of which a (sketch of a) proof is given == + +Banach fixed-point theorem +Banach–Tarski paradox +Basel problem +Bolzano–Weierstrass theorem +Brouwer fixed-point theorem +Buckingham π theorem (proof in progress) +Burnside's lemma +Cantor's theorem +Cantor–Bernstein–Schroeder theorem +Cayley's formula +Cayley's theorem +Clique problem (to do) +Compactness theorem (very compact proof) +Erdős–Ko–Rado theorem +Euler's formula +Euler's four-square identity +Euler's theorem +Five color theorem +Five lemma +Fundamental theorem of arithmetic +Gauss–Markov theorem (brief pointer to proof) +Gödel's incompleteness theorem +Gödel's first incompleteness theorem +Gödel's second incompleteness theorem +Goodstein's theorem +Green's theorem (to do) +Green's theorem when D is a simple region +Heine–Borel theorem +Intermediate value theorem +Itô's lemma +Kőnig's lemma +Kőnig's theorem (set theory) +Kőnig's theorem (graph theory) +Lagrange's theorem (group theory) +Lagrange's theorem (number theory) +Liouville's theorem (complex analysis) +Markov's inequality (proof of a generalization) +Mean value theorem +Multivariate normal distribution (to do) +Holomorphic functions are analytic +Pythagorean theorem +Quadratic equation +Quotient rule +Ramsey's theorem +Rao–Blackwell theorem +Rice's theorem +Rolle's theorem +Splitting lemma +squeeze theorem +Sum rule in differentiation +Sum rule in integration +Sylow theorems +Transcendence of e and π (as corollaries of Lindemann–Weierstrass) +Tychonoff's theorem (to do) +Ultrafilter lemma +Ultraparallel theorem +Urysohn's lemma +Van der Waerden's theorem +Wilson's theorem +Zorn's lemma + + +== Articles devoted to algorithms in which their correctness is proved == +Bellman–Ford algorithm (to do) +Euclidean algorithm +Kruskal's algorithm +Gale–Shapley algorithm +Prim's algorithm +Shor's algorithm (incomplete) + + +== Articles where example statements are proved == + +Basis (linear algebra) +Burrows–Abadi–Needham logic +Direct proof +Generating a vector space +Linear independence +Polynomial +Proof +Pumping lemma +Simpson's rule + + +== Other articles containing proofs == + +Accumulation point +Addition in N +associativity of addition in N +commutativity of addition in N +uniqueness of addition in N +Algorithmic information theory +Boolean ring +commutativity of a boolean ring +Boolean satisfiability problem +NP-completeness of the Boolean satisfiability problem +Cantor's diagonal argument +set is smaller than its power set +uncountability of the real numbers +Cantor's first uncountability proof +uncountability of the real numbers +Combinatorics +Combinatory logic +Co-NP +Coset +Countable +countability of a subset of a countable set (to do) +Angle of parallelism +Galois group +Fundamental theorem of Galois theory (to do) +Gödel number +Gödel's incompleteness theorem +Group (mathematics) +Halting problem +insolubility of the halting problem +Harmonic series (mathematics) +divergence of the (standard) harmonic series +Highly composite number +Area of hyperbolic sector, basis of hyperbolic angle +Infinite series +convergence of the geometric series with first term 1 and ratio 1/2 +Integer partition +Irrational number +irrationality of log23 +irrationality of the square root of 2 +Mathematical induction +sum identity +Power rule +differential of xn +Product and Quotient Rules +Derivation of Product and Quotient rules for differentiating. +Prime number +Infinitude of the prime numbers +Primitive recursive function +Principle of bivalence +no propositions are neither true nor false in intuitionistic logic +Recursion +Relational algebra (to do) +Solvable group +Square root of 2 +Tetris +Algebra of sets +idempotent laws for set union and intersection + + +== Articles which mention dependencies of theorems == +Cauchy's integral formula +Cauchy integral theorem +Computational geometry +Fundamental theorem of algebra +Lambda calculus +Invariance of domain +Minkowski inequality +Nash embedding theorem +Open mapping theorem (functional analysis) +Product topology +Riemann integral +Time hierarchy theorem +Deterministic time hierarchy theorem + + +== Proofs using... == + + +=== topology === +Furstenberg's proof of the infinitude of primes + + +== Articles giving mathematical proofs within a physical model == +No-cloning theorem +Torque + + +== See also == +Gödel's ontological proof +Invalid proof +List of theorems +List of incomplete proofs +List of long proofs \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_properties_of_points-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_properties_of_points-0.md new file mode 100644 index 000000000..a17a98f23 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_properties_of_points-0.md @@ -0,0 +1,81 @@ +--- +title: "List of mathematical properties of points" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_properties_of_points" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:13.832116+00:00" +instance: "kb-cron" +--- + +In mathematics, the following appear: + +Algebraic point +Associated point +Base point +Closed point +Divisor point +Embedded point +Extreme point +Fermat point +Fixed point +Focal point +Geometric point +Hyperbolic equilibrium point +Ideal point +Inflection point +Integral point +Isolated point +Generic point +Heegner point +Lattice hole, Lattice point +Lebesgue point +Midpoint +Napoleon points +Non-singular point +Normal point +Parshin point +Periodic point +Pinch point +Point (geometry) +Point source +Rational point +Recurrent point +Regular point, Regular singular point +Saddle point +Semistable point +Separable point +Simple point +Singular point of a curve +Singular point of an algebraic variety +Smooth point +Special point +Stable point +Torsion point +Vertex (curve) +Weierstrass point + + +== Calculus == +Critical point (aka stationary point), any value v in the domain of a differentiable function of any real or complex variable, such that the derivative of v is 0 or undefined + + +== Geometry == +Antipodal point, the point diametrically opposite to another point on a sphere, such that a line drawn between them passes through the centre of the sphere and forms a true diameter +Conjugate point, any point that can almost be joined to another by a 1-parameter family of geodesics (e.g., the antipodes of a sphere, which are linkable by any meridian +Vertex (geometry), a point that describes a corner or intersection of a geometric shape +Apex (geometry), the vertex that is in some sense the highest of the figure to which it belongs + + +== Topology == +Adherent point, a point x in topological space X such that every open set containing x contains at least one point of a subset A +Condensation point, any point p of a subset S of a topological space, such that every open neighbourhood of p contains uncountably many points of S +Limit point, a set S in a topological space X is a point x (which is in X, but not necessarily in S) that can be approximated by points of S, since every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself +Accumulation point (or cluster point), a point x ∈ X of a sequence (xn)n ∈ N for which there are, for every neighbourhood V of x, infinitely many natural numbers n such that xn ∈ V + + +== See also == +Functor of points +Lists of mathematics topics +Triangle center – Point in a triangle that can be seen as its middle under some criteria +Category:Triangle centers, special points associated with triangles \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_series-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_series-0.md new file mode 100644 index 000000000..230253c89 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_series-0.md @@ -0,0 +1,1718 @@ +--- +title: "List of mathematical series" +chunk: 1/7 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_series" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:50.313770+00:00" +instance: "kb-cron" +--- + +This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. + +Here, + + + + + 0 + + 0 + + + + + {\displaystyle 0^{0}} + + is taken to have the value + + + + 1 + + + {\displaystyle 1} + + + + + + { + x + } + + + {\displaystyle \{x\}} + + denotes the fractional part of + + + + x + + + {\displaystyle x} + + + + + + + B + + n + + + ( + x + ) + + + {\displaystyle B_{n}(x)} + + is a Bernoulli polynomial. + + + + + + B + + n + + + + + {\displaystyle B_{n}} + + is a Bernoulli number, and here, + + + + + B + + 1 + + + = + − + + + 1 + 2 + + + . + + + {\displaystyle B_{1}=-{\frac {1}{2}}.} + + + + + + + E + + n + + + + + {\displaystyle E_{n}} + + is an Euler number. + + + + + ζ + ( + s + ) + + + {\displaystyle \zeta (s)} + + is the Riemann zeta function. + + + + + Γ + ( + z + ) + + + {\displaystyle \Gamma (z)} + + is the gamma function. + + + + + + ψ + + n + + + ( + z + ) + + + {\displaystyle \psi _{n}(z)} + + is a polygamma function. + + + + + + Li + + s + + + ⁡ + ( + z + ) + + + {\displaystyle \operatorname {Li} _{s}(z)} + + is a polylogarithm. + + + + + + ( + + + + n + + k + + + ) + + + + {\displaystyle n \choose k} + + is binomial coefficient + + + + + exp + ⁡ + ( + x + ) + + + {\displaystyle \exp(x)} + + denotes exponential of + + + + x + + + {\displaystyle x} + + +== Sums of powers == +See Faulhaber's formula. + + + + + + ∑ + + k + = + 0 + + + m + + + + k + + n + − + 1 + + + = + + + + + B + + n + + + ( + m + + + 1 + ) + − + + B + + n + + + + n + + + + + {\displaystyle \sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}} + + +The first few values are: + + + + + + ∑ + + k + = + 1 + + + m + + + k + = + + + + m + ( + m + + + 1 + ) + + 2 + + + + + {\displaystyle \sum _{k=1}^{m}k={\frac {m(m+1)}{2}}} + + + + + + + ∑ + + k + = + 1 + + + m + + + + k + + 2 + + + = + + + + m + ( + m + + + 1 + ) + ( + 2 + m + + + 1 + ) + + 6 + + + = + + + + m + + 3 + + + 3 + + + + + + + + m + + 2 + + + 2 + + + + + + + m + 6 + + + + + {\displaystyle \sum _{k=1}^{m}k^{2}={\frac {m(m+1)(2m+1)}{6}}={\frac {m^{3}}{3}}+{\frac {m^{2}}{2}}+{\frac {m}{6}}} + + + + + + + ∑ + + k + = + 1 + + + m + + + + k + + 3 + + + = + + + [ + + + + m + ( + m + + + 1 + ) + + 2 + + + ] + + + 2 + + + = + + + + m + + 4 + + + 4 + + + + + + + + m + + 3 + + + 2 + + + + + + + + m + + 2 + + + 4 + + + + + {\displaystyle \sum _{k=1}^{m}k^{3}=\left[{\frac {m(m+1)}{2}}\right]^{2}={\frac {m^{4}}{4}}+{\frac {m^{3}}{2}}+{\frac {m^{2}}{4}}} + + +See zeta constants. + + + + + ζ + ( + 2 + n + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + k + + 2 + n + + + + + = + ( + − + 1 + + ) + + n + + + 1 + + + + + + + B + + 2 + n + + + ( + 2 + π + + ) + + 2 + n + + + + + 2 + ( + 2 + n + ) + ! + + + + + + {\displaystyle \zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}} + + +The first few values are: + + + + + ζ + ( + 2 + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + k + + 2 + + + + + = + + + + π + + 2 + + + 6 + + + + + {\displaystyle \zeta (2)=\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}} + + (the Basel problem) + + + + + ζ + ( + 4 + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + k + + 4 + + + + + = + + + + π + + 4 + + + 90 + + + + + {\displaystyle \zeta (4)=\sum _{k=1}^{\infty }{\frac {1}{k^{4}}}={\frac {\pi ^{4}}{90}}} + + + + + + ζ + ( + 6 + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + k + + 6 + + + + + = + + + + π + + 6 + + + 945 + + + + + {\displaystyle \zeta (6)=\sum _{k=1}^{\infty }{\frac {1}{k^{6}}}={\frac {\pi ^{6}}{945}}} + + +== Power series == + +=== Low-order polylogarithms === +Finite sums: + + + + + + ∑ + + k + = + m + + + n + + + + z + + k + + + = + + + + + z + + m + + + − + + z + + n + + + 1 + + + + + 1 + − + z + + + + + + {\displaystyle \sum _{k=m}^{n}z^{k}={\frac {z^{m}-z^{n+1}}{1-z}}} + + (geometric series) + + + + + + ∑ + + k + = + 0 + + + n + + + + z + + k + + + = + + + + 1 + − + + z + + n + + + 1 + + + + + 1 + − + z + + + + + + {\displaystyle \sum _{k=0}^{n}z^{k}={\frac {1-z^{n+1}}{1-z}}} + + + + + + + ∑ + + k + = + 1 + + + n + + + + z + + k + + + = + + + + 1 + − + + z + + n + + + 1 + + + + + 1 + − + z + + + + − + 1 + = + + + + z + − + + z + + n + + + 1 + + + + + 1 + − + z + + + + + + {\displaystyle \sum _{k=1}^{n}z^{k}={\frac {1-z^{n+1}}{1-z}}-1={\frac {z-z^{n+1}}{1-z}}} + + + + + + + ∑ + + k + = + 1 + + + n + + + k + + z + + k + + + = + z + + + + 1 + − + ( + n + + + 1 + ) + + z + + n + + + + + n + + z + + n + + + 1 + + + + + ( + 1 + − + z + + ) + + 2 + + + + + + + + {\displaystyle \sum _{k=1}^{n}kz^{k}=z{\frac {1-(n+1)z^{n}+nz^{n+1}}{(1-z)^{2}}}} + + + + + + + ∑ + + k + = + 1 + + + n + + + + k + + 2 + + + + z + + k + + + = + z + + + + 1 + + + z + − + ( + n + + + 1 + + ) + + 2 + + + + z + + n + + + + + ( + 2 + + n + + 2 + + + + + 2 + n + − + 1 + ) + + z + + n + + + 1 + + + − + + n + + 2 + + + + z + + n + + + 2 + + + + + ( + 1 + − + z + + ) + + 3 + + + + + + + + {\displaystyle \sum _{k=1}^{n}k^{2}z^{k}=z{\frac {1+z-(n+1)^{2}z^{n}+(2n^{2}+2n-1)z^{n+1}-n^{2}z^{n+2}}{(1-z)^{3}}}} + + + + + + + ∑ + + k + = + 0 + + + n + + + + k + + m + + + + z + + k + + + = + + + ( + + z + + + d + + d + z + + + + + ) + + + m + + + + + + 1 + − + + z + + n + + + 1 + + + + + 1 + − + z + + + + + + {\displaystyle \sum _{k=0}^{n}k^{m}z^{k}=\left(z{\frac {d}{dz}}\right)^{m}{\frac {1-z^{n+1}}{1-z}}} + + +Infinite sums, valid for + + + + + | + + z + + | + + < + 1 + + + {\displaystyle |z|<1} + + (see polylogarithm): + + + + + + Li + + n + + + ⁡ + ( + z + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + + + z + + k + + + + k + + n + + + + + + + {\displaystyle \operatorname {Li} _{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}} + + +The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form: + + + + + + + + d + + + + d + + z + + + + + Li + + n + + + ⁡ + ( + z + ) + = + + + + + Li + + n + − + 1 + + + ⁡ + ( + z + ) + + z + + + + + {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {Li} _{n}(z)={\frac {\operatorname {Li} _{n-1}(z)}{z}}} + + + + + + + Li + + 1 + + + ⁡ + ( + z + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + + + z + + k + + + k + + + = + − + ln + ⁡ + ( + 1 + − + z + ) + + + {\displaystyle \operatorname {Li} _{1}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k}}=-\ln(1-z)} + + + + + + + Li + + 0 + + + ⁡ + ( + z + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + z + + k + + + = + + + z + + 1 + − + z + + + + + + {\displaystyle \operatorname {Li} _{0}(z)=\sum _{k=1}^{\infty }z^{k}={\frac {z}{1-z}}} + + + + + + + Li + + − + 1 + + + ⁡ + ( + z + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + k + + z + + k + + + = + + + z + + ( + 1 + − + z + + ) + + 2 + + + + + + + + {\displaystyle \operatorname {Li} _{-1}(z)=\sum _{k=1}^{\infty }kz^{k}={\frac {z}{(1-z)^{2}}}} + + + + + + + Li + + − + 2 + + + ⁡ + ( + z + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + k + + 2 + + + + z + + k + + + = + + + + z + ( + 1 + + + z + ) + + + ( + 1 + − + z + + ) + + 3 + + + + + + + + {\displaystyle \operatorname {Li} _{-2}(z)=\sum _{k=1}^{\infty }k^{2}z^{k}={\frac {z(1+z)}{(1-z)^{3}}}} + + + + + + + Li + + − + 3 + + + ⁡ + ( + z + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + k + + 3 + + + + z + + k + + + = + + + + z + ( + 1 + + + 4 + z + + + + z + + 2 + + + ) + + + ( + 1 + − + z + + ) + + 4 + + + + + + + + {\displaystyle \operatorname {Li} _{-3}(z)=\sum _{k=1}^{\infty }k^{3}z^{k}={\frac {z(1+4z+z^{2})}{(1-z)^{4}}}} + + + + + + + Li + + − + 4 + + + ⁡ + ( + z + ) + = + + ∑ + + k + = + 1 + + + ∞ + + + + k + + 4 + + + + z + + k + + + = + + + + z + ( + 1 + + + z + ) + ( + 1 + + + 10 + z + + + + z + + 2 + + + ) + + + ( + 1 + − + z + + ) + + 5 + + + + + + + + {\displaystyle \operatorname {Li} _{-4}(z)=\sum _{k=1}^{\infty }k^{4}z^{k}={\frac {z(1+z)(1+10z+z^{2})}{(1-z)^{5}}}} + + +=== Exponential function === + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + z + + k + + + + k + ! + + + + = + + e + + z + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + k + + + + z + + k + + + + k + ! + + + + = + z + + e + + z + + + + + {\displaystyle \sum _{k=0}^{\infty }k{\frac {z^{k}}{k!}}=ze^{z}} + + (cf. mean of Poisson distribution) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_series-1.md b/data/en.wikipedia.org/wiki/List_of_mathematical_series-1.md new file mode 100644 index 000000000..1ba6be10c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_series-1.md @@ -0,0 +1,1699 @@ +--- +title: "List of mathematical series" +chunk: 2/7 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_series" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:50.313770+00:00" +instance: "kb-cron" +--- + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + k + + 2 + + + + + + z + + k + + + + k + ! + + + + = + ( + z + + + + z + + 2 + + + ) + + e + + z + + + + + {\displaystyle \sum _{k=0}^{\infty }k^{2}{\frac {z^{k}}{k!}}=(z+z^{2})e^{z}} + + (cf. second moment of Poisson distribution) + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + k + + 3 + + + + + + z + + k + + + + k + ! + + + + = + ( + z + + + 3 + + z + + 2 + + + + + + z + + 3 + + + ) + + e + + z + + + + + {\displaystyle \sum _{k=0}^{\infty }k^{3}{\frac {z^{k}}{k!}}=(z+3z^{2}+z^{3})e^{z}} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + k + + 4 + + + + + + z + + k + + + + k + ! + + + + = + ( + z + + + 7 + + z + + 2 + + + + + 6 + + z + + 3 + + + + + + z + + 4 + + + ) + + e + + z + + + + + {\displaystyle \sum _{k=0}^{\infty }k^{4}{\frac {z^{k}}{k!}}=(z+7z^{2}+6z^{3}+z^{4})e^{z}} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + k + + n + + + + + + z + + k + + + + k + ! + + + + = + z + + + d + + d + z + + + + + ∑ + + k + = + 0 + + + ∞ + + + + k + + n + − + 1 + + + + + + z + + k + + + + k + ! + + + + + + = + + e + + z + + + + T + + n + + + ( + z + ) + + + {\displaystyle \sum _{k=0}^{\infty }k^{n}{\frac {z^{k}}{k!}}=z{\frac {d}{dz}}\sum _{k=0}^{\infty }k^{n-1}{\frac {z^{k}}{k!}}\,\!=e^{z}T_{n}(z)} + + +where + + + + + T + + n + + + ( + z + ) + + + {\displaystyle T_{n}(z)} + + is the Touchard polynomials. + +=== Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship === + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + z + + 2 + k + + + 1 + + + + + ( + 2 + k + + + 1 + ) + ! + + + + = + sin + ⁡ + z + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\sin z} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + z + + 2 + k + + + 1 + + + + ( + 2 + k + + + 1 + ) + ! + + + + = + sinh + ⁡ + z + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)!}}=\sinh z} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + z + + 2 + k + + + + + ( + 2 + k + ) + ! + + + + = + cos + ⁡ + z + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k)!}}=\cos z} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + z + + 2 + k + + + + ( + 2 + k + ) + ! + + + + = + cosh + ⁡ + z + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k}}{(2k)!}}=\cosh z} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + 1 + + + ( + + 2 + + 2 + k + + + − + 1 + ) + + 2 + + 2 + k + + + + B + + 2 + k + + + + z + + 2 + k + − + 1 + + + + + ( + 2 + k + ) + ! + + + + = + tan + ⁡ + z + , + + | + + z + + | + + < + + + π + 2 + + + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\tan z,|z|<{\frac {\pi }{2}}} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + + 2 + + 2 + k + + + − + 1 + ) + + 2 + + 2 + k + + + + B + + 2 + k + + + + z + + 2 + k + − + 1 + + + + + ( + 2 + k + ) + ! + + + + = + tanh + ⁡ + z + , + + | + + z + + | + + < + + + π + 2 + + + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\tanh z,|z|<{\frac {\pi }{2}}} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + 2 + + 2 + k + + + + B + + 2 + k + + + + z + + 2 + k + − + 1 + + + + + ( + 2 + k + ) + ! + + + + = + cot + ⁡ + z + , + + | + + z + + | + + < + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\cot z,|z|<\pi } + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + + 2 + + 2 + k + + + + B + + 2 + k + + + + z + + 2 + k + − + 1 + + + + + ( + 2 + k + ) + ! + + + + = + coth + ⁡ + z + , + + | + + z + + | + + < + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\coth z,|z|<\pi } + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + 1 + + + ( + + 2 + + 2 + k + + + − + 2 + ) + + B + + 2 + k + + + + z + + 2 + k + − + 1 + + + + + ( + 2 + k + ) + ! + + + + = + csc + ⁡ + z + , + + | + + z + + | + + < + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}}=\csc z,|z|<\pi } + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + − + ( + + 2 + + 2 + k + + + − + 2 + ) + + B + + 2 + k + + + + z + + 2 + k + − + 1 + + + + + ( + 2 + k + ) + ! + + + + = + csch + ⁡ + z + , + + | + + z + + | + + < + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}}=\operatorname {csch} z,|z|<\pi } + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + E + + 2 + k + + + + z + + 2 + k + + + + + ( + 2 + k + ) + ! + + + + = + sech + ⁡ + z + , + + | + + z + + | + + < + + + π + 2 + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}E_{2k}z^{2k}}{(2k)!}}=\operatorname {sech} z,|z|<{\frac {\pi }{2}}} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + + E + + 2 + k + + + + z + + 2 + k + + + + + ( + 2 + k + ) + ! + + + + = + sec + ⁡ + z + , + + | + + z + + | + + < + + + π + 2 + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {E_{2k}z^{2k}}{(2k)!}}=\sec z,|z|<{\frac {\pi }{2}}} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + 1 + + + + z + + 2 + k + + + + + ( + 2 + k + ) + ! + + + + = + ver + ⁡ + z + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{(2k)!}}=\operatorname {ver} z} + + (versine) + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + 1 + + + + z + + 2 + k + + + + + 2 + ( + 2 + k + ) + ! + + + + = + hav + ⁡ + z + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{2(2k)!}}=\operatorname {hav} z} + + (haversine) + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + 2 + k + ) + ! + + z + + 2 + k + + + 1 + + + + + + 2 + + 2 + k + + + ( + k + ! + + ) + + 2 + + + ( + 2 + k + + + 1 + ) + + + + = + arcsin + ⁡ + z + , + + | + + z + + | + + ≤ + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\arcsin z,|z|\leq 1} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + ( + 2 + k + ) + ! + + z + + 2 + k + + + 1 + + + + + + 2 + + 2 + k + + + ( + k + ! + + ) + + 2 + + + ( + 2 + k + + + 1 + ) + + + + = + arcsinh + ⁡ + + z + + , + + | + + z + + | + + ≤ + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\operatorname {arcsinh} {z},|z|\leq 1} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + z + + 2 + k + + + 1 + + + + + 2 + k + + + 1 + + + + = + arctan + ⁡ + z + , + + | + + z + + | + + < + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2k+1}}=\arctan z,|z|<1} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + z + + 2 + k + + + 1 + + + + 2 + k + + + 1 + + + + = + arctanh + ⁡ + z + , + + | + + z + + | + + < + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k+1}}{2k+1}}=\operatorname {arctanh} z,|z|<1} + + + + + + ln + ⁡ + 2 + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + 1 + + + ( + 2 + k + ) + ! + + z + + 2 + k + + + + + + 2 + + 2 + k + + + 1 + + + k + ( + k + ! + + ) + + 2 + + + + + + = + ln + ⁡ + + ( + + 1 + + + + + 1 + + + + z + + 2 + + + + + + ) + + , + + | + + z + + | + + ≤ + 1 + + + {\displaystyle \ln 2+\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(2k)!z^{2k}}{2^{2k+1}k(k!)^{2}}}=\ln \left(1+{\sqrt {1+z^{2}}}\right),|z|\leq 1} + + + + + + + ∑ + + k + = + 2 + + + ∞ + + + + ( + + k + ⋅ + arctanh + ⁡ + + ( + + + 1 + k + + + ) + + − + 1 + + ) + + = + + + + 3 + − + ln + ⁡ + ( + 4 + π + ) + + 2 + + + + + {\displaystyle \sum _{k=2}^{\infty }\left(k\cdot \operatorname {arctanh} \left({\frac {1}{k}}\right)-1\right)={\frac {3-\ln(4\pi )}{2}}} + + +=== Modified-factorial denominators === \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_series-2.md b/data/en.wikipedia.org/wiki/List_of_mathematical_series-2.md new file mode 100644 index 000000000..9462267a7 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_series-2.md @@ -0,0 +1,1734 @@ +--- +title: "List of mathematical series" +chunk: 3/7 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_series" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:50.313770+00:00" +instance: "kb-cron" +--- + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + 4 + k + ) + ! + + + + 2 + + 4 + k + + + + + 2 + + + ( + 2 + k + ) + ! + ( + 2 + k + + + 1 + ) + ! + + + + + z + + k + + + = + + + + + 1 + − + + + 1 + − + z + + + + z + + + + , + + | + + z + + | + + < + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(4k)!}{2^{4k}{\sqrt {2}}(2k)!(2k+1)!}}z^{k}={\sqrt {\frac {1-{\sqrt {1-z}}}{z}}},|z|<1} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + + 2 + + 2 + k + + + ( + k + ! + + ) + + 2 + + + + + ( + k + + + 1 + ) + ( + 2 + k + + + 1 + ) + ! + + + + + z + + 2 + k + + + 2 + + + = + + + ( + + arcsin + ⁡ + + z + + + ) + + + 2 + + + , + + | + + z + + | + + ≤ + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k}(k!)^{2}}{(k+1)(2k+1)!}}z^{2k+2}=\left(\arcsin {z}\right)^{2},|z|\leq 1} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + + ∏ + + k + = + 0 + + + n + − + 1 + + + ( + 4 + + k + + 2 + + + + + + α + + 2 + + + ) + + + ( + 2 + n + ) + ! + + + + + z + + 2 + n + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + + + α + + ∏ + + k + = + 0 + + + n + − + 1 + + + [ + ( + 2 + k + + + 1 + + ) + + 2 + + + + + + α + + 2 + + + ] + + + ( + 2 + n + + + 1 + ) + ! + + + + + z + + 2 + n + + + 1 + + + = + + e + + α + arcsin + ⁡ + + z + + + + , + + | + + z + + | + + ≤ + 1 + + + {\displaystyle \sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(4k^{2}+\alpha ^{2})}{(2n)!}}z^{2n}+\sum _{n=0}^{\infty }{\frac {\alpha \prod _{k=0}^{n-1}[(2k+1)^{2}+\alpha ^{2}]}{(2n+1)!}}z^{2n+1}=e^{\alpha \arcsin {z}},|z|\leq 1} + + +=== Binomial coefficients === + + + + + ( + 1 + + + z + + ) + + α + + + = + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + + + α + k + + + ) + + + + + z + + k + + + , + + | + + z + + | + + < + 1 + + + {\displaystyle (1+z)^{\alpha }=\sum _{k=0}^{\infty }{\alpha \choose k}z^{k},|z|<1} + + (see Binomial theorem § Newton's generalized binomial theorem) + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + + + + α + + + k + − + 1 + + k + + + ) + + + + + z + + k + + + = + + + 1 + + ( + 1 + − + z + + ) + + α + + + + + + , + + | + + z + + | + + < + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{{\alpha +k-1} \choose k}z^{k}={\frac {1}{(1-z)^{\alpha }}},|z|<1} + + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + k + + + 1 + + + + + + + ( + + + + 2 + k + + k + + + ) + + + + + z + + k + + + = + + + + 1 + − + + + 1 + − + 4 + z + + + + + 2 + z + + + + , + + | + + z + + | + + ≤ + + + 1 + 4 + + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k+1}}{2k \choose k}z^{k}={\frac {1-{\sqrt {1-4z}}}{2z}},|z|\leq {\frac {1}{4}}} + +, generating function of the Catalan numbers + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + + + + 2 + k + + k + + + ) + + + + + z + + k + + + = + + + 1 + + 1 + − + 4 + z + + + + , + + | + + z + + | + + < + + + 1 + 4 + + + + + {\displaystyle \sum _{k=0}^{\infty }{2k \choose k}z^{k}={\frac {1}{\sqrt {1-4z}}},|z|<{\frac {1}{4}}} + +, generating function of the Central binomial coefficients + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + + + + 2 + k + + + α + + k + + + ) + + + + + z + + k + + + = + + + 1 + + 1 + − + 4 + z + + + + + + ( + + + + 1 + − + + + 1 + − + 4 + z + + + + + 2 + z + + + + ) + + + α + + + , + + | + + z + + | + + < + + + 1 + 4 + + + + + {\displaystyle \sum _{k=0}^{\infty }{2k+\alpha \choose k}z^{k}={\frac {1}{\sqrt {1-4z}}}\left({\frac {1-{\sqrt {1-4z}}}{2z}}\right)^{\alpha },|z|<{\frac {1}{4}}} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + + ( + + + + N + + + k + + n + + + ) + + + + + = + + + N + + ( + n + − + 1 + ) + + + + ( + + + N + n + + + ) + + + + + + + , + + N + ≥ + n + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{N+k \choose n}}={\frac {N}{(n-1){N \choose n}}},\quad N\geq n} + + +=== Harmonic numbers === +(See harmonic numbers, themselves defined + + + + + H + + n + + + = + + ∑ + + j + = + 1 + + + n + + + + + 1 + j + + + + + {\textstyle H_{n}=\sum _{j=1}^{n}{\frac {1}{j}}} + +, and + + + + H + ( + x + ) + + + {\displaystyle H(x)} + + generalized to the real numbers) + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + H + + k + + + + z + + k + + + = + + + + − + ln + ⁡ + ( + 1 + − + z + ) + + + 1 + − + z + + + + , + + | + + z + + | + + < + 1 + + + {\displaystyle \sum _{k=1}^{\infty }H_{k}z^{k}={\frac {-\ln(1-z)}{1-z}},|z|<1} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + H + + k + + + + k + + + 1 + + + + + z + + k + + + 1 + + + = + + + 1 + 2 + + + + + [ + + ln + ⁡ + ( + 1 + − + z + ) + + ] + + + 2 + + + , + + + | + + z + + | + + < + 1 + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {H_{k}}{k+1}}z^{k+1}={\frac {1}{2}}\left[\ln(1-z)\right]^{2},\qquad |z|<1} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + 1 + + + + H + + 2 + k + + + + + 2 + k + + + 1 + + + + + z + + 2 + k + + + 1 + + + = + + + 1 + 2 + + + arctan + ⁡ + + z + + log + ⁡ + + ( + 1 + + + + z + + 2 + + + ) + + , + + + | + + z + + | + + < + 1 + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}H_{2k}}{2k+1}}z^{2k+1}={\frac {1}{2}}\arctan {z}\log {(1+z^{2})},\qquad |z|<1} + + + + + + + ∑ + + n + = + 0 + + + ∞ + + + + ∑ + + k + = + 0 + + + 2 + n + + + + + + ( + − + 1 + + ) + + k + + + + + 2 + k + + + 1 + + + + + + + z + + 4 + n + + + 2 + + + + 4 + n + + + 2 + + + + = + + + 1 + 4 + + + arctan + ⁡ + + z + + log + ⁡ + + + + 1 + + + z + + + 1 + − + z + + + + , + + + | + + z + + | + + < + 1 + + + {\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{2n}{\frac {(-1)^{k}}{2k+1}}{\frac {z^{4n+2}}{4n+2}}={\frac {1}{4}}\arctan {z}\log {\frac {1+z}{1-z}},\qquad |z|<1} + + + + + + + ∑ + + n + = + 1 + + + ∞ + + + + + + x + + 2 + + + + + n + + 2 + + + ( + n + + + x + ) + + + + = + x + + + + π + + 2 + + + 6 + + + − + H + ( + x + ) + + + {\displaystyle \sum _{n=1}^{\infty }{\frac {x^{2}}{n^{2}(n+x)}}=x{\frac {\pi ^{2}}{6}}-H(x)} + + +== Binomial coefficients == + + + + + + ∑ + + k + = + 0 + + + n + + + + + + ( + + + n + k + + + ) + + + + = + + 2 + + n + + + + + {\displaystyle \sum _{k=0}^{n}{n \choose k}=2^{n}} + + + + + + + ∑ + + k + = + 0 + + + n + + + + + + + ( + + + n + k + + + ) + + + + + 2 + + + = + + + + ( + + + + 2 + n + + n + + + ) + + + + + + {\displaystyle \sum _{k=0}^{n}{n \choose k}^{2}={2n \choose n}} + + + + + + + ∑ + + k + = + 0 + + + n + + + ( + − + 1 + + ) + + k + + + + + + ( + + + n + k + + + ) + + + + = + 0 + , + + where + + n + ≥ + 1 + + + {\displaystyle \sum _{k=0}^{n}(-1)^{k}{n \choose k}=0,{\text{ where }}n\geq 1} + + + + + + + ∑ + + k + = + 0 + + + n + + + + + + ( + + + k + m + + + ) + + + + = + + + + ( + + + + n + + + 1 + + + m + + + 1 + + + + ) + + + + + + {\displaystyle \sum _{k=0}^{n}{k \choose m}={n+1 \choose m+1}} + + + + + + + ∑ + + k + = + 0 + + + n + + + + + + ( + + + + m + + + k + − + 1 + + k + + + ) + + + + = + + + + ( + + + + n + + + m + + n + + + ) + + + + + + {\displaystyle \sum _{k=0}^{n}{m+k-1 \choose k}={n+m \choose n}} + + (see Multiset) + + + + + + ∑ + + k + = + 0 + + + n + + + + + + ( + + + α + k + + + ) + + + + + + + ( + + + β + + n + − + k + + + + ) + + + + = + + + + ( + + + + α + + + β + + n + + + ) + + + + , + + where + + + α + + + β + ≥ + n + + + {\displaystyle \sum _{k=0}^{n}{\alpha \choose k}{\beta \choose n-k}={\alpha +\beta \choose n},{\text{where}}\ \alpha +\beta \geq n} + + (see Vandermonde identity) + + + + + + ∑ + + A + + ∈ + + + + P + + + ( + E + ) + + + 1 + = + + 2 + + n + + + + , where + + E + + is a finite set, and card( + + E + + ) = n + + + + {\displaystyle \sum _{A\ \in \ {\mathcal {P}}(E)}1=2^{n}{\text{, where }}E{\text{ is a finite set, and card(}}E{\text{) = n}}} + + + + + + + ∑ + + + { + + + + ( + A + , + + B + ) + + ∈ + + ( + + + P + + + ( + E + ) + + ) + + 2 + + + + + + + A + + ⊂ + + B + + + + + + + + 1 + = + + 3 + + n + + + + , where + + E + + is a finite set, and card( + + E + + ) = n + + + + {\displaystyle \sum _{\begin{cases}(A,\ B)\ \in \ ({\mathcal {P}}(E))^{2}\\A\ \subset \ B\end{cases}}1=3^{n}{\text{, where }}E{\text{ is a finite set, and card(}}E{\text{) = n}}} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_series-3.md b/data/en.wikipedia.org/wiki/List_of_mathematical_series-3.md new file mode 100644 index 000000000..8ef095ded --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_series-3.md @@ -0,0 +1,1435 @@ +--- +title: "List of mathematical series" +chunk: 4/7 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_series" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:50.313770+00:00" +instance: "kb-cron" +--- + + + + + + ∑ + + A + + ∈ + + + + P + + + ( + E + ) + + + c + a + r + d + ( + A + ) + = + n + + 2 + + n + − + 1 + + + + , where + + E + + is a finite set, and card( + + E + + ) = n + + + + {\displaystyle \sum _{A\ \in \ {\mathcal {P}}(E)}card(A)=n2^{n-1}{\text{, where }}E{\text{ is a finite set, and card(}}E{\text{) = n}}} + + +== Trigonometric functions == +Sums of sines and cosines arise in Fourier series. + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + cos + ⁡ + ( + k + θ + ) + + k + + + = + − + + + 1 + 2 + + + ln + ⁡ + ( + 2 + − + 2 + cos + ⁡ + θ + ) + = + − + ln + ⁡ + + ( + + 2 + sin + ⁡ + + + θ + 2 + + + + ) + + , + 0 + < + θ + < + 2 + π + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {\cos(k\theta )}{k}}=-{\frac {1}{2}}\ln(2-2\cos \theta )=-\ln \left(2\sin {\frac {\theta }{2}}\right),0<\theta <2\pi } + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + sin + ⁡ + ( + k + θ + ) + + k + + + = + + + + π + − + θ + + 2 + + + , + 0 + < + θ + < + 2 + π + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi } + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + 1 + + + + k + + + cos + ⁡ + ( + k + θ + ) + = + + + 1 + 2 + + + ln + ⁡ + ( + 2 + + + 2 + cos + ⁡ + θ + ) + = + ln + ⁡ + + ( + + 2 + cos + ⁡ + + + θ + 2 + + + + ) + + , + 0 + ≤ + θ + < + π + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}\cos(k\theta )={\frac {1}{2}}\ln(2+2\cos \theta )=\ln \left(2\cos {\frac {\theta }{2}}\right),0\leq \theta <\pi } + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + 1 + + + + k + + + sin + ⁡ + ( + k + θ + ) + = + + + θ + 2 + + + , + − + + + π + 2 + + + ≤ + θ + ≤ + + + π + 2 + + + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}\sin(k\theta )={\frac {\theta }{2}},-{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + cos + ⁡ + ( + 2 + k + θ + ) + + + 2 + k + + + + = + − + + + 1 + 2 + + + ln + ⁡ + ( + 2 + sin + ⁡ + θ + ) + , + 0 + < + θ + < + π + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {\cos(2k\theta )}{2k}}=-{\frac {1}{2}}\ln(2\sin \theta ),0<\theta <\pi } + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + sin + ⁡ + ( + 2 + k + θ + ) + + + 2 + k + + + + = + + + + π + − + 2 + θ + + 4 + + + , + 0 + < + θ + < + π + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(2k\theta )}{2k}}={\frac {\pi -2\theta }{4}},0<\theta <\pi } + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + cos + ⁡ + [ + ( + 2 + k + + + 1 + ) + θ + ] + + + 2 + k + + + 1 + + + + = + + + 1 + 2 + + + ln + ⁡ + + ( + + cot + ⁡ + + + θ + 2 + + + + ) + + , + 0 + < + θ + < + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {\cos[(2k+1)\theta ]}{2k+1}}={\frac {1}{2}}\ln \left(\cot {\frac {\theta }{2}}\right),0<\theta <\pi } + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + sin + ⁡ + [ + ( + 2 + k + + + 1 + ) + θ + ] + + + 2 + k + + + 1 + + + + = + + + π + 4 + + + , + 0 + < + θ + < + π + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {\sin[(2k+1)\theta ]}{2k+1}}={\frac {\pi }{4}},0<\theta <\pi } + +, + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + sin + ⁡ + ( + 2 + π + k + x + ) + + k + + + = + π + + ( + + + + + 1 + 2 + + + + − + { + x + } + + ) + + , + + x + ∈ + + R + + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}=\pi \left({\dfrac {1}{2}}-\{x\}\right),\ x\in \mathbb {R} } + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + sin + ⁡ + + ( + + 2 + π + k + x + + ) + + + + k + + 2 + n + − + 1 + + + + + = + ( + − + 1 + + ) + + n + + + + + + ( + 2 + π + + ) + + 2 + n + − + 1 + + + + + 2 + ( + 2 + n + − + 1 + ) + ! + + + + + B + + 2 + n + − + 1 + + + ( + { + x + } + ) + , + + x + ∈ + + R + + , + + n + ∈ + + N + + + + {\displaystyle \sum \limits _{k=1}^{\infty }{\frac {\sin \left(2\pi kx\right)}{k^{2n-1}}}=(-1)^{n}{\frac {(2\pi )^{2n-1}}{2(2n-1)!}}B_{2n-1}(\{x\}),\ x\in \mathbb {R} ,\ n\in \mathbb {N} } + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + cos + ⁡ + + ( + + 2 + π + k + x + + ) + + + + k + + 2 + n + + + + + = + ( + − + 1 + + ) + + n + − + 1 + + + + + + ( + 2 + π + + ) + + 2 + n + + + + + 2 + ( + 2 + n + ) + ! + + + + + B + + 2 + n + + + ( + { + x + } + ) + , + + x + ∈ + + R + + , + + n + ∈ + + N + + + + {\displaystyle \sum \limits _{k=1}^{\infty }{\frac {\cos \left(2\pi kx\right)}{k^{2n}}}=(-1)^{n-1}{\frac {(2\pi )^{2n}}{2(2n)!}}B_{2n}(\{x\}),\ x\in \mathbb {R} ,\ n\in \mathbb {N} } + + + + + + + B + + n + + + ( + x + ) + = + − + + + + n + ! + + + + 2 + + n + − + 1 + + + + π + + n + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + k + + n + + + + + cos + ⁡ + + ( + + 2 + π + k + x + − + + + + π + n + + 2 + + + + ) + + , + 0 + < + x + < + 1 + + + {\displaystyle B_{n}(x)=-{\frac {n!}{2^{n-1}\pi ^{n}}}\sum _{k=1}^{\infty }{\frac {1}{k^{n}}}\cos \left(2\pi kx-{\frac {\pi n}{2}}\right),0 + 0 + + + {\displaystyle a,b>0} + + are positive whole numbers. Then if + + + + g + c + d + ( + a + , + b + ) + = + c + + + {\displaystyle gcd(a,b)=c} + + we can write + + + + a + = + c + α + + + {\displaystyle a=c\alpha } + + and + + + + b + = + c + β + + + {\displaystyle b=c\beta } + +, where + + + + g + c + d + ( + α + , + β + ) + = + 1 + + + {\displaystyle gcd(\alpha ,\beta )=1} + +, and get: + + + + + S + ( + a + , + b + ) + = + S + ( + c + α + , + c + β + ) + = + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + c + d + k + + + c + e + + + + = + + + 1 + c + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + α + k + + + β + + + + = + + + + S + ( + α + , + β + ) + + c + + + + + {\displaystyle S(a,b)=S(c\alpha ,c\beta )=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{cdk+ce}}={\frac {1}{c}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{\alpha k+\beta }}={\frac {S(\alpha ,\beta )}{c}}} + + +Now if + + + + b + > + a + + + {\displaystyle b>a} + + we can, per Euclid's division lemma, write + + + + b + = + c + a + + + d + + + {\displaystyle b=ca+d} + + where + + + + a + > + d + > + 0 + + + {\displaystyle a>d>0} + + and then + + + + + S + ( + a + , + b + ) + = + S + ( + a + , + c + a + + + d + ) + = + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + a + k + + + c + a + + + d + + + + = + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + a + ( + k + + + c + ) + + + d + + + + = + + ∑ + + k + = + c + + + ∞ + + + + + + ( + − + 1 + + ) + + k + − + c + + + + + a + k + + + d + + + + + + {\displaystyle S(a,b)=S(a,ca+d)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{ak+ca+d}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{a(k+c)+d}}=\sum _{k=c}^{\infty }{\frac {(-1)^{k-c}}{ak+d}}} + + +where we now can add the remaining rows back and subtract them to give us: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_series-5.md b/data/en.wikipedia.org/wiki/List_of_mathematical_series-5.md new file mode 100644 index 000000000..94976e760 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_series-5.md @@ -0,0 +1,1581 @@ +--- +title: "List of mathematical series" +chunk: 6/7 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_series" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:50.313770+00:00" +instance: "kb-cron" +--- + + + + + S + ( + a + , + b + ) + = + ( + − + 1 + + ) + + c + + + + ( + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + a + k + + + d + + + + − + + ∑ + + k + = + 0 + + + c + − + 1 + + + + + + ( + − + 1 + + ) + + k + + + + + a + k + + + d + + + + + ) + + = + ( + − + 1 + + ) + + c + + + + ( + + S + ( + a + , + d + ) + − + + ∑ + + k + = + 0 + + + c + − + 1 + + + + + + ( + − + 1 + + ) + + k + + + + + a + k + + + d + + + + + ) + + + + {\displaystyle S(a,b)=(-1)^{c}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{ak+d}}-\sum _{k=0}^{c-1}{\frac {(-1)^{k}}{ak+d}}\right)=(-1)^{c}\left(S(a,d)-\sum _{k=0}^{c-1}{\frac {(-1)^{k}}{ak+d}}\right)} + + +what that means is that all the infinite choices of + + + + a + + + {\displaystyle a} + + and + + + + b + + + {\displaystyle b} + + can essentially be boiled down to the cases where + + + + g + c + d + ( + a + , + b + ) + = + 1 + + + {\displaystyle gcd(a,b)=1} + + and + + + + a + > + b + > + 0 + + + {\displaystyle a>b>0} + +. If we assume those two things we can then write: + + + + + S + ( + a + , + b + ) + = + + + 1 + a + + + + ( + + + + π + + 2 + sin + ⁡ + + ( + + + + π + b + + a + + + ) + + + + + − + 2 + + ∑ + + m + = + 0 + + + ⌊ + + + a + 2 + + + ⌋ + + + cos + ⁡ + + ( + + π + + + + ( + 2 + m + + + 1 + ) + b + + a + + + + ) + + ln + ⁡ + + ( + + sin + ⁡ + + ( + + π + + + + 2 + m + + + 1 + + + 2 + a + + + + + ) + + + ) + + + ) + + + + {\displaystyle S(a,b)={\frac {1}{a}}\left({\frac {\pi }{2\sin \left({\frac {\pi b}{a}}\right)}}-2\sum _{m=0}^{\lfloor {\frac {a}{2}}\rfloor }\cos \left(\pi {\frac {(2m+1)b}{a}}\right)\ln \left(\sin \left(\pi {\frac {2m+1}{2a}}\right)\right)\right)} + + +and in the case of using a negative sign instead: + + + + + + S + + − + + + ( + a + , + b + ) + = + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + a + k + − + b + + + + + + {\displaystyle S_{-}(a,b)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{ak-b}}} + + +the same two rules apply from above apply and then we can do the following for the case with + + + + a + > + b + > + 0 + + + {\displaystyle a>b>0} + + (since + + + + a + > + a + − + b + > + 0 + + + {\displaystyle a>a-b>0} + +): + + + + + + S + + − + + + ( + a + , + b + ) + = + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + a + k + − + b + + + + = + − + + + 1 + b + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + 1 + + + + + a + ( + k + + + 1 + ) + − + b + + + + = + − + + + 1 + b + + + − + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + a + k + + + ( + a + − + b + ) + + + + = + − + S + ( + a + , + a + − + b + ) + − + + + 1 + b + + + + + {\displaystyle S_{-}(a,b)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{ak-b}}=-{\frac {1}{b}}+\sum _{k=0}^{\infty }{\frac {(-1)^{k+1}}{a(k+1)-b}}=-{\frac {1}{b}}-\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{ak+(a-b)}}=-S(a,a-b)-{\frac {1}{b}}} + + +Let us test out the formula: + + + + + S + ( + 3 + , + 2 + ) + = + + + 1 + 3 + + + + ( + + + + π + + 2 + sin + ⁡ + + ( + + + + 2 + π + + 3 + + + ) + + + + + − + 2 + + ( + + cos + ⁡ + + ( + + + + 2 + π + + 3 + + + ) + + ln + ⁡ + + ( + + sin + ⁡ + + ( + + + π + 6 + + + ) + + + ) + + + + cos + ⁡ + + ( + + 2 + π + + ) + + ln + ⁡ + + ( + + sin + ⁡ + + ( + + + π + 2 + + + ) + + + ) + + + ) + + + ) + + = + + + π + + 3 + + + 3 + + + + + + − + + + + ln + ⁡ + ( + 2 + ) + + 3 + + + + + {\displaystyle S(3,2)={\frac {1}{3}}\left({\frac {\pi }{2\sin \left({\frac {2\pi }{3}}\right)}}-2\left(\cos \left({\frac {2\pi }{3}}\right)\ln \left(\sin \left({\frac {\pi }{6}}\right)\right)+\cos \left(2\pi \right)\ln \left(\sin \left({\frac {\pi }{2}}\right)\right)\right)\right)={\frac {\pi }{3{\sqrt {3}}}}-{\frac {\ln(2)}{3}}} + + +=== Sum of reciprocal of factorials === + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + k + ! + + + + = + + + 1 + + 0 + ! + + + + + + + + 1 + + 1 + ! + + + + + + + + 1 + + 2 + ! + + + + + + + + 1 + + 3 + ! + + + + + + + + 1 + + 4 + ! + + + + + + ⋯ + = + e + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots =e} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + ( + 2 + k + ) + ! + + + + = + + + 1 + + 0 + ! + + + + + + + + 1 + + 2 + ! + + + + + + + + 1 + + 4 + ! + + + + + + + + 1 + + 6 + ! + + + + + + + + 1 + + 8 + ! + + + + + + ⋯ + = + + + 1 + 2 + + + + ( + + e + + + + + 1 + e + + + + ) + + = + cosh + ⁡ + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(2k)!}}={\frac {1}{0!}}+{\frac {1}{2!}}+{\frac {1}{4!}}+{\frac {1}{6!}}+{\frac {1}{8!}}+\cdots ={\frac {1}{2}}\left(e+{\frac {1}{e}}\right)=\cosh 1} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + ( + 3 + k + ) + ! + + + + = + + + 1 + + 0 + ! + + + + + + + + 1 + + 3 + ! + + + + + + + + 1 + + 6 + ! + + + + + + + + 1 + + 9 + ! + + + + + + + + 1 + + 12 + ! + + + + + + ⋯ + = + + + 1 + 3 + + + + ( + + e + + + + + 2 + + e + + + + cos + ⁡ + + + + 3 + + 2 + + + + ) + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(3k)!}}={\frac {1}{0!}}+{\frac {1}{3!}}+{\frac {1}{6!}}+{\frac {1}{9!}}+{\frac {1}{12!}}+\cdots ={\frac {1}{3}}\left(e+{\frac {2}{\sqrt {e}}}\cos {\frac {\sqrt {3}}{2}}\right)} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + ( + 4 + k + ) + ! + + + + = + + + 1 + + 0 + ! + + + + + + + + 1 + + 4 + ! + + + + + + + + 1 + + 8 + ! + + + + + + + + 1 + + 12 + ! + + + + + + + + 1 + + 16 + ! + + + + + + ⋯ + = + + + 1 + 2 + + + + ( + + cos + ⁡ + 1 + + + cosh + ⁡ + 1 + + ) + + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(4k)!}}={\frac {1}{0!}}+{\frac {1}{4!}}+{\frac {1}{8!}}+{\frac {1}{12!}}+{\frac {1}{16!}}+\cdots ={\frac {1}{2}}\left(\cos 1+\cosh 1\right)} + + +=== Trigonometry and π === + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + ( + 2 + k + + + 1 + ) + ! + + + + = + + + 1 + + 1 + ! + + + + − + + + 1 + + 3 + ! + + + + + + + + 1 + + 5 + ! + + + + − + + + 1 + + 7 + ! + + + + + + + + 1 + + 9 + ! + + + + + + ⋯ + = + sin + ⁡ + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}={\frac {1}{1!}}-{\frac {1}{3!}}+{\frac {1}{5!}}-{\frac {1}{7!}}+{\frac {1}{9!}}+\cdots =\sin 1} + + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + ( + 2 + k + ) + ! + + + + = + + + 1 + + 0 + ! + + + + − + + + 1 + + 2 + ! + + + + + + + + 1 + + 4 + ! + + + + − + + + 1 + + 6 + ! + + + + + + + + 1 + + 8 + ! + + + + + + ⋯ + = + cos + ⁡ + 1 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!}}={\frac {1}{0!}}-{\frac {1}{2!}}+{\frac {1}{4!}}-{\frac {1}{6!}}+{\frac {1}{8!}}+\cdots =\cos 1} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + + k + + 2 + + + + + 1 + + + + = + + + 1 + 2 + + + + + + + 1 + 5 + + + + + + + 1 + 10 + + + + + + + 1 + 17 + + + + + ⋯ + = + + + 1 + 2 + + + ( + π + coth + ⁡ + π + − + 1 + ) + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}+1}}={\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{10}}+{\frac {1}{17}}+\cdots ={\frac {1}{2}}(\pi \coth \pi -1)} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + + + + k + + 2 + + + + + 1 + + + + = + − + + + 1 + 2 + + + + + + + 1 + 5 + + + − + + + 1 + 10 + + + + + + + 1 + 17 + + + + + ⋯ + = + + + 1 + 2 + + + ( + π + csch + ⁡ + π + − + 1 + ) + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k^{2}+1}}=-{\frac {1}{2}}+{\frac {1}{5}}-{\frac {1}{10}}+{\frac {1}{17}}+\cdots ={\frac {1}{2}}(\pi \operatorname {csch} \pi -1)} + + + + + + 3 + + + + + 4 + + 2 + × + 3 + × + 4 + + + + − + + + 4 + + 4 + × + 5 + × + 6 + + + + + + + + 4 + + 6 + × + 7 + × + 8 + + + + − + + + 4 + + 8 + × + 9 + × + 10 + + + + + + ⋯ + = + π + + + {\displaystyle 3+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdots =\pi } + + +=== Reciprocal of tetrahedral numbers === \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_series-6.md b/data/en.wikipedia.org/wiki/List_of_mathematical_series-6.md new file mode 100644 index 000000000..9dc257d22 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_series-6.md @@ -0,0 +1,710 @@ +--- +title: "List of mathematical series" +chunk: 7/7 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_series" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:50.313770+00:00" +instance: "kb-cron" +--- + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + T + + e + + k + + + + + + = + + + 1 + 1 + + + + + + + 1 + 4 + + + + + + + 1 + 10 + + + + + + + 1 + 20 + + + + + + + 1 + 35 + + + + + ⋯ + = + + + 3 + 2 + + + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{Te_{k}}}={\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{20}}+{\frac {1}{35}}+\cdots ={\frac {3}{2}}} + + +Where + + + + T + + e + + n + + + = + + ∑ + + k + = + 1 + + + n + + + + T + + k + + + + + {\displaystyle Te_{n}=\sum _{k=1}^{n}T_{k}} + + +=== Exponential and logarithms === + + + + + + ∑ + + k + = + 0 + + + ∞ + + + + + 1 + + ( + 2 + k + + + 1 + ) + ( + 2 + k + + + 2 + ) + + + + = + + + 1 + + 1 + × + 2 + + + + + + + + 1 + + 3 + × + 4 + + + + + + + + 1 + + 5 + × + 6 + + + + + + + + 1 + + 7 + × + 8 + + + + + + + + 1 + + 9 + × + 10 + + + + + + ⋯ + = + ln + ⁡ + 2 + + + {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(2k+1)(2k+2)}}={\frac {1}{1\times 2}}+{\frac {1}{3\times 4}}+{\frac {1}{5\times 6}}+{\frac {1}{7\times 8}}+{\frac {1}{9\times 10}}+\cdots =\ln 2} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + + 2 + + k + + + k + + + + = + + + 1 + 2 + + + + + + + 1 + 8 + + + + + + + 1 + 24 + + + + + + + 1 + 64 + + + + + + + 1 + 160 + + + + + ⋯ + = + ln + ⁡ + 2 + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{2^{k}k}}={\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{24}}+{\frac {1}{64}}+{\frac {1}{160}}+\cdots =\ln 2} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + 1 + + + + + + 2 + + k + + + k + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + + ( + − + 1 + + ) + + k + + + 1 + + + + + + 3 + + k + + + k + + + + = + + + ( + + + + + 1 + 2 + + + + + + + 1 + 3 + + + + + ) + + + − + + + ( + + + + + 1 + 8 + + + + + + + 1 + 18 + + + + + ) + + + + + + + ( + + + + + 1 + 24 + + + + + + + 1 + 81 + + + + + ) + + + − + + + ( + + + + + 1 + 64 + + + + + + + 1 + 324 + + + + + ) + + + + + ⋯ + = + ln + ⁡ + 2 + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{2^{k}k}}+\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{3^{k}k}}={\Bigg (}{\frac {1}{2}}+{\frac {1}{3}}{\Bigg )}-{\Bigg (}{\frac {1}{8}}+{\frac {1}{18}}{\Bigg )}+{\Bigg (}{\frac {1}{24}}+{\frac {1}{81}}{\Bigg )}-{\Bigg (}{\frac {1}{64}}+{\frac {1}{324}}{\Bigg )}+\cdots =\ln 2} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + + 3 + + k + + + k + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + + 4 + + k + + + k + + + + = + + + ( + + + + + 1 + 3 + + + + + + + 1 + 4 + + + + + ) + + + + + + + ( + + + + + 1 + 18 + + + + + + + 1 + 32 + + + + + ) + + + + + + + ( + + + + + 1 + 81 + + + + + + + 1 + 192 + + + + + ) + + + + + + + ( + + + + + 1 + 324 + + + + + + + 1 + 1024 + + + + + ) + + + + + ⋯ + = + ln + ⁡ + 2 + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{3^{k}k}}+\sum _{k=1}^{\infty }{\frac {1}{4^{k}k}}={\Bigg (}{\frac {1}{3}}+{\frac {1}{4}}{\Bigg )}+{\Bigg (}{\frac {1}{18}}+{\frac {1}{32}}{\Bigg )}+{\Bigg (}{\frac {1}{81}}+{\frac {1}{192}}{\Bigg )}+{\Bigg (}{\frac {1}{324}}+{\frac {1}{1024}}{\Bigg )}+\cdots =\ln 2} + + + + + + + ∑ + + k + = + 1 + + + ∞ + + + + + 1 + + + n + + k + + + k + + + + = + ln + ⁡ + + ( + + + n + + n + − + 1 + + + + ) + + + + {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{n^{k}k}}=\ln \left({\frac {n}{n-1}}\right)} + +, that is + + + + ∀ + n + > + 1 + + + {\displaystyle \forall n>1} + + +== See also == + +== Notes == + +== References == +Many books with a list of integrals also have a list of series. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_societies-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_societies-0.md new file mode 100644 index 000000000..704171ecd --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_societies-0.md @@ -0,0 +1,165 @@ +--- +title: "List of mathematical societies" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_societies" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:42.523701+00:00" +instance: "kb-cron" +--- + +This article provides a list of mathematical societies. + + +== International == +African Mathematical Union +Association for Women in Mathematics +Circolo Matematico di Palermo +European Mathematical Society +European Women in Mathematics +Foundations of Computational Mathematics +International Association for Cryptologic Research +International Association of Mathematical Physics +International Linear Algebra Society +International Mathematical Union +International Society for Analysis, its Applications and Computation +International Society for Mathematical Sciences +International Statistical Institute +Kurt Gödel Society +Mathematical Council of the Americas (MCofA) +Mathematical Optimization Society +Mathematical Society of South Eastern Europe (MASSEE) +Quaternion Association +Ramanujan Mathematical Society +Society for Industrial and Applied Mathematics +Southeast Asian Mathematical Society (SEAMS) +Spectra (mathematical association) +Unión Matemática de América Latina y el Caribe (UMALCA) +Young Mathematicians Network + + +== Honor societies == +Kappa Mu Epsilon +Mu Alpha Theta +Pi Mu Epsilon + + +== National and subnational == +This list is sorted by continent. +Country and/or subregion/city is given if not specified in name. + + +=== Africa === +Algeria Mathematical Society +Gabon Mathematical Society +South African Mathematical Society + + +=== Asia === +Bangladesh Mathematical Society +Calcutta Mathematical Society, Kolkata, India +Chinese Mathematical Society +Indian Mathematical Society +Iranian Mathematical Society +Israel Mathematical Union +Kerala Mathematical Association, Kerala State, India +Korean Mathematical Society, South Korea +Mathematical Society of Japan +Mathematical Society of the Philippines +Nepal Mathematical Society +Pakistan Mathematical Society + + +=== Europe === +Albanian Mathematical Association +Armenian Mathematical Union +Austrian Mathematical Society +Catalan Mathematical Society, Spain +Cyprus Mathematical Society +Czech Mathematical Society +Danish Mathematical Society +Edinburgh Mathematical Society, UK +Estonian Mathematical Society +Finnish Mathematical Society +French Mathematical Society +Georgian Mathematical Union +German Mathematical Society +Hellenic Mathematical Society, Greece +Icelandic Mathematical Society +Institute of Mathematics and its Applications, UK +Irish Mathematical Society +Italian Mathematical Union +János Bolyai Mathematical Society, Hungary +Kharkov Mathematical Society, Kharkiv, Ukraine +Kosovar Mathematical Society +Kyiv Mathematical Society, Kyiv, Ukraine +Latvian Mathematical Society +Lithuanian Mathematical Society +London Mathematical Society, UK +Luxembourg Mathematical Society +Malta Mathematical Society +Mathematical Association, UK +Mathematical Society of the Republic of Moldova +Moscow Mathematical Society, Russia +Norwegian Mathematical Society +Norwegian Statistical Association +Polish Mathematical Society +Portuguese Mathematical Society +Romanian Mathematical Society +Royal Dutch Mathematical Society +Royal Spanish Mathematical Society +Royal Statistical Society, UK +Society of Applied Mathematics and Mechanics, Germany +Slovak Mathematical Society +Society of Mathematicians, Physicists and Astronomers of Slovenia +Spanish Society of Statistics and Operations Research +St. Petersburg Mathematical Society, Russia +Swedish Mathematical Society +Swiss Mathematical Society +Trinity Mathematical Society, Cambridge, UK +Turkish Mathematical Society +Union of Bulgarian Mathematicians + + +=== North America === +American Mathematical Society +Canadian Mathematical Society +Mathematical Association of America +National Association of Mathematicians, US +Sociedad Matemática Mexicana (SMM), Mexico + + +=== Central America === +Asociación Matemática Hondureña (ASOMATH), Honduras +Sociedad Cubana de Matemática y Computación (SCMC), Cuba + + +=== South America === +Argentine Mathematical Union +Asociación Argentina de Matemática Aplicada Computacional e Industrial (ASAMACI) +Brazilian Mathematical Society +Colombian Mathematical Society +Sociedad Boliviana de Matemáticas, Bolivia +Sociedad de Matemática de Chile, Chile +Sociedad Ecuatoriana de Matemática, Ecuador +Sociedad Matemática Paraguaya, Paraguay +Venezuelan Mathematical Association + + +=== Oceania === +Australian Mathematical Society +New Zealand Mathematical Society + + +== See also == + +List of academic statistical associations +Category:Mathematical societies + + +== References == + + +== External links == +Member countries, associate members, and affiliate member societies of the International Mathematical Union +Mathematical societies – CIPMA \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory-0.md new file mode 100644 index 000000000..e4ae8fa3c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory-0.md @@ -0,0 +1,178 @@ +--- +title: "List of mathematical topics in quantum theory" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:34.798624+00:00" +instance: "kb-cron" +--- + +This is a list of mathematical topics in quantum theory, by Wikipedia page. See also list of functional analysis topics, list of Lie group topics, list of quantum-mechanical systems with analytical solutions. + + +== Mathematical formulation of quantum mechanics == +bra–ket notation +canonical commutation relation +complete set of commuting observables +Heisenberg picture +Hilbert space +Interaction picture +Measurement in quantum mechanics +quantum field theory +quantum logic +quantum operation +Schrödinger picture +semiclassical +statistical ensemble +wavefunction +wave–particle duality +Wightman axioms +WKB approximation + + +== Schrödinger equation == +quantum mechanics, matrix mechanics, Hamiltonian (quantum mechanics) +particle in a box +particle in a ring +particle in a spherically symmetric potential +quantum harmonic oscillator +hydrogen atom +ring wave guide +particle in a one-dimensional lattice (periodic potential) +Fock symmetry in theory of hydrogen + + +== Symmetry == +identical particles +angular momentum +angular momentum operator +rotational invariance +rotational symmetry +rotation operator +translational symmetry +Lorentz symmetry +Parity transformation +Noether's theorem +Noether charge +Spin (physics) +isospin +Aman matrices +scale invariance +spontaneous symmetry breaking +supersymmetry breaking + + +== Quantum states == +quantum number +Pauli exclusion principle +quantum indeterminacy +uncertainty principle +wavefunction collapse +zero-point energy +bound state +coherent state +squeezed coherent state +density state +Fock state, Fock space +vacuum state +quasinormal mode +no-cloning theorem +quantum entanglement + + +== Dirac equation == +spinor, spinor group, spinor bundle +Dirac sea +Spin foam +Poincaré group +gamma matrices +Dirac adjoint +Wigner's classification +anyon + + +== Interpretations of quantum mechanics == +Copenhagen interpretation +locality principle +Bell's theorem +Bell test loopholes +CHSH inequality +hidden variable theory +path integral formulation, quantum action +Bohm interpretation +many-worlds interpretation +Tsirelson's bound + + +== Quantum field theory == +Feynman diagram +One-loop Feynman diagram +Schwinger's quantum action principle +Propagator +Annihilation operator +S-matrix +Standard Model +Local quantum physics +Nonlocal +Effective field theory +Correlation function (quantum field theory) +Renormalizable +Cutoff +Infrared divergence, infrared fixed point +Ultraviolet divergence +Fermi's interaction +Path-ordering +Landau pole +Higgs mechanism +Wilson line +Wilson loop +Tadpole (physics) +Lattice gauge theory +BRST charge +Anomaly (physics) +Chiral anomaly +Braid statistics +Plekton + + +== Computation == +quantum computing +qubit +qutrit +pure qubit state +quantum dot +Kane quantum computer +quantum cryptography +quantum decoherence +quantum circuit +universal quantum computer +measurement based Quantum Computing +timeline of quantum computing + + +== Supersymmetry == +Lie superalgebra +supergroup (physics) +supercharge +supermultiplet +supergravity + + +== Quantum gravity == +theory of everything +loop quantum gravity +spin network +black hole thermodynamics + + +== Non-commutative geometry == +Quantum group +Hopf algebra +Noncommutative quantum field theory + + +== String theory == +See list of string theory topics + +Matrix model \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity-0.md b/data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity-0.md new file mode 100644 index 000000000..28bc2df3b --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity-0.md @@ -0,0 +1,92 @@ +--- +title: "List of mathematical topics in relativity" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:42.212375+00:00" +instance: "kb-cron" +--- + +This is a list of mathematical topics in relativity, by Wikipedia page. + + +== Special relativity == +Foundational issues +principle of relativity +speed of light +faster-than-light +biquaternion +conjugate diameters +four-vector +four-acceleration +four-force +four-gradient +four-momentum +four-velocity +hyperbolic orthogonality +hyperboloid model +light-like +Lorentz covariance +Lorentz group +Lorentz transformation +Lorentz–FitzGerald contraction hypothesis +Minkowski diagram +Minkowski space +Poincaré group +proper length +proper time +rapidity +relativistic wave equations +relativistic mass +split-complex number +unit hyperbola +world line + + +== General relativity == +black holes +no-hair theorem +Hawking radiation +Hawking temperature +Black hole entropy +charged black hole +rotating black hole +micro black hole +Schwarzschild black hole +Schwarzschild metric +Schwarzschild radius +Reissner–Nordström black hole +Immirzi parameter +closed timelike curve +cosmic censorship hypothesis +chronology protection conjecture +Einstein–Cartan theory +Einstein's field equation +geodesic +gravitational redshift +Penrose–Hawking singularity theorems +Pseudo-Riemannian manifold +stress–energy tensor +worm hole + + +== Cosmology == +anti-de Sitter space +Ashtekar variables +Batalin–Vilkovisky formalism +Big Bang +Cauchy horizon +cosmic inflation +cosmic microwave background +cosmic variance +cosmological constant +dark energy +dark matter +de Sitter space +Friedmann–Lemaître–Robertson–Walker metric +horizon problem +large-scale structure of the cosmos +Randall–Sundrum model +warped geometry +Weyl curvature hypothesis \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematics-based_methods-0.md b/data/en.wikipedia.org/wiki/List_of_mathematics-based_methods-0.md index 7b27e2103..66a87d487 100644 --- a/data/en.wikipedia.org/wiki/List_of_mathematics-based_methods-0.md +++ b/data/en.wikipedia.org/wiki/List_of_mathematics-based_methods-0.md @@ -4,7 +4,7 @@ chunk: 1/1 source: "https://en.wikipedia.org/wiki/List_of_mathematics-based_methods" category: "reference" tags: "science, encyclopedia" -date_saved: "2026-05-05T06:27:54.161701+00:00" +date_saved: "2026-05-05T08:15:46.364138+00:00" instance: "kb-cron" --- diff --git a/data/en.wikipedia.org/wiki/List_of_mathematics_books-0.md b/data/en.wikipedia.org/wiki/List_of_mathematics_books-0.md new file mode 100644 index 000000000..70466567d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematics_books-0.md @@ -0,0 +1,136 @@ +--- +title: "List of mathematics books" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematics_books" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:30.833032+00:00" +instance: "kb-cron" +--- + +This is a list of mathematics books including textbooks, expository works, popular mathematics fields, and historically significant treatises. + + +== General works == +Concrete Mathematics — Ronald Graham, Donald Knuth, and Oren Patashnik +Concepts of Modern Mathematics — Ian Stewart +Mathematics and the Imagination — Edward Kasner and James Newman +Mathematics and Plausible Reasoning — George Pólya +Mathematics: The Loss of Certainty — Morris Kline +The Princeton Companion to Mathematics — Timothy Gowers +What Is Mathematics? — Richard Courant and Herbert Robbins + + +== Popular mathematics and biographies == + +A Mathematician's Apology — G. H. Hardy +The Annotated Turing — Charles Petzold +The Beauty of Fractals — Heinz-Otto Peitgen and Peter Richter +The Emperor's New Mind — Roger Penrose +Fermat's Enigma — Simon Singh +God Created the Integers — Stephen Hawking +Gödel, Escher, Bach — Douglas Hofstadter +How Not to Be Wrong — Jordan Ellenberg +The Man Who Loved Only Numbers — Paul Hoffman +Prime Obsession — John Derbyshire +Where Mathematics Comes From — George Lakoff and Rafael E. Núñez + + +== Algebra == + +Algebra — Serge Lang +Algebra: Chapter 0 — Paolo Aluffi + + +== Calculus and analysis == + +Calculus on Manifolds — Michael Spivak +Principles of Mathematical Analysis — Walter Rudin +Introduction to Analysis — Maxwell Rosenlicht + + +== Geometry and topology == + +Flatland — Edwin Abbott Abbott +Indra's Pearls — David Mumford, Caroline Series, and David Wright +Regular Polytopes — H. S. M. Coxeter +Tilings and patterns — Branko Grünbaum and G. C. Shephard +Topology — James R. Munkres + + +== Number theory == + +An Introduction to the Theory of Numbers — G. H. Hardy and E. M. Wright +A Course in Arithmetic — Jean-Pierre Serre +A Classical Introduction to Modern Number Theory — Michael Rosen + + +== Probability and statistics == + +An Introduction to Probability Theory and Its Applications — William Feller +The Art of Statistics — David Spiegelhalter +Introduction to Probability Models — Sheldon M. Ross + + +== Logic and foundations == + +Proofs and Refutations — Imre Lakatos +The Principles of Mathematics — Bertrand Russell +On Formally Undecidable Propositions of Principia Mathematica and Related Systems — Kurt Gödel + + +== Algorithms == + +Algorithms + Data Structures = Programs — Niklaus Wirth +Algorithms Unlocked — Thomas H. Cormen +The Art of Computer Programming — Donald Knuth +Calendrical Calculations — Nachum Dershowitz and Edward Reingold +Fundamentals of Computer Algorithms — Ellis Horowitz +Hacker's Delight — Henry S. Warren, Jr. +Introduction to Algorithms — Thomas H. Cormen, Charles E. Leiserson, Ronald Rivest, and Clifford Stein +Jewels of Stringology — Maxime Crochemore and Wojciech Rytter +The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World — Pedro Domingos +Numerical Recipes — William H. Press, Saul A. Teukolsky, and Brian P. Flannery +The Preparation of Programs for an Electronic Digital Computer — Maurice Wilkes, David Wheeler, and Stanley Gill + + +== Philosophy and foundations of mathematics == + +Philosophy of Mathematics: Selected Readings — Paul Benacerraf and Hilary Putnam +Philosophy of Mathematics and Natural Science — Hermann Weyl +What Is Mathematics, Really? — Reuben Hersh + + +== Treatises == + +Arithmetica — Diophantus +Disquisitiones Arithmeticae — Carl Friedrich Gauss +Introductio in analysin infinitorum — Leonhard Euler +Mécanique analytique — Joseph-Louis Lagrange +Principia Mathematica — Alfred North Whitehead and Bertrand Russell +The Sand Reckoner — Archimedes +Théorie analytique de la chaleur — Joseph Fourier +The Elements — Euclid + + +== See also == + +Comparison of TeX editors and list of TeX extensions +Computational mathematics +Computer-based mathematics education +List of mathematical software and list of open-source software for mathematics +List of mathematics topics +List of programming books +List of scientific publications by Albert Einstein +List of unsolved problems in mathematics +Lists of books +Lists of mathematicians +MathOverflow +Outline of mathematics +Philosophy of mathematics +Terence Tao publications +The Math(s) Fix – by Conrad Wolfram + + +== External links == +Publications of Joel David Hamkins \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematics_history_topics-0.md b/data/en.wikipedia.org/wiki/List_of_mathematics_history_topics-0.md new file mode 100644 index 000000000..bd9d0cc7b --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematics_history_topics-0.md @@ -0,0 +1,55 @@ +--- +title: "List of mathematics history topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematics_history_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:01.857681+00:00" +instance: "kb-cron" +--- + +This is a list of mathematics history topics, by Wikipedia page. See also list of mathematicians, timeline of mathematics, history of mathematics, list of publications in mathematics. + +1729 (anecdote) +Adequality +Archimedes Palimpsest +Archimedes' use of infinitesimals +Arithmetization of analysis +Brachistochrone curve +Chinese mathematics +Cours d'Analyse +Edinburgh Mathematical Society +Erlangen programme +Fermat's Last Theorem +Greek mathematics +Thomas Little Heath +Hilbert's problems +History of topos theory +Hyperbolic quaternion +Indian mathematics +Islamic mathematics +Italian school of algebraic geometry +Kraków School of Mathematics +Law of Continuity +Lwów School of Mathematics +Nicolas Bourbaki +Non-Euclidean geometry +Scottish Café +Seven bridges of Königsberg +Spectral theory +Synthetic geometry +Tautochrone curve +Unifying theories in mathematics +Waring's problem +Warsaw School of Mathematics + + +== Academic positions == +Lowndean Professor of Astronomy and Geometry +Lucasian professor +Rouse Ball Professor of Mathematics +Sadleirian Chair + + +== See also == + Mathematics portal \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_mathematics_reference_tables-0.md b/data/en.wikipedia.org/wiki/List_of_mathematics_reference_tables-0.md new file mode 100644 index 000000000..130571282 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_mathematics_reference_tables-0.md @@ -0,0 +1,35 @@ +--- +title: "List of mathematics reference tables" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_mathematics_reference_tables" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:41.067246+00:00" +instance: "kb-cron" +--- + +See also: List of reference tables + + +== Mathematics == +List of mathematical topics +List of statistical topics +List of mathematical functions +List of mathematical theorems +List of mathematical proofs +List of matrices +List of numbers +List of relativistic equations +List of small groups +Mathematical constants +Sporadic group +Table of Clebsch-Gordan coefficients +Table of derivatives +Table of divisors +Table of integrals +Table of mathematical symbols +Table of prime factors +Taylor series +Timeline of mathematics +Trigonometric identities +Truth table \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_multivariable_calculus_topics-0.md b/data/en.wikipedia.org/wiki/List_of_multivariable_calculus_topics-0.md new file mode 100644 index 000000000..425a2212f --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_multivariable_calculus_topics-0.md @@ -0,0 +1,74 @@ +--- +title: "List of multivariable calculus topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_multivariable_calculus_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:48.804210+00:00" +instance: "kb-cron" +--- + +This is a list of multivariable calculus topics. See also multivariable calculus, vector calculus, list of real analysis topics, list of calculus topics. + +Closed and exact differential forms +Contact (mathematics) +Contour integral +Contour line +Critical point (mathematics) +Curl (mathematics) +Current (mathematics) +Curvature +Curvilinear coordinates +Del +Differential form +Differential operator +Directional derivative +Divergence +Divergence theorem +Double integral +Equipotential surface +Euler's theorem on homogeneous functions +Exterior derivative +Flux +Frenet–Serret formulas +Gauss's law +Gradient +Green's theorem +Green's identities +Harmonic function +Helmholtz decomposition +Hessian matrix +Hodge star operator +Inverse function theorem +Irrotational vector field +Isoperimetry +Jacobian matrix +Lagrange multiplier +Lamellar vector field +Laplacian +Laplacian vector field +Level set +Line integral +Matrix calculus +Mixed derivatives +Monkey saddle +Multiple integral +Newtonian potential +Parametric equation +Parametric surface +Partial derivative +Partial differential equation +Potential +Real coordinate space +Saddle point +Scalar field +Solenoidal vector field +Stokes' theorem +Submersion +Surface integral +Symmetry of second derivatives +Taylor's theorem +Total derivative +Vector field +Vector operator +Vector potential \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_named_matrices-0.md b/data/en.wikipedia.org/wiki/List_of_named_matrices-0.md new file mode 100644 index 000000000..573ffaff3 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_named_matrices-0.md @@ -0,0 +1,245 @@ +--- +title: "List of named matrices" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/List_of_named_matrices" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:45.006476+00:00" +instance: "kb-cron" +--- + +This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include the identity matrix given by + + + + + + I + + n + + + = + + + [ + + + + 1 + + + 0 + + + ⋯ + + + 0 + + + + + 0 + + + 1 + + + ⋯ + + + 0 + + + + + ⋮ + + + ⋮ + + + ⋱ + + + ⋮ + + + + + 0 + + + 0 + + + ⋯ + + + 1 + + + + ] + + + . + + + {\displaystyle I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.} + + +and the zero matrix of dimension + + + + m + × + n + + + {\displaystyle m\times n} + +. For example: + + + + + + O + + 2 + × + 3 + + + = + + + ( + + + + 0 + + + 0 + + + 0 + + + + + 0 + + + 0 + + + 0 + + + + ) + + + + + {\displaystyle O_{2\times 3}={\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}}} + +. +Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the product of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including physics and chemistry, have particular matrices that are applied chiefly in these areas. + +== Constant matrices == +The list below comprises matrices whose elements are constant for any given dimension (size) of matrix. The matrix entries will be denoted aij. The table below uses the Kronecker delta δij for two integers i and j which is 1 if i = j and 0 else. + +== Specific patterns for entries == +The following lists matrices whose entries are subject to certain conditions. Many of them apply to square matrices only, that is matrices with the same number of columns and rows. The main diagonal of a square matrix is the diagonal joining the upper left corner and the lower right one or equivalently the entries ai,i. The other diagonal is called anti-diagonal (or counter-diagonal). + +== Matrices satisfying some equations == +A number of matrix-related notions is about properties of products or inverses of the given matrix. The matrix product of a m-by-n matrix A and a n-by-k matrix B is the m-by-k matrix C given by + + + + + ( + C + + ) + + i + , + j + + + = + + ∑ + + r + = + 1 + + + n + + + + A + + i + , + r + + + + B + + r + , + j + + + . + + + {\displaystyle (C)_{i,j}=\sum _{r=1}^{n}A_{i,r}B_{r,j}.} + + +This matrix product is denoted AB. Unlike the product of numbers, matrix products are not commutative, that is to say AB need not be equal to BA. A number of notions are concerned with the failure of this commutativity. An inverse of square matrix A is a matrix B (necessarily of the same dimension as A) such that AB = I. Equivalently, BA = I. An inverse need not exist. If it exists, B is uniquely determined, and is also called the inverse of A, denoted A−1. + +== Matrices with conditions on eigenvalues or eigenvectors == + +== Matrices generated by specific data == + +== Matrices used in statistics == +The following matrices find their main application in statistics and probability theory. + +Bernoulli matrix — a square matrix with entries +1, −1, with equal probability of each. +Centering matrix — a matrix which, when multiplied with a vector, has the same effect as subtracting the mean of the components of the vector from every component. +Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables. +Covariance matrix — a symmetric n×n matrix, formed by the pairwise covariances of several random variables. Sometimes called a dispersion matrix. +Dispersion matrix — another name for a covariance matrix. +Doubly stochastic matrix — a non-negative matrix such that each row and each column sums to 1 (thus the matrix is both left stochastic and right stochastic) +Fisher information matrix — a matrix representing the variance of the partial derivative, with respect to a parameter, of the log of the likelihood function of a random variable. +Hat matrix — a square matrix used in statistics to relate fitted values to observed values. +Orthostochastic matrix — doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix +Precision matrix — a symmetric n×n matrix, formed by inverting the covariance matrix. Also called the information matrix. +Stochastic matrix — a non-negative matrix describing a stochastic process. The sum of entries of any row is one. +Transition matrix — a matrix representing the probabilities of conditions changing from one state to another in a Markov chain +Unistochastic matrix — a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix + +== Matrices used in graph theory == +The following matrices find their main application in graph and network theory. + +Adjacency matrix — a square matrix representing a graph, with aij non-zero if vertex i and vertex j are adjacent. +Biadjacency matrix — a special class of adjacency matrix that describes adjacency in bipartite graphs. +Degree matrix — a diagonal matrix defining the degree of each vertex in a graph. +Edmonds matrix — a square matrix of a bipartite graph. +Incidence matrix — a matrix representing a relationship between two classes of objects (usually vertices and edges in the context of graph theory). +Laplacian matrix — a matrix equal to the degree matrix minus the adjacency matrix for a graph, used to find the number of spanning trees in the graph. +Seidel adjacency matrix — a matrix similar to the usual adjacency matrix but with −1 for adjacency; +1 for nonadjacency; 0 on the diagonal. +Skew-adjacency matrix — an adjacency matrix in which each non-zero aij is 1 or −1, accordingly as the direction i → j matches or opposes that of an initially specified orientation. +Tutte matrix — a generalization of the Edmonds matrix for a balanced bipartite graph. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_named_matrices-1.md b/data/en.wikipedia.org/wiki/List_of_named_matrices-1.md new file mode 100644 index 000000000..bfed85279 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_named_matrices-1.md @@ -0,0 +1,46 @@ +--- +title: "List of named matrices" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/List_of_named_matrices" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:45.006476+00:00" +instance: "kb-cron" +--- + +== Matrices used in science and engineering == +Cabibbo–Kobayashi–Maskawa matrix — a unitary matrix used in particle physics to describe the strength of flavour-changing weak decays. +Density matrix — a matrix describing the statistical state of a quantum system. Hermitian, non-negative and with trace 1. +Fundamental matrix (computer vision) — a 3 × 3 matrix in computer vision that relates corresponding points in stereo images. +Fuzzy associative matrix — a matrix in artificial intelligence, used in machine learning processes. +Gamma matrices — 4 × 4 matrices in quantum field theory. +Gell-Mann matrices — a generalization of the Pauli matrices; these matrices are one notable representation of the infinitesimal generators of the special unitary group SU(3). +Hamiltonian matrix — a matrix used in a variety of fields, including quantum mechanics and linear-quadratic regulator (LQR) systems. +Irregular matrix — a matrix used in computer science which has a varying number of elements in each row. +Overlap matrix — a type of Gramian matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system. +S matrix — a matrix in quantum mechanics that connects asymptotic (infinite past and future) particle states. +Scattering matrix - a matrix in Microwave Engineering that describes how the power move in a multiport system. +State transition matrix — exponent of state matrix in control systems. +Substitution matrix — a matrix from bioinformatics, which describes mutation rates of amino acid or DNA sequences. +Supnick matrix — a square matrix used in computer science. +Z-matrix — a matrix in chemistry, representing a molecule in terms of its relative atomic geometry. + +== Specific matrices == +Wilson matrix, a matrix used as an example for test purposes. + +== Other matrix-related terms and definitions == +Jordan canonical form — an 'almost' diagonalised matrix, where the only non-zero elements appear on the lead and superdiagonals. +Linear independence — two or more vectors are linearly independent if there is no way to construct one from linear combinations of the others. +Matrix exponential — defined by the exponential series. +Matrix representation of conic sections +Pseudoinverse — a generalization of the inverse matrix. +Row echelon form — a matrix in this form is the result of applying the forward elimination procedure to a matrix (as used in Gaussian elimination). +Wronskian — the determinant of a matrix of functions and their derivatives such that row n is the (n−1)th derivative of row one. + +== See also == +Perfect matrix + +== Notes == + +== References == +Hogben, Leslie (2006), Handbook of Linear Algebra (Discrete Mathematics and Its Applications), Boca Raton: Chapman & Hall/CRC, ISBN 978-1-58488-510-8 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_number_theory_topics-0.md b/data/en.wikipedia.org/wiki/List_of_number_theory_topics-0.md new file mode 100644 index 000000000..b422e7846 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_number_theory_topics-0.md @@ -0,0 +1,397 @@ +--- +title: "List of number theory topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_number_theory_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:52.778467+00:00" +instance: "kb-cron" +--- + +This is a list of topics in number theory. See also: + +List of recreational number theory topics +Topics in cryptography + + +== Divisibility rule == +Composite number +Highly composite number +Even and odd numbers +Parity +Divisor, aliquot part +Greatest common divisor +Least common multiple +Euclidean algorithm +Coprime +Euclid's lemma +Bézout's identity, Bézout's lemma +Extended Euclidean algorithm +Table of divisors +Prime number, prime power +Bonse's inequality +Prime factor +Table of prime factors +Formula for primes +Factorization +RSA number +Fundamental theorem of arithmetic +Square-free +Square-free integer +Square-free polynomial +Square number +Power of two +Integer-valued polynomial + + +== Fractions == +Rational number +Unit fraction +Irreducible fraction = in lowest terms +Dyadic fraction +Recurring decimal +Cyclic number +Farey sequence +Ford circle +Stern–Brocot tree +Dedekind sum +Egyptian fraction + + +== Modular arithmetic == +Montgomery reduction +Modular exponentiation +Linear congruence theorem +Successive over-relaxation +Chinese remainder theorem +Fermat's little theorem +Proofs of Fermat's little theorem +Fermat quotient +Euler's totient function +Noncototient +Nontotient +Euler's theorem +Wilson's theorem +Primitive root modulo n +Multiplicative order +Discrete logarithm +Quadratic residue +Euler's criterion +Legendre symbol +Gauss's lemma (number theory) +Congruence of squares +Luhn formula +Mod n cryptanalysis + + +== Arithmetic functions == +Multiplicative function +Additive function +Dirichlet convolution +Erdős–Kac theorem +Möbius function +Möbius inversion formula +Divisor function +Liouville function +Partition function (number theory) +Integer partition +Bell numbers +Landau's function +Pentagonal number theorem +Bell series +Lambert series + + +== Analytic number theory: additive problems == +Twin prime +Brun's constant +Cousin prime +Prime triplet +Prime quadruplet +Sexy prime +Sophie Germain prime +Cunningham chain +Goldbach's conjecture +Goldbach's weak conjecture +Second Hardy–Littlewood conjecture +Hardy–Littlewood circle method +Schinzel's hypothesis H +Bateman–Horn conjecture +Waring's problem +Brahmagupta–Fibonacci identity +Euler's four-square identity +Lagrange's four-square theorem +Taxicab number +Generalized taxicab number +Cabtaxi number +Schnirelmann density +Sumset +Landau–Ramanujan constant +Sierpinski number +Seventeen or Bust +Niven's constant + + +== Algebraic number theory == +See list of algebraic number theory topics + + +== Quadratic forms == +Unimodular lattice +Fermat's theorem on sums of two squares +Proofs of Fermat's theorem on sums of two squares + + +== L-functions == +Riemann zeta function +Basel problem on ζ(2) +Hurwitz zeta function +Bernoulli number +Agoh–Giuga conjecture +Von Staudt–Clausen theorem +Dirichlet series +Euler product +Prime number theorem +Prime-counting function +Meissel–Lehmer algorithm +Offset logarithmic integral +Legendre's constant +Skewes' number +Bertrand's postulate +Proof of Bertrand's postulate +Proof that the sum of the reciprocals of the primes diverges +Cramér's conjecture +Riemann hypothesis +Critical line theorem +Hilbert–Pólya conjecture +Generalized Riemann hypothesis +Mertens function, Mertens conjecture, Meissel–Mertens constant +De Bruijn–Newman constant +Dirichlet character +Dirichlet L-series +Siegel zero +Dirichlet's theorem on arithmetic progressions +Linnik's theorem +Elliott–Halberstam conjecture +Functional equation (L-function) +Chebotarev's density theorem +Local zeta function +Weil conjectures +Modular form +modular group +Congruence subgroup +Hecke operator +Cusp form +Eisenstein series +Modular curve +Ramanujan–Petersson conjecture +Birch and Swinnerton-Dyer conjecture +Automorphic form +Selberg trace formula +Artin conjecture +Sato–Tate conjecture +Langlands program +modularity theorem + + +== Diophantine equations == +Pythagorean triple +Pell's equation +Elliptic curve +Nagell–Lutz theorem +Mordell–Weil theorem +Mazur's torsion theorem +Congruent number +Arithmetic of abelian varieties +Elliptic divisibility sequences +Mordell curve +Fermat's Last Theorem +Mordell conjecture +Euler's sum of powers conjecture +abc Conjecture +Catalan's conjecture +Pillai's conjecture +Hasse principle +Diophantine set +Matiyasevich's theorem +Hundred Fowls Problem +1729 + + +== Diophantine approximation == +Davenport–Schmidt theorem +Irrational number +Square root of two +Quadratic irrational +Integer square root +Algebraic number +Pisot–Vijayaraghavan number +Salem number +Transcendental number +e (mathematical constant) +pi, list of topics related to pi +Squaring the circle +Proof that e is irrational +Lindemann–Weierstrass theorem +Hilbert's seventh problem +Gelfond–Schneider theorem +Erdős–Borwein constant +Liouville number +Irrationality measure +Simple continued fraction +Mathematical constant (sorted by continued fraction representation) +Khinchin's constant +Lévy's constant +Lochs' theorem +Gauss–Kuzmin–Wirsing operator +Minkowski's question mark function +Generalized continued fraction +Kronecker's theorem +Thue–Siegel–Roth theorem +Prouhet–Thue–Morse constant +Gelfond–Schneider constant +Equidistribution mod 1 +Beatty's theorem +Littlewood conjecture +Discrepancy function +Low-discrepancy sequence +Illustration of a low-discrepancy sequence +Constructions of low-discrepancy sequences +Halton sequences +Geometry of numbers +Minkowski's theorem +Pick's theorem +Mahler's compactness theorem +Mahler measure +Effective results in number theory +Mahler's theorem + + +== Sieve methods == +Brun sieve +Function field sieve +General number field sieve +Large sieve +Larger sieve +Quadratic sieve +Selberg sieve +Sieve of Atkin +Sieve of Eratosthenes +Sieve of Sundaram +Turán sieve + + +== Named primes == +Chen prime +Cullen prime +Fermat prime +Sophie Germain prime, safe prime +Mersenne prime +New Mersenne conjecture +Great Internet Mersenne Prime Search +Newman–Shanks–Williams prime +Primorial prime +Wagstaff prime +Wall–Sun–Sun prime +Wieferich prime +Wilson prime +Wolstenholme prime +Woodall prime +Prime pages + + +== Combinatorial number theory == +Covering system +Small set (combinatorics) +Erdős–Ginzburg–Ziv theorem +Polynomial method +Van der Waerden's theorem +Szemerédi's theorem +Collatz conjecture +Gilbreath's conjecture +Erdős–Graham conjecture +Znám's problem + + +== Computational number theory == +Note: Computational number theory is also known as algorithmic number theory. + +Residue number system +Cunningham project +Quadratic residuosity problem + + +=== Primality tests === +Prime factorization algorithm +Trial division +Sieve of Eratosthenes +Probabilistic algorithm +Fermat primality test +Pseudoprime +Carmichael number +Euler pseudoprime +Euler–Jacobi pseudoprime +Fibonacci pseudoprime +Probable prime +Baillie–PSW primality test +Miller–Rabin primality test +Lucas–Lehmer primality test +Lucas–Lehmer test for Mersenne numbers +AKS primality test + + +=== Integer factorization === +Pollard's p − 1 algorithm +Pollard's rho algorithm +Lenstra elliptic curve factorization +Quadratic sieve +Special number field sieve +General number field sieve +Shor's algorithm +RSA Factoring Challenge + + +=== Pseudo-random numbers === +Pseudorandom number generator +Pseudorandomness +Cryptographically secure pseudo-random number generator +Middle-square method +Blum Blum Shub +ACORN +ISAAC +Lagged Fibonacci generator +Linear congruential generator +Mersenne twister +Linear-feedback shift register +Shrinking generator +Stream cipher +see also List of random number generators. + + +== Arithmetic dynamics == +Aliquot sequence and Aliquot sum dynamics +Abundant number +Almost perfect number +Amicable number +Betrothed numbers +Deficient number +Quasiperfect number +Perfect number +Sociable number +Collatz conjecture +Digit sum dynamics +Additive persistence +Digital root +Digit product dynamics +Multiplicative digital root +Multiplicative persistence +Lychrel number +Perfect digital invariant +Happy number + + +== History == +Disquisitiones Arithmeticae +"On the Number of Primes Less Than a Given Magnitude" +Vorlesungen über Zahlentheorie +Prime Obsession \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numeral_system_topics-0.md b/data/en.wikipedia.org/wiki/List_of_numeral_system_topics-0.md new file mode 100644 index 000000000..9b6e93e1c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numeral_system_topics-0.md @@ -0,0 +1,75 @@ +--- +title: "List of numeral system topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_numeral_system_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:54.075389+00:00" +instance: "kb-cron" +--- + +This is a list of Wikipedia articles on topics of numeral system and "numeric representations" +See also: computer numbering formats and number names. + + +== Arranged by base == +Radix, radix point, mixed radix, exponentiation +Unary numeral system (base 1) +Tally marks – Numeral form used for counting +Binary numeral system (base 2) +Negative base numeral system (base −2) +Ternary numeral system numeral system (base 3) +Balanced ternary numeral system (base 3) +Negative base numeral system (base −3) +Quaternary numeral system (base 4) +Quater-imaginary base (base 2i) +Quinary numeral system (base 5) +Pentadic numerals – Scandinavian numeral system +Senary numeral system (base 6) +Septenary numeral system (base 7) +Octal numeral system (base 8) +Nonary (novenary) numeral system (base 9) +Decimal (denary) numeral system (base 10) +Bi-quinary coded decimal – Numeral encoding scheme +Negative base numeral system (base −10) +Duodecimal (dozenal) numeral system (base 12) +Hexadecimal numeral system (base 16) +Vigesimal numeral system (base 20) +Sexagesimal numeral system (base 60) + + +== Arranged by culture == +Aegean numbers – Numeral system used by the Minoans and MycenaeansPages displaying short descriptions of redirect targets +Australian Aboriginal enumeration – Counting system used by Australian Aboriginals +Armenian numerals +Babylonian numerals – Numeral systemPages displaying short descriptions of redirect targets +Chinese numerals – Characters used to denote numbers in Chinese +Counting rods – Small bars used for calculating in ancient East Asia +Cyrillic numerals – Numeral system derived from the Cyrillic script +Greek numerals – System of writing numbers using Greek letters +Attic numerals – Symbolic number notation used by the ancient Greeks +Hebrew numerals – Numeral system using letters of the Hebrew alphabet +Hindu–Arabic numeral system – Most common system for writing numbers +Arabic numerals – Symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 +Eastern Arabic numerals – Numerals used in the eastern Arab world and Asia +Indian numerals – Most common system for writing numbersPages displaying short descriptions of redirect targets +Thai numerals – Notation for expressing numbers in Thai +Japanese numerals – Number words used in the Japanese language +Korean numerals – Numbers in traditional Korean writing +Maya numerals – System used by the ancient Mayan civilization to represent numbers and dates +Prehistoric numerals – Numeral form used for countingPages displaying short descriptions of redirect targets +Roman numerals – Numbers in the Roman numeral system +Welsh numerals – Counting system of the Welsh language + + +== Other == +Algorism – Mathematical technique for arithmetic +Goodstein's theorem – Theorem about natural numbers +History of ancient numeral systems +Long and short scales – Different meanings for numbers +Myriad – Order of magnitude name for 10,000 +Non-standard positional numeral systems – Types of numeral system +Quipu – Andean record-keeping system using knotted cords +Tally stick – Memory aid device +Tally mark – Numeral form used for countingPages displaying short descriptions of redirect targets +-yllion – Mathematical notation \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-0.md b/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-0.md new file mode 100644 index 000000000..d1b060105 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-0.md @@ -0,0 +1,47 @@ +--- +title: "List of numerical-analysis software" +chunk: 1/3 +source: "https://en.wikipedia.org/wiki/List_of_numerical-analysis_software" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:56.369537+00:00" +instance: "kb-cron" +--- + +Listed here are notable end-user computer applications intended for use with numerical or data analysis: + +== Numerical-software packages == +Analytica is a widely used proprietary software tool for building and analyzing numerical models. It is a declarative and visual programming language based on influence diagrams. +FlexPro is a program to analyze and present measurement data. It has a rich Excel-like user interface and a built-in vector programming language FPScript has a syntax similar to MATLAB. +FreeMat, an open-source MATLAB-like environment with a GNU General Public License (GPL). +GNU Octave is a high-level programming language, intended for mainly numerical computing. It has a convenient command-line interface to solve linear and nonlinear problems numerically, and to perform other numerical experiments using a language that is compatible mostly with MATLAB. The 4.0 and newer releases of Octave include a GUI. Several independently developed Linux programs (Cantor, KAlgebra) also offer GUI front-ends to Octave. An active community provides technical support to users. +GroovyLab (formerly jLab), a research platform to build an open-source MATLAB-like environment in pure Java and Groovy. Supports interpreted j-Scripts (MATLAB-like) and compiled GroovySci (extension to Groovy) scripts that give direct interfacing to Java code and scripting access to many popular Java scientific libraries (e.g., Weka and JSci) and application Wizards. +Igor Pro is proprietary software to perform complex numerical calculations, statistical analysis, and produce publication-quality graphics. It comes with its own programming language, in which numerical algorithms can be implemented. +Jacket, a proprietary GPU toolbox for MATLAB, enabling some computations to be offloaded to the GPU for acceleration and data visualization. +Julia is a high-level dynamic language with a surface similarity to MATLAB. Packages such as DataFrames.jl are available. +LabVIEW offers both textual and graphical-programming approaches to numerical analysis. Its text-based programming language MathScript uses .m-file-script syntax providing some compatibility with MATLAB and its clones. +LAPACK has Fortran 90 routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems and the associated matrix factorizations (LU, Cholesky, QR, SVD, Schur, and generalized Schur). +MATLAB is a widely used proprietary software to perform numerical computations. It comes with its own programming language, in which numerical algorithms can be implemented. +MCSim a simulation and numerical integration package, with fast Monte Carlo and Markov chain Monte Carlo abilities. +ML.NET is a free software machine learning library for the C# programming language. +NAG Numerical Libraries is an extensive software library of highly optimized numerical-analysis routines for various programming environments. +O-Matrix is a proprietary licensed matrix programming language for mathematics, engineering, science, and financial analysis. +pandas is a BSD-licensed library providing data structures and data analysis tools for the Python programming language. +Perl Data Language has large multidimensional arrays for the Perl programming language, and utilities for image processing and graphical plotting. +ScaLAPACK is a library of high-performance linear algebra routines for parallel distributed-memory machines that features functionality similar to LAPACK (solvers for dense and banded linear systems, least-squares problems, eigenvalue problems, and singular-value problem). +Scilab is advanced numerical analysis package similar to MATLAB or Octave. Comes with a complete GUI and Xcos which is alternative to Simulink. (free software, GPL-compatible CeCILL license) +Sysquake is a computing environment with interactive graphics for mathematics, physics and engineering. Like other applications from Calerga, it is based on a MATLAB-compatible language. +TK Solver is a mathematical modeling and problem-solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc. +Torch is a deep-learning library with support for manipulation, statistical analysis and presentation of Tensors. +XLfit, A plugin to Excel for curve-fitting and statistical analysis. + +== General-purpose computer algebra systems == + +Macsyma, a general-purpose computer algebra system, which has a free GPL-licensed version called Maxima. +Maple, a general-purpose commercial mathematics software package. +Mathcad offers a WYSIWYG interface and the ability to generate publication-quality mathematical equations. +Mathematica offers numerical evaluation, optimization and visualization of a very wide range of numerical functions. It also includes a programming language and computer algebra abilities. +PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves...), but also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers etc., and a lot of transcendental functions. PARI is also available as a C library to allow for faster computations. +SageMath is an open-source math software, with a unified Python interface which is available as a text interface or a graphical web-based one. Includes interfaces for open-source and proprietary general purpose CAS, and other numerical analysis programs, like PARI/GP, GAP, gnuplot, Magma, and Maple. +Speakeasy is an interactive numerical environment also featuring an interpreted programming language. Born in the mid '60s for matrix manipulation and still in continuous evolution, it pioneered the most common paradigms of this kind of tools, featuring dynamic typing of the structured data objects, dynamic allocation and garbage collection, operators overloading, dynamic linking of compiled or interpreted additional modules contributed by the community of the users and so on. +Trilinos is a collection of open-source object-oriented libraries for use in scientific and engineering applications. Trilinos is based on scalable, parallel linear-algebra algorithms. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-1.md b/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-1.md new file mode 100644 index 000000000..7290fb0b5 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-1.md @@ -0,0 +1,36 @@ +--- +title: "List of numerical-analysis software" +chunk: 2/3 +source: "https://en.wikipedia.org/wiki/List_of_numerical-analysis_software" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:56.369537+00:00" +instance: "kb-cron" +--- + +== Interface-oriented == +Baudline is a time-frequency browser for numerical signals analysis and scientific visualization. +COMSOL Multiphysics is a finite-element analysis, solver and simulation software / FEA Software package for various physics and engineering applications, especially coupled phenomena, or multiphysics. +Dataplot is provided by NIST. +DADiSP is a commercial program focused on digital signal processing (DSP) that combines the numerical ability of MATLAB with a spreadsheet-like interface. +Easy Java Simulations (EJS) is an open-source software tool, written in Java, for generating simulations. +Euler Mathematical Toolbox is a powerful numerical laboratory with a programming language that can handle real, complex and interval numbers, vectors and matrices. It can produce 2D/3D plots. +FEATool Multiphysics is a MATLAB GUI toolbox for finite element FEM and PDE multiphysics simulations. +FEniCS Project is a collection of project for automated solutions to partial differential equations (PDEs). +Hermes is a C++ library of advanced adaptive finite element algorithms to solve PDEs and multiphysics coupled problems. +Fityk is a curve fitting and data-analysis program. Primarily used for peak fitting and analyzing peak data. +FlexPro is a commercial program for interactive and automated analysis and presentation of mainly measurement data. It supports many binary instrument data formats and has its own vectorized programming language. +IGOR Pro, a software package with emphasis on time series, image analysis, and curve fitting. It comes with its own programming language and can be used interactively. +LabPlot is a data analysis and visualization application built on the KDE Platform. +MFEM is a free, lightweight, scalable C++ library for finite element methods. +Origin, a software package that is widely used for making scientific graphs. It comes with its own C/C++ compiler that conforms quite closely to ANSI standard. +PAW is a free data analysis package developed at CERN. +SPSS, an application for statistical analysis. +QtiPlot is a data analysis and scientific visualisation program, similar to Origin. +ROOT is a free object-oriented multi-purpose data-analysis package, developed at CERN. +Salome is a free software tool that is a generic platform for pre- and post-processing for numerical simulation. +Shogun, an open-source large-scale machine learning toolbox that gives several SVM implementations (like libSVM, SVMlight) under a common framework and interfaces to MATLAB, Octave, Python, R +Waffles is a free-software collection of command-line tools designed for scripting machine-learning operations in automated experiments and processes. +Weka is a suite of machine learning software written at the University of Waikato. + +== Language-oriented == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-2.md b/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-2.md new file mode 100644 index 000000000..34d98eb1b --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-2.md @@ -0,0 +1,56 @@ +--- +title: "List of numerical-analysis software" +chunk: 3/3 +source: "https://en.wikipedia.org/wiki/List_of_numerical-analysis_software" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:56.369537+00:00" +instance: "kb-cron" +--- + +acslX is a software application for modeling and evaluating the performance of continuous systems described by time-dependent, nonlinear differential equations. +ADMB is a software suite for non-linear statistical modeling based on C++ which uses automatic differentiation. +AMPL is a mathematical modeling language for describing and solving high complexity problems for large-scale optimization. +Calcpad is an open-source numerical software for engineering calculations. Supports numerical differentiation and integration, root and extrema finding, vector and matrix calculations. +Ch, a commercial C/C++-based interpreted language with computational array for scientific numerical computation and visualization. +APMonitor: APMonitor is a mathematical modeling language for describing and solving representations of physical systems in the form of differential and algebraic equations. +Armadillo is C++ template library for linear algebra; includes various decompositions, factorisations, and statistics functions; its syntax (application programming interface (API) is similar to MATLAB. +Clojure with numeric libraries Neanderthal, ClojureCUDA, and ClojureCL to call optimized matrix and linear algebra functions on CPU and GPU. +Julia is designed for cloud parallel scientific computing in mind on LLVM-based just-in-time compilation (JIT) as a backend. Lightweight green threading (coroutines). Direct calls of C functions from code (no wrappers or special APIs needed), support for Unicode. Powerful shell-like abilities to manage other processes. Lisp-like macros and other metaprogramming facilities. +Environment for DeveLoping KDD-Applications Supported by Index-Structures (ELKI) a software framework for developing data mining algorithms in Java. +GAUSS, a matrix programming language for mathematics and statistics. +GNU Data Language, a free compiler designed as a drop-in replacement for IDL. +IDL, a commercial interpreted language based on FORTRAN with some vectorization. Widely used in the solar physics, fusion power, atmospheric sciences and medical communities. The GNU Data Language is a free alternative. +ILNumerics, a C# math library that brings numeric computing functions for science, engineering and financial analysis to the .NET framework. +Kinetic PreProcessor (KPP) generates Fortran 90, FORTRAN 77, C, or MATLAB code for the integration of ordinary differential equations (ODEs) resulting from chemical reaction mechanisms. +Madagascar, an open-source software package for multidimensional data analysis and reproducible computational experiments. +mlpack is an open-source library for machine learning, providing a simple and consistent API, while exploiting C++ language features to provide maximum performance and flexibility +NCAR Command Language is an interpreted language designed specifically for scientific data analysis and visualization. +O-Matrix - a matrix programming language for mathematics, engineering, science, and financial analysis. +OptimJ is a mathematical Java-based modeling language for describing and solving high-complexity problems for large-scale optimization. +Perl Data Language, also known as PDL, an array extension to Perl ver.5, used for data manipulation, statistics, numerical simulation and visualization. +Python with well-known scientific computing packages: NumPy, SymPy and SciPy. +R is a widely used system with a focus on data manipulation and statistics which implements the S language. Many add-on packages are available (free software, GNU GPL license). +SAS, a system of software products for statistics. It includes SAS/IML, a matrix programming language. +Stata is a general-purpose statistical software package for data manipulation, visualization, statistics, and automated reporting. It is used by researchers in many fields, including biomedicine, economics, epidemiology, and sociology. +VisSim is a visual block-diagram language for simulation of nonlinear dynamic systems and model-based embedded development. Its fast ODE engine supports real-time simulation of complex large-scale models. The highly efficient fixed-point code generator allows targeting of low-cost fixed-point embedded processors. +Wolfram Language which is used within many Wolfram technologies such as Mathematica and the Wolfram Cloud +World Programming System (WPS), supports mixing Python, R and SAS programming languages in a single-user program for statistical analysis and data manipulation +Yorick is an interpreted programming language designed for numerics, graph plotting and simulation. + +== Historically significant == +Expensive Desk Calculator written for the TX-0 and PDP-1 in the late 1950s or early 1960s. +S is an (array-based) programming language with strong numerical support. R is an implementation of the S language. + +== See also == +Comparison of deep learning software +List of information graphics software +List of numerical analysis topics +List of numerical libraries +List of open-source mathematical libraries +List of statistical software +Outline of software +Mathematical software +Web-based simulation + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-0.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-0.md new file mode 100644 index 000000000..8c1c35df2 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-0.md @@ -0,0 +1,140 @@ +--- +title: "List of numerical analysis topics" +chunk: 1/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +This is a list of numerical analysis topics. + +== General == +Validated numerics +Iterative method +Rate of convergence — the speed at which a convergent sequence approaches its limit +Order of accuracy — rate at which numerical solution of differential equation converges to exact solution +Series acceleration — methods to accelerate the speed of convergence of a series +Aitken's delta-squared process — most useful for linearly converging sequences +Minimum polynomial extrapolation — for vector sequences +Richardson extrapolation +Shanks transformation — similar to Aitken's delta-squared process, but applied to the partial sums +Van Wijngaarden transformation — for accelerating the convergence of an alternating series +Abramowitz and Stegun — book containing formulas and tables of many special functions +Digital Library of Mathematical Functions — successor of book by Abramowitz and Stegun +Curse of dimensionality +Local convergence and global convergence — whether you need a good initial guess to get convergence +Superconvergence +Discretization +Difference quotient +Complexity: +Computational complexity of mathematical operations +Smoothed analysis — measuring the expected performance of algorithms under slight random perturbations of worst-case inputs +Symbolic-numeric computation — combination of symbolic and numeric methods +Cultural and historical aspects: +History of numerical solution of differential equations using computers +Hundred-dollar, Hundred-digit Challenge problems — list of ten problems proposed by Nick Trefethen in 2002 +Timeline of numerical analysis after 1945 +General classes of methods: +Collocation method — discretizes a continuous equation by requiring it only to hold at certain points +Level-set method +Level set (data structures) — data structures for representing level sets +Sinc numerical methods — methods based on the sinc function, sinc(x) = sin(x) / x +ABS methods + +== Error == +Error analysis (mathematics) + +Approximation +Approximation error +Catastrophic cancellation +Condition number +Discretization error +Floating point number +Guard digit — extra precision introduced during a computation to reduce round-off error +Truncation — rounding a floating-point number by discarding all digits after a certain digit +Round-off error +Numeric precision in Microsoft Excel +Arbitrary-precision arithmetic +Interval arithmetic — represent every number by two floating-point numbers guaranteed to have the unknown number between them +Interval contractor — maps interval to subinterval which still contains the unknown exact answer +Interval propagation — contracting interval domains without removing any value consistent with the constraints +See also: Interval boundary element method, Interval finite element +Loss of significance +Numerical error +Numerical stability +Error propagation: +Propagation of uncertainty +Residual (numerical analysis) +Relative change and difference — the relative difference between x and y is |x − y| / max(|x|, |y|) +Significant figures +Artificial precision — when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user input +False precision — giving more significant figures than appropriate +Sterbenz lemma +Truncation error — error committed by doing only a finite numbers of steps +Well-posed problem +Affine arithmetic + +== Elementary and special functions == +Unrestricted algorithm +Summation: +Kahan summation algorithm +Pairwise summation — slightly worse than Kahan summation but cheaper +Binary splitting +2Sum +Multiplication: +Multiplication algorithm — general discussion, simple methods +Karatsuba algorithm — the first algorithm which is faster than straightforward multiplication +Toom–Cook multiplication — generalization of Karatsuba multiplication +Schönhage–Strassen algorithm — based on Fourier transform, asymptotically very fast +Fürer's algorithm — asymptotically slightly faster than Schönhage–Strassen +Division algorithm — for computing quotient and/or remainder of two numbers +Long division +Restoring division +Non-restoring division +SRT division +Newton–Raphson division: uses Newton's method to find the reciprocal of D, and multiply that reciprocal by N to find the final quotient Q. +Goldschmidt division +Exponentiation: +Exponentiation by squaring +Addition-chain exponentiation +Multiplicative inverse Algorithms: for computing a number's multiplicative inverse (reciprocal). +Newton's method +Polynomials: +Horner's method +Estrin's scheme — modification of the Horner scheme with more possibilities for parallelization +Clenshaw algorithm +De Casteljau's algorithm +Square roots and other roots: +Integer square root +Methods of computing square roots +nth root algorithm +hypot — the function (x2 + y2)1/2 +Alpha max plus beta min algorithm — approximates hypot(x,y) +Fast inverse square root — calculates 1 / √x using details of the IEEE floating-point system +Elementary functions (exponential, logarithm, trigonometric functions): +Trigonometric tables — different methods for generating them +CORDIC — shift-and-add algorithm using a table of arc tangents +BKM algorithm — shift-and-add algorithm using a table of logarithms and complex numbers +Gamma function: +Lanczos approximation +Spouge's approximation — modification of Stirling's approximation; easier to apply than Lanczos +AGM method — computes arithmetic–geometric mean; related methods compute special functions +FEE method (Fast E-function Evaluation) — fast summation of series like the power series for ex +Gal's accurate tables — table of function values with unequal spacing to reduce round-off error +Spigot algorithm — algorithms that can compute individual digits of a real number +Approximations of π: +Liu Hui's π algorithm — first algorithm that can compute π to arbitrary precision +Leibniz formula for π — alternating series with very slow convergence +Wallis product — infinite product converging slowly to π/2 +Viète's formula — more complicated infinite product which converges faster +Gauss–Legendre algorithm — iteration which converges quadratically to π, based on arithmetic–geometric mean +Borwein's algorithm — iteration which converges quartically to 1/π, and other algorithms +Chudnovsky algorithm — fast algorithm that calculates a hypergeometric series +Bailey–Borwein–Plouffe formula — can be used to compute individual hexadecimal digits of π +Bellard's formula — faster version of Bailey–Borwein–Plouffe formula +List of formulae involving π + +== Numerical linear algebra == +Numerical linear algebra — study of numerical algorithms for linear algebra problems \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-1.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-1.md new file mode 100644 index 000000000..e5ccf2071 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-1.md @@ -0,0 +1,155 @@ +--- +title: "List of numerical analysis topics" +chunk: 2/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +=== Basic concepts === +Types of matrices appearing in numerical analysis: +Sparse matrix +Band matrix +Bidiagonal matrix +Tridiagonal matrix +Pentadiagonal matrix +Skyline matrix +Circulant matrix +Triangular matrix +Diagonally dominant matrix +Block matrix — matrix composed of smaller matrices +Stieltjes matrix — symmetric positive definite with non-positive off-diagonal entries +Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle) +Wilkinson matrix — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues +Convergent matrix — square matrix whose successive powers approach the zero matrix +Algorithms for matrix multiplication: +Strassen algorithm +Coppersmith–Winograd algorithm +Cannon's algorithm — a distributed algorithm, especially suitable for processors laid out in a 2d grid +Freivalds' algorithm — a randomized algorithm for checking the result of a multiplication +Matrix decompositions: +LU decomposition — lower triangular times upper triangular +QR decomposition — orthogonal matrix times triangular matrix +RRQR factorization — rank-revealing QR factorization, can be used to compute rank of a matrix +Polar decomposition — unitary matrix times positive-semidefinite Hermitian matrix +Decompositions by similarity: +Eigendecomposition — decomposition in terms of eigenvectors and eigenvalues +Jordan normal form — bidiagonal matrix of a certain form; generalizes the eigendecomposition +Weyr canonical form — permutation of Jordan normal form +Jordan–Chevalley decomposition — sum of commuting nilpotent matrix and diagonalizable matrix +Schur decomposition — similarity transform bringing the matrix to a triangular matrix +Singular value decomposition — unitary matrix times diagonal matrix times unitary matrix +Matrix splitting — expressing a given matrix as a sum or difference of matrices + +=== Solving systems of linear equations === +Gaussian elimination +Row echelon form — matrix in which all entries below a nonzero entry are zero +Bareiss algorithm — variant which ensures that all entries remain integers if the initial matrix has integer entries +Tridiagonal matrix algorithm — simplified form of Gaussian elimination for tridiagonal matrices +LU decomposition — write a matrix as a product of an upper- and a lower-triangular matrix +Crout matrix decomposition +LU reduction — a special parallelized version of a LU decomposition algorithm +Block LU decomposition +Cholesky decomposition — for solving a system with a positive definite matrix +Minimum degree algorithm +Symbolic Cholesky decomposition +Iterative refinement — procedure to turn an inaccurate solution in a more accurate one +Direct methods for sparse matrices: +Frontal solver — used in finite element methods +Nested dissection — for symmetric matrices, based on graph partitioning +Levinson recursion — for Toeplitz matrices +SPIKE algorithm — hybrid parallel solver for narrow-banded matrices +Cyclic reduction — eliminate even or odd rows or columns, repeat +Iterative methods: +Jacobi method +Gauss–Seidel method +Successive over-relaxation (SOR) — a technique to accelerate the Gauss–Seidel method +Symmetric successive over-relaxation (SSOR) — variant of SOR for symmetric matrices +Backfitting algorithm — iterative procedure used to fit a generalized additive model, often equivalent to Gauss–Seidel +Modified Richardson iteration +Conjugate gradient method (CG) — assumes that the matrix is positive definite +Derivation of the conjugate gradient method +Nonlinear conjugate gradient method — generalization for nonlinear optimization problems +Biconjugate gradient method (BiCG) +Biconjugate gradient stabilized method (BiCGSTAB) — variant of BiCG with better convergence +Conjugate residual method — similar to CG but only assumed that the matrix is symmetric +Generalized minimal residual method (GMRES) — based on the Arnoldi iteration +Chebyshev iteration — avoids inner products but needs bounds on the spectrum +Stone's method (SIP — Strongly Implicit Procedure) — uses an incomplete LU decomposition +Kaczmarz method +Preconditioner +Incomplete Cholesky factorization — sparse approximation to the Cholesky factorization +Incomplete LU factorization — sparse approximation to the LU factorization +Uzawa iteration — for saddle node problems +Underdetermined and overdetermined systems (systems that have no or more than one solution): +Numerical computation of null space — find all solutions of an underdetermined system +Moore–Penrose pseudoinverse — for finding solution with smallest 2-norm (for underdetermined systems) or smallest residual +Sparse approximation — for finding the sparsest solution (i.e., the solution with as many zeros as possible) + +=== Eigenvalue algorithms === +Eigenvalue algorithm — a numerical algorithm for locating the eigenvalues of a matrix + +Power iteration +Inverse iteration +Rayleigh quotient iteration +Arnoldi iteration — based on Krylov subspaces +Lanczos algorithm — Arnoldi, specialized for positive-definite matrices +Block Lanczos algorithm — for when matrix is over a finite field +QR algorithm +Jacobi eigenvalue algorithm — select a small submatrix which can be diagonalized exactly, and repeat +Jacobi rotation — the building block, almost a Givens rotation +Jacobi method for complex Hermitian matrices +Divide-and-conquer eigenvalue algorithm +Folded spectrum method +LOBPCG — Locally Optimal Block Preconditioned Conjugate Gradient Method +Eigenvalue perturbation — stability of eigenvalues under perturbations of the matrix + +=== Other concepts and algorithms === +Orthogonalization algorithms: +Gram–Schmidt process +Householder transformation +Householder operator — analogue of Householder transformation for general inner product spaces +Givens rotation +Krylov subspace +Block matrix pseudoinverse +Bidiagonalization +Cuthill–McKee algorithm — permutes rows/columns in sparse matrix to yield a narrow band matrix +In-place matrix transposition — computing the transpose of a matrix without using much additional storage +Pivot element — entry in a matrix on which the algorithm concentrates +Matrix-free methods — methods that only access the matrix by evaluating matrix-vector products + +== Interpolation and approximation == +Interpolation — construct a function going through some given data points + +Nearest-neighbor interpolation — takes the value of the nearest neighbor + +=== Polynomial interpolation === +Polynomial interpolation — interpolation by polynomials + +Linear interpolation +Runge's phenomenon +Vandermonde matrix +Chebyshev polynomials +Chebyshev nodes +Lebesgue constants +Different forms for the interpolant: +Newton polynomial +Divided differences +Neville's algorithm — for evaluating the interpolant; based on the Newton form +Lagrange polynomial +Bernstein polynomial — especially useful for approximation +Brahmagupta's interpolation formula — seventh-century formula for quadratic interpolation +Extensions to multiple dimensions: +Bilinear interpolation +Trilinear interpolation +Bicubic interpolation +Tricubic interpolation +Padua points — set of points in R2 with unique polynomial interpolant and minimal growth of Lebesgue constant +Hermite interpolation +Birkhoff interpolation +Abel–Goncharov interpolation + +=== Spline interpolation === +Spline interpolation — interpolation by piecewise polynomials \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-2.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-2.md new file mode 100644 index 000000000..a9938eb76 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-2.md @@ -0,0 +1,93 @@ +--- +title: "List of numerical analysis topics" +chunk: 3/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +Spline (mathematics) — the piecewise polynomials used as interpolants +Perfect spline — polynomial spline of degree m whose mth derivate is ±1 +Cubic Hermite spline +Centripetal Catmull–Rom spline — special case of cubic Hermite splines without self-intersections or cusps +Monotone cubic interpolation +Hermite spline +Bézier curve +De Casteljau's algorithm +composite Bézier curve +Generalizations to more dimensions: +Bézier triangle — maps a triangle to R3 +Bézier surface — maps a square to R3 +B-spline +Box spline — multivariate generalization of B-splines +Truncated power function +De Boor's algorithm — generalizes De Casteljau's algorithm +Non-uniform rational B-spline (NURBS) +T-spline — can be thought of as a NURBS surface for which a row of control points is allowed to terminate +Kochanek–Bartels spline +Coons patch — type of manifold parametrization used to smoothly join other surfaces together +M-spline — a non-negative spline +I-spline — a monotone spline, defined in terms of M-splines +Smoothing spline — a spline fitted smoothly to noisy data +Blossom (functional) — a unique, affine, symmetric map associated to a polynomial or spline +See also: List of numerical computational geometry topics + +=== Trigonometric interpolation === +Trigonometric interpolation — interpolation by trigonometric polynomials + +Discrete Fourier transform — can be viewed as trigonometric interpolation at equidistant points +Relations between Fourier transforms and Fourier series +Fast Fourier transform (FFT) — a fast method for computing the discrete Fourier transform +Bluestein's FFT algorithm +Bruun's FFT algorithm +Cooley–Tukey FFT algorithm +Split-radix FFT algorithm — variant of Cooley–Tukey that uses a blend of radices 2 and 4 +Goertzel algorithm +Prime-factor FFT algorithm +Rader's FFT algorithm +Bit-reversal permutation — particular permutation of vectors with 2m entries used in many FFTs. +Butterfly diagram +Twiddle factor — the trigonometric constant coefficients that are multiplied by the data +Cyclotomic fast Fourier transform — for FFT over finite fields +Methods for computing discrete convolutions with finite impulse response filters using the FFT: +Overlap–add method +Overlap–save method +Sigma approximation +Dirichlet kernel — convolving any function with the Dirichlet kernel yields its trigonometric interpolant +Gibbs phenomenon + +=== Other interpolants === +Simple rational approximation +Polynomial and rational function modeling — comparison of polynomial and rational interpolation +Wavelet +Continuous wavelet +Transfer matrix +See also: List of functional analysis topics, List of wavelet-related transforms +Inverse distance weighting +Radial basis function (RBF) — a function of the form ƒ(x) = φ(|x−x0|) +Polyharmonic spline — a commonly used radial basis function +Thin plate spline — a specific polyharmonic spline: r2 log r +Hierarchical RBF +Subdivision surface — constructed by recursively subdividing a piecewise linear interpolant +Catmull–Clark subdivision surface +Doo–Sabin subdivision surface +Loop subdivision surface +Slerp (spherical linear interpolation) — interpolation between two points on a sphere +Generalized quaternion interpolation — generalizes slerp for interpolation between more than two quaternions +Irrational base discrete weighted transform +Nevanlinna–Pick interpolation — interpolation by analytic functions in the unit disc subject to a bound +Pick matrix — the Nevanlinna–Pick interpolation has a solution if this matrix is positive semi-definite +Multivariate interpolation — the function being interpolated depends on more than one variable +Barnes interpolation — method for two-dimensional functions using Gaussians common in meteorology +Coons surface — combination of linear interpolation and bilinear interpolation +Lanczos resampling — based on convolution with a sinc function +Natural neighbor interpolation +PDE surface +Transfinite interpolation — constructs function on planar domain given its values on the boundary +Trend surface analysis — based on low-order polynomials of spatial coordinates; uses scattered observations +Method based on polynomials are listed under Polynomial interpolation + +=== Approximation theory === +Approximation theory \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-3.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-3.md new file mode 100644 index 000000000..2a0a4f064 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-3.md @@ -0,0 +1,125 @@ +--- +title: "List of numerical analysis topics" +chunk: 4/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +Orders of approximation +Lebesgue's lemma +Curve fitting +Vector field reconstruction +Modulus of continuity — measures smoothness of a function +Least squares (function approximation) — minimizes the error in the L2-norm +Minimax approximation algorithm — minimizes the maximum error over an interval (the L∞-norm) +Equioscillation theorem — characterizes the best approximation in the L∞-norm +Unisolvent point set — function from given function space is determined uniquely by values on such a set of points +Stone–Weierstrass theorem — continuous functions can be approximated uniformly by polynomials, or certain other function spaces +Approximation by polynomials: +Linear approximation +Bernstein polynomial — basis of polynomials useful for approximating a function +Bernstein's constant — error when approximating |x| by a polynomial +Remez algorithm — for constructing the best polynomial approximation in the L∞-norm +Bernstein's inequality (mathematical analysis) — bound on maximum of derivative of polynomial in unit disk +Mergelyan's theorem — generalization of Stone–Weierstrass theorem for polynomials +Müntz–Szász theorem — variant of Stone–Weierstrass theorem for polynomials if some coefficients have to be zero +Bramble–Hilbert lemma — upper bound on Lp error of polynomial approximation in multiple dimensions +Discrete Chebyshev polynomials — polynomials orthogonal with respect to a discrete measure +Favard's theorem — polynomials satisfying suitable 3-term recurrence relations are orthogonal polynomials +Approximation by Fourier series / trigonometric polynomials: +Jackson's inequality — upper bound for best approximation by a trigonometric polynomial +Bernstein's theorem (approximation theory) — a converse to Jackson's inequality +Fejér's theorem — Cesàro means of partial sums of Fourier series converge uniformly for continuous periodic functions +Erdős–Turán inequality — bounds distance between probability and Lebesgue measure in terms of Fourier coefficients +Different approximations: +Moving least squares +Padé approximant +Padé table — table of Padé approximants +Hartogs–Rosenthal theorem — continuous functions can be approximated uniformly by rational functions on a set of Lebesgue measure zero +Szász–Mirakyan operator — approximation by e−n xk on a semi-infinite interval +Szász–Mirakjan–Kantorovich operator +Baskakov operator — generalize Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators +Favard operator — approximation by sums of Gaussians +Surrogate model — application: replacing a function that is hard to evaluate by a simpler function +Constructive function theory — field that studies connection between degree of approximation and smoothness +Universal differential equation — differential–algebraic equation whose solutions can approximate any continuous function +Fekete problem — find N points on a sphere that minimize some kind of energy +Carleman's condition — condition guaranteeing that a measure is uniquely determined by its moments +Krein's condition — condition that exponential sums are dense in weighted L2 space +Lethargy theorem — about distance of points in a metric space from members of a sequence of subspaces +Wirtinger's representation and projection theorem +Journals: +Constructive Approximation +Journal of Approximation Theory + +=== Miscellaneous === +Extrapolation +Linear predictive analysis — linear extrapolation +Unisolvent functions — functions for which the interpolation problem has a unique solution +Regression analysis +Isotonic regression +Curve-fitting compaction +Interpolation (computer graphics) + +== Finding roots of nonlinear equations == +See #Numerical linear algebra for linear equations +Root-finding algorithm — algorithms for solving the equation f(x) = 0 + +General methods: +Bisection method — simple and robust; linear convergence +Lehmer–Schur algorithm — variant for complex functions +Fixed-point iteration +Newton's method — based on linear approximation around the current iterate; quadratic convergence +Kantorovich theorem — gives a region around solution such that Newton's method converges +Newton fractal — indicates which initial condition converges to which root under Newton iteration +Quasi-Newton method — uses an approximation of the Jacobian: +Broyden's method — uses a rank-one update for the Jacobian +Symmetric rank-one — a symmetric (but not necessarily positive definite) rank-one update of the Jacobian +Davidon–Fletcher–Powell formula — update of the Jacobian in which the matrix remains positive definite +Broyden–Fletcher–Goldfarb–Shanno algorithm — rank-two update of the Jacobian in which the matrix remains positive definite +Limited-memory BFGS method — truncated, matrix-free variant of BFGS method suitable for large problems +Steffensen's method — uses divided differences instead of the derivative +Secant method — based on linear interpolation at last two iterates +False position method — secant method with ideas from the bisection method +Muller's method — based on quadratic interpolation at last three iterates +Sidi's generalized secant method — higher-order variants of secant method +Inverse quadratic interpolation — similar to Muller's method, but interpolates the inverse +Brent's method — combines bisection method, secant method and inverse quadratic interpolation +Ridders' method — fits a linear function times an exponential to last two iterates and their midpoint +Halley's method — uses f, f' and f''; achieves the cubic convergence +Householder's method — uses first d derivatives to achieve order d + 1; generalizes Newton's and Halley's method +Methods for polynomials: +Aberth method +Bairstow's method +Durand–Kerner method +Graeffe's method +Jenkins–Traub algorithm — fast, reliable, and widely used +Laguerre's method +Splitting circle method +Analysis: +Wilkinson's polynomial +Numerical continuation — tracking a root as one parameter in the equation changes +Piecewise linear continuation + +== Optimization == +Mathematical optimization — algorithm for finding maxima or minima of a given function + +=== Basic concepts === +Active set +Candidate solution +Constraint (mathematics) +Constrained optimization — studies optimization problems with constraints +Binary constraint — a constraint that involves exactly two variables +Corner solution +Feasible region — contains all solutions that satisfy the constraints but may not be optimal +Global optimum and Local optimum +Maxima and minima +Slack variable +Continuous optimization +Discrete optimization + +=== Linear programming === +Linear programming (also treats integer programming) — objective function and constraints are linear \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-4.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-4.md new file mode 100644 index 000000000..7ac19dbd4 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-4.md @@ -0,0 +1,147 @@ +--- +title: "List of numerical analysis topics" +chunk: 5/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +Algorithms for linear programming: +Simplex algorithm +Bland's rule — rule to avoid cycling in the simplex method +Klee–Minty cube — perturbed (hyper)cube; simplex method has exponential complexity on such a domain +Criss-cross algorithm — similar to the simplex algorithm +Big M method — variation of simplex algorithm for problems with both "less than" and "greater than" constraints +Interior point method +Ellipsoid method +Karmarkar's algorithm +Mehrotra predictor–corrector method +Column generation +k-approximation of k-hitting set — algorithm for specific LP problems (to find a weighted hitting set) +Linear complementarity problem +Decompositions: +Benders' decomposition +Dantzig–Wolfe decomposition +Theory of two-level planning +Variable splitting +Basic solution (linear programming) — solution at vertex of feasible region +Fourier–Motzkin elimination +Hilbert basis (linear programming) — set of integer vectors in a convex cone which generate all integer vectors in the cone +LP-type problem +Linear inequality +Vertex enumeration problem — list all vertices of the feasible set + +=== Convex optimization === +Convex optimization + +Quadratic programming +Linear least squares (mathematics) +Total least squares +Frank–Wolfe algorithm +Sequential minimal optimization — breaks up large QP problems into a series of smallest possible QP problems +Bilinear program +Basis pursuit — minimize L1-norm of vector subject to linear constraints +Basis pursuit denoising (BPDN) — regularized version of basis pursuit +In-crowd algorithm — algorithm for solving basis pursuit denoising +Linear matrix inequality +Conic optimization +Semidefinite programming +Second-order cone programming +Sum-of-squares optimization +Quadratic programming (see above) +Bregman method — row-action method for strictly convex optimization problems +Proximal gradient method — use splitting of objective function in sum of possible non-differentiable pieces +Subgradient method — extension of steepest descent for problems with a non-differentiable objective function +Biconvex optimization — generalization where objective function and constraint set can be biconvex + +=== Nonlinear programming === +Nonlinear programming — the most general optimization problem in the usual framework + +Special cases of nonlinear programming: +See Linear programming and Convex optimization above +Geometric programming — problems involving signomials or posynomials +Signomial — similar to polynomials, but exponents need not be integers +Posynomial — a signomial with positive coefficients +Quadratically constrained quadratic program +Linear-fractional programming — objective is ratio of linear functions, constraints are linear +Fractional programming — objective is ratio of nonlinear functions, constraints are linear +Nonlinear complementarity problem (NCP) — find x such that x ≥ 0, f(x) ≥ 0 and xT f(x) = 0 +Least squares — the objective function is a sum of squares +Non-linear least squares +Gauss–Newton algorithm +BHHH algorithm — variant of Gauss–Newton in econometrics +Generalized Gauss–Newton method — for constrained nonlinear least-squares problems +Levenberg–Marquardt algorithm +Iteratively reweighted least squares (IRLS) — solves a weighted least-squares problem at every iteration +Partial least squares — statistical techniques similar to principal components analysis +Non-linear iterative partial least squares (NIPLS) +Mathematical programming with equilibrium constraints — constraints include variational inequalities or complementarities +Univariate optimization: +Golden section search +Successive parabolic interpolation — based on quadratic interpolation through the last three iterates +General algorithms: +Concepts: +Descent direction +Guess value — the initial guess for a solution with which an algorithm starts +Line search +Backtracking line search +Wolfe conditions +Gradient method — method that uses the gradient as the search direction +Gradient descent +Stochastic gradient descent +Landweber iteration — mainly used for ill-posed problems +Successive linear programming (SLP) — replace problem by a linear programming problem, solve that, and repeat +Sequential quadratic programming (SQP) — replace problem by a quadratic programming problem, solve that, and repeat +Newton's method in optimization +See also under Newton algorithm in the section Finding roots of nonlinear equations +Nonlinear conjugate gradient method +Derivative-free methods +Coordinate descent — move in one of the coordinate directions +Adaptive coordinate descent — adapt coordinate directions to objective function +Random coordinate descent — randomized version +Nelder–Mead method +Pattern search (optimization) +Powell's method — based on conjugate gradient descent +Rosenbrock methods — derivative-free method, similar to Nelder–Mead but with guaranteed convergence +Augmented Lagrangian method — replaces constrained problems by unconstrained problems with a term added to the objective function +Ternary search +Tabu search +Guided Local Search — modification of search algorithms which builds up penalties during a search +Reactive search optimization (RSO) — the algorithm adapts its parameters automatically +MM algorithm — majorize-minimization, a wide framework of methods +Least absolute deviations +Expectation–maximization algorithm +Ordered subset expectation maximization +Nearest neighbor search +Space mapping — uses "coarse" (ideal or low-fidelity) and "fine" (practical or high-fidelity) models + +=== Optimal control and infinite-dimensional optimization === +Optimal control + +Pontryagin's minimum principle — infinite-dimensional version of Lagrange multipliers +Costate equations — equation for the "Lagrange multipliers" in Pontryagin's minimum principle +Hamiltonian (control theory) — minimum principle says that this function should be minimized +Types of problems: +Linear-quadratic regulator — system dynamics is a linear differential equation, objective is quadratic +Linear-quadratic-Gaussian control (LQG) — system dynamics is a linear SDE with additive noise, objective is quadratic +Optimal projection equations — method for reducing dimension of LQG control problem +Algebraic Riccati equation — matrix equation occurring in many optimal control problems +Bang–bang control — control that switches abruptly between two states +Covector mapping principle +Differential dynamic programming — uses locally-quadratic models of the dynamics and cost functions +DNSS point — initial state for certain optimal control problems with multiple optimal solutions +Legendre–Clebsch condition — second-order condition for solution of optimal control problem +Pseudospectral optimal control +Bellman pseudospectral method — based on Bellman's principle of optimality +Chebyshev pseudospectral method — uses Chebyshev polynomials (of the first kind) +Flat pseudospectral method — combines Ross–Fahroo pseudospectral method with differential flatness +Gauss pseudospectral method — uses collocation at the Legendre–Gauss points +Legendre pseudospectral method — uses Legendre polynomials +Pseudospectral knotting method — generalization of pseudospectral methods in optimal control +Ross–Fahroo pseudospectral method — class of pseudospectral method including Chebyshev, Legendre and knotting +Ross–Fahroo lemma — condition to make discretization and duality operations commute +Ross' π lemma — there is fundamental time constant within which a control solution must be computed for controllability and stability +Sethi model — optimal control problem modelling advertising +Infinite-dimensional optimization \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-5.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-5.md new file mode 100644 index 000000000..2a7ea9026 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-5.md @@ -0,0 +1,199 @@ +--- +title: "List of numerical analysis topics" +chunk: 6/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +Semi-infinite programming — infinite number of variables and finite number of constraints, or other way around +Shape optimization, Topology optimization — optimization over a set of regions +Topological derivative — derivative with respect to changing in the shape +Generalized semi-infinite programming — finite number of variables, infinite number of constraints + +=== Uncertainty and randomness === +Approaches to deal with uncertainty: +Markov decision process +Partially observable Markov decision process +Robust optimization +Wald's maximin model +Scenario optimization — constraints are uncertain +Stochastic approximation +Stochastic optimization +Stochastic programming +Stochastic gradient descent +Random optimization algorithms: +Random search — choose a point randomly in ball around current iterate +Simulated annealing +Adaptive simulated annealing — variant in which the algorithm parameters are adjusted during the computation. +Great Deluge algorithm +Mean field annealing — deterministic variant of simulated annealing +Bayesian optimization — treats objective function as a random function and places a prior over it +Evolutionary algorithm +Differential evolution +Evolutionary programming +Genetic algorithm, Genetic programming +Genetic algorithms in economics +MCACEA (Multiple Coordinated Agents Coevolution Evolutionary Algorithm) — uses an evolutionary algorithm for every agent +Simultaneous perturbation stochastic approximation (SPSA) +Luus–Jaakola +Particle swarm optimization +Stochastic tunneling +Harmony search — mimicks the improvisation process of musicians +see also the section Monte Carlo method + +=== Theoretical aspects === +Convex analysis — function f such that f(tx + (1 − t)y) ≥ tf(x) + (1 − t)f(y) for t ∈ [0,1] +Pseudoconvex function — function f such that ∇f · (y − x) ≥ 0 implies f(y) ≥ f(x) +Quasiconvex function — function f such that f(tx + (1 − t)y) ≤ max(f(x), f(y)) for t ∈ [0,1] +Subderivative +Geodesic convexity — convexity for functions defined on a Riemannian manifold +Duality (optimization) +Weak duality — dual solution gives a bound on the primal solution +Strong duality — primal and dual solutions are equivalent +Shadow price +Dual cone and polar cone +Duality gap — difference between primal and dual solution +Fenchel's duality theorem — relates minimization problems with maximization problems of convex conjugates +Perturbation function — any function which relates to primal and dual problems +Slater's condition — sufficient condition for strong duality to hold in a convex optimization problem +Total dual integrality — concept of duality for integer linear programming +Wolfe duality — for when objective function and constraints are differentiable +Farkas' lemma +Karush–Kuhn–Tucker conditions (KKT) — sufficient conditions for a solution to be optimal +Fritz John conditions — variant of KKT conditions +Lagrange multiplier +Lagrange multipliers on Banach spaces +Semi-continuity +Complementarity theory — study of problems with constraints of the form ⟨u, v⟩ = 0 +Mixed complementarity problem +Mixed linear complementarity problem +Lemke's algorithm — method for solving (mixed) linear complementarity problems +Danskin's theorem — used in the analysis of minimax problems +Maximum theorem — the maximum and maximizer are continuous as function of parameters, under some conditions +No free lunch in search and optimization +Relaxation (approximation) — approximating a given problem by an easier problem by relaxing some constraints +Lagrangian relaxation +Linear programming relaxation — ignoring the integrality constraints in a linear programming problem +Self-concordant function +Reduced cost — cost for increasing a variable by a small amount +Hardness of approximation — computational complexity of getting an approximate solution + +=== Applications === +In geometry: +Geometric median — the point minimizing the sum of distances to a given set of points +Chebyshev center — the centre of the smallest ball containing a given set of points +In statistics: +Iterated conditional modes — maximizing joint probability of Markov random field +Response surface methodology — used in the design of experiments +Automatic label placement +Compressed sensing — reconstruct a signal from knowledge that it is sparse or compressible +Cutting stock problem +Demand optimization +Destination dispatch — an optimization technique for dispatching elevators +Energy minimization +Entropy maximization +Highly optimized tolerance +Hyperparameter optimization +Inventory control problem +Newsvendor model +Extended newsvendor model +Assemble-to-order system +Linear programming decoding +Linear search problem — find a point on a line by moving along the line +Low-rank approximation — find best approximation, constraint is that rank of some matrix is smaller than a given number +Meta-optimization — optimization of the parameters in an optimization method +Multidisciplinary design optimization +Optimal computing budget allocation — maximize the overall simulation efficiency for finding an optimal decision +Paper bag problem +Process optimization +Recursive economics — individuals make a series of two-period optimization decisions over time. +Stigler diet +Space allocation problem +Stress majorization +Trajectory optimization +Transportation theory +Wing-shape optimization + +=== Miscellaneous === +Combinatorial optimization +Dynamic programming +Bellman equation +Hamilton–Jacobi–Bellman equation — continuous-time analogue of Bellman equation +Backward induction — solving dynamic programming problems by reasoning backwards in time +Optimal stopping — choosing the optimal time to take a particular action +Odds algorithm +Robbins' problem +Global optimization: +BRST algorithm +MCS algorithm +Multi-objective optimization — there are multiple conflicting objectives +Benson's algorithm — for linear vector optimization problems +Bilevel optimization — studies problems in which one problem is embedded in another +Optimal substructure +Dykstra's projection algorithm — finds a point in intersection of two convex sets +Algorithmic concepts: +Barrier function +Penalty method +Trust region +Test functions for optimization: +Rosenbrock function — two-dimensional function with a banana-shaped valley +Himmelblau's function — two-dimensional with four local minima, defined by + + + + f + ( + x + , + y + ) + = + ( + + x + + 2 + + + + + y + − + 11 + + ) + + 2 + + + + + ( + x + + + + y + + 2 + + + − + 7 + + ) + + 2 + + + + + {\displaystyle f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}} + + +Rastrigin function — two-dimensional function with many local minima +Shekel function — multimodal and multidimensional +Mathematical Optimization Society + +== Numerical quadrature (integration) == +Numerical integration — the numerical evaluation of an integral \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-6.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-6.md new file mode 100644 index 000000000..bcf8e5096 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-6.md @@ -0,0 +1,120 @@ +--- +title: "List of numerical analysis topics" +chunk: 7/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +Rectangle method — first-order method, based on (piecewise) constant approximation +Trapezoidal rule — second-order method, based on (piecewise) linear approximation +Simpson's rule — fourth-order method, based on (piecewise) quadratic approximation +Adaptive Simpson's method +Boole's rule — sixth-order method, based on the values at five equidistant points +Newton–Cotes formulas — generalizes the above methods +Romberg's method — Richardson extrapolation applied to trapezium rule +Gaussian quadrature — highest possible degree with given number of points +Chebyshev–Gauss quadrature — extension of Gaussian quadrature for integrals with weight (1 − x2)±1/2 on [−1, 1] +Gauss–Hermite quadrature — extension of Gaussian quadrature for integrals with weight exp(−x2) on [−∞, ∞] +Gauss–Jacobi quadrature — extension of Gaussian quadrature for integrals with weight (1 − x)α (1 + x)β on [−1, 1] +Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight exp(−x) on [0, ∞] +Gauss–Kronrod quadrature formula — nested rule based on Gaussian quadrature +Gauss–Kronrod rules +Tanh-sinh quadrature — variant of Gaussian quadrature which works well with singularities at the end points +Clenshaw–Curtis quadrature — based on expanding the integrand in terms of Chebyshev polynomials +Adaptive quadrature — adapting the subintervals in which the integration interval is divided depending on the integrand +Monte Carlo integration — takes random samples of the integrand +See also #Monte Carlo method +Quantized state systems method (QSS) — based on the idea of state quantization +Lebedev quadrature — uses a grid on a sphere with octahedral symmetry +Sparse grid +Coopmans approximation +Numerical differentiation — for fractional-order integrals +Numerical smoothing and differentiation +Adjoint state method — approximates gradient of a function in an optimization problem +Euler–Maclaurin formula + +== Numerical methods for ordinary differential equations == +Numerical methods for ordinary differential equations — the numerical solution of ordinary differential equations (ODEs) + +Euler method — the most basic method for solving an ODE +Explicit and implicit methods — implicit methods need to solve an equation at every step +Backward Euler method — implicit variant of the Euler method +Trapezoidal rule — second-order implicit method +Runge–Kutta methods — one of the two main classes of methods for initial-value problems +Midpoint method — a second-order method with two stages +Heun's method — either a second-order method with two stages, or a third-order method with three stages +Bogacki–Shampine method — a third-order method with four stages (FSAL) and an embedded fourth-order method +Cash–Karp method — a fifth-order method with six stages and an embedded fourth-order method +Dormand–Prince method — a fifth-order method with seven stages (FSAL) and an embedded fourth-order method +Runge–Kutta–Fehlberg method — a fifth-order method with six stages and an embedded fourth-order method +Gauss–Legendre method — family of A-stable method with optimal order based on Gaussian quadrature +Butcher group — algebraic formalism involving rooted trees for analysing Runge–Kutta methods +List of Runge–Kutta methods +Linear multistep method — the other main class of methods for initial-value problems +Backward differentiation formula — implicit methods of order 2 to 6; especially suitable for stiff equations +Numerov's method — fourth-order method for equations of the form + + + + + y + ″ + + = + f + ( + t + , + y + ) + + + {\displaystyle y''=f(t,y)} + + +Predictor–corrector method — uses one method to approximate solution and another one to increase accuracy +General linear methods — a class of methods encapsulating linear multistep and Runge-Kutta methods +Bulirsch–Stoer algorithm — combines the midpoint method with Richardson extrapolation to attain arbitrary order +Exponential integrator — based on splitting ODE in a linear part, which is solved exactly, and a nonlinear part +Methods designed for the solution of ODEs from classical physics: +Newmark-beta method — based on the extended mean-value theorem +Verlet integration — a popular second-order method +Leapfrog integration — another name for Verlet integration +Beeman's algorithm — a two-step method extending the Verlet method +Dynamic relaxation +Geometric integrator — a method that preserves some geometric structure of the equation +Symplectic integrator — a method for the solution of Hamilton's equations that preserves the symplectic structure +Variational integrator — symplectic integrators derived using the underlying variational principle +Semi-implicit Euler method — variant of Euler method which is symplectic when applied to separable Hamiltonians +Energy drift — phenomenon that energy, which should be conserved, drifts away due to numerical errors +Other methods for initial value problems (IVPs): +Bi-directional delay line +Partial element equivalent circuit +Methods for solving two-point boundary value problems (BVPs): +Shooting method +Direct multiple shooting method — divides interval in several subintervals and applies the shooting method on each subinterval +Methods for solving differential-algebraic equations (DAEs), i.e., ODEs with constraints: +Constraint algorithm — for solving Newton's equations with constraints +Pantelides algorithm — for reducing the index of a DEA +Methods for solving stochastic differential equations (SDEs): +Euler–Maruyama method — generalization of the Euler method for SDEs +Milstein method — a method with strong order one +Runge–Kutta method (SDE) — generalization of the family of Runge–Kutta methods for SDEs +Methods for solving integral equations: +Nyström method — replaces the integral with a quadrature rule +Analysis: +Truncation error (numerical integration) — local and global truncation errors, and their relationships +Lady Windermere's Fan (mathematics) — telescopic identity relating local and global truncation errors +Stiff equation — roughly, an ODE for which unstable methods need a very short step size, but stable methods do not +L-stability — method is A-stable and stability function vanishes at infinity +Adaptive stepsize — automatically changing the step size when that seems advantageous +Parareal -- a parallel-in-time integration algorithm + +== Numerical methods for partial differential equations == +Numerical partial differential equations — the numerical solution of partial differential equations (PDEs) + +=== Finite difference methods === +Finite difference method — based on approximating differential operators with difference operators \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-7.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-7.md new file mode 100644 index 000000000..fce8bc147 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-7.md @@ -0,0 +1,78 @@ +--- +title: "List of numerical analysis topics" +chunk: 8/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +Finite difference — the discrete analogue of a differential operator +Finite difference coefficient — table of coefficients of finite-difference approximations to derivatives +Discrete Laplace operator — finite-difference approximation of the Laplace operator +Eigenvalues and eigenvectors of the second derivative — includes eigenvalues of discrete Laplace operator +Kronecker sum of discrete Laplacians — used for Laplace operator in multiple dimensions +Discrete Poisson equation — discrete analogue of the Poisson equation using the discrete Laplace operator +Stencil (numerical analysis) — the geometric arrangements of grid points affected by a basic step of the algorithm +Compact stencil — stencil which only uses a few grid points, usually only the immediate and diagonal neighbours +Higher-order compact finite difference scheme +Non-compact stencil — any stencil that is not compact +Five-point stencil — two-dimensional stencil consisting of a point and its four immediate neighbours on a rectangular grid +Finite difference methods for heat equation and related PDEs: +FTCS scheme (forward-time central-space) — first-order explicit +Crank–Nicolson method — second-order implicit +Finite difference methods for hyperbolic PDEs like the wave equation: +Lax–Friedrichs method — first-order explicit +Lax–Wendroff method — second-order explicit +MacCormack method — second-order explicit +Upwind scheme +Upwind differencing scheme for convection — first-order scheme for convection–diffusion problems +Lax–Wendroff theorem — conservative scheme for hyperbolic system of conservation laws converges to the weak solution +Alternating direction implicit method (ADI) — update using the flow in x-direction and then using flow in y-direction +Nonstandard finite difference scheme +Specific applications: +Finite difference methods for option pricing +Finite-difference time-domain method — a finite-difference method for electrodynamics + +=== Finite element methods, gradient discretisation methods === +Finite element method — based on a discretization of the space of solutions +gradient discretisation method — based on both the discretization of the solution and of its gradient + +Finite element method in structural mechanics — a physical approach to finite element methods +Galerkin method — a finite element method in which the residual is orthogonal to the finite element space +Discontinuous Galerkin method — a Galerkin method in which the approximate solution is not continuous +Rayleigh–Ritz method — a finite element method based on variational principles +Spectral element method — high-order finite element methods +hp-FEM — variant in which both the size and the order of the elements are automatically adapted +Examples of finite elements: +Bilinear quadrilateral element — also known as the Q4 element +Constant strain triangle element (CST) — also known as the T3 element +Quadratic quadrilateral element — also known as the Q8 element +Barsoum elements +Direct stiffness method — a particular implementation of the finite element method, often used in structural analysis +Trefftz method +Finite element updating +Extended finite element method — puts functions tailored to the problem in the approximation space +Functionally graded elements — elements for describing functionally graded materials +Superelement — particular grouping of finite elements, employed as a single element +Interval finite element method — combination of finite elements with interval arithmetic +Discrete exterior calculus — discrete form of the exterior calculus of differential geometry +Modal analysis using FEM — solution of eigenvalue problems to find natural vibrations +Céa's lemma — solution in the finite-element space is an almost best approximation in that space of the true solution +Patch test (finite elements) — simple test for the quality of a finite element +MAFELAP (MAthematics of Finite ELements and APplications) — international conference held at Brunel University +NAFEMS — not-for-profit organisation that sets and maintains standards in computer-aided engineering analysis +Multiphase topology optimisation — technique based on finite elements for determining optimal composition of a mixture +Interval finite element +Applied element method — for simulation of cracks and structural collapse +Wood–Armer method — structural analysis method based on finite elements used to design reinforcement for concrete slabs +Isogeometric analysis — integrates finite elements into conventional NURBS-based CAD design tools +Loubignac iteration +Stiffness matrix — finite-dimensional analogue of differential operator +Combination with meshfree methods: +Weakened weak form — form of a PDE that is weaker than the standard weak form +G space — functional space used in formulating the weakened weak form +Smoothed finite element method +Variational multiscale method +List of finite element software packages \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-8.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-8.md new file mode 100644 index 000000000..097df13c0 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-8.md @@ -0,0 +1,81 @@ +--- +title: "List of numerical analysis topics" +chunk: 9/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +=== Other methods === +Spectral method — based on the Fourier transformation +Pseudo-spectral method +Method of lines — reduces the PDE to a large system of ordinary differential equations +Boundary element method (BEM) — based on transforming the PDE to an integral equation on the boundary of the domain +Interval boundary element method — a version using interval arithmetics +Analytic element method — similar to the boundary element method, but the integral equation is evaluated analytically +Finite volume method — based on dividing the domain in many small domains; popular in computational fluid dynamics +Godunov's scheme — first-order conservative scheme for fluid flow, based on piecewise constant approximation +MUSCL scheme — second-order variant of Godunov's scheme +AUSM — advection upstream splitting method +Flux limiter — limits spatial derivatives (fluxes) in order to avoid spurious oscillations +Riemann solver — a solver for Riemann problems (a conservation law with piecewise constant data) +Properties of discretization schemes — finite volume methods can be conservative, bounded, etc. +Discrete element method — a method in which the elements can move freely relative to each other +Extended discrete element method — adds properties such as strain to each particle +Movable cellular automaton — combination of cellular automata with discrete elements +Meshfree methods — does not use a mesh, but uses a particle view of the field +Discrete least squares meshless method — based on minimization of weighted summation of the squared residual +Diffuse element method +Finite pointset method — represent continuum by a point cloud +Moving Particle Semi-implicit Method +Method of fundamental solutions (MFS) — represents solution as linear combination of fundamental solutions +Variants of MFS with source points on the physical boundary: +Boundary knot method (BKM) +Boundary particle method (BPM) +Regularized meshless method (RMM) +Singular boundary method (SBM) +Methods designed for problems from electromagnetics: +Finite-difference time-domain method — a finite-difference method +Rigorous coupled-wave analysis — semi-analytical Fourier-space method based on Floquet's theorem +Transmission-line matrix method (TLM) — based on analogy between electromagnetic field and mesh of transmission lines +Uniform theory of diffraction — specifically designed for scattering problems +Particle-in-cell — used especially in fluid dynamics +Multiphase particle-in-cell method — considers solid particles as both numerical particles and fluid +High-resolution scheme +Shock capturing method +Vorticity confinement — for vortex-dominated flows in fluid dynamics, similar to shock capturing +Split-step method +Fast marching method +Orthogonal collocation +Lattice Boltzmann methods — for the solution of the Navier-Stokes equations +Roe solver — for the solution of the Euler equation +Relaxation (iterative method) — a method for solving elliptic PDEs by converting them to evolution equations +Broad classes of methods: +Mimetic methods — methods that respect in some sense the structure of the original problem +Multiphysics — models consisting of various submodels with different physics +Immersed boundary method — for simulating elastic structures immersed within fluids +Multisymplectic integrator — extension of symplectic integrators, which are for ODEs +Stretched grid method — for problems solution that can be related to an elastic grid behavior. + +=== Techniques for improving these methods === +Multigrid method — uses a hierarchy of nested meshes to speed up the methods +Domain decomposition methods — divides the domain in a few subdomains and solves the PDE on these subdomains +Additive Schwarz method +Abstract additive Schwarz method — abstract version of additive Schwarz without reference to geometric information +Balancing domain decomposition method (BDD) — preconditioner for symmetric positive definite matrices +Balancing domain decomposition by constraints (BDDC) — further development of BDD +Finite element tearing and interconnect (FETI) +FETI-DP — further development of FETI +Fictitious domain method — preconditioner constructed with a structured mesh on a fictitious domain of simple shape +Mortar methods — meshes on subdomain do not mesh +Neumann–Dirichlet method — combines Neumann problem on one subdomain with Dirichlet problem on other subdomain +Neumann–Neumann methods — domain decomposition methods that use Neumann problems on the subdomains +Poincaré–Steklov operator — maps tangential electric field onto the equivalent electric current +Schur complement method — early and basic method on subdomains that do not overlap +Schwarz alternating method — early and basic method on subdomains that overlap +Coarse space — variant of the problem which uses a discretization with fewer degrees of freedom +Adaptive mesh refinement — uses the computed solution to refine the mesh only where necessary +Fast multipole method — hierarchical method for evaluating particle-particle interactions +Perfectly matched layer — artificial absorbing layer for wave equations, used to implement absorbing boundary conditions \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-9.md b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-9.md new file mode 100644 index 000000000..a263c9dc6 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-9.md @@ -0,0 +1,182 @@ +--- +title: "List of numerical analysis topics" +chunk: 10/10 +source: "https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:57.787804+00:00" +instance: "kb-cron" +--- + +=== Grids and meshes === +Grid classification / Types of mesh: +Polygon mesh — consists of polygons in 2D or 3D +Triangle mesh — consists of triangles in 2D or 3D +Triangulation (geometry) — subdivision of given region in triangles, or higher-dimensional analogue +Nonobtuse mesh — mesh in which all angles are less than or equal to 90° +Point-set triangulation — triangle mesh such that given set of point are all a vertex of a triangle +Polygon triangulation — triangle mesh inside a polygon +Delaunay triangulation — triangulation such that no vertex is inside the circumcentre of a triangle +Constrained Delaunay triangulation — generalization of the Delaunay triangulation that forces certain required segments into the triangulation +Pitteway triangulation — for any point, triangle containing it has nearest neighbour of the point as a vertex +Minimum-weight triangulation — triangulation of minimum total edge length +Kinetic triangulation — a triangulation that moves over time +Triangulated irregular network +Quasi-triangulation — subdivision into simplices, where vertices are not points but arbitrary sloped line segments +Volume mesh — consists of three-dimensional shapes +Regular grid — consists of congruent parallelograms, or higher-dimensional analogue +Unstructured grid +Geodesic grid — isotropic grid on a sphere +Mesh generation +Image-based meshing — automatic procedure of generating meshes from 3D image data +Marching cubes — extracts a polygon mesh from a scalar field +Parallel mesh generation +Ruppert's algorithm — creates quality Delauney triangularization from piecewise linear data +Subdivisions: +Apollonian network — undirected graph formed by recursively subdividing a triangle +Barycentric subdivision — standard way of dividing arbitrary convex polygons into triangles, or the higher-dimensional analogue +Improving an existing mesh: +Chew's second algorithm — improves Delauney triangularization by refining poor-quality triangles +Laplacian smoothing — improves polynomial meshes by moving the vertices +Jump-and-Walk algorithm — for finding triangle in a mesh containing a given point +Spatial twist continuum — dual representation of a mesh consisting of hexahedra +Pseudotriangle — simply connected region between any three mutually tangent convex sets +Simplicial complex — all vertices, line segments, triangles, tetrahedra, ..., making up a mesh + +=== Analysis === +Lax equivalence theorem — a consistent method is convergent if and only if it is stable +Courant–Friedrichs–Lewy condition — stability condition for hyperbolic PDEs +Von Neumann stability analysis — all Fourier components of the error should be stable +Numerical diffusion — diffusion introduced by the numerical method, above to that which is naturally present +False diffusion +Numerical dispersion +Numerical resistivity — the same, with resistivity instead of diffusion +Weak formulation — a functional-analytic reformulation of the PDE necessary for some methods +Total variation diminishing — property of schemes that do not introduce spurious oscillations +Godunov's theorem — linear monotone schemes can only be of first order +Motz's problem — benchmark problem for singularity problems + +== Monte Carlo method == +Variants of the Monte Carlo method: +Direct simulation Monte Carlo +Quasi-Monte Carlo method +Markov chain Monte Carlo +Metropolis–Hastings algorithm +Multiple-try Metropolis — modification which allows larger step sizes +Wang and Landau algorithm — extension of Metropolis Monte Carlo +Equation of State Calculations by Fast Computing Machines — 1953 article proposing the Metropolis Monte Carlo algorithm +Multicanonical ensemble — sampling technique that uses Metropolis–Hastings to compute integrals +Gibbs sampling +Coupling from the past +Reversible-jump Markov chain Monte Carlo +Dynamic Monte Carlo method +Kinetic Monte Carlo +Gillespie algorithm +Particle filter +Auxiliary particle filter +Reverse Monte Carlo +Demon algorithm +Pseudo-random number sampling +Inverse transform sampling — general and straightforward method but computationally expensive +Rejection sampling — sample from a simpler distribution but reject some of the samples +Ziggurat algorithm — uses a pre-computed table covering the probability distribution with rectangular segments +For sampling from a normal distribution: +Box–Muller transform +Marsaglia polar method +Convolution random number generator — generates a random variable as a sum of other random variables +Indexed search +Variance reduction techniques: +Antithetic variates +Control variates +Importance sampling +Stratified sampling +VEGAS algorithm +Low-discrepancy sequence +Constructions of low-discrepancy sequences +Event generator +Parallel tempering +Umbrella sampling — improves sampling in physical systems with significant energy barriers +Hybrid Monte Carlo +Ensemble Kalman filter — recursive filter suitable for problems with a large number of variables +Transition path sampling +Walk-on-spheres method — to generate exit-points of Brownian motion from bounded domains +Applications: +Ensemble forecasting — produce multiple numerical predictions from slightly initial conditions or parameters +Bond fluctuation model — for simulating the conformation and dynamics of polymer systems +Iterated filtering +Metropolis light transport +Monte Carlo localization — estimates the position and orientation of a robot +Monte Carlo methods for electron transport +Monte Carlo method for photon transport +Monte Carlo methods in finance +Monte Carlo methods for option pricing +Quasi-Monte Carlo methods in finance +Monte Carlo molecular modeling +Path integral molecular dynamics — incorporates Feynman path integrals +Quantum Monte Carlo +Diffusion Monte Carlo — uses a Green function to solve the Schrödinger equation +Gaussian quantum Monte Carlo +Path integral Monte Carlo +Reptation Monte Carlo +Variational Monte Carlo +Methods for simulating the Ising model: +Swendsen–Wang algorithm — entire sample is divided into equal-spin clusters +Wolff algorithm — improvement of the Swendsen–Wang algorithm +Metropolis–Hastings algorithm +Auxiliary field Monte Carlo — computes averages of operators in many-body quantum mechanical problems +Cross-entropy method — for multi-extremal optimization and importance sampling +Also see the list of statistics topics + +== Applications == +Computational physics +Computational electromagnetics +Computational fluid dynamics (CFD) +Numerical methods in fluid mechanics +Large eddy simulation +Smoothed-particle hydrodynamics +Aeroacoustic analogy — used in numerical aeroacoustics to reduce sound sources to simple emitter types +Stochastic Eulerian Lagrangian method — uses Eulerian description for fluids and Lagrangian for structures +Explicit algebraic stress model +Computational magnetohydrodynamics (CMHD) — studies electrically conducting fluids +Climate model +Numerical weather prediction +Geodesic grid +Celestial mechanics +Numerical model of the Solar System +Quantum jump method — used for simulating open quantum systems, operates on wave function +Dynamic design analysis method (DDAM) — for evaluating effect of underwater explosions on equipment +Computational chemistry +Cell lists +Coupled cluster +Density functional theory +DIIS — direct inversion in (or of) the iterative subspace +Computational sociology +Computational statistics + +== Software == +For a large list of software, see the list of numerical-analysis software. + +== Journals == +Acta Numerica +Mathematics of Computation (published by the American Mathematical Society) +Journal of Computational and Applied Mathematics +BIT Numerical Mathematics +Numerische Mathematik +Journals from the Society for Industrial and Applied Mathematics +SIAM Journal on Numerical Analysis +SIAM Journal on Scientific Computing + +== Researchers == +Cleve Moler +Gene H. Golub +James H. Wilkinson +Margaret H. Wright +Nicholas J. Higham +Nick Trefethen +Peter Lax +Richard S. Varga +Ulrich W. Kulisch +Vladik Kreinovich +Chi-Wang Shu + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics-0.md b/data/en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics-0.md new file mode 100644 index 000000000..06171c983 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics-0.md @@ -0,0 +1,37 @@ +--- +title: "List of numerical computational geometry topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:58.970321+00:00" +instance: "kb-cron" +--- + +List of numerical computational geometry topics enumerates the topics of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis. This area is also called "machine geometry", computer-aided geometric design, and geometric modelling. +See List of combinatorial computational geometry topics for another flavor of computational geometry that states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character. + + +== Curves == +In the list of curves topics, the following ones are fundamental to geometric modelling. + +Parametric curve +Bézier curve +Spline +Hermite spline +Beta spline +B-spline +Higher-order spline +NURBS +Contour line + + +== Surfaces == +Bézier surface +Isosurface +Parametric surface + + +== Other == +Level-set method +Computational topology \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_numerical_libraries-0.md b/data/en.wikipedia.org/wiki/List_of_numerical_libraries-0.md new file mode 100644 index 000000000..a4f4804f7 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_numerical_libraries-0.md @@ -0,0 +1,67 @@ +--- +title: "List of numerical libraries" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_numerical_libraries" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:55.235107+00:00" +instance: "kb-cron" +--- + +This is a list of numerical libraries, which are libraries used in software development for performing numerical calculations. It is not a complete listing but is instead a list of numerical libraries with articles on Wikipedia, with few exceptions. +The choice of a typical library depends on a range of requirements such as: desired features (e.g. large dimensional linear algebra, parallel computation, partial differential equations), licensing, readability of API, portability or platform/compiler dependence (e.g. Linux, Windows, Visual C++, GCC), performance, ease-of-use, continued support from developers, standard compliance, specialized optimization in code for specific application scenarios or even the size of the code-base to be installed. + + +== Multi-language == + + +== C == + + +== C++ == + + +== Delphi == +ALGLIB - an open source numerical analysis library. + + +== .NET Framework languages: C#, F#, VB.NET and PowerShell == + + +== Fortran == + + +== Java == + + +== OCaml == +OCaml programming language has support for array programming in the standard library, also with a specific module named bigarrays for multi-dimensional, numerical arrays, with both C and Fortran layout options. A comprehensive support of numerical computations is provided by the library Owl Scientific Computing which provides methods for statistics, linear algebra (using OpenBLAS), differential equations, algorithmic differentiation, Fourier fast transform, or deep neural networks. Other numerical libraries in OCaml are Lacaml that interfaces BLAS and LAPACK Fortran/C libraries, L-BFGS-ocaml (OCaml bindings for L-BFGS). For visualization there are libraries for plotting using PLplot, gnuplot or matplotlib. + + +== Perl == +Perl Data Language gives standard Perl the ability to compactly store and speedily manipulate the large N-dimensional data arrays. It can perform complex and matrix maths, and has interfaces for the GNU Scientific Library, LINPACK, PROJ, and plotting with PGPLOT. There are libraries on CPAN adding support for the linear algebra library LAPACK, the Fourier transform library FFTW, and plotting with gnuplot, and PLplot. + + +== Python == + + +== Others == +XNUMBERS – multi-precision floating-Point computing and numerical methods for Microsoft Excel. +INTLAB – interval arithmetic library for MATLAB. + + +== See also == +List of computer algebra systems +List of information graphics software +List of numerical analysis programming languages +List of numerical-analysis software +List of open source code libraries +List of optimization software +List of statistical software + + +== References == + + +== External links == +The Math Forum - Math Libraries, an extensive list of mathematical libraries with short descriptions \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_operator_splitting_topics-0.md b/data/en.wikipedia.org/wiki/List_of_operator_splitting_topics-0.md new file mode 100644 index 000000000..dbd5aa76e --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_operator_splitting_topics-0.md @@ -0,0 +1,23 @@ +--- +title: "List of operator splitting topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_operator_splitting_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:00.110845+00:00" +instance: "kb-cron" +--- + +This is a list of operator splitting topics. + + +== General == +Alternating direction implicit method — finite difference method for parabolic, hyperbolic, and elliptic partial differential equations +GRADELA — simple gradient elasticity model +Matrix splitting — general method of splitting a matrix operator into a sum or difference of matrices +Paul Tseng — resolved question on convergence of matrix splitting algorithms +PISO algorithm — pressure-velocity calculation for Navier-Stokes equations +Projection method (fluid dynamics) — computational fluid dynamics method +Reactive transport modeling in porous media — modeling of chemical reactions and fluid flow through the Earth's crust +Richard S. Varga — developed matrix splitting +Strang splitting — specific numerical method for solving differential equations using operator splitting \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_order_structures_in_mathematics-0.md b/data/en.wikipedia.org/wiki/List_of_order_structures_in_mathematics-0.md new file mode 100644 index 000000000..166d371f4 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_order_structures_in_mathematics-0.md @@ -0,0 +1,27 @@ +--- +title: "List of order structures in mathematics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_order_structures_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:03.849108+00:00" +instance: "kb-cron" +--- + +In mathematics, and more specifically in order theory, several different types of ordered set have been studied. +They include: + +Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise +Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. +Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be +Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities) +Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions +Total orders, orderings that specify, for every two distinct elements, which one is less than the other +Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities) +Well-orders, total orders in which every non-empty subset has a least element +Well-quasi-orderings, a class of preorders generalizing the well-orders + + +== See also == +Glossary of order theory +List of order theory topics \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_order_theory_topics-0.md b/data/en.wikipedia.org/wiki/List_of_order_theory_topics-0.md new file mode 100644 index 000000000..dcc5ac98f --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_order_theory_topics-0.md @@ -0,0 +1,170 @@ +--- +title: "List of order theory topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_order_theory_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:02.632494+00:00" +instance: "kb-cron" +--- + +Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. +An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value and mathematical optimization. + + +== Overview == +Partially ordered set +Preorder +Totally ordered set +Total preorder +Chain +Trichotomy +Extended real number line +Antichain +Strict order +Hasse diagram +Directed acyclic graph +Duality (order theory) +Product order + + +== Distinguished elements of partial orders == +Greatest element (maximum, top, unit), Least element (minimum, bottom, zero) +Maximal element, minimal element +Upper bound +Least upper bound (supremum, join) +Greatest lower bound (infimum, meet) +Limit superior and limit inferior +Irreducible element +Prime element +Compact element + + +== Subsets of partial orders == +Cofinal and coinitial set, sometimes also called dense +Meet-dense set and join-dense set +Linked set (upwards and downwards) +Directed set (upwards and downwards) +centered and σ-centered set +Net (mathematics) +Upper set and lower set +Ideal and filter +Ultrafilter + + +== Special types of partial orders == +Completeness (order theory) +Dense order +Distributivity (order theory) +Modular lattice +Distributive lattice +Completely distributive lattice +Ascending chain condition +Infinite descending chain +Countable chain condition, often abbreviated as ccc +Knaster's condition, sometimes denoted property (K) + + +=== Well-orders === +Well-founded relation +Ordinal number +Well-quasi-ordering + + +=== Completeness properties === +Semilattice +Lattice +(Directed) complete partial order, (d)cpo +Bounded complete +Complete lattice +Knaster–Tarski theorem +Infinite divisibility + + +=== Orders with further algebraic operations === +Heyting algebra +Relatively complemented lattice +Complete Heyting algebra +Pointless topology +MV-algebra +Ockham algebras: +Stone algebra +De Morgan algebra +Kleene algebra (with involution) +Łukasiewicz–Moisil algebra +Boolean algebra (structure) +Boolean ring +Complete Boolean algebra +Orthocomplemented lattice +Quantale + + +=== Orders in algebra === +Partially ordered monoid +Ordered group +Archimedean property +Ordered ring +Ordered field +Artinian ring +Noetherian +Linearly ordered group +Monomial order +Weak order of permutations +Bruhat order on a Coxeter group +Incidence algebra + + +== Functions between partial orders == +Monotonic +Pointwise order of functions +Galois connection +Order embedding +Order isomorphism +Closure operator +Functions that preserve suprema/infima + + +== Completions and free constructions == +Dedekind completion +Ideal completion + + +== Domain theory == + +Way-below relation +Continuous poset +Continuous lattice +Algebraic poset +Scott domain +Algebraic lattice +Scott information system +Powerdomain +Scott topology +Scott continuity + + +== Orders in mathematical logic == +Lindenbaum algebra +Zorn's lemma +Hausdorff maximality theorem +Boolean prime ideal theorem +Ultrafilter +Ultrafilter lemma +Tree (set theory) +Tree (descriptive set theory) +Suslin's problem +Absorption law +Prewellordering + + +== Orders in topology == +Stone duality +Stone's representation theorem for Boolean algebras +Specialization (pre)order +Order topology of a total order (open interval topology) +Alexandrov topology +Upper topology +Scott topology +Scott continuity +Lawson topology +Finer topology \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_partial_differential_equation_topics-0.md b/data/en.wikipedia.org/wiki/List_of_partial_differential_equation_topics-0.md new file mode 100644 index 000000000..66852164e --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_partial_differential_equation_topics-0.md @@ -0,0 +1,75 @@ +--- +title: "List of partial differential equation topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_partial_differential_equation_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:05.112454+00:00" +instance: "kb-cron" +--- + +This is a list of partial differential equation topics. + + +== General topics == +Partial differential equation +Nonlinear partial differential equation +list of nonlinear partial differential equations +Boundary condition +Boundary value problem +Dirichlet problem, Dirichlet boundary condition +Neumann boundary condition +Stefan problem +Wiener–Hopf problem +Separation of variables +Green's function +Elliptic partial differential equation +Singular perturbation +Cauchy–Kovalevskaya theorem +H-principle +Atiyah–Singer index theorem +Bäcklund transform +Viscosity solution +Weak solution +Loewy decomposition of linear differential equations + + +== Specific partial differential equations == +Broer–Kaup equations +Burgers' equation +Euler equations +Fokker–Planck equation +Hamilton–Jacobi equation, Hamilton–Jacobi–Bellman equation +Heat equation +Laplace's equation +Laplace operator +Harmonic function +Spherical harmonic +Poisson integral formula +Klein–Gordon equation +Korteweg–de Vries equation +Modified KdV–Burgers equation +Maxwell's equations +Navier–Stokes equations +Poisson's equation +Primitive equations (hydrodynamics) +Schrödinger equation +Wave equation + + +== Numerical methods for PDEs == +Finite difference +Finite element method +Finite volume method +Boundary element method +Multigrid +Spectral method +Computational fluid dynamics +Alternating direction implicit + + +== Related areas of mathematics == +Calculus of variations +Harmonic analysis +Ordinary differential equation +Sobolev space \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_partition_topics-0.md b/data/en.wikipedia.org/wiki/List_of_partition_topics-0.md new file mode 100644 index 000000000..5c82c56ed --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_partition_topics-0.md @@ -0,0 +1,90 @@ +--- +title: "List of partition topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_partition_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:06.339911+00:00" +instance: "kb-cron" +--- + +Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are + +partition of a set or an ordered partition of a set, +partition of a graph, +partition of an integer, +partition of an interval, +partition of unity, +partition of a matrix; see block matrix, and +partition of the sum of squares in statistics problems, especially in the analysis of variance, +quotition and partition, two ways of viewing the operation of division of integers. + + +== Integer partitions == + +Composition (combinatorics) +Ewens's sampling formula +Ferrers graph +Glaisher's theorem +Landau's function +Partition function (number theory) +Pentagonal number theorem +Plane partition +Quotition and partition +Rank of a partition +Crank of a partition +Solid partition +Young tableau +Young's lattice + + +== Set partitions == + +Bell number +Bell polynomials +Dobinski's formula +Cumulant +Data clustering +Equivalence relation +Exact cover +Knuth's Algorithm X +Dancing Links +Exponential formula +Faà di Bruno's formula +Feshbach–Fano partitioning +Foliation +Frequency partition +Graph partition +Kernel of a function +Lamination (topology) +Matroid partitioning +Multipartition +Multiplicative partition +Noncrossing partition +Ordered partition of a set +Partition calculus +Partition function (quantum field theory) +Partition function (statistical mechanics) +Derivation of the partition function +Partition of an interval +Partition of a set +Ordered partition +Partition refinement +Disjoint-set data structure +Partition problem +3-partition problem +Partition topology +Quotition and partition +Recursive partitioning +Stirling number +Stirling transform +Stratification (mathematics) +Tverberg partition +Twelvefold way + + +=== In probability and stochastic processes === +Chinese restaurant process +Dobinski's formula +Ewens's sampling formula +Law of total cumulance \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_periodic_functions-0.md b/data/en.wikipedia.org/wiki/List_of_periodic_functions-0.md new file mode 100644 index 000000000..5e00a0943 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_periodic_functions-0.md @@ -0,0 +1,133 @@ +--- +title: "List of periodic functions" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_periodic_functions" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:07.555333+00:00" +instance: "kb-cron" +--- + +This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions. + + +== Smooth functions == +All trigonometric functions listed have period + + + + 2 + π + + + {\displaystyle 2\pi } + +, unless otherwise stated. For the following trigonometric functions: + +Un is the nth up/down number, +Bn is the nth Bernoulli number +in Jacobi elliptic functions, + + + + q + = + + e + + − + π + + + + K + ( + 1 + − + m + ) + + + K + ( + m + ) + + + + + + + + {\displaystyle q=e^{-\pi {\frac {K(1-m)}{K(m)}}}} + + + +== Non-smooth functions == +The following functions have period + + + + p + + + {\displaystyle p} + + and take + + + + x + + + {\displaystyle x} + + as their argument. The symbol + + + + ⌊ + n + ⌋ + + + {\displaystyle \lfloor n\rfloor } + + is the floor function of + + + + n + + + {\displaystyle n} + + and + + + + sgn + + + {\displaystyle \operatorname {sgn} } + + is the sign function. + +K means Elliptic integral K(m) + + +== Vector-valued functions == +Epitrochoid +Epicycloid (special case of the epitrochoid) +Limaçon (special case of the epitrochoid) +Hypotrochoid +Hypocycloid (special case of the hypotrochoid) +Spirograph (special case of the hypotrochoid) + + +== Doubly periodic functions == +Jacobi's elliptic functions +Weierstrass's elliptic function + + +== Notes == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_permutation_topics-0.md b/data/en.wikipedia.org/wiki/List_of_permutation_topics-0.md new file mode 100644 index 000000000..fdf1fad97 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_permutation_topics-0.md @@ -0,0 +1,164 @@ +--- +title: "List of permutation topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_permutation_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:08.751910+00:00" +instance: "kb-cron" +--- + +This is a list of topics on mathematical permutations. + + +== Particular kinds of permutations == +Alternating permutation +Circular shift +Cyclic permutation +Derangement +Even and odd permutations—see Parity of a permutation +Josephus permutation +Parity of a permutation +Separable permutation +Stirling permutation +Superpattern +Transposition (mathematics) +Unpredictable permutation + + +== Combinatorics of permutations == +Bijection +Combination +Costas array +Cycle index +Cycle notation +Cycles and fixed points +Cyclic order +Direct sum of permutations +Enumerations of specific permutation classes +Factorial +Falling factorial +Permutation matrix +Generalized permutation matrix +Inversion (discrete mathematics) +Major index +Ménage problem +Permutation graph +Permutation pattern +Permutation polynomial +Permutohedron +Rencontres numbers +Robinson–Schensted correspondence +Sum of permutations: +Direct sum of permutations +Skew sum of permutations +Stanley–Wilf conjecture +Symmetric function +Szymanski's conjecture +Twelvefold way + + +== Permutation groups and other algebraic structures == + + +=== Groups === + +Alternating group +Automorphisms of the symmetric and alternating groups +Block (permutation group theory) +Cayley's theorem +Cycle index +Frobenius group +Galois group of a polynomial +Jucys–Murphy element +Landau's function +Oligomorphic group +O'Nan–Scott theorem +Parker vector +Permutation group +Place-permutation action +Primitive permutation group +Rank 3 permutation group +Representation theory of the symmetric group +Schreier vector +Strong generating set +Symmetric group +Symmetric inverse semigroup +Weak order of permutations +Wreath product +Young symmetrizer +Zassenhaus group +Zolotarev's lemma + + +=== Other algebraic structures === +Burnside ring + + +== Mathematical analysis == +Conditionally convergent series +Riemann series theorem +Lévy–Steinitz theorem + + +== Mathematics applicable to physical sciences == +Antisymmetrizer +Identical particles +Levi-Civita symbol + + +== Number theory == +Permutable prime + + +== Algorithms and information processing == +Bit-reversal permutation +Claw-free permutation +Heap's algorithm +Permutation automaton +Schreier vector +Sorting algorithm +Sorting network +Substitution–permutation network +Steinhaus–Johnson–Trotter algorithm +Tompkins–Paige algorithm + + +=== Cryptography === +Permutation box +Substitution box +Permutation cipher +Substitution cipher +Transposition cipher + + +== Probability, stochastic processes, and statistics == +Combinatorial data analysis +Ewens' sampling formula +Fisher–Yates shuffle +Order statistic +Permutation test +Permutational analysis of variance +Rankit +Resampling (statistics) +Seriation (statistics) + + +=== Random permutations === + +Golomb–Dickman constant +Random permutation +Random permutation statistics + + +== Music == + +Change ringing +Method ringing +Permutation (music) + + +== Games == +Faro shuffle +Fifteen puzzle +Shuffling \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_planar_symmetry_groups-0.md b/data/en.wikipedia.org/wiki/List_of_planar_symmetry_groups-0.md new file mode 100644 index 000000000..272f91fda --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_planar_symmetry_groups-0.md @@ -0,0 +1,56 @@ +--- +title: "List of planar symmetry groups" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_planar_symmetry_groups" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:12.595642+00:00" +instance: "kb-cron" +--- + +This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. +There are three kinds of symmetry groups of the plane: + +2 families of rosette groups – 2D point groups +7 frieze groups – 2D line groups +17 wallpaper groups – 2D space groups. + + +== Rosette groups == +There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group. + + +== Frieze groups == +The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each. + + +== Wallpaper groups == +The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular). +The p1 and p2 groups, with no reflectional symmetry, are repeated in all classes. The related pure reflectional Coxeter group are given with all classes except oblique. + + +== Wallpaper subgroup relationships == + + +== See also == +List of spherical symmetry groups +Orbifold notation#Hyperbolic plane - Hyperbolic symmetry groups + + +== Notes == + + +== References == +The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5 (Orbifold notation for polyhedra, Euclidean and hyperbolic tilings) +On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5 +Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2] +(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] +(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591] +(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45] +Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9. +N. W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 12: Euclidean Symmetry Groups + + +== External links == +"Conway's manuscript" on Orbifold notation (Notation changed from this original, x is now used in place of open-dot, and o is used in place of the closed dot) +The 17 Wallpaper Groups \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_polygons-0.md b/data/en.wikipedia.org/wiki/List_of_polygons-0.md new file mode 100644 index 000000000..84646d612 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_polygons-0.md @@ -0,0 +1,43 @@ +--- +title: "List of polygons" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_polygons" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:15.076521+00:00" +instance: "kb-cron" +--- + +In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides, and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners. +The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well. + + +== Greek numbers == +Polygons are primarily named by prefixes from Ancient Greek numbers. + + +== Systematic polygon names == +To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" connector is not included by some authors. + +Extending the system up to 999 is expressed with these prefixes. + + +== List of n-gons by Greek numerical prefixes == + + +== See also == +Platonic solid +Dice +List of polygons, polyhedra and polytopes +Circle +Ellipse +Shape + + +== Notes == + + +== References == + +NAMING POLYGONS +Benjamin Franklin Finkel, A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems, 1888 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-0.md b/data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-0.md new file mode 100644 index 000000000..9453ae8f8 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-0.md @@ -0,0 +1,48 @@ +--- +title: "List of polyhedral stellations" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/List_of_polyhedral_stellations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:16.272155+00:00" +instance: "kb-cron" +--- + +In three-dimensional space, applying the operation of stellation to a polyhedron extends its faces (or edges and planes) until they generate new vertices that bound a newly formed figure. Stellation represents the dual action to faceting a polyhedron. +Originating from studies of star polyhedra in 14th century Europe, a proper mathematical account of polyhedral stellations was given by Johannes Kepler in his 1619 classic work, Harmonices Mundi. Progress later ensued on detailing and enumerating stellations of prominent stars, such as the regular Kepler-Poinsot polyhedra, with developments on different stellation methods occurring in the 1900s, principally from Coxeter et al. (1938) and soon afterward, Pawley (1975). +A short generalized table of the most notable polyhedral stellations belonging to convex uniform polyhedra is provided, with complete sets of stellations for the Platonic solids (including the fifty-nine icosahedral stellations), as well as for select Catalan solids (e.g., the rhombic dodecahedron and the rhombic triacontahedron). Stellations of non-convex uniform polyhedra of structure with facial planes passing through their centers (i.e., hemipolyhedral), which render unbounded vertices, are also included; these are stellations to infinity — per Wenninger (1983) — conforming to extensions on traditional definitions of polyhedra. + +== Background == + +=== Star polytopes === + +Experimentation with star polygons and star polyhedra since the fourteenth century AD led the way to formal theories for stellating polyhedra: + +It was in 1619 that the first geometric description of a stellation was given, by Johannes Kepler in his landmark book, Harmonices Mundi: the process of extending the edges (or faces) of a figure until new vertices are generated, which collectively form a new figure. Using this method, Kepler was able to discover the small stellated dodecahedron and the great stellated dodecahedron. In 1809, Louis Poinsot rediscovered Kepler's star figures and discovered a further two, the great icosahedron and great dodecahedron; he achieved this by experimenting assembling regular star polygons and convex regular polygons on vertices of the regular icosahedron and dodecahedron (i.e., pentagons, pentagrams and equilateral triangles). Three years later, Augustin-Louis Cauchy proved, using concepts of symmetry, that these four stellations are the only regular star-polyhedra, eventually termed the Kepler–Poinsot polyhedra. As with most non-convex polyhedra including stellations and other star polyhedra, the Kepler-Poinsot polyhedra, with regular self-intersecting faces, are now known to be inequivalent to the topological sphere as a simple connected surface (this is in contrast with the traditional convex uniform polyhedra and their corresponding homotopy invariance). + +=== Stellation process === + +Coxeter et al. (1938) details, for the first time, all stellations of the regular icosahedron with specific rules proposed by J. C. P. Miller. Generalizing these (Miller's rules) for stellating any uniform polyhedron yields the following: + +These rules are ideal for stellating smaller uniform solids, such as the regular polyhedra; however, when assessing stellations of other larger uniform polyhedra, this method can quickly become overwhelming. (For example, there are a total of 358,833,072 stellations to the rhombic triacontahedron using this set of rules.) To address this, Pawley (1973) proposed a set of rules that restrict the number of stellations to a more manageable set of fully supported stellations that are radially convex, such that an outward ray from the center of the original polyhedron (in any direction) crosses the stellation surface only once (that is to say, all visible parts of a face are seen from the same side). +In the 1948 first edition of Regular Polytopes, H. S. M. Coxeter describes the stellation process as the reciprocal action to faceting, identifying the four Kepler-Poinsot polyhedra as stellations and facetings of the regular dodecahedron and icosahedron. He specifies the construction of a star polyhedron as a stellation of its core (with congruent face-planes), or by faceting its case — the former requires the addition of solid pieces that generate new vertices, while the latter involves the removal of solid pieces, without forming any new vertices (the core of a star polyhedron or compound is the largest convex solid that can be drawn inside them, while their case is the smallest convex solid that contains them). + +== Lists == +Lists for polyhedral stellations contain non-convex polyhedra; some of the most notable examples include: + +Stellations that topologically do not fit into standard definitions of uniform polyhedra are listed further down (i.e. stellations of hemipolyhedra). + +KEY +* Kepler-Poinsot polyhedron (star polyhedron with regular facets) + +† Regular compound polyhedron (vertex, edge, and face-transitive compound) + +‡ Compound of dual regular polyhedra (Platonic or Kepler-Poinsot duals) + +¶ First/outermost stellation of stellation core +"Stellation core" describes a stellated regular (Platonic), semi-regular (Archimedean), or dual to a semi-regular (Catalan) figure."Face diagram" represents the lines of intersection from extended polyhedral edges that are used in the stellation process."Refs." (references) such as indexes found in Coxeter et al. (1999) using the Crennells' illustration notation (C), and Wenninger (1989) (W). + +=== Enumerations === +The table below is adapted from research by Robert Webb, using his program Stella. It enumerates fully supported stellations and stellations per Miller's process, of the regular Platonic solids as well as the semi-regular Archimedean solids and their Catalan duals. In this list, the elongated square gyrobicupola and its dual polyhedron are not included (these are sometimes considered a fourteenth Archimedean and Catalan solid, respectively). The base polyhedron stellation core is included as a zeroth convex stellation following the Crennells' indexing, with stellation totals the sum of chiral and reflexible stellations (a "chiral" stellation is enantiomorphous, while a "reflexible" stellation maintains the same group symmetry as its stellation core, yet remains achiral – for a count of these separately, visit the parent source). + +"Cell types" are sets of symmetrically equivalent stellation cells, where "stellation cells" are the minimal 3D spaces enclosed on all sides by the original polyhedron's extended facial planes."?" denotes an unknown total number of stellations; however, the number of reflexible stellations are sometimes known for these (where chiral stellations are excluded). \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-1.md b/data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-1.md new file mode 100644 index 000000000..007a0e58a --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-1.md @@ -0,0 +1,60 @@ +--- +title: "List of polyhedral stellations" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/List_of_polyhedral_stellations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:16.272155+00:00" +instance: "kb-cron" +--- + +== Stellations of Platonic solids == +Only three of the five Platonic solids produce stellations: the regular octahedron, regular dodecahedron, and regular icosahedron. The regular tetrahedron and cube are unable to generate stellations when extending their faces, since extending their vertices only form one possible convex hull. + +=== Stellations of the octahedron === + +The stella octangula (or stellated octahedron) is the only stellation of the regular octahedron. This stellation is made of self-dual tetrahedra, as the simplest regular polyhedral compound: + +=== Stellations of the dodecahedron === +All stellations of the regular dodecahedron are Kepler-Poinsot polyhedra: + +=== Stellations of the icosahedron === + +Coxeter et al. (1938) detailed the stellations of the regular icosahedron with rules proposed by J. C. P. Miller. As found in Coxeter et al. (1999), the following table lists all stellations of the icosahedron per the Crennells' indexing (in it, the regular icosahedron (or snub octahedron) stellation core is indexed as "1"): + +"Cells" (du Val notation) correspond to the internal congruent spaces formed by extending face-planes of the regular icosahedron. +A subset of these are illustrated in Wenninger (1989), alongside constructions for physical models (W19–W66). + +== Stellations of Catalan solids == + +=== Stellations of the rhombic dodecahedron === + +The rhombic dodecahedron produces three fully supported stellations, described in Luke (1957): + +An additional fourth stellation is possible under Miller's rules. The first stellation of the rhombic dodecahedron is notable for being able to form a honeycomb in three-dimensional space, using copies of itself. + +=== Stellations of the rhombic triacontahedron === +Pawley (1975) shows the rhombic triacontahedron produces 227 fully supported stellations, including the rhombic triacontahedron itself. Some of these are shown in the table below: + +Of these, the compound of five cubes is notable for being a regular compound polyhedron. The medial rhombic triacontahedron and the great rhombic triacontahedron are also notable for being star (non-convex) isotoxal polyhedra. + +== Hemipolychrons == + +In Wenninger (1983), a unique family of stellations with unbounded vertices are identified. These originate from orthogonal edges of faces that pass through centers of their corresponding dual hemipolyhedra. The following is a list of these stellations; specifically, of non-convex uniform hemipolyhedra (with coincidental figures in parentheses): + +This family of stellations does not strictly fulfill the definition of a polyhedron that is bound by vertices, and Wenninger notes that at the limit their facets can be interpreted as forming unbounded elongated pyramids, or equivalently, prisms (indistinguishably). As with their dual polyhedra, these hemipolyhedral stellations are isotoxal polyhedra (in their case, at infinity). The final polyhedron on this list, the great dirhombicosidodecacron, is the only stellation whose dual figure — the last-indexed and most complex uniform polyhedron, the great dirhombicosidodecahedron (U75) — is constructed using a spherical quadrilateral Wythoff construction (rather than with spherical triangles). +The tetrahemihexahedron is the only hemipolyhedron to produce a dual hemipolychron without a coincidental figure, the tetrahemihexacron. + +== Notes == + +== References == + +=== Works cited === + +==== Secondary sources ==== + +==== Primary sources ==== + +== External links == +Stellation and Facetting - a Brief History from Guy's Polyhedral Pages (Guy Inchbald) for a brief chronological listing regarding stellation +Stellations of the Rhombic Triacontahedron from Virtual Polyhedra (The Encyclopedia of Polyhedra) (George W. Hart) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_polynomial_topics-0.md b/data/en.wikipedia.org/wiki/List_of_polynomial_topics-0.md new file mode 100644 index 000000000..8fae94d21 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_polynomial_topics-0.md @@ -0,0 +1,204 @@ +--- +title: "List of polynomial topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_polynomial_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:17.484027+00:00" +instance: "kb-cron" +--- + +This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics. + + +== Terminology == +Degree: The maximum exponents among the monomials. +Factor: An expression being multiplied. +Linear factor: A factor of degree one. +Coefficient: An expression multiplying one of the monomials of the polynomial. +Root (or zero) of a polynomial: Given a polynomial p(x), the x values that satisfy p(x) = 0 are called roots (or zeroes) of the polynomial p. +Graphing +End behaviour – +Concavity – +Orientation – +Tangency point – +Inflection point – Point where concavity changes. + + +== Basics == +Polynomial +Coefficient +Monomial +Polynomial long division +Synthetic division +Polynomial factorization +Rational function +Partial fraction +Partial fraction decomposition over R +Vieta's formulas +Integer-valued polynomial +Algebraic equation +Factor theorem +Polynomial remainder theorem + + +=== Elementary abstract algebra === +See also Theory of equations below. +Polynomial ring +Greatest common divisior of two polynomials +Symmetric function +Homogeneous polynomial +Polynomial SOS (sum of squares) + + +== Theory of equations == +Polynomial family +Quadratic function +Cubic function +Quartic function +Quintic function +Sextic function +Septic function +Octic function +Completing the square +Abel–Ruffini theorem +Bring radical +Binomial theorem +Blossom (functional) +Root of a function +nth root (radical) +Surd +Square root +Methods of computing square roots +Cube root +Root of unity +Constructible number +Complex conjugate root theorem +Algebraic element +Horner scheme +Rational root theorem +Gauss's lemma (polynomial) +Irreducible polynomial +Eisenstein's criterion +Primitive polynomial +Fundamental theorem of algebra +Hurwitz polynomial +Polynomial transformation +Tschirnhaus transformation +Galois theory +Discriminant of a polynomial +Resultant +Elimination theory +Gröbner basis +Regular chain +Triangular decomposition +Sturm's theorem +Descartes' rule of signs +Carlitz–Wan conjecture +Polynomial decomposition, factorization under functional composition + + +== Calculus with polynomials == +Delta operator +Bernstein–Sato polynomial + + +== Polynomial interpolation == +Lagrange polynomial +Runge's phenomenon +Spline (mathematics) + + +== Weierstrass approximation theorem == +Bernstein polynomial + + +== Linear algebra == +Characteristic polynomial +Minimal polynomial +Invariant polynomial + + +== Named polynomials and polynomial sequences == +Abel polynomials +Actuarial polynomials +Additive polynomials +All one polynomials +Appell sequence +Askey–Wilson polynomials +Bell polynomials +Bernoulli polynomials +Bernstein polynomial +Bessel polynomials +Binomial type +Brahmagupta polynomials +Caloric polynomial +Charlier polynomials +Chebyshev polynomials +Chihara–Ismail polynomials +Cyclotomic polynomials +Dickson polynomial +Ehrhart polynomial +Exponential polynomials +Favard's theorem +Fibonacci polynomials +Gegenbauer polynomials +Gottlieb polynomials +Hahn polynomials +Hall–Littlewood polynomials +Heat polynomial — see caloric polynomial +Heckman–Opdam polynomials +Hermite polynomials +Hurwitz polynomial +Jack function +Jacobi polynomials +Koornwinder polynomials +Kostka polynomial +Kravchuk polynomials +Laguerre polynomials +Laurent polynomial +Linearised polynomial +Littlewood polynomial +Legendre polynomials +Associated Legendre polynomials +Spherical harmonic +Lucas polynomials +Macdonald polynomials +Meixner polynomials +Necklace polynomial +Newton polynomial +Orthogonal polynomials +Orthogonal polynomials on the unit circle +Permutation polynomial +Racah polynomials +Rogers polynomials +Rogers–Szegő polynomials +Rook polynomial +Schur polynomials +Shapiro polynomials +Sheffer sequence +Spread polynomials +Tricomi–Carlitz polynomials +Touchard polynomials +Wilkinson's polynomial +Wilson polynomials +Zernike polynomials +Pseudo-Zernike polynomials + + +== Knot polynomials == +Alexander polynomial +HOMFLY polynomial +Jones polynomial + + +== Algorithms == +Karatsuba multiplication +Lenstra–Lenstra–Lovász lattice basis reduction algorithm (for polynomial factorization) +Lindsey–Fox algorithm +Remez algorithm (to find best approximating polynomials) +Schönhage–Strassen algorithm + + +== Other == +Polynomial mapping \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_prime_knots-0.md b/data/en.wikipedia.org/wiki/List_of_prime_knots-0.md new file mode 100644 index 000000000..77c8b788d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_prime_knots-0.md @@ -0,0 +1,59 @@ +--- +title: "List of prime knots" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_prime_knots" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:22.748133+00:00" +instance: "kb-cron" +--- + +In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes. + + +== Table of prime knots == + + +=== Six or fewer crossings === + + +=== Seven crossings === + + +=== Eight crossings === + + +=== Nine crossings === + + +=== Ten crossings === + + +=== Higher === + +Conway knot 11n34 +Kinoshita–Terasaka knot 11n42 + + +== Table of prime links == + + +=== Eight or fewer crossings === + + +=== Higher === + + +== See also == +List of knots +List of mathematical knots and links +Knot tabulation +(−2,3,7) pretzel knot + + +== Notes == + + +== External links == +"The Rolfsen Knot Table", The Knot Atlas. +"KnotInfo", Indiana.edu. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_prime_numbers-0.md b/data/en.wikipedia.org/wiki/List_of_prime_numbers-0.md new file mode 100644 index 000000000..92fc9bfcc --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_prime_numbers-0.md @@ -0,0 +1,180 @@ +--- +title: "List of prime numbers" +chunk: 1/6 +source: "https://en.wikipedia.org/wiki/List_of_prime_numbers" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:23.923371+00:00" +instance: "kb-cron" +--- + +This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. +The first 1,000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. The number 1 is neither prime nor composite. + +== The first 1,000 prime numbers == +The following table lists the first 1,000 primes, with 20 columns of consecutive primes in each of the 50 rows. + +(sequence A000040 in the OEIS). +The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×1018. That means 95,676,260,903,887,607 primes (nearly 1017), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2×1021) smaller than 1023. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2×1022) smaller than 1024, if the Riemann hypothesis is true. + +== Lists of primes by type == +Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. n is a natural number (including 0) in the definitions. + +=== Balanced primes === + +Balanced primes are primes with equal-sized prime gaps before and after them, making them the arithmetic mean of their next larger and next smaller prime. + +5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393 (OEIS: A006562). + +=== Bell primes === + +Bell primes are primes that are also the number of partitions of some finite set. +2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. +The next term has 6,539 digits. (OEIS: A051131) + +=== Chen primes === + +Chen primes are primes p such that p+2 is either a prime or semiprime. +2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409 (OEIS: A109611) + +=== Circular primes === + +A circular prime is a number that remains prime on any cyclic rotation of its base 10 digits. +2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331 (OEIS: A068652) +Some sources only include the smallest prime in each cycle. For example, listing 13, but omitting 31. +2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111 (OEIS: A016114) + +=== Cluster primes === + +A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p. +3, 5, 7, 11, 13, 17, 19, 23, ... (OEIS: A038134) +All primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are: +2, 97, 127, 149, 191, 211, 223, 227, 229, 251. + +=== Cousin primes === + +Cousin primes are pairs of primes that differ by four. +(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281) (OEIS: A023200, OEIS: A046132) + +=== Cuban primes === + +Cuban primes are primes + + + + p + + + {\displaystyle p} + + of the form + + + + p + = + + k + + 3 + + + − + ( + k + − + 1 + + ) + + 3 + + + , + + + {\displaystyle p=k^{3}-(k-1)^{3},} + + where + + + + k + + + {\displaystyle k} + + is a natural number. +7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, 27361, 33391, 35317 (OEIS: A002407) +The term is also used to refer to primes + + + + p + + + {\displaystyle p} + + of the form + + + + p + = + ( + + k + + 3 + + + − + ( + k + − + 2 + + ) + + 3 + + + ) + + / + + 2 + , + + + {\displaystyle p=(k^{3}-(k-2)^{3})/2,} + + where + + + + k + + + {\displaystyle k} + + is a natural number. +13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249 (OEIS: A002648) + +=== Cullen primes === + +Cullen primes are primes p of the form p=k2k + 1, for some natural number k. +3, 393050634124102232869567034555427371542904833 (OEIS: A050920) + +=== Delicate primes === + +Delicate primes are those primes that always become a composite number when any of their base 10 digit is changed. +294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 (OEIS: A050249) + +=== Dihedral primes === + +Dihedral primes are primes that satisfy 180° rotational symmetry and mirror symmetry on a seven-segment display. +2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, +121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 (OEIS: A134996) + +=== Real Eisenstein primes === \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_prime_numbers-1.md b/data/en.wikipedia.org/wiki/List_of_prime_numbers-1.md new file mode 100644 index 000000000..f96343153 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_prime_numbers-1.md @@ -0,0 +1,139 @@ +--- +title: "List of prime numbers" +chunk: 2/6 +source: "https://en.wikipedia.org/wiki/List_of_prime_numbers" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:23.923371+00:00" +instance: "kb-cron" +--- + +Real Eisenstein primes are real Eisenstein integers that are irreducible. Equivalently, they are primes of the form 3k − 1, for a positive integer k. +2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401 (OEIS: A003627) + +=== Emirps === + +Emirps are primes that become a different prime after their base 10 digits have been reversed. The name "emirp" is the reverse of the word "prime". +13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991 (OEIS: A006567) + +=== Euclid primes === + +Euclid primes are primes p such that p−1 is a primorial. +3, 7, 31, 211, 2311, 200560490131 (OEIS: A018239) + +=== Euler irregular primes === + +Euler irregular primes are primes + + + + p + + + {\displaystyle p} + + that divide an Euler number + + + + + E + + 2 + n + + + , + + + {\displaystyle E_{2n},} + + for some + + + + 0 + ≤ + 2 + n + ≤ + p + − + 3. + + + {\displaystyle 0\leq 2n\leq p-3.} + + +19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587 (OEIS: A120337) + +==== Euler (p, p − 3) irregular primes ==== +Euler (p, p - 3) irregular primes are primes p that divide the (p + 3)rd Euler number. +149, 241, 2946901 (OEIS: A198245) + +=== Factorial primes === + +Factorial primes are of the form n! ± 1. +2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 (OEIS: A088054) + +=== Fermat primes === + +Fermat primes are primes p of the form p = 22k + 1, for a non-negative integer k. As of June 2024 only five Fermat primes have been discovered. +3, 5, 17, 257, 65537 (OEIS: A019434) + +==== Generalized Fermat primes ==== + +Generalized Fermat primes are primes p of the form p = a2k + 1, for a non-negative integer k and even natural number a. + +=== Fibonacci primes === + +Fibonacci primes are primes that appear in the Fibonacci sequence. +2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 (OEIS: A005478) + +=== Fortunate primes === + +Fortunate primes are primes that are also Fortunate numbers. There are no known composite Fortunate numbers. +3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397 (OEIS: A046066) + +=== Gaussian primes === + +Gaussian primes are primes p of the form p = 4k + 3, for a non-negative integer k. +3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503 (OEIS: A002145) + +=== Good primes === + +Good primes are primes p satisfying ab < p2, for all primes a and b such that a,b < p +5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 (OEIS: A028388) + +=== Happy primes === + +Happy primes are primes that are also happy numbers. +7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881, 907, 937, 1009, 1033, 1039, 1093 (OEIS: A035497) + +=== Harmonic primes === + +Harmonic primes are primes p for which there are no solutions to Hk ≡ 0 (mod p) and Hk ≡ −ωp (mod p), for 1 ≤ k ≤ p−2, where Hk denotes the k-th harmonic number and ωp denotes the Wolstenholme quotient. +5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349 (OEIS: A092101) + +=== Higgs primes === + +Higgs primes are primes p for which p − 1 divides the square of the product of all smaller Higgs primes. +2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349 (OEIS: A007459) + +=== Highly cototient primes === + +Highly cototient primes are primes that are a cototient more often than any integer below it except 1. +2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889 (OEIS: A105440) + +=== Home primes === + +For n ≥ 2, write the prime factorization of n in base 10 and concatenate the factors; iterate until a prime is reached. +For a non-negative integer, its home prime is obtained by concatenating its prime factors in increasing order repeatedly, until a prime is achieved. +2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277 (OEIS: A037274) + +=== Irregular primes === + +Irregular primes are odd primes p that divide the class number of the p-th cyclotomic field. +37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 607, 613 (OEIS: A000928) + +==== (p, p − 3) irregular primes ==== \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_prime_numbers-2.md b/data/en.wikipedia.org/wiki/List_of_prime_numbers-2.md new file mode 100644 index 000000000..52698dc9d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_prime_numbers-2.md @@ -0,0 +1,134 @@ +--- +title: "List of prime numbers" +chunk: 3/6 +source: "https://en.wikipedia.org/wiki/List_of_prime_numbers" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:23.923371+00:00" +instance: "kb-cron" +--- + +The (p, p - 3) irregular primes are primes p such that (p, p − 3) is an irregular pair. +16843, 2124679 (OEIS: A088164) + +==== (p, p − 5) irregular primes ==== + +The (p, p - 5) irregular primes are primes p such that (p, p − 5) is an irregular pair. +37 + +==== (p, p − 9) irregular primes ==== + +The (p, p - 9) irregular primes are primes p such that (p, p − 9) is an irregular pair. +67, 877 (OEIS: A212557) + +=== Isolated primes === + +Isolated primes are primes p such that both p − 2 and p + 2 are both composite. +2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 839, 853, 863, 877, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 (OEIS: A007510) + +=== Leyland primes === + +Leyland primes are primes p of the form p = ab + ba, where a and b are integers larger than one. +17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEIS: A094133) + +=== Long primes === + +Long primes, or full reptend primes, are odd primes p for which + + + + ( + + 10 + + p + − + 1 + + + − + 1 + ) + + / + + p + + + {\displaystyle (10^{p-1}-1)/p} + + is a cyclic number. Bases other than 10 are also used. +7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593 (OEIS: A001913) + +=== Lucas primes === + +Lucas primes are primes that appear in the Lucas sequence. +2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 (OEIS: A005479) + +=== Lucky primes === + +Lucky primes are primes that are also lucky numbers. +3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997 (OEIS: A031157) + +=== Mersenne primes === + +Mersenne primes are primes p of the form p = 2k − 1, for some non-negative integer k. +3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (OEIS: A000668) +As of 2024, there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits. The largest known primes since 1992 have all been Mersenne primes, with the largest as of 2026 being 2136,279,841−1, the 52nd Mersenne prime. + +==== Mersenne divisors ==== +Mersenne divisors are primes that divide 2k − 1, for some prime k. Every Mersenne prime p is also a Mersenne divisor, with k = p. +3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343 (OEIS: A122094) + +==== Mersenne prime exponents ==== +Primes p such that 2p − 1 is prime. +2, 3, 5, 7, 13, 17, 19, 31, 61, 89, +107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, +9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, +216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, +24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917 (OEIS: A000043) +As of September 2025, two more are known to be in the sequence, but it is not known whether they are the next: +82589933, 136279841 + +==== Double Mersenne primes ==== + +A subset of Mersenne primes of the form 22p−1 − 1 for prime p. +7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in OEIS: A077586) + +==== Generalized repunit primes ==== +Of the form (an − 1) / (a − 1) for fixed integer a. +For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below: +a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (OEIS: A076481) +a = 4: 5 (the only prime for a = 4) +a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 (OEIS: A086122) +a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 (OEIS: A165210) +a = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457 +a = 8: 73 (the only prime for a = 8) +a = 9: none exist + +==== Other generalizations and variations ==== +Many generalizations of Mersenne primes have been defined. This include the following: + +Primes of the form bn − (b − 1)n, including the Mersenne primes and the cuban primes as special cases +Williams primes, of the form (b − 1)·bn − 1 + +=== Mills primes === + +Of the form ⌊θ3n⌋, where θ is Mills' constant. This form is prime for all positive integers n. +2, 11, 1361, 2521008887, 16022236204009818131831320183 (OEIS: A051254) + +=== Minimal primes === + +Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes: +2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 (OEIS: A071062) + +=== Newman–Shanks–Williams primes === + +Newman–Shanks–Williams numbers that are prime. +7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 (OEIS: A088165) + +=== Non-generous primes === +Primes p for which the least positive primitive root is not a primitive root of p2. Three such primes are known; it is not known whether there are more. +2, 40487, 6692367337 (OEIS: A055578) + +=== Palindromic primes === \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_prime_numbers-3.md b/data/en.wikipedia.org/wiki/List_of_prime_numbers-3.md new file mode 100644 index 000000000..6bb6d1c19 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_prime_numbers-3.md @@ -0,0 +1,166 @@ +--- +title: "List of prime numbers" +chunk: 4/6 +source: "https://en.wikipedia.org/wiki/List_of_prime_numbers" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:23.923371+00:00" +instance: "kb-cron" +--- + +Primes that remain the same when their decimal digits are read backwards. +2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 (OEIS: A002385) + +=== Palindromic wing primes === +Primes of the form + + + + + + + a + + + ( + + + + 10 + + m + + + − + 1 + + + ) + + + + 9 + + + ± + b + × + + 10 + + + + m + − + 1 + + 2 + + + + + + {\displaystyle {\frac {a{\big (}10^{m}-1{\big )}}{9}}\pm b\times 10^{\frac {m-1}{2}}} + + with + + + + 0 + ≤ + a + ± + b + < + 10 + + + {\displaystyle 0\leq a\pm b<10} + +. This means all digits except the middle digit are equal. +101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 (OEIS: A077798) + +=== Partition primes === + +Partition function values that are prime. +2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 (OEIS: A049575) + +=== Pell primes === + +Primes in the Pell number sequence P0 = 0, P1 = 1, +Pn = 2Pn−1 + Pn−2. +2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 (OEIS: A086383) + +=== Permutable primes === + +Any permutation of the decimal digits is a prime. +2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 (OEIS: A003459) + +=== Perrin primes === + +Primes in the Perrin number sequence P(0) = 3, P(1) = 0, P(2) = 2, +P(n) = P(n−2) + P(n−3). +2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 (OEIS: A074788) + +=== Pierpont primes === + +Of the form 2u3v + 1 for some integers u,v ≥ 0. +These are also class 1- primes. +2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457 (OEIS: A005109) + +=== Pillai primes === + +Primes p for which there exist n > 0 such that p divides n! + 1 and n does not divide p − 1. +23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499 (OEIS: A063980) + +=== Primes of the form n4 + 1 === +Of the form n4 + 1. +2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 (OEIS: A037896) + +=== Primeval primes === + +Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number. +2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079 (OEIS: A119535) + +=== Primorial primes === + +Of the form pn# ± 1. +3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of OEIS: A057705 and OEIS: A018239) + +=== Proth primes === + +Of the form k×2n + 1, with odd k and k < 2n. +3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (OEIS: A080076) + +=== Pythagorean primes === + +Of the form 4n + 1. +5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449 (OEIS: A002144) + +=== Prime quadruplets === + +Where (p, p+2, p+6, p+8) are all prime. +(5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109), (191, 193, 197, 199), (821, 823, 827, 829), (1481, 1483, 1487, 1489), (1871, 1873, 1877, 1879), (2081, 2083, 2087, 2089), (3251, 3253, 3257, 3259), (3461, 3463, 3467, 3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) (OEIS: A007530, OEIS: A136720, OEIS: A136721, OEIS: A090258) + +=== Quartan primes === + +Of the form x4 + y4, where x,y > 0. +2, 17, 97, 257, 337, 641, 881 (OEIS: A002645) + +=== Ramanujan primes === + +Integers Rn that are the smallest to give at least n primes from x/2 to x for all x ≥ Rn (all such integers are primes). +2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491 (OEIS: A104272) + +=== Regular primes === + +Primes p that do not divide the class number of the p-th cyclotomic field. +3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281 (OEIS: A007703) + +=== Repunit primes === + +Primes containing only the decimal digit 1. +11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) (OEIS: A004022) +The next have 317, 1031, 49081, 86453, 109297, and 270343 digits, respectively (OEIS: A004023). + +=== Residue classes of primes === \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_prime_numbers-4.md b/data/en.wikipedia.org/wiki/List_of_prime_numbers-4.md new file mode 100644 index 000000000..2ef01296d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_prime_numbers-4.md @@ -0,0 +1,94 @@ +--- +title: "List of prime numbers" +chunk: 5/6 +source: "https://en.wikipedia.org/wiki/List_of_prime_numbers" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:23.923371+00:00" +instance: "kb-cron" +--- + +Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a. +The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are the Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d. +If a and d are relatively prime, the arithmetic progression contains infinitely many primes. +2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 (OEIS: A065091) +4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 (OEIS: A002144) +4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 (OEIS: A002145) +6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 (OEIS: A002476) +6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (OEIS: A007528) +8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 (OEIS: A007519) +8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 (OEIS: A007520) +8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 (OEIS: A007521) +8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 (OEIS: A007522) +10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 (OEIS: A030430) +10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 (OEIS: A030431) +10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 (OEIS: A030432) +10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 (OEIS: A030433) +12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 (OEIS: A068228) +12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 (OEIS: A040117) +12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 (OEIS: A068229) +12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 (OEIS: A068231) + +=== Safe primes === + +Where p and (p−1) / 2 are both prime. +5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907 (OEIS: A005385) + +=== Self primes in base 10 === + +Primes that cannot be generated by any integer added to the sum of its decimal digits. +3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873 (OEIS: A006378) + +=== Sexy primes === + +Where (p, p + 6) are both prime. +(5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), (53, 59), (61, 67), (67, 73), (73, 79), (83, 89), (97, 103), (101, 107), (103, 109), (107, 113), (131, 137), (151, 157), (157, 163), (167, 173), (173, 179), (191, 197), (193, 199) (OEIS: A023201, OEIS: A046117) + +=== Smarandache–Wellin primes === + +Primes that are the concatenation of the first n primes written in decimal. +2, 23, 2357 (OEIS: A069151) +The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719. + +=== Solinas primes === + +Of the form 2k − c1·2k−1 − c2·2k−2 − ... − ck. + +3, 5, 7, 11, 13 (OEIS: A165255) +232 − 5, the largest prime that fits into 32 bits of memory. +264 − 59, the largest prime that fits into 64 bits of memory. + +=== Sophie Germain primes === + +Where p and 2p + 1 are both prime. A Sophie Germain prime has a corresponding safe prime. +2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953 (OEIS: A005384) + +=== Stern primes === + +Primes that are not the sum of a smaller prime and twice the square of a nonzero integer. +2, 3, 17, 137, 227, 977, 1187, 1493 (OEIS: A042978) +As of 2011, these are the only known Stern primes, and possibly the only existing. + +=== Super-primes === + +Primes with prime-numbered indexes in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime). +3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991 (OEIS: A006450) + +=== Supersingular primes === + +There are exactly fifteen supersingular primes: +2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 (OEIS: A002267) + +=== Thabit primes === + +Of the form 3×2n − 1. +2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 (OEIS: A007505) +The primes of the form 3×2n + 1 are related. +7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 (OEIS: A039687) + +=== Prime triplets === + +Where (p, p+2, p+6) or (p, p+4, p+6) are all prime. +(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353) (OEIS: A007529, OEIS: A098414, OEIS: A098415) + +=== Truncatable prime === \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_prime_numbers-5.md b/data/en.wikipedia.org/wiki/List_of_prime_numbers-5.md new file mode 100644 index 000000000..805678e51 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_prime_numbers-5.md @@ -0,0 +1,302 @@ +--- +title: "List of prime numbers" +chunk: 6/6 +source: "https://en.wikipedia.org/wiki/List_of_prime_numbers" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:23.923371+00:00" +instance: "kb-cron" +--- + +==== Left-truncatable ==== +Primes that remain prime when the leading decimal digit is successively removed. +2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683 (OEIS: A024785) + +==== Right-truncatable ==== +Primes that remain prime when the least significant decimal digit is successively removed. +2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797 (OEIS: A024770) + +==== Two-sided ==== +Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes: +2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (OEIS: A020994) + +=== Twin primes === + +Where (p, p+2) are both prime. +(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349), (419, 421), (431, 433), (461, 463) (OEIS: A001359, OEIS: A006512) + +=== Unique primes === + +The list of primes p for which the period length of the decimal expansion of 1/p is unique (no other prime gives the same period). +3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 (OEIS: A040017) + +=== Wagstaff primes === + +Of the form (2n + 1) / 3. +3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 (OEIS: A000979) +Values of n: +3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 (OEIS: A000978) + +=== Wall–Sun–Sun primes === + +A prime p > 5, if p2 divides the Fibonacci number + + + + + F + + p + − + + ( + + + p + 5 + + + ) + + + + + + {\displaystyle F_{p-\left({\frac {p}{5}}\right)}} + +, where the Legendre symbol + + + + + ( + + + p + 5 + + + ) + + + + {\displaystyle \left({\frac {p}{5}}\right)} + + is defined as + + + + + + ( + + + p + 5 + + + ) + + = + + + { + + + + 1 + + + + + if + + + + p + ≡ + ± + 1 + + + ( + mod + + 5 + ) + + + + + + − + 1 + + + + + if + + + + p + ≡ + ± + 2 + + + ( + mod + + 5 + ) + + . + + + + + + + + + {\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}1&{\textrm {if}}\;p\equiv \pm 1{\pmod {5}}\\-1&{\textrm {if}}\;p\equiv \pm 2{\pmod {5}}.\end{cases}}} + + +As of 2022, no Wall-Sun-Sun primes have been found below + + + + + 2 + + 64 + + + + + {\displaystyle 2^{64}} + + (about + + + + 18 + ⋅ + + 10 + + 18 + + + + + {\displaystyle 18\cdot 10^{18}} + +). + +=== Wieferich primes === + +Primes p such that ap − 1 ≡ 1 (mod p2) for fixed integer a > 1. +2p − 1 ≡ 1 (mod p2): 1093, 3511 (OEIS: A001220) +3p − 1 ≡ 1 (mod p2): 11, 1006003 (OEIS: A014127) +4p − 1 ≡ 1 (mod p2): 1093, 3511 +5p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 (OEIS: A123692) +6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573 (OEIS: A212583) +7p − 1 ≡ 1 (mod p2): 5, 491531 (OEIS: A123693) +8p − 1 ≡ 1 (mod p2): 3, 1093, 3511 +9p − 1 ≡ 1 (mod p2): 2, 11, 1006003 +10p − 1 ≡ 1 (mod p2): 3, 487, 56598313 (OEIS: A045616) +11p − 1 ≡ 1 (mod p2): 71 +12p − 1 ≡ 1 (mod p2): 2693, 123653 (OEIS: A111027) +13p − 1 ≡ 1 (mod p2): 2, 863, 1747591 (OEIS: A128667) +14p − 1 ≡ 1 (mod p2): 29, 353, 7596952219 (OEIS: A234810) +15p − 1 ≡ 1 (mod p2): 29131, 119327070011 (OEIS: A242741) +16p − 1 ≡ 1 (mod p2): 1093, 3511 +17p − 1 ≡ 1 (mod p2): 2, 3, 46021, 48947 (OEIS: A128668) +18p − 1 ≡ 1 (mod p2): 5, 7, 37, 331, 33923, 1284043 (OEIS: A244260) +19p − 1 ≡ 1 (mod p2): 3, 7, 13, 43, 137, 63061489 (OEIS: A090968) +20p − 1 ≡ 1 (mod p2): 281, 46457, 9377747, 122959073 (OEIS: A242982) +21p − 1 ≡ 1 (mod p2): 2 +22p − 1 ≡ 1 (mod p2): 13, 673, 1595813, 492366587, 9809862296159 (OEIS: A298951) +23p − 1 ≡ 1 (mod p2): 13, 2481757, 13703077, 15546404183, 2549536629329 (OEIS: A128669) +24p − 1 ≡ 1 (mod p2): 5, 25633 +25p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 +As of 2018, these are all known Wieferich primes with a ≤ 25. + +=== Wilson primes === + +Primes p for which p2 divides (p−1)! + 1. +5, 13, 563 (OEIS: A007540) +As of 2018, these are the only known Wilson primes. + +=== Wolstenholme primes === + +Primes p for which the binomial coefficient + + + + + + + ( + + + + 2 + p + − + 1 + + + p + − + 1 + + + + ) + + + + ≡ + 1 + + + ( + mod + + + p + + 4 + + + ) + + . + + + {\displaystyle {{2p-1} \choose {p-1}}\equiv 1{\pmod {p^{4}}}.} + + +16843, 2124679 (OEIS: A088164) +As of 2018, these are the only known Wolstenholme primes. + +=== Woodall primes === + +Of the form n×2n − 1. +7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 (OEIS: A050918) + +== See also == + +== References == + +== External links == +Lists of Primes at the Prime Pages. +The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. +Interface to a list of the first 98 million primes (primes less than 2,000,000,000) +Weisstein, Eric W. "Prime Number Sequences". MathWorld. +Selected prime related sequences in OEIS. +Fischer, R. Thema: Fermatquotient B^(P−1) == 1 (mod P^2) (in German) (Lists Wieferich primes in all bases up to 1052) +Padilla, Tony (7 February 2013). "New Largest Known Prime Number". Numberphile. Brady Haran. Archived from the original on 2 November 2021. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals-0.md b/data/en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals-0.md new file mode 100644 index 000000000..52727e412 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals-0.md @@ -0,0 +1,41 @@ +--- +title: "List of properties of sets of reals" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:38.430648+00:00" +instance: "kb-cron" +--- + +This article lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory, which has a rather different emphasis from general topology. + + +== Definability properties == +Borel set +Analytic set +C-measurable set +Projective set +Inductive set +Infinity-Borel set +Suslin set +Homogeneously Suslin set +Weakly homogeneously Suslin set +Set of uniqueness + + +== Regularity properties == +Property of Baire +Lebesgue measurable +Universally measurable set +Perfect set property +Universally Baire set + + +== Largeness and smallness properties == +Meager set +Comeager set - A comeager set is one whose complement is meager. +Null set +Conull set +Dense set +Nowhere dense set \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-0.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-0.md new file mode 100644 index 000000000..f235c93a2 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-0.md @@ -0,0 +1,119 @@ +--- +title: "List of publications in mathematics" +chunk: 1/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +This is a list of publications in mathematics, organized by field. +Some reasons a particular publication might be regarded as important: + +Topic creator – A publication that created a new topic +Breakthrough – A publication that changed scientific knowledge significantly +Influence – A publication which has significantly influenced the world or has had a massive impact on the teaching of mathematics. +Among published compilations of important publications in mathematics are Landmark writings in Western mathematics 1640–1940 by Ivor Grattan-Guinness and A Source Book in Mathematics by David Eugene Smith. + +== Algebra == + +=== Theory of equations === + +==== Baudhayana Sulba Sutra ==== +Baudhayana (8th century BCE) +With linguistic evidence suggesting this text to have been written between the 8th and 5th centuries century BCE, this is one of the oldest mathematical texts. It laid the foundations of Indian mathematics and was influential in South Asia. It was primarily a geometrical text and also contained some important developments, including the list of Pythagorean triples, geometric solutions of linear and quadratic equations and square root of 2. + +==== The Nine Chapters on the Mathematical Art ==== +The Nine Chapters on the Mathematical Art (10th–2nd century BCE) +Contains the earliest description of Gaussian elimination for solving system of linear equations, it also contains method for finding square root and cubic root. + +==== Arithmetica ==== +Diophantus (3rd century CE) +Contains the collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations. + +==== Haidao Suanjing ==== +Liu Hui (220-280 CE) +Contains the application of right angle triangles for survey of depth or height of distant objects. + +==== Sunzi Suanjing ==== +Sunzi (5th century CE) +Contains the earliest description of Chinese remainder theorem. + +==== Aryabhatiya ==== +Aryabhata (499 CE) +The text contains 33 verses covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations. It also gave the modern standard algorithm for solving first-order diophantine equations. + +==== Jigu Suanjing ==== +Jigu Suanjing (626 CE) +This book by Tang dynasty mathematician Wang Xiaotong contains the world's earliest third order equation. + +==== Brāhmasphuṭasiddhānta ==== +Brahmagupta (628 CE) +Contained rules for manipulating both negative and positive numbers, rules for dealing with the number zero, a method for computing square roots, and general methods of solving linear and some quadratic equations, solution to Pell's equation. + +==== Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala ==== +Muhammad ibn Mūsā al-Khwārizmī (820 CE) +The first book on the systematic algebraic solutions of linear and quadratic equations by the Persian scholar Muhammad ibn Mūsā al-Khwārizmī. The book is considered to be the foundation of modern algebra and Islamic mathematics. The word "algebra" itself is derived from the al-Jabr in the title of the book. + +=== Līlāvatī, Siddhānta Shiromani and Bijaganita === +One of the major treatises on mathematics by Bhāskara II provides the solution for indeterminate equations of 1st and 2nd order. + +==== Yigu yanduan ==== +Li Ye (13th century) +Contains the earliest invention of 4th order polynomial equation. + +==== Mathematical Treatise in Nine Sections ==== +Qin Jiushao (1247) +This 13th-century book contains the earliest complete solution of 19th-century Horner's method of solving high order polynomial equations (up to 10th order). It also contains a complete solution of Chinese remainder theorem, which predates Euler and Gauss by several centuries. + +==== Ceyuan haijing ==== +Li Zhi (1248) +Contains the application of high order polynomial equation in solving complex geometry problems. + +==== Jade Mirror of the Four Unknowns ==== +Zhu Shijie (1303) +Contains the method of establishing system of high order polynomial equations of up to four unknowns. + +==== Ars Magna ==== +Gerolamo Cardano (1545) +Otherwise known as The Great Art, provided the first published methods for solving cubic and quartic equations (due to Scipione del Ferro, Niccolò Fontana Tartaglia, and Lodovico Ferrari), and exhibited the first published calculations involving non-real complex numbers. + +==== Vollständige Anleitung zur Algebra ==== +Leonhard Euler (1770) +Also known as Elements of Algebra, Euler's textbook on elementary algebra is one of the first to set out algebra in the modern form we would recognize today. The first volume deals with determinate equations, while the second part deals with Diophantine equations. The last section contains a proof of Fermat's Last Theorem for the case n = 3, making some valid assumptions regarding + + + + + Q + + ( + + + − + 3 + + + ) + + + {\displaystyle \mathbb {Q} ({\sqrt {-3}})} + + that Euler did not prove. + +==== Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse ==== +Carl Friedrich Gauss (1799) +Gauss's doctoral dissertation, which contained a widely accepted (at the time) but incomplete proof of the fundamental theorem of algebra. + +=== Abstract algebra === + +==== Group theory ==== + +===== Réflexions sur la résolution algébrique des équations ===== +Joseph Louis Lagrange (1770) +The title means "Reflections on the algebraic solutions of equations". Made the prescient observation that the roots of the Lagrange resolvent of a polynomial equation are tied to permutations of the roots of the original equation, laying a more general foundation for what had previously been an ad hoc analysis and helping motivate the later development of the theory of permutation groups, group theory, and Galois theory. The Lagrange resolvent also introduced the discrete Fourier transform of order 3. + +==== Articles Publiés par Galois dans les Annales de Mathématiques ==== +Journal de Mathematiques pures et Appliquées, II (1846) +Posthumous publication of the mathematical manuscripts of Évariste Galois by Joseph Liouville. Included are Galois' papers Mémoire sur les conditions de résolubilité des équations par radicaux and Des équations primitives qui sont solubles par radicaux. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-1.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-1.md new file mode 100644 index 000000000..ea94fc112 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-1.md @@ -0,0 +1,57 @@ +--- +title: "List of publications in mathematics" +chunk: 2/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +==== Traité des substitutions et des équations algébriques ==== +Camille Jordan (1870) +Online version: Online version +Traité des substitutions et des équations algébriques (Treatise on Substitutions and Algebraic Equations). The first book on group theory, giving a then-comprehensive study of permutation groups and Galois theory. In this book, Jordan introduced the notion of a simple group and epimorphism (which he called isomorphisme mériédrique), proved part of the Jordan–Hölder theorem, and discussed matrix groups over finite fields as well as the Jordan normal form. + +==== Theorie der Transformationsgruppen ==== +Sophus Lie, Friedrich Engel (1888–1893). +Publication data: 3 volumes, B.G. Teubner, Verlagsgesellschaft, mbH, Leipzig, 1888–1893. Volume 1, Volume 2, Volume 3. +The first comprehensive work on transformation groups, serving as the foundation for the modern theory of Lie groups. + +==== Solvability of groups of odd order ==== +Walter Feit and John Thompson (1960) +Description: Gave a complete proof of the solvability of finite groups of odd order, establishing the long-standing Burnside conjecture that all finite non-abelian simple groups are of even order. Many of the original techniques used in this paper were used in the eventual classification of finite simple groups. + +==== Homological algebra ==== + +==== Homological Algebra ==== +Henri Cartan and Samuel Eilenberg (1956) +Provided the first fully worked out treatment of abstract homological algebra, unifying previously disparate presentations of homology and cohomology for associative algebras, Lie algebras, and groups into a single theory. + +==== "Sur Quelques Points d'Algèbre Homologique" ==== +Alexander Grothendieck (1957) +Often referred to as the "Tôhoku paper", it revolutionized homological algebra by introducing abelian categories and providing a general framework for Cartan and Eilenberg's notion of derived functors. + +== Algebraic geometry == + +=== Theorie der Abelschen Functionen === +Bernhard Riemann (1857) +Publication data: Journal für die Reine und Angewandte Mathematik +Developed the concept of Riemann surfaces and their topological properties beyond Riemann's 1851 thesis work, proved an index theorem for the genus (the original formulation of the Riemann–Hurwitz formula), proved the Riemann inequality for the dimension of the space of meromorphic functions with prescribed poles (the original formulation of the Riemann–Roch theorem), discussed birational transformations of a given curve and the dimension of the corresponding moduli space of inequivalent curves of a given genus, and solved more general inversion problems than those investigated by Abel and Jacobi. André Weil once wrote that this paper "is one of the greatest pieces of mathematics that has ever been written; there is not a single word in it that is not of consequence." + +=== Faisceaux Algébriques Cohérents === +Jean-Pierre Serre +Publication data: Annals of Mathematics, 1955 +FAC, as it is usually called, was foundational for the use of sheaves in algebraic geometry, extending beyond the case of complex manifolds. Serre introduced Čech cohomology of sheaves in this paper, and, despite some technical deficiencies, revolutionized formulations of algebraic geometry. For example, the long exact sequence in sheaf cohomology allows one to show that some surjective maps of sheaves induce surjective maps on sections; specifically, these are the maps whose kernel (as a sheaf) has a vanishing first cohomology group. The dimension of a vector space of sections of a coherent sheaf is finite, in projective geometry, and such dimensions include many discrete invariants of varieties, for example Hodge numbers. While Grothendieck's derived functor cohomology has replaced Čech cohomology for technical reasons, actual calculations, such as of the cohomology of projective space, are usually carried out by Čech techniques, and for this reason Serre's paper remains important. + +=== Géométrie Algébrique et Géométrie Analytique === +Jean-Pierre Serre (1956) +In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. (NB: While analytic geometry as use of Cartesian coordinates is also in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this article.) The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre, now usually referred to as GAGA. A GAGA-style result would now mean any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, and a well-defined subcategory of analytic geometry objects and holomorphic mappings. + +=== Le théorème de Riemann–Roch, d'après A. Grothendieck === +Armand Borel, Jean-Pierre Serre (1958) +Borel and Serre's exposition of Grothendieck's version of the Riemann–Roch theorem, published after Grothendieck made it clear that he was not interested in writing up his own result. Grothendieck reinterpreted both sides of the formula that Hirzebruch proved in 1953 in the framework of morphisms between varieties, resulting in a sweeping generalization. In his proof, Grothendieck broke new ground with his concept of Grothendieck groups, which led to the development of K-theory. + +=== Éléments de géométrie algébrique === +Alexander Grothendieck (1960–1967) +Written with the assistance of Jean Dieudonné, this is Grothendieck's exposition of his reworking of the foundations of algebraic geometry. It has become the most important foundational work in modern algebraic geometry. The approach expounded in EGA, as these books are known, transformed the field and led to monumental advances. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-2.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-2.md new file mode 100644 index 000000000..571ea42fb --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-2.md @@ -0,0 +1,79 @@ +--- +title: "List of publications in mathematics" +chunk: 3/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +=== Séminaire de géométrie algébrique === +Alexander Grothendieck et al. +These seminar notes on Grothendieck's reworking of the foundations of algebraic geometry report on work done at IHÉS starting in the 1960s. SGA 1 dates from the seminars of 1960–1961, and the last in the series, SGA 7, dates from 1967 to 1969. In contrast to EGA, which is intended to set foundations, SGA describes ongoing research as it unfolded in Grothendieck's seminar; as a result, it is quite difficult to read, since many of the more elementary and foundational results were relegated to EGA. One of the major results building on the results in SGA is Pierre Deligne's proof of the last of the open Weil conjectures in the early 1970s. Other authors who worked on one or several volumes of SGA include Michel Raynaud, Michael Artin, Jean-Pierre Serre, Jean-Louis Verdier, Pierre Deligne, and Nicholas Katz. + +== Number theory == + +=== Brāhmasphuṭasiddhānta === +Brahmagupta (628) +Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. The current system of the four fundamental operations (addition, subtraction, multiplication and division) based on the Hindu-Arabic number system also first appeared in Brahmasphutasiddhanta. It was also one of the first texts to provide concrete ideas on positive and negative numbers. + +=== De fractionibus continuis dissertatio === +Leonhard Euler (1744) +First presented in 1737, this paper provided the first then-comprehensive account of the properties of continued fractions. It also contains the first proof that the number e is irrational. + +=== Recherches d'Arithmétique === +Joseph Louis Lagrange (1775) +Developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form + + + + a + + x + + 2 + + + + + b + + y + + 2 + + + + + c + x + y + + + {\displaystyle ax^{2}+by^{2}+cxy} + +. This included a reduction theory for binary quadratic forms, where he proved that every form is equivalent to a certain canonically chosen reduced form. + +=== Disquisitiones Arithmeticae === +Carl Friedrich Gauss (1801) +The Disquisitiones Arithmeticae is a profound and masterful book on number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds many important new results of his own. Among his contributions was the first complete proof known of the Fundamental theorem of arithmetic, the first two published proofs of the law of quadratic reciprocity, a deep investigation of binary quadratic forms going beyond Lagrange's work in Recherches d'Arithmétique, a first appearance of Gauss sums, cyclotomy, and the theory of constructible polygons with a particular application to the constructibility of the regular 17-gon. Of note, in section V, article 303 of Disquisitiones, Gauss summarized his calculations of class numbers of imaginary quadratic number fields, and in fact found all imaginary quadratic number fields of class numbers 1, 2, and 3 (confirmed in 1986) as he had conjectured. In section VII, article 358, Gauss proved what can be interpreted as the first non-trivial case of the Riemann Hypothesis for curves over finite fields (the Hasse–Weil theorem). + +=== "Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält" === +Peter Gustav Lejeune Dirichlet (1837) +Pioneering paper in analytic number theory, which introduced Dirichlet characters and their L-functions to establish Dirichlet's theorem on arithmetic progressions. In subsequent publications, Dirichlet used these tools to determine, among other things, the class number for quadratic forms. + +=== "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" === +Bernhard Riemann (1859) +"Über die Anzahl der Primzahlen unter einer gegebenen Grösse" (or "On the Number of Primes Less Than a Given Magnitude") is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. Although it is the only paper he ever published on number theory, it contains ideas which influenced dozens of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory. It also contains the famous Riemann Hypothesis, one of the most important open problems in mathematics. + +=== Vorlesungen über Zahlentheorie === +Peter Gustav Lejeune Dirichlet and Richard Dedekind +Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P. G. Lejeune Dirichlet and R. Dedekind, and published in 1863. +The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert. Dirichlet does not explicitly recognise the concept of the group that is central to modern algebra, but many of his proofs show an implicit understanding of group theory. + +=== Zahlbericht === +David Hilbert (1897) +Unified and made accessible many of the developments in algebraic number theory made during the nineteenth century. Although criticized by André Weil (who stated "more than half of his famous Zahlbericht is little more than an account of Kummer's number-theoretical work, with inessential improvements") and Emmy Noether, it was highly influential for many years following its publication. + +=== Fourier Analysis in Number Fields and Hecke's Zeta-Functions === +John Tate (1950) +Generally referred to simply as Tate's Thesis, Tate's Princeton PhD thesis, under Emil Artin, is a reworking of Erich Hecke's theory of zeta- and L-functions in terms of Fourier analysis on the adeles. The introduction of these methods into number theory made it possible to formulate extensions of Hecke's results to more general L-functions such as those arising from automorphic forms. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-3.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-3.md new file mode 100644 index 000000000..7d68f616c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-3.md @@ -0,0 +1,82 @@ +--- +title: "List of publications in mathematics" +chunk: 4/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +=== "Automorphic Forms on GL(2)" === +Hervé Jacquet and Robert Langlands (1970) +This publication offers evidence towards Langlands' conjectures by reworking and expanding the classical theory of modular forms and their L-functions through the introduction of representation theory. + +=== "La conjecture de Weil. I." === +Pierre Deligne (1974) +Proved the Riemann hypothesis for varieties over finite fields, settling the last of the open Weil conjectures. + +=== "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" === +Gerd Faltings (1983) +Faltings proves a collection of important results in this paper, the most famous of which is the first proof of the Mordell conjecture (a conjecture dating back to 1922). Other theorems proved in this paper include an instance of the Tate conjecture (relating the homomorphisms between two abelian varieties over a number field to the homomorphisms between their Tate modules) and some finiteness results concerning abelian varieties over number fields with certain properties. + +=== "Modular Elliptic Curves and Fermat's Last Theorem" === +Andrew Wiles (1995) +This article proceeds to prove a special case of the Shimura–Taniyama conjecture through the study of the deformation theory of Galois representations. This in turn implies the famed Fermat's Last Theorem. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory. + +=== The geometry and cohomology of some simple Shimura varieties === +Michael Harris and Richard Taylor (2001) +Harris and Taylor provide the first proof of the local Langlands conjecture for GL(n). As part of the proof, this monograph also makes an in depth study of the geometry and cohomology of certain Shimura varieties at primes of bad reduction. + +=== "Le lemme fondamental pour les algèbres de Lie" === +Ngô Bảo Châu (2008) +Ngô Bảo Châu proved a long-standing unsolved problem in the classical Langlands program, using methods from the Geometric Langlands program. + +=== "Perfectoid space" === +Peter Scholze (2012) +Peter Scholze introduced Perfectoid space. + +== Analysis == + +=== Introductio in analysin infinitorum === +Leonhard Euler (1748) +The eminent historian of mathematics Carl Boyer once called Euler's Introductio in analysin infinitorum the greatest modern textbook in mathematics. Published in two volumes, this book more than any other work succeeded in establishing analysis as a major branch of mathematics, with a focus and approach distinct from that used in geometry and algebra. Notably, Euler identified functions rather than curves to be the central focus in his book. Logarithmic, exponential, trigonometric, and transcendental functions were covered, as were expansions into partial fractions, evaluations of ζ(2k) for k a positive integer between 1 and 13, infinite series and infinite product formulas, continued fractions, and partitions of integers. In this work, Euler proved that every rational number can be written as a finite continued fraction, that the continued fraction of an irrational number is infinite, and derived continued fraction expansions for e and + + + + + + + e + + + + + + {\displaystyle \textstyle {\sqrt {e}}} + +. This work also contains a statement of Euler's formula and a statement of the pentagonal number theorem, which he had discovered earlier and would publish a proof for in 1751. + +==== Yuktibhāṣā ==== +Jyeshtadeva (1501) +Written in India in 1530, + and served as a summary of the Kerala School's achievements in infinite series, trigonometry and mathematical analysis, most of which were earlier discovered by the 14th century mathematician Madhava. Some of its important developments includes infinite series and Taylor series expansion of some trigonometry functions and π approximation. + +=== Calculus === + +==== Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus ==== +Gottfried Leibniz (1684) +Leibniz's first publication on differential calculus, containing the now familiar notation for differentials as well as rules for computing the derivatives of powers, products and quotients. + +==== Philosophiae Naturalis Principia Mathematica ==== +Isaac Newton (1687) +The Philosophiae Naturalis Principia Mathematica (Latin: "mathematical principles of natural philosophy", often Principia or Principia Mathematica for short) is a three-volume work by Isaac Newton published on 5 July 1687. Perhaps the most influential scientific book ever published, it contains the statement of Newton's laws of motion forming the foundation of classical mechanics as well as his law of universal gravitation, and derives Kepler's laws for the motion of the planets (which were first obtained empirically). Here was born the practice, now so standard we identify it with science, of explaining nature by postulating mathematical axioms and demonstrating that their conclusion are observable phenomena. In formulating his physical theories, Newton freely used his unpublished work on calculus. When he submitted Principia for publication, however, Newton chose to recast the majority of his proofs as geometric arguments. + +==== Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum ==== + +Leonhard Euler (1755) +Published in two books, Euler's textbook on differential calculus presented the subject in terms of the function concept, which he had introduced in his 1748 Introductio in analysin infinitorum. This work opens with a study of the calculus of finite differences and makes a thorough investigation of how differentiation behaves under substitutions. Also included is a systematic study of Bernoulli polynomials and the Bernoulli numbers (naming them as such), a demonstration of how the Bernoulli numbers are related to the coefficients in the Euler–Maclaurin formula and the values of ζ(2n), a further study of Euler's constant (including its connection to the gamma function), and an application of partial fractions to differentiation. + +==== Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe ==== +Bernhard Riemann (1867) +Written in 1853, Riemann's work on trigonometric series was published posthumously. In it, he extended Cauchy's definition of the integral to that of the Riemann integral, allowing some functions with dense subsets of discontinuities on an interval to be integrated (which he demonstrated by an example). He also stated the Riemann series theorem, proved the Riemann–Lebesgue lemma for the case of bounded Riemann integrable functions, and developed the Riemann localization principle. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-4.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-4.md new file mode 100644 index 000000000..6f3e6da5c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-4.md @@ -0,0 +1,241 @@ +--- +title: "List of publications in mathematics" +chunk: 5/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +==== Intégrale, longueur, aire ==== +Henri Lebesgue (1901) +Lebesgue's doctoral dissertation, summarizing and extending his research to date regarding his development of measure theory and the Lebesgue integral. + +=== Complex analysis === + +==== Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse ==== +Bernhard Riemann (1851) +Riemann's doctoral dissertation introduced the notion of a Riemann surface, conformal mapping, simple connectivity, the Riemann sphere, the Laurent series expansion for functions having poles and branch points, and the Riemann mapping theorem. + +=== Functional analysis === + +==== Théorie des opérations linéaires ==== +Stefan Banach (1932; originally published 1931 in Polish under the title Teorja operacyj.) +Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 11 January 2014. Retrieved 11 July 2020. +The first mathematical monograph on the subject of linear metric spaces, bringing the abstract study of functional analysis to the wider mathematical community. The book introduced the ideas of a normed space and the notion of a so-called B-space, a complete normed space. The B-spaces are now called Banach spaces and are one of the basic objects of study in all areas of modern mathematical analysis. Banach also gave proofs of versions of the open mapping theorem, closed graph theorem, and Hahn–Banach theorem. + +==== Produits Tensoriels Topologiques et Espaces Nucléaires ==== +Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. MR 0075539. OCLC 9308061. +Grothendieck's thesis introduced the notion of a nuclear space, tensor products of locally convex topological vector spaces, and the start of Grothendieck's work on tensor products of Banach spaces. +Alexander Grothendieck also wrote a textbook on topological vector spaces: + +Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. + +==== Sur certains espaces vectoriels topologiques ==== +Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190. + +=== Fourier analysis === + +==== Mémoire sur la propagation de la chaleur dans les corps solides ==== +Joseph Fourier (1807) +Introduced Fourier analysis, specifically Fourier series. Key contribution was to not simply use trigonometric series, but to model all functions by trigonometric series: + + + + + φ + ( + y + ) + = + a + cos + ⁡ + + + + π + y + + 2 + + + + + + a + ′ + + cos + ⁡ + 3 + + + + π + y + + 2 + + + + + + a + ″ + + cos + ⁡ + 5 + + + + π + y + + 2 + + + + + ⋯ + . + + + {\displaystyle \varphi (y)=a\cos {\frac {\pi y}{2}}+a'\cos 3{\frac {\pi y}{2}}+a''\cos 5{\frac {\pi y}{2}}+\cdots .} + + +Multiplying both sides by + + + + cos + ⁡ + ( + 2 + i + + + 1 + ) + + + + π + y + + 2 + + + + + {\displaystyle \cos(2i+1){\frac {\pi y}{2}}} + +, and then integrating from + + + + y + = + − + 1 + + + {\displaystyle y=-1} + + to + + + + y + = + + + 1 + + + {\displaystyle y=+1} + + yields: + + + + + + a + + i + + + = + + ∫ + + − + 1 + + + 1 + + + φ + ( + y + ) + cos + ⁡ + ( + 2 + i + + + 1 + ) + + + + π + y + + 2 + + + + d + y + . + + + {\displaystyle a_{i}=\int _{-1}^{1}\varphi (y)\cos(2i+1){\frac {\pi y}{2}}\,dy.} + + +When Fourier submitted his paper in 1807, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and [...] his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. Making Fourier series rigorous, which in detail took over a century, led directly to a number of developments in analysis, notably the rigorous statement of the integral via the Dirichlet integral and later the Lebesgue integral. + +==== Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données ==== +Peter Gustav Lejeune Dirichlet (1829, expanded German edition in 1837) +In his habilitation thesis on Fourier series, Riemann characterized this work of Dirichlet as "the first profound paper about the subject". This paper gave the first rigorous proof of the convergence of Fourier series under fairly general conditions (piecewise continuity and monotonicity) by considering partial sums, which Dirichlet transformed into a particular Dirichlet integral involving what is now called the Dirichlet kernel. This paper introduced the nowhere continuous Dirichlet function and an early version of the Riemann–Lebesgue lemma. + +==== On convergence and growth of partial sums of Fourier series ==== +Lennart Carleson (1966) +Settled Lusin's conjecture that the Fourier expansion of any + + + + + L + + 2 + + + + + {\displaystyle L^{2}} + + function converges almost everywhere. + +== Geometry == + +=== Baudhayana Sulba Sutra === +Baudhayana +Believed to have been written around the 8th century BCE, this is one of the oldest mathematical texts. It laid the foundations of Indian mathematics and was influential in South Asia . Though this was primarily a geometrical text, it also contained some important algebraic developments, including the list of Pythagorean triples discovered algebraically, geometric solutions of linear equations, the use of quadratic equations and square root of 2. + +=== Euclid's Elements === +Euclid +Publication data: c. 300 BC +Online version: Interactive Java version +This is often regarded as not only the most important work in geometry but one of the most important works in mathematics. It contains many important results in plane and solid geometry, algebra (books II and V), and number theory (book VII, VIII, and IX). More than any specific result in the publication, it seems that the major achievement of this publication is the promotion of an axiomatic approach as a means for proving results. Euclid's Elements has been referred to as the most successful and influential textbook ever written. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-5.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-5.md new file mode 100644 index 000000000..439f57d66 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-5.md @@ -0,0 +1,67 @@ +--- +title: "List of publications in mathematics" +chunk: 6/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +=== The Nine Chapters on the Mathematical Art === +Unknown author +This was a Chinese mathematics book, mostly geometric, composed during the Han dynasty, perhaps as early as 200 BC. It remained the most important textbook in China and East Asia for over a thousand years, similar to the position of Euclid's Elements in Europe. Among its contents: Linear problems solved using the principle known later in the West as the rule of false position. Problems with several unknowns, solved by a principle similar to Gaussian elimination. Problems involving the principle known in the West as the Pythagorean theorem. The earliest solution of a matrix using a method equivalent to the modern method. + +=== The Conics === +Apollonius of Perga +The Conics was written by Apollonius of Perga, a Greek mathematician. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes. It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which we know them. + +=== Surya Siddhanta === +Unknown (400 CE) +It describes the archeo-astronomy theories, principles and methods of the ancient Hindus. This siddhanta is supposed to be the knowledge that the Sun god gave to an Asura called Maya. It uses sine (jya), cosine (kojya or "perpendicular sine") and inverse sine (otkram jya) for the first time . Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East. + +=== Aryabhatiya === +Aryabhata (499 CE) +This was a highly influential text during the Golden Age of mathematics in India. The text was highly concise and therefore elaborated upon in commentaries by later mathematicians. It made significant contributions to geometry and astronomy, including introduction of sine/ cosine, determination of the approximate value of pi and accurate calculation of the earth's circumference. + +=== La Géométrie === +René Descartes +La Géométrie was published in 1637 and written by René Descartes. The book was influential in developing the Cartesian coordinate system and specifically discussed the representation of points of a plane, via real numbers; and the representation of curves, via equations. + +=== Grundlagen der Geometrie === +David Hilbert +Online version: English +Publication data: Hilbert, David (1899). Grundlagen der Geometrie. Teubner-Verlag Leipzig. ISBN 978-1-4020-2777-2. {{cite book}}: ISBN / Date incompatibility (help) +Hilbert's axiomatization of geometry, whose primary influence was in its pioneering approach to metamathematical questions including the use of models to prove axiom independence and the importance of establishing the consistency and completeness of an axiomatic system. + +=== Regular Polytopes === +H.S.M. Coxeter +Regular Polytopes is a comprehensive survey of the geometry of regular polytopes, the generalisation of regular polygons and regular polyhedra to higher dimensions. Originating with an essay entitled Dimensional Analogy written in 1923, the first edition of the book took Coxeter 24 years to complete. Originally written in 1947, the book was updated and republished in 1963 and 1973. + +=== Differential geometry === + +==== Recherches sur la courbure des surfaces ==== +Leonhard Euler (1760) +Publication data: Mémoires de l'académie des sciences de Berlin 16 (1760) pp. 119–143; published 1767. (Full text and an English translation available from the Dartmouth Euler archive.) +Established the theory of surfaces, and introduced the idea of principal curvatures, laying the foundation for subsequent developments in the differential geometry of surfaces. + +==== Disquisitiones generales circa superficies curvas ==== +Carl Friedrich Gauss (1827) +Publication data: "Disquisitiones generales circa superficies curvas", Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores Vol. VI (1827), pp. 99–146; "General Investigations of Curved Surfaces" (published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead. +Groundbreaking work in differential geometry, introducing the notion of Gaussian curvature and Gauss's celebrated Theorema Egregium. + +==== Über die Hypothesen, welche der Geometrie zu Grunde Liegen ==== +Bernhard Riemann (1854) +Publication data: "Über die Hypothesen, welche der Geometrie zu Grunde Liegen", Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Vol. 13, 1867. English translation +Riemann's famous Habiltationsvortrag, in which he introduced the notions of a manifold, Riemannian metric, and curvature tensor. Richard Dedekind reported on the reaction of the then 77 year old Gauss to Riemann's presentation, stating that it had "surpassed all his expectations" and that he spoke "with the greatest appreciation, and with an excitement rare for him, about the depth of the ideas presented by Riemann." + +==== Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal ==== +Gaston Darboux +Publication data: Darboux, Gaston (1887,1889,1896) (1890). Leçons sur la théorie génerale des surfaces. Gauthier-Villars.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) Volume I, Volume II, Volume III, Volume IV +Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal (on the General Theory of Surfaces and the Geometric Applications of Infinitesimal Calculus). A treatise covering virtually every aspect of the 19th century differential geometry of surfaces. + +== Topology == + +=== Analysis situs === +Henri Poincaré (1895, 1899–1905) +Description: Poincaré's Analysis Situs and his Compléments à l'Analysis Situs laid the general foundations for algebraic topology. In these papers, Poincaré introduced the notions of homology and the fundamental group, provided an early formulation of Poincaré duality, gave the Euler–Poincaré characteristic for chain complexes, and mentioned several important conjectures including the Poincaré conjecture, demonstrated by Grigori Perelman in 2003. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-6.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-6.md new file mode 100644 index 000000000..c41582c22 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-6.md @@ -0,0 +1,71 @@ +--- +title: "List of publications in mathematics" +chunk: 7/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +=== L'anneau d'homologie d'une représentation, Structure de l'anneau d'homologie d'une représentation === +Jean Leray (1946) +These two Comptes Rendus notes of Leray from 1946 introduced the novel concepts of sheafs, sheaf cohomology, and spectral sequences, which he had developed during his years of captivity as a prisoner of war. Leray's announcements and applications (published in other Comptes Rendus notes from 1946) drew immediate attention from other mathematicians. Subsequent clarification, development, and generalization by Henri Cartan, Jean-Louis Koszul, Armand Borel, Jean-Pierre Serre, and Leray himself allowed these concepts to be understood and applied to many other areas of mathematics. Dieudonné would later write that these notions created by Leray "undoubtedly rank at the same level in the history of mathematics as the methods invented by Poincaré and Brouwer". + +=== Quelques propriétés globales des variétés differentiables === +René Thom (1954) +In this paper, Thom proved the Thom transversality theorem, introduced the notions of oriented and unoriented cobordism, and demonstrated that cobordism groups could be computed as the homotopy groups of certain Thom spaces. Thom completely characterized the unoriented cobordism ring and achieved strong results for several problems, including Steenrod's problem on the realization of cycles. + +== Category theory == + +=== "General Theory of Natural Equivalences" === +Samuel Eilenberg and Saunders Mac Lane (1945) +The first paper on category theory. Mac Lane later wrote in Categories for the Working Mathematician that he and Eilenberg introduced categories so that they could introduce functors, and they introduced functors so that they could introduce natural equivalences. Prior to this paper, "natural" was used in an informal and imprecise way to designate constructions that could be made without making any choices. Afterwards, "natural" had a precise meaning which occurred in a wide variety of contexts and had powerful and important consequences. + +=== Categories for the Working Mathematician === +Saunders Mac Lane (1971, second edition 1998) +Saunders Mac Lane, one of the founders of category theory, wrote this exposition to bring categories to the masses. Mac Lane brings to the fore the important concepts that make category theory useful, such as adjoint functors and universal properties. + +=== Higher Topos Theory === +Jacob Lurie (2010) +This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included. (see arXiv.) + +== Set theory == + +=== "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" === +Georg Cantor (1874) +Online version: Online version +Contains the first published proof that the set of all real numbers is uncountable; also contains a proof that the set of algebraic numbers is countable. Both published proofs were supplied by Richard Dedekind in correspondence with Cantor, but left uncredited by Cantor. (See Georg Cantor's first set theory article.) + +=== Grundzüge der Mengenlehre === +Felix Hausdorff +First published in 1914, this was the first comprehensive introduction to set theory. Besides the systematic treatment of known results in set theory, the book also contains chapters on measure theory and topology, which were then still considered parts of set theory. Here Hausdorff presents and develops highly original material which was later to become the basis for those areas. + +=== "The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory" === +Kurt Gödel (1938) +Gödel proves the results of the title. Also, in the process, introduces the class L of constructible sets, a major influence in the development of axiomatic set theory. + +=== "The Independence of the Continuum Hypothesis" === +Paul J. Cohen (1963, 1964) +Cohen's breakthrough work proved the independence of the continuum hypothesis and axiom of choice with respect to Zermelo–Fraenkel set theory. In proving this Cohen introduced the concept of forcing which led to many other major results in axiomatic set theory. + +== Logic == + +=== The Laws of Thought === +George Boole (1854) +Published in 1854, The Laws of Thought was the first book to provide a mathematical foundation for logic. Its aim was a complete re-expression and extension of Aristotle's logic in the language of mathematics. Boole's work founded the discipline of algebraic logic and would later be central for Claude Shannon in the development of digital logic. + +=== Begriffsschrift === +Gottlob Frege (1879) +Published in 1879, the title Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formula language, modelled on that of arithmetic, of pure thought". Frege's motivation for developing his formal logical system was similar to Leibniz's desire for a calculus ratiocinator. Frege defines a logical calculus to support his research in the foundations of mathematics. Begriffsschrift is both the name of the book and the calculus defined therein. It was arguably the most significant publication in logic since Aristotle. + +=== Formulario mathematico === +Giuseppe Peano (1895) +First published in 1895, the Formulario mathematico was the first mathematical book written entirely in a formalized language. It contained a description of mathematical logic and many important theorems in other branches of mathematics. Many of the notations introduced in the book are now in common use. + +=== Principia Mathematica === +Bertrand Russell and Alfred North Whitehead (1910–1913) +The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Bertrand Russell and Alfred North Whitehead and published in 1910–1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. These questions were settled, in a rather surprising way, by Gödel's incompleteness theorem in 1931. + +=== "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" === +(On Formally Undecidable Propositions of Principia Mathematica and Related Systems) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-7.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-7.md new file mode 100644 index 000000000..9a20e5b90 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-7.md @@ -0,0 +1,114 @@ +--- +title: "List of publications in mathematics" +chunk: 8/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +Kurt Gödel (1931) +Online version: Online version +In mathematical logic, Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. +The first incompleteness theorem states: + +For any formal system such that (1) it is + + + + ω + + + {\displaystyle \omega } + +-consistent (omega-consistent), (2) it has a recursively definable set of axioms and rules of derivation, and (3) every recursive relation of natural numbers is definable in it, there exists a formula of the system such that, according to the intended interpretation of the system, it expresses a truth about natural numbers and yet it is not a theorem of the system. + +=== Systems of Logic Based on Ordinals === +Alan Turing's PhD thesis (1938) + +== Combinatorics == + +=== "On sets of integers containing no k elements in arithmetic progression" === +Endre Szemerédi (1975) +Settled a conjecture of Paul Erdős and Pál Turán (now known as Szemerédi's theorem) that if a sequence of natural numbers has positive upper density then it contains arbitrarily long arithmetic progressions. Szemerédi's solution has been described as a "masterpiece of combinatorics" and it introduced new ideas and tools to the field including a weak form of the Szemerédi regularity lemma. + +=== Graph theory === + +==== Solutio problematis ad geometriam situs pertinentis ==== +Leonhard Euler (1741) +Euler's original publication (in Latin) +Euler's solution of the Königsberg bridge problem in Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position) is considered to be the first theorem of graph theory. + +==== "On the evolution of random graphs" ==== +Paul Erdős and Alfréd Rényi (1960) +Provides a detailed discussion of sparse random graphs, including distribution of components, occurrence of small subgraphs, and phase transitions. + +==== "Network Flows and General Matchings" ==== +L. R. Ford, Jr. & D. R. Fulkerson +Flows in Networks. Prentice-Hall, 1962. +Presents the Ford–Fulkerson algorithm for solving the maximum flow problem, along with many ideas on flow-based models. + +== Probability theory and statistics == +See list of important publications in statistics. + +== Game theory == + +=== "Zur Theorie der Gesellschaftsspiele" === +John von Neumann (1928) +Went well beyond Émile Borel's initial investigations into strategic two-person game theory by proving the minimax theorem for two-person, zero-sum games. + +=== Theory of Games and Economic Behavior === +Oskar Morgenstern, John von Neumann (1944) +This book led to the investigation of modern game theory as a prominent branch of mathematics. This work contained the method for finding optimal solutions for two-person zero-sum games. + +=== "Equilibrium Points in N-person Games" === +Nash, John F. (January 1950). "Equilibrium Points in N-person Games". Proceedings of the National Academy of Sciences of the United States of America. 36 (1): 48–9. Bibcode:1950PNAS...36...48N. doi:10.1073/pnas.36.1.48. MR 0031701. PMC 1063129. PMID 16588946. +Nash equilibrium + +=== On Numbers and Games === +John Horton Conway (1976) +The book is in two, {0,1|}, parts. The zeroth part is about numbers, the first part about games – both the values of games and also some real games that can be played such as Nim, Hackenbush, Col and Snort amongst the many described. + +=== Winning Ways for your Mathematical Plays === +Elwyn Berlekamp, John Conway and Richard K. Guy (1982) +A compendium of information on mathematical games. It was first published in 1982 in two volumes, one focusing on Combinatorial game theory and surreal numbers, and the other concentrating on a number of specific games. + +== Information theory == + +=== A Mathematical Theory of Communication === +Claude Shannon (1948) +An article, later expanded into a book, which developed the concepts of information entropy and redundancy, and introduced the term bit (which Shannon credited to John Tukey) as a unit of information. + +== Fractals == + +=== How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension === +Benoît Mandelbrot (1967) +A discussion of self-similar curves that have fractional dimensions between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. +Shows Mandelbrot's early thinking on fractals, and is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work. + +== Numerical analysis == + +=== Optimization === + +==== Method of Fluxions ==== +Isaac Newton (1736) +Method of Fluxions was a book written by Isaac Newton. The book was completed in 1671, and published in 1736. Within this book, Newton describes a method (the Newton–Raphson method) for finding the real zeroes of a function. + +==== Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies ==== +Joseph Louis Lagrange (1761) +Major early work on the calculus of variations, building upon some of Lagrange's prior investigations as well as those of Euler. Contains investigations of minimal surface determination as well as the initial appearance of Lagrange multipliers. + +==== "Математические методы организации и планирования производства" ==== +Leonid Kantorovich (1939) "[The Mathematical Method of Production Planning and Organization]" (in Russian). +Kantorovich wrote the first paper on production planning, which used Linear Programs as the model. He received the Nobel prize for this work in 1975. + +==== "Decomposition Principle for Linear Programs" ==== +George Dantzig and P. Wolfe +Operations Research 8:101–111, 1960. +Dantzig's is considered the father of linear programming in the western world. He independently invented the simplex algorithm. Dantzig and Wolfe worked on decomposition algorithms for large-scale linear programs in factory and production planning. + +==== "How Good is the Simplex Algorithm?" ==== +Victor Klee and George J. Minty +Klee, Victor; Minty, George J. (1972). "How good is the simplex algorithm?". In Shisha, Oved (ed.). Inequalities III (Proceedings of the Third Symposium on Inequalities held at the University of California, Los Angeles, Calif., September 1–9, 1969, dedicated to the memory of Theodore S. Motzkin). New York-London: Academic Press. pp. 159–175. MR 0332165. +Klee and Minty gave an example showing that the simplex algorithm can take exponentially many steps to solve a linear program. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-8.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-8.md new file mode 100644 index 000000000..593f2604e --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-8.md @@ -0,0 +1,87 @@ +--- +title: "List of publications in mathematics" +chunk: 9/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +==== "Полиномиальный алгоритм в линейном программировании" ==== +Khachiyan, Leonid Genrikhovich (1979). Полиномиальный алгоритм в линейном программировании [A polynomial algorithm for linear programming]. Doklady Akademii Nauk SSSR (in Russian). 244: 1093–1096.. +Khachiyan's work on the ellipsoid method. This was the first polynomial time algorithm for linear programming. + +== Early manuscripts == + +These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the history of mathematics. + +=== Moscow Mathematical Papyrus === +This is one of the earliest mathematical treatises that still survives today. The Papyrus contains 25 problems involving arithmetic, geometry, and algebra, each with a solution given. Written in Ancient Egypt at approximately 1850 BC. + +=== Rhind Mathematical Papyrus === +Ahmes (scribe) +One of the oldest mathematical texts, dating to the Second Intermediate Period of ancient Egypt. It was copied by the scribe Ahmes (properly Ahmose) from an older Middle Kingdom papyrus. It laid the foundations of Egyptian mathematics and in turn, later influenced Greek and Hellenistic mathematics. Besides describing how to obtain an approximation of π only missing the mark by less than one per cent, Ahmes make use of a kind of an analogue of the cotangent. + +=== Archimedes Palimpsest === +Archimedes of Syracuse +Although the only mathematical tools at its author's disposal were what we might now consider secondary-school geometry, he used those methods with rare brilliance, explicitly using infinitesimals to solve problems that would now be treated by integral calculus. Among those problems were that of the center of gravity of a solid hemisphere, that of the center of gravity of a frustum of a circular paraboloid, and that of the area of a region bounded by a parabola and one of its secant lines. For explicit details of the method used, see Archimedes' use of infinitesimals. + +=== The Sand Reckoner === +Archimedes of Syracuse +Online version: Online version +The first known (European) system of number-naming that can be expanded beyond the needs of everyday life. + +== Textbooks == + +=== Abstract Algebra === +David Dummit and Richard Foote +Dummit and Foote has become the modern dominant abstract algebra textbook following Jacobson's Basic Algebra. + +=== Arithmetika Horvatzka === +Mihalj Šilobod Bolšić +Arithmetika Horvatzka (1758) was the first Croatian language arithmetic textbook, written in the vernacular Kajkavian dialect of Croatian language. It established a complete system of arithmetic terminology in Croatian, and vividly used examples from everyday life in Croatia to present mathematical operations. Although it was clear that Šilobod had made use of words that were in dictionaries, this was clearly insufficient for his purposes; and he made up some names by adapting Latin terminology to Kaikavian use. Full text of Arithmetika Horvatszka is available via archive.org. + +=== Synopsis of Pure Mathematics === +G. S. Carr +Contains over 6000 theorems of mathematics, assembled by George Shoobridge Carr for the purpose of training his students for the Cambridge Mathematical Tripos exams. Studied extensively by Ramanujan. (first half here) + +=== Éléments de mathématique === +Nicolas Bourbaki +One of the most influential books in French mathematical literature. It introduces some of the notations and definitions that are now usual (the symbol ∅ or the term bijective for example). Characterized by an extreme level of rigour, formalism and generality (up to the point of being highly criticized for that), its publication started in 1939 and is still unfinished today. + +=== Arithmetick: or, The Grounde of Arts === +Robert Recorde +Written in 1542, it was the first really popular arithmetic book written in the English Language. + +=== Cocker's Arithmetick === +Edward Cocker (authorship disputed) +Textbook of arithmetic published in 1678 by John Hawkins, who claimed to have edited manuscripts left by Edward Cocker, who had died in 1676. This influential mathematics textbook used to teach arithmetic in schools in the United Kingdom for over 150 years. + +=== The Schoolmaster's Assistant, Being a Compendium of Arithmetic both Practical and Theoretical === +Thomas Dilworth +An early and popular English arithmetic textbook published in America in the 18th century. The book reached from the introductory topics to the advanced in five sections. + +=== Geometry === +Andrei Kiselyov +Publication data: 1892 +The most widely used and influential textbook in Russian mathematics. (See Kiselyov page.) + +=== A Course of Pure Mathematics === +G. H. Hardy +A classic textbook in introductory mathematical analysis, written by G. H. Hardy. It was first published in 1908, and went through many editions. It was intended to help reform mathematics teaching in the UK, and more specifically in the University of Cambridge, and in schools preparing pupils to study mathematics at Cambridge. As such, it was aimed directly at "scholarship level" students – the top 10% to 20% by ability. The book contains a large number of difficult problems. The content covers introductory calculus and the theory of infinite series. + +=== Moderne Algebra === +B. L. van der Waerden +The first introductory textbook (graduate level) expounding the abstract approach to algebra developed by Emil Artin and Emmy Noether. First published in German in 1931 by Springer Verlag. A later English translation was published in 1949 by Frederick Ungar Publishing Company. + +=== Algebra === +Saunders Mac Lane and Garrett Birkhoff +A definitive introductory text for abstract algebra using a category theoretic approach. Both a rigorous introduction from first principles, and a reasonably comprehensive survey of the field. + +=== Calculus, Vol. 1 === +Tom M. Apostol + +=== Algebraic Geometry === +Robin Hartshorne +The first comprehensive introductory (graduate level) text in algebraic geometry that used the language of schemes and cohomology. Published in 1977, it lacks aspects of the scheme language which are nowadays considered central, like the functor of points. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-9.md b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-9.md new file mode 100644 index 000000000..acb512e2d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-9.md @@ -0,0 +1,80 @@ +--- +title: "List of publications in mathematics" +chunk: 10/10 +source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:32.440767+00:00" +instance: "kb-cron" +--- + +=== Naive Set Theory === +Paul Halmos +An undergraduate introduction to not-very-naive set theory which has lasted for decades. It is still considered by many to be the best introduction to set theory for beginners. While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of Zermelo–Fraenkel set theory and gives correct and rigorous definitions for basic objects. Where it differs from a "true" axiomatic set theory book is its character: There are no long-winded discussions of axiomatic minutiae, and there is next to nothing about topics like large cardinals. Instead it aims, and succeeds, in being intelligible to someone who has never thought about set theory before. + +=== Cardinal and Ordinal Numbers === +Wacław Sierpiński +The nec plus ultra reference for basic facts about cardinal and ordinal numbers. If you have a question about the cardinality of sets occurring in everyday mathematics, the first place to look is this book, first published in the early 1950s but based on the author's lectures on the subject over the preceding 40 years. + +=== Set Theory: An Introduction to Independence Proofs === +Kenneth Kunen +This book is not really for beginners, but graduate students with some minimal experience in set theory and formal logic will find it a valuable self-teaching tool, particularly in regard to forcing. It is far easier to read than a true reference work such as Jech, Set Theory. It may be the best textbook from which to learn forcing, though it has the disadvantage that the exposition of forcing relies somewhat on the earlier presentation of Martin's axiom. + +=== Topologie === +Pavel Sergeevich Alexandrov +Heinz Hopf +First published round 1935, this text was a pioneering "reference" text book in topology, already incorporating many modern concepts from set-theoretic topology, homological algebra and homotopy theory. + +=== General Topology === +John L. Kelley +First published in 1955, for many years the only introductory graduate level textbook in the US, teaching the basics of point set, as opposed to algebraic, topology. Prior to this the material, essential for advanced study in many fields, was only available in bits and pieces from texts on other topics or journal articles. + +=== Topology from the Differentiable Viewpoint === +John Milnor +This short book introduces the main concepts of differential topology in Milnor's lucid and concise style. While the book does not cover very much, its topics are explained beautifully in a way that illuminates all their details. + +=== Number Theory, An approach through history from Hammurapi to Legendre === +André Weil +An historical study of number theory, written by one of the 20th century's greatest researchers in the field. The book covers some thirty six centuries of arithmetical work but the bulk of it is devoted to a detailed study and exposition of the work of Fermat, Euler, Lagrange, and Legendre. The author wishes to take the reader into the workshop of his subjects to share their successes and failures. A rare opportunity to see the historical development of a subject through the mind of one of its greatest practitioners. + +=== An Introduction to the Theory of Numbers === +G. H. Hardy and E. M. Wright +An Introduction to the Theory of Numbers was first published in 1938, and is still in print, with the latest edition being the 6th (2008). It is likely that almost every serious student and researcher into number theory has consulted this book, and probably has it on their bookshelf. It was not intended to be a textbook, and is rather an introduction to a wide range of differing areas of number theory which would now almost certainly be covered in separate volumes. The writing style has long been regarded as exemplary, and the approach gives insight into a variety of areas without requiring much more than a good grounding in algebra, calculus and complex numbers. + +=== Foundations of Differential Geometry === +Shoshichi Kobayashi and Katsumi Nomizu (1963; 1969) + +=== Hodge Theory and Complex Algebraic Geometry I === + +=== Hodge Theory and Complex Algebraic Geometry II === +Claire Voisin + +== Handbooks == + +=== Bronshtein and Semendyayev === +Ilya Nikolaevich Bronshtein and Konstantin Adolfovic Semendyayev +Bronshtein and Semendyayev is the informal name of a comprehensive handbook of fundamental working knowledge of mathematics and table of formulas originally compiled by the Russian mathematician Ilya Nikolaevich Bronshtein and engineer Konstantin Adolfovic Semendyayev. The work was first published in 1945 in Russia and soon became a "standard" and frequently used guide for scientists, engineers, and technical university students. It has been translated into German, English, and many other languages. The latest edition was published in 2015 by Springer. + +=== CRC Standard Mathematical Tables === +Editors-in-chief: Charles D. Hodgman (14th edition and earlier); Samuel M. Selby (15th–23rd editions); William H. Beyer (24th–29th editions); Daniel Zwillinger (30th–33rd editions) +CRC Standard Mathematical Tables is a comprehensive one-volume handbook of fundamental working knowledge of mathematics and table of formulas. The handbook was originally published in 1928. The latest edition was published in 2018 by CRC Press, with Daniel Zwillinger as the editor-in-chief. + +== Popular writings == + +=== Gödel, Escher, Bach === +Douglas Hofstadter +Gödel, Escher, Bach: an Eternal Golden Braid is a Pulitzer Prize-winning book, first published in 1979 by Basic Books. +It is a book about how the creative achievements of logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach interweave. As the author states: "I realized that to me, Gödel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book." + +=== The World of Mathematics === +James R. Newman +The World of Mathematics was specially designed to make mathematics more accessible to the inexperienced. It comprises nontechnical essays on every aspect of the vast subject, including articles by and about scores of eminent mathematicians, as well as literary figures, economists, biologists, and many other eminent thinkers. Includes the work of Archimedes, Galileo, Descartes, Newton, Gregor Mendel, Edmund Halley, Jonathan Swift, John Maynard Keynes, Henri Poincaré, Lewis Carroll, George Boole, Bertrand Russell, Alfred North Whitehead, John von Neumann, and many others. In addition, an informative commentary by distinguished scholar James R. Newman precedes each essay or group of essays, explaining their relevance and context in the history and development of mathematics. Originally published in 1956, it does not include many of the exciting discoveries of the later years of the 20th century but it has no equal as a general historical survey of important topics and applications. + +== See also == + Mathematics portal +List of mathematics journals + +== References == + +== External links == +MAA Basic Library List - list of books recommended by the MAA for undergraduate libraries, including books considered essential \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_q-analogs-0.md b/data/en.wikipedia.org/wiki/List_of_q-analogs-0.md new file mode 100644 index 000000000..9419c392a --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_q-analogs-0.md @@ -0,0 +1,70 @@ +--- +title: "List of q-analogs" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_q-analogs" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:33.618494+00:00" +instance: "kb-cron" +--- + +This is a list of q-analogs in mathematics and related fields. + + +== Algebra == +Iwahori–Hecke algebra +Quantum affine algebra +Quantum enveloping algebra +Quantum group + + +== Analysis == +Jackson integral +q-derivative +q-difference polynomial +Quantum calculus + + +== Combinatorics == +LLT polynomial +q-binomial coefficient +q-Pochhammer symbol +q-Vandermonde identity + + +== Orthogonal polynomials == +q-Bessel polynomials +q-Charlier polynomials +q-Hahn polynomials +q-Jacobi polynomials: +Big q-Jacobi polynomials +Continuous q-Jacobi polynomials +Little q-Jacobi polynomials +q-Krawtchouk polynomials +q-Laguerre polynomials +Continuous q-Legendre polynomials +q-Meixner polynomials +q-Meixner–Pollaczek polynomials +q-Racah polynomials + + +== Probability and statistics == +Gaussian q-distribution +q-exponential distribution +q-Weibull distribution +Tsallis q-Gaussian +Tsallis entropy + + +== Special functions == +Basic hypergeometric series +Elliptic gamma function +Hahn–Exton q-Bessel function +Jackson q-Bessel function +q-exponential +q-gamma function +q-theta function + + +== See also == +Lists of mathematics topics \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_random_number_generators-0.md b/data/en.wikipedia.org/wiki/List_of_random_number_generators-0.md index e13fd121f..2c6ad2a02 100644 --- a/data/en.wikipedia.org/wiki/List_of_random_number_generators-0.md +++ b/data/en.wikipedia.org/wiki/List_of_random_number_generators-0.md @@ -4,7 +4,7 @@ chunk: 1/1 source: "https://en.wikipedia.org/wiki/List_of_random_number_generators" category: "reference" tags: "science, encyclopedia" -date_saved: "2026-05-05T08:04:18.399851+00:00" +date_saved: "2026-05-05T08:16:35.976218+00:00" instance: "kb-cron" --- diff --git a/data/en.wikipedia.org/wiki/List_of_real_analysis_topics-0.md b/data/en.wikipedia.org/wiki/List_of_real_analysis_topics-0.md new file mode 100644 index 000000000..712d4e5c2 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_real_analysis_topics-0.md @@ -0,0 +1,377 @@ +--- +title: "List of real analysis topics" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/List_of_real_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:37.250027+00:00" +instance: "kb-cron" +--- + +This is a list of articles that are considered real analysis topics. +See also: glossary of real and complex analysis. + +== General topics == + +=== Limits === +Limit of a sequence +Subsequential limit – the limit of some subsequence +Limit of a function (see List of limits for a list of limits of common functions) +One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below +Squeeze theorem – confirms the limit of a function via comparison with two other functions +Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions + +=== Sequences and series === +(see also list of mathematical series) + +Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant +Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants +Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number +Harmonic progression – a sequence formed by taking the reciprocals of the terms of an arithmetic progression +Finite sequence – see sequence +Infinite sequence – see sequence +Divergent sequence – see limit of a sequence or divergent series +Convergent sequence – see limit of a sequence or convergent series +Cauchy sequence – a sequence whose elements become arbitrarily close to each other as the sequence progresses +Convergent series – a series whose sequence of partial sums converges +Divergent series – a series whose sequence of partial sums diverges +Power series – a series of the form + + + + f + ( + x + ) + = + + ∑ + + n + = + 0 + + + ∞ + + + + a + + n + + + + + ( + + x + − + c + + ) + + + n + + + = + + a + + 0 + + + + + + a + + 1 + + + ( + x + − + c + + ) + + 1 + + + + + + a + + 2 + + + ( + x + − + c + + ) + + 2 + + + + + + a + + 3 + + + ( + x + − + c + + ) + + 3 + + + + + ⋯ + + + {\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)^{1}+a_{2}(x-c)^{2}+a_{3}(x-c)^{3}+\cdots } + + +Taylor series – a series of the form + + + + f + ( + a + ) + + + + + + + f + ′ + + ( + a + ) + + + 1 + ! + + + + ( + x + − + a + ) + + + + + + + f + ″ + + ( + a + ) + + + 2 + ! + + + + ( + x + − + a + + ) + + 2 + + + + + + + + + f + + ( + 3 + ) + + + ( + a + ) + + + 3 + ! + + + + ( + x + − + a + + ) + + 3 + + + + + ⋯ + . + + + {\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f^{(3)}(a)}{3!}}(x-a)^{3}+\cdots .} + + +Maclaurin series – see Taylor series +Binomial series – the Maclaurin series of the function f given by f(x) = (1 + x) α +Telescoping series +Alternating series +Geometric series +Divergent geometric series +Harmonic series +Fourier series +Lambert series + +==== Summation methods ==== +Cesàro summation +Euler summation +Lambert summation +Borel summation +Summation by parts – transforms the summation of products of into other summations +Cesàro mean +Abel's summation formula + +==== More advanced topics ==== +Convolution +Cauchy product –is the discrete convolution of two sequences +Farey sequence – the sequence of completely reduced fractions between 0 and 1 +Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that. +Indeterminate forms – algebraic expressions gained in the context of limits. The indeterminate forms include 00, 0/0, 1∞, ∞ − ∞, ∞/∞, 0 × ∞, and ∞0. + +=== Convergence === +Pointwise convergence, Uniform convergence +Absolute convergence, Conditional convergence +Normal convergence +Radius of convergence + +==== Convergence tests ==== +Integral test for convergence +Cauchy's convergence test +Ratio test +Direct comparison test +Limit comparison test +Root test +Alternating series test +Dirichlet's test +Stolz–Cesàro theorem – is a criterion for proving the convergence of a sequence + +=== Functions === +Function of a real variable +Real multivariable function +Continuous function +Nowhere continuous function +Weierstrass function +Smooth function +Analytic function +Quasi-analytic function +Non-analytic smooth function +Flat function +Bump function +Differentiable function +Integrable function +Square-integrable function, p-integrable function +Monotonic function +Bernstein's theorem on monotone functions – states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions +Inverse function +Convex function, Concave function +Singular function +Harmonic function +Weakly harmonic function +Proper convex function +Rational function +Orthogonal function +Implicit and explicit functions +Implicit function theorem – allows relations to be converted to functions +Measurable function +Baire one star function +Symmetric function +Domain +Codomain +Image +Support +Differential of a function + +==== Continuity ==== +Uniform continuity +Modulus of continuity +Lipschitz continuity +Semi-continuity +Equicontinuous +Absolute continuity +Hölder condition – condition for Hölder continuity + +==== Distributions ==== +Dirac delta function +Heaviside step function +Hilbert transform +Green's function + +==== Variation ==== +Bounded variation +Total variation + +=== Derivatives === +Second derivative +Inflection point – found using second derivatives +Directional derivative, Total derivative, Partial derivative + +==== Differentiation rules ==== +Linearity of differentiation +Product rule +Quotient rule +Chain rule +Inverse function theorem – gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function + +==== Differentiation in geometry and topology ==== +see also List of differential geometry topics + +Differentiable manifold +Differentiable structure +Submersion – a differentiable map between differentiable manifolds whose differential is everywhere surjective + +=== Integrals === +(see also Lists of integrals) + +Antiderivative +Fundamental theorem of calculus – a theorem of antiderivatives +Multiple integral +Iterated integral +Improper integral +Cauchy principal value – method for assigning values to certain improper integrals +Line integral +Anderson's theorem – says that the integral of an integrable, symmetric, unimodal, non-negative function over an n-dimensional convex body (K) does not decrease if K is translated inwards towards the origin + +==== Integration and measure theory ==== +see also List of integration and measure theory topics + +Riemann integral, Riemann sum +Riemann–Stieltjes integral +Darboux integral +Lebesgue integration \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_real_analysis_topics-1.md b/data/en.wikipedia.org/wiki/List_of_real_analysis_topics-1.md new file mode 100644 index 000000000..814fe64af --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_real_analysis_topics-1.md @@ -0,0 +1,223 @@ +--- +title: "List of real analysis topics" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/List_of_real_analysis_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:37.250027+00:00" +instance: "kb-cron" +--- + +== Fundamental theorems == +Monotone convergence theorem – relates monotonicity with convergence +Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value +Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero +Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc +Taylor's theorem – gives an approximation of a + + + + k + + + {\displaystyle k} + + times differentiable function around a given point by a + + + + k + + + {\displaystyle k} + +-th order Taylor-polynomial. +L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms +Abel's theorem – relates the limit of a power series to the sum of its coefficients +Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function +Darboux's theorem – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval +Heine–Borel theorem – sometimes used as the defining property of compactness +Bolzano–Weierstrass theorem – states that each bounded sequence in + + + + + + R + + + n + + + + + {\displaystyle \mathbb {R} ^{n}} + + has a convergent subsequence +Extreme value theorem - states that if a function + + + + f + + + {\displaystyle f} + + is continuous in the closed and bounded interval + + + + [ + a + , + b + ] + + + {\displaystyle [a,b]} + +, then it must attain a maximum and a minimum + +== Foundational topics == + +=== Numbers === + +==== Real numbers ==== +Construction of the real numbers +Natural number +Integer +Rational number +Irrational number +Completeness of the real numbers +Least-upper-bound property +Real line +Extended real number line +Dedekind cut + +==== Specific numbers ==== +0 +1 +0.999... +Infinity + +=== Sets === +Open set +Neighbourhood +Cantor set +Derived set (mathematics) +Completeness +Limit superior and limit inferior +Supremum +Infimum +Interval +Partition of an interval + +=== Maps === +Contraction mapping +Metric map +Fixed point – a point of a function that maps to itself + +== Applied mathematical tools == + +=== Infinite expressions === +Continued fraction +Series +Infinite products + +=== Inequalities === +See list of inequalities + +Triangle inequality +Bernoulli's inequality +Cauchy–Schwarz inequality +Hölder's inequality +Minkowski inequality +Jensen's inequality +Chebyshev's inequality +Inequality of arithmetic and geometric means + +=== Means === +Generalized mean +Pythagorean means +Arithmetic mean +Geometric mean +Harmonic mean +Geometric–harmonic mean +Arithmetic–geometric mean +Weighted mean +Quasi-arithmetic mean + +=== Orthogonal polynomials === +Classical orthogonal polynomials +Hermite polynomials +Laguerre polynomials +Jacobi polynomials +Gegenbauer polynomials +Legendre polynomials + +=== Spaces === +Euclidean space +Metric space +Banach fixed point theorem – guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them +Complete metric space +Topological space +Function space +Sequence space +Compact space + +=== Measures === +Lebesgue measure +Outer measure +Hausdorff measure +Dominated convergence theorem – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions. + +=== Field of sets === +Sigma-algebra + +== Historical figures == +Michel Rolle (1652–1719) +Brook Taylor (1685–1731) +Leonhard Euler (1707–1783) +Joseph-Louis Lagrange (1736–1813) +Joseph Fourier (1768–1830) +Bernard Bolzano (1781–1848) +Augustin Cauchy (1789–1857) +Niels Henrik Abel (1802–1829) +Peter Gustav Lejeune Dirichlet (1805–1859) +Karl Weierstrass (1815–1897) +Eduard Heine (1821–1881) +Pafnuty Chebyshev (1821–1894) +Leopold Kronecker (1823–1891) +Bernhard Riemann (1826–1866) +Richard Dedekind (1831–1916) +Rudolf Lipschitz (1832–1903) +Camille Jordan (1838–1922) +Jean Gaston Darboux (1842–1917) +Georg Cantor (1845–1918) +Ernesto Cesàro (1859–1906) +Otto Hölder (1859–1937) +Hermann Minkowski (1864–1909) +Alfred Tauber (1866–1942) +Felix Hausdorff (1868–1942) +Émile Borel (1871–1956) +Henri Lebesgue (1875–1941) +Wacław Sierpiński (1882–1969) +Johann Radon (1887–1956) +Karl Menger (1902–1985) + +== Related fields of analysis == +Asymptotic analysis – studies a method of describing limiting behaviour +Convex analysis – studies the properties of convex functions and convex sets +List of convexity topics +Harmonic analysis – studies the representation of functions or signals as superpositions of basic waves +List of harmonic analysis topics +Fourier analysis – studies Fourier series and Fourier transforms +List of Fourier analysis topics +List of Fourier-related transforms +Complex analysis – studies the extension of real analysis to include complex numbers +Functional analysis – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces +Nonstandard analysis – studies mathematical analysis using a rigorous treatment of infinitesimals. + +== See also == +Calculus, the classical calculus of Newton and Leibniz. +Non-standard calculus, a rigorous application of infinitesimals, in the sense of non-standard analysis, to the classical calculus of Newton and Leibniz. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_recreational_number_theory_topics-0.md b/data/en.wikipedia.org/wiki/List_of_recreational_number_theory_topics-0.md new file mode 100644 index 000000000..5ab2f053e --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_recreational_number_theory_topics-0.md @@ -0,0 +1,141 @@ +--- +title: "List of recreational number theory topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_recreational_number_theory_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:39.664179+00:00" +instance: "kb-cron" +--- + +This is a list of recreational number theory topics (see number theory, recreational mathematics). Listing here is not pejorative: many famous topics in number theory have origins in challenging problems posed purely for their own sake. +See list of number theory topics for pages dealing with aspects of number theory with more consolidated theories. + + +== Number sequences == + +Integer sequence +Fibonacci sequence +Golden mean base +Fibonacci coding +Lucas sequence +Padovan sequence +Figurate numbers +Polygonal number +Triangular number +Square number +Pentagonal number +Hexagonal number +Heptagonal number +Octagonal number +Nonagonal number +Decagonal number +Centered polygonal number +Centered square number +Centered pentagonal number +Centered hexagonal number +Tetrahedral number +Pyramidal number +Triangular pyramidal number +Square pyramidal number +Pentagonal pyramidal number +Hexagonal pyramidal number +Heptagonal pyramidal number +Octahedral number +Star number +Perfect number +Quasiperfect number +Almost perfect number +Multiply perfect number +Hyperperfect number +Semiperfect number +Primitive semiperfect number +Unitary perfect number +Weird number +Untouchable number +Amicable number +Sociable number +Abundant number +Deficient number +Amenable number +Aliquot sequence +Super-Poulet number +Lucky number +Powerful number +Primeval number +Palindromic number +Telephone number +Triangular square number +Harmonic divisor number +Sphenic number +Smith number +Double Mersenne number +Zeisel number +Heteromecic number +Niven numbers +Superparticular number +Highly composite number +Highly totient number +Practical number +Juggler sequence +Look-and-say sequence + + +== Digits == +Polydivisible number +Automorphic number +Armstrong number +Self number +Harshad number +Keith number +Kaprekar number +Digit sum +Persistence of a number +Perfect digital invariant +Happy number +Perfect digit-to-digit invariant +Factorion +Emirp +Palindromic prime +Home prime +Normal number +Stoneham number +Champernowne constant +Absolutely normal number +Repunit +Repdigit + + +== Prime and related sequences == +Semiprime +Almost prime +Unique prime +Factorial prime +Permutable prime +Palindromic prime +Cuban prime +Lucky prime + + +== Magic squares, etc. == +Ulam spiral +Magic star +Magic square +Frénicle standard form +Prime reciprocal magic square +Trimagic square +Multimagic square +Panmagic square +Satanic square +Most-perfect magic square +Geometric magic square +Conway's Lux method for magic squares +Magic cube +Perfect magic cube +Semiperfect magic cube +Bimagic cube +Trimagic cube +Multimagic cube +Magic hypercube +Magic constant +Squaring the square \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-0.md b/data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-0.md new file mode 100644 index 000000000..352bf3cc5 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-0.md @@ -0,0 +1,44 @@ +--- +title: "List of regular polytope compounds" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/List_of_regular_polytope_compounds" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:32.110805+00:00" +instance: "kb-cron" +--- + +This article lists the regular polytope compounds in Euclidean, spherical and hyperbolic spaces. + +== Two dimensional compounds == +For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon. +In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m. +A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form k{n/m}, as 2{5/2}, rather than the commonly used {10/4}. +Coxeter's extended notation for compounds is of the form c{m,n,...}[d{p,q,...}]e{s,t,...}, indicating that d distinct {p,q,...}'s together cover the vertices of {m,n,...} c times and the facets of {s,t,...} e times. If no regular {m,n,...} exists, the first part of the notation is removed, leaving [d{p,q,...}]e{s,t,...}; the opposite holds if no regular {s,t,...} exists. The dual of c{m,n,...}[d{p,q,...}]e{s,t,...} is e{t,s,...}[d{q,p,...}]c{n,m,...}. If c or e are 1, they may be omitted. For compound polygons, this notation reduces to {nk}[k{n/m}]{nk}: for example, the hexagram may be written thus as {6}[2{3}]{6}. + +Regular skew polygons also create compounds, seen in the edges of prismatic compound of antiprisms, for instance: + +== Three dimensional compounds == + +A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds. + +Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols. The material inside the square brackets, [d{p,q}], denotes the components of the compound: d separate {p,q}'s. The material before the square brackets denotes the vertex arrangement of the compound: c{m,n}[d{p,q}] is a compound of d {p,q}'s sharing the vertices of an {m,n} counted c times. The material after the square brackets denotes the facet arrangement of the compound: [d{p,q}]e{s,t} is a compound of d {p,q}'s sharing the faces of {s,t} counted e times. These may be combined: thus c{m,n}[d{p,q}]e{s,t} is a compound of d {p,q}'s sharing the vertices of {m,n} counted c times and the faces of {s,t} counted e times. This notation can be generalised to compounds in any number of dimensions. +If improper regular polyhedra (dihedra and hosohedra) are allowed, then two more compounds are possible: 2{3,4}[3{4,2}]{4,3} and its dual {3,4}[3{2,4}]2{4,3}. + +=== Euclidean and hyperbolic plane compounds === +There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not yet been proven. + +A distinction must be made when an integer can be expressed in the forms b2+c2 or b2+bc+c2 in two different ways, e.g. 145 = 122 + 12 = 92 + 82, or 91 = 92 + 9 ⋅ 1 + 12 = 62 + 6 ⋅ 5 + 52. In such cases, Coxeter notates the sum explicitly, e.g. {4,4}[(144+1){4,4}]{4,4} as opposed to {4,4}[(81+64){4,4}]{4,4}. +The following compounds of compact or paracompact hyperbolic tessellations were known to Coxeter in 1964, though a proof of completeness was not then known: + +The Euclidean and hyperbolic compound families {4,q}[2{q,q}]{q,4} appear because h{4,q} = {q,q}, i.e. taking alternate vertices of a {4,q} results in a {q,q}. They are thus the Euclidean and hyperbolic analogues of the spherical stella octangula, which is the q=3 case. +It is also the case that h{2q,q} = {q,2q}, yielding the compound {2q,q}[2{q,2q}] and its dual [2{2q,q}]{q,2q}. Now if we take the dual of the {2q,q}, we obtain a third {q,2q} whose vertices are at the centres of alternate faces of the other two {q,2q}; this gives the compound {3,2q}[3{q,2q}]2{2q,3} and its dual 2{3,2q}[3{2q,q}]{2q,3}. These compounds are hyperbolic if q > 3 and Euclidean if q = 3. These compounds show an analogy to the spherical compounds {4,3,3}[2{3,3,4}], [2{4,3,3}]{3,3,4}, {3,4,3}[3{3,3,4}]2{3,4,3}, and 2{3,4,3}[3{4,3,3}]{3,4,3}. +If one sets q = 8 in {4,q}[2{q,q}]{q,4}, and q = 4 in {3,2q}[3{q,2q}]2{2q,3}, then one obtains the special cases {4,8}[2{8,8}]{8,4} and {3,8}[3{4,8}]2{8,3}. The latter's {4,8}'s can be replaced by pairs of {8,8}'s according to the former, giving the self-dual compound {3,8}[6{8,8}]{8,3}. + +== Four dimensional compounds == + +Coxeter lists 46 regular compounds of regular 4-polytopes in his book Regular Polytopes. McMullen adds six in his paper New Regular Compounds of 4-Polytopes, in which he also proves that the list is now complete. In the following tables, the superscript (var) indicates that the labeled compounds are distinct from the other compounds with the same symbols. + +There are two different compounds of 75 tesseracts: one shares the vertices of a 120-cell, while the other shares the vertices of a 600-cell. It immediately follows therefore that the corresponding dual compounds of 75 16-cells are also different. + +There are also fourteen partially regular compounds, that are either vertex-transitive or cell-transitive but not both. The seven vertex-transitive partially regular compounds are the duals of the seven cell-transitive partially regular compounds. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-1.md b/data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-1.md new file mode 100644 index 000000000..220d80cfa --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-1.md @@ -0,0 +1,51 @@ +--- +title: "List of regular polytope compounds" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/List_of_regular_polytope_compounds" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:32.110805+00:00" +instance: "kb-cron" +--- + +Although the 5-cell and 24-cell are both self-dual, their dual compounds (the compound of two 5-cells and compound of two 24-cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any regular 4-polytope. However, they are vertex-, edge-, face-, and cell-transitive. + +=== Euclidean 3-space compounds === +The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[d{4,3,4}]{4,3,4}. + +=== Hyperbolic 3-space compounds === +C. W. L. Garner described two dual pairs of regular hyperbolic compound honeycombs in 1970: the compact pair 2{5,3,4}[5{4,3,5}] and [5{5,3,4}]2{4,3,5}, and the paracompact pair {6,3,3}[5{6,3,4}] and [5{4,3,6}]{3,3,6}. He did not consider vertex-regular compounds where the vertices are at infinity, or (reciprocally) cell-regular compounds where the cells are centred at infinity. In 2019, Peter McMullen (who focused only on the compact case) pointed out and filled a gap in Garner's proof of completeness, so that it is now proven that 2{5,3,4}[5{4,3,5}] and [5{5,3,4}]2{4,3,5} are the only compact regular hyperbolic honeycomb compounds. + +== Five dimensions and higher compounds == +There are no regular compounds in five or six dimensions. There are three known seven-dimensional compounds (16, 240, or 480 7-simplices), and six known eight-dimensional ones (16, 240, or 480 8-cubes or 8-orthoplexes). There is also one compound of n-simplices in n-dimensional space provided that n is one less than a power of two, and also two compounds (one of n-cubes and a dual one of n-orthoplexes) in n-dimensional space if n is a power of two. +The Coxeter notation for these compounds are (using αn = {3n−1}, βn = {3n−2,4}, γn = {4,3n−2}): + +7-simplexes: cγ7[16cα7]cβ7, where c = 1, 15, or 30 +8-orthoplexes: cγ8[16cβ8] +8-cubes: [16cγ8]cβ8 +The general cases (where n = 2k and d = 22k − k − 1, k = 2, 3, 4, ...): + +Simplexes: γn−1[dαn−1]βn−1 +Orthoplexes: γn[dβn] +Hypercubes: [dγn]βn + +=== Euclidean honeycomb compound === +A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. The Coxeter notation is δn[dδn]δn where δn = {∞} when n = 2 and {4,3n−3,4} when n ≥ 3. + +=== Hyperbolic honeycomb compounds === +In four dimensions, Garner (1970) asserted the existence of {3,3,3,5}[26{5,3,3,5}]{5,3,3,3}; although neither justification nor construction was given, McMullen (2019) proved that this claim is correct. McMullen showed the existence of the following compact compounds: + +2{3,3,3,5}[17{4,3,3,5}] and dual [17{5,3,3,4}]2{5,3,3,3}; +10{3,3,3,5}[85{4,3,3,5}] and dual [85{5,3,3,4}]10{5,3,3,3}; +{3,3,3,5}[26{5,3,3,5}]{5,3,3,3} (self-dual, comes in left- and right-handed forms); +6{3,3,3,5}[156{5,3,3,5}]6{5,3,3,3} (self-dual, comes in left- and right-handed forms); +12{3,3,3,5}[312{5,3,3,5}]12{5,3,3,3} (self-dual). +McMullen conjectures that this list is complete regarding the compact compounds. If any more compact compounds exist, they must involve {4,3,3,5} or {5,3,3,5} being inscribed in {5,3,3,3} (the only case not yet excluded). +In five dimensions, there is only one regular hyperbolic honeycomb whose vertices are not at infinity: {3,4,3,3,3}. Thus there are no regular compounds conforming to Garner's restriction that the vertices of a vertex-regular compound should not be at infinity. In six dimensions or higher, there are no compact or paracompact regular hyperbolic honeycombs at all, and thus no compact or paracompact compounds exist. + +== See also == +List of regular polytopes + +== References == + +== Bibliography == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_regular_polytopes-0.md b/data/en.wikipedia.org/wiki/List_of_regular_polytopes-0.md new file mode 100644 index 000000000..41618cc2a --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_regular_polytopes-0.md @@ -0,0 +1,174 @@ +--- +title: "List of regular polytopes" +chunk: 1/5 +source: "https://en.wikipedia.org/wiki/List_of_regular_polytopes" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:18.735925+00:00" +instance: "kb-cron" +--- + +This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. + +== Overview == +This table shows a summary of regular polytope counts by rank. + +== 1-polytopes == + +There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol { }, or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion and gives it the Schläfli symbol { }. +Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. + +== 2-polytopes (polygons) == +The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}. +Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed. + +=== Convex === +The Schläfli symbol {p} represents a regular p-gon. + +==== Spherical ==== +The regular digon {2} can be considered to be a degenerate regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a sphere or torus. For example, digon can be realised non-degenerately as a spherical lune. A monogon {1} could also be realised on the sphere as a single point with a great circle through it. However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two. + +=== Stars === +There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons. +In general, for any natural number n, there are regular n-pointed stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m} = {n/(n − m)}) and m and n are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols where m and n are not coprime may be used to represent compound polygons. + +Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist (for example: {3/2}, {5/3}, {5/4}, {7/4}, {9/5}), however these have not been studied in detail. +There also exist failed star polygons, such as the piangle, which do not cover the surface of a circle finitely many times. + +=== Skew polygons === +In addition to the planar regular polygons there are infinitely many regular skew polygons. Skew polygons can be created via the blending operation. +The blend of two polygons P and Q, written P#Q, can be constructed as follows: + +take the cartesian product of their vertices VP × VQ. +add edges (p0 × q0, p1 × q1) where (p0, p1) is an edge of P and (q0, q1) is an edge of Q. +select an arbitrary connected component of the result. +Alternatively, the blend is the polygon ⟨ρ0σ0, ρ1σ1⟩ where ρ and σ are the generating mirrors of P and Q placed in orthogonal subspaces. +The blending operation is commutative, associative and idempotent. +Every regular skew polygon can be expressed as the blend of a unique set of planar polygons. If P and Q share no factors then Dim(P#Q) = Dim(P) + Dim(Q). + +==== In 3 space ==== +The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism ({n/m}#{} where n is odd) or an antiprism ({n/m}#{} where n is even). All polygons in 3 space have an even number of vertices and edges. +Several of these appear as the Petrie polygons of regular polyhedra. + +==== In 4 space ==== +The regular finite polygons in 4 dimensions are exactly the polygons formed as a blend of two distinct planar polygons. They have vertices lying on a Clifford torus and related by a Clifford displacement. Unlike 3-dimensional polygons, skew polygons on double rotations can include an odd-number of sides. + +== 3-polytopes (polyhedra) == +Polytopes of dimension 3 are called polyhedra: +A regular polyhedron with Schläfli symbol {p, q}, Coxeter diagrams , has a regular face type {p}, and regular vertex figure {q}. +A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon. +Existence of a regular polyhedron {p, q} is constrained by an inequality, related to the vertex figure's angle defect: + + + + + + + + + + + + 1 + p + + + + + + + 1 + q + + + > + + + 1 + 2 + + + : + + Polyhedron (existing in Euclidean 3-space) + + + + + + + + + 1 + p + + + + + + + 1 + q + + + = + + + 1 + 2 + + + : + + Euclidean plane tiling + + + + + + + + + 1 + p + + + + + + + 1 + q + + + < + + + 1 + 2 + + + : + + Hyperbolic plane tiling + + + + + + + + {\displaystyle {\begin{aligned}&{\frac {1}{p}}+{\frac {1}{q}}>{\frac {1}{2}}:{\text{Polyhedron (existing in Euclidean 3-space)}}\\[6pt]&{\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{2}}:{\text{Euclidean plane tiling}}\\[6pt]&{\frac {1}{p}}+{\frac {1}{q}}<{\frac {1}{2}}:{\text{Hyperbolic plane tiling}}\end{aligned}}} + + +By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons {p} and {q} limited to: {3}, {4}, {5}, {5/2}, and {6}. +Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings. + +=== Convex === +The five convex regular polyhedra are called the Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic ( + + + + χ + + + {\displaystyle \chi } + +) of 2. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_regular_polytopes-1.md b/data/en.wikipedia.org/wiki/List_of_regular_polytopes-1.md new file mode 100644 index 000000000..4c1cae1a2 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_regular_polytopes-1.md @@ -0,0 +1,384 @@ +--- +title: "List of regular polytopes" +chunk: 2/5 +source: "https://en.wikipedia.org/wiki/List_of_regular_polytopes" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:18.735925+00:00" +instance: "kb-cron" +--- + +==== Spherical ==== +In spherical geometry, regular spherical polyhedra (tilings of the sphere) exist that would otherwise be degenerate as polytopes. These are the hosohedra {2,n} and their dual dihedra {n,2}. Coxeter calls these cases "improper" tessellations. +The first few cases (n from 2 to 6) are listed below. + +Star-dihedra and hosohedra {p/q, 2} and {2, p/q} also exist for any star polygon {p/q}. + +=== Stars === +The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}: +As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow. + +There are infinitely many failed star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are {5/2,4}, {5/2,9}, {7/2,3}, {5/2,5/2}, {7/2,7/3}, {4,5/2}, and {3,7/3}. + +=== Skew polyhedra === +Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures. +For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes. +The regular skew polyhedra, represented by {l,m|n}, follow this equation: + + + + + 2 + sin + ⁡ + + ( + + + π + l + + + ) + + sin + ⁡ + + ( + + + π + m + + + ) + + = + cos + ⁡ + + ( + + + π + n + + + ) + + + + {\displaystyle 2\sin \left({\frac {\pi }{l}}\right)\sin \left({\frac {\pi }{m}}\right)=\cos \left({\frac {\pi }{n}}\right)} + + +Four of them can be seen in 4-dimensions as a subset of faces of four regular 4-polytopes, sharing the same vertex arrangement and edge arrangement: + +== 4-polytopes (polychora) == +Regular 4-polytopes with Schläfli symbol + + + + { + p + , + q + , + r + } + + + {\displaystyle \{p,q,r\}} + + have cells of type + + + + { + p + , + q + } + + + {\displaystyle \{p,q\}} + +, faces of type + + + + { + p + } + + + {\displaystyle \{p\}} + +, edge figures + + + + + { + r + } + + + {\displaystyle \{r\}} + +, and vertex figures + + + + { + q + , + r + } + + + {\displaystyle \{q,r\}} + +. + +A vertex figure (of a 4-polytope) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular 4-polytopes, this vertex figure is a regular polyhedron. +An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular 4-polytopes, this edge figure will always be a regular polygon. +The existence of a regular 4-polytope + + + + { + p + , + q + , + r + } + + + {\displaystyle \{p,q,r\}} + + is constrained by the existence of the regular polyhedra + + + + { + p + , + q + } + , + { + q + , + r + } + + + {\displaystyle \{p,q\},\{q,r\}} + +. A suggested name for 4-polytopes is "polychoron". +Each will exist in a space dependent upon this expression: + + + + + sin + ⁡ + + ( + + + π + p + + + ) + + sin + ⁡ + + ( + + + π + r + + + ) + + − + cos + ⁡ + + ( + + + π + q + + + ) + + + + {\displaystyle \sin \left({\frac {\pi }{p}}\right)\sin \left({\frac {\pi }{r}}\right)-\cos \left({\frac {\pi }{q}}\right)} + + + + + + > + 0 + + + {\displaystyle >0} + + : Hyperspherical 3-space honeycomb or 4-polytope + + + + + = + 0 + + + {\displaystyle =0} + + : Euclidean 3-space honeycomb + + + + + < + 0 + + + {\displaystyle <0} + + : Hyperbolic 3-space honeycomb +These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs. + +=== Convex === +The 6 convex regular 4-polytopes are shown in the table below. All these 4-polytopes have an Euler characteristic ( + + + + χ + + + {\displaystyle \chi } + +) of 0. + +==== Spherical ==== +Di-4-topes and hoso-4-topes exist as regular tessellations of the 3-sphere. +Regular di-4-topes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hoso-4-tope duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. 4-polytopes of the form {2,p,2} are the same as {2,2,p}. There are also the cases {p,2,q} which have dihedral cells and hosohedral vertex figures. + +=== Stars === +There are ten regular star 4-polytopes, which are called the Schläfli–Hess 4-polytopes. Their vertices are based on the convex 120-cell {5,3,3} and 600-cell {3,3,5}. +Ludwig Schläfli found four of them and skipped the last six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F+V−E=2). Edmund Hess (1843–1903) completed the full list of ten in his German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder (1883)[1]. +There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4-polytopes, shown as orthogonal projections: + +There are 4 failed potential regular star 4-polytopes permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions. + +=== Skew 4-polytopes === + +In addition to the 16 planar 4-polytopes above there are 18 finite skew polytopes. One of these is obtained as the Petrie dual of the tesseract, and the other 17 can be formed by applying the kappa operation to the planar polytopes and the Petrie dual of the tesseract. + +== Dimension 5 and higher == +5-polytopes can be given the symbol + + + + { + p + , + q + , + r + , + s + } + + + {\displaystyle \{p,q,r,s\}} + + where + + + + { + p + , + q + , + r + } + + + {\displaystyle \{p,q,r\}} + + is the 4-face type, + + + + { + p + , + q + } + + + {\displaystyle \{p,q\}} + + is the cell type, + + + + { + p + } + + + {\displaystyle \{p\}} + + is the face type, and + + + + { + s + } + + + {\displaystyle \{s\}} + + is the face figure, + + + + { + r + , + s + } + + + {\displaystyle \{r,s\}} + + is the edge figure, and + + + + { + q + , + r + , + s + } + + + {\displaystyle \{q,r,s\}} + + is the vertex figure. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_regular_polytopes-2.md b/data/en.wikipedia.org/wiki/List_of_regular_polytopes-2.md new file mode 100644 index 000000000..2f38e6b07 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_regular_polytopes-2.md @@ -0,0 +1,254 @@ +--- +title: "List of regular polytopes" +chunk: 3/5 +source: "https://en.wikipedia.org/wiki/List_of_regular_polytopes" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:18.735925+00:00" +instance: "kb-cron" +--- + +A vertex figure (of a 5-polytope) is a 4-polytope, seen by the arrangement of neighboring vertices to each vertex. +An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge. +A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face. +A regular 5-polytope + + + + { + p + , + q + , + r + , + s + } + + + {\displaystyle \{p,q,r,s\}} + + exists only if + + + + { + p + , + q + , + r + } + + + {\displaystyle \{p,q,r\}} + + and + + + + { + q + , + r + , + s + } + + + {\displaystyle \{q,r,s\}} + + are regular 4-polytopes. +The space it fits in is based on the expression: + + + + + + + + + cos + + 2 + + + ⁡ + + ( + + + π + q + + + ) + + + + + sin + + 2 + + + ⁡ + + ( + + + π + p + + + ) + + + + + + + + + + + cos + + 2 + + + ⁡ + + ( + + + π + r + + + ) + + + + + sin + + 2 + + + ⁡ + + ( + + + π + s + + + ) + + + + + + + {\displaystyle {\frac {\cos ^{2}\left({\frac {\pi }{q}}\right)}{\sin ^{2}\left({\frac {\pi }{p}}\right)}}+{\frac {\cos ^{2}\left({\frac {\pi }{r}}\right)}{\sin ^{2}\left({\frac {\pi }{s}}\right)}}} + + + + + + < + 1 + + + {\displaystyle <1} + + : Spherical 4-space tessellation or 5-space polytope + + + + + = + 1 + + + {\displaystyle =1} + + : Euclidean 4-space tessellation + + + + + > + 1 + + + {\displaystyle >1} + + : hyperbolic 4-space tessellation +Enumeration of these constraints produce 3 convex polytopes, no star polytopes, 3 tessellations of Euclidean 4-space, and 5 tessellations of paracompact hyperbolic 4-space. The only non-convex regular polytopes for ranks 5 and higher are skews. + +=== Convex === +In dimensions 5 and higher, there are only three kinds of convex regular polytopes. + +There are also improper cases where some numbers in the Schläfli symbol are 2. For example, {p,q,r,...2} is an improper regular spherical polytope whenever {p,q,r...} is a regular spherical polytope, and {2,...p,q,r} is an improper regular spherical polytope whenever {...p,q,r} is a regular spherical polytope. Such polytopes may also be used as facets, yielding forms such as {p,q,...2...y,z}. + +==== 5 dimensions ==== + +==== 6 dimensions ==== + +==== 7 dimensions ==== + +==== 8 dimensions ==== + +==== 9 dimensions ==== + +==== 10 dimensions ==== + +=== Star polytopes === +There are no regular star polytopes of rank 5 or higher, with the exception of degenerate polytopes created by the star product of lower rank star polytopes. e.g. hosotopes and ditopes. + +== Regular projective polytopes == +A projective regular (n+1)-polytope exists when an original regular n-spherical tessellation, {p,q,...}, is centrally symmetric. Such a polytope is named hemi-{p,q,...}, and contain half as many elements. Coxeter gives a symbol {p,q,...}/2, while McMullen writes {p,q,...}h/2 with h as the coxeter number. +Even-sided regular polygons have hemi-2n-gon projective polygons, {2p}/2. +There are 4 regular projective polyhedra related to 4 of 5 Platonic solids. +The hemi-cube and hemi-octahedron generalize as hemi-n-cubes and hemi-n-orthoplexes to any rank. + +=== Regular projective polyhedra === + +=== Regular projective 4-polytopes === +5 of 6 convex regular 4-polytopes are centrally symmetric generating projective 4-polytopes. The 3 special cases are hemi-24-cell, hemi-600-cell, and hemi-120-cell. + +=== Regular projective 5-polytopes === +Only 2 of 3 regular spherical polytopes are centrally symmetric for ranks 5 or higher. The corresponding regular projective polytopes are the hemi versions of the regular hypercube and orthoplex. They are tabulated below for rank 5, for example: + +== Apeirotopes == +An apeirotope or infinite polytope is a polytope which has infinitely many facets. An n-apeirotope is an infinite n-polytope: a 2-apeirotope or apeirogon is an infinite polygon, a 3-apeirotope or apeirohedron is an infinite polyhedron, etc. +There are two main geometric classes of apeirotope: + +Regular honeycombs in n dimensions, which completely fill an n-dimensional space. +Regular skew apeirotopes, comprising an n-dimensional manifold in a higher space. + +=== 2-apeirotopes (apeirogons) === +The straight apeirogon is a regular tessellation of the line, subdividing it into infinitely many equal segments. It has infinitely many vertices and edges. Its Schläfli symbol is {∞}, and Coxeter diagram . +...... +It exists as the limit of the p-gon as p tends to infinity, as follows: + +Apeirogons in the hyperbolic plane, most notably the regular apeirogon, {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles. +Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles, while more generally they can exist on hypercycles. + +Above are two regular hyperbolic apeirogons in the Poincaré disk model, the right one shows perpendicular reflection lines of divergent fundamental domains, separated by length λ. + +==== Skew apeirogons ==== +A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular. +Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed. + +=== 3-apeirotopes (apeirohedra) === + +==== Euclidean tilings ==== +There are six regular tessellations of the plane: the three listed below, and their corresponding Petrials. + +There are two improper regular tilings: {∞,2}, an apeirogonal dihedron, made from two apeirogons, each filling half the plane; and secondly, its dual, {2,∞}, an apeirogonal hosohedron, seen as an infinite set of parallel lines. + +==== Euclidean star-tilings ==== +There are no regular plane tilings of star polygons. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_regular_polytopes-3.md b/data/en.wikipedia.org/wiki/List_of_regular_polytopes-3.md new file mode 100644 index 000000000..35758066b --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_regular_polytopes-3.md @@ -0,0 +1,83 @@ +--- +title: "List of regular polytopes" +chunk: 4/5 +source: "https://en.wikipedia.org/wiki/List_of_regular_polytopes" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:18.735925+00:00" +instance: "kb-cron" +--- + +==== Hyperbolic tilings ==== +Tessellations of hyperbolic 2-space are hyperbolic tilings. There are infinitely many regular tilings in H2. As stated above, every positive integer pair {p,q} such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle (p, q, r) the same holds true for 1/p + 1/q + 1/r < 1. +There are a number of different ways to display the hyperbolic plane, including the Poincaré disk model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens. +There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q + 2 + + + {\displaystyle p>2} + + such that + + + + + R + + p + + + ( + b + ) + + + {\displaystyle R_{p}(b)} + + is prime are (start with + + + + b + = + 2 + + + {\displaystyle b=2} + +, 0 if no such + + + + p + + + {\displaystyle p} + + exists) + +3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, 25667, 19, 3, 3, 5, 5, 3, 0, 7, 3, 5, 5, 5, 7, 0, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 3, 41, 3, 7, 7, 3, 5, 0, 19, 3, 19, 5, 3, 29, 3, 7, 5, 5, 3, 41, 3, 3, 5, 3, 0, 23, 5, 17, 5, 11, 7, 61, 3, 3, 4421, 439, 7, 5, 7, 3343, 17, 13, 3, 0, 3, ... (sequence A128164 in the OEIS) +Smallest prime + + + + p + > + 2 + + + {\displaystyle p>2} + + such that + + + + + R + + p + + + ( + − + b + ) + + + {\displaystyle R_{p}(-b)} + + is prime are (start with + + + + b + = + 2 + + + {\displaystyle b=2} + +, 0 if no such + + + + p + + + {\displaystyle p} + + exists) + +3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, 7, 3, 3, 7, 3, 17, 5, 3, 3, 11, 7, 3, 11, 0, 3, 7, 139, 109, 0, 5, 3, 11, 31, 5, 5, 3, 53, 17, 3, 5, 7, 103, 7, 5, 5, 7, 1153, 3, 7, 21943, 7, 3, 37, 53, 3, 17, 3, 7, 11, 3, 0, 19, 7, 3, 757, 11, 3, 5, 3, 7, 13, 5, 3, 37, 3, 3, 5, 3, 293, 19, 7, 167, 7, 7, 709, 13, 3, 3, 37, 89, 71, 43, 37, (>800000), 19, 7, 3, 7, ... (sequence A084742 in the OEIS) + +* Repunits with negative base and even n are negative. If their absolute value is prime then they are included above and marked with an asterisk. They are not included in the corresponding OEIS sequences. +For more information, see. + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_rules_of_inference-0.md b/data/en.wikipedia.org/wiki/List_of_rules_of_inference-0.md new file mode 100644 index 000000000..3f8dfa0dc --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_rules_of_inference-0.md @@ -0,0 +1,1218 @@ +--- +title: "List of rules of inference" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/List_of_rules_of_inference" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:46.303073+00:00" +instance: "kb-cron" +--- + +This is a list of rules of inference, logical laws that relate to mathematical formulae. + +== Introduction == +Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules. +Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation + + + + + φ + ⊢ + ψ + + + {\displaystyle \varphi \vdash \psi } + + +indicates such a subderivation from the temporary assumption + + + + φ + + + {\displaystyle \varphi } + + to + + + + ψ + + + {\displaystyle \psi } + +. + +== Rules for propositional calculus == + +=== Rules for negations === +Reductio ad absurdum (or Negation Introduction) + + + + + φ + ⊢ + ψ + + + {\displaystyle \varphi \vdash \psi } + + + + + + + + + φ + ⊢ + ¬ + ψ + + _ + + + + + {\displaystyle {\underline {\varphi \vdash \lnot \psi }}} + + + + + + ¬ + φ + + + {\displaystyle \lnot \varphi } + + +Reductio ad absurdum (related to the law of excluded middle) + + + + + ¬ + φ + ⊢ + ψ + + + {\displaystyle \lnot \varphi \vdash \psi } + + + + + + + + + ¬ + φ + ⊢ + ¬ + ψ + + _ + + + + + {\displaystyle {\underline {\lnot \varphi \vdash \lnot \psi }}} + + + + + + φ + + + {\displaystyle \varphi } + + +Ex contradictione quodlibet + + + + + φ + + + {\displaystyle \varphi } + + + + + + + + + ¬ + φ + + _ + + + + + {\displaystyle {\underline {\lnot \varphi }}} + + + + + + ψ + + + {\displaystyle \psi } + + +=== Rules for conditionals === +Deduction theorem (or Conditional Introduction) + + + + + + + + φ + ⊢ + ψ + + _ + + + + + {\displaystyle {\underline {\varphi \vdash \psi }}} + + + + + + φ + → + ψ + + + {\displaystyle \varphi \rightarrow \psi } + + +Modus ponens (a type of Conditional Elimination) + + + + + φ + → + ψ + + + {\displaystyle \varphi \rightarrow \psi } + + + + + + + + + φ + + + + + _ + + + + + {\displaystyle {\underline {\varphi \quad \quad \quad }}} + + + + + + ψ + + + {\displaystyle \psi } + + +Modus tollens (a type of Conditional Elimination) + + + + + φ + → + ψ + + + {\displaystyle \varphi \rightarrow \psi } + + + + + + + + + ¬ + ψ + + + + + _ + + + + + {\displaystyle {\underline {\lnot \psi \quad \quad \quad }}} + + + + + + ¬ + φ + + + {\displaystyle \lnot \varphi } + + +=== Rules for conjunctions === +Adjunction (or Conjunction Introduction) + + + + + φ + + + {\displaystyle \varphi } + + + + + + + + + ψ + + + + + + _ + + + + + {\displaystyle {\underline {\psi \quad \quad \ \ }}} + + + + + + φ + ∧ + ψ + + + {\displaystyle \varphi \land \psi } + + +Simplification (or Conjunction Elimination) + + + + + + + + φ + ∧ + ψ + + _ + + + + + {\displaystyle {\underline {\varphi \land \psi }}} + + + + + + φ + + + {\displaystyle \varphi } + + + + + + + + + φ + ∧ + ψ + + _ + + + + + {\displaystyle {\underline {\varphi \land \psi }}} + + + + + + ψ + + + {\displaystyle \psi } + + +=== Rules for disjunctions === +Addition (or Disjunction Introduction) + + + + + + + + φ + + + + + + _ + + + + + {\displaystyle {\underline {\varphi \quad \quad \ \ }}} + + + + + + φ + ∨ + ψ + + + {\displaystyle \varphi \lor \psi } + + + + + + + + + ψ + + + + + + _ + + + + + {\displaystyle {\underline {\psi \quad \quad \ \ }}} + + + + + + φ + ∨ + ψ + + + {\displaystyle \varphi \lor \psi } + + +Case analysis (or Proof by Cases or Argument by Cases or Disjunction elimination) + + + + + φ + → + χ + + + {\displaystyle \varphi \rightarrow \chi } + + + + + + ψ + → + χ + + + {\displaystyle \psi \rightarrow \chi } + + + + + + + + + φ + ∨ + ψ + + _ + + + + + {\displaystyle {\underline {\varphi \lor \psi }}} + + + + + + χ + + + {\displaystyle \chi } + + +Disjunctive syllogism + + + + + φ + ∨ + ψ + + + {\displaystyle \varphi \lor \psi } + + + + + + + + + ¬ + φ + + + + _ + + + + + {\displaystyle {\underline {\lnot \varphi \quad \quad }}} + + + + + + ψ + + + {\displaystyle \psi } + + + + + + φ + ∨ + ψ + + + {\displaystyle \varphi \lor \psi } + + + + + + + + + ¬ + ψ + + + + _ + + + + + {\displaystyle {\underline {\lnot \psi \quad \quad }}} + + + + + + φ + + + {\displaystyle \varphi } + + +Constructive dilemma + + + + + φ + → + χ + + + {\displaystyle \varphi \rightarrow \chi } + + + + + + ψ + → + ξ + + + {\displaystyle \psi \rightarrow \xi } + + + + + + + + + φ + ∨ + ψ + + _ + + + + + {\displaystyle {\underline {\varphi \lor \psi }}} + + + + + + χ + ∨ + ξ + + + {\displaystyle \chi \lor \xi } + + +=== Rules for biconditionals === +Biconditional introduction + + + + + φ + → + ψ + + + {\displaystyle \varphi \rightarrow \psi } + + + + + + + + + ψ + → + φ + + _ + + + + + {\displaystyle {\underline {\psi \rightarrow \varphi }}} + + + + + + φ + ↔ + ψ + + + {\displaystyle \varphi \leftrightarrow \psi } + + +Biconditional elimination + + + + + φ + ↔ + ψ + + + {\displaystyle \varphi \leftrightarrow \psi } + + + + + + + + + φ + + + + _ + + + + + {\displaystyle {\underline {\varphi \quad \quad }}} + + + + + + ψ + + + {\displaystyle \psi } + + + + + + φ + ↔ + ψ + + + {\displaystyle \varphi \leftrightarrow \psi } + + + + + + + + + ψ + + + + _ + + + + + {\displaystyle {\underline {\psi \quad \quad }}} + + + + + + φ + + + {\displaystyle \varphi } + + + + + + φ + ↔ + ψ + + + {\displaystyle \varphi \leftrightarrow \psi } + + + + + + + + + ¬ + φ + + + + _ + + + + + {\displaystyle {\underline {\lnot \varphi \quad \quad }}} + + + + + + ¬ + ψ + + + {\displaystyle \lnot \psi } + + + + + + φ + ↔ + ψ + + + {\displaystyle \varphi \leftrightarrow \psi } + + + + + + + + + ¬ + ψ + + + + _ + + + + + {\displaystyle {\underline {\lnot \psi \quad \quad }}} + + + + + + ¬ + φ + + + {\displaystyle \lnot \varphi } + + + + + + φ + ↔ + ψ + + + {\displaystyle \varphi \leftrightarrow \psi } + + + + + + + + + ψ + ∨ + φ + + _ + + + + + {\displaystyle {\underline {\psi \lor \varphi }}} + + + + + + ψ + ∧ + φ + + + {\displaystyle \psi \land \varphi } + + + + + + φ + ↔ + ψ + + + {\displaystyle \varphi \leftrightarrow \psi } + + + + + + + + + ¬ + ψ + ∨ + ¬ + φ + + _ + + + + + {\displaystyle {\underline {\lnot \psi \lor \lnot \varphi }}} + + + + + + ¬ + ψ + ∧ + ¬ + φ + + + {\displaystyle \lnot \psi \land \lnot \varphi } + + +== Rules of classical predicate calculus == +In the following rules, + + + + φ + ( + β + + / + + α + ) + + + {\displaystyle \varphi (\beta /\alpha )} + + is exactly like + + + + φ + + + {\displaystyle \varphi } + + except for having the term + + + + β + + + {\displaystyle \beta } + + wherever + + + + φ + + + {\displaystyle \varphi } + + has the free variable + + + + α + + + {\displaystyle \alpha } + +. + +Universal Generalization (or Universal Introduction) + + + + + + + + φ + + ( + β + + / + + α + ) + + + _ + + + + + {\displaystyle {\underline {\varphi {(\beta /\alpha )}}}} + + + + + + ∀ + α + + φ + + + {\displaystyle \forall \alpha \,\varphi } + + +Restriction 1: + + + + β + + + {\displaystyle \beta } + + is a variable which does not occur in + + + + φ + + + {\displaystyle \varphi } + +. + +Restriction 2: + + + + β + + + {\displaystyle \beta } + + is not mentioned in any hypothesis or undischarged assumptions. + +Universal Instantiation (or Universal Elimination) + + + + + ∀ + α + + φ + + + {\displaystyle \forall \alpha \,\varphi } + + + + + + + + + φ + + ( + β + + / + + α + ) + + + ¯ + + + + + {\displaystyle {\overline {\varphi {(\beta /\alpha )}}}} + + +Restriction: No free occurrence of + + + + α + + + {\displaystyle \alpha } + + in + + + + φ + + + {\displaystyle \varphi } + + falls within the scope of a quantifier quantifying a variable occurring in + + + + β + + + {\displaystyle \beta } + +. + +Existential Generalization (or Existential Introduction) + + + + + + + + φ + ( + β + + / + + α + ) + + _ + + + + + {\displaystyle {\underline {\varphi (\beta /\alpha )}}} + + + + + + ∃ + α + + φ + + + {\displaystyle \exists \alpha \,\varphi } + + +Restriction: No free occurrence of + + + + α + + + {\displaystyle \alpha } + + in + + + + φ + + + {\displaystyle \varphi } + + falls within the scope of a quantifier quantifying a variable occurring in + + + + β + + + {\displaystyle \beta } + +. + +Existential Instantiation (or Existential Elimination) + + + + + ∃ + α + + φ + + + {\displaystyle \exists \alpha \,\varphi } + + + + + + + + + φ + ( + β + + / + + α + ) + ⊢ + ψ + + _ + + + + + {\displaystyle {\underline {\varphi (\beta /\alpha )\vdash \psi }}} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_rules_of_inference-1.md b/data/en.wikipedia.org/wiki/List_of_rules_of_inference-1.md new file mode 100644 index 000000000..65997d137 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_rules_of_inference-1.md @@ -0,0 +1,352 @@ +--- +title: "List of rules of inference" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/List_of_rules_of_inference" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:46.303073+00:00" +instance: "kb-cron" +--- + + + + + ψ + + + {\displaystyle \psi } + + +Restriction 1: + + + + β + + + {\displaystyle \beta } + + is a variable which does not occur in + + + + φ + + + {\displaystyle \varphi } + +. + +Restriction 2: There is no occurrence, free or bound, of + + + + β + + + {\displaystyle \beta } + + in + + + + ψ + + + {\displaystyle \psi } + +. + +Restriction 3: + + + + β + + + {\displaystyle \beta } + + is not mentioned in any hypothesis or undischarged assumptions. + +== Rules of substructural logic == +The following are special cases of universal generalization and existential elimination; these occur in substructural logics, such as linear logic. + +Rule of weakening (or monotonicity of entailment) (aka no-cloning theorem) + + + + + α + ⊢ + β + + + {\displaystyle \alpha \vdash \beta } + + + + + + + + + α + , + α + ⊢ + β + + ¯ + + + + + {\displaystyle {\overline {\alpha ,\alpha \vdash \beta }}} + + +Rule of contraction (or idempotency of entailment) (aka no-deleting theorem) + + + + + + + + α + , + α + , + γ + ⊢ + β + + _ + + + + + {\displaystyle {\underline {\alpha ,\alpha ,\gamma \vdash \beta }}} + + + + + + α + , + γ + ⊢ + β + + + {\displaystyle \alpha ,\gamma \vdash \beta } + + +== Table: Rules of Inference == +The rules above can be summed up in the following table. The "Tautology" column shows how to interpret the notation of a given rule. + +All rules use the basic logic operators. A complete table of "logic operators" is shown by a truth table, giving definitions of all the possible (16) truth functions of 2 boolean variables (p, q): + +where T = true and F = false, and, the columns are the logical operators: + +0, false, Contradiction; +1, NOR, Logical NOR (Peirce's arrow); +2, Converse nonimplication; +3, ¬p, Negation; +4, Material nonimplication; +5, ¬q, Negation; +6, XOR, Exclusive disjunction; +7, NAND, Logical NAND (Sheffer stroke); +8, AND, Logical conjunction; +9, XNOR, If and only if, Logical biconditional; +10, q, Projection function; +11, if/then, Material conditional; +12, p, Projection function; +13, then/if, Converse implication; +14, OR, Logical disjunction; +15, true, Tautology. +Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples: + +The column-14 operator (OR), shows Addition rule: when p=T (the hypothesis selects the first two lines of the table), we see (at column-14) that p∨q=T. +We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15 are T. +The column-8 operator (AND), shows Simplification rule: when p∧q=T (first line of the table), we see that p=T. +With this premise, we also conclude that q=T, p∨q=T, etc. as shown by columns 9–15. +The column-11 operator (IF/THEN), shows Modus ponens rule: when p→q=T and p=T only one line of the truth table (the first) satisfies these two conditions. On this line, q is also true. Therefore, whenever p → q is true and p is true, q must also be true. +Machines and well-trained people use this look at table approach to do basic inferences, and to check if other inferences (for the same premises) can be obtained. + +=== Example 1 === +Consider the following assumptions: "If it rains today, then we will not go on a canoe today. If we do not go on a canoe trip today, then we will go on a canoe trip tomorrow. Therefore (Mathematical symbol for "therefore" is + + + + ∴ + + + {\displaystyle \therefore } + +), if it rains today, we will go on a canoe trip tomorrow". +To make use of the rules of inference in the above table we let + + + + p + + + {\displaystyle p} + + be the proposition "If it rains today", + + + + q + + + {\displaystyle q} + + be "We will not go on a canoe today" and let + + + + r + + + {\displaystyle r} + + be "We will go on a canoe trip tomorrow". Then this argument is of the form: + + + + + + + + + p + → + q + + + + + q + → + r + + + + + ∴ + + + + p + → + r + + ¯ + + + + + + + + + {\displaystyle {\begin{aligned}p\rightarrow q\\q\rightarrow r\\\therefore {\overline {p\rightarrow r}}\\\end{aligned}}} + + +=== Example 2 === +Consider a more complex set of assumptions: "It is not sunny today and it is colder than yesterday". "We will go swimming only if it is sunny", "If we do not go swimming, then we will have a barbecue", and "If we will have a barbecue, then we will be home by sunset" lead to the conclusion "We will be home by sunset." +Proof by rules of inference: Let + + + + p + + + {\displaystyle p} + + be the proposition "It is sunny today", + + + + q + + + {\displaystyle q} + + the proposition "It is colder than yesterday", + + + + r + + + {\displaystyle r} + + the proposition "We will go swimming", + + + + s + + + {\displaystyle s} + + the proposition "We will have a barbecue", and + + + + t + + + {\displaystyle t} + + the proposition "We will be home by sunset". Then the hypotheses become + + + + ¬ + p + ∧ + q + , + r + → + p + , + ¬ + r + → + s + + + {\displaystyle \neg p\wedge q,r\rightarrow p,\neg r\rightarrow s} + + and + + + + s + → + t + + + {\displaystyle s\rightarrow t} + +. Using our intuition we conjecture that the conclusion might be + + + + t + + + {\displaystyle t} + +. Using the Rules of Inference table we can prove the conjecture easily: + +== See also == + +List of logic systems +Modus ponendo tollens + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_self-intersecting_polygons-0.md b/data/en.wikipedia.org/wiki/List_of_self-intersecting_polygons-0.md new file mode 100644 index 000000000..254ac5c69 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_self-intersecting_polygons-0.md @@ -0,0 +1,32 @@ +--- +title: "List of self-intersecting polygons" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_self-intersecting_polygons" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:48.847957+00:00" +instance: "kb-cron" +--- + +Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross. +Some types of self-intersecting polygons are: + +the crossed quadrilateral, with four edges +the antiparallelogram, a crossed quadrilateral with alternate edges of equal length +the crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle, hence having two edges parallel +Star polygons +pentagram, with five edges +hexagram, with six edges +heptagram, with seven edges +octagram, with eight edges +enneagram or nonagram, with nine edges +decagram, with ten edges +hendecagram, with eleven edges +dodecagram, with twelve edges +icositetragram, with twenty four edges +257-gram, with two hundred and fifty seven edges + + +== See also == +List of regular polytopes and compounds § Stars +Complex polygon \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-0.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-0.md new file mode 100644 index 000000000..6e5d21989 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-0.md @@ -0,0 +1,1230 @@ +--- +title: "List of set identities and relations" +chunk: 1/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. +The binary operations of set union ( + + + + ∪ + + + {\displaystyle \cup } + +) and intersection ( + + + + ∩ + + + {\displaystyle \cap } + +) satisfy many identities. Several of these identities or "laws" have well established names. + +== Notation == +Throughout this article, capital letters (such as + + + + A + , + B + , + C + , + L + , + M + , + R + , + S + , + + + {\displaystyle A,B,C,L,M,R,S,} + + and + + + + X + + + {\displaystyle X} + +) will denote sets. On the left hand side of an identity, typically, + + + + + L + + + {\displaystyle L} + + will be the leftmost set, + + + + + M + + + {\displaystyle M} + + will be the middle set, and + + + + + R + + + {\displaystyle R} + + will be the rightmost set. +This is to facilitate applying identities to expressions that are complicated or use the same symbols as the identity. +For example, the identity + + + + + ( + L + + ∖ + + M + ) + + ∖ + + R + + = + + ( + L + + ∖ + + R + ) + + ∖ + + ( + M + + ∖ + + R + ) + + + {\displaystyle (L\,\setminus \,M)\,\setminus \,R~=~(L\,\setminus \,R)\,\setminus \,(M\,\setminus \,R)} + + +may be read as: + + + + + ( + + Left set + + + ∖ + + + Middle set + + ) + + ∖ + + + Right set + + + = + + ( + + Left set + + + ∖ + + + Right set + + ) + + ∖ + + ( + + Middle set + + + ∖ + + + Right set + + ) + . + + + {\displaystyle ({\text{Left set}}\,\setminus \,{\text{Middle set}})\,\setminus \,{\text{Right set}}~=~({\text{Left set}}\,\setminus \,{\text{Right set}})\,\setminus \,({\text{Middle set}}\,\setminus \,{\text{Right set}}).} + + +=== Elementary set operations === +For sets + + + + L + + + {\displaystyle L} + + and + + + + R + , + + + {\displaystyle R,} + + define: + + + + + + + + + L + ∪ + R + + + + + + + + + = + + + + + def + + + + + + + + { + + x + + : + + x + ∈ + L + + + + + + or + + + + + + + + x + ∈ + R + + } + + + + + L + ∩ + R + + + + + + + + + = + + + + + def + + + + + + + + { + + x + + : + + x + ∈ + L + + + + + + and + + + + + + x + ∈ + R + + } + + + + + L + ∖ + R + + + + + + + + + = + + + + + def + + + + + + + + { + + x + + : + + x + ∈ + L + + + + + + and + + + + + + x + ∉ + R + + } + + + + + + + {\displaystyle {\begin{alignedat}{4}L\cup R&&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~x~:~x\in L\;&&{\text{ or }}\;\,&&\;x\in R~\}\\L\cap R&&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~x~:~x\in L\;&&{\text{ and }}&&\;x\in R~\}\\L\setminus R&&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~x~:~x\in L\;&&{\text{ and }}&&\;x\notin R~\}\\\end{alignedat}}} + + +and + + + + + L + △ + R + + + + + + = + + + + + def + + + + + + + + { + + x + + : + + x + + belongs to exactly one of + + L + + and + + R + + } + + + {\displaystyle L\triangle R~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~x~:~x{\text{ belongs to exactly one of }}L{\text{ and }}R~\}} + + +where the symmetric difference + + + + L + △ + R + + + {\displaystyle L\triangle R} + + is sometimes denoted by + + + + L + ⊖ + R + + + {\displaystyle L\ominus R} + + and equals: + + + + + + + + + L + + △ + + R + + + + + = + + ( + L + + ∖ + + + + + R + ) + + ∪ + + + + + + ( + R + + ∖ + + + + + L + ) + + + + + + + + + = + + ( + L + + ∪ + + + + + R + ) + + ∖ + + + + + + ( + L + + ∩ + + + + + R + ) + . + + + + + + + {\displaystyle {\begin{alignedat}{4}L\;\triangle \;R~&=~(L~\setminus ~&&R)~\cup ~&&(R~\setminus ~&&L)\\~&=~(L~\cup ~&&R)~\setminus ~&&(L~\cap ~&&R).\end{alignedat}}} + + +One set + + + + L + + + {\displaystyle L} + + is said to intersect another set + + + + R + + + {\displaystyle R} + + if + + + + L + ∩ + R + ≠ + ∅ + . + + + {\displaystyle L\cap R\neq \varnothing .} + + Sets that do not intersect are said to be disjoint. +The power set of + + + + X + + + {\displaystyle X} + + is the set of all subsets of + + + + X + + + {\displaystyle X} + + and will be denoted by + + + + + + + P + + + ( + X + ) + + + + + + = + + + + + def + + + + + + + + { + + L + + : + + L + ⊆ + X + + } + . + + + {\displaystyle {\mathcal {P}}(X)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~L~:~L\subseteq X~\}.} + + +Universe set and complement notation +The notation + + + + + + L + + ∁ + + + + + + + + = + + + + + def + + + + + + + + X + ∖ + L + . + + + {\displaystyle L^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X\setminus L.} + + +may be used if + + + + L + + + {\displaystyle L} + + is a subset of some set + + + + X + + + {\displaystyle X} + + that is understood (say from context, or because it is clearly stated what the superset + + + + X + + + {\displaystyle X} + + is). +It is emphasized that the definition of + + + + + L + + ∁ + + + + + {\displaystyle L^{\complement }} + + depends on context. For instance, had + + + + L + + + {\displaystyle L} + + been declared as a subset of + + + + Y + , + + + {\displaystyle Y,} + + with the sets + + + + Y + + + {\displaystyle Y} + + and + + + + X + + + {\displaystyle X} + + not necessarily related to each other in any way, then + + + + + L + + ∁ + + + + + {\displaystyle L^{\complement }} + + would likely mean + + + + Y + ∖ + L + + + {\displaystyle Y\setminus L} + + instead of + + + + X + ∖ + L + . + + + {\displaystyle X\setminus L.} + + +If it is needed then unless indicated otherwise, it should be assumed that + + + + X + + + {\displaystyle X} + + denotes the universe set, which means that all sets that are used in the formula are subsets of + + + + X + . + + + {\displaystyle X.} + + +In particular, the complement of a set + + + + L + + + {\displaystyle L} + + will be denoted by + + + + + L + + ∁ + + + + + {\displaystyle L^{\complement }} + + where unless indicated otherwise, it should be assumed that + + + + + L + + ∁ + + + + + {\displaystyle L^{\complement }} + + denotes the complement of + + + + L + + + {\displaystyle L} + + in (the universe) + + + + X + . + + + {\displaystyle X.} + + +== One subset involved == +Assume + + + + L + ⊆ + X + . + + + {\displaystyle L\subseteq X.} + + +Identity: +Definition: + + + + e + + + {\displaystyle e} + + is called a left identity element of a binary operator + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + if + + + + e + + ∗ + + R + = + R + + + {\displaystyle e\,\ast \,R=R} + + for all + + + + R + + + {\displaystyle R} + + and it is called a right identity element of + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + if + + + + L + + ∗ + + e + = + L + + + {\displaystyle L\,\ast \,e=L} + + for all + + + + L + . + + + {\displaystyle L.} + + A left identity element that is also a right identity element if called an identity element. +The empty set + + + + ∅ + + + {\displaystyle \varnothing } + + is an identity element of binary union + + + + ∪ + + + {\displaystyle \cup } + + and symmetric difference + + + + △ + , + + + {\displaystyle \triangle ,} + + and it is also a right identity element of set subtraction + + + + + ∖ + : + + + {\displaystyle \,\setminus :} + + + + + + + + + + + L + ∩ + X + + + + + = + + + + + L + + + + = + + + + X + ∩ + L + + + + + + where + + L + ⊆ + X + + + + + L + ∪ + ∅ + + + + + = + + + + + L + + + + = + + + + ∅ + ∪ + L + + + + + L + + △ + ∅ + + + + + = + + + + + L + + + + = + + + + ∅ + + △ + L + + + + + L + ∖ + ∅ + + + + + = + + + + + L + + + + + + + {\displaystyle {\begin{alignedat}{10}L\cap X&\;=\;&&L&\;=\;&X\cap L~~~~{\text{ where }}L\subseteq X\\[1.4ex]L\cup \varnothing &\;=\;&&L&\;=\;&\varnothing \cup L\\[1.4ex]L\,\triangle \varnothing &\;=\;&&L&\;=\;&\varnothing \,\triangle L\\[1.4ex]L\setminus \varnothing &\;=\;&&L\\[1.4ex]\end{alignedat}}} + + +but + + + + ∅ + + + {\displaystyle \varnothing } + + is not a left identity element of + + + + + ∖ + + + + {\displaystyle \,\setminus \,} + + since + + + + + ∅ + ∖ + L + = + ∅ + + + {\displaystyle \varnothing \setminus L=\varnothing } + + +so + + + + ∅ + ∖ + L + = + L + + + {\textstyle \varnothing \setminus L=L} + + if and only if + + + + L + = + ∅ + . + + + {\displaystyle L=\varnothing .} + + +Idempotence + + + + L + ∗ + L + = + L + + + {\displaystyle L\ast L=L} + + and Nilpotence + + + + L + ∗ + L + = + ∅ + + + {\displaystyle L\ast L=\varnothing } + +: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-1.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-1.md new file mode 100644 index 000000000..882840246 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-1.md @@ -0,0 +1,1616 @@ +--- +title: "List of set identities and relations" +chunk: 2/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + L + ∪ + L + + + + + = + + + + + L + + + + + + (Idempotence) + + + + + + L + ∩ + L + + + + + = + + + + + L + + + + + + (Idempotence) + + + + + + L + + △ + + L + + + + + = + + + + + ∅ + + + + + + (Nilpotence of index 2) + + + + + + L + ∖ + L + + + + + = + + + + + ∅ + + + + + + (Nilpotence of index 2) + + + + + + + + {\displaystyle {\begin{alignedat}{10}L\cup L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\cap L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\,\triangle \,L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]L\setminus L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]\end{alignedat}}} + + +Domination/Absorbing element: +Definition: + + + + z + + + {\displaystyle z} + + is called a left absorbing element of a binary operator + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + if + + + + z + + ∗ + + R + = + z + + + {\displaystyle z\,\ast \,R=z} + + for all + + + + R + + + {\displaystyle R} + + and it is called a right absorbing element of + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + if + + + + L + + ∗ + + z + = + z + + + {\displaystyle L\,\ast \,z=z} + + for all + + + + L + . + + + {\displaystyle L.} + + A left absorbing element that is also a right absorbing element if called an absorbing element. Absorbing elements are also sometime called annihilating elements or zero elements. +A universe set is an absorbing element of binary union + + + + ∪ + . + + + {\displaystyle \cup .} + + The empty set + + + + ∅ + + + {\displaystyle \varnothing } + + is an absorbing element of binary intersection + + + + ∩ + + + {\displaystyle \cap } + + and binary Cartesian product + + + + × + , + + + {\displaystyle \times ,} + + and it is also a left absorbing element of set subtraction + + + + + ∖ + : + + + {\displaystyle \,\setminus :} + + + + + + + + + + + X + ∪ + L + + + + + = + + + + + X + + + + = + + + + L + ∪ + X + + + + + + where + + L + ⊆ + X + + + + + ∅ + ∩ + L + + + + + = + + + + + ∅ + + + + = + + + + L + ∩ + ∅ + + + + + ∅ + × + L + + + + + = + + + + + ∅ + + + + = + + + + L + × + ∅ + + + + + ∅ + ∖ + L + + + + + = + + + + + ∅ + + + + + + + + + + + + {\displaystyle {\begin{alignedat}{10}X\cup L&\;=\;&&X&\;=\;&L\cup X~~~~{\text{ where }}L\subseteq X\\[1.4ex]\varnothing \cap L&\;=\;&&\varnothing &\;=\;&L\cap \varnothing \\[1.4ex]\varnothing \times L&\;=\;&&\varnothing &\;=\;&L\times \varnothing \\[1.4ex]\varnothing \setminus L&\;=\;&&\varnothing &\;\;&\\[1.4ex]\end{alignedat}}} + + +but + + + + ∅ + + + {\displaystyle \varnothing } + + is not a right absorbing element of set subtraction since + + + + + L + ∖ + ∅ + = + L + + + {\displaystyle L\setminus \varnothing =L} + + +where + + + + L + ∖ + ∅ + = + ∅ + + + {\textstyle L\setminus \varnothing =\varnothing } + + if and only if + + + + L + = + ∅ + . + + + {\textstyle L=\varnothing .} + + +Double complement or involution law: + + + + + + + + + X + ∖ + ( + X + ∖ + L + ) + + + + = + L + + + + + + Also written + + + + + + + + ( + + L + + ∁ + + + ) + + + ∁ + + + = + L + + + + + + + + + where + + L + ⊆ + X + + + (Double complement/Involution law) + + + + + + + + {\displaystyle {\begin{alignedat}{10}X\setminus (X\setminus L)&=L&&\qquad {\text{ Also written }}\quad &&\left(L^{\complement }\right)^{\complement }=L&&\quad &&{\text{ where }}L\subseteq X\quad {\text{ (Double complement/Involution law)}}\\[1.4ex]\end{alignedat}}} + + + + + + L + ∖ + ∅ + = + L + + + {\displaystyle L\setminus \varnothing =L} + + + + + + + + + + ∅ + + + + = + L + + + + + ∖ + L + + + + + + + = + ∅ + + + + + ∖ + L + + + + + + + = + L + + + + + ∖ + X + + + + + + where + + L + ⊆ + X + + + + + + + {\displaystyle {\begin{alignedat}{4}\varnothing &=L&&\setminus L\\&=\varnothing &&\setminus L\\&=L&&\setminus X~~~~{\text{ where }}L\subseteq X\\\end{alignedat}}} + + + + + + + L + + ∁ + + + = + X + ∖ + L + + + (definition of notation) + + + + {\displaystyle L^{\complement }=X\setminus L\quad {\text{ (definition of notation)}}} + + + + + + + + + + L + + ∪ + ( + X + ∖ + L + ) + + + + = + X + + + + + + Also written + + + + + + L + ∪ + + L + + ∁ + + + = + X + + + + + + + + + where + + L + ⊆ + X + + + + + L + + △ + ( + X + ∖ + L + ) + + + + = + X + + + + + + Also written + + + + + + L + + △ + + L + + ∁ + + + = + X + + + + + + + + + where + + L + ⊆ + X + + + + + L + + ∩ + ( + X + ∖ + L + ) + + + + = + ∅ + + + + + + Also written + + + + + + L + ∩ + + L + + ∁ + + + = + ∅ + + + + + + + + + + + + + {\displaystyle {\begin{alignedat}{10}L\,\cup (X\setminus L)&=X&&\qquad {\text{ Also written }}\quad &&L\cup L^{\complement }=X&&\quad &&{\text{ where }}L\subseteq X\\[1.4ex]L\,\triangle (X\setminus L)&=X&&\qquad {\text{ Also written }}\quad &&L\,\triangle L^{\complement }=X&&\quad &&{\text{ where }}L\subseteq X\\[1.4ex]L\,\cap (X\setminus L)&=\varnothing &&\qquad {\text{ Also written }}\quad &&L\cap L^{\complement }=\varnothing &&\quad &&\\[1.4ex]\end{alignedat}}} + + + + + + + + + + X + ∖ + ∅ + + + + = + X + + + + + + Also written + + + + + + + ∅ + + ∁ + + + = + X + + + + + + + + + (Complement laws for the empty set)) + + + + + + X + ∖ + X + + + + = + ∅ + + + + + + Also written + + + + + + + X + + ∁ + + + = + ∅ + + + + + + + + + (Complement laws for the universe set) + + + + + + + + {\displaystyle {\begin{alignedat}{10}X\setminus \varnothing &=X&&\qquad {\text{ Also written }}\quad &&\varnothing ^{\complement }=X&&\quad &&{\text{ (Complement laws for the empty set))}}\\[1.4ex]X\setminus X&=\varnothing &&\qquad {\text{ Also written }}\quad &&X^{\complement }=\varnothing &&\quad &&{\text{ (Complement laws for the universe set)}}\\[1.4ex]\end{alignedat}}} + + +== Two sets involved == +In the left hand sides of the following identities, + + + + L + + + {\displaystyle L} + + is the L eft most set and + + + + R + + + {\displaystyle R} + + is the R ight most set. +Assume both + + + + L + + and + + R + + + {\displaystyle L{\text{ and }}R} + + are subsets of some universe set + + + + X + . + + + {\displaystyle X.} + + +=== Formulas for binary set operations ⋂, ⋃, \, and ∆ === +In the left hand sides of the following identities, L is the L eft most set and R is the R ight most set. Whenever necessary, both L and R should be assumed to be subsets of some universe set X, so that + + + + + L + + ∁ + + + := + X + ∖ + L + + and + + + R + + ∁ + + + := + X + ∖ + R + . + + + {\displaystyle L^{\complement }:=X\setminus L{\text{ and }}R^{\complement }:=X\setminus R.} + + + + + + + + + + L + ∩ + R + + + + = + L + + + + + + + ∖ + + + + + + ( + L + + + + + + + ∖ + + + + R + ) + + + + + + + = + R + + + + + + + ∖ + + + + + + ( + R + + + + + + + ∖ + + + + L + ) + + + + + + + = + L + + + + + + + ∖ + + + + + + ( + L + + + + + △ + + + + + R + ) + + + + + + + = + L + + + + + △ + + + + + + ( + L + + + + + + + ∖ + + + + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{9}L\cap R&=L&&\,\,\setminus \,&&(L&&\,\,\setminus &&R)\\&=R&&\,\,\setminus \,&&(R&&\,\,\setminus &&L)\\&=L&&\,\,\setminus \,&&(L&&\,\triangle \,&&R)\\&=L&&\,\triangle \,&&(L&&\,\,\setminus &&R)\\\end{alignedat}}} + + + + + + + + + + L + ∪ + R + + + + = + ( + + + + L + + △ + + R + ) + + + + + + + ∪ + + + + + + L + + + + + + + + + + + = + ( + + + + L + + △ + + R + ) + + + + + △ + + + + + + ( + + + + L + + + + + ∩ + + + + + R + ) + + + + + + + = + ( + + + + R + + ∖ + + L + ) + + + + + + + ∪ + + + + + + L + + + + + + + + + + + + (union is disjoint) + + + + + + + + {\displaystyle {\begin{alignedat}{9}L\cup R&=(&&L\,\triangle \,R)&&\,\,\cup &&&&L&&&&\\&=(&&L\,\triangle \,R)&&\,\triangle \,&&(&&L&&\cap \,&&R)\\&=(&&R\,\setminus \,L)&&\,\,\cup &&&&L&&&&~~~~~{\text{ (union is disjoint)}}\\\end{alignedat}}} + + + + + + + + + + L + + △ + + R + + + + = + + + + R + + △ + + L + + + + + + + + + + + + + + + = + ( + + + + L + + ∪ + + R + ) + + + + + + ∖ + + + + + + ( + + + + L + + + ∩ + + R + ) + + + + + + + + + = + ( + + + + L + + ∖ + + R + ) + + + + + ∪ + + + + + + ( + + + + R + + + ∖ + + L + ) + + + + + + + + + + (union is disjoint) + + + + + + + + = + ( + + + + L + + △ + + M + ) + + + + + △ + + + + + + ( + + + + M + + △ + + R + ) + + + + + + + + + + where + + M + + is an arbitrary set. + + + + + + + + = + ( + + + + + L + + ∁ + + + ) + + + + + △ + + + + + + ( + + + + + R + + ∁ + + + ) + + + + + + + + + {\displaystyle {\begin{alignedat}{9}L\,\triangle \,R&=&&R\,\triangle \,L&&&&&&&&\\&=(&&L\,\cup \,R)&&\,\setminus \,&&(&&L\,\,\cap \,R)&&\\&=(&&L\,\setminus \,R)&&\cup \,&&(&&R\,\,\setminus \,L)&&~~~~~{\text{ (union is disjoint)}}\\&=(&&L\,\triangle \,M)&&\,\triangle \,&&(&&M\,\triangle \,R)&&~~~~~{\text{ where }}M{\text{ is an arbitrary set. }}\\&=(&&L^{\complement })&&\,\triangle \,&&(&&R^{\complement })&&\\\end{alignedat}}} + + + + + + + + + + L + ∖ + R + + + + = + + + + L + + + + + + + ∖ + + + + + ( + L + + + + + + + ∩ + + + + R + ) + + + + + + + = + + + + L + + + + + + + ∩ + + + + + ( + L + + + + + △ + + + + + R + ) + + + + + + + = + + + + L + + + + + △ + + + + + + ( + L + + + + + + + ∩ + + + + R + ) + + + + + + + = + + + + R + + + + + △ + + + + + + ( + L + + + + + + + ∪ + + + + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{9}L\setminus R&=&&L&&\,\,\setminus &&(L&&\,\,\cap &&R)\\&=&&L&&\,\,\cap &&(L&&\,\triangle \,&&R)\\&=&&L&&\,\triangle \,&&(L&&\,\,\cap &&R)\\&=&&R&&\,\triangle \,&&(L&&\,\,\cup &&R)\\\end{alignedat}}} + + +=== De Morgan's laws === +De Morgan's laws state that for + + + + L + , + R + ⊆ + X + : + + + {\displaystyle L,R\subseteq X:} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-10.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-10.md new file mode 100644 index 000000000..c8e3f15ba --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-10.md @@ -0,0 +1,1335 @@ +--- +title: "List of set identities and relations" +chunk: 11/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + ( + L + + ∖ + M + ) + + + + + ∪ + + + + + + ( + + + + L + + ∖ + R + ) + + + + + + = + + + + + L + + ∖ + + ( + M + + ∩ + + R + ) + + + + + ( + L + + ∖ + M + ) + + + + + ∩ + + + + + + ( + + + + L + + ∖ + R + ) + + + + + + = + + + + + L + + ∖ + + ( + M + + ∪ + + R + ) + + + + + ( + L + + ∖ + M + ) + + + + + ∖ + + + + + + ( + + + + L + + ∖ + R + ) + + + + + + = + + + + + + ( + L + + ∩ + + R + ) + + ∖ + + M + + + + + ( + L + + ∖ + M + ) + + + + △ + + + + + + ( + + + + L + + ∖ + R + ) + + + + + + = + + + + + L + + ∩ + + ( + M + + △ + + R + ) + + + + + + + + + + + + + + + + + + + + = + + + + + + ( + L + ∩ + M + ) + + △ + + ( + L + ∩ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{9}(L\,\setminus M)&\,\cup \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cap \,R)\\[1.4ex](L\,\setminus M)&\,\cap \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cup \,R)\\[1.4ex](L\,\setminus M)&\,\setminus \,&&(&&L\,\setminus R)&&\;=\;&&(L\,\cap \,R)\,\setminus \,M\\[1.4ex](L\,\setminus M)&\,\triangle \,&&(&&L\,\setminus R)&&\;=\;&&L\,\cap \,(M\,\triangle \,R)\\[1.4ex]&\,&&\,&&\,&&\;=\;&&(L\cap M)\,\triangle \,(L\cap R)\\[1.4ex]\end{alignedat}}} + + +=== Other simplifications === +Other properties: + + + + + L + ∩ + M + = + R + + + and + + + L + ∩ + R + = + M + + + if and only if + + + M + = + R + ⊆ + L + . + + + {\displaystyle L\cap M=R\;{\text{ and }}\;L\cap R=M\qquad {\text{ if and only if }}\qquad M=R\subseteq L.} + + + +If + + + + L + ⊆ + M + + + {\displaystyle L\subseteq M} + + then + + + + L + ∖ + R + = + L + ∩ + ( + M + ∖ + R + ) + . + + + {\displaystyle L\setminus R=L\cap (M\setminus R).} + + + + + + L + × + ( + M + + ∖ + R + ) + = + ( + L + × + M + ) + + ∖ + ( + L + × + R + ) + + + {\displaystyle L\times (M\,\setminus R)=(L\times M)\,\setminus (L\times R)} + + +If + + + + L + ⊆ + R + + + {\displaystyle L\subseteq R} + + then + + + + M + ∖ + R + ⊆ + M + ∖ + L + . + + + {\displaystyle M\setminus R\subseteq M\setminus L.} + + + + + + L + ∩ + M + ∩ + R + = + ∅ + + + {\displaystyle L\cap M\cap R=\varnothing } + + if and only if for any + + + + x + ∈ + L + ∪ + M + ∪ + R + , + + + {\displaystyle x\in L\cup M\cup R,} + + + + + + x + + + {\displaystyle x} + + belongs to at most two of the sets + + + + L + , + M + , + + and + + R + . + + + {\displaystyle L,M,{\text{ and }}R.} + + +== Symmetric difference ∆ of finitely many sets == +Given finitely many sets + + + + + L + + 1 + + + , + … + , + + L + + n + + + , + + + {\displaystyle L_{1},\ldots ,L_{n},} + + something belongs to their symmetric difference if and only if it belongs to an odd number of these sets. Explicitly, for any + + + + x + , + + + {\displaystyle x,} + + + + + + x + ∈ + + L + + 1 + + + △ + ⋯ + △ + + L + + n + + + + + {\displaystyle x\in L_{1}\triangle \cdots \triangle L_{n}} + + if and only if the cardinality + + + + + | + + { + + i + : + x + ∈ + + L + + i + + + + } + + | + + + + {\displaystyle \left|\left\{i:x\in L_{i}\right\}\right|} + + is odd. (Recall that symmetric difference is associative so parentheses are not needed for the set + + + + + L + + 1 + + + △ + ⋯ + △ + + L + + n + + + + + {\displaystyle L_{1}\triangle \cdots \triangle L_{n}} + +). +Consequently, the symmetric difference of three sets satisfies: + + + + + + + + + L + + △ + + M + + △ + + R + + + + = + ( + L + ∩ + M + ∩ + R + ) + ∪ + { + x + : + x + + belongs to exactly one of the sets + + L + , + M + , + R + } + + + + + + + + (the union is disjoint) + + + + + + + + = + [ + L + ∩ + M + ∩ + R + ] + ∪ + [ + L + ∖ + ( + M + ∪ + R + ) + ] + ∪ + [ + M + ∖ + ( + L + ∪ + R + ) + ] + ∪ + [ + R + ∖ + ( + L + ∪ + M + ) + ] + + + + + + + + + + + (all 4 sets enclosed by [ ] are pairwise disjoint) + + + + + + + + {\displaystyle {\begin{alignedat}{4}L\,\triangle \,M\,\triangle \,R&=(L\cap M\cap R)\cup \{x:x{\text{ belongs to exactly one of the sets }}L,M,R\}~~~~~~{\text{ (the union is disjoint) }}\\&=[L\cap M\cap R]\cup [L\setminus (M\cup R)]\cup [M\setminus (L\cup R)]\cup [R\setminus (L\cup M)]~~~~~~~~~{\text{ (all 4 sets enclosed by [ ] are pairwise disjoint) }}\\\end{alignedat}}} + + +== Cartesian products ⨯ of finitely many sets == + +=== Binary ⨯ distributes over ⋃ and ⋂ and \ and ∆ === +The binary Cartesian product ⨯ distributes over unions, intersections, set subtraction, and symmetric difference: + + + + + + + + + ( + L + + ∩ + + M + ) + + × + + R + + + + + + = + + + + + + + ( + L + + × + + R + ) + + + + + + ∩ + + + + + + ( + M + + × + + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + ∩ + + + ) + + + + + + ( + L + + ∪ + + M + ) + + × + + R + + + + + + = + + + + + + + ( + L + + × + + R + ) + + + + + + ∪ + + + + + + ( + M + + × + + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + ∪ + + + ) + + + + + + ( + L + + ∖ + + M + ) + + × + + R + + + + + + = + + + + + + + ( + L + + × + + R + ) + + + + + + ∖ + + + + + + ( + M + + × + + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + ∖ + + + ) + + + + + + ( + L + + △ + + M + ) + + × + + R + + + + + + = + + + + + + + ( + L + + × + + R + ) + + + + + △ + + + + + + ( + M + + × + + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + △ + + + ) + + + + + + + + {\displaystyle {\begin{alignedat}{9}(L\,\cap \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\cap \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\cup \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\setminus \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\setminus \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex](L\,\triangle \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\triangle \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex]\end{alignedat}}} + + + + + + + + + + L + × + ( + M + ∩ + R + ) + + + + + = + + + + + + + ( + L + × + M + ) + ∩ + ( + L + × + R + ) + + + + + + (Left-distributivity of + + + × + + + over + + + ∩ + + + ) + + + + + + L + × + ( + M + ∪ + R + ) + + + + + = + + + + + + + ( + L + × + M + ) + ∪ + ( + L + × + R + ) + + + + + (Left-distributivity of + + + × + + + over + + + ∪ + + + ) + + + + + + L + × + ( + M + ∖ + R + ) + + + + + = + + + + + + + ( + L + × + M + ) + ∖ + ( + L + × + R + ) + + + + + (Left-distributivity of + + + × + + + over + + + ∖ + + + ) + + + + + + L + × + ( + M + △ + R + ) + + + + + = + + + + + + + ( + L + × + M + ) + △ + ( + L + × + R + ) + + + + + (Left-distributivity of + + + × + + + over + + + △ + + + ) + + + + + + + + {\displaystyle {\begin{alignedat}{5}L\times (M\cap R)&\;=\;\;&&(L\times M)\cap (L\times R)\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex]L\times (M\cup R)&\;=\;\;&&(L\times M)\cup (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times (M\setminus R)&\;=\;\;&&(L\times M)\setminus (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex]L\times (M\triangle R)&\;=\;\;&&(L\times M)\triangle (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex]\end{alignedat}}} + + +But in general, ⨯ does not distribute over itself: + + + + + L + × + ( + M + × + R + ) + + + + ≠ + + + + + + + ( + L + × + M + ) + × + ( + L + × + R + ) + + + + + {\displaystyle L\times (M\times R)~\color {Red}{\neq }\color {Black}{}~(L\times M)\times (L\times R)} + + + + + + ( + L + × + M + ) + × + R + + + + ≠ + + + + + + + ( + L + × + R + ) + × + ( + M + × + R + ) + . + + + + + {\displaystyle (L\times M)\times R~\color {Red}{\neq }\color {Black}{}~(L\times R)\times (M\times R).} + + +=== Binary ⋂ of finite ⨯ === + + + + + ( + L + × + R + ) + ∩ + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + + = + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + + {\displaystyle (L\times R)\cap \left(L_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(R\cap R_{2}\right)} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-11.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-11.md new file mode 100644 index 000000000..e63a8b788 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-11.md @@ -0,0 +1,1869 @@ +--- +title: "List of set identities and relations" +chunk: 12/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + ( + L + × + M + × + R + ) + ∩ + + ( + + + L + + 2 + + + × + + M + + 2 + + + × + + R + + 2 + + + + ) + + + = + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + + ( + + M + ∩ + + M + + 2 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + + {\displaystyle (L\times M\times R)\cap \left(L_{2}\times M_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)} + + +=== Binary ⋃ of finite ⨯ === + + + + + + + + + + ( + + L + × + R + + ) + + + ∪ + + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + + + + + = + + + [ + + + ( + + L + ∖ + + L + + 2 + + + + ) + + × + R + + ] + + + ∪ + + + [ + + + ( + + + L + + 2 + + + ∖ + L + + ) + + × + + R + + 2 + + + + ] + + + ∪ + + + [ + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + + ( + + R + ∪ + + R + + 2 + + + + ) + + + ] + + + + + + + + + + = + + + [ + + L + × + + ( + + R + ∖ + + R + + 2 + + + + ) + + + ] + + + ∪ + + + [ + + + L + + 2 + + + × + + ( + + + R + + 2 + + + ∖ + R + + ) + + + ] + + + ∪ + + + [ + + + ( + + L + ∪ + + L + + 2 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + ] + + + + + + + + {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\cup ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\setminus L_{2}\right)\times R\right]~\cup ~\left[\left(L_{2}\setminus L\right)\times R_{2}\right]~\cup ~\left[\left(L\cap L_{2}\right)\times \left(R\cup R_{2}\right)\right]\\[0.5ex]~&=~\left[L\times \left(R\setminus R_{2}\right)\right]~\cup ~\left[L_{2}\times \left(R_{2}\setminus R\right)\right]~\cup ~\left[\left(L\cup L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\\end{alignedat}}} + + +=== Difference \ of finite ⨯ === + + + + + + + + + + ( + + L + × + R + + ) + + + ∖ + + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + + + + + = + + + [ + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + R + + ] + + + ∪ + + + [ + + L + × + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + ] + + + + + + + + {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\setminus ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]~\cup ~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}} + + +and + + + + + ( + L + × + M + × + R + ) + + ∖ + + + ( + + + L + + 2 + + + × + + M + + 2 + + + × + + R + + 2 + + + + ) + + + = + + + [ + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + M + × + R + + ] + + + ∪ + + + [ + + L + × + + ( + + M + + ∖ + + + M + + 2 + + + + ) + + × + R + + ] + + + ∪ + + + [ + + L + × + M + × + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + ] + + + + {\displaystyle (L\times M\times R)~\setminus ~\left(L_{2}\times M_{2}\times R_{2}\right)~=~\left[\left(L\,\setminus \,L_{2}\right)\times M\times R\right]~\cup ~\left[L\times \left(M\,\setminus \,M_{2}\right)\times R\right]~\cup ~\left[L\times M\times \left(R\,\setminus \,R_{2}\right)\right]} + + +=== Finite ⨯ of differences \ === + + + + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + = + + + ( + + L + × + R + + ) + + + ∖ + + + [ + + + ( + + + L + + 2 + + + × + R + + ) + + ∪ + + ( + + L + × + + R + + 2 + + + + ) + + + ] + + + + {\displaystyle \left(L\,\setminus \,L_{2}\right)\times \left(R\,\setminus \,R_{2}\right)~=~\left(L\times R\right)\,\setminus \,\left[\left(L_{2}\times R\right)\cup \left(L\times R_{2}\right)\right]} + + + + + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + + ( + + M + + ∖ + + + M + + 2 + + + + ) + + × + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + = + + + ( + + L + × + M + × + R + + ) + + + ∖ + + + [ + + + ( + + + L + + 2 + + + × + M + × + R + + ) + + ∪ + + ( + + L + × + + M + + 2 + + + × + R + + ) + + ∪ + + ( + + L + × + M + × + + R + + 2 + + + + ) + + + ] + + + + {\displaystyle \left(L\,\setminus \,L_{2}\right)\times \left(M\,\setminus \,M_{2}\right)\times \left(R\,\setminus \,R_{2}\right)~=~\left(L\times M\times R\right)\,\setminus \,\left[\left(L_{2}\times M\times R\right)\cup \left(L\times M_{2}\times R\right)\cup \left(L\times M\times R_{2}\right)\right]} + + +=== Symmetric difference ∆ and finite ⨯ === + + + + + L + × + + ( + + R + + △ + + + R + + 2 + + + + ) + + + = + + + [ + + L + × + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + ] + + + ∪ + + + [ + + L + × + + ( + + + R + + 2 + + + + ∖ + + R + + ) + + + ] + + + + {\displaystyle L\times \left(R\,\triangle \,R_{2}\right)~=~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\,\cup \,\left[L\times \left(R_{2}\,\setminus \,R\right)\right]} + + + + + + + ( + + L + + △ + + + L + + 2 + + + + ) + + × + R + + = + + + [ + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + R + + ] + + + ∪ + + + [ + + + ( + + + L + + 2 + + + + ∖ + + L + + ) + + × + R + + ] + + + + {\displaystyle \left(L\,\triangle \,L_{2}\right)\times R~=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]\,\cup \,\left[\left(L_{2}\,\setminus \,L\right)\times R\right]} + + + + + + + + + + + ( + + L + + △ + + + L + + 2 + + + + ) + + × + + ( + + R + + △ + + + R + + 2 + + + + ) + + + + + + = + + + + + + + + + [ + + + ( + + L + ∪ + + L + + 2 + + + + ) + + × + + ( + + R + ∪ + + R + + 2 + + + + ) + + + ] + + + ∖ + + + [ + + + ( + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + R + + ) + + + ∪ + + + ( + + L + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + ) + + + ] + + + + + + + + = + + + + + + + + + [ + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + + ( + + + R + + 2 + + + + ∖ + + R + + ) + + + ] + + + ∪ + + + [ + + + ( + + + L + + 2 + + + + ∖ + + L + + ) + + × + + ( + + + R + + 2 + + + + ∖ + + R + + ) + + + ] + + + ∪ + + + [ + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + ] + + + ∪ + + + [ + + + ( + + + L + + 2 + + + + ∖ + + L + + ) + + ∪ + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + ] + + + + + + + + {\displaystyle {\begin{alignedat}{4}\left(L\,\triangle \,L_{2}\right)\times \left(R\,\triangle \,R_{2}\right)~&=~&&&&\,\left[\left(L\cup L_{2}\right)\times \left(R\cup R_{2}\right)\right]\;\setminus \;\left[\left(\left(L\cap L_{2}\right)\times R\right)\;\cup \;\left(L\times \left(R\cap R_{2}\right)\right)\right]\\[0.7ex]&=~&&&&\,\left[\left(L\,\setminus \,L_{2}\right)\times \left(R_{2}\,\setminus \,R\right)\right]\,\cup \,\left[\left(L_{2}\,\setminus \,L\right)\times \left(R_{2}\,\setminus \,R\right)\right]\,\cup \,\left[\left(L\,\setminus \,L_{2}\right)\times \left(R\,\setminus \,R_{2}\right)\right]\,\cup \,\left[\left(L_{2}\,\setminus \,L\right)\cup \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}} + + + + + + + + + + + ( + + L + + △ + + + L + + 2 + + + + ) + + × + + ( + + M + + △ + + + M + + 2 + + + + ) + + × + + ( + + R + + △ + + + R + + 2 + + + + ) + + + + + + = + + + [ + + + ( + + L + ∪ + + L + + 2 + + + + ) + + × + + ( + + M + ∪ + + M + + 2 + + + + ) + + × + + ( + + R + ∪ + + R + + 2 + + + + ) + + + ] + + + ∖ + + + [ + + + ( + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + M + × + R + + ) + + + ∪ + + + ( + + L + × + + ( + + M + ∩ + + M + + 2 + + + + ) + + × + R + + ) + + + ∪ + + + ( + + L + × + M + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + ) + + + ] + + + + + + + + {\displaystyle {\begin{alignedat}{4}\left(L\,\triangle \,L_{2}\right)\times \left(M\,\triangle \,M_{2}\right)\times \left(R\,\triangle \,R_{2}\right)~&=~\left[\left(L\cup L_{2}\right)\times \left(M\cup M_{2}\right)\times \left(R\cup R_{2}\right)\right]\;\setminus \;\left[\left(\left(L\cap L_{2}\right)\times M\times R\right)\;\cup \;\left(L\times \left(M\cap M_{2}\right)\times R\right)\;\cup \;\left(L\times M\times \left(R\cap R_{2}\right)\right)\right]\\\end{alignedat}}} + + +In general, + + + + + ( + + L + + △ + + + L + + 2 + + + + ) + + × + + ( + + R + + △ + + + R + + 2 + + + + ) + + + + {\displaystyle \left(L\,\triangle \,L_{2}\right)\times \left(R\,\triangle \,R_{2}\right)} + + need not be a subset nor a superset of + + + + + ( + + L + × + R + + ) + + + △ + + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + . + + + {\displaystyle \left(L\times R\right)\,\triangle \,\left(L_{2}\times R_{2}\right).} + + + + + + + + + + + ( + + L + × + R + + ) + + + △ + + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + + + + + = + + + + + + ( + + L + × + R + + ) + + ∪ + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + + ∖ + + + [ + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + ] + + + + + + + + {\displaystyle {\begin{alignedat}{4}\left(L\times R\right)\,\triangle \,\left(L_{2}\times R_{2}\right)~&=~&&\left(L\times R\right)\cup \left(L_{2}\times R_{2}\right)\;\setminus \;\left[\left(L\cap L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\[0.7ex]\end{alignedat}}} + + + + + + + + + + + ( + + L + × + M + × + R + + ) + + + △ + + + ( + + + L + + 2 + + + × + + M + + 2 + + + × + + R + + 2 + + + + ) + + + + + + = + + + + + + ( + + L + × + M + × + R + + ) + + ∪ + + ( + + + L + + 2 + + + × + + M + + 2 + + + × + + R + + 2 + + + + ) + + + ∖ + + + [ + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + + ( + + M + ∩ + + M + + 2 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + ] + + + + + + + + {\displaystyle {\begin{alignedat}{4}\left(L\times M\times R\right)\,\triangle \,\left(L_{2}\times M_{2}\times R_{2}\right)~&=~&&\left(L\times M\times R\right)\cup \left(L_{2}\times M_{2}\times R_{2}\right)\;\setminus \;\left[\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)\right]\\[0.7ex]\end{alignedat}}} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-12.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-12.md new file mode 100644 index 000000000..445519b83 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-12.md @@ -0,0 +1,1139 @@ +--- +title: "List of set identities and relations" +chunk: 13/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +== Arbitrary families of sets == +Let + + + + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + , + + + {\displaystyle \left(L_{i}\right)_{i\in I},} + + + + + + + + ( + + R + + j + + + ) + + + j + ∈ + J + + + , + + + {\displaystyle \left(R_{j}\right)_{j\in J},} + + and + + + + + + ( + + S + + i + , + j + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + + + {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} + + be indexed families of sets. Whenever the assumption is needed, then all indexing sets, such as + + + + I + + + {\displaystyle I} + + and + + + + J + , + + + {\displaystyle J,} + + are assumed to be non-empty. + +=== Definitions === +A family of sets or (more briefly) a family refers to a set whose elements are sets. +An indexed family of sets is a function from some set, called its indexing set, into some family of sets. +An indexed family of sets will be denoted by + + + + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + , + + + {\displaystyle \left(L_{i}\right)_{i\in I},} + + where this notation assigns the symbol + + + + I + + + {\displaystyle I} + + for the indexing set and for every index + + + + i + ∈ + I + , + + + {\displaystyle i\in I,} + + assigns the symbol + + + + + L + + i + + + + + {\displaystyle L_{i}} + + to the value of the function at + + + + i + . + + + {\displaystyle i.} + + +The function itself may then be denoted by the symbol + + + + + L + + ∙ + + + , + + + {\displaystyle L_{\bullet },} + + which is obtained from the notation + + + + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(L_{i}\right)_{i\in I}} + + by replacing the index + + + + i + + + {\displaystyle i} + + with a bullet symbol + + + + ∙ + + ; + + + {\displaystyle \bullet \,;} + + explicitly, + + + + + L + + ∙ + + + + + {\displaystyle L_{\bullet }} + + is the function: + + + + + + + + + + L + + ∙ + + + : + + + + + I + + + + + → + + + + + { + + + L + + i + + + : + i + ∈ + I + + } + + + + + + + + i + + + + + ↦ + + + + + L + + i + + + + + + + + + {\displaystyle {\begin{alignedat}{4}L_{\bullet }:\;&&I&&\;\to \;&\left\{L_{i}:i\in I\right\}\\[0.3ex]&&i&&\;\mapsto \;&L_{i}\\\end{alignedat}}} + + +which may be summarized by writing + + + + + L + + ∙ + + + = + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + . + + + {\displaystyle L_{\bullet }=\left(L_{i}\right)_{i\in I}.} + + +Any given indexed family of sets + + + + + L + + ∙ + + + = + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle L_{\bullet }=\left(L_{i}\right)_{i\in I}} + + (which is a function) can be canonically associated with its image/range + + + + Im + ⁡ + + L + + ∙ + + + + + + + + = + + + + + def + + + + + + + + + { + + + L + + i + + + : + i + ∈ + I + + } + + + + {\displaystyle \operatorname {Im} L_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{L_{i}:i\in I\right\}} + + (which is a family of sets). +Conversely, any given family of sets + + + + + + B + + + + + {\displaystyle {\mathcal {B}}} + + may be associated with the + + + + + + B + + + + + {\displaystyle {\mathcal {B}}} + +-indexed family of sets + + + + ( + B + + ) + + B + ∈ + + + B + + + + + , + + + {\displaystyle (B)_{B\in {\mathcal {B}}},} + + which is technically the identity map + + + + + + B + + + → + + + B + + + . + + + {\displaystyle {\mathcal {B}}\to {\mathcal {B}}.} + + +However, this is not a bijective correspondence because an indexed family of sets + + + + + L + + ∙ + + + = + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle L_{\bullet }=\left(L_{i}\right)_{i\in I}} + + is not required to be injective (that is, there may exist distinct indices + + + + i + ≠ + j + + + {\displaystyle i\neq j} + + such as + + + + + L + + i + + + = + + L + + j + + + + + {\displaystyle L_{i}=L_{j}} + +), which in particular means that it is possible for distinct indexed families of sets (which are functions) to be associated with the same family of sets (by having the same image/range). +Arbitrary unions defined + +If + + + + I + = + ∅ + + + {\displaystyle I=\varnothing } + + then + + + + + ⋃ + + i + ∈ + ∅ + + + + L + + i + + + = + { + x + + : + + + there exists + + i + ∈ + ∅ + + such that + + x + ∈ + + L + + i + + + } + = + ∅ + , + + + {\displaystyle \bigcup _{i\in \varnothing }L_{i}=\{x~:~{\text{ there exists }}i\in \varnothing {\text{ such that }}x\in L_{i}\}=\varnothing ,} + + which is somethings called the nullary union convention (despite being called a convention, this equality follows from the definition). +If + + + + + + B + + + + + {\displaystyle {\mathcal {B}}} + + is a family of sets then + + + + ∪ + + + B + + + + + {\displaystyle \cup {\mathcal {B}}} + + denotes the set: + + + + + ⋃ + + + B + + + + + + + + + = + + + + + def + + + + + + + + + ⋃ + + B + ∈ + + + B + + + + + B + + + + + + + = + + + + + def + + + + + + + + { + x + + : + + + there exists + + B + ∈ + + + B + + + + such that + + x + ∈ + B + } + . + + + {\displaystyle \bigcup {\mathcal {B}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcup _{B\in {\mathcal {B}}}B~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{x~:~{\text{ there exists }}B\in {\mathcal {B}}{\text{ such that }}x\in B\}.} + + +Arbitrary intersections defined +If + + + + I + ≠ + ∅ + + + {\displaystyle I\neq \varnothing } + + then + +If + + + + + + B + + + ≠ + ∅ + + + {\displaystyle {\mathcal {B}}\neq \varnothing } + + is a non-empty family of sets then + + + + ∩ + + + B + + + + + {\displaystyle \cap {\mathcal {B}}} + + denotes the set: + + + + + ⋂ + + + B + + + + + + + + + = + + + + + def + + + + + + + + + ⋂ + + B + ∈ + B + + + B + + + + + + + = + + + + + def + + + + + + + + { + x + + : + + x + ∈ + B + + for every + + B + ∈ + + + B + + + } + + = + + { + x + + : + + + for all + + B + , + + if + + B + ∈ + + + B + + + + then + + x + ∈ + B + } + . + + + {\displaystyle \bigcap {\mathcal {B}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcap _{B\in B}B~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{x~:~x\in B{\text{ for every }}B\in {\mathcal {B}}\}~=~\{x~:~{\text{ for all }}B,{\text{ if }}B\in {\mathcal {B}}{\text{ then }}x\in B\}.} + + +Nullary intersections +If + + + + I + = + ∅ + + + {\displaystyle I=\varnothing } + + then + + + + + + ⋂ + + i + ∈ + ∅ + + + + L + + i + + + = + { + x + + : + + + for all + + i + , + + if + + i + ∈ + ∅ + + then + + x + ∈ + + L + + i + + + } + + + {\displaystyle \bigcap _{i\in \varnothing }L_{i}=\{x~:~{\text{ for all }}i,{\text{ if }}i\in \varnothing {\text{ then }}x\in L_{i}\}} + + +where every possible thing + + + + x + + + {\displaystyle x} + + in the universe vacuously satisfied the condition: "if + + + + i + ∈ + ∅ + + + {\displaystyle i\in \varnothing } + + then + + + + x + ∈ + + L + + i + + + + + {\displaystyle x\in L_{i}} + +". Consequently, + + + + + + + ⋂ + + i + ∈ + ∅ + + + + + + L + + i + + + = + { + x + : + + true + + } + + + {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}=\{x:{\text{ true }}\}} + + consists of everything in the universe. +So if + + + + I + = + ∅ + + + {\displaystyle I=\varnothing } + + and: + +if you are working in a model in which there exists some universe set + + + + X + + + {\displaystyle X} + + then + + + + + + + ⋂ + + i + ∈ + ∅ + + + + + + L + + i + + + = + { + x + + : + + x + ∈ + + L + + i + + + + for every + + i + ∈ + ∅ + } + + = + + X + . + + + {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}=\{x~:~x\in L_{i}{\text{ for every }}i\in \varnothing \}~=~X.} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-13.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-13.md new file mode 100644 index 000000000..9573cd37a --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-13.md @@ -0,0 +1,1205 @@ +--- +title: "List of set identities and relations" +chunk: 14/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +otherwise, if you are working in a model in which "the class of all things + + + + x + + + {\displaystyle x} + +" is not a set (by far the most common situation) then + + + + + + + ⋂ + + i + ∈ + ∅ + + + + + + L + + i + + + + + {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} + + is undefined because + + + + + + + ⋂ + + i + ∈ + ∅ + + + + + + L + + i + + + + + {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} + + consists of everything, which makes + + + + + + + ⋂ + + i + ∈ + ∅ + + + + + + L + + i + + + + + {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} + + a proper class and not a set. +Assumption: Henceforth, whenever a formula requires some indexing set to be non-empty in order for an arbitrary intersection to be well-defined, then this will automatically be assumed without mention. +A consequence of this is the following assumption/definition: + +A finite intersection of sets or an intersection of finitely many sets refers to the intersection of a finite collection of one or more sets. +Some authors adopt the so called nullary intersection convention, which is the convention that an empty intersection of sets is equal to some canonical set. In particular, if all sets are subsets of some set + + + + X + + + {\displaystyle X} + + then some author may declare that the empty intersection of these sets be equal to + + + + X + . + + + {\displaystyle X.} + + However, the nullary intersection convention is not as commonly accepted as the nullary union convention and this article will not adopt it (this is due to the fact that unlike the empty union, the value of the empty intersection depends on + + + + X + + + {\displaystyle X} + + so if there are multiple sets under consideration, which is commonly the case, then the value of the empty intersection risks becoming ambiguous). +Multiple index sets + + + + + + ⋃ + + + + + j + ∈ + J + + + i + ∈ + I + , + + + + + + + S + + i + , + j + + + + + + + + + = + + + + + def + + + + + + + + + ⋃ + + ( + i + , + j + ) + ∈ + I + × + J + + + + S + + i + , + j + + + + + {\displaystyle \bigcup _{\stackrel {i\in I,}{j\in J}}S_{i,j}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcup _{(i,j)\in I\times J}S_{i,j}} + + + + + + + ⋂ + + + + + j + ∈ + J + + + i + ∈ + I + , + + + + + + + S + + i + , + j + + + + + + + + + = + + + + + def + + + + + + + + + ⋂ + + ( + i + , + j + ) + ∈ + I + × + J + + + + S + + i + , + j + + + + + {\displaystyle \bigcap _{\stackrel {i\in I,}{j\in J}}S_{i,j}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcap _{(i,j)\in I\times J}S_{i,j}} + + +=== Distributing unions and intersections === + +==== Binary ⋂ of arbitrary ⋃'s ==== + +and + +If all + + + + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(L_{i}\right)_{i\in I}} + + are pairwise disjoint and all + + + + + + ( + + R + + j + + + ) + + + j + ∈ + J + + + + + {\displaystyle \left(R_{j}\right)_{j\in J}} + + are also pairwise disjoint, then so are all + + + + + + ( + + + L + + i + + + ∩ + + R + + j + + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + + + {\displaystyle \left(L_{i}\cap R_{j}\right)_{(i,j)\in I\times J}} + + (that is, if + + + + ( + i + , + j + ) + ≠ + + ( + + + i + + 2 + + + , + + j + + 2 + + + + ) + + + + {\displaystyle (i,j)\neq \left(i_{2},j_{2}\right)} + + then + + + + + ( + + + L + + i + + + ∩ + + R + + j + + + + ) + + ∩ + + ( + + + L + + + i + + 2 + + + + + ∩ + + R + + + j + + 2 + + + + + + ) + + = + ∅ + + + {\displaystyle \left(L_{i}\cap R_{j}\right)\cap \left(L_{i_{2}}\cap R_{j_{2}}\right)=\varnothing } + +). + +Importantly, if + + + + I + = + J + + + {\displaystyle I=J} + + then in general, + + + + + + ( + + + ⋃ + + i + ∈ + I + + + + L + + i + + + + ) + + ∩ + + ( + + + ⋃ + + i + ∈ + I + + + + R + + i + + + + ) + + + + + + ≠ + + + + + + + + + ⋃ + + i + ∈ + I + + + + ( + + + L + + i + + + ∩ + + R + + i + + + + ) + + + + + + + {\displaystyle ~\left(\bigcup _{i\in I}L_{i}\right)\cap \left(\bigcup _{i\in I}R_{i}\right)~~\color {Red}{\neq }\color {Black}{}~~\bigcup _{i\in I}\left(L_{i}\cap R_{i}\right)~} + + (an example of this is given below). The single union on the right hand side must be over all pairs + + + + ( + i + , + j + ) + ∈ + I + × + I + : + + + {\displaystyle (i,j)\in I\times I:} + + + + + + + + ( + + + ⋃ + + i + ∈ + I + + + + L + + i + + + + ) + + ∩ + + ( + + + ⋃ + + i + ∈ + I + + + + R + + i + + + + ) + + + + = + + + + ⋃ + + + + + j + ∈ + I + + + i + ∈ + I + , + + + + + + + ( + + + L + + i + + + ∩ + + R + + j + + + + ) + + . + + + + {\displaystyle ~\left(\bigcup _{i\in I}L_{i}\right)\cap \left(\bigcup _{i\in I}R_{i}\right)~~=~~\bigcup _{\stackrel {i\in I,}{j\in I}}\left(L_{i}\cap R_{j}\right).~} + + The same is usually true for other similar non-trivial set equalities and relations that depend on two (potentially unrelated) indexing sets + + + + I + + + {\displaystyle I} + + and + + + + J + + + {\displaystyle J} + + (such as Eq. 4b or Eq. 7g). Two exceptions are Eq. 2c (unions of unions) and Eq. 2d (intersections of intersections), but both of these are among the most trivial of set equalities (although even for these equalities there is still something that must be proven). +Example where equality fails: Let + + + + X + ≠ + ∅ + + + {\displaystyle X\neq \varnothing } + + and let + + + + I + = + { + 1 + , + 2 + } + . + + + {\displaystyle I=\{1,2\}.} + + Let + + + + + L + + 1 + + + : + = + + R + + 2 + + + : + = + X + + + {\displaystyle L_{1}\colon =R_{2}\colon =X} + + and let + + + + + L + + 2 + + + : + = + + R + + 1 + + + : + = + ∅ + . + + + {\displaystyle L_{2}\colon =R_{1}\colon =\varnothing .} + + Then + + + + X + = + X + ∩ + X + = + + ( + + + L + + 1 + + + ∪ + + L + + 2 + + + + ) + + ∩ + + ( + + + R + + 2 + + + ∪ + + R + + 2 + + + + ) + + = + + ( + + + ⋃ + + i + ∈ + I + + + + L + + i + + + + ) + + ∩ + + ( + + + ⋃ + + i + ∈ + I + + + + R + + i + + + + ) + + + ≠ + + + ⋃ + + i + ∈ + I + + + + ( + + + L + + i + + + ∩ + + R + + i + + + + ) + + = + + ( + + + L + + 1 + + + ∩ + + R + + 1 + + + + ) + + ∪ + + ( + + + L + + 2 + + + ∩ + + R + + 2 + + + + ) + + = + ∅ + ∪ + ∅ + = + ∅ + . + + + {\displaystyle X=X\cap X=\left(L_{1}\cup L_{2}\right)\cap \left(R_{2}\cup R_{2}\right)=\left(\bigcup _{i\in I}L_{i}\right)\cap \left(\bigcup _{i\in I}R_{i}\right)~\neq ~\bigcup _{i\in I}\left(L_{i}\cap R_{i}\right)=\left(L_{1}\cap R_{1}\right)\cup \left(L_{2}\cap R_{2}\right)=\varnothing \cup \varnothing =\varnothing .} + + Furthermore, + + + + ∅ + = + ∅ + ∪ + ∅ + = + + ( + + + L + + 1 + + + ∩ + + L + + 2 + + + + ) + + ∪ + + ( + + + R + + 2 + + + ∩ + + R + + 2 + + + + ) + + = + + ( + + + ⋂ + + i + ∈ + I + + + + L + + i + + + + ) + + ∪ + + ( + + + ⋂ + + i + ∈ + I + + + + R + + i + + + + ) + + + ≠ + + + ⋂ + + i + ∈ + I + + + + ( + + + L + + i + + + ∪ + + R + + i + + + + ) + + = + + ( + + + L + + 1 + + + ∪ + + R + + 1 + + + + ) + + ∩ + + ( + + + L + + 2 + + + ∪ + + R + + 2 + + + + ) + + = + X + ∩ + X + = + X + . + + + {\displaystyle \varnothing =\varnothing \cup \varnothing =\left(L_{1}\cap L_{2}\right)\cup \left(R_{2}\cap R_{2}\right)=\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~\neq ~\bigcap _{i\in I}\left(L_{i}\cup R_{i}\right)=\left(L_{1}\cup R_{1}\right)\cap \left(L_{2}\cup R_{2}\right)=X\cap X=X.} + + +==== Binary ⋃ of arbitrary ⋂'s ==== + +and + +Importantly, if + + + + I + = + J + + + {\displaystyle I=J} + + then in general, + + + + + + ( + + + ⋂ + + i + ∈ + I + + + + L + + i + + + + ) + + ∪ + + ( + + + ⋂ + + i + ∈ + I + + + + R + + i + + + + ) + + + + + + ≠ + + + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-14.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-14.md new file mode 100644 index 000000000..93d6f0be1 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-14.md @@ -0,0 +1,1204 @@ +--- +title: "List of set identities and relations" +chunk: 15/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + ⋂ + + i + ∈ + I + + + + ( + + + L + + i + + + ∪ + + R + + i + + + + ) + + + + + + + {\displaystyle ~\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~~\color {Red}{\neq }\color {Black}{}~~\bigcap _{i\in I}\left(L_{i}\cup R_{i}\right)~} + + (an example of this is given above). The single intersection on the right hand side must be over all pairs + + + + ( + i + , + j + ) + ∈ + I + × + I + : + + + {\displaystyle (i,j)\in I\times I:} + + + + + + + + ( + + + ⋂ + + i + ∈ + I + + + + L + + i + + + + ) + + ∪ + + ( + + + ⋂ + + i + ∈ + I + + + + R + + i + + + + ) + + + + = + + + + ⋂ + + + + + j + ∈ + I + + + i + ∈ + I + , + + + + + + + ( + + + L + + i + + + ∪ + + R + + j + + + + ) + + . + + + + {\displaystyle ~\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~~=~~\bigcap _{\stackrel {i\in I,}{j\in I}}\left(L_{i}\cup R_{j}\right).~} + + +==== Arbitrary ⋂'s and arbitrary ⋃'s ==== + +===== Incorrectly distributing by swapping ⋂ and ⋃ ===== +Naively swapping + + + + + + + + ⋃ + + i + ∈ + I + + + + + + + + {\displaystyle \;{\textstyle \bigcup \limits _{i\in I}}\;} + + and + + + + + + + + ⋂ + + j + ∈ + J + + + + + + + + {\displaystyle \;{\textstyle \bigcap \limits _{j\in J}}\;} + + may produce a different set +The following inclusion always holds: + +In general, equality need not hold and moreover, the right hand side depends on how for each fixed + + + + i + ∈ + I + , + + + {\displaystyle i\in I,} + + the sets + + + + + + ( + + S + + i + , + j + + + ) + + + j + ∈ + J + + + + + {\displaystyle \left(S_{i,j}\right)_{j\in J}} + + are labelled; and analogously, the left hand side depends on how for each fixed + + + + j + ∈ + J + , + + + {\displaystyle j\in J,} + + the sets + + + + + + ( + + S + + i + , + j + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(S_{i,j}\right)_{i\in I}} + + are labelled. An example demonstrating this is now given. + +Example of dependence on labeling and failure of equality: To see why equality need not hold when + + + + ∪ + + + {\displaystyle \cup } + + and + + + + ∩ + + + {\displaystyle \cap } + + are swapped, let + + + + I + : + = + J + : + = + { + 1 + , + 2 + } + , + + + {\displaystyle I\colon =J\colon =\{1,2\},} + + and let + + + + + S + + 11 + + + = + { + 1 + , + 2 + } + , + + + S + + 12 + + + = + { + 1 + , + 3 + } + , + + + S + + 21 + + + = + { + 3 + , + 4 + } + , + + + {\displaystyle S_{11}=\{1,2\},~S_{12}=\{1,3\},~S_{21}=\{3,4\},} + + and + + + + + S + + 22 + + + = + { + 2 + , + 4 + } + . + + + {\displaystyle S_{22}=\{2,4\}.} + + Then + + + + { + 1 + , + 4 + } + = + { + 1 + } + ∪ + { + 4 + } + = + + ( + + + S + + 11 + + + ∩ + + S + + 12 + + + + ) + + ∪ + + ( + + + S + + 21 + + + ∩ + + S + + 22 + + + + ) + + = + + ⋃ + + i + ∈ + I + + + + ( + + + ⋂ + + j + ∈ + J + + + + S + + i + , + j + + + + ) + + + ≠ + + + ⋂ + + j + ∈ + J + + + + ( + + + ⋃ + + i + ∈ + I + + + + S + + i + , + j + + + + ) + + = + + ( + + + S + + 11 + + + ∪ + + S + + 21 + + + + ) + + ∩ + + ( + + + S + + 12 + + + ∪ + + S + + 22 + + + + ) + + = + { + 1 + , + 2 + , + 3 + , + 4 + } + . + + + {\displaystyle \{1,4\}=\{1\}\cup \{4\}=\left(S_{11}\cap S_{12}\right)\cup \left(S_{21}\cap S_{22}\right)=\bigcup _{i\in I}\left(\bigcap _{j\in J}S_{i,j}\right)~\neq ~\bigcap _{j\in J}\left(\bigcup _{i\in I}S_{i,j}\right)=\left(S_{11}\cup S_{21}\right)\cap \left(S_{12}\cup S_{22}\right)=\{1,2,3,4\}.} + + +If + + + + + S + + 11 + + + + + {\displaystyle S_{11}} + + and + + + + + S + + 21 + + + + + {\displaystyle S_{21}} + + are swapped while + + + + + S + + 12 + + + + + {\displaystyle S_{12}} + + and + + + + + S + + 22 + + + + + {\displaystyle S_{22}} + + are unchanged, which gives rise to the sets + + + + + + + + S + ^ + + + + + 11 + + + : + = + { + 3 + , + 4 + } + , + + + + + + S + ^ + + + + + 12 + + + : + = + { + 1 + , + 3 + } + , + + + + + + S + ^ + + + + + 21 + + + : + = + { + 1 + , + 2 + } + , + + + {\displaystyle {\hat {S}}_{11}\colon =\{3,4\},~{\hat {S}}_{12}\colon =\{1,3\},~{\hat {S}}_{21}\colon =\{1,2\},} + + and + + + + + + + + S + ^ + + + + + 22 + + + : + = + { + 2 + , + 4 + } + , + + + {\displaystyle {\hat {S}}_{22}\colon =\{2,4\},} + + then + + + + + { + 2 + , + 3 + } + = + { + 3 + } + ∪ + { + 2 + } + = + + ( + + + + + + S + ^ + + + + + 11 + + + ∩ + + + + + S + ^ + + + + + 12 + + + + ) + + ∪ + + ( + + + + + + S + ^ + + + + + 21 + + + ∩ + + + + + S + ^ + + + + + 22 + + + + ) + + = + + ⋃ + + i + ∈ + I + + + + ( + + + ⋂ + + j + ∈ + J + + + + + + + S + ^ + + + + + i + , + j + + + + ) + + + ≠ + + + ⋂ + + j + ∈ + J + + + + ( + + + ⋃ + + i + ∈ + I + + + + + + + S + ^ + + + + + i + , + j + + + + ) + + = + + ( + + + + + + S + ^ + + + + + 11 + + + ∪ + + + + + S + ^ + + + + + 21 + + + + ) + + ∩ + + ( + + + + + + S + ^ + + + + + 12 + + + ∪ + + + + + S + ^ + + + + + 22 + + + + ) + + = + { + 1 + , + 2 + , + 3 + , + 4 + } + . + + + {\displaystyle \{2,3\}=\{3\}\cup \{2\}=\left({\hat {S}}_{11}\cap {\hat {S}}_{12}\right)\cup \left({\hat {S}}_{21}\cap {\hat {S}}_{22}\right)=\bigcup _{i\in I}\left(\bigcap _{j\in J}{\hat {S}}_{i,j}\right)~\neq ~\bigcap _{j\in J}\left(\bigcup _{i\in I}{\hat {S}}_{i,j}\right)=\left({\hat {S}}_{11}\cup {\hat {S}}_{21}\right)\cap \left({\hat {S}}_{12}\cup {\hat {S}}_{22}\right)=\{1,2,3,4\}.} + + +In particular, the left hand side is no longer + + + + { + 1 + , + 4 + } + , + + + {\displaystyle \{1,4\},} + + which shows that the left hand side + + + + + + + ⋃ + + i + ∈ + I + + + + + + + + + ⋂ + + j + ∈ + J + + + + + + S + + i + , + j + + + + + {\displaystyle {\textstyle \bigcup \limits _{i\in I}}\;{\textstyle \bigcap \limits _{j\in J}}S_{i,j}} + + depends on how the sets are labelled. +If instead + + + + + S + + 11 + + + + + {\displaystyle S_{11}} + + and + + + + + S + + 12 + + + + + {\displaystyle S_{12}} + + are swapped while + + + + + S + + 21 + + + + + {\displaystyle S_{21}} + + and + + + + + S + + 22 + + + + + {\displaystyle S_{22}} + + are unchanged, which gives rise to the sets + + + + + + + S + ¯ + + + + 11 + + + : + = + { + 1 + , + 3 + } + , + + + + + S + ¯ + + + + 12 + + + : + = + { + 1 + , + 2 + } + , + + + + + S + ¯ + + + + 21 + + + : + = + { + 3 + , + 4 + } + , + + + {\displaystyle {\overline {S}}_{11}\colon =\{1,3\},~{\overline {S}}_{12}\colon =\{1,2\},~{\overline {S}}_{21}\colon =\{3,4\},} + + and + + + + + + + S + ¯ + + + + 22 + + + : + = + { + 2 + , + 4 + } + , + + + {\displaystyle {\overline {S}}_{22}\colon =\{2,4\},} + + then both the left hand side and right hand side are equal to + + + + { + 1 + , + 4 + } + , + + + {\displaystyle \{1,4\},} + + which shows that the right hand side also depends on how the sets are labeled. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-15.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-15.md new file mode 100644 index 000000000..460f050db --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-15.md @@ -0,0 +1,1432 @@ +--- +title: "List of set identities and relations" +chunk: 16/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +Equality in Inclusion 1 ∪∩ is a subset of ∩∪ can hold under certain circumstances, such as in 7e, which is the special case where + + + + + + ( + + S + + i + , + j + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + + + {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} + + is + + + + + + ( + + + L + + i + + + ∖ + + R + + j + + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + + + {\displaystyle \left(L_{i}\setminus R_{j}\right)_{(i,j)\in I\times J}} + + (that is, + + + + + S + + i + , + j + + + : + = + + L + + i + + + ∖ + + R + + j + + + + + {\displaystyle S_{i,j}\colon =L_{i}\setminus R_{j}} + + with the same indexing sets + + + + I + + + {\displaystyle I} + + and + + + + J + + + {\displaystyle J} + +), or such as in 7f, which is the special case where + + + + + + ( + + S + + i + , + j + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + + + {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} + + is + + + + + + ( + + + L + + i + + + ∖ + + R + + j + + + + ) + + + ( + j + , + i + ) + ∈ + J + × + I + + + + + {\displaystyle \left(L_{i}\setminus R_{j}\right)_{(j,i)\in J\times I}} + + (that is, + + + + + + + + S + ^ + + + + + j + , + i + + + : + = + + L + + i + + + ∖ + + R + + j + + + + + {\displaystyle {\hat {S}}_{j,i}\colon =L_{i}\setminus R_{j}} + + with the indexing sets + + + + I + + + {\displaystyle I} + + and + + + + J + + + {\displaystyle J} + + swapped). +For a correct formula that extends the distributive laws, an approach other than just switching + + + + ∪ + + + {\displaystyle \cup } + + and + + + + ∩ + + + {\displaystyle \cap } + + is needed. + +===== Correct distributive laws ===== +Suppose that for each + + + + i + ∈ + I + , + + + {\displaystyle i\in I,} + + + + + + + J + + i + + + + + {\displaystyle J_{i}} + + is a non-empty index set and for each + + + + j + ∈ + + J + + i + + + , + + + {\displaystyle j\in J_{i},} + + let + + + + + T + + i + , + j + + + + + {\displaystyle T_{i,j}} + + be any set (for example, to apply this law to + + + + + + ( + + S + + i + , + j + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + , + + + {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J},} + + use + + + + + J + + i + + + : + = + J + + + {\displaystyle J_{i}\colon =J} + + for all + + + + i + ∈ + I + + + {\displaystyle i\in I} + + and use + + + + + T + + i + , + j + + + : + = + + S + + i + , + j + + + + + {\displaystyle T_{i,j}\colon =S_{i,j}} + + for all + + + + i + ∈ + I + + + {\displaystyle i\in I} + + and all + + + + j + ∈ + + J + + i + + + = + J + + + {\displaystyle j\in J_{i}=J} + +). Let + + + + + + + ∏ + + + + J + + ∙ + + + + + + + + = + + + + + def + + + + + + + + + ∏ + + i + ∈ + I + + + + J + + i + + + + + {\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\prod _{i\in I}J_{i}} + + +denote the Cartesian product, which can be interpreted as the set of all functions + + + + f + + : + + I + + → + + + + + ⋃ + + i + ∈ + I + + + + + + J + + i + + + + + {\displaystyle f~:~I~\to ~{\textstyle \bigcup \limits _{i\in I}}J_{i}} + + such that + + + + f + ( + i + ) + ∈ + + J + + i + + + + + {\displaystyle f(i)\in J_{i}} + + for every + + + + i + ∈ + I + . + + + {\displaystyle i\in I.} + + Such a function may also be denoted using the tuple notation + + + + + + ( + + f + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(f_{i}\right)_{i\in I}} + + where + + + + + f + + i + + + + + + + + = + + + + + def + + + + + + + + f + ( + i + ) + + + {\displaystyle f_{i}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(i)} + + for every + + + + i + ∈ + I + + + {\displaystyle i\in I} + + and conversely, a tuple + + + + + + ( + + f + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(f_{i}\right)_{i\in I}} + + is just notation for the function with domain + + + + I + + + {\displaystyle I} + + whose value at + + + + i + ∈ + I + + + {\displaystyle i\in I} + + is + + + + + f + + i + + + ; + + + {\displaystyle f_{i};} + + both notations can be used to denote the elements of + + + + + + ∏ + + + + J + + ∙ + + + . + + + {\displaystyle {\textstyle \prod }J_{\bullet }.} + + +Then + +where + + + + + + ∏ + + + + J + + ∙ + + + + + + + + = + + + + + def + + + + + + + + + + + ∏ + + i + ∈ + I + + + + + + J + + i + + + . + + + {\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}.} + + + +===== Applying the distributive laws ===== +Example application: In the particular case where all + + + + + J + + i + + + + + {\displaystyle J_{i}} + + are equal (that is, + + + + + J + + i + + + = + + J + + + i + + 2 + + + + + + + {\displaystyle J_{i}=J_{i_{2}}} + + for all + + + + i + , + + i + + 2 + + + ∈ + I + , + + + {\displaystyle i,i_{2}\in I,} + + which is the case with the family + + + + + + ( + + S + + i + , + j + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + , + + + {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J},} + + for example), then letting + + + + J + + + {\displaystyle J} + + denote this common set, the Cartesian product will be + + + + + + ∏ + + + + J + + ∙ + + + + + + + + = + + + + + def + + + + + + + + + + + ∏ + + i + ∈ + I + + + + + + J + + i + + + = + + + + ∏ + + i + ∈ + I + + + + + J + = + + J + + I + + + , + + + {\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}={\textstyle \prod \limits _{i\in I}}J=J^{I},} + + which is the set of all functions of the form + + + + f + + : + + I + + → + + J + . + + + {\displaystyle f~:~I~\to ~J.} + + The above set equalities Eq. 5 ∩∪ to ∪∩ and Eq. 6 ∪∩ to ∩∪, respectively become: + + + + + + ⋂ + + i + ∈ + I + + + + + ⋃ + + j + ∈ + J + + + + S + + i + , + j + + + = + + ⋃ + + f + ∈ + + J + + I + + + + + + + ⋂ + + i + ∈ + I + + + + S + + i + , + f + ( + i + ) + + + + + {\displaystyle \bigcap _{i\in I}\;\bigcup _{j\in J}S_{i,j}=\bigcup _{f\in J^{I}}\;\bigcap _{i\in I}S_{i,f(i)}} + + + + + + + ⋃ + + i + ∈ + I + + + + + ⋂ + + j + ∈ + J + + + + S + + i + , + j + + + = + + ⋂ + + f + ∈ + + J + + I + + + + + + + ⋃ + + i + ∈ + I + + + + S + + i + , + f + ( + i + ) + + + + + {\displaystyle \bigcup _{i\in I}\;\bigcap _{j\in J}S_{i,j}=\bigcap _{f\in J^{I}}\;\bigcup _{i\in I}S_{i,f(i)}} + + +which when combined with Inclusion 1 ∪∩ is a subset of ∩∪ implies: + + + + + + ⋃ + + i + ∈ + I + + + + + ⋂ + + j + ∈ + J + + + + S + + i + , + j + + + + = + + + ⋂ + + f + ∈ + + J + + I + + + + + + + ⋃ + + i + ∈ + I + + + + S + + i + , + f + ( + i + ) + + + + + + + ⊆ + + + + + + + + + ⋃ + + g + ∈ + + I + + J + + + + + + + ⋂ + + j + ∈ + J + + + + S + + g + ( + j + ) + , + j + + + + = + + + ⋂ + + j + ∈ + J + + + + + ⋃ + + i + ∈ + I + + + + S + + i + , + j + + + + + + + {\displaystyle \bigcup _{i\in I}\;\bigcap _{j\in J}S_{i,j}~=~\bigcap _{f\in J^{I}}\;\bigcup _{i\in I}S_{i,f(i)}~~\color {Red}{\subseteq }\color {Black}{}~~\bigcup _{g\in I^{J}}\;\bigcap _{j\in J}S_{g(j),j}~=~\bigcap _{j\in J}\;\bigcup _{i\in I}S_{i,j}} + + +where + +on the left hand side, the indices + + + + f + + and + + i + + + {\displaystyle f{\text{ and }}i} + + range over + + + + f + ∈ + + J + + I + + + + and + + i + ∈ + I + + + {\displaystyle f\in J^{I}{\text{ and }}i\in I} + + (so the subscripts of + + + + + S + + i + , + f + ( + i + ) + + + + + {\displaystyle S_{i,f(i)}} + + range over + + + + i + ∈ + I + + and + + f + ( + i + ) + ∈ + f + ( + I + ) + ⊆ + J + + + {\displaystyle i\in I{\text{ and }}f(i)\in f(I)\subseteq J} + +) +on the right hand side, the indices + + + + g + + and + + j + + + {\displaystyle g{\text{ and }}j} + + range over + + + + g + ∈ + + I + + J + + + + and + + j + ∈ + J + + + {\displaystyle g\in I^{J}{\text{ and }}j\in J} + + (so the subscripts of + + + + + S + + g + ( + j + ) + , + j + + + + + {\displaystyle S_{g(j),j}} + + range over + + + + j + ∈ + J + + and + + g + ( + j + ) + ∈ + g + ( + J + ) + ⊆ + I + + + {\displaystyle j\in J{\text{ and }}g(j)\in g(J)\subseteq I} + +). \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-16.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-16.md new file mode 100644 index 000000000..2e6c77d8e --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-16.md @@ -0,0 +1,1547 @@ +--- +title: "List of set identities and relations" +chunk: 17/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +Example application: To apply the general formula to the case of + + + + + + ( + + C + + k + + + ) + + + k + ∈ + K + + + + + {\displaystyle \left(C_{k}\right)_{k\in K}} + + and + + + + + + ( + + D + + l + + + ) + + + l + ∈ + L + + + , + + + {\displaystyle \left(D_{l}\right)_{l\in L},} + + use + + + + I + : + = + { + 1 + , + 2 + } + , + + + {\displaystyle I\colon =\{1,2\},} + + + + + + + J + + 1 + + + : + = + K + , + + + {\displaystyle J_{1}\colon =K,} + + + + + + + J + + 2 + + + : + = + L + , + + + {\displaystyle J_{2}\colon =L,} + + and let + + + + + T + + 1 + , + k + + + : + = + + C + + k + + + + + {\displaystyle T_{1,k}\colon =C_{k}} + + for all + + + + k + ∈ + + J + + 1 + + + + + {\displaystyle k\in J_{1}} + + and let + + + + + T + + 2 + , + l + + + : + = + + D + + l + + + + + {\displaystyle T_{2,l}\colon =D_{l}} + + for all + + + + l + ∈ + + J + + 2 + + + . + + + {\displaystyle l\in J_{2}.} + + +Every map + + + + f + ∈ + + + ∏ + + + + J + + ∙ + + + + + + + + = + + + + + def + + + + + + + + + + + ∏ + + i + ∈ + I + + + + + + J + + i + + + = + + J + + 1 + + + × + + J + + 2 + + + = + K + × + L + + + {\displaystyle f\in {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}=J_{1}\times J_{2}=K\times L} + + can be bijectively identified with the pair + + + + + ( + + f + ( + 1 + ) + , + f + ( + 2 + ) + + ) + + ∈ + K + × + L + + + {\displaystyle \left(f(1),f(2)\right)\in K\times L} + + (the inverse sends + + + + ( + k + , + l + ) + ∈ + K + × + L + + + {\displaystyle (k,l)\in K\times L} + + to the map + + + + + f + + ( + k + , + l + ) + + + ∈ + + + ∏ + + + + J + + ∙ + + + + + {\displaystyle f_{(k,l)}\in {\textstyle \prod }J_{\bullet }} + + defined by + + + + 1 + ↦ + k + + + {\displaystyle 1\mapsto k} + + and + + + + 2 + ↦ + l + ; + + + {\displaystyle 2\mapsto l;} + + this is technically just a change of notation). Recall that Eq. 5 ∩∪ to ∪∩ was + + + + + + + ⋂ + + i + ∈ + I + + + + + ⋃ + + j + ∈ + + J + + i + + + + + + T + + i + , + j + + + = + + ⋃ + + f + ∈ + + + ∏ + + + + J + + ∙ + + + + + + + ⋂ + + i + ∈ + I + + + + T + + i + , + f + ( + i + ) + + + . + + + + {\displaystyle ~\bigcap _{i\in I}\;\bigcup _{j\in J_{i}}T_{i,j}=\bigcup _{f\in {\textstyle \prod }J_{\bullet }}\;\bigcap _{i\in I}T_{i,f(i)}.~} + + +Expanding and simplifying the left hand side gives + + + + + + ⋂ + + i + ∈ + I + + + + + ⋃ + + j + ∈ + + J + + i + + + + + + T + + i + , + j + + + = + + ( + + + ⋃ + + j + ∈ + + J + + 1 + + + + + + T + + 1 + , + j + + + + ) + + ∩ + + ( + + + + ⋃ + + j + ∈ + + J + + 2 + + + + + + T + + 2 + , + j + + + + ) + + = + + ( + + + ⋃ + + k + ∈ + K + + + + T + + 1 + , + k + + + + ) + + ∩ + + ( + + + + ⋃ + + l + ∈ + L + + + + T + + 2 + , + l + + + + ) + + = + + ( + + + ⋃ + + k + ∈ + K + + + + C + + k + + + + ) + + ∩ + + ( + + + + ⋃ + + l + ∈ + L + + + + D + + l + + + + ) + + + + {\displaystyle \bigcap _{i\in I}\;\bigcup _{j\in J_{i}}T_{i,j}=\left(\bigcup _{j\in J_{1}}T_{1,j}\right)\cap \left(\;\bigcup _{j\in J_{2}}T_{2,j}\right)=\left(\bigcup _{k\in K}T_{1,k}\right)\cap \left(\;\bigcup _{l\in L}T_{2,l}\right)=\left(\bigcup _{k\in K}C_{k}\right)\cap \left(\;\bigcup _{l\in L}D_{l}\right)} + + +and doing the same to the right hand side gives: + + + + + + ⋃ + + f + ∈ + ∏ + + J + + ∙ + + + + + + + ⋂ + + i + ∈ + I + + + + T + + i + , + f + ( + i + ) + + + = + + ⋃ + + f + ∈ + ∏ + + J + + ∙ + + + + + + ( + + + T + + 1 + , + f + ( + 1 + ) + + + ∩ + + T + + 2 + , + f + ( + 2 + ) + + + + ) + + = + + ⋃ + + f + ∈ + ∏ + + J + + ∙ + + + + + + ( + + + C + + f + ( + 1 + ) + + + ∩ + + D + + f + ( + 2 + ) + + + + ) + + = + + ⋃ + + ( + k + , + l + ) + ∈ + K + × + L + + + + ( + + + C + + k + + + ∩ + + D + + l + + + + ) + + = + + ⋃ + + + + + l + ∈ + L + + + k + ∈ + K + , + + + + + + + ( + + + C + + k + + + ∩ + + D + + l + + + + ) + + . + + + {\displaystyle \bigcup _{f\in \prod J_{\bullet }}\;\bigcap _{i\in I}T_{i,f(i)}=\bigcup _{f\in \prod J_{\bullet }}\left(T_{1,f(1)}\cap T_{2,f(2)}\right)=\bigcup _{f\in \prod J_{\bullet }}\left(C_{f(1)}\cap D_{f(2)}\right)=\bigcup _{(k,l)\in K\times L}\left(C_{k}\cap D_{l}\right)=\bigcup _{\stackrel {k\in K,}{l\in L}}\left(C_{k}\cap D_{l}\right).} + + +Thus the general identity Eq. 5 ∩∪ to ∪∩ reduces down to the previously given set equality Eq. 3b: + + + + + + ( + + + ⋃ + + k + ∈ + K + + + + C + + k + + + + ) + + ∩ + + + ⋃ + + l + ∈ + L + + + + D + + l + + + = + + ⋃ + + + + + l + ∈ + L + + + k + ∈ + K + , + + + + + + + ( + + + C + + k + + + ∩ + + D + + l + + + + ) + + . + + + {\displaystyle \left(\bigcup _{k\in K}C_{k}\right)\cap \;\bigcup _{l\in L}D_{l}=\bigcup _{\stackrel {k\in K,}{l\in L}}\left(C_{k}\cap D_{l}\right).} + + +=== Distributing subtraction over ⋃ and ⋂ === + +The next identities are known as De Morgan's laws. + +The following four set equalities can be deduced from the equalities 7a - 7d above. + +In general, naively swapping + + + + + ∪ + + + + {\displaystyle \;\cup \;} + + and + + + + + ∩ + + + + {\displaystyle \;\cap \;} + + may produce a different set (see this note for more details). +The equalities + + + + + + ⋃ + + i + ∈ + I + + + + + ⋂ + + j + ∈ + J + + + + ( + + + L + + i + + + ∖ + + R + + j + + + + ) + + + = + + + ⋂ + + j + ∈ + J + + + + + ⋃ + + i + ∈ + I + + + + ( + + + L + + i + + + ∖ + + R + + j + + + + ) + + + + and + + + + ⋃ + + j + ∈ + J + + + + + ⋂ + + i + ∈ + I + + + + ( + + + L + + i + + + ∖ + + R + + j + + + + ) + + + = + + + ⋂ + + i + ∈ + I + + + + + ⋃ + + j + ∈ + J + + + + ( + + + L + + i + + + ∖ + + R + + j + + + + ) + + + + {\displaystyle \bigcup _{i\in I}\;\bigcap _{j\in J}\left(L_{i}\setminus R_{j}\right)~=~\bigcap _{j\in J}\;\bigcup _{i\in I}\left(L_{i}\setminus R_{j}\right)\quad {\text{ and }}\quad \bigcup _{j\in J}\;\bigcap _{i\in I}\left(L_{i}\setminus R_{j}\right)~=~\bigcap _{i\in I}\;\bigcup _{j\in J}\left(L_{i}\setminus R_{j}\right)} + + +found in Eq. 7e and Eq. 7f are thus unusual in that they state exactly that swapping + + + + + ∪ + + + + {\displaystyle \;\cup \;} + + and + + + + + ∩ + + + + {\displaystyle \;\cap \;} + + will not change the resulting set. + +=== Commutativity and associativity of ⋃ and ⋂ === +Commutativity: + + + + + + ⋃ + + + + + j + ∈ + J + + + i + ∈ + I + , + + + + + + + S + + i + , + j + + + + + + + + + = + + + + + def + + + + + + + + + ⋃ + + ( + i + , + j + ) + ∈ + I + × + J + + + + S + + i + , + j + + + + = + + + ⋃ + + i + ∈ + I + + + + ( + + + ⋃ + + j + ∈ + J + + + + S + + i + , + j + + + + ) + + + = + + + ⋃ + + j + ∈ + J + + + + ( + + + ⋃ + + i + ∈ + I + + + + S + + i + , + j + + + + ) + + + + {\displaystyle \bigcup _{\stackrel {i\in I,}{j\in J}}S_{i,j}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcup _{(i,j)\in I\times J}S_{i,j}~=~\bigcup _{i\in I}\left(\bigcup _{j\in J}S_{i,j}\right)~=~\bigcup _{j\in J}\left(\bigcup _{i\in I}S_{i,j}\right)} + + + + + + + ⋂ + + + + + j + ∈ + J + + + i + ∈ + I + , + + + + + + + S + + i + , + j + + + + + + + + + = + + + + + def + + + + + + + + + ⋂ + + ( + i + , + j + ) + ∈ + I + × + J + + + + S + + i + , + j + + + + = + + + ⋂ + + i + ∈ + I + + + + ( + + + ⋂ + + j + ∈ + J + + + + S + + i + , + j + + + + ) + + + = + + + ⋂ + + j + ∈ + J + + + + ( + + + ⋂ + + i + ∈ + I + + + + S + + i + , + j + + + + ) + + + + {\displaystyle \bigcap _{\stackrel {i\in I,}{j\in J}}S_{i,j}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\bigcap _{(i,j)\in I\times J}S_{i,j}~=~\bigcap _{i\in I}\left(\bigcap _{j\in J}S_{i,j}\right)~=~\bigcap _{j\in J}\left(\bigcap _{i\in I}S_{i,j}\right)} + + +Unions of unions and intersections of intersections: + + + + + + ( + + + ⋃ + + i + ∈ + I + + + + L + + i + + + + ) + + ∪ + R + + = + + + ⋃ + + i + ∈ + I + + + + ( + + + L + + i + + + ∪ + R + + ) + + + + {\displaystyle \left(\bigcup _{i\in I}L_{i}\right)\cup R~=~\bigcup _{i\in I}\left(L_{i}\cup R\right)} + + + + + + + ( + + + ⋂ + + i + ∈ + I + + + + L + + i + + + + ) + + ∩ + R + + = + + + ⋂ + + i + ∈ + I + + + + ( + + + L + + i + + + ∩ + R + + ) + + + + {\displaystyle \left(\bigcap _{i\in I}L_{i}\right)\cap R~=~\bigcap _{i\in I}\left(L_{i}\cap R\right)} + + +and \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-17.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-17.md new file mode 100644 index 000000000..7e120045a --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-17.md @@ -0,0 +1,1314 @@ +--- +title: "List of set identities and relations" +chunk: 18/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +and if + + + + I + = + J + + + {\displaystyle I=J} + + then also: + +=== Cartesian products Π of arbitrarily many sets === + +==== Intersections ⋂ of Π ==== +If + + + + + + ( + + S + + i + , + j + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + + + {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} + + is a family of sets then + +Moreover, a tuple + + + + + + ( + + x + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(x_{i}\right)_{i\in I}} + + belongs to the set in Eq. 8 above if and only if + + + + + x + + i + + + ∈ + + S + + i + , + j + + + + + {\displaystyle x_{i}\in S_{i,j}} + + for all + + + + i + ∈ + I + + + {\displaystyle i\in I} + + and all + + + + j + ∈ + J + . + + + {\displaystyle j\in J.} + + +In particular, if + + + + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(L_{i}\right)_{i\in I}} + + and + + + + + + ( + + R + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(R_{i}\right)_{i\in I}} + + are two families indexed by the same set then + + + + + + ( + + + ∏ + + i + ∈ + I + + + + L + + i + + + + ) + + ∩ + + ∏ + + i + ∈ + I + + + + R + + i + + + + = + + + ∏ + + i + ∈ + I + + + + ( + + + L + + i + + + ∩ + + R + + i + + + + ) + + + + {\displaystyle \left(\prod _{i\in I}L_{i}\right)\cap \prod _{i\in I}R_{i}~=~\prod _{i\in I}\left(L_{i}\cap R_{i}\right)} + + +So for instance, + + + + + ( + L + × + R + ) + ∩ + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + + = + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + + {\displaystyle (L\times R)\cap \left(L_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(R\cap R_{2}\right)} + + + + + + ( + L + × + R + ) + ∩ + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + ∩ + + ( + + + L + + 3 + + + × + + R + + 3 + + + + ) + + + = + + + ( + + L + ∩ + + L + + 2 + + + ∩ + + L + + 3 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + ∩ + + R + + 3 + + + + ) + + + + {\displaystyle (L\times R)\cap \left(L_{2}\times R_{2}\right)\cap \left(L_{3}\times R_{3}\right)~=~\left(L\cap L_{2}\cap L_{3}\right)\times \left(R\cap R_{2}\cap R_{3}\right)} + + and + + + + + ( + L + × + M + × + R + ) + ∩ + + ( + + + L + + 2 + + + × + + M + + 2 + + + × + + R + + 2 + + + + ) + + + = + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + + ( + + M + ∩ + + M + + 2 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + + {\displaystyle (L\times M\times R)\cap \left(L_{2}\times M_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)} + + +Intersections of products indexed by different sets +Let + + + + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(L_{i}\right)_{i\in I}} + + and + + + + + + ( + + R + + j + + + ) + + + j + ∈ + J + + + + + {\displaystyle \left(R_{j}\right)_{j\in J}} + + be two families indexed by different sets. +Technically, + + + + I + ≠ + J + + + {\displaystyle I\neq J} + + implies + + + + + ( + + + + + ∏ + + i + ∈ + I + + + + + + L + + i + + + + ) + + ∩ + + + + ∏ + + j + ∈ + J + + + + + + R + + j + + + = + ∅ + . + + + {\displaystyle \left({\textstyle \prod \limits _{i\in I}}L_{i}\right)\cap {\textstyle \prod \limits _{j\in J}}R_{j}=\varnothing .} + + +However, sometimes these products are somehow identified as the same set through some bijection or one of these products is identified as a subset of the other via some injective map, in which case (by abuse of notation) this intersection may be equal to some other (possibly non-empty) set. + +For example, if + + + + I + := + { + 1 + , + 2 + } + + + {\displaystyle I:=\{1,2\}} + + and + + + + J + := + { + 1 + , + 2 + , + 3 + } + + + {\displaystyle J:=\{1,2,3\}} + + with all sets equal to + + + + + R + + + + {\displaystyle \mathbb {R} } + + then + + + + + + + ∏ + + i + ∈ + I + + + + + + L + + i + + + = + + + + ∏ + + i + ∈ + { + 1 + , + 2 + } + + + + + + R + + = + + + R + + + 2 + + + + + {\displaystyle {\textstyle \prod \limits _{i\in I}}L_{i}={\textstyle \prod \limits _{i\in \{1,2\}}}\mathbb {R} =\mathbb {R} ^{2}} + + and + + + + + + + ∏ + + j + ∈ + J + + + + + + R + + j + + + = + + + + ∏ + + j + ∈ + { + 1 + , + 2 + , + 3 + } + + + + + + R + + = + + + R + + + 3 + + + + + {\displaystyle {\textstyle \prod \limits _{j\in J}}R_{j}={\textstyle \prod \limits _{j\in \{1,2,3\}}}\mathbb {R} =\mathbb {R} ^{3}} + + where + + + + + + R + + + 2 + + + ∩ + + + R + + + 3 + + + = + ∅ + + + {\displaystyle \mathbb {R} ^{2}\cap \mathbb {R} ^{3}=\varnothing } + + unless, for example, + + + + + + + ∏ + + i + ∈ + { + 1 + , + 2 + } + + + + + + R + + = + + + R + + + 2 + + + + + {\displaystyle {\textstyle \prod \limits _{i\in \{1,2\}}}\mathbb {R} =\mathbb {R} ^{2}} + + is identified as a subset of + + + + + + + ∏ + + j + ∈ + { + 1 + , + 2 + , + 3 + } + + + + + + R + + = + + + R + + + 3 + + + + + {\displaystyle {\textstyle \prod \limits _{j\in \{1,2,3\}}}\mathbb {R} =\mathbb {R} ^{3}} + + through some injection, such as maybe + + + + ( + x + , + y + ) + ↦ + ( + x + , + y + , + 0 + ) + + + {\displaystyle (x,y)\mapsto (x,y,0)} + + for instance; however, in this particular case the product + + + + + + + ∏ + + i + ∈ + I + = + { + 1 + , + 2 + } + + + + + + L + + i + + + + + {\displaystyle {\textstyle \prod \limits _{i\in I=\{1,2\}}}L_{i}} + + actually represents the + + + + J + + + {\displaystyle J} + +-indexed product + + + + + + + ∏ + + j + ∈ + J + = + { + 1 + , + 2 + , + 3 + } + + + + + + L + + i + + + + + {\displaystyle {\textstyle \prod \limits _{j\in J=\{1,2,3\}}}L_{i}} + + where + + + + + L + + 3 + + + := + { + 0 + } + . + + + {\displaystyle L_{3}:=\{0\}.} + + +For another example, take + + + + I + := + { + 1 + , + 2 + } + + + {\displaystyle I:=\{1,2\}} + + and + + + + J + := + { + 1 + , + 2 + , + 3 + } + + + {\displaystyle J:=\{1,2,3\}} + + with + + + + + L + + 1 + + + := + + + R + + + 2 + + + + + {\displaystyle L_{1}:=\mathbb {R} ^{2}} + + and + + + + + L + + 2 + + + , + + R + + 1 + + + , + + R + + 2 + + + , + + and + + + R + + 3 + + + + + {\displaystyle L_{2},R_{1},R_{2},{\text{ and }}R_{3}} + + all equal to + + + + + R + + . + + + {\displaystyle \mathbb {R} .} + + Then + + + + + + + ∏ + + i + ∈ + I + + + + + + L + + i + + + = + + + R + + + 2 + + + × + + R + + + + {\displaystyle {\textstyle \prod \limits _{i\in I}}L_{i}=\mathbb {R} ^{2}\times \mathbb {R} } + + and + + + + + + + ∏ + + j + ∈ + J + + + + + + R + + j + + + = + + R + + × + + R + + × + + R + + , + + + {\displaystyle {\textstyle \prod \limits _{j\in J}}R_{j}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} ,} + + which can both be identified as the same set via the bijection that sends + + + + ( + ( + x + , + y + ) + , + z + ) + ∈ + + + R + + + 2 + + + × + + R + + + + {\displaystyle ((x,y),z)\in \mathbb {R} ^{2}\times \mathbb {R} } + + to + + + + ( + x + , + y + , + z + ) + ∈ + + R + + × + + R + + × + + R + + . + + + {\displaystyle (x,y,z)\in \mathbb {R} \times \mathbb {R} \times \mathbb {R} .} + + Under this identification, + + + + + ( + + + + + ∏ + + i + ∈ + I + + + + + + L + + i + + + + ) + + ∩ + + + + + ∏ + + j + ∈ + J + + + + + + R + + j + + + + = + + + + R + + + 3 + + + . + + + {\displaystyle \left({\textstyle \prod \limits _{i\in I}}L_{i}\right)\cap \,{\textstyle \prod \limits _{j\in J}}R_{j}~=~\mathbb {R} ^{3}.} + + +==== Binary ⨯ distributes over arbitrary ⋃ and ⋂ ==== +The binary Cartesian product ⨯ distributes over arbitrary intersections (when the indexing set is not empty) and over arbitrary unions: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-18.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-18.md new file mode 100644 index 000000000..d67027273 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-18.md @@ -0,0 +1,1622 @@ +--- +title: "List of set identities and relations" +chunk: 19/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + L + × + + ( + + + ⋃ + + i + ∈ + I + + + + R + + i + + + + ) + + + + + + = + + + + + + + + ⋃ + + i + ∈ + I + + + ( + L + × + + R + + i + + + ) + + + + + + (Left-distributivity of + + + × + + + over + + + ∪ + + + ) + + + + + + L + × + + ( + + + ⋂ + + i + ∈ + I + + + + R + + i + + + + ) + + + + + + = + + + + + + + + ⋂ + + i + ∈ + I + + + ( + L + × + + R + + i + + + ) + + + + + + (Left-distributivity of + + + × + + + over + + + + ⋂ + + i + ∈ + I + + + + + when + + I + ≠ + ∅ + + + ) + + + + + + + ( + + + ⋃ + + i + ∈ + I + + + + L + + i + + + + ) + + × + R + + + + + = + + + + + + + + ⋃ + + i + ∈ + I + + + ( + + L + + i + + + × + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + ∪ + + + ) + + + + + + + ( + + + ⋂ + + i + ∈ + I + + + + L + + i + + + + ) + + × + R + + + + + = + + + + + + + + ⋂ + + i + ∈ + I + + + ( + + L + + i + + + × + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + + ⋂ + + i + ∈ + I + + + + + when + + I + ≠ + ∅ + + + ) + + + + + + + + {\displaystyle {\begin{alignedat}{5}L\times \left(\bigcup _{i\in I}R_{i}\right)&\;=\;\;&&\bigcup _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times \left(\bigcap _{i\in I}R_{i}\right)&\;=\;\;&&\bigcap _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\left(\bigcup _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcup _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]\left(\bigcap _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcap _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\end{alignedat}}} + + +==== Distributing arbitrary Π over arbitrary ⋃ ==== +Suppose that for each + + + + i + ∈ + I + , + + + {\displaystyle i\in I,} + + + + + + + J + + i + + + + + {\displaystyle J_{i}} + + is a non-empty index set and for each + + + + j + ∈ + + J + + i + + + , + + + {\displaystyle j\in J_{i},} + + let + + + + + T + + i + , + j + + + + + {\displaystyle T_{i,j}} + + be any set (for example, to apply this law to + + + + + + ( + + S + + i + , + j + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + , + + + {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J},} + + use + + + + + J + + i + + + : + = + J + + + {\displaystyle J_{i}\colon =J} + + for all + + + + i + ∈ + I + + + {\displaystyle i\in I} + + and use + + + + + T + + i + , + j + + + : + = + + S + + i + , + j + + + + + {\displaystyle T_{i,j}\colon =S_{i,j}} + + for all + + + + i + ∈ + I + + + {\displaystyle i\in I} + + and all + + + + j + ∈ + + J + + i + + + = + J + + + {\displaystyle j\in J_{i}=J} + +). Let + + + + + + + ∏ + + + + J + + ∙ + + + + + + + + = + + + + + def + + + + + + + + + ∏ + + i + ∈ + I + + + + J + + i + + + + + {\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\prod _{i\in I}J_{i}} + + +denote the Cartesian product, which (as mentioned above) can be interpreted as the set of all functions + + + + f + + : + + I + + → + + + + + ⋃ + + i + ∈ + I + + + + + + J + + i + + + + + {\displaystyle f~:~I~\to ~{\textstyle \bigcup \limits _{i\in I}}J_{i}} + + such that + + + + f + ( + i + ) + ∈ + + J + + i + + + + + {\displaystyle f(i)\in J_{i}} + + for every + + + + i + ∈ + I + + + {\displaystyle i\in I} + +. +Then + +where + + + + + + ∏ + + + + J + + ∙ + + + + + + + + = + + + + + def + + + + + + + + + + + ∏ + + i + ∈ + I + + + + + + J + + i + + + . + + + {\displaystyle {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}.} + + +==== Unions ⋃ of Π ==== +For unions, only the following is guaranteed in general: + + + + + + ⋃ + + j + ∈ + J + + + + + ∏ + + i + ∈ + I + + + + S + + i + , + j + + + + + + + ⊆ + + + + + + + + + ∏ + + i + ∈ + I + + + + + ⋃ + + j + ∈ + J + + + + S + + i + , + j + + + + + and + + + + ⋃ + + i + ∈ + I + + + + + ∏ + + j + ∈ + J + + + + S + + i + , + j + + + + + + + ⊆ + + + + + + + + + ∏ + + j + ∈ + J + + + + + ⋃ + + i + ∈ + I + + + + S + + i + , + j + + + + + + + + + {\displaystyle \bigcup _{j\in J}\;\prod _{i\in I}S_{i,j}~~\color {Red}{\subseteq }\color {Black}{}~~\prod _{i\in I}\;\bigcup _{j\in J}S_{i,j}\qquad {\text{ and }}\qquad \bigcup _{i\in I}\;\prod _{j\in J}S_{i,j}~~\color {Red}{\subseteq }\color {Black}{}~~\prod _{j\in J}\;\bigcup _{i\in I}S_{i,j}} + + +where + + + + + + ( + + S + + i + , + j + + + ) + + + ( + i + , + j + ) + ∈ + I + × + J + + + + + {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} + + is a family of sets. + +Example where equality fails: Let + + + + I + = + J + = + { + 1 + , + 2 + } + , + + + {\displaystyle I=J=\{1,2\},} + + let + + + + + S + + 1 + , + 1 + + + = + + S + + 2 + , + 2 + + + = + ∅ + , + + + {\displaystyle S_{1,1}=S_{2,2}=\varnothing ,} + + let + + + + X + ≠ + ∅ + , + + + {\displaystyle X\neq \varnothing ,} + + and let + + + + + S + + 1 + , + 2 + + + = + + S + + 2 + , + 1 + + + = + X + . + + + {\displaystyle S_{1,2}=S_{2,1}=X.} + + Then + + + + ∅ + = + ∅ + ∪ + ∅ + = + + ( + + + ∏ + + i + ∈ + I + + + + S + + i + , + 1 + + + + ) + + ∪ + + ( + + + ∏ + + i + ∈ + I + + + + S + + i + , + 2 + + + + ) + + = + + ⋃ + + j + ∈ + J + + + + + ∏ + + i + ∈ + I + + + + S + + i + , + j + + + + + + + ≠ + + + + + + + + + ∏ + + i + ∈ + I + + + + + ⋃ + + j + ∈ + J + + + + S + + i + , + j + + + = + + ( + + + ⋃ + + j + ∈ + J + + + + S + + 1 + , + j + + + + ) + + × + + ( + + + ⋃ + + j + ∈ + J + + + + S + + 2 + , + j + + + + ) + + = + X + × + X + . + + + + + {\displaystyle \varnothing =\varnothing \cup \varnothing =\left(\prod _{i\in I}S_{i,1}\right)\cup \left(\prod _{i\in I}S_{i,2}\right)=\bigcup _{j\in J}\;\prod _{i\in I}S_{i,j}~~\color {Red}{\neq }\color {Black}{}~~\prod _{i\in I}\;\bigcup _{j\in J}S_{i,j}=\left(\bigcup _{j\in J}S_{1,j}\right)\times \left(\bigcup _{j\in J}S_{2,j}\right)=X\times X.} + + More generally, + + + + ∅ + = + + ⋃ + + j + ∈ + J + + + + + ∏ + + i + ∈ + I + + + + S + + i + , + j + + + + + {\textstyle \varnothing =\bigcup _{j\in J}\;\prod _{i\in I}S_{i,j}} + + if and only if for each + + + + j + ∈ + J + , + + + {\displaystyle j\in J,} + + at least one of the sets in the + + + + I + + + {\displaystyle I} + +-indexed collections of sets + + + + + S + + ∙ + , + j + + + = + + + ( + + S + + i + , + j + + + ) + + + i + ∈ + I + + + + + {\displaystyle S_{\bullet ,j}=\left(S_{i,j}\right)_{i\in I}} + + is empty, while + + + + + ∏ + + i + ∈ + I + + + + + ⋃ + + j + ∈ + J + + + + S + + i + , + j + + + ≠ + ∅ + + + {\textstyle \prod _{i\in I}\;\bigcup _{j\in J}S_{i,j}\neq \varnothing } + + if and only if for each + + + + i + ∈ + I + , + + + {\displaystyle i\in I,} + + at least one of the sets in the + + + + J + + + {\displaystyle J} + +-indexed collections of sets + + + + + S + + i + , + ∙ + + + = + + + ( + + S + + i + , + j + + + ) + + + j + ∈ + J + + + + + {\displaystyle S_{i,\bullet }=\left(S_{i,j}\right)_{j\in J}} + + is not empty. +However, + + + + + + + + + + ( + + L + × + R + + ) + + + ∪ + + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + + + + + = + + + [ + + + ( + + L + ∖ + + L + + 2 + + + + ) + + × + R + + ] + + + ∪ + + + [ + + + ( + + + L + + 2 + + + ∖ + L + + ) + + × + + R + + 2 + + + + ] + + + ∪ + + + [ + + + ( + + L + ∩ + + L + + 2 + + + + ) + + × + + ( + + R + ∪ + + R + + 2 + + + + ) + + + ] + + + + + + + + + + = + + + [ + + L + × + + ( + + R + ∖ + + R + + 2 + + + + ) + + + ] + + + ∪ + + + [ + + + L + + 2 + + + × + + ( + + + R + + 2 + + + ∖ + R + + ) + + + ] + + + ∪ + + + [ + + + ( + + L + ∪ + + L + + 2 + + + + ) + + × + + ( + + R + ∩ + + R + + 2 + + + + ) + + + ] + + + + + + + + {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\cup ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\setminus L_{2}\right)\times R\right]~\cup ~\left[\left(L_{2}\setminus L\right)\times R_{2}\right]~\cup ~\left[\left(L\cap L_{2}\right)\times \left(R\cup R_{2}\right)\right]\\[0.5ex]~&=~\left[L\times \left(R\setminus R_{2}\right)\right]~\cup ~\left[L_{2}\times \left(R_{2}\setminus R\right)\right]~\cup ~\left[\left(L\cup L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\\end{alignedat}}} + + +==== Difference \ of Π ==== +If + + + + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(L_{i}\right)_{i\in I}} + + and + + + + + + ( + + R + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(R_{i}\right)_{i\in I}} + + are two families of sets then: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-19.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-19.md new file mode 100644 index 000000000..e430c0739 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-19.md @@ -0,0 +1,1491 @@ +--- +title: "List of set identities and relations" +chunk: 20/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + + ( + + + ∏ + + i + ∈ + I + + + + L + + i + + + + ) + + + ∖ + + + ∏ + + i + ∈ + I + + + + R + + i + + + + + + + = + + + + + ⋃ + + j + ∈ + I + + + + + + ∏ + + i + ∈ + I + + + + + { + + + + + L + + j + + + + ∖ + + + R + + j + + + + + + if + + i + = + j + + + + + + L + + i + + + + + + if + + i + ≠ + j + + + + + + + + + + + + + + + = + + + + + ⋃ + + j + ∈ + I + + + + + + + [ + + + + ( + + + L + + j + + + + ∖ + + + R + + j + + + + ) + + + × + + + ∏ + + + + + j + ≠ + i + + + i + ∈ + I + , + + + + + + + L + + i + + + + + ] + + + + + + + + + + + = + + + ⋃ + + + + + + L + + j + + + ⊈ + + R + + j + + + + + j + ∈ + I + , + + + + + + + + [ + + + + ( + + + L + + j + + + + ∖ + + + R + + j + + + + ) + + + × + + + ∏ + + + + + j + ≠ + i + + + i + ∈ + I + , + + + + + + + L + + i + + + + + ] + + + + + + + + + {\displaystyle {\begin{alignedat}{9}\left(\prod _{i\in I}L_{i}\right)~\setminus ~\prod _{i\in I}R_{i}~&=~\;~\bigcup _{j\in I}\;~\prod _{i\in I}{\begin{cases}L_{j}\,\setminus \,R_{j}&{\text{ if }}i=j\\L_{i}&{\text{ if }}i\neq j\\\end{cases}}\\[0.5ex]~&=~\;~\bigcup _{j\in I}\;~{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.5ex]~&=~\bigcup _{\stackrel {j\in I,}{L_{j}\not \subseteq R_{j}}}{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.3ex]\end{alignedat}}} + + +so for instance, + + + + + + + + + + ( + + L + × + R + + ) + + + ∖ + + + ( + + + L + + 2 + + + × + + R + + 2 + + + + ) + + + + + + = + + + [ + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + R + + ] + + + ∪ + + + [ + + L + × + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + ] + + + + + + + + {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\setminus ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]~\cup ~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}} + + +and + + + + + ( + L + × + M + × + R + ) + + ∖ + + + ( + + + L + + 2 + + + × + + M + + 2 + + + × + + R + + 2 + + + + ) + + + = + + + [ + + + ( + + L + + ∖ + + + L + + 2 + + + + ) + + × + M + × + R + + ] + + + ∪ + + + [ + + L + × + + ( + + M + + ∖ + + + M + + 2 + + + + ) + + × + R + + ] + + + ∪ + + + [ + + L + × + M + × + + ( + + R + + ∖ + + + R + + 2 + + + + ) + + + ] + + + + {\displaystyle (L\times M\times R)~\setminus ~\left(L_{2}\times M_{2}\times R_{2}\right)~=~\left[\left(L\,\setminus \,L_{2}\right)\times M\times R\right]~\cup ~\left[L\times \left(M\,\setminus \,M_{2}\right)\times R\right]~\cup ~\left[L\times M\times \left(R\,\setminus \,R_{2}\right)\right]} + + +==== Symmetric difference ∆ of Π ==== + + + + + + + + + + ( + + + ∏ + + i + ∈ + I + + + + L + + i + + + + ) + + + △ + + + ( + + + ∏ + + i + ∈ + I + + + + R + + i + + + + ) + + + + + + = + + + + + ( + + + ∏ + + i + ∈ + I + + + + L + + i + + + + ) + + + ∪ + + + ( + + + ∏ + + i + ∈ + I + + + + R + + i + + + + ) + + + ∖ + + + ∏ + + i + ∈ + I + + + + L + + i + + + ∩ + + R + + i + + + + + + + + + {\displaystyle {\begin{alignedat}{9}\left(\prod _{i\in I}L_{i}\right)~\triangle ~\left(\prod _{i\in I}R_{i}\right)~&=~\;~\left(\prod _{i\in I}L_{i}\right)~\cup ~\left(\prod _{i\in I}R_{i}\right)\;\setminus \;\prod _{i\in I}L_{i}\cap R_{i}\\[0.5ex]\end{alignedat}}} + + +== Functions and sets == + +Let + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + be any function. +Let + + + + L + + and + + R + + + {\displaystyle L{\text{ and }}R} + + be completely arbitrary sets. Assume + + + + A + ⊆ + X + + and + + C + ⊆ + Y + . + + + {\displaystyle A\subseteq X{\text{ and }}C\subseteq Y.} + + + +=== Definitions === +Let + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + be any function, where we denote its domain + + + + X + + + {\displaystyle X} + + by + + + + domain + ⁡ + f + + + {\displaystyle \operatorname {domain} f} + + and denote its codomain + + + + Y + + + {\displaystyle Y} + + by + + + + codomain + ⁡ + f + . + + + {\displaystyle \operatorname {codomain} f.} + + +Many of the identities below do not actually require that the sets be somehow related to + + + + f + + + {\displaystyle f} + +'s domain or codomain (that is, to + + + + X + + + {\displaystyle X} + + or + + + + Y + + + {\displaystyle Y} + +) so when some kind of relationship is necessary then it will be clearly indicated. +Because of this, in this article, if + + + + L + + + {\displaystyle L} + + is declared to be "any set," and it is not indicated that + + + + L + + + {\displaystyle L} + + must be somehow related to + + + + X + + + {\displaystyle X} + + or + + + + Y + + + {\displaystyle Y} + + (say for instance, that it be a subset + + + + X + + + {\displaystyle X} + + or + + + + Y + + + {\displaystyle Y} + +) then it is meant that + + + + L + + + {\displaystyle L} + + is truly arbitrary. +This generality is useful in situations where + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + is a map between two subsets + + + + X + ⊆ + U + + + {\displaystyle X\subseteq U} + + and + + + + Y + ⊆ + V + + + {\displaystyle Y\subseteq V} + + of some larger sets + + + + U + + + {\displaystyle U} + + and + + + + V + , + + + {\displaystyle V,} + + and where the set + + + + L + + + {\displaystyle L} + + might not be entirely contained in + + + + X + = + domain + ⁡ + f + + + {\displaystyle X=\operatorname {domain} f} + + and/or + + + + Y + = + codomain + ⁡ + f + + + {\displaystyle Y=\operatorname {codomain} f} + + (e.g. if all that is known about + + + + L + + + {\displaystyle L} + + is that + + + + L + ⊆ + U + + + {\displaystyle L\subseteq U} + +); in such a situation it may be useful to know what can and cannot be said about + + + + f + ( + L + ) + + + {\displaystyle f(L)} + + and/or + + + + + f + + − + 1 + + + ( + L + ) + + + {\displaystyle f^{-1}(L)} + + without having to introduce a (potentially unnecessary) intersection such as: + + + + f + ( + L + ∩ + X + ) + + + {\displaystyle f(L\cap X)} + + and/or + + + + + f + + − + 1 + + + ( + L + ∩ + Y + ) + . + + + {\displaystyle f^{-1}(L\cap Y).} + + +Images and preimages of sets +If + + + + L + + + {\displaystyle L} + + is any set then the image of + + + + L + + + {\displaystyle L} + + under + + + + f + + + {\displaystyle f} + + is defined to be the set: + + + + + f + ( + L + ) + + + + + + = + + + + + def + + + + + + + + { + + f + ( + l + ) + + : + + l + ∈ + L + ∩ + domain + ⁡ + f + + } + + + {\displaystyle f(L)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{\,f(l)~:~l\in L\cap \operatorname {domain} f\,\}} + + +while the preimage of + + + + L + + + {\displaystyle L} + + under + + + + f + + + {\displaystyle f} + + is: + + + + + + f + + − + 1 + + + ( + L + ) + + + + + + = + + + + + def + + + + + + + + { + + x + ∈ + domain + ⁡ + f + + : + + f + ( + x + ) + ∈ + L + + } + + + {\displaystyle f^{-1}(L)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{\,x\in \operatorname {domain} f~:~f(x)\in L\,\}} + + +where if + + + + L + = + { + s + } + + + {\displaystyle L=\{s\}} + + is a singleton set then the fiber or preimage of + + + + s + + + {\displaystyle s} + + under + + + + f + + + {\displaystyle f} + + is + + + + + + f + + − + 1 + + + ( + s + ) + + + + + + = + + + + + def + + + + + + + + + f + + − + 1 + + + ( + { + s + } + ) + + = + + { + + x + ∈ + domain + ⁡ + f + + : + + f + ( + x + ) + = + s + + } + . + + + {\displaystyle f^{-1}(s)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f^{-1}(\{s\})~=~\{\,x\in \operatorname {domain} f~:~f(x)=s\,\}.} + + +Denote by + + + + Im + ⁡ + f + + + {\displaystyle \operatorname {Im} f} + + or + + + + image + ⁡ + f + + + {\displaystyle \operatorname {image} f} + + the image or range of + + + + f + : + X + → + Y + , + + + {\displaystyle f:X\to Y,} + + which is the set: + + + + + Im + ⁡ + f + + + + + + = + + + + + def + + + + + + + + f + ( + X + ) + + + + + + = + + + + + def + + + + + + + + f + ( + domain + ⁡ + f + ) + + = + + { + f + ( + x + ) + + : + + x + ∈ + domain + ⁡ + f + } + . + + + {\displaystyle \operatorname {Im} f~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(X)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(\operatorname {domain} f)~=~\{f(x)~:~x\in \operatorname {domain} f\}.} + + +Saturated sets + +A set + + + + A + + + {\displaystyle A} + + is said to be + + + + f + + + {\displaystyle f} + +-saturated or a saturated set if any of the following equivalent conditions are satisfied: + +There exists a set + + + + R + + + {\displaystyle R} + + such that + + + + A + = + + f + + − + 1 + + + ( + R + ) + . + + + {\displaystyle A=f^{-1}(R).} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-2.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-2.md new file mode 100644 index 000000000..af04f71cf --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-2.md @@ -0,0 +1,1164 @@ +--- +title: "List of set identities and relations" +chunk: 3/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + X + ∖ + ( + L + ∩ + R + ) + + + + = + ( + X + ∖ + L + ) + ∪ + ( + X + ∖ + R + ) + + + + + + Also written + + + + + + + ( + L + ∩ + R + + ) + + ∁ + + + = + + L + + ∁ + + + ∪ + + R + + ∁ + + + + + + + + + + + (De Morgan's law) + + + + + + X + ∖ + ( + L + ∪ + R + ) + + + + = + ( + X + ∖ + L + ) + ∩ + ( + X + ∖ + R + ) + + + + + + Also written + + + + + + + ( + L + ∪ + R + + ) + + ∁ + + + = + + L + + ∁ + + + ∩ + + R + + ∁ + + + + + + + + + + + (De Morgan's law) + + + + + + + + {\displaystyle {\begin{alignedat}{10}X\setminus (L\cap R)&=(X\setminus L)\cup (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cap R)^{\complement }=L^{\complement }\cup R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]X\setminus (L\cup R)&=(X\setminus L)\cap (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cup R)^{\complement }=L^{\complement }\cap R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]\end{alignedat}}} + + +=== Commutativity === +Unions, intersection, and symmetric difference are commutative operations: + + + + + + + + + L + ∪ + R + + + + + = + + + + + R + ∪ + L + + + + + + (Commutativity) + + + + + + L + ∩ + R + + + + + = + + + + + R + ∩ + L + + + + + + (Commutativity) + + + + + + L + + △ + R + + + + + = + + + + + R + + △ + L + + + + + + (Commutativity) + + + + + + + + {\displaystyle {\begin{alignedat}{10}L\cup R&\;=\;&&R\cup L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\cap R&\;=\;&&R\cap L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\,\triangle R&\;=\;&&R\,\triangle L&&\quad {\text{ (Commutativity)}}\\[1.4ex]\end{alignedat}}} + + +Set subtraction is not commutative. However, the commutativity of set subtraction can be characterized: from + + + + ( + L + + ∖ + + R + ) + ∩ + ( + R + + ∖ + + L + ) + = + ∅ + + + {\displaystyle (L\,\setminus \,R)\cap (R\,\setminus \,L)=\varnothing } + + it follows that: + + + + + L + + ∖ + + R + = + R + + ∖ + + L + + + if and only if + + + L + = + R + . + + + {\displaystyle L\,\setminus \,R=R\,\setminus \,L\quad {\text{ if and only if }}\quad L=R.} + + +Said differently, if distinct symbols always represented distinct sets, then the only true formulas of the form + + + + + ⋅ + + + ∖ + + + ⋅ + + = + + ⋅ + + + ∖ + + + ⋅ + + + + {\displaystyle \,\cdot \,\,\setminus \,\,\cdot \,=\,\cdot \,\,\setminus \,\,\cdot \,} + + that could be written would be those involving a single symbol; that is, those of the form: + + + + S + + ∖ + + S + = + S + + ∖ + + S + . + + + {\displaystyle S\,\setminus \,S=S\,\setminus \,S.} + + +But such formulas are necessarily true for every binary operation + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + (because + + + + x + + ∗ + + x + = + x + + ∗ + + x + + + {\displaystyle x\,\ast \,x=x\,\ast \,x} + + must hold by definition of equality), and so in this sense, set subtraction is as diametrically opposite to being commutative as is possible for a binary operation. +Set subtraction is also neither left alternative nor right alternative; instead, + + + + ( + L + ∖ + L + ) + ∖ + R + = + L + ∖ + ( + L + ∖ + R + ) + + + {\displaystyle (L\setminus L)\setminus R=L\setminus (L\setminus R)} + + if and only if + + + + L + ∩ + R + = + ∅ + + + {\displaystyle L\cap R=\varnothing } + + if and only if + + + + ( + R + ∖ + L + ) + ∖ + L + = + R + ∖ + ( + L + ∖ + L + ) + . + + + {\displaystyle (R\setminus L)\setminus L=R\setminus (L\setminus L).} + + +Set subtraction is quasi-commutative and satisfies the Jordan identity. + +=== Other identities involving two sets === +Absorption laws: + + + + + + + + + L + ∪ + ( + L + ∩ + R + ) + + + + + = + + + + + L + + + + + + (Absorption) + + + + + + L + ∩ + ( + L + ∪ + R + ) + + + + + = + + + + + L + + + + + + (Absorption) + + + + + + + + {\displaystyle {\begin{alignedat}{4}L\cup (L\cap R)&\;=\;&&L&&\quad {\text{ (Absorption)}}\\[1.4ex]L\cap (L\cup R)&\;=\;&&L&&\quad {\text{ (Absorption)}}\\[1.4ex]\end{alignedat}}} + + +Other properties + + + + + + + + + L + ∖ + R + + + + = + L + ∩ + ( + X + ∖ + R + ) + + + + + + Also written + + + + + + L + ∖ + R + = + L + ∩ + + R + + ∁ + + + + + + + + + + + where + + L + , + R + ⊆ + X + + + + + X + ∖ + ( + L + ∖ + R + ) + + + + = + ( + X + ∖ + L + ) + ∪ + R + + + + + + Also written + + + + + + + ( + L + ∖ + R + + ) + + ∁ + + + = + + L + + ∁ + + + ∪ + R + + + + + + + + + where + + R + ⊆ + X + + + + + L + ∖ + R + + + + = + ( + X + ∖ + R + ) + ∖ + ( + X + ∖ + L + ) + + + + + + Also written + + + + + + L + ∖ + R + = + + R + + ∁ + + + ∖ + + L + + ∁ + + + + + + + + + + + where + + L + , + R + ⊆ + X + + + + + + + {\displaystyle {\begin{alignedat}{10}L\setminus R&=L\cap (X\setminus R)&&\qquad {\text{ Also written }}\quad &&L\setminus R=L\cap R^{\complement }&&\quad &&{\text{ where }}L,R\subseteq X\\[1.4ex]X\setminus (L\setminus R)&=(X\setminus L)\cup R&&\qquad {\text{ Also written }}\quad &&(L\setminus R)^{\complement }=L^{\complement }\cup R&&\quad &&{\text{ where }}R\subseteq X\\[1.4ex]L\setminus R&=(X\setminus R)\setminus (X\setminus L)&&\qquad {\text{ Also written }}\quad &&L\setminus R=R^{\complement }\setminus L^{\complement }&&\quad &&{\text{ where }}L,R\subseteq X\\[1.4ex]\end{alignedat}}} + + +Intervals: + + + + + ( + a + , + b + ) + ∩ + ( + c + , + d + ) + = + ( + max + { + a + , + c + } + , + min + { + b + , + d + } + ) + + + {\displaystyle (a,b)\cap (c,d)=(\max\{a,c\},\min\{b,d\})} + + + + + + [ + a + , + b + ) + ∩ + [ + c + , + d + ) + = + [ + max + { + a + , + c + } + , + min + { + b + , + d + } + ) + + + {\displaystyle [a,b)\cap [c,d)=[\max\{a,c\},\min\{b,d\})} + + +=== Subsets ⊆ and supersets ⊇ === +The following statements are equivalent for any + + + + L + , + R + ⊆ + X + : + + + {\displaystyle L,R\subseteq X:} + + + + + + L + ⊆ + R + + + {\displaystyle L\subseteq R} + + +Definition of subset: if + + + + l + ∈ + L + + + {\displaystyle l\in L} + + then + + + + l + ∈ + R + + + {\displaystyle l\in R} + + + + + + L + ∩ + R + = + L + + + {\displaystyle L\cap R=L} + + + + + + L + ∪ + R + = + R + + + {\displaystyle L\cup R=R} + + + + + + L + + △ + + R + = + R + ∖ + L + + + {\displaystyle L\,\triangle \,R=R\setminus L} + + + + + + L + + △ + + R + ⊆ + R + ∖ + L + + + {\displaystyle L\,\triangle \,R\subseteq R\setminus L} + + + + + + L + ∖ + R + = + ∅ + + + {\displaystyle L\setminus R=\varnothing } + + + + + + L + + + {\displaystyle L} + + and + + + + X + ∖ + R + + + {\displaystyle X\setminus R} + + are disjoint (that is, + + + + L + ∩ + ( + X + ∖ + R + ) + = + ∅ + + + {\displaystyle L\cap (X\setminus R)=\varnothing } + +) + + + + + X + ∖ + R + ⊆ + X + ∖ + L + + + + {\displaystyle X\setminus R\subseteq X\setminus L\qquad } + + (that is, + + + + + R + + ∁ + + + ⊆ + + L + + ∁ + + + + + {\displaystyle R^{\complement }\subseteq L^{\complement }} + +) + +The following statements are equivalent for any + + + + L + , + R + ⊆ + X + : + + + {\displaystyle L,R\subseteq X:} + + + + + + L + ⊈ + R + + + {\displaystyle L\not \subseteq R} + + +There exists some + + + + l + ∈ + L + ∖ + R + . + + + {\displaystyle l\in L\setminus R.} + + +==== Set equality ==== + +The following statements are equivalent: + + + + + L + = + R + + + {\displaystyle L=R} + + + + + + L + + △ + + R + = + ∅ + + + {\displaystyle L\,\triangle \,R=\varnothing } + + + + + + L + + ∖ + + R + = + R + + ∖ + + L + + + {\displaystyle L\,\setminus \,R=R\,\setminus \,L} + + +If + + + + L + ∩ + R + = + ∅ + + + {\displaystyle L\cap R=\varnothing } + + then + + + + L + = + R + + + {\displaystyle L=R} + + if and only if + + + + L + = + ∅ + = + R + . + + + {\displaystyle L=\varnothing =R.} + + +Uniqueness of complements: If + + + + L + ∪ + R + = + X + + and + + L + ∩ + R + = + ∅ + + + {\textstyle L\cup R=X{\text{ and }}L\cap R=\varnothing } + + then + + + + R + = + X + ∖ + L + + + {\displaystyle R=X\setminus L} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-20.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-20.md new file mode 100644 index 000000000..82f34c3a5 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-20.md @@ -0,0 +1,1344 @@ +--- +title: "List of set identities and relations" +chunk: 21/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +Any such set + + + + R + + + {\displaystyle R} + + necessarily contains + + + + f + ( + A + ) + + + {\displaystyle f(A)} + + as a subset. +Any set not entirely contained in the domain of + + + + f + + + {\displaystyle f} + + cannot be + + + + f + + + {\displaystyle f} + +-saturated. + + + + + A + = + + f + + − + 1 + + + ( + f + ( + A + ) + ) + . + + + {\displaystyle A=f^{-1}(f(A)).} + + + + + + A + ⊇ + + f + + − + 1 + + + ( + f + ( + A + ) + ) + + + {\displaystyle A\supseteq f^{-1}(f(A))} + + and + + + + A + ⊆ + domain + ⁡ + f + . + + + {\displaystyle A\subseteq \operatorname {domain} f.} + + +The inclusion + + + + L + ∩ + domain + ⁡ + f + ⊆ + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + + {\displaystyle L\cap \operatorname {domain} f\subseteq f^{-1}(f(L))} + + always holds, where if + + + + A + ⊆ + domain + ⁡ + f + + + {\displaystyle A\subseteq \operatorname {domain} f} + + then this becomes + + + + A + ⊆ + + f + + − + 1 + + + ( + f + ( + A + ) + ) + . + + + {\displaystyle A\subseteq f^{-1}(f(A)).} + + + + + + A + ⊆ + domain + ⁡ + f + + + {\displaystyle A\subseteq \operatorname {domain} f} + + and if + + + + a + ∈ + A + + + {\displaystyle a\in A} + + and + + + + x + ∈ + domain + ⁡ + f + + + {\displaystyle x\in \operatorname {domain} f} + + satisfy + + + + f + ( + x + ) + = + f + ( + a + ) + , + + + {\displaystyle f(x)=f(a),} + + then + + + + x + ∈ + A + . + + + {\displaystyle x\in A.} + + +Whenever a fiber of + + + + f + + + {\displaystyle f} + + intersects + + + + A + , + + + {\displaystyle A,} + + then + + + + A + + + {\displaystyle A} + + contains the entire fiber. In other words, + + + + A + + + {\displaystyle A} + + contains every + + + + f + + + {\displaystyle f} + +-fiber that intersects it. +Explicitly: whenever + + + + y + ∈ + Im + ⁡ + f + + + {\displaystyle y\in \operatorname {Im} f} + + is such that + + + + A + ∩ + + f + + − + 1 + + + ( + y + ) + ≠ + ∅ + , + + + {\displaystyle A\cap f^{-1}(y)\neq \varnothing ,} + + then + + + + + f + + − + 1 + + + ( + y + ) + ⊆ + A + . + + + {\displaystyle f^{-1}(y)\subseteq A.} + + +In both this statement and the next, the set + + + + Im + ⁡ + f + + + {\displaystyle \operatorname {Im} f} + + may be replaced with any superset of + + + + Im + ⁡ + f + + + {\displaystyle \operatorname {Im} f} + + (such as + + + + codomain + ⁡ + f + + + {\displaystyle \operatorname {codomain} f} + +) and the resulting statement will still be equivalent to the rest. +The intersection of + + + + A + + + {\displaystyle A} + + with a fiber of + + + + f + + + {\displaystyle f} + + is equal to the empty set or to the fiber itself. +Explicitly: for every + + + + y + ∈ + Im + ⁡ + f + , + + + {\displaystyle y\in \operatorname {Im} f,} + + the intersection + + + + A + ∩ + + f + + − + 1 + + + ( + y + ) + + + {\displaystyle A\cap f^{-1}(y)} + + is equal to the empty set + + + + ∅ + + + {\displaystyle \varnothing } + + or to + + + + + f + + − + 1 + + + ( + y + ) + + + {\displaystyle f^{-1}(y)} + + (that is, + + + + A + ∩ + + f + + − + 1 + + + ( + y + ) + = + ∅ + + + {\displaystyle A\cap f^{-1}(y)=\varnothing } + + or + + + + A + ∩ + + f + + − + 1 + + + ( + y + ) + = + + f + + − + 1 + + + ( + y + ) + + + {\displaystyle A\cap f^{-1}(y)=f^{-1}(y)} + +). + +For a set + + + + A + + + {\displaystyle A} + + to be + + + + f + + + {\displaystyle f} + +-saturated, it is necessary that + + + + A + ⊆ + domain + ⁡ + f + . + + + {\displaystyle A\subseteq \operatorname {domain} f.} + + +Compositions and restrictions of functions +If + + + + f + + + {\displaystyle f} + + and + + + + g + + + {\displaystyle g} + + are maps then + + + + g + ∘ + f + + + {\displaystyle g\circ f} + + denotes the composition map + + + + + g + ∘ + f + + : + + { + + x + ∈ + domain + ⁡ + f + + : + + f + ( + x + ) + ∈ + domain + ⁡ + g + + } + + → + + codomain + ⁡ + g + + + {\displaystyle g\circ f~:~\{\,x\in \operatorname {domain} f~:~f(x)\in \operatorname {domain} g\,\}~\to ~\operatorname {codomain} g} + + +with domain and codomain + + + + + + + + + domain + ⁡ + ( + g + ∘ + f + ) + + + + = + { + + x + ∈ + domain + ⁡ + f + + : + + f + ( + x + ) + ∈ + domain + ⁡ + g + + } + + + + + codomain + ⁡ + ( + g + ∘ + f + ) + + + + = + codomain + ⁡ + g + + + + + + + {\displaystyle {\begin{alignedat}{4}\operatorname {domain} (g\circ f)&=\{\,x\in \operatorname {domain} f~:~f(x)\in \operatorname {domain} g\,\}\\[0.4ex]\operatorname {codomain} (g\circ f)&=\operatorname {codomain} g\\[0.7ex]\end{alignedat}}} + + +defined by + + + + + ( + g + ∘ + f + ) + ( + x + ) + + + + + + = + + + + + def + + + + + + + + g + ( + f + ( + x + ) + ) + . + + + {\displaystyle (g\circ f)(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~g(f(x)).} + + +The restriction of + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + to + + + + L + , + + + {\displaystyle L,} + + denoted by + + + + f + + + + | + + + + L + + + , + + + {\displaystyle f{\big \vert }_{L},} + + is the map + + + + + f + + + + | + + + + L + + + + : + + L + ∩ + domain + ⁡ + f + + → + + Y + + + {\displaystyle f{\big \vert }_{L}~:~L\cap \operatorname {domain} f~\to ~Y} + + +with + + + + domain + ⁡ + f + + + + | + + + + L + + + + = + + L + ∩ + domain + ⁡ + f + + + {\displaystyle \operatorname {domain} f{\big \vert }_{L}~=~L\cap \operatorname {domain} f} + + defined by sending + + + + x + ∈ + L + ∩ + domain + ⁡ + f + + + {\displaystyle x\in L\cap \operatorname {domain} f} + + to + + + + f + ( + x + ) + ; + + + {\displaystyle f(x);} + + that is, + + + + + f + + + + | + + + + L + + + ( + x + ) + + + + + + = + + + + + def + + + + + + + + f + ( + x + ) + . + + + {\displaystyle f{\big \vert }_{L}(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(x).} + + +Alternatively, + + + + + f + + + + | + + + + L + + + + = + + f + ∘ + In + ⁡ + + + + {\displaystyle ~f{\big \vert }_{L}~=~f\circ \operatorname {In} ~} + + where + + + + + In + ⁡ + + : + + L + ∩ + X + → + X + + + + {\displaystyle ~\operatorname {In} ~:~L\cap X\to X~} + + denotes the inclusion map, which is defined by + + + + In + ⁡ + ( + s + ) + + + + + + = + + + + + def + + + + + + + + s + . + + + {\displaystyle \operatorname {In} (s)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~s.} + + +=== (Pre)Images of arbitrary unions ⋃'s and intersections ⋂'s === +If + + + + + + ( + + L + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(L_{i}\right)_{i\in I}} + + is a family of arbitrary sets indexed by + + + + I + ≠ + ∅ + + + {\displaystyle I\neq \varnothing } + + then: + + + + + + + + + f + + ( + + + ⋂ + + i + ∈ + I + + + + L + + i + + + + ) + + + + + + + + + ⊆ + + + + + + + + + + + ⋂ + + i + ∈ + I + + + f + + ( + + L + + i + + + ) + + + + + + + + f + + ( + + + ⋃ + + i + ∈ + I + + + + L + + i + + + + ) + + + + + + = + + + + ⋃ + + i + ∈ + I + + + f + + ( + + L + + i + + + ) + + + + + + + f + + − + 1 + + + + ( + + + ⋃ + + i + ∈ + I + + + + L + + i + + + + ) + + + + + + = + + + + ⋃ + + i + ∈ + I + + + + f + + − + 1 + + + + ( + + L + + i + + + ) + + + + + + + f + + − + 1 + + + + ( + + + ⋂ + + i + ∈ + I + + + + L + + i + + + + ) + + + + + + = + + + + ⋂ + + i + ∈ + I + + + + f + + − + 1 + + + + ( + + L + + i + + + ) + + + + + + + + {\displaystyle {\begin{alignedat}{4}f\left(\bigcap _{i\in I}L_{i}\right)\;&~\;\color {Red}{\subseteq }\color {Black}{}~\;\;\;\bigcap _{i\in I}f\left(L_{i}\right)\\f\left(\bigcup _{i\in I}L_{i}\right)\;&~=~\;\bigcup _{i\in I}f\left(L_{i}\right)\\f^{-1}\left(\bigcup _{i\in I}L_{i}\right)\;&~=~\;\bigcup _{i\in I}f^{-1}\left(L_{i}\right)\\f^{-1}\left(\bigcap _{i\in I}L_{i}\right)\;&~=~\;\bigcap _{i\in I}f^{-1}\left(L_{i}\right)\\\end{alignedat}}} + + +So of these four identities, it is only images of intersections that are not always preserved. Preimages preserve all basic set operations. Unions are preserved by both images and preimages. +If all + + + + + L + + i + + + + + {\displaystyle L_{i}} + + are + + + + f + + + {\displaystyle f} + +-saturated then + + + + + ⋂ + + i + ∈ + I + + + + L + + i + + + + + {\displaystyle \bigcap _{i\in I}L_{i}} + + be will be + + + + f + + + {\displaystyle f} + +-saturated and equality will hold in the first relation above; explicitly, this means: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-21.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-21.md new file mode 100644 index 000000000..55ef66779 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-21.md @@ -0,0 +1,1136 @@ +--- +title: "List of set identities and relations" +chunk: 22/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +If + + + + + + ( + + A + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle \left(A_{i}\right)_{i\in I}} + + is a family of arbitrary subsets of + + + + X + = + domain + ⁡ + f + , + + + {\displaystyle X=\operatorname {domain} f,} + + which means that + + + + + A + + i + + + ⊆ + X + + + {\displaystyle A_{i}\subseteq X} + + for all + + + + i + , + + + {\displaystyle i,} + + then Conditional Equality 10a becomes: + +=== (Pre)Images of binary set operations === +Throughout, let + + + + L + + + {\displaystyle L} + + and + + + + R + + + {\displaystyle R} + + be any sets and let + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + be any function. +Summary +As the table below shows, set equality is not guaranteed only for images of: intersections, set subtractions, and symmetric differences. + +Preimages preserve set operations +Preimages of sets are well-behaved with respect to all basic set operations: + + + + + + + + + + f + + − + 1 + + + ( + L + ∪ + R + ) + + + + + = + + + f + + − + 1 + + + ( + L + ) + ∪ + + f + + − + 1 + + + ( + R + ) + + + + + + f + + − + 1 + + + ( + L + ∩ + R + ) + + + + + = + + + f + + − + 1 + + + ( + L + ) + ∩ + + f + + − + 1 + + + ( + R + ) + + + + + + f + + − + 1 + + + ( + L + ∖ + + R + ) + + + + + = + + + f + + − + 1 + + + ( + L + ) + ∖ + + + f + + − + 1 + + + ( + R + ) + + + + + + f + + − + 1 + + + ( + L + + △ + + R + ) + + + + + = + + + f + + − + 1 + + + ( + L + ) + + △ + + + f + + − + 1 + + + ( + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}f^{-1}(L\cup R)~&=~f^{-1}(L)\cup f^{-1}(R)\\f^{-1}(L\cap R)~&=~f^{-1}(L)\cap f^{-1}(R)\\f^{-1}(L\setminus \,R)~&=~f^{-1}(L)\setminus \,f^{-1}(R)\\f^{-1}(L\,\triangle \,R)~&=~f^{-1}(L)\,\triangle \,f^{-1}(R)\\\end{alignedat}}} + + +In words, preimages distribute over unions, intersections, set subtraction, and symmetric difference. +Images only preserve unions +Images of unions are well-behaved: + + + + + + + + + f + ( + L + ∪ + R + ) + + + + + = + + f + ( + L + ) + ∪ + f + ( + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}f(L\cup R)~&=~f(L)\cup f(R)\\\end{alignedat}}} + + +but images of the other basic set operations are not since only the following are guaranteed in general: + + + + + + + + + f + ( + L + ∩ + R + ) + + + + + ⊆ + + f + ( + L + ) + ∩ + f + ( + R + ) + + + + + f + ( + L + ∖ + R + ) + + + + + ⊇ + + f + ( + L + ) + ∖ + f + ( + R + ) + + + + + f + ( + L + △ + R + ) + + + + + ⊇ + + f + ( + L + ) + + △ + + f + ( + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}f(L\cap R)~&\subseteq ~f(L)\cap f(R)\\f(L\setminus R)~&\supseteq ~f(L)\setminus f(R)\\f(L\triangle R)~&\supseteq ~f(L)\,\triangle \,f(R)\\\end{alignedat}}} + + +In words, images distribute over unions but not necessarily over intersections, set subtraction, or symmetric difference. What these latter three operations have in common is set subtraction: they either are set subtraction + + + + L + ∖ + R + + + {\displaystyle L\setminus R} + + or else they can naturally be defined as the set subtraction of two sets: + + + + + L + ∩ + R + = + L + ∖ + ( + L + ∖ + R + ) + + + and + + + L + △ + R + = + ( + L + ∪ + R + ) + ∖ + ( + L + ∩ + R + ) + . + + + {\displaystyle L\cap R=L\setminus (L\setminus R)\quad {\text{ and }}\quad L\triangle R=(L\cup R)\setminus (L\cap R).} + + +If + + + + L + = + X + + + {\displaystyle L=X} + + then + + + + f + ( + X + ∖ + R + ) + ⊇ + f + ( + X + ) + ∖ + f + ( + R + ) + + + {\displaystyle f(X\setminus R)\supseteq f(X)\setminus f(R)} + + where as in the more general case, equality is not guaranteed. If + + + + f + + + {\displaystyle f} + + is surjective then + + + + f + ( + X + ∖ + R + ) + + ⊇ + + Y + ∖ + f + ( + R + ) + , + + + {\displaystyle f(X\setminus R)~\supseteq ~Y\setminus f(R),} + + which can be rewritten as: + + + + f + + ( + + R + + ∁ + + + ) + + + ⊇ + + f + ( + R + + ) + + ∁ + + + + + {\displaystyle f\left(R^{\complement }\right)~\supseteq ~f(R)^{\complement }} + + if + + + + + R + + ∁ + + + + + + + + = + + + + + def + + + + + + + + X + ∖ + R + + + {\displaystyle R^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X\setminus R} + + and + + + + f + ( + R + + ) + + ∁ + + + + + + + + = + + + + + def + + + + + + + + Y + ∖ + f + ( + R + ) + . + + + {\displaystyle f(R)^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~Y\setminus f(R).} + + +==== Counter-examples: images of operations not distributing ==== + +If + + + + f + : + { + 1 + , + 2 + } + → + Y + + + {\displaystyle f:\{1,2\}\to Y} + + is constant, + + + + L + = + { + 1 + } + , + + + {\displaystyle L=\{1\},} + + and + + + + R + = + { + 2 + } + + + {\displaystyle R=\{2\}} + + then all four of the set containments + + + + + + + + + f + ( + L + ∩ + R + ) + + + + + ⊊ + + f + ( + L + ) + ∩ + f + ( + R + ) + + + + + f + ( + L + ∖ + R + ) + + + + + ⊋ + + f + ( + L + ) + ∖ + f + ( + R + ) + + + + + f + ( + X + ∖ + R + ) + + + + + ⊋ + + f + ( + X + ) + ∖ + f + ( + R + ) + + + + + f + ( + L + △ + R + ) + + + + + ⊋ + + f + ( + L + ) + △ + f + ( + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}f(L\cap R)~&\subsetneq ~f(L)\cap f(R)\\f(L\setminus R)~&\supsetneq ~f(L)\setminus f(R)\\f(X\setminus R)~&\supsetneq ~f(X)\setminus f(R)\\f(L\triangle R)~&\supsetneq ~f(L)\triangle f(R)\\\end{alignedat}}} + + +are strict/proper (that is, the sets are not equal) since one side is the empty set while the other is non-empty. Thus equality is not guaranteed for even the simplest of functions. +The example above is now generalized to show that these four set equalities can fail for any constant function whose domain contains at least two (distinct) points. +Example: Let + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + be any constant function with image + + + + f + ( + X + ) + = + { + y + } + + + {\displaystyle f(X)=\{y\}} + + and suppose that + + + + L + , + R + ⊆ + X + + + {\displaystyle L,R\subseteq X} + + are non-empty disjoint subsets; that is, + + + + L + ≠ + ∅ + , + R + ≠ + ∅ + , + + + {\displaystyle L\neq \varnothing ,R\neq \varnothing ,} + + and + + + + L + ∩ + R + = + ∅ + , + + + {\displaystyle L\cap R=\varnothing ,} + + which implies that all of the sets + + + + L + + △ + + R + = + L + ∪ + R + , + + + {\displaystyle L~\triangle ~R=L\cup R,} + + + + + + + L + ∖ + R + = + L + , + + + {\displaystyle \,L\setminus R=L,} + + and + + + + X + ∖ + R + ⊇ + L + ∖ + R + + + {\displaystyle X\setminus R\supseteq L\setminus R} + + are not empty and so consequently, their images under + + + + f + + + {\displaystyle f} + + are all equal to + + + + { + y + } + . + + + {\displaystyle \{y\}.} + + + +The containment + + + + + f + ( + L + ∖ + R + ) + + ⊋ + + f + ( + L + ) + ∖ + f + ( + R + ) + + + + {\displaystyle ~f(L\setminus R)~\supsetneq ~f(L)\setminus f(R)~} + + is strict: + + + + { + y + } + + = + + f + ( + L + ∖ + R + ) + + ≠ + + f + ( + L + ) + ∖ + f + ( + R + ) + + = + + { + y + } + ∖ + { + y + } + + = + + ∅ + + + {\displaystyle \{y\}~=~f(L\setminus R)~\neq ~f(L)\setminus f(R)~=~\{y\}\setminus \{y\}~=~\varnothing } + + +In words: functions might not distribute over set subtraction + + + + + ∖ + + + + {\displaystyle \,\setminus \,} + + + +The containment + + + + + f + ( + X + ∖ + R + ) + + ⊋ + + f + ( + X + ) + ∖ + f + ( + R + ) + + + + {\displaystyle ~f(X\setminus R)~\supsetneq ~f(X)\setminus f(R)~} + + is strict: + + + + { + y + } + + = + + f + ( + X + ∖ + R + ) + + ≠ + + f + ( + X + ) + ∖ + f + ( + R + ) + + = + + { + y + } + ∖ + { + y + } + + = + + ∅ + . + + + {\displaystyle \{y\}~=~f(X\setminus R)~\neq ~f(X)\setminus f(R)~=~\{y\}\setminus \{y\}~=~\varnothing .} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-22.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-22.md new file mode 100644 index 000000000..b14f2709b --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-22.md @@ -0,0 +1,932 @@ +--- +title: "List of set identities and relations" +chunk: 23/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +The containment + + + + + f + ( + L + + △ + + R + ) + + ⊋ + + f + ( + L + ) + + △ + + f + ( + R + ) + + + + {\displaystyle ~f(L~\triangle ~R)~\supsetneq ~f(L)~\triangle ~f(R)~} + + is strict: + + + + { + y + } + + = + + f + + ( + + L + + △ + + R + + ) + + + ≠ + + f + ( + L + ) + + △ + + f + ( + R + ) + + = + + { + y + } + △ + { + y + } + + = + + ∅ + + + {\displaystyle \{y\}~=~f\left(L~\triangle ~R\right)~\neq ~f(L)~\triangle ~f(R)~=~\{y\}\triangle \{y\}~=~\varnothing } + + +In words: functions might not distribute over symmetric difference + + + + + △ + + + + {\displaystyle \,\triangle \,} + + (which can be defined as the set subtraction of two sets: + + + + L + △ + R + = + ( + L + ∪ + R + ) + ∖ + ( + L + ∩ + R + ) + + + {\displaystyle L\triangle R=(L\cup R)\setminus (L\cap R)} + +). + +The containment + + + + + f + ( + L + ∩ + R + ) + + ⊊ + + f + ( + L + ) + ∩ + f + ( + R + ) + + + + {\displaystyle ~f(L\cap R)~\subsetneq ~f(L)\cap f(R)~} + + is strict: + + + + ∅ + + = + + f + ( + ∅ + ) + + = + + f + ( + L + ∩ + R + ) + + ≠ + + f + ( + L + ) + ∩ + f + ( + R + ) + + = + + { + y + } + ∩ + { + y + } + + = + + { + y + } + + + {\displaystyle \varnothing ~=~f(\varnothing )~=~f(L\cap R)~\neq ~f(L)\cap f(R)~=~\{y\}\cap \{y\}~=~\{y\}} + + +In words: functions might not distribute over set intersection + + + + + ∩ + + + + {\displaystyle \,\cap \,} + + (which can be defined as the set subtraction of two sets: + + + + L + ∩ + R + = + L + ∖ + ( + L + ∖ + R + ) + + + {\displaystyle L\cap R=L\setminus (L\setminus R)} + +). + +What the set operations in these four examples have in common is that they either are set subtraction + + + + ∖ + + + {\displaystyle \setminus } + + (examples (1) and (2)) or else they can naturally be defined as the set subtraction of two sets (examples (3) and (4)). +Mnemonic: In fact, for each of the above four set formulas for which equality is not guaranteed, the direction of the containment (that is, whether to use + + + + + ⊆ + + or + + ⊇ + + + + {\displaystyle \,\subseteq {\text{ or }}\supseteq \,} + +) can always be deduced by imagining the function + + + + f + + + {\displaystyle f} + + as being constant and the two sets ( + + + + L + + + {\displaystyle L} + + and + + + + R + + + {\displaystyle R} + +) as being non-empty disjoint subsets of its domain. This is because every equality fails for such a function and sets: one side will be always be + + + + ∅ + + + {\displaystyle \varnothing } + + and the other non-empty − from this fact, the correct choice of + + + + + ⊆ + + or + + ⊇ + + + + {\displaystyle \,\subseteq {\text{ or }}\supseteq \,} + + can be deduced by answering: "which side is empty?" For example, to decide if the + + + + ? + + + {\displaystyle ?} + + in + + + + + f + ( + L + △ + R + ) + ∖ + f + ( + R + ) + + + + ? + + + + f + ( + ( + L + △ + R + ) + ∖ + R + ) + + + {\displaystyle f(L\triangle R)\setminus f(R)~\;~?~\;~f((L\triangle R)\setminus R)} + + +should be + + + + + ⊆ + + or + + ⊇ + , + + + + {\displaystyle \,\subseteq {\text{ or }}\supseteq ,\,} + + pretend +that + + + + f + + + {\displaystyle f} + + is constant and that + + + + L + △ + R + + + {\displaystyle L\triangle R} + + and + + + + R + + + {\displaystyle R} + + are non-empty disjoint subsets of + + + + f + + + {\displaystyle f} + +'s domain; then the left hand side would be empty (since + + + + f + ( + L + △ + R + ) + ∖ + f + ( + R + ) + = + { + f + + 's single value + + } + ∖ + { + f + + 's single value + + } + = + ∅ + + + {\displaystyle f(L\triangle R)\setminus f(R)=\{f{\text{'s single value}}\}\setminus \{f{\text{'s single value}}\}=\varnothing } + +), which indicates that + + + + + ? + + + + {\displaystyle \,?\,} + + should be + + + + + ⊆ + + + + {\displaystyle \,\subseteq \,} + + (the resulting statement is always guaranteed to be true) because this is the choice that will make + + + + + ∅ + = + + left hand side + + + + + ? + + + + + right hand side + + + + {\displaystyle \varnothing ={\text{left hand side}}~\;~?~\;~{\text{right hand side}}} + + +true. +Alternatively, the correct direction of containment can also be deduced by consideration of any constant + + + + f + : + { + 1 + , + 2 + } + → + Y + + + {\displaystyle f:\{1,2\}\to Y} + + with + + + + L + = + { + 1 + } + + + {\displaystyle L=\{1\}} + + and + + + + R + = + { + 2 + } + . + + + {\displaystyle R=\{2\}.} + + +Furthermore, this mnemonic can also be used to correctly deduce whether or not a set operation always distribute over images or preimages; for example, to determine whether or not + + + + f + ( + L + ∩ + R + ) + + + {\displaystyle f(L\cap R)} + + always equals + + + + f + ( + L + ) + ∩ + f + ( + R + ) + , + + + {\displaystyle f(L)\cap f(R),} + + or alternatively, whether or not + + + + + f + + − + 1 + + + ( + L + ∩ + R + ) + + + {\displaystyle f^{-1}(L\cap R)} + + always equals + + + + + f + + − + 1 + + + ( + L + ) + ∩ + + f + + − + 1 + + + ( + R + ) + + + {\displaystyle f^{-1}(L)\cap f^{-1}(R)} + + (although + + + + + ∩ + + + + {\displaystyle \,\cap \,} + + was used here, it can replaced by + + + + + ∪ + , + + ∖ + , + + or + + + △ + + + {\displaystyle \,\cup ,\,\setminus ,{\text{ or }}\,\triangle } + +). The answer to such a question can, as before, be deduced by consideration of this constant function: the answer for the general case (that is, for arbitrary + + + + f + , + L + , + + + {\displaystyle f,L,} + + and + + + + R + + + {\displaystyle R} + +) is always the same as the answer for this choice of (constant) function and disjoint non-empty sets. + +==== Conditions guaranteeing that images distribute over set operations ==== +Characterizations of when equality holds for all sets: +For any function + + + + f + : + X + → + Y + , + + + {\displaystyle f:X\to Y,} + + the following statements are equivalent: + + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + is injective. +This means: + + + + f + ( + x + ) + ≠ + f + ( + y + ) + + + {\displaystyle f(x)\neq f(y)} + + for all distinct + + + + x + , + y + ∈ + X + . + + + {\displaystyle x,y\in X.} + + + + + + f + ( + L + ∩ + R + ) + = + f + ( + L + ) + + ∩ + + f + ( + R + ) + + + for all + + + L + , + R + ⊆ + X + . + + + {\displaystyle f(L\cap R)=f(L)\,\cap \,f(R)\,{\text{ for all }}\,L,R\subseteq X.} + + (The equals sign + + + + + = + + + + {\displaystyle \,=\,} + + can be replaced with + + + + + ⊇ + + + + {\displaystyle \,\supseteq \,} + +). + + + + + f + ( + L + + ∖ + R + ) + = + f + ( + L + ) + + ∖ + + f + ( + R + ) + + + for all + + + L + , + R + ⊆ + X + . + + + {\displaystyle f(L\,\setminus R)=f(L)\,\setminus \,f(R)\;{\text{ for all }}\,L,R\subseteq X.} + + (The equals sign + + + + + = + + + + {\displaystyle \,=\,} + + can be replaced with + + + + + ⊆ + + + + {\displaystyle \,\subseteq \,} + +). + + + + + f + ( + X + ∖ + R + ) + = + f + ( + X + ) + ∖ + + f + ( + R + ) + + + for all + + + + + + + + R + ⊆ + X + . + + + {\displaystyle f(X\setminus R)=f(X)\setminus \,f(R)\;{\text{ for all }}\,~~~~~R\subseteq X.} + + (The equals sign + + + + + = + + + + {\displaystyle \,=\,} + + can be replaced with + + + + + ⊆ + + + + {\displaystyle \,\subseteq \,} + +). \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-23.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-23.md new file mode 100644 index 000000000..4f45f2287 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-23.md @@ -0,0 +1,1094 @@ +--- +title: "List of set identities and relations" +chunk: 24/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + f + ( + L + + △ + + R + ) + = + f + ( + L + ) + + △ + + f + ( + R + ) + + + for all + + + L + , + R + ⊆ + X + . + + + {\displaystyle f(L\,\triangle \,R)=f(L)\,\triangle \,f(R)\;{\text{ for all }}\,L,R\subseteq X.} + + (The equals sign + + + + + = + + + + {\displaystyle \,=\,} + + can be replaced with + + + + + ⊆ + + + + {\displaystyle \,\subseteq \,} + +). + +Any one of the four statements (b) - (e) but with the words "for all" replaced with any one of the following: + +"for all singleton subsets" +In particular, the statement that results from (d) gives a characterization of injectivity that explicitly involves only one point (rather than two): + + + + f + + + {\displaystyle f} + + is injective if and only if + + + + f + ( + x + ) + ∉ + f + ( + X + ∖ + { + x + } + ) + + + for every + + + x + ∈ + X + . + + + {\displaystyle f(x)\not \in f(X\setminus \{x\})\;{\text{ for every }}\,x\in X.} + + +"for all disjoint singleton subsets" +For statement (d), this is the same as: "for all singleton subsets" (because the definition of "pairwise disjoint" is satisfies vacuously by any family that consists of exactly 1 set). +"for all disjoint subsets" + +In particular, if a map is not known to be injective then barring additional information, there is no guarantee that any of the equalities in statements (b) - (e) hold. +An example above can be used to help prove this characterization. Indeed, comparison of that example with such a proof suggests that the example is representative of the fundamental reason why one of these four equalities in statements (b) - (e) might not hold (that is, representative of "what goes wrong" when a set equality does not hold). + +===== Conditions for f(L⋂R) = f(L)⋂f(R) ===== + + + + + f + ( + L + ∩ + R + ) + + ⊆ + + f + ( + L + ) + ∩ + f + ( + R + ) + + + + always holds + + + + {\displaystyle f(L\cap R)~\subseteq ~f(L)\cap f(R)\qquad \qquad {\text{ always holds}}} + + +Characterizations of equality: The following statements are equivalent: + + + + + f + ( + L + ∩ + R + ) + + = + + f + ( + L + ) + ∩ + f + ( + R + ) + + + {\displaystyle f(L\cap R)~=~f(L)\cap f(R)} + + + + + + f + ( + L + ∩ + R + ) + + ⊇ + + f + ( + L + ) + ∩ + f + ( + R + ) + + + {\displaystyle f(L\cap R)~\supseteq ~f(L)\cap f(R)} + + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + + {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~f^{-1}(f(L\cap R))} + + +The left hand side + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + + {\displaystyle L\cap f^{-1}(f(R))} + + is always equal to + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + L + ) + ∩ + f + ( + R + ) + ) + + + {\displaystyle L\cap f^{-1}(f(L)\cap f(R))} + + (because + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + + {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~f^{-1}(f(L))} + + always holds). + + + + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + ⊆ + + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + + {\displaystyle R\cap f^{-1}(f(L))~\subseteq ~f^{-1}(f(L\cap R))} + + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + = + + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + ∩ + L + + + {\displaystyle L\cap f^{-1}(f(R))~=~f^{-1}(f(L\cap R))\cap L} + + + + + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + = + + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + ∩ + R + + + {\displaystyle R\cap f^{-1}(f(L))~=~f^{-1}(f(L\cap R))\cap R} + + +If + + + + l + ∈ + L + + + {\displaystyle l\in L} + + satisfies + + + + f + ( + l + ) + ∈ + f + ( + R + ) + + + {\displaystyle f(l)\in f(R)} + + then + + + + f + ( + l + ) + ∈ + f + ( + L + ∩ + R + ) + . + + + {\displaystyle f(l)\in f(L\cap R).} + + +If + + + + y + ∈ + f + ( + L + ) + + + {\displaystyle y\in f(L)} + + but + + + + y + ∉ + f + ( + L + ∩ + R + ) + + + {\displaystyle y\notin f(L\cap R)} + + then + + + + y + ∉ + f + ( + R + ) + . + + + {\displaystyle y\notin f(R).} + + + + + + f + ( + L + ) + + ∖ + + f + ( + L + ∩ + R + ) + + ⊆ + + f + ( + L + ) + + ∖ + + f + ( + R + ) + + + {\displaystyle f(L)\,\setminus \,f(L\cap R)~\subseteq ~f(L)\,\setminus \,f(R)} + + + + + + f + ( + R + ) + + ∖ + + f + ( + L + ∩ + R + ) + + ⊆ + + f + ( + R + ) + + ∖ + + f + ( + L + ) + + + {\displaystyle f(R)\,\setminus \,f(L\cap R)~\subseteq ~f(R)\,\setminus \,f(L)} + + + + + + f + ( + L + ∪ + R + ) + ∖ + f + ( + L + ∩ + R + ) + + ⊆ + + f + ( + L + ) + + △ + + f + ( + R + ) + + + {\displaystyle f(L\cup R)\setminus f(L\cap R)~\subseteq ~f(L)\,\triangle \,f(R)} + + +Any of the above three conditions (i) - (k) but with the subset symbol + + + + + ⊆ + + + + {\displaystyle \,\subseteq \,} + + replaced with an equals sign + + + + + = + . + + + + {\displaystyle \,=.\,} + + +Sufficient conditions for equality: Equality holds if any of the following are true: + + + + + f + + + {\displaystyle f} + + is injective. +The restriction + + + + f + + + + | + + + + L + ∪ + R + + + + + {\displaystyle f{\big \vert }_{L\cup R}} + + is injective. + + + + + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + R + + + {\displaystyle f^{-1}(f(R))~\subseteq ~R} + + + + + + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + ⊆ + + L + + + {\displaystyle f^{-1}(f(L))~\subseteq ~L} + + + + + + R + + + {\displaystyle R} + + is + + + + f + + + {\displaystyle f} + +-saturated; that is, + + + + + f + + − + 1 + + + ( + f + ( + R + ) + ) + = + R + + + {\displaystyle f^{-1}(f(R))=R} + + + + + + L + + + {\displaystyle L} + + is + + + + f + + + {\displaystyle f} + +-saturated; that is, + + + + + f + + − + 1 + + + ( + f + ( + L + ) + ) + = + L + + + {\displaystyle f^{-1}(f(L))=L} + + + + + + f + ( + L + ) + + ⊆ + + f + ( + L + ∩ + R + ) + + + {\displaystyle f(L)~\subseteq ~f(L\cap R)} + + + + + + f + ( + R + ) + + ⊆ + + f + ( + L + ∩ + R + ) + + + {\displaystyle f(R)~\subseteq ~f(L\cap R)} + + + + + + f + ( + L + + ∖ + + R + ) + + ⊆ + + f + ( + L + ) + ∖ + + f + ( + R + ) + + + {\displaystyle f(L\,\setminus \,R)~\subseteq ~f(L)\setminus \,f(R)} + + or equivalently, + + + + f + ( + L + + ∖ + + R + ) + + = + + f + ( + L + ) + ∖ + f + ( + R + ) + + + {\displaystyle f(L\,\setminus \,R)~=~f(L)\setminus f(R)} + + + + + + f + ( + R + + ∖ + + L + ) + + ⊆ + + f + ( + R + ) + ∖ + + f + ( + L + ) + + + {\displaystyle f(R\,\setminus \,L)~\subseteq ~f(R)\setminus \,f(L)} + + or equivalently, + + + + f + ( + R + + ∖ + + L + ) + + = + + f + ( + R + ) + ∖ + f + ( + L + ) + + + {\displaystyle f(R\,\setminus \,L)~=~f(R)\setminus f(L)} + + + + + + f + + ( + + L + + △ + + R + + ) + + ⊆ + f + ( + L + ) + + △ + + f + ( + R + ) + + + {\displaystyle f\left(L~\triangle ~R\right)\subseteq f(L)~\triangle ~f(R)} + + or equivalently, + + + + f + + ( + + L + + △ + + R + + ) + + = + f + ( + L + ) + + △ + + f + ( + R + ) + + + {\displaystyle f\left(L~\triangle ~R\right)=f(L)~\triangle ~f(R)} + + + + + + R + ∩ + domain + ⁡ + f + + ⊆ + L + + + {\displaystyle R\cap \operatorname {domain} f\,\subseteq L} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-24.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-24.md new file mode 100644 index 000000000..fc71ae4b3 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-24.md @@ -0,0 +1,1284 @@ +--- +title: "List of set identities and relations" +chunk: 25/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + L + ∩ + domain + ⁡ + f + + ⊆ + R + + + {\displaystyle L\cap \operatorname {domain} f\,\subseteq R} + + + + + + R + ⊆ + L + + + {\displaystyle R\subseteq L} + + + + + + L + ⊆ + R + + + {\displaystyle L\subseteq R} + + +In addition, the following always hold: + + + + + f + + ( + + + f + + − + 1 + + + ( + L + ) + ∩ + R + + ) + + + = + + L + ∩ + f + ( + R + ) + + + {\displaystyle f\left(f^{-1}(L)\cap R\right)~=~L\cap f(R)} + + + + + + f + + ( + + + f + + − + 1 + + + ( + L + ) + ∪ + R + + ) + + + = + + ( + L + ∩ + Im + ⁡ + f + ) + ∪ + f + ( + R + ) + + + {\displaystyle f\left(f^{-1}(L)\cup R\right)~=~(L\cap \operatorname {Im} f)\cup f(R)} + + +===== Conditions for f(L\R) = f(L)\f(R) ===== + + + + + f + ( + L + ∖ + R + ) + + ⊇ + + f + ( + L + ) + ∖ + f + ( + R + ) + + + + always holds + + + + {\displaystyle f(L\setminus R)~\supseteq ~f(L)\setminus f(R)\qquad \qquad {\text{ always holds}}} + + +Characterizations of equality: The following statements are equivalent: + + + + + f + ( + L + ∖ + R + ) + + = + + f + ( + L + ) + ∖ + f + ( + R + ) + + + {\displaystyle f(L\setminus R)~=~f(L)\setminus f(R)} + + + + + + f + ( + L + ∖ + R + ) + + ⊆ + + f + ( + L + ) + ∖ + f + ( + R + ) + + + {\displaystyle f(L\setminus R)~\subseteq ~f(L)\setminus f(R)} + + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + R + + + {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~R} + + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + = + + L + ∩ + R + ∩ + domain + ⁡ + f + + + {\displaystyle L\cap f^{-1}(f(R))~=~L\cap R\cap \operatorname {domain} f} + + +Whenever + + + + y + ∈ + f + ( + L + ) + ∩ + f + ( + R + ) + + + {\displaystyle y\in f(L)\cap f(R)} + + then + + + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + . + + + {\displaystyle L\cap f^{-1}(y)\subseteq R.} + + + + + + f + ( + L + ) + ∩ + f + ( + R + ) + + ⊆ + + + { + + y + ∈ + f + ( + L + ) + : + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + {\textstyle f(L)\cap f(R)~\subseteq ~\left\{y\in f(L):L\cap f^{-1}(y)\subseteq R\right\}} + + +The set on the right hand side is always equal to + + + + + { + + y + ∈ + f + ( + L + ∩ + R + ) + : + L + ∩ + + f + + − + 1 + + + ( + y + ) + + ⊆ + R + + } + + . + + + {\displaystyle \left\{y\in f(L\cap R):L\cap f^{-1}(y)\,\subseteq R\right\}.} + + + + + + f + ( + L + ) + ∩ + f + ( + R + ) + + = + + + { + + y + ∈ + f + ( + L + ) + : + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + {\textstyle f(L)\cap f(R)~=~\left\{y\in f(L):L\cap f^{-1}(y)\subseteq R\right\}} + + +This is the above condition (f) but with the subset symbol + + + + + ⊆ + + + + {\displaystyle \,\subseteq \,} + + replaced with an equals sign + + + + + = + . + + + + {\displaystyle \,=.\,} + + +Necessary conditions for equality (excluding characterizations): If equality holds then the following are necessarily true: + + + + + f + ( + L + ∩ + R + ) + = + f + ( + L + ) + ∩ + f + ( + R + ) + , + + + {\displaystyle f(L\cap R)=f(L)\cap f(R),} + + or equivalently + + + + f + ( + L + ∩ + R + ) + ⊇ + f + ( + L + ) + ∩ + f + ( + R + ) + . + + + {\displaystyle f(L\cap R)\supseteq f(L)\cap f(R).} + + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + = + + L + ∩ + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + + {\displaystyle L\cap f^{-1}(f(R))~=~L\cap f^{-1}(f(L\cap R))} + + or equivalently, + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + + {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~f^{-1}(f(L\cap R))} + + + + + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + = + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + + {\displaystyle R\cap f^{-1}(f(L))~=~R\cap f^{-1}(f(L\cap R))} + + +Sufficient conditions for equality: Equality holds if any of the following are true: + + + + + f + + + {\displaystyle f} + + is injective. +The restriction + + + + f + + + + | + + + + L + ∪ + R + + + + + {\displaystyle f{\big \vert }_{L\cup R}} + + is injective. + + + + + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + R + + + {\displaystyle f^{-1}(f(R))~\subseteq ~R} + + or equivalently, + + + + R + ∩ + domain + ⁡ + f + + = + + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + + {\displaystyle R\cap \operatorname {domain} f~=~f^{-1}(f(R))} + + + + + + R + + + {\displaystyle R} + + is + + + + f + + + {\displaystyle f} + +-saturated; that is, + + + + R + = + + f + + − + 1 + + + ( + f + ( + R + ) + ) + . + + + {\displaystyle R=f^{-1}(f(R)).} + + + + + + f + + ( + + L + + △ + + R + + ) + + ⊆ + f + ( + L + ) + + △ + + f + ( + R + ) + + + {\displaystyle f\left(L~\triangle ~R\right)\subseteq f(L)~\triangle ~f(R)} + + or equivalently, + + + + f + + ( + + L + + △ + + R + + ) + + = + f + ( + L + ) + + △ + + f + ( + R + ) + + + {\displaystyle f\left(L~\triangle ~R\right)=f(L)~\triangle ~f(R)} + + +===== Conditions for f(X\R) = f(X)\f(R) ===== + + + + + f + ( + X + ∖ + R + ) + + ⊇ + + f + ( + X + ) + ∖ + f + ( + R + ) + + + + always holds, where + + f + : + X + → + Y + + + {\displaystyle f(X\setminus R)~\supseteq ~f(X)\setminus f(R)\qquad \qquad {\text{ always holds, where }}f:X\to Y} + + +Characterizations of equality: The following statements are equivalent: + + + + + f + ( + X + ∖ + R + ) + + = + + f + ( + X + ) + ∖ + f + ( + R + ) + + + {\displaystyle f(X\setminus R)~=~f(X)\setminus f(R)} + + + + + + f + ( + X + ∖ + R + ) + + ⊆ + + f + ( + X + ) + ∖ + f + ( + R + ) + + + {\displaystyle f(X\setminus R)~\subseteq ~f(X)\setminus f(R)} + + + + + + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + R + + + {\displaystyle f^{-1}(f(R))\,\subseteq \,R} + + + + + + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + = + + R + ∩ + domain + ⁡ + f + + + {\displaystyle f^{-1}(f(R))\,=\,R\cap \operatorname {domain} f} + + + + + + R + ∩ + domain + ⁡ + f + + + {\displaystyle R\cap \operatorname {domain} f} + + is + + + + f + + + {\displaystyle f} + +-saturated. +Whenever + + + + y + ∈ + f + ( + R + ) + + + {\displaystyle y\in f(R)} + + then + + + + + f + + − + 1 + + + ( + y + ) + ⊆ + R + . + + + {\displaystyle f^{-1}(y)\subseteq R.} + + + + + + f + ( + R + ) + + ⊆ + + + { + + y + ∈ + f + ( + R + ) + : + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + {\textstyle f(R)~\subseteq ~\left\{y\in f(R):f^{-1}(y)\subseteq R\right\}} + + + + + + f + ( + R + ) + + = + + + { + + y + ∈ + f + ( + R + ) + : + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + {\textstyle f(R)~=~\left\{y\in f(R):f^{-1}(y)\subseteq R\right\}} + + + where if + + + + R + ⊆ + domain + ⁡ + f + + + {\displaystyle R\subseteq \operatorname {domain} f} + + then this list can be extended to include: + + + + + R + + + {\displaystyle R} + + is + + + + f + + + {\displaystyle f} + +-saturated; that is, + + + + R + = + + f + + − + 1 + + + ( + f + ( + R + ) + ) + . + + + {\displaystyle R=f^{-1}(f(R)).} + + +Sufficient conditions for equality: Equality holds if any of the following are true: + + + + + f + + + {\displaystyle f} + + is injective. + + + + + R + + + {\displaystyle R} + + is + + + + f + + + {\displaystyle f} + +-saturated; that is, + + + + R + = + + f + + − + 1 + + + ( + f + ( + R + ) + ) + . + + + {\displaystyle R=f^{-1}(f(R)).} + + +===== Conditions for f(L∆R) = f(L)∆f(R) ===== + + + + + f + + ( + + L + + △ + + R + + ) + + + ⊇ + + f + ( + L + ) + + △ + + f + ( + R + ) + + + + always holds + + + + {\displaystyle f\left(L~\triangle ~R\right)~\supseteq ~f(L)~\triangle ~f(R)\qquad \qquad {\text{ always holds}}} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-25.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-25.md new file mode 100644 index 000000000..0243ba33d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-25.md @@ -0,0 +1,1286 @@ +--- +title: "List of set identities and relations" +chunk: 26/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +Characterizations of equality: The following statements are equivalent: + + + + + f + + ( + + L + + △ + + R + + ) + + = + f + ( + L + ) + + △ + + f + ( + R + ) + + + {\displaystyle f\left(L~\triangle ~R\right)=f(L)~\triangle ~f(R)} + + + + + + f + + ( + + L + + △ + + R + + ) + + ⊆ + f + ( + L + ) + + △ + + f + ( + R + ) + + + {\displaystyle f\left(L~\triangle ~R\right)\subseteq f(L)~\triangle ~f(R)} + + + + + + f + ( + L + + ∖ + + R + ) + = + f + ( + L + ) + + ∖ + + f + ( + R + ) + + + {\displaystyle f(L\,\setminus \,R)=f(L)\,\setminus \,f(R)} + + and + + + + f + ( + R + + ∖ + + L + ) + = + f + ( + R + ) + + ∖ + + f + ( + L + ) + + + {\displaystyle f(R\,\setminus \,L)=f(R)\,\setminus \,f(L)} + + + + + + f + ( + L + + ∖ + + R + ) + ⊆ + f + ( + L + ) + + ∖ + + f + ( + R + ) + + + {\displaystyle f(L\,\setminus \,R)\subseteq f(L)\,\setminus \,f(R)} + + and + + + + f + ( + R + + ∖ + + L + ) + ⊆ + f + ( + R + ) + + ∖ + + f + ( + L + ) + + + {\displaystyle f(R\,\setminus \,L)\subseteq f(R)\,\setminus \,f(L)} + + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + R + + + {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~R} + + and + + + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + ⊆ + + L + + + {\displaystyle R\cap f^{-1}(f(L))~\subseteq ~L} + + +The inclusions + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + ⊆ + + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + + {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~f^{-1}(f(L))} + + and + + + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + ⊆ + + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + + {\displaystyle R\cap f^{-1}(f(L))~\subseteq ~f^{-1}(f(R))} + + always hold. + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + R + ) + ) + + = + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + + {\displaystyle L\cap f^{-1}(f(R))~=~R\cap f^{-1}(f(L))} + + +If this above set equality holds, then this set will also be equal to both + + + + L + ∩ + R + ∩ + domain + ⁡ + f + + + {\displaystyle L\cap R\cap \operatorname {domain} f} + + and + + + + L + ∩ + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + . + + + {\displaystyle L\cap R\cap f^{-1}(f(L\cap R)).} + + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + = + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + + {\displaystyle L\cap f^{-1}(f(L\cap R))~=~R\cap f^{-1}(f(L\cap R))} + + and + + + + f + ( + L + ∩ + R + ) + + ⊇ + + f + ( + L + ) + ∩ + f + ( + R + ) + . + + + {\displaystyle f(L\cap R)~\supseteq ~f(L)\cap f(R).} + + +Necessary conditions for equality (excluding characterizations): If equality holds then the following are necessarily true: + + + + + f + ( + L + ∩ + R + ) + = + f + ( + L + ) + ∩ + f + ( + R + ) + , + + + {\displaystyle f(L\cap R)=f(L)\cap f(R),} + + or equivalently + + + + f + ( + L + ∩ + R + ) + ⊇ + f + ( + L + ) + ∩ + f + ( + R + ) + . + + + {\displaystyle f(L\cap R)\supseteq f(L)\cap f(R).} + + + + + + L + ∩ + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + = + + R + ∩ + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + + {\displaystyle L\cap f^{-1}(f(L\cap R))~=~R\cap f^{-1}(f(L\cap R))} + + +Sufficient conditions for equality: Equality holds if any of the following are true: + + + + + f + + + {\displaystyle f} + + is injective. +The restriction + + + + f + + + + | + + + + L + ∪ + R + + + + + {\displaystyle f{\big \vert }_{L\cup R}} + + is injective. + +==== Exact formulas/equalities for images of set operations ==== + +===== Formulas for f(L\R) = ===== +For any function + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + and any sets + + + + L + + + {\displaystyle L} + + and + + + + R + , + + + {\displaystyle R,} + + + + + + + + + + f + ( + L + ∖ + R + ) + + + + = + Y + + + + + + + ∖ + + { + + y + ∈ + Y + + + + + + + + + + + + + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + + + + = + f + ( + L + ) + ∖ + + { + + y + ∈ + f + ( + L + ) + + + + + + + + + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + + + + = + f + ( + L + ) + ∖ + + { + + y + ∈ + f + ( + L + ∩ + R + ) + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + + + + = + f + ( + L + ) + ∖ + + { + + y + ∈ + V + + + + + + + + + + + + + + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + + + for any superset + + + V + ⊇ + f + ( + L + ∩ + R + ) + + + + + + + = + f + ( + S + ) + ∖ + + { + + y + ∈ + f + ( + S + ) + + + + + + + + + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + + + for any superset + + + S + ⊇ + L + ∩ + X + . + + + + + + + {\displaystyle {\begin{alignedat}{4}f(L\setminus R)&=Y~~~\;\,\,\setminus \left\{y\in Y~~~~~~~~~~\;\,~:~L\cap f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(L)\setminus \left\{y\in f(L)~~~~~~~\,~:~L\cap f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(L)\setminus \left\{y\in f(L\cap R)~:~L\cap f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(L)\setminus \left\{y\in V~~~~~~~~~~~~\,~:~L\cap f^{-1}(y)\subseteq R\right\}\qquad &&{\text{ for any superset }}\quad V\supseteq f(L\cap R)\\[0.4ex]&=f(S)\setminus \left\{y\in f(S)~~~~~~~\,~:~L\cap f^{-1}(y)\subseteq R\right\}\qquad &&{\text{ for any superset }}\quad S\supseteq L\cap X.\\[0.7ex]\end{alignedat}}} + + +===== Formulas for f(X\R) = ===== +Taking + + + + L + := + X + = + domain + ⁡ + f + + + {\displaystyle L:=X=\operatorname {domain} f} + + in the above formulas gives: + + + + + + + + + f + ( + X + ∖ + R + ) + + + + = + Y + + + + + + + ∖ + + { + + y + ∈ + Y + + + + + + + + : + + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + + + + = + f + ( + X + ) + ∖ + + { + + y + ∈ + f + ( + X + ) + + : + + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + + + + = + f + ( + X + ) + ∖ + + { + + y + ∈ + f + ( + R + ) + + : + + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + + + + = + f + ( + X + ) + ∖ + + { + + y + ∈ + W + + + + + + + : + + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + for any superset + + + W + ⊇ + f + ( + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}f(X\setminus R)&=Y~~~\;\,\,\setminus \left\{y\in Y~~~~\;\,\,:~f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(X)\setminus \left\{y\in f(X)~:~f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(X)\setminus \left\{y\in f(R)~:~f^{-1}(y)\subseteq R\right\}\\[0.4ex]&=f(X)\setminus \left\{y\in W~~~\;\,\,:~f^{-1}(y)\subseteq R\right\}\qquad {\text{ for any superset }}\quad W\supseteq f(R)\\[0.4ex]\end{alignedat}}} + + +where the set + + + + + { + + y + ∈ + f + ( + R + ) + : + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + {\displaystyle \left\{y\in f(R):f^{-1}(y)\subseteq R\right\}} + + is equal to the image under + + + + f + + + {\displaystyle f} + + of the largest + + + + f + + + {\displaystyle f} + +-saturated subset of + + + + R + . + + + {\displaystyle R.} + + +In general, only + + + + f + ( + X + ∖ + R + ) + + ⊇ + + f + ( + X + ) + ∖ + f + ( + R + ) + + + {\displaystyle f(X\setminus R)\,\supseteq \,f(X)\setminus f(R)} + + always holds and equality is not guaranteed; but replacing " + + + + f + ( + R + ) + + + {\displaystyle f(R)} + +" with its subset " + + + + + { + + y + ∈ + f + ( + R + ) + : + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + {\displaystyle \left\{y\in f(R):f^{-1}(y)\subseteq R\right\}} + +" results in a formula in which equality is always guaranteed: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-26.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-26.md new file mode 100644 index 000000000..77341d05b --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-26.md @@ -0,0 +1,1294 @@ +--- +title: "List of set identities and relations" +chunk: 27/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + f + ( + X + ∖ + R + ) + + = + + f + ( + X + ) + ∖ + + { + + y + ∈ + f + ( + R + ) + : + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + . + + + {\displaystyle f(X\setminus R)\,=\,f(X)\setminus \left\{y\in f(R):f^{-1}(y)\subseteq R\right\}.} + + +From this it follows that: + + + + + f + ( + X + ∖ + R + ) + = + f + ( + X + ) + ∖ + f + ( + R + ) + + + if and only if + + + f + ( + R + ) + = + + { + + y + ∈ + f + ( + R + ) + : + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + if and only if + + + + f + + − + 1 + + + ( + f + ( + R + ) + ) + ⊆ + R + . + + + {\displaystyle f(X\setminus R)=f(X)\setminus f(R)\quad {\text{ if and only if }}\quad f(R)=\left\{y\in f(R):f^{-1}(y)\subseteq R\right\}\quad {\text{ if and only if }}\quad f^{-1}(f(R))\subseteq R.} + + +If + + + + + f + + R + + + := + + { + + y + ∈ + f + ( + X + ) + : + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + {\displaystyle f_{R}:=\left\{y\in f(X):f^{-1}(y)\subseteq R\right\}} + + then + + + + f + ( + X + ∖ + R + ) + = + f + ( + X + ) + ∖ + + f + + R + + + , + + + {\displaystyle f(X\setminus R)=f(X)\setminus f_{R},} + + which can be written more symmetrically as + + + + f + ( + X + ∖ + R + ) + = + + f + + X + + + ∖ + + f + + R + + + + + {\displaystyle f(X\setminus R)=f_{X}\setminus f_{R}} + + (since + + + + + f + + X + + + = + f + ( + X + ) + + + {\displaystyle f_{X}=f(X)} + +). + +===== Formulas for f(L∆R) = ===== +It follows from + + + + L + + △ + + R + = + ( + L + ∪ + R + ) + ∖ + ( + L + ∩ + R + ) + + + {\displaystyle L\,\triangle \,R=(L\cup R)\setminus (L\cap R)} + + and the above formulas for the image of a set subtraction that for any function + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + and any sets + + + + L + + + {\displaystyle L} + + and + + + + R + , + + + {\displaystyle R,} + + + + + + + + + + f + ( + L + + △ + + R + ) + + + + = + Y + + + + + + + + + + + + + ∖ + + { + + y + ∈ + Y + + + + + + + + + + + + + + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + = + R + ∩ + + f + + − + 1 + + + ( + y + ) + + } + + + + + + + + = + f + ( + L + ∪ + R + ) + ∖ + + { + + y + ∈ + f + ( + L + ∪ + R + ) + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + = + R + ∩ + + f + + − + 1 + + + ( + y + ) + + } + + + + + + + + = + f + ( + L + ∪ + R + ) + ∖ + + { + + y + ∈ + f + ( + L + ∩ + R + ) + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + = + R + ∩ + + f + + − + 1 + + + ( + y + ) + + } + + + + + + + + = + f + ( + L + ∪ + R + ) + ∖ + + { + + y + ∈ + V + + + + + + + + + + + + + + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + = + R + ∩ + + f + + − + 1 + + + ( + y + ) + + } + + + + + + + for any superset + + + V + ⊇ + f + ( + L + ∩ + R + ) + + + + + + + = + f + ( + S + ) + + + + + + + + + + ∖ + + { + + y + ∈ + f + ( + S + ) + + + + + + + + + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + = + R + ∩ + + f + + − + 1 + + + ( + y + ) + + } + + + + + + + for any superset + + + S + ⊇ + ( + L + ∪ + R + ) + ∩ + X + . + + + + + + + {\displaystyle {\begin{alignedat}{4}f(L\,\triangle \,R)&=Y~~~\;~~~\;~~~\;\setminus \left\{y\in Y~~~\,~~~\;~~~\,~~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\\[0.4ex]&=f(L\cup R)\setminus \left\{y\in f(L\cup R)~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\\[0.4ex]&=f(L\cup R)\setminus \left\{y\in f(L\cap R)~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\\[0.4ex]&=f(L\cup R)\setminus \left\{y\in V~~~\,~~~~~~~~~~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\qquad &&{\text{ for any superset }}\quad V\supseteq f(L\cap R)\\[0.4ex]&=f(S)~~\,~~~\,~\,\setminus \left\{y\in f(S)~~~\,~~~\;~:~L\cap f^{-1}(y)=R\cap f^{-1}(y)\right\}\qquad &&{\text{ for any superset }}\quad S\supseteq (L\cup R)\cap X.\\[0.7ex]\end{alignedat}}} + + +===== Formulas for f(L) = ===== +It follows from the above formulas for the image of a set subtraction that for any function + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + and any set + + + + L + , + + + {\displaystyle L,} + + + + + + + + + + f + ( + L + ) + + + + = + Y + + + + + + ∖ + + { + + y + ∈ + Y + + + + + + + : + + + f + + − + 1 + + + ( + y + ) + ∩ + L + = + ∅ + + } + + + + + + + + = + Im + ⁡ + f + ∖ + + { + + y + ∈ + Im + ⁡ + f + + : + + + f + + − + 1 + + + ( + y + ) + ∩ + L + = + ∅ + + } + + + + + + + + = + W + + + + + ∖ + + { + + y + ∈ + W + + + + + + : + + + f + + − + 1 + + + ( + y + ) + ∩ + L + = + ∅ + + } + + + + for any superset + + + W + ⊇ + f + ( + L + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}f(L)&=Y~~~\;\,\setminus \left\{y\in Y~~~\;\,~:~f^{-1}(y)\cap L=\varnothing \right\}\\[0.4ex]&=\operatorname {Im} f\setminus \left\{y\in \operatorname {Im} f~:~f^{-1}(y)\cap L=\varnothing \right\}\\[0.4ex]&=W~~~\,\setminus \left\{y\in W~~\;\,~:~f^{-1}(y)\cap L=\varnothing \right\}\qquad {\text{ for any superset }}\quad W\supseteq f(L)\\[0.7ex]\end{alignedat}}} + + +This is more easily seen as being a consequence of the fact that for any + + + + y + ∈ + Y + , + + + {\displaystyle y\in Y,} + + + + + + + f + + − + 1 + + + ( + y + ) + ∩ + L + = + ∅ + + + {\displaystyle f^{-1}(y)\cap L=\varnothing } + + if and only if + + + + y + ∉ + f + ( + L + ) + . + + + {\displaystyle y\not \in f(L).} + + +===== Formulas for f(L⋂R) = ===== +It follows from the above formulas for the image of a set that for any function + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + and any sets + + + + L + + + {\displaystyle L} + + and + + + + R + , + + + {\displaystyle R,} + + + + + + + + + + f + ( + L + ∩ + R + ) + + + + = + Y + + + + + + ∖ + + { + + y + ∈ + Y + + + + + + + : + + L + ∩ + R + ∩ + + f + + − + 1 + + + ( + y + ) + = + ∅ + + } + + + + + + + + + + = + f + ( + L + ) + ∖ + + { + + y + ∈ + f + ( + L + ) + + : + + L + ∩ + R + ∩ + + f + + − + 1 + + + ( + y + ) + = + ∅ + + } + + + + + + + + + + = + f + ( + L + ) + ∖ + + { + + y + ∈ + U + + + + + + + : + + L + ∩ + R + ∩ + + f + + − + 1 + + + ( + y + ) + = + ∅ + + } + + + + + + + for any superset + + + U + ⊇ + f + ( + L + ) + + + + + + + = + f + ( + R + ) + ∖ + + { + + y + ∈ + f + ( + R + ) + + : + + L + ∩ + R + ∩ + + f + + − + 1 + + + ( + y + ) + = + ∅ + + } + + + + + + + + + + = + f + ( + R + ) + ∖ + + { + + y + ∈ + V + + + + + + + : + + L + ∩ + R + ∩ + + f + + − + 1 + + + ( + y + ) + = + ∅ + + } + + + + + + + for any superset + + + V + ⊇ + f + ( + R + ) + + + + + + + = + f + ( + L + ) + ∩ + f + ( + R + ) + ∖ + + { + + y + ∈ + f + ( + L + ) + ∩ + f + ( + R + ) + + : + + L + ∩ + R + ∩ + + f + + − + 1 + + + ( + y + ) + = + ∅ + + } + + + + + + + + + + {\displaystyle {\begin{alignedat}{4}f(L\cap R)&=Y~~~~~\setminus \left\{y\in Y~~~~~~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}&&\\[0.4ex]&=f(L)\setminus \left\{y\in f(L)~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}&&\\[0.4ex]&=f(L)\setminus \left\{y\in U~~~~~~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}\qquad &&{\text{ for any superset }}\quad U\supseteq f(L)\\[0.4ex]&=f(R)\setminus \left\{y\in f(R)~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}&&\\[0.4ex]&=f(R)\setminus \left\{y\in V~~~~~~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}\qquad &&{\text{ for any superset }}\quad V\supseteq f(R)\\[0.4ex]&=f(L)\cap f(R)\setminus \left\{y\in f(L)\cap f(R)~:~L\cap R\cap f^{-1}(y)=\varnothing \right\}&&\\[0.7ex]\end{alignedat}}} + + +where moreover, for any + + + + y + ∈ + Y + , + + + {\displaystyle y\in Y,} + + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-27.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-27.md new file mode 100644 index 000000000..62e872258 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-27.md @@ -0,0 +1,1576 @@ +--- +title: "List of set identities and relations" +chunk: 28/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + L + ∖ + R + + + + {\displaystyle L\cap f^{-1}(y)\subseteq L\setminus R~} + + if and only if + + + + + L + ∩ + R + ∩ + + f + + − + 1 + + + ( + y + ) + = + ∅ + + + + {\displaystyle ~L\cap R\cap f^{-1}(y)=\varnothing ~} + + if and only if + + + + + R + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + ∖ + L + + + + {\displaystyle ~R\cap f^{-1}(y)\subseteq R\setminus L~} + + if and only if + + + + + y + ∉ + f + ( + L + ∩ + R + ) + . + + + {\displaystyle ~y\not \in f(L\cap R).} + + +The sets + + + + U + + + {\displaystyle U} + + and + + + + V + + + {\displaystyle V} + + mentioned above could, in particular, be any of the sets + + + + f + ( + L + ∪ + R + ) + , + + Im + ⁡ + f + , + + + {\displaystyle f(L\cup R),\;\operatorname {Im} f,} + + or + + + + Y + , + + + {\displaystyle Y,} + + for example. + +=== (Pre)Images of set operations on (pre)images === +Let + + + + L + + + {\displaystyle L} + + and + + + + R + + + {\displaystyle R} + + be arbitrary sets, + + + + f + : + X + → + Y + + + {\displaystyle f:X\to Y} + + be any map, and let + + + + A + ⊆ + X + + + {\displaystyle A\subseteq X} + + and + + + + C + ⊆ + Y + . + + + {\displaystyle C\subseteq Y.} + + + +(Pre)Images of operations on images +Since + + + + f + ( + L + ) + ∖ + f + ( + L + ∖ + R + ) + + = + + + { + + y + ∈ + f + ( + L + ∩ + R + ) + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + , + + + {\displaystyle f(L)\setminus f(L\setminus R)~=~\left\{y\in f(L\cap R)~:~L\cap f^{-1}(y)\subseteq R\right\},} + + + + + + + + + + + f + + − + 1 + + + ( + f + ( + L + ) + ∖ + f + ( + L + ∖ + R + ) + ) + + + + = + + + + + f + + − + 1 + + + + ( + + { + + y + ∈ + f + ( + L + ∩ + R + ) + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + ) + + + + + + + + = + + + + + { + + x + ∈ + + f + + − + 1 + + + ( + f + ( + L + ∩ + R + ) + ) + + : + + L + ∩ + + f + + − + 1 + + + ( + f + ( + x + ) + ) + ⊆ + R + + } + + + + + + + + {\displaystyle {\begin{alignedat}{4}f^{-1}(f(L)\setminus f(L\setminus R))&=&&f^{-1}\left(\left\{y\in f(L\cap R)~:~L\cap f^{-1}(y)\subseteq R\right\}\right)\\&=&&\left\{x\in f^{-1}(f(L\cap R))~:~L\cap f^{-1}(f(x))\subseteq R\right\}\\\end{alignedat}}} + + +Since + + + + f + ( + X + ) + ∖ + f + ( + L + ∖ + R + ) + + = + + + { + + y + ∈ + f + ( + X + ) + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + , + + + {\displaystyle f(X)\setminus f(L\setminus R)~=~\left\{y\in f(X)~:~L\cap f^{-1}(y)\subseteq R\right\},} + + + + + + + + + + + f + + − + 1 + + + ( + Y + ∖ + f + ( + L + ∖ + R + ) + ) + + + + = + + + + + + f + + − + 1 + + + ( + f + ( + X + ) + ∖ + f + ( + L + ∖ + R + ) + ) + + + + + + + = + + + + + f + + − + 1 + + + + ( + + { + + y + ∈ + f + ( + X + ) + + : + + L + ∩ + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + ) + + + + + + + + = + + + + + { + + x + ∈ + X + + : + + L + ∩ + + f + + − + 1 + + + ( + f + ( + x + ) + ) + ⊆ + R + + } + + + + + + + + = + + + + + X + ∖ + + f + + − + 1 + + + ( + f + ( + L + ∖ + R + ) + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}f^{-1}(Y\setminus f(L\setminus R))&~=~&&f^{-1}(f(X)\setminus f(L\setminus R))\\&=&&f^{-1}\left(\left\{y\in f(X)~:~L\cap f^{-1}(y)\subseteq R\right\}\right)\\&=&&\left\{x\in X~:~L\cap f^{-1}(f(x))\subseteq R\right\}\\&~=~&&X\setminus f^{-1}(f(L\setminus R))\\\end{alignedat}}} + + +Using + + + + L + := + X + , + + + {\displaystyle L:=X,} + + this becomes + + + + + f + ( + X + ) + ∖ + f + ( + X + ∖ + R + ) + + = + + + { + + y + ∈ + f + ( + R + ) + + : + + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + + + + {\displaystyle ~f(X)\setminus f(X\setminus R)~=~\left\{y\in f(R)~:~f^{-1}(y)\subseteq R\right\}~} + + and + + + + + + + + + + f + + − + 1 + + + ( + Y + ∖ + f + ( + X + ∖ + R + ) + ) + + + + = + + + + + + f + + − + 1 + + + ( + f + ( + X + ) + ∖ + f + ( + X + ∖ + R + ) + ) + + + + + + + = + + + + + f + + − + 1 + + + + ( + + { + + y + ∈ + f + ( + R + ) + + : + + + f + + − + 1 + + + ( + y + ) + ⊆ + R + + } + + ) + + + + + + + + = + + + + + { + + r + ∈ + R + ∩ + X + + : + + + f + + − + 1 + + + ( + f + ( + r + ) + ) + ⊆ + R + + } + + + + + + + + ⊆ + + + + R + + + + + + + {\displaystyle {\begin{alignedat}{4}f^{-1}(Y\setminus f(X\setminus R))&~=~&&f^{-1}(f(X)\setminus f(X\setminus R))\\&=&&f^{-1}\left(\left\{y\in f(R)~:~f^{-1}(y)\subseteq R\right\}\right)\\&=&&\left\{r\in R\cap X~:~f^{-1}(f(r))\subseteq R\right\}\\&\subseteq &&R\\\end{alignedat}}} + + +and so + + + + + + + + + + f + + − + 1 + + + ( + Y + ∖ + f + ( + L + ) + ) + + + + = + + + + + + f + + − + 1 + + + ( + f + ( + X + ) + ∖ + f + ( + L + ) + ) + + + + + + + = + + + + + f + + − + 1 + + + + ( + + { + + y + ∈ + f + ( + X + ∖ + L + ) + + : + + + f + + − + 1 + + + ( + y + ) + ∩ + L + = + ∅ + + } + + ) + + + + + + + + = + + + + + { + x + ∈ + X + ∖ + L + + : + + f + ( + x + ) + ∉ + f + ( + L + ) + } + + + + + + + = + + + + X + ∖ + + f + + − + 1 + + + ( + f + ( + L + ) + ) + + + + + + + ⊆ + + + + X + ∖ + L + + + + + + + {\displaystyle {\begin{alignedat}{4}f^{-1}(Y\setminus f(L))&~=~&&f^{-1}(f(X)\setminus f(L))\\&=&&f^{-1}\left(\left\{y\in f(X\setminus L)~:~f^{-1}(y)\cap L=\varnothing \right\}\right)\\&=&&\{x\in X\setminus L~:~f(x)\not \in f(L)\}\\&=&&X\setminus f^{-1}(f(L))\\&\subseteq &&X\setminus L\\\end{alignedat}}} + + +=== (Pre)Images and Cartesian products Π === +Let + + + + ∏ + + Y + + ∙ + + + + + + + + = + + + + + def + + + + + + + + + ∏ + + j + ∈ + J + + + + Y + + j + + + + + {\displaystyle \prod Y_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\prod _{j\in J}Y_{j}} + + and for every + + + + k + ∈ + J + , + + + {\displaystyle k\in J,} + + let + + + + + + π + + k + + + + : + + + ∏ + + j + ∈ + J + + + + Y + + j + + + + → + + + Y + + k + + + + + {\displaystyle \pi _{k}~:~\prod _{j\in J}Y_{j}~\to ~Y_{k}} + + +denote the canonical projection onto + + + + + Y + + k + + + . + + + {\displaystyle Y_{k}.} + + +Definitions +Given a collection of maps + + + + + F + + j + + + : + X + → + + Y + + j + + + + + {\displaystyle F_{j}:X\to Y_{j}} + + indexed by + + + + j + ∈ + J + , + + + {\displaystyle j\in J,} + + define the map + + + + + + + + + + + ( + + F + + j + + + ) + + + j + ∈ + J + + + : + + + + + X + + + + + → + + + + + + ∏ + + j + ∈ + J + + + + Y + + j + + + + + + + + + x + + + + + ↦ + + + + + + ( + + + F + + j + + + + ( + + x + + j + + + ) + + + ) + + + j + ∈ + J + + + , + + + + + + + {\displaystyle {\begin{alignedat}{4}\left(F_{j}\right)_{j\in J}:\;&&X&&\;\to \;&\prod _{j\in J}Y_{j}\\[0.3ex]&&x&&\;\mapsto \;&\left(F_{j}\left(x_{j}\right)\right)_{j\in J},\\\end{alignedat}}} + + +which is also denoted by + + + + + F + + ∙ + + + = + + + ( + + F + + j + + + ) + + + j + ∈ + J + + + . + + + {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}.} + + This is the unique map satisfying + + + + + + π + + j + + + ∘ + + F + + ∙ + + + = + + F + + j + + + + + for all + + j + ∈ + J + . + + + {\displaystyle \pi _{j}\circ F_{\bullet }=F_{j}\quad {\text{ for all }}j\in J.} + + +Conversely, if given a map + + + + F + + : + + X + + → + + + ∏ + + j + ∈ + J + + + + Y + + j + + + + + {\displaystyle F~:~X~\to ~\prod _{j\in J}Y_{j}} + + then + + + + F + = + + + ( + + + π + + j + + + ∘ + F + + ) + + + j + ∈ + J + + + . + + + {\displaystyle F=\left(\pi _{j}\circ F\right)_{j\in J}.} + + +Explicitly, what this means is that if + + + + + + F + + k + + + + + + + + = + + + + + def + + + + + + + + + π + + k + + + ∘ + F + + : + + X + + → + + + Y + + k + + + + + {\displaystyle F_{k}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\pi _{k}\circ F~:~X~\to ~Y_{k}} + + +is defined for every + + + + k + ∈ + J + , + + + {\displaystyle k\in J,} + + then + + + + F + + + {\displaystyle F} + + the unique map satisfying: + + + + + π + + j + + + ∘ + F + = + + F + + j + + + + + {\displaystyle \pi _{j}\circ F=F_{j}} + + for all + + + + j + ∈ + J + ; + + + {\displaystyle j\in J;} + + or said more briefly, + + + + F + = + + + ( + + F + + j + + + ) + + + j + ∈ + J + + + . + + + {\displaystyle F=\left(F_{j}\right)_{j\in J}.} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-28.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-28.md new file mode 100644 index 000000000..b169a5431 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-28.md @@ -0,0 +1,991 @@ +--- +title: "List of set identities and relations" +chunk: 29/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +The map + + + + + F + + ∙ + + + = + + + ( + + F + + j + + + ) + + + j + ∈ + J + + + + : + + X + + → + + + ∏ + + j + ∈ + J + + + + Y + + j + + + + + {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}} + + should not be confused with the Cartesian product + + + + + ∏ + + j + ∈ + J + + + + F + + j + + + + + {\displaystyle \prod _{j\in J}F_{j}} + + of these maps, which is by definition is the map + + + + + + + + + + ∏ + + j + ∈ + J + + + + F + + j + + + : + + + + + + ∏ + + j + ∈ + J + + + X + + + + + + → + + + + + + + ∏ + + j + ∈ + J + + + + Y + + j + + + + + + + + + + + ( + + x + + j + + + ) + + + j + ∈ + J + + + + + + + + ↦ + + + + + + + ( + + + F + + j + + + + ( + + x + + j + + + ) + + + ) + + + j + ∈ + J + + + + + + + + + {\displaystyle {\begin{alignedat}{4}\prod _{j\in J}F_{j}:\;&&\prod _{j\in J}X&&~\;\to \;~&\prod _{j\in J}Y_{j}\\[0.3ex]&&\left(x_{j}\right)_{j\in J}&&~\;\mapsto \;~&\left(F_{j}\left(x_{j}\right)\right)_{j\in J}\\\end{alignedat}}} + + +with domain + + + + + ∏ + + j + ∈ + J + + + X + = + + X + + J + + + + + {\displaystyle \prod _{j\in J}X=X^{J}} + + rather than + + + + X + . + + + {\displaystyle X.} + + +Preimage and images of a Cartesian product +Suppose + + + + + F + + ∙ + + + = + + + ( + + F + + j + + + ) + + + j + ∈ + J + + + + : + + X + + → + + + ∏ + + j + ∈ + J + + + + Y + + j + + + . + + + {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}.} + + +If + + + + A + + ⊆ + + X + + + {\displaystyle A~\subseteq ~X} + + then + + + + + + F + + ∙ + + + ( + A + ) + + + + + + ⊆ + + + + + + + + + + ∏ + + j + ∈ + J + + + + F + + j + + + ( + A + ) + . + + + + + {\displaystyle F_{\bullet }(A)~~\;\color {Red}{\subseteq }\color {Black}{}\;~~\prod _{j\in J}F_{j}(A).} + + +If + + + + B + + ⊆ + + + ∏ + + j + ∈ + J + + + + Y + + j + + + + + {\displaystyle B~\subseteq ~\prod _{j\in J}Y_{j}} + + then + + + + + + F + + ∙ + + + − + 1 + + + ( + B + ) + + + + + + ⊆ + + + + + + + + + + ⋂ + + j + ∈ + J + + + + F + + j + + + − + 1 + + + + ( + + + π + + j + + + ( + B + ) + + ) + + + + + + {\displaystyle F_{\bullet }^{-1}(B)~~\;\color {Red}{\subseteq }\color {Black}{}\;~~\bigcap _{j\in J}F_{j}^{-1}\left(\pi _{j}(B)\right)} + + +where equality will hold if + + + + B + = + + ∏ + + j + ∈ + J + + + + π + + j + + + ( + B + ) + , + + + {\displaystyle B=\prod _{j\in J}\pi _{j}(B),} + + in which case + + + + + F + + ∙ + + + − + 1 + + + ( + B + ) + = + + + ⋂ + + j + ∈ + J + + + + F + + j + + + − + 1 + + + + ( + + + π + + j + + + ( + B + ) + + ) + + + + + {\textstyle F_{\bullet }^{-1}(B)=\displaystyle \bigcap _{j\in J}F_{j}^{-1}\left(\pi _{j}(B)\right)} + + and + +For equality to hold, it suffices for there to exist a family + + + + + + ( + + B + + j + + + ) + + + j + ∈ + J + + + + + {\displaystyle \left(B_{j}\right)_{j\in J}} + + of subsets + + + + + B + + j + + + ⊆ + + Y + + j + + + + + {\displaystyle B_{j}\subseteq Y_{j}} + + such that + + + + B + = + + ∏ + + j + ∈ + J + + + + B + + j + + + , + + + {\displaystyle B=\prod _{j\in J}B_{j},} + + in which case: + +and + + + + + π + + j + + + ( + B + ) + = + + B + + j + + + + + {\displaystyle \pi _{j}(B)=B_{j}} + + for all + + + + j + ∈ + J + . + + + {\displaystyle j\in J.} + + +=== (Pre)Image of a single set === + +==== Containments ⊆ and intersections ⋂ of images and preimages ==== +Equivalences and implications of images and preimages + +Intersection of a set and a (pre)image +The following statements are equivalent: + + + + + ∅ + = + f + ( + L + ) + ∩ + R + + + {\displaystyle \varnothing =f(L)\cap R} + + + + + + ∅ + = + L + ∩ + + f + + − + 1 + + + ( + R + ) + + + {\displaystyle \varnothing =L\cap f^{-1}(R)} + + + + + + ∅ + = + + f + + − + 1 + + + ( + f + ( + L + ) + ) + ∩ + + f + + − + 1 + + + ( + R + ) + + + {\displaystyle \varnothing =f^{-1}(f(L))\cap f^{-1}(R)} + + + + + + ∅ + = + + f + + − + 1 + + + ( + f + ( + L + ) + ∩ + R + ) + + + {\displaystyle \varnothing =f^{-1}(f(L)\cap R)} + + +Thus for any + + + + t + , + + + {\displaystyle t,} + + + + + + t + ∉ + f + ( + L + ) + + + if and only if + + + L + ∩ + + f + + − + 1 + + + ( + t + ) + = + ∅ + . + + + {\displaystyle t\not \in f(L)\quad {\text{ if and only if }}\quad L\cap f^{-1}(t)=\varnothing .} + + +== Sequences and collections of families of sets == + +=== Definitions === +A family of sets or simply a family is a set whose elements are sets. +A family over + + + + X + + + {\displaystyle X} + + is a family of subsets of + + + + X + . + + + {\displaystyle X.} + + +The power set of a set + + + + X + + + {\displaystyle X} + + is the set of all subsets of + + + + X + + + {\displaystyle X} + +: + + + + + + + P + + + ( + X + ) + + : + = + + { + + S + + : + + S + ⊆ + X + + } + . + + + {\displaystyle {\mathcal {P}}(X)~\colon =~\{\;S~:~S\subseteq X\;\}.} + + +Notation for sequences of sets +Throughout, + + + + S + + and + + T + + + {\displaystyle S{\text{ and }}T} + + will be arbitrary sets and + + + + + S + + ∙ + + + + + {\displaystyle S_{\bullet }} + + and will denote a net or a sequence of sets where if it is a sequence then this will be indicated by either of the notations + + + + + + S + + ∙ + + + = + + + ( + + S + + i + + + ) + + + i + = + 1 + + + ∞ + + + + + or + + + + S + + ∙ + + + = + + + ( + + S + + i + + + ) + + + i + ∈ + + N + + + + + + {\displaystyle S_{\bullet }=\left(S_{i}\right)_{i=1}^{\infty }\qquad {\text{ or }}\qquad S_{\bullet }=\left(S_{i}\right)_{i\in \mathbb {N} }} + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-29.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-29.md new file mode 100644 index 000000000..3a06b878c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-29.md @@ -0,0 +1,1301 @@ +--- +title: "List of set identities and relations" +chunk: 30/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +where + + + + + N + + + + {\displaystyle \mathbb {N} } + + denotes the natural numbers. +A notation + + + + + S + + ∙ + + + = + + + ( + + S + + i + + + ) + + + i + ∈ + I + + + + + {\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}} + + indicates that + + + + + S + + ∙ + + + + + {\displaystyle S_{\bullet }} + + is a net directed by + + + + ( + I + , + ≤ + ) + , + + + {\displaystyle (I,\leq ),} + + which (by definition) is a sequence if the set + + + + I + , + + + {\displaystyle I,} + + which is called the net's indexing set, is the natural numbers (that is, if + + + + I + = + + N + + + + {\displaystyle I=\mathbb {N} } + +) and + + + + + ≤ + + + + {\displaystyle \,\leq \,} + + is the natural order on + + + + + N + + . + + + {\displaystyle \mathbb {N} .} + + +Disjoint and monotone sequences of sets +If + + + + + S + + i + + + ∩ + + S + + j + + + = + ∅ + + + {\displaystyle S_{i}\cap S_{j}=\varnothing } + + for all distinct indices + + + + i + ≠ + j + + + {\displaystyle i\neq j} + + then + + + + + S + + ∙ + + + + + {\displaystyle S_{\bullet }} + + is called a pairwise disjoint or simply a disjoint. +A sequence or net + + + + + S + + ∙ + + + + + {\displaystyle S_{\bullet }} + + of set is called increasing or non-decreasing if (resp. decreasing or non-increasing) if for all indices + + + + i + ≤ + j + , + + + {\displaystyle i\leq j,} + + + + + + + S + + i + + + ⊆ + + S + + j + + + + + {\displaystyle S_{i}\subseteq S_{j}} + + (resp. + + + + + S + + i + + + ⊇ + + S + + j + + + + + {\displaystyle S_{i}\supseteq S_{j}} + +). +A sequence or net + + + + + S + + ∙ + + + + + {\displaystyle S_{\bullet }} + + of set is called strictly increasing (resp. strictly decreasing) if it is non-decreasing (resp. is non-increasing) and also + + + + + S + + i + + + ≠ + + S + + j + + + + + {\displaystyle S_{i}\neq S_{j}} + + for all distinct indices + + + + i + + and + + j + . + + + {\displaystyle i{\text{ and }}j.} + + +It is called monotone if it is non-decreasing or non-increasing and it is called strictly monotone if it is strictly increasing or strictly decreasing. +A sequences or net + + + + + S + + ∙ + + + + + {\displaystyle S_{\bullet }} + + is said to increase to + + + + S + , + + + {\displaystyle S,} + + denoted by + + + + + S + + ∙ + + + ↑ + S + + + {\displaystyle S_{\bullet }\uparrow S} + + or + + + + + S + + ∙ + + + ↗ + S + , + + + {\displaystyle S_{\bullet }\nearrow S,} + + if + + + + + S + + ∙ + + + + + {\displaystyle S_{\bullet }} + + is increasing and the union of all + + + + + S + + i + + + + + {\displaystyle S_{i}} + + is + + + + S + ; + + + {\displaystyle S;} + + that is, if + + + + + ⋃ + + n + + + + S + + n + + + = + S + + + and + + + + S + + i + + + ⊆ + + S + + j + + + + + whenever + + i + ≤ + j + . + + + {\displaystyle \bigcup _{n}S_{n}=S\qquad {\text{ and }}\qquad S_{i}\subseteq S_{j}\quad {\text{ whenever }}i\leq j.} + + +It is said to decrease to + + + + S + , + + + {\displaystyle S,} + + denoted by + + + + + S + + ∙ + + + ↓ + S + + + {\displaystyle S_{\bullet }\downarrow S} + + or + + + + + S + + ∙ + + + ↘ + S + , + + + {\displaystyle S_{\bullet }\searrow S,} + + if + + + + + S + + ∙ + + + + + {\displaystyle S_{\bullet }} + + is increasing and the intersection of all + + + + + S + + i + + + + + {\displaystyle S_{i}} + + is + + + + S + + + {\displaystyle S} + + that is, if + + + + + ⋂ + + n + + + + S + + n + + + = + S + + + and + + + + S + + i + + + ⊇ + + S + + j + + + + + whenever + + i + ≤ + j + . + + + {\displaystyle \bigcap _{n}S_{n}=S\qquad {\text{ and }}\qquad S_{i}\supseteq S_{j}\quad {\text{ whenever }}i\leq j.} + + +Definitions of elementwise operations on families +If + + + + + + L + + + + and + + + + R + + + + + {\displaystyle {\mathcal {L}}{\text{ and }}{\mathcal {R}}} + + are families of sets and if + + + + S + + + {\displaystyle S} + + is any set then define: + + + + + + + L + + + + ( + ∪ + ) + + + + R + + + + : + = + + { + + L + ∪ + R + + : + + L + ∈ + + + L + + + + + and + + + R + ∈ + + + R + + + + } + + + {\displaystyle {\mathcal {L}}\;(\cup )\;{\mathcal {R}}~\colon =~\{~L\cup R~:~L\in {\mathcal {L}}~{\text{ and }}~R\in {\mathcal {R}}~\}} + + + + + + + + L + + + + ( + ∩ + ) + + + + R + + + + : + = + + { + + L + ∩ + R + + : + + L + ∈ + + + L + + + + + and + + + R + ∈ + + + R + + + + } + + + {\displaystyle {\mathcal {L}}\;(\cap )\;{\mathcal {R}}~\colon =~\{~L\cap R~:~L\in {\mathcal {L}}~{\text{ and }}~R\in {\mathcal {R}}~\}} + + + + + + + + L + + + + ( + ∖ + ) + + + + R + + + + : + = + + { + + L + ∖ + R + + : + + L + ∈ + + + L + + + + + and + + + R + ∈ + + + R + + + + } + + + {\displaystyle {\mathcal {L}}\;(\setminus )\;{\mathcal {R}}~\colon =~\{~L\setminus R~:~L\in {\mathcal {L}}~{\text{ and }}~R\in {\mathcal {R}}~\}} + + + + + + + + L + + + + ( + △ + ) + + + + R + + + + : + = + + { + + L + + △ + + R + + : + + L + ∈ + + + L + + + + + and + + + R + ∈ + + + R + + + + } + + + {\displaystyle {\mathcal {L}}\;(\triangle )\;{\mathcal {R}}~\colon =~\{~L\;\triangle \;R~:~L\in {\mathcal {L}}~{\text{ and }}~R\in {\mathcal {R}}~\}} + + + + + + + + L + + + + + + | + + + + S + + + + : + = + + { + L + ∩ + S + + : + + L + ∈ + + + L + + + } + = + + + L + + + + ( + ∩ + ) + + { + S + } + + + {\displaystyle {\mathcal {L}}{\big \vert }_{S}~\colon =~\{L\cap S~:~L\in {\mathcal {L}}\}={\mathcal {L}}\;(\cap )\;\{S\}} + + +which are respectively called elementwise union, elementwise intersection, elementwise (set) difference, elementwise symmetric difference, and the trace/restriction of + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + to + + + + S + . + + + {\displaystyle S.} + + The regular union, intersection, and set difference are all defined as usual and are denoted with their usual notation: + + + + + + L + + + ∪ + + + R + + + , + + + L + + + ∩ + + + R + + + , + + + L + + + + △ + + + + R + + + , + + + {\displaystyle {\mathcal {L}}\cup {\mathcal {R}},{\mathcal {L}}\cap {\mathcal {R}},{\mathcal {L}}\;\triangle \;{\mathcal {R}},} + + and + + + + + + L + + + ∖ + + + R + + + , + + + {\displaystyle {\mathcal {L}}\setminus {\mathcal {R}},} + + respectively. +These elementwise operations on families of sets play an important role in, among other subjects, the theory of filters and prefilters on sets. +The upward closure in + + + + X + + + {\displaystyle X} + + of a family + + + + + + L + + + ⊆ + + + P + + + ( + X + ) + + + {\displaystyle {\mathcal {L}}\subseteq {\mathcal {P}}(X)} + + is the family: + + + + + + + + L + + + + ↑ + X + + + + : + = + + + ⋃ + + L + ∈ + + + L + + + + + { + + S + + : + + L + ⊆ + S + ⊆ + X + + } + + = + + { + + S + ⊆ + X + + : + + + there exists + + L + ∈ + + + L + + + + such that + + L + ⊆ + S + + } + + + {\displaystyle {\mathcal {L}}^{\uparrow X}~\colon =~\bigcup _{L\in {\mathcal {L}}}\{\;S~:~L\subseteq S\subseteq X\;\}~=~\{\;S\subseteq X~:~{\text{ there exists }}L\in {\mathcal {L}}{\text{ such that }}L\subseteq S\;\}} + + +and the downward closure of + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + is the family: + + + + + + + + L + + + + ↓ + + + + : + = + + + ⋃ + + L + ∈ + + + L + + + + + + + P + + + ( + L + ) + + = + + { + + S + + : + + + there exists + + L + ∈ + + + L + + + + such that + + S + ⊆ + L + + } + . + + + {\displaystyle {\mathcal {L}}^{\downarrow }~\colon =~\bigcup _{L\in {\mathcal {L}}}{\mathcal {P}}(L)~=~\{\;S~:~{\text{ there exists }}L\in {\mathcal {L}}{\text{ such that }}S\subseteq L\;\}.} + + + +==== Definitions of categories of families of sets ==== +The following table lists some well-known categories of families of sets having applications in general topology and measure theory. + +A family + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + is called isotone, ascending, or upward closed in + + + + X + + + {\displaystyle X} + + if + + + + + + L + + + ⊆ + + + P + + + ( + X + ) + + + {\displaystyle {\mathcal {L}}\subseteq {\mathcal {P}}(X)} + + and + + + + + + L + + + = + + + + L + + + + ↑ + X + + + . + + + {\displaystyle {\mathcal {L}}={\mathcal {L}}^{\uparrow X}.} + + +A family + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + is called downward closed if + + + + + + L + + + = + + + + L + + + + ↓ + + + . + + + {\displaystyle {\mathcal {L}}={\mathcal {L}}^{\downarrow }.} + + +A family + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + is said to be: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-3.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-3.md new file mode 100644 index 000000000..0953ae563 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-3.md @@ -0,0 +1,1021 @@ +--- +title: "List of set identities and relations" +chunk: 4/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +===== Empty set ===== +A set + + + + L + + + {\displaystyle L} + + is empty if the sentence + + + + ∀ + x + ( + x + ∉ + L + ) + + + {\displaystyle \forall x(x\not \in L)} + + is true, where the notation + + + + x + ∉ + L + + + {\displaystyle x\not \in L} + + is shorthand for + + + + ¬ + ( + x + ∈ + L + ) + . + + + {\displaystyle \lnot (x\in L).} + + +If + + + + L + + + {\displaystyle L} + + is any set then the following are equivalent: + + + + + L + + + {\displaystyle L} + + is not empty, meaning that the sentence + + + + ¬ + [ + ∀ + x + ( + x + ∉ + L + ) + ] + + + {\displaystyle \lnot [\forall x(x\not \in L)]} + + is true (literally, the logical negation of " + + + + L + + + {\displaystyle L} + + is empty" holds true). +(In classical mathematics) + + + + L + + + {\displaystyle L} + + is inhabited, meaning: + + + + ∃ + x + ( + x + ∈ + L + ) + + + {\displaystyle \exists x(x\in L)} + + +In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set + + + + L + + + {\displaystyle L} + + that is not empty (where by definition, " + + + + L + + + {\displaystyle L} + + is empty" means that the statement + + + + ∀ + x + ( + x + ∉ + L + ) + + + {\displaystyle \forall x(x\not \in L)} + + is true) might not have an inhabitant (which is an + + + + x + + + {\displaystyle x} + + such that + + + + x + ∈ + L + + + {\displaystyle x\in L} + +). + + + + + L + ⊈ + R + + + {\displaystyle L\not \subseteq R} + + for some set + + + + R + + + {\displaystyle R} + + +If + + + + L + + + {\displaystyle L} + + is any set then the following are equivalent: + + + + + L + + + {\displaystyle L} + + is empty ( + + + + L + = + ∅ + + + {\displaystyle L=\varnothing } + +), meaning: + + + + ∀ + x + ( + x + ∉ + L + ) + + + {\displaystyle \forall x(x\not \in L)} + + + + + + L + ∪ + R + ⊆ + R + + + {\displaystyle L\cup R\subseteq R} + + for every set + + + + R + + + {\displaystyle R} + + + + + + L + ⊆ + R + + + {\displaystyle L\subseteq R} + + for every set + + + + R + + + {\displaystyle R} + + + + + + L + ⊆ + R + ∖ + L + + + {\displaystyle L\subseteq R\setminus L} + + for some/every set + + + + R + + + {\displaystyle R} + + + + + + ∅ + + / + + L + = + L + + + {\displaystyle \varnothing /L=L} + + +Given any + + + + x + , + + + {\displaystyle x,} + + the following are equivalent: + + + + + x + ∉ + L + ∖ + R + + + {\textstyle x\not \in L\setminus R} + + + + + + x + ∈ + L + ∩ + R + + + or + + + x + ∉ + L + . + + + {\textstyle x\in L\cap R\;{\text{ or }}\;x\not \in L.} + + + + + + x + ∈ + R + + + or + + + x + ∉ + L + . + + + {\textstyle x\in R\;{\text{ or }}\;x\not \in L.} + + +Moreover, + + + + + ( + L + ∖ + R + ) + ∩ + R + = + ∅ + + + always holds + + . + + + {\displaystyle (L\setminus R)\cap R=\varnothing \qquad {\text{ always holds}}.} + + +==== Meets, Joins, and lattice properties ==== +Inclusion is a partial order: +Explicitly, this means that inclusion + + + + + ⊆ + , + + + + {\displaystyle \,\subseteq ,\,} + + which is a binary operation, has the following three properties: + +Reflexivity: + + + + + L + ⊆ + L + + + {\textstyle L\subseteq L} + + +Antisymmetry: + + + + + ( + L + ⊆ + R + + and + + R + ⊆ + L + ) + + if and only if + + L + = + R + + + {\textstyle (L\subseteq R{\text{ and }}R\subseteq L){\text{ if and only if }}L=R} + + +Transitivity: + + + + + + If + + L + ⊆ + M + + and + + M + ⊆ + R + + then + + L + ⊆ + R + + + {\textstyle {\text{If }}L\subseteq M{\text{ and }}M\subseteq R{\text{ then }}L\subseteq R} + + +The following proposition says that for any set + + + + S + , + + + {\displaystyle S,} + + the power set of + + + + S + , + + + {\displaystyle S,} + + ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra. +Existence of a least element and a greatest element: + + + + + ∅ + ⊆ + L + ⊆ + X + + + {\displaystyle \varnothing \subseteq L\subseteq X} + + +Joins/supremums exist: + + + + + L + ⊆ + L + ∪ + R + + + {\displaystyle L\subseteq L\cup R} + + +The union + + + + L + ∪ + R + + + {\displaystyle L\cup R} + + is the join/supremum of + + + + L + + + {\displaystyle L} + + and + + + + R + + + {\displaystyle R} + + with respect to + + + + + ⊆ + + + + {\displaystyle \,\subseteq \,} + + because: + + + + + L + ⊆ + L + ∪ + R + + + {\displaystyle L\subseteq L\cup R} + + and + + + + R + ⊆ + L + ∪ + R + , + + + {\displaystyle R\subseteq L\cup R,} + + and +if + + + + Z + + + {\displaystyle Z} + + is a set such that + + + + L + ⊆ + Z + + + {\displaystyle L\subseteq Z} + + and + + + + R + ⊆ + Z + + + {\displaystyle R\subseteq Z} + + then + + + + L + ∪ + R + ⊆ + Z + . + + + {\displaystyle L\cup R\subseteq Z.} + + +The intersection + + + + L + ∩ + R + + + {\displaystyle L\cap R} + + is the join/supremum of + + + + L + + + {\displaystyle L} + + and + + + + R + + + {\displaystyle R} + + with respect to + + + + + ⊇ + . + + + + {\displaystyle \,\supseteq .\,} + + +Meets/infimums exist: + + + + + L + ∩ + R + ⊆ + L + + + {\displaystyle L\cap R\subseteq L} + + +The intersection + + + + L + ∩ + R + + + {\displaystyle L\cap R} + + is the meet/infimum of + + + + L + + + {\displaystyle L} + + and + + + + R + + + {\displaystyle R} + + with respect to + + + + + ⊆ + + + + {\displaystyle \,\subseteq \,} + + because: + +if + + + + L + ∩ + R + ⊆ + L + + + {\displaystyle L\cap R\subseteq L} + + and + + + + L + ∩ + R + ⊆ + R + , + + + {\displaystyle L\cap R\subseteq R,} + + and +if + + + + Z + + + {\displaystyle Z} + + is a set such that + + + + Z + ⊆ + L + + + {\displaystyle Z\subseteq L} + + and + + + + Z + ⊆ + R + + + {\displaystyle Z\subseteq R} + + then + + + + Z + ⊆ + L + ∩ + R + . + + + {\displaystyle Z\subseteq L\cap R.} + + +The union + + + + L + ∪ + R + + + {\displaystyle L\cup R} + + is the meet/infimum of + + + + L + + + {\displaystyle L} + + and + + + + R + + + {\displaystyle R} + + with respect to + + + + + ⊇ + . + + + + {\displaystyle \,\supseteq .\,} + + +Other inclusion properties: + + + + + L + ∖ + R + ⊆ + L + + + {\displaystyle L\setminus R\subseteq L} + + + + + + ( + L + ∖ + R + ) + ∩ + L + = + L + ∖ + R + + + {\displaystyle (L\setminus R)\cap L=L\setminus R} + + +If + + + + L + ⊆ + R + + + {\displaystyle L\subseteq R} + + then + + + + L + + △ + + R + = + R + ∖ + L + . + + + {\displaystyle L\,\triangle \,R=R\setminus L.} + + +If + + + + L + ⊆ + X + + + {\displaystyle L\subseteq X} + + and + + + + R + ⊆ + Y + + + {\displaystyle R\subseteq Y} + + then + + + + L + × + R + ⊆ + X + × + Y + + + {\displaystyle L\times R\subseteq X\times Y} + + +== Three sets involved == +In the left hand sides of the following identities, + + + + L + + + {\displaystyle L} + + is the L eft most set, + + + + M + + + {\displaystyle M} + + is the M iddle set, and + + + + R + + + {\displaystyle R} + + is the R ight most set. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-30.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-30.md new file mode 100644 index 000000000..336b35a4f --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-30.md @@ -0,0 +1,1259 @@ +--- +title: "List of set identities and relations" +chunk: 31/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +closed under finite intersections (resp. closed under finite unions) if whenever + + + + L + , + R + ∈ + + + L + + + + + {\displaystyle L,R\in {\mathcal {L}}} + + then + + + + L + ∩ + R + ∈ + + + L + + + + + {\displaystyle L\cap R\in {\mathcal {L}}} + + (respectively, + + + + L + ∪ + R + ∈ + + + L + + + + + {\displaystyle L\cup R\in {\mathcal {L}}} + +). +closed under countable intersections (resp. closed under countable unions) if whenever + + + + + L + + 1 + + + , + + L + + 2 + + + , + + L + + 3 + + + , + … + + + {\displaystyle L_{1},L_{2},L_{3},\ldots } + + are elements of + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + then so is their intersections + + + + + ⋂ + + i + = + 1 + + + ∞ + + + + L + + i + + + := + + L + + 1 + + + ∩ + + L + + 2 + + + ∩ + + L + + 3 + + + ∩ + ⋯ + + + {\displaystyle \bigcap _{i=1}^{\infty }L_{i}:=L_{1}\cap L_{2}\cap L_{3}\cap \cdots } + + (resp. so is their union + + + + + ⋃ + + i + = + 1 + + + ∞ + + + + L + + i + + + := + + L + + 1 + + + ∪ + + L + + 2 + + + ∪ + + L + + 3 + + + ∪ + ⋯ + + + {\displaystyle \bigcup _{i=1}^{\infty }L_{i}:=L_{1}\cup L_{2}\cup L_{3}\cup \cdots } + +). +closed under complementation in (or with respect to) + + + + X + + + {\displaystyle X} + + if whenever + + + + L + ∈ + + + L + + + + + {\displaystyle L\in {\mathcal {L}}} + + then + + + + X + ∖ + L + ∈ + + + L + + + . + + + {\displaystyle X\setminus L\in {\mathcal {L}}.} + + +A family + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + of sets is called a/an: + +π−system if + + + + + + L + + + ≠ + ∅ + + + {\displaystyle {\mathcal {L}}\neq \varnothing } + + and + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + is closed under finite-intersections. +Every non-empty family + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + is contained in a unique smallest (with respect to + + + + ⊆ + + + {\displaystyle \subseteq } + +) π−system that is denoted by + + + + π + ( + + + L + + + ) + + + {\displaystyle \pi ({\mathcal {L}})} + + and called the π−system generated by + + + + + + L + + + . + + + {\displaystyle {\mathcal {L}}.} + + +filter subbase and is said to have the finite intersection property if + + + + + + L + + + ≠ + ∅ + + + {\displaystyle {\mathcal {L}}\neq \varnothing } + + and + + + + ∅ + ∉ + π + ( + + + L + + + ) + . + + + {\displaystyle \varnothing \not \in \pi ({\mathcal {L}}).} + + +filter on + + + + X + + + {\displaystyle X} + + if + + + + + + L + + + ≠ + ∅ + + + {\displaystyle {\mathcal {L}}\neq \varnothing } + + is a family of subsets of + + + + X + + + {\displaystyle X} + + that is a π−system, is upward closed in + + + + X + , + + + {\displaystyle X,} + + and is also proper, which by definition means that it does not contain the empty set as an element. +prefilter or filter base if it is a non-empty family of subsets of some set + + + + X + + + {\displaystyle X} + + whose upward closure in + + + + X + + + {\displaystyle X} + + is a filter on + + + + X + . + + + {\displaystyle X.} + + +algebra on + + + + X + + + {\displaystyle X} + + is a non-empty family of subsets of + + + + X + + + {\displaystyle X} + + that contains the empty set, forms a π−system, and is also closed under complementation with respect to + + + + X + . + + + {\displaystyle X.} + + +σ-algebra on + + + + X + + + {\displaystyle X} + + is an algebra on + + + + X + + + {\displaystyle X} + + that is closed under countable unions (or equivalently, closed under countable intersections). +Sequences of sets often arise in measure theory. +Algebra of sets + +A family + + + + Φ + + + {\displaystyle \Phi } + + of subsets of a set + + + + X + + + {\displaystyle X} + + is said to be an algebra of sets if + + + + ∅ + ∈ + Φ + + + {\displaystyle \varnothing \in \Phi } + + and for all + + + + L + , + R + ∈ + Φ + , + + + {\displaystyle L,R\in \Phi ,} + + all three of the sets + + + + X + ∖ + R + , + + L + ∩ + R + , + + + {\displaystyle X\setminus R,\,L\cap R,} + + and + + + + L + ∪ + R + + + {\displaystyle L\cup R} + + are elements of + + + + Φ + . + + + {\displaystyle \Phi .} + + +The article on this topic lists set identities and other relationships these three operations. +Every algebra of sets is also a ring of sets and a π-system. +Algebra generated by a family of sets +Given any family + + + + + + S + + + + + {\displaystyle {\mathcal {S}}} + + of subsets of + + + + X + , + + + {\displaystyle X,} + + there is a unique smallest algebra of sets in + + + + X + + + {\displaystyle X} + + containing + + + + + + S + + + . + + + {\displaystyle {\mathcal {S}}.} + + +It is called the algebra generated by + + + + + + S + + + + + {\displaystyle {\mathcal {S}}} + + and it will be denote it by + + + + + Φ + + + S + + + + . + + + {\displaystyle \Phi _{\mathcal {S}}.} + + +This algebra can be constructed as follows: + +If + + + + + + S + + + = + ∅ + + + {\displaystyle {\mathcal {S}}=\varnothing } + + then + + + + + Φ + + + S + + + + = + { + ∅ + , + X + } + + + {\displaystyle \Phi _{\mathcal {S}}=\{\varnothing ,X\}} + + and we are done. Alternatively, if + + + + + + S + + + + + {\displaystyle {\mathcal {S}}} + + is empty then + + + + + + S + + + + + {\displaystyle {\mathcal {S}}} + + may be replaced with + + + + { + ∅ + } + , + { + X + } + , + + or + + { + ∅ + , + X + } + + + {\displaystyle \{\varnothing \},\{X\},{\text{ or }}\{\varnothing ,X\}} + + and continue with the construction. +Let + + + + + + + S + + + + 0 + + + + + {\displaystyle {\mathcal {S}}_{0}} + + be the family of all sets in + + + + + + S + + + + + {\displaystyle {\mathcal {S}}} + + together with their complements (taken in + + + + X + + + {\displaystyle X} + +). +Let + + + + + + + S + + + + 1 + + + + + {\displaystyle {\mathcal {S}}_{1}} + + be the family of all possible finite intersections of sets in + + + + + + + S + + + + 0 + + + . + + + {\displaystyle {\mathcal {S}}_{0}.} + + +Then the algebra generated by + + + + + + S + + + + + {\displaystyle {\mathcal {S}}} + + is the set + + + + + Φ + + + S + + + + + + {\displaystyle \Phi _{\mathcal {S}}} + + consisting of all possible finite unions of sets in + + + + + + + S + + + + 1 + + + . + + + {\displaystyle {\mathcal {S}}_{1}.} + + +==== Elementwise operations on families ==== +Let + + + + + + L + + + , + + + M + + + , + + + {\displaystyle {\mathcal {L}},{\mathcal {M}},} + + and + + + + + + R + + + + + {\displaystyle {\mathcal {R}}} + + be families of sets over + + + + X + . + + + {\displaystyle X.} + + +On the left hand sides of the following identities, + + + + + + L + + + + + {\displaystyle {\mathcal {L}}} + + is the L eft most family, + + + + + + M + + + + + {\displaystyle {\mathcal {M}}} + + is in the M iddle, and + + + + + + R + + + + + {\displaystyle {\mathcal {R}}} + + is the R ight most set. +Commutativity: + + + + + + + L + + + + ( + ∪ + ) + + + + R + + + = + + + R + + + + ( + ∪ + ) + + + + L + + + + + {\displaystyle {\mathcal {L}}\;(\cup )\;{\mathcal {R}}={\mathcal {R}}\;(\cup )\;{\mathcal {L}}} + + + + + + + + L + + + + ( + ∩ + ) + + + + R + + + = + + + R + + + + ( + ∩ + ) + + + + L + + + + + {\displaystyle {\mathcal {L}}\;(\cap )\;{\mathcal {R}}={\mathcal {R}}\;(\cap )\;{\mathcal {L}}} + + +Associativity: + + + + + [ + + + L + + + + ( + ∪ + ) + + + + M + + + ] + + ( + ∪ + ) + + + + R + + + = + + + L + + + + ( + ∪ + ) + + [ + + + M + + + + ( + ∪ + ) + + + + R + + + ] + + + {\displaystyle [{\mathcal {L}}\;(\cup )\;{\mathcal {M}}]\;(\cup )\;{\mathcal {R}}={\mathcal {L}}\;(\cup )\;[{\mathcal {M}}\;(\cup )\;{\mathcal {R}}]} + + + + + + [ + + + L + + + + ( + ∩ + ) + + + + M + + + ] + + ( + ∩ + ) + + + + R + + + = + + + L + + + + ( + ∩ + ) + + [ + + + M + + + + ( + ∩ + ) + + + + R + + + ] + + + {\displaystyle [{\mathcal {L}}\;(\cap )\;{\mathcal {M}}]\;(\cap )\;{\mathcal {R}}={\mathcal {L}}\;(\cap )\;[{\mathcal {M}}\;(\cap )\;{\mathcal {R}}]} + + +Identity: + + + + + + + L + + + + ( + ∪ + ) + + { + ∅ + } + = + + + L + + + + + {\displaystyle {\mathcal {L}}\;(\cup )\;\{\varnothing \}={\mathcal {L}}} + + + + + + + + L + + + + ( + ∩ + ) + + { + X + } + = + + + L + + + + + {\displaystyle {\mathcal {L}}\;(\cap )\;\{X\}={\mathcal {L}}} + + + + + + + + L + + + + ( + ∖ + ) + + { + ∅ + } + = + + + L + + + + + {\displaystyle {\mathcal {L}}\;(\setminus )\;\{\varnothing \}={\mathcal {L}}} + + +Domination: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-31.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-31.md new file mode 100644 index 000000000..227c05862 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-31.md @@ -0,0 +1,1005 @@ +--- +title: "List of set identities and relations" +chunk: 32/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + L + + + + ( + ∪ + ) + + { + X + } + = + { + X + } + + + + + + if + + + + L + + + ≠ + ∅ + + + {\displaystyle {\mathcal {L}}\;(\cup )\;\{X\}=\{X\}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing } + + + + + + + + L + + + + ( + ∩ + ) + + { + ∅ + } + = + { + ∅ + } + + + + + + if + + + + L + + + ≠ + ∅ + + + {\displaystyle {\mathcal {L}}\;(\cap )\;\{\varnothing \}=\{\varnothing \}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing } + + + + + + + + L + + + + ( + ∪ + ) + + ∅ + = + ∅ + + + {\displaystyle {\mathcal {L}}\;(\cup )\;\varnothing =\varnothing } + + + + + + + + L + + + + ( + ∩ + ) + + ∅ + = + ∅ + + + {\displaystyle {\mathcal {L}}\;(\cap )\;\varnothing =\varnothing } + + + + + + + + L + + + + ( + ∖ + ) + + ∅ + = + ∅ + + + {\displaystyle {\mathcal {L}}\;(\setminus )\;\varnothing =\varnothing } + + + + + + ∅ + + ( + ∖ + ) + + + + R + + + = + ∅ + + + {\displaystyle \varnothing \;(\setminus )\;{\mathcal {R}}=\varnothing } + + +=== Power set === + + + + + + + P + + + ( + L + ∩ + R + ) + + = + + + + P + + + ( + L + ) + ∩ + + + P + + + ( + R + ) + + + {\displaystyle {\mathcal {P}}(L\cap R)~=~{\mathcal {P}}(L)\cap {\mathcal {P}}(R)} + + + + + + + + P + + + ( + L + ∪ + R + ) + + = + + + + P + + + ( + L + ) + + ( + ∪ + ) + + + + P + + + ( + R + ) + + ⊇ + + + + P + + + ( + L + ) + ∪ + + + P + + + ( + R + ) + . + + + {\displaystyle {\mathcal {P}}(L\cup R)~=~{\mathcal {P}}(L)\ (\cup )\ {\mathcal {P}}(R)~\supseteq ~{\mathcal {P}}(L)\cup {\mathcal {P}}(R).} + + +If + + + + L + + + {\displaystyle L} + + and + + + + R + + + {\displaystyle R} + + are subsets of a vector space + + + + X + + + {\displaystyle X} + + and if + + + + s + + + {\displaystyle s} + + is a scalar then + + + + + + + P + + + ( + s + L + ) + + = + + s + + + P + + + ( + L + ) + + + {\displaystyle {\mathcal {P}}(sL)~=~s{\mathcal {P}}(L)} + + + + + + + + P + + + ( + L + + + R + ) + + ⊇ + + + + P + + + ( + L + ) + + + + + P + + + ( + R + ) + . + + + {\displaystyle {\mathcal {P}}(L+R)~\supseteq ~{\mathcal {P}}(L)+{\mathcal {P}}(R).} + + +=== Sequences of sets === +Suppose that + + + + L + + + {\displaystyle L} + + is any set such that + + + + L + ⊇ + + R + + i + + + + + {\displaystyle L\supseteq R_{i}} + + for every index + + + + i + . + + + {\displaystyle i.} + + +If + + + + + R + + ∙ + + + + + {\displaystyle R_{\bullet }} + + decreases to + + + + R + + + {\displaystyle R} + + then + + + + L + ∖ + + R + + ∙ + + + := + + + ( + + L + ∖ + + R + + i + + + + ) + + + i + + + + + {\displaystyle L\setminus R_{\bullet }:=\left(L\setminus R_{i}\right)_{i}} + + increases to + + + + L + ∖ + R + + + {\displaystyle L\setminus R} + + +whereas if instead + + + + + R + + ∙ + + + + + {\displaystyle R_{\bullet }} + + increases to + + + + R + + + {\displaystyle R} + + then + + + + L + ∖ + + R + + ∙ + + + + + {\displaystyle L\setminus R_{\bullet }} + + decreases to + + + + L + ∖ + R + . + + + {\displaystyle L\setminus R.} + + +If + + + + L + + and + + R + + + {\displaystyle L{\text{ and }}R} + + are arbitrary sets and if + + + + + L + + ∙ + + + = + + + ( + + L + + i + + + ) + + + i + + + + + {\displaystyle L_{\bullet }=\left(L_{i}\right)_{i}} + + increases (resp. decreases) to + + + + L + + + {\displaystyle L} + + then + + + + + + ( + + + L + + i + + + ∖ + R + + ) + + + i + + + + + {\displaystyle \left(L_{i}\setminus R\right)_{i}} + + increase (resp. decreases) to + + + + L + ∖ + R + . + + + {\displaystyle L\setminus R.} + + + +==== Partitions ==== +Suppose that + + + + + S + + ∙ + + + = + + + ( + + S + + i + + + ) + + + i + = + 1 + + + ∞ + + + + + {\displaystyle S_{\bullet }=\left(S_{i}\right)_{i=1}^{\infty }} + + is any sequence of sets, that + + + + S + ⊆ + + ⋃ + + i + + + + S + + i + + + + + {\displaystyle S\subseteq \bigcup _{i}S_{i}} + + is any subset, and for every index + + + + i + , + + + {\displaystyle i,} + + let + + + + + D + + i + + + = + + ( + + + S + + i + + + ∩ + S + + ) + + ∖ + + ⋃ + + m + = + 1 + + + i + + + + ( + + + S + + m + + + ∩ + S + + ) + + . + + + {\displaystyle D_{i}=\left(S_{i}\cap S\right)\setminus \bigcup _{m=1}^{i}\left(S_{m}\cap S\right).} + + +Then + + + + S + = + + ⋃ + + i + + + + D + + i + + + + + {\displaystyle S=\bigcup _{i}D_{i}} + + and + + + + + D + + ∙ + + + := + + + ( + + D + + i + + + ) + + + i + = + 1 + + + ∞ + + + + + {\displaystyle D_{\bullet }:=\left(D_{i}\right)_{i=1}^{\infty }} + + is a sequence of pairwise disjoint sets. +Suppose that + + + + + S + + ∙ + + + = + + + ( + + S + + i + + + ) + + + i + = + 1 + + + ∞ + + + + + {\displaystyle S_{\bullet }=\left(S_{i}\right)_{i=1}^{\infty }} + + is non-decreasing, let + + + + + S + + 0 + + + = + ∅ + , + + + {\displaystyle S_{0}=\varnothing ,} + + and let + + + + + D + + i + + + = + + S + + i + + + ∖ + + S + + i + − + 1 + + + + + {\displaystyle D_{i}=S_{i}\setminus S_{i-1}} + + for every + + + + i + = + 1 + , + 2 + , + … + . + + + {\displaystyle i=1,2,\ldots .} + + Then + + + + + ⋃ + + i + + + + S + + i + + + = + + ⋃ + + i + + + + D + + i + + + + + {\displaystyle \bigcup _{i}S_{i}=\bigcup _{i}D_{i}} + + and + + + + + D + + ∙ + + + = + + + ( + + D + + i + + + ) + + + i + = + 1 + + + ∞ + + + + + {\displaystyle D_{\bullet }=\left(D_{i}\right)_{i=1}^{\infty }} + + is a sequence of pairwise disjoint sets. + +== See also == +Algebra of sets – Identities and relationships involving sets +Complement (set theory) – Set of the elements not in a given subset +Image (mathematics)#Properties – Set of the values of a function +Inclusion–exclusion principle – Counting technique in combinatorics +Intersection (set theory) – Set of elements common to all of some sets +List of mathematical identities +Naive set theory – Informal set theories +Pigeonhole principle – If there are more items than boxes holding them, one box must contain at least two items +Set (mathematics) – Collection of mathematical objects +Simple theorems in the algebra of sets – Basic set identities like commutative and associative laws +Symmetric difference (set theory) – Elements in exactly one of two setsPages displaying short descriptions of redirect targets +Union (set theory) – Set of elements in any of some sets + +== Notes == +Notes + +Proofs + +== Citations == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-32.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-32.md new file mode 100644 index 000000000..ffa0bd0d7 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-32.md @@ -0,0 +1,37 @@ +--- +title: "List of set identities and relations" +chunk: 33/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +== References == +Artin, Michael (1991). Algebra. Prentice Hall. ISBN 81-203-0871-9. +Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures. Springer. ISBN 1-85233-905-5.. +Bylinski, Czeslaw (2004). "Some Basic Properties of Sets". Journal of Formalized Mathematics. 1. Retrieved 5 October 2021. +Courant, Richard, Herbert Robbins, Ian Stewart, What is mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. "SUPPLEMENT TO CHAPTER II THE ALGEBRA OF SETS". +Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011. +Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303. +Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. +Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. +Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020. +Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. {{cite book}}: ISBN / Date incompatibility (help) +Joshi, K. D. (1983). Introduction to General Topology. New York: John Wiley and Sons Ltd. ISBN 978-0-85226-444-7. OCLC 9218750. +Kelley, John L. (1985). General Topology. Graduate Texts in Mathematics. Vol. 27 (2 ed.). Birkhäuser. ISBN 978-0-387-90125-1. +Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. +Monk, James Donald (1969). Introduction to Set Theory (PDF). International series in pure and applied mathematics. New York: McGraw-Hill. ISBN 978-0-07-042715-0. OCLC 1102. +Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. (accessible to patrons with print disabilities) +Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. +Padlewska, Beata (1990). "Families of Sets". Journal of Formalized Mathematics. 1: 1. Retrieved 5 October 2021. +Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. +Schubert, Horst (1968). Topology. London: Macdonald & Co. ISBN 978-0-356-02077-8. OCLC 463753. +Stoll, Robert R.; Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4. "The Algebra of Sets", pp 16—23. +Trybulec, Zinaida (2002). "Properties of subsets" (PDF). Journal of Formalized Mathematics. 1: 1. Retrieved 5 October 2021. +Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. +Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240. + +== External links == +Operations on Sets at ProvenMath \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-4.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-4.md new file mode 100644 index 000000000..03a44a42f --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-4.md @@ -0,0 +1,925 @@ +--- +title: "List of set identities and relations" +chunk: 5/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + +=== Precedence rules === +There is no universal agreement on the order of precedence of the basic set operators. +Nevertheless, many authors use precedence rules for set operators, although these rules vary with the author. +One common convention is to associate intersection + + + + L + ∩ + R + = + { + x + : + ( + x + ∈ + L + ) + ∧ + ( + x + ∈ + R + ) + } + + + {\displaystyle L\cap R=\{x:(x\in L)\land (x\in R)\}} + + with logical conjunction (and) + + + + L + ∧ + R + + + {\displaystyle L\land R} + + and associate union + + + + L + ∪ + R + = + { + x + : + ( + x + ∈ + L + ) + ∨ + ( + x + ∈ + R + ) + } + + + {\displaystyle L\cup R=\{x:(x\in L)\lor (x\in R)\}} + + with logical disjunction (or) + + + + L + ∨ + R + , + + + {\displaystyle L\lor R,} + + and then transfer the precedence of these logical operators (where + + + + + ∧ + + + + {\displaystyle \,\land \,} + + has precedence over + + + + + ∨ + + + + {\displaystyle \,\lor \,} + +) to these set operators, thereby giving + + + + + ∩ + + + + {\displaystyle \,\cap \,} + + precedence over + + + + + ∪ + . + + + + {\displaystyle \,\cup .\,} + + +So for example, + + + + L + ∪ + M + ∩ + R + + + {\displaystyle L\cup M\cap R} + + would mean + + + + L + ∪ + ( + M + ∩ + R + ) + + + {\displaystyle L\cup (M\cap R)} + + since it would be associated with the logical statement + + + + L + ∨ + M + ∧ + R + + = + + L + ∨ + ( + M + ∧ + R + ) + + + {\displaystyle L\lor M\land R~=~L\lor (M\land R)} + + and similarly, + + + + L + ∪ + M + ∩ + R + ∪ + Z + + + {\displaystyle L\cup M\cap R\cup Z} + + would mean + + + + L + ∪ + ( + M + ∩ + R + ) + ∪ + Z + + + {\displaystyle L\cup (M\cap R)\cup Z} + + since it would be associated with + + + + L + ∨ + M + ∧ + R + ∨ + Z + + = + + L + ∨ + ( + M + ∧ + R + ) + ∨ + Z + . + + + {\displaystyle L\lor M\land R\lor Z~=~L\lor (M\land R)\lor Z.} + + +Sometimes, set complement (subtraction) + + + + + ∖ + + + + {\displaystyle \,\setminus \,} + + is also associated with logical complement (not) + + + + + ¬ + , + + + + {\displaystyle \,\lnot ,\,} + + in which case it will have the highest precedence. +More specifically, + + + + L + ∖ + R + = + { + x + : + ( + x + ∈ + L + ) + ∧ + ¬ + ( + x + ∈ + R + ) + } + + + {\displaystyle L\setminus R=\{x:(x\in L)\land \lnot (x\in R)\}} + + is rewritten + + + + L + ∧ + ¬ + R + + + {\displaystyle L\land \lnot R} + + so that for example, + + + + L + ∪ + M + ∖ + R + + + {\displaystyle L\cup M\setminus R} + + would mean + + + + L + ∪ + ( + M + ∖ + R + ) + + + {\displaystyle L\cup (M\setminus R)} + + since it would be rewritten as the logical statement + + + + L + ∨ + M + ∧ + ¬ + R + + + {\displaystyle L\lor M\land \lnot R} + + which is equal to + + + + L + ∨ + ( + M + ∧ + ¬ + R + ) + . + + + {\displaystyle L\lor (M\land \lnot R).} + + +For another example, because + + + + L + ∧ + ¬ + M + ∧ + R + + + {\displaystyle L\land \lnot M\land R} + + means + + + + L + ∧ + ( + ¬ + M + ) + ∧ + R + , + + + {\displaystyle L\land (\lnot M)\land R,} + + which is equal to both + + + + ( + L + ∧ + ( + ¬ + M + ) + ) + ∧ + R + + + {\displaystyle (L\land (\lnot M))\land R} + + and + + + + L + ∧ + ( + ( + ¬ + M + ) + ∧ + R + ) + + = + + L + ∧ + ( + R + ∧ + ( + ¬ + M + ) + ) + + + {\displaystyle L\land ((\lnot M)\land R)~=~L\land (R\land (\lnot M))} + + (where + + + + ( + ¬ + M + ) + ∧ + R + + + {\displaystyle (\lnot M)\land R} + + was rewritten as + + + + R + ∧ + ( + ¬ + M + ) + + + {\displaystyle R\land (\lnot M)} + +), the formula + + + + L + ∖ + M + ∩ + R + + + {\displaystyle L\setminus M\cap R} + + would refer to the set + + + + ( + L + ∖ + M + ) + ∩ + R + = + L + ∩ + ( + R + ∖ + M + ) + ; + + + {\displaystyle (L\setminus M)\cap R=L\cap (R\setminus M);} + + +moreover, since + + + + L + ∧ + ( + ¬ + M + ) + ∧ + R + = + ( + L + ∧ + R + ) + ∧ + ¬ + M + , + + + {\displaystyle L\land (\lnot M)\land R=(L\land R)\land \lnot M,} + + this set is also equal to + + + + ( + L + ∩ + R + ) + ∖ + M + + + {\displaystyle (L\cap R)\setminus M} + + (other set identities can similarly be deduced from propositional calculus identities in this way). +However, because set subtraction is not associative + + + + ( + L + ∖ + M + ) + ∖ + R + ≠ + L + ∖ + ( + M + ∖ + R + ) + , + + + {\displaystyle (L\setminus M)\setminus R\neq L\setminus (M\setminus R),} + + a formula such as + + + + L + ∖ + M + ∖ + R + + + {\displaystyle L\setminus M\setminus R} + + would be ambiguous; for this reason, among others, set subtraction is often not assigned any precedence at all. +Symmetric difference + + + + L + △ + R + = + { + x + : + ( + x + ∈ + L + ) + ⊕ + ( + x + ∈ + R + ) + } + + + {\displaystyle L\triangle R=\{x:(x\in L)\oplus (x\in R)\}} + + is sometimes associated with exclusive or (xor) + + + + L + ⊕ + R + + + {\displaystyle L\oplus R} + + (also sometimes denoted by + + + + + ⊻ + + + {\displaystyle \,\veebar } + +), in which case if the order of precedence from highest to lowest is + + + + + ¬ + , + + ⊕ + , + + ∧ + , + + ∨ + + + + {\displaystyle \,\lnot ,\,\oplus ,\,\land ,\,\lor \,} + + then the order of precedence (from highest to lowest) for the set operators would be + + + + + ∖ + , + + △ + , + + ∩ + , + + ∪ + . + + + {\displaystyle \,\setminus ,\,\triangle ,\,\cap ,\,\cup .} + + +There is no universal agreement on the precedence of exclusive disjunction + + + + + ⊕ + + + + {\displaystyle \,\oplus \,} + + with respect to the other logical connectives, which is why symmetric difference + + + + + △ + + + + {\displaystyle \,\triangle \,} + + is not often assigned a precedence. + +=== Associativity === +Definition: A binary operator + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + is called associative if + + + + ( + L + + ∗ + + M + ) + + ∗ + + R + = + L + + ∗ + + ( + M + + ∗ + + R + ) + + + {\displaystyle (L\,\ast \,M)\,\ast \,R=L\,\ast \,(M\,\ast \,R)} + + always holds. +The following set operators are associative: + + + + + + + + + ( + L + ∪ + M + ) + ∪ + R + + + + + = + + + + + + L + ∪ + ( + M + ∪ + R + ) + + + + + ( + L + ∩ + M + ) + ∩ + R + + + + + = + + + + + + L + ∩ + ( + M + ∩ + R + ) + + + + + ( + L + + △ + M + ) + + △ + R + + + + + = + + + + + + L + + △ + ( + M + + △ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{5}(L\cup M)\cup R&\;=\;\;&&L\cup (M\cup R)\\[1.4ex](L\cap M)\cap R&\;=\;\;&&L\cap (M\cap R)\\[1.4ex](L\,\triangle M)\,\triangle R&\;=\;\;&&L\,\triangle (M\,\triangle R)\\[1.4ex]\end{alignedat}}} + + +For set subtraction, instead of associativity, only the following is always guaranteed: + + + + + ( + L + + ∖ + + M + ) + + ∖ + + R + + + + + + + ⊆ + + + + + + + L + + ∖ + + ( + M + + ∖ + + R + ) + + + {\displaystyle (L\,\setminus \,M)\,\setminus \,R\;~~{\color {red}{\subseteq }}~~\;L\,\setminus \,(M\,\setminus \,R)} + + +where equality holds if and only if + + + + L + ∩ + R + = + ∅ + + + {\displaystyle L\cap R=\varnothing } + + (this condition does not depend on + + + + M + + + {\displaystyle M} + +). Thus \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-5.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-5.md new file mode 100644 index 000000000..f0707fd06 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-5.md @@ -0,0 +1,1249 @@ +--- +title: "List of set identities and relations" +chunk: 6/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + ( + L + ∖ + M + ) + ∖ + R + = + L + ∖ + ( + M + ∖ + R + ) + + + + {\textstyle \;(L\setminus M)\setminus R=L\setminus (M\setminus R)\;} + + if and only if + + + + + ( + R + ∖ + M + ) + ∖ + L + = + R + ∖ + ( + M + ∖ + L + ) + , + + + + {\displaystyle \;(R\setminus M)\setminus L=R\setminus (M\setminus L),\;} + + +where the only difference between the left and right hand side set equalities is that the locations of + + + + L + + and + + R + + + {\displaystyle L{\text{ and }}R} + + have been swapped. + +=== Distributivity === +Definition: If + + + + ∗ + + and + + ∙ + + + {\displaystyle \ast {\text{ and }}\bullet } + + are binary operators then + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + left distributes over + + + + + ∙ + + + + {\displaystyle \,\bullet \,} + + if + + + + L + + ∗ + + ( + M + + ∙ + + R + ) + + = + + ( + L + + ∗ + + M + ) + + ∙ + + ( + L + + ∗ + + R + ) + + + + for all + + L + , + M + , + R + + + {\displaystyle L\,\ast \,(M\,\bullet \,R)~=~(L\,\ast \,M)\,\bullet \,(L\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R} + + +while + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + right distributes over + + + + + ∙ + + + + {\displaystyle \,\bullet \,} + + if + + + + ( + L + + ∙ + + M + ) + + ∗ + + R + + = + + ( + L + + ∗ + + R + ) + + ∙ + + ( + M + + ∗ + + R + ) + + + + for all + + L + , + M + , + R + . + + + {\displaystyle (L\,\bullet \,M)\,\ast \,R~=~(L\,\ast \,R)\,\bullet \,(M\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R.} + + +The operator + + + + + ∗ + + + + {\displaystyle \,\ast \,} + + distributes over + + + + + ∙ + + + + {\displaystyle \,\bullet \,} + + if it both left distributes and right distributes over + + + + + ∙ + + . + + + + {\displaystyle \,\bullet \,.\,} + + +In the definitions above, to transform one side to the other, the innermost operator (the operator inside the parentheses) becomes the outermost operator and the outermost operator becomes the innermost operator. +Right distributivity: + + + + + + + + + ( + L + + ∩ + + M + ) + + ∪ + + R + + + + + + = + + + + + + + ( + L + + ∪ + + R + ) + + + + + + ∩ + + + + + + ( + M + + ∪ + + R + ) + + + + + + (Right-distributivity of + + + ∪ + + + over + + + ∩ + + + ) + + + + + + ( + L + + ∪ + + M + ) + + ∪ + + R + + + + + + = + + + + + + + ( + L + + ∪ + + R + ) + + + + + + ∪ + + + + + + ( + M + + ∪ + + R + ) + + + + + + (Right-distributivity of + + + ∪ + + + over + + + ∪ + + + ) + + + + + + ( + L + + ∪ + + M + ) + + ∩ + + R + + + + + + = + + + + + + + ( + L + + ∩ + + R + ) + + + + + + ∪ + + + + + + ( + M + + ∩ + + R + ) + + + + + + (Right-distributivity of + + + ∩ + + + over + + + ∪ + + + ) + + + + + + ( + L + + ∩ + + M + ) + + ∩ + + R + + + + + + = + + + + + + + ( + L + + ∩ + + R + ) + + + + + + ∩ + + + + + + ( + M + + ∩ + + R + ) + + + + + + (Right-distributivity of + + + ∩ + + + over + + + ∩ + + + ) + + + + + + ( + L + + △ + + M + ) + + ∩ + + R + + + + + + = + + + + + + + ( + L + + ∩ + + R + ) + + + + + △ + + + + + + ( + M + + ∩ + + R + ) + + + + + + (Right-distributivity of + + + ∩ + + + over + + + △ + + + ) + + + + + + ( + L + + ∩ + + M + ) + + × + + R + + + + + + = + + + + + + + ( + L + + × + + R + ) + + + + + + ∩ + + + + + + ( + M + + × + + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + ∩ + + + ) + + + + + + ( + L + + ∪ + + M + ) + + × + + R + + + + + + = + + + + + + + ( + L + + × + + R + ) + + + + + + ∪ + + + + + + ( + M + + × + + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + ∪ + + + ) + + + + + + ( + L + + ∖ + + M + ) + + × + + R + + + + + + = + + + + + + + ( + L + + × + + R + ) + + + + + + ∖ + + + + + + ( + M + + × + + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + ∖ + + + ) + + + + + + ( + L + + △ + + M + ) + + × + + R + + + + + + = + + + + + + + ( + L + + × + + R + ) + + + + + △ + + + + + + ( + M + + × + + R + ) + + + + + + (Right-distributivity of + + + × + + + over + + + △ + + + ) + + + + + + ( + L + + ∪ + + M + ) + + ∖ + + R + + + + + + = + + + + + + + ( + L + + ∖ + + R + ) + + + + + + ∪ + + + + + + ( + M + + ∖ + + R + ) + + + + + + (Right-distributivity of + + + ∖ + + + over + + + ∪ + + + ) + + + + + + ( + L + + ∩ + + M + ) + + ∖ + + R + + + + + + = + + + + + + + ( + L + + ∖ + + R + ) + + + + + + ∩ + + + + + + ( + M + + ∖ + + R + ) + + + + + + (Right-distributivity of + + + ∖ + + + over + + + ∩ + + + ) + + + + + + ( + L + + △ + + M + ) + + ∖ + + R + + + + + + = + + + + + + + ( + L + + ∖ + + R + ) + + + + + △ + + + + + + ( + M + + ∖ + + R + ) + + + + + + (Right-distributivity of + + + ∖ + + + over + + + △ + + + ) + + + + + + ( + L + + ∖ + + M + ) + + ∖ + + R + + + + + + = + + + + + + + ( + L + + ∖ + + R + ) + + + + + + ∖ + + + + + + ( + M + + ∖ + + R + ) + + + + + + (Right-distributivity of + + + ∖ + + + over + + + ∖ + + + ) + + + + + + + + + + + = + + + + + + + + + + + + + + + + L + + + + + + ∖ + + + + + + ( + M + ∪ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{9}(L\,\cap \,M)\,\cup \,R~&~~=~~&&(L\,\cup \,R)\,&&\cap \,&&(M\,\cup \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cup \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\cup \,R~&~~=~~&&(L\,\cup \,R)\,&&\cup \,&&(M\,\cup \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cup \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\cap \,R~&~~=~~&&(L\,\cap \,R)\,&&\cup \,&&(M\,\cap \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cap \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\cap \,M)\,\cap \,R~&~~=~~&&(L\,\cap \,R)\,&&\cap \,&&(M\,\cap \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cap \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\triangle \,M)\,\cap \,R~&~~=~~&&(L\,\cap \,R)\,&&\triangle \,&&(M\,\cap \,R)\qquad &&{\text{ (Right-distributivity of }}\,\cap \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex](L\,\cap \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\cap \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\cup \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\setminus \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\setminus \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex](L\,\triangle \,M)\,\times \,R~&~~=~~&&(L\,\times \,R)\,&&\triangle \,&&(M\,\times \,R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex](L\,\cup \,M)\,\setminus \,R~&~~=~~&&(L\,\setminus \,R)\,&&\cup \,&&(M\,\setminus \,R)\qquad &&{\text{ (Right-distributivity of }}\,\setminus \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex](L\,\cap \,M)\,\setminus \,R~&~~=~~&&(L\,\setminus \,R)\,&&\cap \,&&(M\,\setminus \,R)\qquad &&{\text{ (Right-distributivity of }}\,\setminus \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex](L\,\triangle \,M)\,\setminus \,R~&~~=~~&&(L\,\setminus \,R)&&\,\triangle \,&&(M\,\setminus \,R)\qquad &&{\text{ (Right-distributivity of }}\,\setminus \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex](L\,\setminus \,M)\,\setminus \,R~&~~=~~&&(L\,\setminus \,R)&&\,\setminus \,&&(M\,\setminus \,R)\qquad &&{\text{ (Right-distributivity of }}\,\setminus \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex]~&~~=~~&&~~\;~~\;~~\;~L&&\,\setminus \,&&(M\cup R)\\[1.4ex]\end{alignedat}}} + + +Left distributivity: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-6.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-6.md new file mode 100644 index 000000000..b8fe62e5c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-6.md @@ -0,0 +1,1298 @@ +--- +title: "List of set identities and relations" +chunk: 7/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + L + ∪ + ( + M + ∩ + R + ) + + + + + = + + + + + + + ( + L + ∪ + M + ) + ∩ + ( + L + ∪ + R + ) + + + + + + (Left-distributivity of + + + ∪ + + + over + + + ∩ + + + ) + + + + + + L + ∪ + ( + M + ∪ + R + ) + + + + + = + + + + + + + ( + L + ∪ + M + ) + ∪ + ( + L + ∪ + R + ) + + + + + (Left-distributivity of + + + ∪ + + + over + + + ∪ + + + ) + + + + + + L + ∩ + ( + M + ∪ + R + ) + + + + + = + + + + + + + ( + L + ∩ + M + ) + ∪ + ( + L + ∩ + R + ) + + + + + (Left-distributivity of + + + ∩ + + + over + + + ∪ + + + ) + + + + + + L + ∩ + ( + M + ∩ + R + ) + + + + + = + + + + + + + ( + L + ∩ + M + ) + ∩ + ( + L + ∩ + R + ) + + + + + (Left-distributivity of + + + ∩ + + + over + + + ∩ + + + ) + + + + + + L + ∩ + ( + M + + △ + + R + ) + + + + + = + + + + + + + ( + L + ∩ + M + ) + + △ + + ( + L + ∩ + R + ) + + + + + (Left-distributivity of + + + ∩ + + + over + + + △ + + + ) + + + + + + L + × + ( + M + ∩ + R + ) + + + + + = + + + + + + + ( + L + × + M + ) + ∩ + ( + L + × + R + ) + + + + + (Left-distributivity of + + + × + + + over + + + ∩ + + + ) + + + + + + L + × + ( + M + ∪ + R + ) + + + + + = + + + + + + + ( + L + × + M + ) + ∪ + ( + L + × + R + ) + + + + + (Left-distributivity of + + + × + + + over + + + ∪ + + + ) + + + + + + L + × + ( + M + + ∖ + R + ) + + + + + = + + + + + + + ( + L + × + M + ) + + ∖ + ( + L + × + R + ) + + + + + (Left-distributivity of + + + × + + + over + + + ∖ + + + ) + + + + + + L + × + ( + M + + △ + R + ) + + + + + = + + + + + + + ( + L + × + M + ) + + △ + ( + L + × + R + ) + + + + + (Left-distributivity of + + + × + + + over + + + △ + + + ) + + + + + + + + {\displaystyle {\begin{alignedat}{5}L\cup (M\cap R)&\;=\;\;&&(L\cup M)\cap (L\cup R)\qquad &&{\text{ (Left-distributivity of }}\,\cup \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex]L\cup (M\cup R)&\;=\;\;&&(L\cup M)\cup (L\cup R)&&{\text{ (Left-distributivity of }}\,\cup \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\cap (M\cup R)&\;=\;\;&&(L\cap M)\cup (L\cap R)&&{\text{ (Left-distributivity of }}\,\cap \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\cap (M\cap R)&\;=\;\;&&(L\cap M)\cap (L\cap R)&&{\text{ (Left-distributivity of }}\,\cap \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex]L\cap (M\,\triangle \,R)&\;=\;\;&&(L\cap M)\,\triangle \,(L\cap R)&&{\text{ (Left-distributivity of }}\,\cap \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex]L\times (M\cap R)&\;=\;\;&&(L\times M)\cap (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cap \,{\text{)}}\\[1.4ex]L\times (M\cup R)&\;=\;\;&&(L\times M)\cup (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times (M\,\setminus R)&\;=\;\;&&(L\times M)\,\setminus (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\setminus \,{\text{)}}\\[1.4ex]L\times (M\,\triangle R)&\;=\;\;&&(L\times M)\,\triangle (L\times R)&&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\triangle \,{\text{)}}\\[1.4ex]\end{alignedat}}} + + +==== Distributivity and symmetric difference ∆ ==== +Intersection distributes over symmetric difference: + + + + + + + + + L + + ∩ + + ( + M + + △ + + R + ) + + + + + + = + + + + + + + ( + L + + ∩ + + M + ) + + △ + + ( + L + + ∩ + + R + ) + + + + + + + + + + + + {\displaystyle {\begin{alignedat}{5}L\,\cap \,(M\,\triangle \,R)~&~~=~~&&(L\,\cap \,M)\,\triangle \,(L\,\cap \,R)~&&~\\[1.4ex]\end{alignedat}}} + + + + + + + + + + ( + L + + △ + + M + ) + + ∩ + + R + + + + + + = + + + + + + + ( + L + + ∩ + + R + ) + + △ + + ( + M + + ∩ + + R + ) + + + + + + + + + + + + {\displaystyle {\begin{alignedat}{5}(L\,\triangle \,M)\,\cap \,R~&~~=~~&&(L\,\cap \,R)\,\triangle \,(M\,\cap \,R)~&&~\\[1.4ex]\end{alignedat}}} + + +Union does not distribute over symmetric difference because only the following is guaranteed in general: + + + + + + + + + L + ∪ + ( + M + + △ + + R + ) + + + + + + ⊇ + + + + + + + + + + ( + L + ∪ + M + ) + + △ + + ( + L + ∪ + R + ) + + + + + + = + + + + + + ( + M + + △ + + R + ) + + ∖ + + L + + + + = + + + + + ( + M + + ∖ + + L + ) + + △ + + ( + R + + ∖ + + L + ) + + + + + + + {\displaystyle {\begin{alignedat}{5}L\cup (M\,\triangle \,R)~~{\color {red}{\supseteq }}~~\color {black}{\,}(L\cup M)\,\triangle \,(L\cup R)~&~=~&&(M\,\triangle \,R)\,\setminus \,L&~=~&&(M\,\setminus \,L)\,\triangle \,(R\,\setminus \,L)\\[1.4ex]\end{alignedat}}} + + +Symmetric difference does not distribute over itself: + + + + + L + + △ + + ( + M + + △ + + R + ) + + + + + + ≠ + + + + + + + + + + ( + L + + △ + + M + ) + + △ + + ( + L + + △ + + R + ) + + = + + M + + △ + + R + + + + {\displaystyle L\,\triangle \,(M\,\triangle \,R)~~{\color {red}{\neq }}~~\color {black}{\,}(L\,\triangle \,M)\,\triangle \,(L\,\triangle \,R)~=~M\,\triangle \,R} + + +and in general, for any sets + + + + L + + and + + A + + + {\displaystyle L{\text{ and }}A} + + (where + + + + A + + + {\displaystyle A} + + represents + + + + M + + △ + + R + + + {\displaystyle M\,\triangle \,R} + +), + + + + L + + △ + + A + + + {\displaystyle L\,\triangle \,A} + + might not be a subset, nor a superset, of + + + + L + + + {\displaystyle L} + + (and the same is true for + + + + A + + + {\displaystyle A} + +). + +==== Distributivity and set subtraction \ ==== +Failure of set subtraction to left distribute: +Set subtraction is right distributive over itself. However, set subtraction is not left distributive over itself because only the following is guaranteed in general: + + + + + + + + + L + + ∖ + + ( + M + + ∖ + + R + ) + + + + + + + + ⊇ + + + + + + + + + + + + + ( + L + + ∖ + + M + ) + + ∖ + + ( + L + + ∖ + + R + ) + + + = + + + L + ∩ + R + + ∖ + + M + + + + + + + + {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\setminus \,R)&~~{\color {red}{\supseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\setminus \,(L\,\setminus \,R)~~=~~L\cap R\,\setminus \,M\\[1.4ex]\end{alignedat}}} + + +where equality holds if and only if + + + + L + + ∖ + + M + = + L + + ∩ + + R + , + + + {\displaystyle L\,\setminus \,M=L\,\cap \,R,} + + which happens if and only if + + + + L + ∩ + M + ∩ + R + = + ∅ + + and + + L + ∖ + M + ⊆ + R + . + + + {\displaystyle L\cap M\cap R=\varnothing {\text{ and }}L\setminus M\subseteq R.} + + +For symmetric difference, the sets + + + + L + + ∖ + + ( + M + + △ + + R + ) + + + {\displaystyle L\,\setminus \,(M\,\triangle \,R)} + + and + + + + ( + L + + ∖ + + M + ) + + △ + + ( + L + + ∖ + + R + ) + = + L + + ∩ + + ( + M + + △ + + R + ) + + + {\displaystyle (L\,\setminus \,M)\,\triangle \,(L\,\setminus \,R)=L\,\cap \,(M\,\triangle \,R)} + + are always disjoint. +So these two sets are equal if and only if they are both equal to + + + + ∅ + . + + + {\displaystyle \varnothing .} + + +Moreover, + + + + L + + ∖ + + ( + M + + △ + + R + ) + = + ∅ + + + {\displaystyle L\,\setminus \,(M\,\triangle \,R)=\varnothing } + + if and only if + + + + L + ∩ + M + ∩ + R + = + ∅ + + and + + L + ⊆ + M + ∪ + R + . + + + {\displaystyle L\cap M\cap R=\varnothing {\text{ and }}L\subseteq M\cup R.} + + +To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related: + + + + + + + + + ( + L + + ∖ + + M + ) + + ∩ + + ( + L + + ∖ + + R + ) + + + = + + + L + + ∖ + + ( + M + + ∪ + + R + ) + + + + + + + + + ⊆ + + + + + + + + + + + + + L + + ∖ + + ( + M + + ∩ + + R + ) + + + = + + + ( + L + + ∖ + + M + ) + + ∪ + + ( + L + + ∖ + + R + ) + + + + + + + + {\displaystyle {\begin{alignedat}{5}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}L\,\setminus \,(M\,\cap \,R)~~=~~(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)\\[1.4ex]\end{alignedat}}} + + +always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment + + + + + + + ⊆ + + + + + + {\displaystyle {\color {red}{\subseteq }}} + + might be strict). +Equality holds if and only if + + + + L + + ∖ + + ( + M + + ∩ + + R + ) + + ⊆ + + L + + ∖ + + ( + M + + ∪ + + R + ) + , + + + {\displaystyle L\,\setminus \,(M\,\cap \,R)\;\subseteq \;L\,\setminus \,(M\,\cup \,R),} + + which happens if and only if + + + + L + + ∩ + + M + = + L + + ∩ + + R + . + + + {\displaystyle L\,\cap \,M=L\,\cap \,R.} + + +This observation about De Morgan's laws shows that + + + + + ∖ + + + + {\displaystyle \,\setminus \,} + + is not left distributive over + + + + + ∪ + + + + {\displaystyle \,\cup \,} + + or + + + + + ∩ + + + + {\displaystyle \,\cap \,} + + because only the following are guaranteed in general: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-7.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-7.md new file mode 100644 index 000000000..6ad0e706a --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-7.md @@ -0,0 +1,1272 @@ +--- +title: "List of set identities and relations" +chunk: 8/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + L + + ∖ + + ( + M + + ∪ + + R + ) + + + + + + + + + ⊆ + + + + + + + + + + + + + ( + L + + ∖ + + M + ) + + ∪ + + ( + L + + ∖ + + R + ) + + + = + + + L + + ∖ + + ( + M + + ∩ + + R + ) + + + + + + + + {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cap \,R)\\[1.4ex]\end{alignedat}}} + + + + + + + + + + L + + ∖ + + ( + M + + ∩ + + R + ) + + + + + + + + + ⊇ + + + + + + + + + + + + + ( + L + + ∖ + + M + ) + + ∩ + + ( + L + + ∖ + + R + ) + + + = + + + L + + ∖ + + ( + M + + ∪ + + R + ) + + + + + + + + {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cap \,R)~&~~{\color {red}{\supseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)\\[1.4ex]\end{alignedat}}} + + +where equality holds for one (or equivalently, for both) of the above two inclusion formulas if and only if + + + + L + + ∩ + + M + = + L + + ∩ + + R + . + + + {\displaystyle L\,\cap \,M=L\,\cap \,R.} + + +The following statements are equivalent: + + + + + L + ∩ + M + + = + + L + ∩ + R + + + {\displaystyle L\cap M\,=\,L\cap R} + + + + + + L + + ∖ + + M + + = + + L + + ∖ + + R + + + {\displaystyle L\,\setminus \,M\,=\,L\,\setminus \,R} + + + + + + L + + ∖ + + ( + M + + ∩ + + R + ) + = + ( + L + + ∖ + + M + ) + + ∩ + + ( + L + + ∖ + + R + ) + ; + + + {\displaystyle L\,\setminus \,(M\,\cap \,R)=(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R);} + + that is, + + + + + ∖ + + + + {\displaystyle \,\setminus \,} + + left distributes over + + + + + ∩ + + + + {\displaystyle \,\cap \,} + + for these three particular sets + + + + + L + + ∖ + + ( + M + + ∪ + + R + ) + = + ( + L + + ∖ + + M + ) + + ∪ + + ( + L + + ∖ + + R + ) + ; + + + {\displaystyle L\,\setminus \,(M\,\cup \,R)=(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R);} + + that is, + + + + + ∖ + + + + {\displaystyle \,\setminus \,} + + left distributes over + + + + + ∪ + + + + {\displaystyle \,\cup \,} + + for these three particular sets + + + + + L + + ∖ + + ( + M + + ∩ + + R + ) + + = + + L + + ∖ + + ( + M + + ∪ + + R + ) + + + {\displaystyle L\,\setminus \,(M\,\cap \,R)\,=\,L\,\setminus \,(M\,\cup \,R)} + + + + + + L + ∩ + ( + M + ∪ + R + ) + + = + + L + ∩ + M + ∩ + R + + + {\displaystyle L\cap (M\cup R)\,=\,L\cap M\cap R} + + + + + + L + ∩ + ( + M + ∪ + R + ) + + ⊆ + + M + ∩ + R + + + {\displaystyle L\cap (M\cup R)~\subseteq ~M\cap R} + + + + + + L + ∩ + R + + ⊆ + + M + + + + {\displaystyle L\cap R~\subseteq ~M\;} + + and + + + + + L + ∩ + M + + ⊆ + + R + + + {\displaystyle \;L\cap M~\subseteq ~R} + + + + + + L + ∖ + ( + M + ∖ + R + ) + + = + + L + ∖ + ( + R + ∖ + M + ) + + + {\displaystyle L\setminus (M\setminus R)\,=\,L\setminus (R\setminus M)} + + + + + + L + ∖ + ( + M + ∖ + R + ) + + = + + L + ∖ + ( + R + ∖ + M + ) + + = + + L + + + {\displaystyle L\setminus (M\setminus R)\,=\,L\setminus (R\setminus M)\,=\,L} + + +Quasi-commutativity: + + + + + ( + L + ∖ + M + ) + ∖ + R + + = + + ( + L + ∖ + R + ) + ∖ + M + + + (Quasi-commutative) + + + + {\displaystyle (L\setminus M)\setminus R~=~(L\setminus R)\setminus M\qquad {\text{ (Quasi-commutative)}}} + + +always holds but in general, + + + + + L + ∖ + ( + M + ∖ + R + ) + + + + + + ≠ + + + + + + L + ∖ + ( + R + ∖ + M + ) + . + + + {\displaystyle L\setminus (M\setminus R)~~{\color {red}{\neq }}~~L\setminus (R\setminus M).} + + +However, + + + + L + ∖ + ( + M + ∖ + R + ) + + ⊆ + + L + ∖ + ( + R + ∖ + M + ) + + + {\displaystyle L\setminus (M\setminus R)~\subseteq ~L\setminus (R\setminus M)} + + if and only if + + + + L + ∩ + R + + ⊆ + + M + + + {\displaystyle L\cap R~\subseteq ~M} + + if and only if + + + + L + ∖ + ( + R + ∖ + M + ) + + = + + L + . + + + {\displaystyle L\setminus (R\setminus M)~=~L.} + + +Set subtraction complexity: To manage the many identities involving set subtraction, this section is divided based on where the set subtraction operation and parentheses are located on the left hand side of the identity. The great variety and (relative) complexity of formulas involving set subtraction (compared to those without it) is in part due to the fact that unlike + + + + + ∪ + , + + ∩ + , + + + {\displaystyle \,\cup ,\,\cap ,} + + and + + + + △ + , + + + + {\displaystyle \triangle ,\,} + + set subtraction is neither associative nor commutative and it also is not left distributive over + + + + + ∪ + , + + ∩ + , + + △ + , + + + {\displaystyle \,\cup ,\,\cap ,\,\triangle ,} + + or even over itself. + +=== Two set subtractions === +Set subtraction is not associative in general: + + + + + ( + L + + ∖ + + M + ) + + ∖ + + R + + + + + + + ≠ + + + + + + + L + + ∖ + + ( + M + + ∖ + + R + ) + + + {\displaystyle (L\,\setminus \,M)\,\setminus \,R\;~~{\color {red}{\neq }}~~\;L\,\setminus \,(M\,\setminus \,R)} + + +since only the following is always guaranteed: + + + + + ( + L + + ∖ + + M + ) + + ∖ + + R + + + + + + + ⊆ + + + + + + + L + + ∖ + + ( + M + + ∖ + + R + ) + . + + + {\displaystyle (L\,\setminus \,M)\,\setminus \,R\;~~{\color {red}{\subseteq }}~~\;L\,\setminus \,(M\,\setminus \,R).} + + +==== (L\M)\R ==== + + + + + + + + + ( + L + ∖ + M + ) + ∖ + R + + + + = + + + + L + ∖ + ( + M + ∪ + R + ) + + + + + + + = + ( + + + + L + ∖ + R + ) + ∖ + M + + + + + + + = + ( + + + + L + ∖ + M + ) + ∩ + ( + L + ∖ + R + ) + + + + + + + = + ( + + + + L + ∖ + R + ) + ∖ + M + + + + + + + = + ( + + + + L + + ∖ + + R + ) + + ∖ + + ( + M + + ∖ + + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}(L\setminus M)\setminus R&=&&L\setminus (M\cup R)\\[0.6ex]&=(&&L\setminus R)\setminus M\\[0.6ex]&=(&&L\setminus M)\cap (L\setminus R)\\[0.6ex]&=(&&L\setminus R)\setminus M\\[0.6ex]&=(&&L\,\setminus \,R)\,\setminus \,(M\,\setminus \,R)\\[1.4ex]\end{alignedat}}} + + +==== L\(M\R) ==== + + + + + + + + + L + ∖ + ( + M + ∖ + R + ) + + + + = + ( + L + ∖ + M + ) + ∪ + ( + L + ∩ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}L\setminus (M\setminus R)&=(L\setminus M)\cup (L\cap R)\\[1.4ex]\end{alignedat}}} + + +If + + + + L + ⊆ + M + + then + + L + ∖ + ( + M + ∖ + R + ) + = + L + ∩ + R + + + {\displaystyle L\subseteq M{\text{ then }}L\setminus (M\setminus R)=L\cap R} + + + + + + L + ∖ + ( + M + ∖ + R + ) + ⊆ + ( + L + ∖ + M + ) + ∪ + R + + + {\textstyle L\setminus (M\setminus R)\subseteq (L\setminus M)\cup R} + + with equality if and only if + + + + R + ⊆ + L + . + + + {\displaystyle R\subseteq L.} + + +=== One set subtraction === + +==== (L\M) ⁎ R ==== +Set subtraction on the left, and parentheses on the left + + + + + + + + + + ( + + L + ∖ + M + + ) + + ∪ + R + + + + = + ( + L + ∪ + R + ) + ∖ + ( + M + ∖ + R + ) + + + + + + + = + ( + L + ∖ + ( + M + ∪ + R + ) + ) + ∪ + R + + + + + + + (the outermost union is disjoint) + + + + + + + + {\displaystyle {\begin{alignedat}{4}\left(L\setminus M\right)\cup R&=(L\cup R)\setminus (M\setminus R)\\&=(L\setminus (M\cup R))\cup R~~~~~{\text{ (the outermost union is disjoint) }}\\\end{alignedat}}} + + + + + + + + + + ( + L + ∖ + M + ) + ∩ + R + + + + = + ( + + + + L + ∩ + R + ) + ∖ + ( + M + ∩ + R + ) + + + + + (Distributive law of + + ∩ + + over + + ∖ + + ) + + + + + + + + = + ( + + + + L + ∩ + R + ) + ∖ + M + + + + + + + = + + + + L + ∩ + ( + R + ∖ + M + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}(L\setminus M)\cap R&=(&&L\cap R)\setminus (M\cap R)~~~{\text{ (Distributive law of }}\cap {\text{ over }}\setminus {\text{ )}}\\&=(&&L\cap R)\setminus M\\&=&&L\cap (R\setminus M)\\\end{alignedat}}} + + + + + + + + + + ( + L + + ∖ + + M + ) + + ∩ + + ( + L + + ∖ + + R + ) + + + = + + + L + + ∖ + + ( + M + + ∪ + + R + ) + + + + + + + + + ⊆ + + + + + + + + + + + + + L + + ∖ + + ( + M + + ∩ + + R + ) + + + = + + + ( + L + + ∖ + + M + ) + + ∪ + + ( + L + + ∖ + + R + ) + + + + + + + + {\displaystyle {\begin{alignedat}{5}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}L\,\setminus \,(M\,\cap \,R)~~=~~(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)\\[1.4ex]\end{alignedat}}} + + \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-8.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-8.md new file mode 100644 index 000000000..d450dbccf --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-8.md @@ -0,0 +1,906 @@ +--- +title: "List of set identities and relations" +chunk: 9/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + ( + L + ∖ + M + ) + + △ + + R + + + + = + ( + L + ∖ + ( + M + ∪ + R + ) + ) + ∪ + ( + R + ∖ + L + ) + ∪ + ( + L + ∩ + M + ∩ + R + ) + + + + + (the three outermost sets are pairwise disjoint) + + + + + + + + {\displaystyle {\begin{alignedat}{4}(L\setminus M)~\triangle ~R&=(L\setminus (M\cup R))\cup (R\setminus L)\cup (L\cap M\cap R)~~~{\text{ (the three outermost sets are pairwise disjoint) }}\\\end{alignedat}}} + + + + + + ( + L + + ∖ + M + ) + × + R + = + ( + L + × + R + ) + + ∖ + ( + M + × + R + ) + + + + + + + (Distributivity) + + + + {\displaystyle (L\,\setminus M)\times R=(L\times R)\,\setminus (M\times R)~~~~~{\text{ (Distributivity)}}} + + +==== L\(M ⁎ R) ==== +Set subtraction on the left, and parentheses on the right + + + + + + + + + L + ∖ + ( + M + ∪ + R + ) + + + + = + ( + L + ∖ + M + ) + + + + + + ∩ + + ( + + + + L + ∖ + R + ) + + + + + + (De Morgan's law) + + + + + + + + = + ( + L + ∖ + M + ) + + + + + + + ∖ + + + + R + + + + + + + = + ( + L + ∖ + R + ) + + + + + + + ∖ + + + + M + + + + + + + {\displaystyle {\begin{alignedat}{3}L\setminus (M\cup R)&=(L\setminus M)&&\,\cap \,(&&L\setminus R)~~~~{\text{ (De Morgan's law) }}\\&=(L\setminus M)&&\,\,\setminus &&R\\&=(L\setminus R)&&\,\,\setminus &&M\\\end{alignedat}}} + + + + + + + + + + L + ∖ + ( + M + ∩ + R + ) + + + + = + ( + L + ∖ + M + ) + ∪ + ( + L + ∖ + R + ) + + + + + + (De Morgan's law) + + + + + + + + {\displaystyle {\begin{alignedat}{4}L\setminus (M\cap R)&=(L\setminus M)\cup (L\setminus R)~~~~{\text{ (De Morgan's law) }}\\\end{alignedat}}} + + +where the above two sets that are the subjects of De Morgan's laws always satisfy + + + + L + + ∖ + + ( + M + + ∪ + + R + ) + + + + + + ⊆ + + + + + + + + + + L + + ∖ + + ( + M + + ∩ + + R + ) + . + + + + {\displaystyle L\,\setminus \,(M\,\cup \,R)~~{\color {red}{\subseteq }}~~\color {black}{\,}L\,\setminus \,(M\,\cap \,R).} + + + + + + + + + + L + ∖ + ( + M + + △ + + R + ) + + + + = + ( + L + ∖ + ( + M + ∪ + R + ) + ) + ∪ + ( + L + ∩ + M + ∩ + R + ) + + + + + (the outermost union is disjoint) + + + + + + + + {\displaystyle {\begin{alignedat}{4}L\setminus (M~\triangle ~R)&=(L\setminus (M\cup R))\cup (L\cap M\cap R)~~~{\text{ (the outermost union is disjoint) }}\\\end{alignedat}}} + + +==== (L ⁎ M)\R ==== +Set subtraction on the right, and parentheses on the left + + + + + + + + + ( + L + ∪ + M + ) + ∖ + R + + + + = + ( + L + ∖ + R + ) + ∪ + ( + M + ∖ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}(L\cup M)\setminus R&=(L\setminus R)\cup (M\setminus R)\\\end{alignedat}}} + + + + + + + + + + ( + L + ∩ + M + ) + ∖ + R + + + + = + ( + + + + L + ∖ + R + ) + + + + + ∩ + ( + M + ∖ + R + ) + + + + + + + = + + + + L + + + + + ∩ + ( + M + ∖ + R + ) + + + + + + + = + + + + M + + + + + ∩ + ( + L + ∖ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}(L\cap M)\setminus R&=(&&L\setminus R)&&\cap (M\setminus R)\\&=&&L&&\cap (M\setminus R)\\&=&&M&&\cap (L\setminus R)\\\end{alignedat}}} + + + + + + + + + + ( + L + + △ + + M + ) + ∖ + R + + + + = + ( + L + ∖ + R + ) + + + + + △ + + ( + M + ∖ + R + ) + + + + + + + = + ( + L + ∪ + R + ) + + + + + △ + + ( + M + ∪ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}(L\,\triangle \,M)\setminus R&=(L\setminus R)~&&\triangle ~(M\setminus R)\\&=(L\cup R)~&&\triangle ~(M\cup R)\\\end{alignedat}}} + + +==== L ⁎ (M\R) ==== +Set subtraction on the right, and parentheses on the right + + + + + + + + + L + ∪ + ( + M + ∖ + R + ) + + + + = + + + + + + L + + + + + ∪ + + + + + + ( + M + ∖ + ( + R + ∪ + L + ) + ) + + + + + + + + (the outermost union is disjoint) + + + + + + + + = + [ + + + + + ( + + + + L + ∖ + M + ) + + + + + ∪ + + + + + + ( + R + ∩ + L + ) + ] + ∪ + ( + M + ∖ + R + ) + + + + + + + + (the outermost union is disjoint) + + + + + + + + = + + + + + ( + + + + L + ∖ + ( + M + ∪ + R + ) + ) + + + + + + + ∪ + + + + + ( + R + ∩ + L + ) + + + ∪ + ( + M + ∖ + R + ) + + + + + + + + (the three outermost sets are pairwise disjoint) + + + + + + + + {\displaystyle {\begin{alignedat}{3}L\cup (M\setminus R)&=&&&&L&&\cup \;&&(M\setminus (R\cup L))&&~~~{\text{ (the outermost union is disjoint) }}\\&=[&&(&&L\setminus M)&&\cup \;&&(R\cap L)]\cup (M\setminus R)&&~~~{\text{ (the outermost union is disjoint) }}\\&=&&(&&L\setminus (M\cup R))\;&&\;\cup &&(R\cap L)\,\,\cup (M\setminus R)&&~~~{\text{ (the three outermost sets are pairwise disjoint) }}\\\end{alignedat}}} + + + + + + + + + + L + ∩ + ( + M + ∖ + R + ) + + + + = + ( + + + + L + ∩ + M + ) + + + + + ∖ + ( + L + ∩ + R + ) + + + + + (Distributive law of + + ∩ + + over + + ∖ + + ) + + + + + + + + = + ( + + + + L + ∩ + M + ) + + + + + ∖ + R + + + + + + + = + + + + M + + + + + ∩ + ( + L + ∖ + R + ) + + + + + + + = + ( + + + + L + ∖ + R + ) + + + + + ∩ + ( + M + ∖ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{4}L\cap (M\setminus R)&=(&&L\cap M)&&\setminus (L\cap R)~~~{\text{ (Distributive law of }}\cap {\text{ over }}\setminus {\text{ )}}\\&=(&&L\cap M)&&\setminus R\\&=&&M&&\cap (L\setminus R)\\&=(&&L\setminus R)&&\cap (M\setminus R)\\\end{alignedat}}} + + + + + + L + × + ( + M + + ∖ + R + ) + = + ( + L + × + M + ) + + ∖ + ( + L + × + R + ) + + + + + + + (Distributivity) + + + + {\displaystyle L\times (M\,\setminus R)=(L\times M)\,\setminus (L\times R)~~~~~{\text{ (Distributivity)}}} + + +=== Three operations on three sets === + +==== (L • M) ⁎ (M • R) ==== +Operations of the form + + + + ( + L + ∙ + M + ) + ∗ + ( + M + ∙ + R + ) + + + {\displaystyle (L\bullet M)\ast (M\bullet R)} + +: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-9.md b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-9.md new file mode 100644 index 000000000..b630f0c97 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-9.md @@ -0,0 +1,1993 @@ +--- +title: "List of set identities and relations" +chunk: 10/33 +source: "https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:53.246995+00:00" +instance: "kb-cron" +--- + + + + + + + + + ( + L + ∪ + M + ) + + + + + ∪ + + + + + + ( + + + + M + ∪ + R + ) + + + + + + + + = + + + + + + L + ∪ + M + ∪ + R + + + + + ( + L + ∪ + M + ) + + + + + ∩ + + + + + + ( + + + + M + ∪ + R + ) + + + + + + + + = + + + + + + M + ∪ + ( + L + ∩ + R + ) + + + + + ( + L + ∪ + M + ) + + + + + ∖ + + + + + + ( + + + + M + ∪ + R + ) + + + + + + + + = + + + + + + L + + ∖ + + ( + M + ∪ + R + ) + + + + + ( + L + ∪ + M + ) + + + + △ + + + + + + ( + + + + M + ∪ + R + ) + + + + + + + + = + + + + + + + ( + L + + ∖ + + ( + M + ∪ + R + ) + ) + + ∪ + + ( + R + + ∖ + + ( + L + ∪ + M + ) + ) + + + + + + + + + + + + + + + + + + + + + + = + + + + + + + ( + L + + △ + + R + ) + + ∖ + + M + + + + + ( + L + ∩ + M + ) + + + + + ∪ + + + + + + ( + + + + M + ∩ + R + ) + + + + + + + + = + + + + + + M + ∩ + ( + L + ∪ + R + ) + + + + + ( + L + ∩ + M + ) + + + + + ∩ + + + + + + ( + + + + M + ∩ + R + ) + + + + + + + + = + + + + + + L + ∩ + M + ∩ + R + + + + + ( + L + ∩ + M + ) + + + + + ∖ + + + + + + ( + + + + M + ∩ + R + ) + + + + + + + + = + + + + + + + ( + L + ∩ + M + ) + + ∖ + + R + + + + + ( + L + ∩ + M + ) + + + + △ + + + + + + ( + + + + M + ∩ + R + ) + + + + + + + + = + + + + + + + [ + ( + L + + ∩ + M + ) + ∪ + ( + M + + ∩ + R + ) + ] + + ∖ + + ( + L + + ∩ + M + + ∩ + R + ) + + + + + ( + L + + ∖ + M + ) + + + + + ∪ + + + + + + ( + + + + M + + ∖ + R + ) + + + + + + + + = + + + + + + + ( + L + + ∪ + M + ) + + ∖ + ( + M + + ∩ + + R + ) + + + + + ( + L + + ∖ + M + ) + + + + + ∩ + + + + + + ( + + + + M + + ∖ + R + ) + + + + + + + + = + + + + + + ∅ + + + + + ( + L + + ∖ + M + ) + + + + + ∖ + + + + + + ( + + + + M + + ∖ + R + ) + + + + + + + + = + + + + + + L + + ∖ + M + + + + + ( + L + + ∖ + M + ) + + + + △ + + + + + + ( + + + + M + + ∖ + R + ) + + + + + + + + = + + + + + + + ( + L + + ∖ + M + ) + ∪ + ( + M + + ∖ + R + ) + + + + + + + + + + + + + + + + + + + + + + = + + + + + + + ( + L + + ∪ + M + ) + ∖ + ( + M + + ∩ + R + ) + + + + + ( + L + + △ + + M + ) + + + + + ∪ + + + + + + ( + + + + M + + △ + + R + ) + + + + + + + + = + + + + + + + ( + L + + ∪ + + M + + ∪ + + R + ) + + ∖ + + ( + L + + ∩ + + M + + ∩ + + R + ) + + + + + ( + L + + △ + + M + ) + + + + + ∩ + + + + + + ( + + + + M + + △ + + R + ) + + + + + + + + = + + + + + + + ( + ( + L + + ∩ + + R + ) + + ∖ + + M + ) + + ∪ + + ( + M + + ∖ + + ( + L + + ∪ + + R + ) + ) + + + + + ( + L + + △ + + M + ) + + + + + ∖ + + + + + + ( + + + + M + + △ + + R + ) + + + + + + + + = + + + + + + + ( + L + + ∖ + + ( + M + + ∪ + + R + ) + ) + + ∪ + + ( + ( + M + + ∩ + + R + ) + + ∖ + + L + ) + + + + + ( + L + + △ + + M + ) + + + + △ + + + + + + ( + + + + M + + △ + + R + ) + + + + + + + + = + + + + + + L + + △ + + R + + + + + + + {\displaystyle {\begin{alignedat}{9}(L\cup M)&\,\cup \,&&(&&M\cup R)&&&&\;=\;\;&&L\cup M\cup R\\[1.4ex](L\cup M)&\,\cap \,&&(&&M\cup R)&&&&\;=\;\;&&M\cup (L\cap R)\\[1.4ex](L\cup M)&\,\setminus \,&&(&&M\cup R)&&&&\;=\;\;&&L\,\setminus \,(M\cup R)\\[1.4ex](L\cup M)&\,\triangle \,&&(&&M\cup R)&&&&\;=\;\;&&(L\,\setminus \,(M\cup R))\,\cup \,(R\,\setminus \,(L\cup M))\\[1.4ex]&\,&&\,&&\,&&&&\;=\;\;&&(L\,\triangle \,R)\,\setminus \,M\\[1.4ex](L\cap M)&\,\cup \,&&(&&M\cap R)&&&&\;=\;\;&&M\cap (L\cup R)\\[1.4ex](L\cap M)&\,\cap \,&&(&&M\cap R)&&&&\;=\;\;&&L\cap M\cap R\\[1.4ex](L\cap M)&\,\setminus \,&&(&&M\cap R)&&&&\;=\;\;&&(L\cap M)\,\setminus \,R\\[1.4ex](L\cap M)&\,\triangle \,&&(&&M\cap R)&&&&\;=\;\;&&[(L\,\cap M)\cup (M\,\cap R)]\,\setminus \,(L\,\cap M\,\cap R)\\[1.4ex](L\,\setminus M)&\,\cup \,&&(&&M\,\setminus R)&&&&\;=\;\;&&(L\,\cup M)\,\setminus (M\,\cap \,R)\\[1.4ex](L\,\setminus M)&\,\cap \,&&(&&M\,\setminus R)&&&&\;=\;\;&&\varnothing \\[1.4ex](L\,\setminus M)&\,\setminus \,&&(&&M\,\setminus R)&&&&\;=\;\;&&L\,\setminus M\\[1.4ex](L\,\setminus M)&\,\triangle \,&&(&&M\,\setminus R)&&&&\;=\;\;&&(L\,\setminus M)\cup (M\,\setminus R)\\[1.4ex]&\,&&\,&&\,&&&&\;=\;\;&&(L\,\cup M)\setminus (M\,\cap R)\\[1.4ex](L\,\triangle \,M)&\,\cup \,&&(&&M\,\triangle \,R)&&&&\;=\;\;&&(L\,\cup \,M\,\cup \,R)\,\setminus \,(L\,\cap \,M\,\cap \,R)\\[1.4ex](L\,\triangle \,M)&\,\cap \,&&(&&M\,\triangle \,R)&&&&\;=\;\;&&((L\,\cap \,R)\,\setminus \,M)\,\cup \,(M\,\setminus \,(L\,\cup \,R))\\[1.4ex](L\,\triangle \,M)&\,\setminus \,&&(&&M\,\triangle \,R)&&&&\;=\;\;&&(L\,\setminus \,(M\,\cup \,R))\,\cup \,((M\,\cap \,R)\,\setminus \,L)\\[1.4ex](L\,\triangle \,M)&\,\triangle \,&&(&&M\,\triangle \,R)&&&&\;=\;\;&&L\,\triangle \,R\\[1.7ex]\end{alignedat}}} + + +==== (L • M) ⁎ (R\M) ==== +Operations of the form + + + + ( + L + ∙ + M + ) + ∗ + ( + R + + ∖ + + M + ) + + + {\displaystyle (L\bullet M)\ast (R\,\setminus \,M)} + +: + + + + + + + + + ( + L + ∪ + M + ) + + + + + ∪ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + L + ∪ + M + ∪ + R + + + + + ( + L + ∪ + M + ) + + + + + ∩ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + + ( + L + ∩ + R + ) + + ∖ + + M + + + + + ( + L + ∪ + M + ) + + + + + ∖ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + M + ∪ + ( + L + + ∖ + + R + ) + + + + + ( + L + ∪ + M + ) + + + + △ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + M + ∪ + ( + L + + △ + + R + ) + + + + + ( + L + ∩ + M + ) + + + + + ∪ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + + [ + L + ∩ + ( + M + ∪ + R + ) + ] + ∪ + [ + R + + ∖ + + ( + L + ∪ + M + ) + ] + + + (disjoint union) + + + + + + + + + + + + + + + + + + + + + + + = + + + + + + + ( + L + ∩ + M + ) + + △ + + ( + R + + ∖ + + M + ) + + + + + ( + L + ∩ + M + ) + + + + + ∩ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + ∅ + + + + + ( + L + ∩ + M + ) + + + + + ∖ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + L + ∩ + M + + + + + ( + L + ∩ + M + ) + + + + △ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + + ( + L + ∩ + M + ) + ∪ + ( + R + + ∖ + + M + ) + + + (disjoint union) + + + + + + ( + L + + ∖ + + M + ) + + + + + ∪ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + L + ∪ + R + + ∖ + + M + + + + + ( + L + + ∖ + + M + ) + + + + + ∩ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + + ( + L + ∩ + R + ) + + ∖ + + M + + + + + ( + L + + ∖ + + M + ) + + + + + ∖ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + L + + ∖ + + ( + M + ∪ + R + ) + + + + + ( + L + + ∖ + + M + ) + + + + △ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + + ( + L + + △ + + R + ) + + ∖ + + M + + + + + ( + L + + △ + + M + ) + + + + + ∪ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + + ( + L + ∪ + M + ∪ + R + ) + + ∖ + + ( + L + ∩ + M + ) + + + + + ( + L + + △ + + M + ) + + + + + ∩ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + + ( + L + ∩ + R + ) + + ∖ + + M + + + + + ( + L + + △ + + M + ) + + + + + ∖ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + + [ + L + + ∖ + + ( + M + ∪ + R + ) + ] + ∪ + ( + M + + ∖ + + L + ) + + + (disjoint union) + + + + + + + + + + + + + + + + + + + + + + + = + + + + + + + ( + L + + △ + + M + ) + ∖ + ( + L + + ∩ + R + ) + + + + + ( + L + + △ + + M + ) + + + + △ + + + + + + ( + + + + R + + ∖ + + M + ) + + + + + + + + = + + + + + + L + + △ + + ( + M + ∪ + R + ) + + + + + + + {\displaystyle {\begin{alignedat}{9}(L\cup M)&\,\cup \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\cup M\cup R\\[1.4ex](L\cup M)&\,\cap \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cap R)\,\setminus \,M\\[1.4ex](L\cup M)&\,\setminus \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&M\cup (L\,\setminus \,R)\\[1.4ex](L\cup M)&\,\triangle \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&M\cup (L\,\triangle \,R)\\[1.4ex](L\cap M)&\,\cup \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&[L\cap (M\cup R)]\cup [R\,\setminus \,(L\cup M)]\qquad {\text{ (disjoint union)}}\\[1.4ex]&\,&&\,&&\,&&&&\;=\;\;&&(L\cap M)\,\triangle \,(R\,\setminus \,M)\\[1.4ex](L\cap M)&\,\cap \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&\varnothing \\[1.4ex](L\cap M)&\,\setminus \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\cap M\\[1.4ex](L\cap M)&\,\triangle \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cap M)\cup (R\,\setminus \,M)\qquad {\text{ (disjoint union)}}\\[1.4ex](L\,\setminus \,M)&\,\cup \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\cup R\,\setminus \,M\\[1.4ex](L\,\setminus \,M)&\,\cap \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cap R)\,\setminus \,M\\[1.4ex](L\,\setminus \,M)&\,\setminus \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\,\setminus \,(M\cup R)\\[1.4ex](L\,\setminus \,M)&\,\triangle \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\,\triangle \,R)\,\setminus \,M\\[1.4ex](L\,\triangle \,M)&\,\cup \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cup M\cup R)\,\setminus \,(L\cap M)\\[1.4ex](L\,\triangle \,M)&\,\cap \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&(L\cap R)\,\setminus \,M\\[1.4ex](L\,\triangle \,M)&\,\setminus \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&[L\,\setminus \,(M\cup R)]\cup (M\,\setminus \,L)\qquad {\text{ (disjoint union)}}\\[1.4ex]&\,&&\,&&\,&&&&\;=\;\;&&(L\,\triangle \,M)\setminus (L\,\cap R)\\[1.4ex](L\,\triangle \,M)&\,\triangle \,&&(&&R\,\setminus \,M)&&&&\;=\;\;&&L\,\triangle \,(M\cup R)\\[1.7ex]\end{alignedat}}} + + +==== (L\M) ⁎ (L\R) ==== +Operations of the form + + + + ( + L + + ∖ + + M + ) + ∗ + ( + L + + ∖ + + R + ) + + + {\displaystyle (L\,\setminus \,M)\ast (L\,\setminus \,R)} + +: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_set_theory_topics-0.md b/data/en.wikipedia.org/wiki/List_of_set_theory_topics-0.md new file mode 100644 index 000000000..b59c75aa7 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_set_theory_topics-0.md @@ -0,0 +1,29 @@ +--- +title: "List of set theory topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_set_theory_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:55.603082+00:00" +instance: "kb-cron" +--- + +This page is a list of articles related to set theory. + + +== Articles on individual set theory topics == + + +== Lists related to set theory == +Glossary of set theory +List of large cardinal properties +List of properties of sets of reals +List of set identities and relations + + +== Set theorists == + + +== Societies and organizations == +Association for Symbolic Logic +The Cabal \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant-0.md b/data/en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant-0.md new file mode 100644 index 000000000..e81b3ee0d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant-0.md @@ -0,0 +1,14 @@ +--- +title: "List of shapes with known packing constant" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:58.046518+00:00" +instance: "kb-cron" +--- + +The packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown. The following is a list of bodies in Euclidean spaces whose packing constant is known. Fejes Tóth proved that in the plane, a point symmetric body has a packing constant that is equal to its translative packing constant and its lattice packing constant. Therefore, any such body for which the lattice packing constant was previously known, such as any ellipse, consequently has a known packing constant. In addition to these bodies, the packing constants of hyperspheres in 8 and 24 dimensions are almost exactly known. + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere-0.md b/data/en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere-0.md new file mode 100644 index 000000000..539aea702 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere-0.md @@ -0,0 +1,35 @@ +--- +title: "List of solids derived from the sphere" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:59.236190+00:00" +instance: "kb-cron" +--- + +This page lists solids derived from a sphere. + + +== Solids from cutting a sphere with one or more planes == +Dome +Spherical cap +Spherical sector +Spherical segment +Spherical shell +Spherical wedge + + +== Solids from deforming a sphere == +Ellipsoid +Spheroid +Solid bounded by Morin surface +Any Genus 0 surface + + +== Solids from intersecting a sphere with other solids or curved planes == +Reuleaux tetrahedron +Spherical lens + + +== Notes == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_spherical_symmetry_groups-0.md b/data/en.wikipedia.org/wiki/List_of_spherical_symmetry_groups-0.md new file mode 100644 index 000000000..b1792609c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_spherical_symmetry_groups-0.md @@ -0,0 +1,63 @@ +--- +title: "List of spherical symmetry groups" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:17:01.639486+00:00" +instance: "kb-cron" +--- + +Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. +This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway used a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. +Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6. + + +== Involutional symmetry == +There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and central point symmetry (Ci). + + +== Cyclic symmetry == +There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry) + + +== Dihedral symmetry == +There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case). + + +== Polyhedral symmetry == + +There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries. + + +== Continuous symmetries == +All of the discrete point symmetries are subgroups of certain continuous symmetries. They can be classified as products of orthogonal groups O(n) or special orthogonal groups SO(n). O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih1. SO(1) is just the identity. Half turns, C2, are needed to complete. + + +== See also == +Crystallographic point group +List of planar symmetry groups +Point groups in two dimensions +Triangle group + + +== References == + + +== Further reading == +Peter R. Cromwell, Polyhedra (1997), Appendix I +Sands, Donald E. (1993). "Crystal Systems and Geometry". Introduction to Crystallography. Mineola, New York: Dover Publications, Inc. p. 165. ISBN 0-486-67839-3. +On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5 +The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5 +Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] +(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] +(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591] +(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45] +N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, Table 11.4 Finite Groups of Isometries in 3-space + + +== External links == +Finite spherical symmetry groups +Weisstein, Eric W. "Schoenflies symbol". MathWorld. +Weisstein, Eric W. "Crystallographic point groups". MathWorld. +Simplest Canonical Polyhedra of Each Symmetry Type, by David I. McCooey \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_stochastic_processes_topics-0.md b/data/en.wikipedia.org/wiki/List_of_stochastic_processes_topics-0.md new file mode 100644 index 000000000..550637730 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_stochastic_processes_topics-0.md @@ -0,0 +1,89 @@ +--- +title: "List of stochastic processes topics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_stochastic_processes_topics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:43.740008+00:00" +instance: "kb-cron" +--- + +In the mathematics of probability, a stochastic process is a random function. In practical applications, the domain over which the function is defined is a time interval (time series) or a region of space (random field). +Familiar examples of time series include stock market and exchange rate fluctuations, signals such as speech, audio and video; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as Brownian motion or random walks. +Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogeneous material. + + +== Stochastic processes topics == +This list is currently incomplete. See also Category:Stochastic processes +Basic affine jump diffusion +Bernoulli process: discrete-time processes with two possible states. +Bernoulli schemes: discrete-time processes with N possible states; every stationary process in N outcomes is a Bernoulli scheme, and vice versa. +Bessel process +Birth–death process +Branching process +Branching random walk +Brownian bridge +Brownian motion +Chinese restaurant process +CIR process +Continuous stochastic process +Cox process +Dirichlet processes +Finite-dimensional distribution +First passage time +Galton–Watson process +Gamma process +Gaussian process – a process where all linear combinations of coordinates are normally distributed random variables. +Gauss–Markov process (cf. below) +GenI process +Girsanov's theorem +Hawkes process +Homogeneous processes: processes where the domain has some symmetry and the finite-dimensional probability distributions also have that symmetry. Special cases include stationary processes, also called time-homogeneous. +Karhunen–Loève theorem +Lévy process +Local time (mathematics) +Loop-erased random walk +Markov processes are those in which the future is conditionally independent of the past given the present. +Markov chain +Markov chain central limit theorem +Continuous-time Markov process +Markov process +Semi-Markov process +Gauss–Markov processes: processes that are both Gaussian and Markov +Martingales – processes with constraints on the expectation +Onsager–Machlup function +Ornstein–Uhlenbeck process +Percolation theory +Point processes: random arrangements of points in a space + + + + S + + + {\displaystyle S} + +. They can be modelled as stochastic processes where the domain is a sufficiently large family of subsets of S, ordered by inclusion; the range is the set of natural numbers; and, if A is a subset of B, ƒ(A) ≤ ƒ(B) with probability 1. +Poisson process +Compound Poisson process +Population process +Probabilistic cellular automaton +Queueing theory +Queue +Random field +Gaussian random field +Markov random field +Sample-continuous process +Stationary process +Stochastic calculus +Itô calculus +Malliavin calculus +Semimartingale +Stratonovich integral +Stochastic control +Stochastic differential equation +Stochastic process +Telegraph process +Time series +Wald's martingale +Wiener process \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_textbooks_in_thermodynamics_and_statistical_mechanics-0.md b/data/en.wikipedia.org/wiki/List_of_textbooks_in_thermodynamics_and_statistical_mechanics-0.md new file mode 100644 index 000000000..6be8213b4 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_textbooks_in_thermodynamics_and_statistical_mechanics-0.md @@ -0,0 +1,115 @@ +--- +title: "List of textbooks in thermodynamics and statistical mechanics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_textbooks_in_thermodynamics_and_statistical_mechanics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:17:02.992953+00:00" +instance: "kb-cron" +--- + +A list of notable textbooks in thermodynamics and statistical mechanics, arranged by category and date. + + +== Only or mainly thermodynamics == + +Fermi, Enrico (1956). Thermodynamics (New ed.). Dover Publications. ISBN 978-0486603612. {{cite book}}: ISBN / Date incompatibility (help) +Van Ness, H. C. (1983). Understanding Thermodynamics. Dover Publications. ISBN 978-0486632773. +Callen, Herbert (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8. +Finn, C. B. P. (1993). Thermal Physics (2nd ed.). United States of America: CRC Press. ISBN 0-7487-4379-0. +Zemansky, Mark W.; Dittman, Richard (1997). Heat and Thermodynamics: An Intermediate Textbook (7th ed.). McGraw-Hill. ISBN 978-0-070-17059-9. +Hanson, Robert M.; Green, Susan (2008). Introduction to Molecular Thermodynamics. University Science Books. ISBN 978-1891389498, 978-1-938787-63-8. +Pokrovskii, Vladimir (2020). Thermodynamics of Complex Systems: Principles and applications. IOP Publishing, Bristol, UK. ISBN 978-0-7503-3451-8. + + +== Both thermodynamics and statistical mechanics == +Reif, Frederick (1965). Fundamentals of Statistical and Thermal Physics. McGraw-Hill. ISBN 0-07-051800-9. +Sears, Francis W. (1975). Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. Addison Wesley. ISBN 020106894X. +Kittel, Charles (1969). Thermal Physics. Chichester: Wiley. ISBN 0-471-49030-X. 2e Kittel, Charles; and Kroemer, Herbert (1980) New York: W.H. Freeman ISBN 0-7167-1088-9 +Mandl, Franz (1971). Statistical physics. Chichester: Wiley. ISBN 0-471-56658-6. 2e (1988) Chichester: Wiley ISBN 0-471-91532-7, ISBN 0-471-91533-5. +Landsberg, P. T. (1978). Thermodynamics and statistical mechanics. Oxford University Press. ISBN 0-19-851142-6. (1990) New York: Dover ISBN 0-486-66493-7 +Stowe, Keith (1983). Introduction to Statistical Mechanics and Thermodynamics (1st ed.). John Wiley & Sons. ISBN 0-471-87058-7. +Waldram, J. R. (1985). The theory of thermodynamics. Cambridge: University Press. ISBN 0-521-28796-0. +Schroeder, Daniel (2000). An Introduction to Thermal Physics. United States of America: Addison Wesley Longman. ISBN 0-201-38027-7. +Blundell, Stephen; Blundell, Katherine (2006). Concepts in Thermal Physics. United Kingdom: Oxford University Press. ISBN 978-0-19-856769-1. +Gould, Harvey and Tobochnik, Jan (2010). Statistical and Thermal Physics. Princeton University Press. ISBN 978-0-691-13744-5.{{cite book}}: CS1 maint: multiple names: authors list (link) +Swendsen, Robert (2012). An Introduction to Statistical Mechanics and Thermodynamics. Oxford University Press. ISBN 978-0-19-964694-4. +Stephen G. Brush (1976) The Kind of Motion We Call Heat I-II North-Holland ISBN 0-444-87008-3 + + +== Statistical mechanics == +Fowler, R. H. (1929). Statistical mechanics : the theory of the properties of matter in equilibrium. Cambridge: University Press.. 2e (1936) Cambridge: University Press; (1980) Cambridge University Press. ISBN 0-521-09377-5 +Tolman, Richard C. (1938). The principles of statistical mechanics. Oxford: Clarendon Press.; (1979) New York: Dover ISBN 0-486-63896-0 +Landau, Lev Davidovich; and Lifshitz, Evgeny Mikhailovich (1969). Statistical Physics. Internet Archive.{{cite book}}: CS1 maint: multiple names: authors list (link) Vol. 5 of the Course of Theoretical Physics. 3e (1976) Translated by J.B. Sykes and M.J. Kearsley (1980) Oxford : Pergamon Press. ISBN 0-7506-3372-7 +ter Haar, Dirk (1954). Elements of statistical mechanics. London: Constable.. 3e (1995) Oxford: Butterworth-Heinemann ISBN 0-7506-2347-0 +Huang, Kerson (1963). Statistical mechanics. New York: Wiley. ISBN 0-471-41760-2. {{cite book}}: ISBN / Date incompatibility (help). 2e (1987) New York: Wiley ISBN 0-471-81518-7 +Kubo, Ryogo; et al. (1965). Statistical mechanics. Amsterdam: North-Holland. ISBN 0-444-10637-5.. 2e (1988) Amsterdam: North-Holland ISBN 0-444-87103-9. 2e (1991) Berlin: Springer Verlag ISBN 0-387-53662-0, ISBN 3-540-53662-0 +Penrose, Oliver (1970). Foundations of statistical mechanics : a deductive treatment. Oxford: Pergamon. ISBN 0-08-013314-2.; (2005) New York: Dover ISBN 0-486-43870-8 +McQuarrie, Donald A. (1975). Statistical mechanics. New York: Harper & Row. ISBN 0-06-044366-9. +2e (2000) Sausalito, Calif.: University Science ISBN 1-891389-15-7 +Reichl, Linda E (1980). A modern course in statistical physics. London: Edward Arnold. ISBN 0-7131-2777-5. +2e (1998) Chichester: Wiley ISBN 0-471-59520-9 +Ma, Shang-keng (1985). Statistical mechanics. Singapore: World Scientific. ISBN 9971-966-06-9. +Chandler, David (1987). Introduction to Modern Statistical Mechanics. Oxford University Press. ISBN 0-19-504277-8. +W.A. Wassam, Jr. (2002). Statistical Mechanics : Encyclopedia of Physical Science and Technology, Third Edition, Volume 15. Academic Press. ISBN 978-0-12-227410-7. +Bowley, Roger and Sanchez, Mariana (2000). Introductory Statistical Mechanics. Oxford University Press. ISBN 978-0-19-850576-1.{{cite book}}: CS1 maint: multiple names: authors list (link) +S. R. De Groot, P. Mazur (2011) Non-Equilibrium Thermodynamics, Dover Books on Physics, ISBN 978-0486647418. +Van Vliet, Carolyne M. (2008). Equilibrium and Non-equilibrium Statistical Mechanics. World Scientific Publishing Company. p. 982. ISBN 978-981-270-477-1. +Peliti, Luca (2011). Statistical Mechanics in a Nutshell. Princeton University Press. p. 417. ISBN 978-0-691-14529-7. +Pathria, P. K.; Beale, Paul (2021). Statistical Mechanics (4th ed.). United States: Elsevier/Academic Press. ISBN 978-0081026922. +Müller-Kirsten, Harald J.W. (2022). Basics of Statistical Physics, 3rd ed. World Scientific. ISBN 978-981-125-609-7. + + +== Specialized topics == + + +=== Kinetic theory === +Lifshitz, E. M. & Pitaevskii, L. P. (1981). Physical kinetics. Vol. 10 of the Course of Theoretical Physics (3rd Ed). Translated by J.B. Sykes and R.N. Franklin (1981) London: Pergamon ISBN 0-08-026480-8, ISBN 0-7506-2635-6 +Zubarev, D. N. (1974). Nonequilibrium Statistical Thermodynamics. New York: Consultants Bureau. ISBN 978-0-306-10895-2. +Zubarev, D. N.; Morozov V.; Ropke G. (1996). Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory. John Wiley & Sons. ISBN 3-05-501708-0. +Zubarev, D. N.; Morozov V.; Ropke G. (1997). Statistical Mechanics of Nonequilibrium Processes: Relaxation and Hydrodynamic Processes. John Wiley & Sons. ISBN 3-527-40084-2. + + +=== Quantum statistical mechanics === +Bogoliubov, N. N. (1967–1970). Lectures on Quantum Statistics. Problems of Statistical Mechanics of Quantum Systems. New York: Gordon and Breach. +Bogoliubov, N. N.; N. N. Bogolubov, Jnr. (1992). Introduction to Quantum Statistical Mechanics. New York: Gordon and Breach. ISBN 2-88124-879-9. + + +=== Mathematics of statistical mechanics === +Khinchin, Aleksandr Ya. (1943). Mathematical Foundations of Statistical Mechanics. Translated by G. Gamow (1949) New York: Dover ISBN 0-486-60147-1 +Ruelle, David (1969). Statistical Mechanics: Rigorous Results. New York: Benjamin. ISBN 9780805383607. ISBN 0-8053-8361-1. Reissued (1974), (1989); (1999) Singapore: World Scientific ISBN 981-02-3862-2 +Ruelle, David (1978). Thermodynamic formalism : the mathematical structures of classical equilibrium statistical mechanics. Addison-Wesley. ISBN 0-201-13504-3.; (1984) Cambridge: University Press ISBN 0-521-30225-0. 2e (2004) Cambridge: University Press ISBN 0-521-54649-4 +Minlos, Robert Adol'fovich (2000). Introduction to Mathematical Statistical Mechanics. Providence, RI: American Mathematical Society. ISBN 978-0-8218-1337-9. +Friedli, Sacha; Velenik, Yvan (2017). Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge: Cambridge University Press. ISBN 978-1-107-18482-4. + + +=== Miscellaneous === +Hoover, Wm. G. (1991). Computational Statistical Mechanics. Elsevier. ISBN 0-444-88192-1. +(available online here) +Sethna, James (2006). Statistical Mechanics: Entropy, Order Parameters, and Complexity. Oxford University Press. ISBN 0-19-856677-8. +Kardar, Mehran (2007). Statistical Physics of Fields. Cambridge University Press. ISBN 978-0-521-87341-3. + + +== Historical == +Boltzmann, Ludwig. Lectures on gas theory. (1896, 1898) Translated by Stephen G. Brush (1964) Berkeley: University of California Press; (1995) New York: Dover ISBN 0-486-68455-5 +Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics. New York: Charles Scribner's Sons. +Sommerfeld, Arnold; ed: F. Bopp, J. Meixner (1952). Thermodynamics and statistical mechanics.{{cite book}}: CS1 maint: multiple names: authors list (link) Translated by J. Kestin (1956) New York: Academic Press. +Ehrenfest, Paul and Tatiana (1912). The conceptual foundations of the statistical approach in mechanics. German Encyclopedia of Mathematical Sciences. Translated by Michael J. Moravcsik (1959) Ithaca: Cornell University Press; (1990) New York: Dover ISBN 0-486-66250-0 + + +== See also == + +List of textbooks on classical mechanics and quantum mechanics +List of textbooks in electromagnetism +List of books on general relativity + + +== Further reading == +Thorne, Kip S.; Blandford, Roger D. (2017). Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics. Princeton University Press. ISBN 978-0691159027. + + +== References == + + +== External links == +Statistical Mechanics and Thermodynamics Texts Clark University curriculum development project \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff-0.md b/data/en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff-0.md new file mode 100644 index 000000000..ed81a6131 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff-0.md @@ -0,0 +1,39 @@ +--- +title: "List of things named after Felix Hausdorff" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:33.394906+00:00" +instance: "kb-cron" +--- + +Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician. He made significant contributions to mathematics, specifically in the fields of set theory and topology. + + +== Mathematics == +Baker–Campbell–Hausdorff formula +Gromov–Hausdorff convergence +Hausdorff–Young inequality +Hausdorff completion +Hausdorff dimension +Hausdorff distance +Hausdorff gap +Hausdorff maximal principle +Hausdorff measure +Hausdorff Medal +Hausdorff moment problem +Hausdorff paradox +Hausdorff space + + +== Places == +Felix Hausdorff International Meeting Centre at the University of Greifswald +Hausdorff Center for Mathematics + + +== Space == +24947 Hausdorff + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_topics_related_to_π-0.md b/data/en.wikipedia.org/wiki/List_of_topics_related_to_π-0.md new file mode 100644 index 000000000..9eabb6bca --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_topics_related_to_π-0.md @@ -0,0 +1,60 @@ +--- +title: "List of topics related to π" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_topics_related_to_π" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:11.270763+00:00" +instance: "kb-cron" +--- + +This is a list of topics related to pi (π), the fundamental mathematical constant. + +2π theorem +Approximations of π +Arithmetic–geometric mean +Bailey–Borwein–Plouffe formula +Basel problem +Borwein's algorithm +Buffon's needle +Cadaeic Cadenza +Chronology of computation of π +Circle +Euler's identity +Six nines in pi +Gauss–Legendre algorithm +Gaussian function +History of π +A History of Pi +Indiana Pi Bill +Leibniz formula for pi +Lindemann–Weierstrass theorem (Proof that π is transcendental) +List of circle topics +List of formulae involving π +Liu Hui's π algorithm +Mathematical constant (sorted by continued fraction representation) +Mathematical constants and functions +Method of exhaustion +Milü +Pi +Pi (art project) +Pi (letter) +Pi Day +PiFast +PiHex +Pi in the Sky +Pilish +Pimania +Piphilology +Proof that π is irrational +Proof that 22/7 exceeds π +Proof of Wallis product +Rabbi Nehemiah +Radian +Ramanujan–Sato series +Rhind Mathematical Papyrus +Salamin–Brent algorithm +Software for calculating π +Squaring the circle +Turn (geometry) +Viète's formula \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_types_of_sets-0.md b/data/en.wikipedia.org/wiki/List_of_types_of_sets-0.md new file mode 100644 index 000000000..fd54a5139 --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_types_of_sets-0.md @@ -0,0 +1,82 @@ +--- +title: "List of types of sets" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/List_of_types_of_sets" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:56.774657+00:00" +instance: "kb-cron" +--- + +Sets can be classified according to the properties they have. + + +== Relative to set theory == +Empty set +Finite set, Infinite set +Countable set, Uncountable set +Power set + + +== Relative to a topology == +Closed set +Open set +Clopen set +Fσ set +Gδ set +Compact set +Relatively compact set +Regular open set, regular closed set +Connected set +Perfect set +Meagre set +Nowhere dense set + + +== Relative to a metric == +Bounded set +Totally bounded set + + +== Relative to measurability == +Borel set +Baire set +Measurable set, Non-measurable set +Universally measurable set + + +== Relative to a measure == +Negligible set +Null set +Haar null set + + +== In a linear space == +Convex set +Balanced set, Absolutely convex set + + +== Relative to the real/complex numbers == +Fractal set + + +== Ways of defining sets/Relation to descriptive set theory == +Recursive set +Recursively enumerable set +Arithmetical set +Diophantine set +Hyperarithmetical set +Analytical set +Analytic set, Coanalytic set +Suslin set +Projective set +Inhabited set + + +== More general objects still called sets == +Multiset + + +== See also == +List of set identities and relations – Equalities for combinations of sets +List of types of functions \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-0.md b/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-0.md new file mode 100644 index 000000000..ac9243d9b --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-0.md @@ -0,0 +1,19 @@ +--- +title: "List of works by Nicolas Minorsky" +chunk: 1/4 +source: "https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:47.573957+00:00" +instance: "kb-cron" +--- + +List of works by Nicolas Minorsky. + +== Books == +Minorsky, N. (1947). Introduction to non-linear mechanics: Topological methods, analytical methods, non-linear resonance, relaxation oscillations. J.W. Edwards. ASIN B0007DXRVY. +Minorsky, Nicolai (1955). Influence d'Henri Poincaré sur l'évolution moderne de la théorie des oscillations non linéaires [Influence of Henri Poincaré on the modern development of the theory of nonlinear oscillations]. Le livre du centenaire de la naissance de Henri Poincaré, 1854-1954 (in French). Paris, Gauthier-Villars. pp. 120–126. OCLC 10571426. +Minorsky, N. (1958). Dynamics and Nonlinear Mechanics: The Theory of Oscillations. Surveys in Applied Mathematics. Wiley. pp. 110–206. ASIN B0006AVKQW. +Minorsky, N. (1962). G. Szego; et al. (eds.). On some aspects of non-linear oscillations. Studies in Mathematical Analysis and Related Topics. Stanford Univ Pr (April 1962). ISBN 978-0804701402. {{cite book}}: ISBN / Date incompatibility (help) +Minorsky, N. (1967). Théorie des Oscillations [Theory of Oscillations]. Memorial des Sciences Mathematiques, Fascicule CLXIII (in French). ASIN B000LQD7KI. +Minorsky, Nicolai; Hetnarski, Richard B.; Mróz, Zenon (1967). Drgania nieliniowe [Nonlinear oscillations] (in Polish). Warszawa : Państwowe Wydawnictwo Naukowe. OCLC 749191659.{{cite book}}: CS1 maint: publisher location (link) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-1.md b/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-1.md new file mode 100644 index 000000000..12b8f603d --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-1.md @@ -0,0 +1,12 @@ +--- +title: "List of works by Nicolas Minorsky" +chunk: 2/4 +source: "https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:47.573957+00:00" +instance: "kb-cron" +--- + +== Papers == +Minorsky, N. (May 1922). "Directional stability of automatically steered bodies". Journal of the American Society of Naval Engineers. 34 (2): 280–309. doi:10.1111/j.1559-3584.1922.tb04958.x. Minorsky, N. (February 1927). "Phenomenon of direct-current self excitation in vacuum tubes circuits and in applications". J. Franklin Inst. 203 (2): 181–209. doi:10.1016/S0016-0032(27)92437-5. Minorsky, N. (April 1928). "La rotation de l'arc électrique dans champ magnétique radial" [The rotation of an arc in a radial magnetic field] (PDF). J. Phys. Radium (in French). 9 (4): 127–136. doi:10.1051/jphysrad:0192800904012700. Minorsky, N. (1928). "Mesure de la vitesse d'un aéronef par rapport au sol en l'absence supposée de tout repère extérieur" [Measuring the speed of an aircraft relative to the ground in the absence of any assumed approximate]. L'Aéronautique (in French). 113. Minorsky, N. (May 1930). "Automatic steering test". Journal of the American Society for Naval Engineers. 42 (2): 285–310. doi:10.1111/j.1559-3584.1930.tb05036.x. Minorsky, N. (June 1930). "Electronic conduction and ionization in crossed electric and magnetic fields". Journal of the Franklin Institute. 209 (6): 757–775. doi:10.1016/S0016-0032(30)91472-X. ISSN 0016-0032. Minorsky, N. (August 1934). "Ship stabilization by activated tank: An experimental investigation". The Engineer. 158: 154. Minorsky, N. (27 January 1936). "Une méthode d'intégration de quelques équations différentielles par un procédé électrique" [A method of integration of the differential equations by some electrical method]. C. R. Acad. Sci. (in French). 202: 293–295. Retrieved 8 July 2014. Minorsky, N. (30 May 1936). "Application des circuits électroniques à l'intégration graphique de quelques équations différentielles" [Application of the electronic circuits to the graphic integration of some differential equations]. Rev Gén Elec (in French). 34. Minorsky, N. (March 1937). "The principles and practice of automatic control". The Engineer. 9. Minorsky, N. (September 1941). "Note on the angular motion of ships". Journal of Applied Mechanics. 45. Minorsky, N. (November 1941). "Control Problems". Journal of the Franklin Institute. 232 (5): 451–487. doi:10.1016/S0016-0032(41)90069-8. Minorsky, N. (December 1941). "Control Problems (cont.)". Journal of the Franklin Institute. 232 (6): 519–551. doi:10.1016/S0016-0032(41)90178-3. Minorsky, N. (June 1942). "Self-excited in dynamical systems possessing retarded actions". J. Appl. Mech. 9. Minorsky, N. (15 October 1944). "On mechanical self-excited oscillations". Proceedings of the National Academy of Sciences of the United States of America. 30 (10): 308–314. Bibcode:1944PNAS...30..308M. doi:10.1073/pnas.30.10.308. PMC 1078718. PMID 16578134. Minorsky, N. (1 November 1945). "On non-linear phenomenon of self-rolling". Proceedings of the National Academy of Sciences of the United States of America. 31 (11): 346–349. Bibcode:1945PNAS...31..346M. doi:10.1073/pnas.31.11.346. PMC 1078842. PMID 16578177. Minorsky, N. (July 1945). "On parametric excitation". J. Franklin Inst. 240 (1): 25–46. doi:10.1016/0016-0032(45)90217-1. Minorsky, N. (February 1947). "A dynamical analogue". J. Franklin Inst. 243 (2): 131–149. doi:10.1016/0016-0032(74)90312-3. Minorsky, N. (October 1947). "Experiments with activated tanks". Trans. ASME. 69: 735. Minorsky, N. (April 1948). "Self-excited mechanical oscillations". Journal of Applied Physics. 19 (4): 332–338. Bibcode:1948JAP....19..332M. doi:10.1063/1.1715068. ISSN 0021-8979. Minorsky, N. (April 1948). "Sur une classe d'oscillations auto-entretenues" [On a class of self-sustained oscillations]. C. R. Acad. Sci. Mécanique (in French). 226. Minorsky, N. (January 1949). "Sur l'oscillateur de van der Pol" [On the van der Pol oscillator]. C. R. Acad. Sci. Physique mathématique (in French). 228. Minorsky, N. (September 1949). "Energy fluctuations in a van der Pol oscillator". J. Franklin Inst. 248 (3): 205–223. doi:10.1016/0016-0032(49)90210-0. Minorsky, N. (December 1950). "Meccanica non-lineare" [Non-Linear Mechanics]. Bollettino dell'Unione Matematica Italiana. 3 (in Italian). 5: 313–330. Retrieved 9 July 2014. Minorsky, N. (18 December 1950). "Sur l'excitation paramétrique" [On the parametric excitation]. C. R. Acad. Sci. Mécanique (in French). 231: 1417–1419. Retrieved 9 July 2014. Minorsky, N. (January 1951). "Parametric excitation". J. Appl. Phys. 22 (1): 49–54. Bibcode:1951JAP....22...49M. doi:10.1063/1.1699819. Minorsky, N. (1951). "Sur une équation différentielle de la physique" [On a differential equation of the physical]. C. R. Acad. Sci. Mécanique (in French). 232. Minorsky, N. (1 October 1951). "Sur le pendule entretenu par un courant alternatif" [On the clock maintained by an alternating current]. C. R. Acad. Sci. Mécanique (in French). 233: 728, 729. Retrieved 10 July 2014. Minorsky, N. (March 1951). "Modern nonlinear trends in engineering". Applied Mechanics Reviews. 4. ISSN 0003-6900. Minorsky, N. (June 1951). "Sur l'oscillateur non linéaire de Mathieu" [The Mathieu nonlinear oscillator]. C. R. Acad. Sci. (in French). 232. Minorsky, N. (January 1952). "Sur l'interaction des oscillations non linéaires" [On the interaction of nonlinear oscillations]. C. R. Acad. Sci. (in French). 234. Minorsky, N. (March 1952). "Sur les systèmes à l'action retardée le pendule entretenu par un courant alternatif" [Systems to share the delayed clock maintained by an alternating current]. C. R. Acad. Sci. Mécanique (in French). 234. Minorsky, N. (July 1952). "Stationary solutions of certain nonlinear differential equations". J. Franklin Inst. 254 (1): 21–42. doi:10.1016/0016-0032(52)90003-3. Minorsky, N. (September 1952). "Sur des systèmes oscillatoires contenant des paramètres à inertie" [On oscillatory systems containing inertia parameters]. C. R. Acad. Sci. (in French). 235. Minorsky, N. (August 1953). "On interaction of non-linear oscillations". J. Franklin Inst. 256 (2): 147–165. doi:10.1016/0016-0032(53)90941-7. Minorsky, N. (1953). "Sur l'extinction asynchrone". C. R. Acad. Sci. 237. Minorsky, N. (December 1953). "Sur quelques applications des équations differentielles aux différences" [On some applications of differential equations to differences]. Rendiconti del Seminario Matematico e Fisico di Milano (in French). 23 (1): 36–47. doi:10.1007/BF02922522. S2CID 124405883. Minorsky, N. (1953). "Sur les systèmes non linéaires à deux degrés de liberté". Rend. Semin. Mal. Torino. 13. Minorsky, N. (1953). "Sur l'interaction des oscillation non linéaires". Rendiconti del Seminario Matematico e Fisico di Milano. 25 (1): 145–163. doi:10.1007/BF02923816. S2CID 121303719. Minorsky, N. (1954). "La méthode stroboscopique et ses applications". Bull. Soc. Math. Fr. 13. Minorsky, N. (1954). "Sur les systèmes non linéaires à deux degrés de liberté". C. R. Acad. Sci. 238. Minorsky, N. (1955). "On synchronous actions". J. Franklin Inst. 259 (3): 209–219. doi:10.1016/0016-0032(55)90825-5. Minorsky, N. (1955). "Sur la méthode stroboscopique et ses applications". Proc. 8th Int. Congr. Theorel. Appl. Mech. Minorsky, N. (1955). "Sur la résonance non linéaires". C. R. Acad. Sci. 240. Minorsky, N. (1955). "Sur l'espace paramétrique de l'équation de M. Liénard". C. R. Acad. Sci. Mécanique. 240. Minorsky, N. (1957). "Structure topolgique de l'équation de M. Liénard" (PDF). J. Phys. Phys. Appl. 18 (s12): 121–130. doi:10.1051/jphysap:019570018012012100. Minorsky, N. (1958). "Equations différentielles - Sur l'excitation paramétrique". C. R. Acad. Sci. 247. Minorsky, N. (1958). "Sur la synchronisation". C. R. Acad. Sci. Mécanique. 247. Minorsky, N. (1959). "Sur les phénomènes paramétriques". Atti. Accad. Sci. Ist. Bologna, Cl. Sci. Fis. 247. Minorsky, N. (1959). "Sur les phénomènes paramétriques". Rend. Semin. Mat. Fis. Milano. 11. 6. Minorsky, N. (1959). "Sur l'action asynchrone". C. R. Acad. Sci. Mécanique. 248. Minorsky, N. (1960). "Sur l'interaction des oscillations non linéaires". C. R. Acad. Sci. Mécanique. 250. Minorsky, N. (1960). "Méthode stroboscopique et ses applications". Cah. Phys. 119. Minorsky, N. (1960). "Theoretical aspects of nonlinear oscillations". IRE Transactions on Circuit Theory. 7 (4): 368–381. doi:10.1109/TCT.1960.1086717. Minorsky, N. (1961). "Sur quelques aspects des oscillation non linéaires". C. R. Acad. Sci. 253. Minorsky, N. (1962). "Sur les oscillation quasi discontinues". C. R. Acad. Sci. 255. Minorsky, N. (1962). "Sur la résonance non linéaires". C. R. Acad. Sci. 254. Minorsky, N. (1962). "Sur la résonance non linéaires". C. R. Acad. Sci. 254. Minorsky, N. (1963). "Sur la méthode stroboscopique". C. R. Acad. Sci. 256. Minorsky, N. (1963). "Existence et stabilité de certaines solutions périodiques multiformes d'une équation de Duffing". C. R. Acad. Sci. 256. Minorsky, N. (1964). "Sur la synchronisation". C. R. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-2.md b/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-2.md new file mode 100644 index 000000000..ee86ed81c --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-2.md @@ -0,0 +1,11 @@ +--- +title: "List of works by Nicolas Minorsky" +chunk: 3/4 +source: "https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:47.573957+00:00" +instance: "kb-cron" +--- + +Acad. Sci. 259. Minorsky, N. (1965). "Sur les oscillations quasi discontinues". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur les phénomènes paramétriques". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur ll'interaction des oscillations non linéaires". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur les oscillations quasi discontinues". C. R. Acad. Sci. Mécanique. 261. Minorsky, N. (1966). "Sur la méthode stroboscopique". C. R. Acad. Sci. 263. Minorsky, N. (1967). "Comments "On asynchronous quenching"". IEEE Transactions on Automatic Control. 12 (2): 225–227. doi:10.1109/TAC.1967.1098559. Minorsky, N. (1968). "Sur les équations différentielles aux différences". C. R. Acad. Sci. 266. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-3.md b/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-3.md new file mode 100644 index 000000000..bb14da83e --- /dev/null +++ b/data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-3.md @@ -0,0 +1,40 @@ +--- +title: "List of works by Nicolas Minorsky" +chunk: 4/4 +source: "https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:47.573957+00:00" +instance: "kb-cron" +--- + +== Conferences == +Minorsky, N. (1948). Self-excited oscillations in systems possessing retarded actions. 7th Int. Congr. Appl. Math, London, England. +Minorsky, N. (1951). Sur le phénomène Béthenod. Actes Colloq. Int. Vibrations linéaires, Porqurolles, France. +Minorsky, N. (1952). Non-linear control systems. Conf. Automat. Contr., London:Butterworth. +Minorsky, N. (1952). Sur la méthode stroboscopique. Mem. lett. Accad. Sci. Ist., Bologna. +Minorsky, N. (1952). Sur quelques oscillatoires contenant les paramètres à inertie. Ann. Fac. Sci. Univ. d'Aix, Marseilles, France. +Minorsky, N. (1957). Nouvelles méthodes de la théorie des oscillations. Conf. Semin. Mat. Univ. Bari. +Minorsky, N. (1961). On synchronization. Int. Union Theor. Appl. Mech. Symp. Nonlinear Vibrations, Kiev, U.S.S.R. +Minorsky, N. (1963). On synchronization. Tr. Mizh. Symp. po Nelin. Kolebelina, vol. 1, Kiev, U.S.S.R. +Minorsky, N. (1964). Les vibrations forcées dans les systèmes non linéaires. Coll. Int. Centre Nat. Rech. Sci., no. 148, Marseilles, France. +Minorsky, N. (1965). Sur les systèmes non autonomes. Colloques Inl. Centre Nat. Rech. Sci., vol 148. + +== Patents == +US patent 1306552, N. Minorsky, "Gyrometer", published 1919-06-10, assigned to The Sperry Gyroscope company +US patent 1372184, N. Minorsky, "Angular-velocity-indicating apparatus", published 1921-03-22, assigned to N. Minorsky +US patent 1436280, N. Minorsky, "Automatic steering device", published 1922-11-21, assigned to N. Minorsky +US patent 1633822, N. Minorsky, "System of motor control", published 1927-06-28, assigned to The General Electric Company +US patent 1703280, N. Minorsky, "Directional stabilizer", published 1929-02-26, assigned to N. Minorsky +US patent 1703317, N. Minorsky, "Automatic steering device", published 1929-02-26, assigned to N. Minorsky +US patent 1840911, N. Minorsky, "Induction compass", published 1932-01-12, assigned to N. Minorsky +US patent 1853069, N. Minorsky, "Stabilizing apparatus", published 1932-04-12, assigned to N. Minorsky +US patent RE19038, N. Minorsky, "Induction compass", published 1934-01-02, assigned to Bendix Aviation Corporation +US patent 1950946, N. Minorsky, "Navigational instrument", published 1934-03-13, assigned to Pioneer Instr. Co. Inc. +US patent 2017072, N. Minorsky, "Stabilizing apparatus", published 1935-10-15, assigned to N. Minorsky +US patent 2202162, N. Minorsky, "Antirolling stabilization of ships", published 1940-05-28, assigned to N. Minorsky +US patent 2071759, N. Minorsky, "Electron discharge tube system", published 1937-02-23, assigned to RCA Corp. +US patent 2449563, N. Minorsky, "Balancing machine", published 1948-09-21, assigned to Gyro Balance Corp. +A US patent 2590029 A, N. Minorsky, "Torque amplifying system", published 1952-03-18, assigned to Lear Inc. + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Outline_of_logic-0.md b/data/en.wikipedia.org/wiki/Outline_of_logic-0.md new file mode 100644 index 000000000..c42a69e5e --- /dev/null +++ b/data/en.wikipedia.org/wiki/Outline_of_logic-0.md @@ -0,0 +1,385 @@ +--- +title: "Outline of logic" +chunk: 1/3 +source: "https://en.wikipedia.org/wiki/Outline_of_logic" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:37.128800+00:00" +instance: "kb-cron" +--- + +Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct (or valid) and incorrect (or fallacious) inferences. Logicians study the criteria for the evaluation of arguments. + +== Foundations of logic == +Philosophy of logic + +Analytic-synthetic distinction +Antinomy +A priori and a posteriori +Definition +Description +Entailment +Identity (philosophy) +Inference +Logical form +Logical implication +Logical truth +Logical consequence +Name +Necessity +Material conditional +Meaning (linguistic) +Meaning (non-linguistic) +Paradox (list) +Possible world +Presupposition +Probability +Quantification +Reason +Reasoning +Reference +Semantics +Strict conditional +Syntax (logic) +Truth +Truth value +Validity + +== Branches of logic == +Affine logic +Alethic logic +Aristotelian logic +Boolean logic +Buddhist logic +Bunched logic +Categorical logic +Classical logic +Computability logic +Deontic logic +Dependence logic +Description logic +Deviant logic +Doxastic logic +Epistemic logic +First-order logic +Formal logic +Free logic +Fuzzy logic +Higher-order logic +Infinitary logic +Informal logic +Intensional logic +Intermediate logic +Interpretability logic +Intuitionistic logic +Linear logic +Many-valued logic +Mathematical logic +Metalogic +Minimal logic +Modal logic +Non-Aristotelian logic +Non-classical logic +Noncommutative logic +Non-monotonic logic +Ordered logic +Paraconsistent logic +Philosophical logic +Predicate logic +Propositional logic +Provability logic +Quantum logic +Relevance logic +Sequential logic +Spatial logic +Strict logic +Substructural logic +Syllogistic logic +Symbolic logic +Temporal logic +Term logic +Topical logic +Traditional logic +Zeroth-order logic + +== Philosophical logic == + +=== Informal logic and critical thinking === +Informal logic +Critical thinking +Argumentation theory + +Argument +Argument map +Accuracy and precision +Ad hoc hypothesis +Ambiguity +Analysis +Attacking Faulty Reasoning +Belief +Belief bias +Bias +Cognitive bias +Confirmation bias +Credibility +Critical reading +Critical thinking +Decidophobia +Decision making +Dispositional and occurrent belief +Emotional reasoning +Evidence +Expert +Explanation +Explanatory power +Fact +Fallacy +Higher-order thinking +Inquiry +Interpretive discussion +Occam's razor +Opinion +Practical syllogism +Precision questioning +Propaganda +Propaganda techniques +Problem Solving +Prudence +Pseudophilosophy +Reasoning +Relevance +Rhetoric +Rigour +Socratic questioning +Source credibility +Source criticism +Theory of justification +Topical logic +Vagueness + +=== Deductive reasoning === + +==== Theories of deduction ==== +Anti-psychologism +Conceptualism +Constructivism +Conventionalism +Counterpart theory +Deflationary theory of truth +Dialetheism +Fictionalism +Formalism (philosophy) +Game theory +Illuminationist philosophy +Logical atomism +Logical holism +Logicism +Modal fictionalism +Nominalism +Polylogism +Pragmatism +Preintuitionism +Proof theory +Psychologism +Ramism +Semantic theory of truth +Sophism +Trivialism +Ultrafinitism + +=== Fallacies === +Fallacy (list) – incorrect argumentation in reasoning resulting in a misconception or presumption. By accident or design, fallacies may exploit emotional triggers in the listener or interlocutor (appeal to emotion), or take advantage of social relationships between people (e.g. argument from authority). Fallacious arguments are often structured using rhetorical patterns that obscure any logical argument. Fallacies can be used to win arguments regardless of the merits. There are dozens of types of fallacies. + +== Formal logic == +Formal logic – Mathematical logic, symbolic logic and formal logic are largely, if not completely synonymous. The essential feature of this field is the use of formal languages to express the ideas whose logical validity is being studied. +List of mathematical logic topics + +=== Symbols and strings of symbols === + +==== Logical symbols ==== + +Logical variables +Propositional variable +Predicate variable +Literal +Metavariable +Logical constants +Logical connective +Quantifier +Identity +Brackets + +===== Logical connectives ===== +Logical connective + +Converse implication +Converse nonimplication +Exclusive or +Logical NOR +Logical biconditional +Logical conjunction +Logical disjunction +Material implication +Material nonimplication +Negation +Sheffer stroke + +==== Strings of symbols ==== + +Atomic formula +Open sentence + +==== Types of propositions ==== +Proposition + +Analytic proposition +Axiom +Atomic sentence +Clause (logic) +Contingent proposition +Contradiction +Logical truth +Propositional formula +Rule of inference +Sentence (mathematical logic) +Sequent +Statement (logic) +Subalternation +Tautology +Theorem + +===== Rules of inference ===== +Rule of inference (list) + +Biconditional elimination +Biconditional introduction +Case analysis +Commutativity of conjunction +Conjunction introduction +Constructive dilemma +Contraposition (traditional logic) +Conversion (logic) +De Morgan's laws +Destructive dilemma +Disjunction elimination +Disjunction introduction +Disjunctive syllogism +Double negation elimination +Generalization (logic) +Hypothetical syllogism +Law of excluded middle +Law of identity +Modus ponendo tollens +Modus ponens +Modus tollens +Obversion +Principle of contradiction +Resolution (logic) +Simplification +Transposition (logic) + +==== Formal theories ==== + +Formal proof +List of first-order theories + +==== Expressions in a metalanguage ==== +Metalanguage + +Metalinguistic variable +Deductive system +Metatheorem +Metatheory +Interpretation + +=== Propositional and Boolean logic === + +==== Propositional logic ==== + +Absorption law +Clause (logic) +Deductive closure +Distributive property +Entailment +Formation rule +Functional completeness +Intermediate logic +Literal (mathematical logic) +Logical connective +Logical consequence +Negation normal form +Open sentence +Propositional calculus +Propositional formula +Propositional variable +Rule of inference +Strict conditional +Substitution instance +Truth table +Zeroth-order logic + +==== Boolean logic ==== +Boolean algebra (list) +Boolean logic +Boolean algebra (structure) +Boolean algebras canonically defined +Introduction to Boolean algebra +Complete Boolean algebra +Free Boolean algebra +Monadic Boolean algebra +Residuated Boolean algebra +Two-element Boolean algebra +Modal algebra +Derivative algebra (abstract algebra) +Relation algebra +Absorption law +Laws of Form +De Morgan's laws +Algebraic normal form +Canonical form (Boolean algebra) +Boolean conjunctive query +Boolean-valued model +Boolean domain +Boolean expression +Boolean ring +Boolean function +Boolean-valued function +Parity function +Symmetric Boolean function +Conditioned disjunction +Field of sets +Functional completeness +Implicant +Logic alphabet +Logic redundancy +Logical connective +Logical matrix +Product term +True quantified Boolean formula +Truth table + +=== Predicate logic and relations === + +==== Predicate logic ==== + +Atomic formula +Atomic sentence +Domain of discourse +Empty domain +Extension (predicate logic) +First-order logic +First-order predicate +Formation rule +Free variables and bound variables +Generalization (logic) +Monadic predicate calculus +Predicate (mathematical logic) +Predicate logic +Predicate variable +Quantification +Second-order predicate +Sentence (mathematical logic) +Universal instantiation + +==== Relations ==== +Mathematical relation \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Outline_of_logic-1.md b/data/en.wikipedia.org/wiki/Outline_of_logic-1.md new file mode 100644 index 000000000..aee8ba74d --- /dev/null +++ b/data/en.wikipedia.org/wiki/Outline_of_logic-1.md @@ -0,0 +1,434 @@ +--- +title: "Outline of logic" +chunk: 2/3 +source: "https://en.wikipedia.org/wiki/Outline_of_logic" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:37.128800+00:00" +instance: "kb-cron" +--- + +Finitary relation +Antisymmetric relation +Asymmetric relation +Bijection +Bijection, injection and surjection +Binary relation +Composition of relations +Congruence relation +Connected relation +Converse relation +Coreflexive relation +Covering relation +Cyclic order +Dense relation +Dependence relation +Dependency relation +Directed set +Equivalence relation +Euclidean relation +Homogeneous relation +Idempotence +Intransitivity +Involutive relation +Partial equivalence relation +Partial function +Partially ordered set +Preorder +Prewellordering +Propositional function +Quasitransitive relation +Reflexive relation +Serial relation +Surjective function +Symmetric relation +Ternary relation +Transitive relation +Trichotomy (mathematics) +Well-founded relation + +== Mathematical logic == +Mathematical logic + +=== Set theory === +Set theory list +Aleph null +Bijection, injection and surjection +Binary set +Cantor's diagonal argument +Cantor's first uncountability proof +Cantor's theorem +Cardinality of the continuum +Cardinal number +Codomain +Complement (set theory) +Constructible universe +Continuum hypothesis +Countable set +Decidable set +Denumerable set +Disjoint sets +Disjoint union +Domain of a function +Effective enumeration +Element (mathematics) +Empty function +Empty set +Enumeration +Extensionality +Finite set +Forcing (mathematics) +Function (set theory) +Function composition +Generalized continuum hypothesis +Index set +Infinite set +Intension +Intersection (set theory) +Inverse function +Large cardinal +Löwenheim–Skolem theorem +Map (mathematics) +Multiset +Morse–Kelley set theory +Naïve set theory +One-to-one correspondence +Ordered pair +Partition of a set +Pointed set +Power set +Projection (set theory) +Proper subset +Proper superset +Range of a function +Russell's paradox +Sequence (mathematics) +Set (mathematics) +Set of all sets +Simple theorems in the algebra of sets +Singleton (mathematics) +Skolem paradox +Subset +Superset +Tuple +Uncountable set +Union (set theory) +Von Neumann–Bernays–Gödel set theory +Zermelo set theory +Zermelo–Fraenkel set theory + +=== Metalogic === +Metalogic – The study of the metatheory of logic. + +Completeness (logic) +Syntax (logic) +Consistency +Decidability (logic) +Deductive system +Interpretation (logic) +Cantor's theorem +Church's theorem +Church's thesis +Effective method +Formal system +Gödel's completeness theorem +Gödel's first incompleteness theorem +Gödel's second incompleteness theorem +Independence (mathematical logic) +Logical consequence +Löwenheim–Skolem theorem +Metalanguage +Metasyntactic variable +Metatheorem +Object language – see metalanguage +Symbol (formal) +Type–token distinction +Use–mention distinction +Well-formed formula + +==== Proof theory ==== +Proof theory – The study of deductive systems. + +Axiom +Deductive system +Formal proof +Formal system +Formal theorem +Syntactic consequence +Syntax (logic) +Transformation rules + +==== Model theory ==== +Model theory – The study of interpretation of formal systems. + +Interpretation (logic) +Logical validity +Non-standard model +Normal model +Model +Semantic consequence +Truth value + +=== Computability theory === +Computability theory – branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. The basic questions addressed by recursion theory are "What does it mean for a function from the natural numbers to themselves to be computable?" and "How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?". The answers to these questions have led to a rich theory that is still being actively researched. + +Alpha recursion theory +Arithmetical set +Church–Turing thesis +Computability logic +Computable function +Computation +Decision problem +Effective method +Entscheidungsproblem +Enumeration +Forcing (computability) +Halting problem +History of the Church–Turing thesis +Lambda calculus +List of undecidable problems +Post correspondence problem +Post's theorem +Primitive recursive function +Recursion (computer science) +Recursive language +Recursive set +Recursively enumerable language +Recursively enumerable set +Reduction (recursion theory) +Turing machine + +== Semantics of natural language == +Formal semantics (natural language) + +Formal systems +Alternative semantics +Categorial grammar +Combinatory categorial grammar +Discourse representation theory +Dynamic semantics +Inquisitive semantics +Montague grammar +Situation semantics +Concepts +Compositionality +Counterfactuals +Generalized quantifier +Logic translation +Mereology +Modality (natural language) +Opaque context +Presupposition +Propositional attitudes +Scope (formal semantics) +Type shifter +Vagueness + +== Classical logic == +Classical logic + +Properties of classical logics: +Law of the excluded middle +Double negation elimination +Law of noncontradiction +Principle of explosion +Monotonicity of entailment +Idempotency of entailment +Commutativity of conjunction +De Morgan duality – every logical operator is dual to another +Term logic +Outline of logic +Baralipton +Baroco +Bivalence +Boolean logic +Boolean-valued function +Categorical proposition +Distribution of terms +End term +Enthymeme +Immediate inference +Law of contraries +Logical connective +Logical cube +Logical hexagon +Major term +Middle term +Minor term +Octagon of Prophecies +Organon +Polysyllogism +Port-Royal Logic +Premise +Prior Analytics +Absolute and relative terms +Sorites paradox +Square of opposition +Triangle of opposition +Sum of Logic +Syllogism +Tetralemma +Truth function + +=== Modal logic === +Modal logic + +Alethic logic +Deontic logic +Doxastic logic +Epistemic logic +Temporal logic + +== Non-classical logic == +Non-classical logic + +Affine logic +Bunched logic +Computability logic +Decision theory +Description logic +Deviant logic +Free logic +Fuzzy logic +Game theory +Intensional logic +Intuitionistic logic +Linear logic +Many-valued logic +Minimal logic +Non-monotonic logic +Noncommutative logic +Paraconsistent logic +Probability theory +Quantum logic +Relevance logic +Strict conditional +Substructural logic + +== Concepts of logic == +Deductive reasoning +Inductive reasoning +Abductive reasoning +Mathematical logic + +Proof theory +Set theory +Formal system +Predicate logic +Predicate +Higher-order logic +Propositional calculus +Proposition +Boolean algebra +Boolean logic +Truth value +Venn diagram +Peirce's law +Aristotelian logic +Non-Aristotelian logic +Informal logic +Fuzzy logic +Infinitary logic +Infinity +Categorical logic +Linear logic +Metalogic +Order +Ordered logic +Temporal logic +Linear temporal logic +Linear temporal logic to Büchi automaton +Sequential logic +Provability logic +Interpretability logic +Interpretability +Quantum logic +Relevant logic +Consequent +Affirming the consequent +Antecedent +Denying the antecedent +Theorem +Axiom +Axiomatic system +Axiomatization +Conditional proof +Invalid proof +Degree of truth +Truth +Truth condition +Truth function +Double negation +Double negation elimination +Fallacy +Existential fallacy +Logical fallacy +Syllogistic fallacy +Type theory +Game theory +Game semantics +Rule of inference +Inference procedure +Inference rule +Introduction rule +Law of excluded middle +Law of non-contradiction +Logical constant +Logical connective +Quantifier +Logic gate +Boolean Function +Quantum logic gate +Tautology +Logical assertion +Logical conditional +Logical biconditional +Logical equivalence +Logical AND +Negation +Logical OR +Logical NAND +Logical NOR +Contradiction +Subalternation +Logicism +Polysyllogism +Syllogism +Hypothetical syllogism +Major premise +Minor premise +Term +Singular term +Major term +Middle term +Quantification +Plural quantification +Logical argument +Validity +Soundness +Inverse (logic) +Non sequitur +Tolerance +Satisfiability +Logical language +Paradox +Polish notation +Principia Mathematica +Quod erat demonstrandum +Reductio ad absurdum +Rhetoric +Self-reference +Necessary and sufficient +Sufficient condition +Nonfirstorderizability +Occam's Razor +Socratic dialogue +Socratic method +Argument form +Logic programming +Unification + +== History of logic == +History of logic + +== Literature about logic == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Outline_of_logic-2.md b/data/en.wikipedia.org/wiki/Outline_of_logic-2.md new file mode 100644 index 000000000..e8c920e89 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Outline_of_logic-2.md @@ -0,0 +1,74 @@ +--- +title: "Outline of logic" +chunk: 3/3 +source: "https://en.wikipedia.org/wiki/Outline_of_logic" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:37.128800+00:00" +instance: "kb-cron" +--- + +=== Journals === +Journal of Logic, Language and Information +Journal of Philosophical Logic +Linguistics and Philosophy + +=== Books === +A System of Logic +Attacking Faulty Reasoning +Begriffsschrift +Categories (Aristotle) +Charles Sanders Peirce bibliography +De Interpretatione +Gödel, Escher, Bach +Introduction to Mathematical Philosophy +Language, Truth, and Logic +Laws of Form +Novum Organum +On Formally Undecidable Propositions of Principia Mathematica and Related Systems +Organon +Philosophy of Arithmetic +Polish Logic +Port-Royal Logic +Posterior Analytics +Principia Mathematica +Principles of Mathematical Logic +Prior Analytics +Rhetoric (Aristotle) +Sophistical Refutations +Sum of Logic +The Art of Being Right +The Foundations of Arithmetic +Topics (Aristotle) +Tractatus Logico-Philosophicus + +== Logic organizations == +Association for Symbolic Logic + +== Logicians == +List of logicians +List of philosophers of language + +== See also == + +Glossary of logic +Index of logic articles +Mathematics +List of basic mathematics topics +List of mathematics articles +Philosophy +List of basic philosophy topics +List of philosophy topics +Outline of philosophy +Outline of discrete mathematics – for introductory set theory and other supporting material + +== References == + +== External links == + +Taxonomy of Logical Fallacies +forall x: an introduction to formal logic, by P.D. Magnus, covers sentential and quantified logic +Translation Tips, by Peter Suber, for translating from English into logical notation +Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. Archived 2009-09-08 at the Wayback Machine In The Dictionary of the History of Ideas. +Logic test Test your logic skills +Logic Self-Taught: A Workbook (originally prepared for on-line logic instruction) \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Probability_and_statistics-0.md b/data/en.wikipedia.org/wiki/Probability_and_statistics-0.md new file mode 100644 index 000000000..68006826a --- /dev/null +++ b/data/en.wikipedia.org/wiki/Probability_and_statistics-0.md @@ -0,0 +1,26 @@ +--- +title: "Probability and statistics" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Probability_and_statistics" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:27.641847+00:00" +instance: "kb-cron" +--- + +Probability and statistics are two closely related fields in mathematics that are sometimes combined for academic purposes. They are covered in multiple articles and lists: + +Probability +Statistics +Glossary of probability and statistics +Notation in probability and statistics +Timeline of probability and statistics +Publications named for both fields include the following: + +Brazilian Journal of Probability and Statistics +Counterexamples in Probability and Statistics +Probability and Mathematical Statistics +Theory of Probability and Mathematical Statistics + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Table_of_Lie_groups-0.md b/data/en.wikipedia.org/wiki/Table_of_Lie_groups-0.md new file mode 100644 index 000000000..22d7a4846 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Table_of_Lie_groups-0.md @@ -0,0 +1,74 @@ +--- +title: "Table of Lie groups" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Table_of_Lie_groups" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:27.102675+00:00" +instance: "kb-cron" +--- + +This article gives a table of some common Lie groups and their associated Lie algebras. +The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). +For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; see classification of low-dimensional real Lie algebras for up to four dimensions; and the list of Lie group topics. + + +== Real Lie groups and their algebras == +Column legend + +Cpt: Is this group G compact? (Yes or No) + + + + + + π + + 0 + + + + + {\displaystyle \pi _{0}} + +: Gives the group of components of G. The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0). + + + + + + π + + 1 + + + + + {\displaystyle \pi _{1}} + +: Gives the fundamental group of G whenever G is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0). +UC: If G is not simply connected, gives the universal cover of G. + + +== Real Lie algebras == + + +== Complex Lie groups and their algebras == + +Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension. + + +== Complex Lie algebras == + +The dimensions given are dimensions over C. Note that every complex Lie algebra can also be viewed as a real Lie algebra of twice the dimension. + +The Lie algebra of affine transformations of dimension two, in fact, exist for any field. An instance has already been listed in the first table for real Lie algebras. + + +== See also == +Classification of low-dimensional real Lie algebras +Simple Lie group#Full classification + + +== References == +Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Table_of_Newtonian_series-0.md b/data/en.wikipedia.org/wiki/Table_of_Newtonian_series-0.md new file mode 100644 index 000000000..c3b46cba8 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Table_of_Newtonian_series-0.md @@ -0,0 +1,1489 @@ +--- +title: "Table of Newtonian series" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Table_of_Newtonian_series" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:15:50.124582+00:00" +instance: "kb-cron" +--- + +In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence + + + + + a + + n + + + + + {\displaystyle a_{n}} + + written in the form + + + + + f + ( + s + ) + = + + ∑ + + n + = + 0 + + + ∞ + + + ( + − + 1 + + ) + + n + + + + + + ( + + + s + n + + + ) + + + + + a + + n + + + = + + ∑ + + n + = + 0 + + + ∞ + + + + + + ( + − + s + + ) + + n + + + + + n + ! + + + + + a + + n + + + + + {\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n}}{n!}}a_{n}} + + +where + + + + + + + + ( + + + s + n + + + ) + + + + + + {\displaystyle {s \choose n}} + + +is the binomial coefficient and + + + + ( + s + + ) + + n + + + + + {\displaystyle (s)_{n}} + + is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus. + + +== List == +The generalized binomial theorem gives + + + + + ( + 1 + + + z + + ) + + s + + + = + + ∑ + + n + = + 0 + + + ∞ + + + + + + ( + + + s + n + + + ) + + + + + z + + n + + + = + 1 + + + + + + ( + + + s + 1 + + + ) + + + + z + + + + + + ( + + + s + 2 + + + ) + + + + + z + + 2 + + + + + ⋯ + . + + + {\displaystyle (1+z)^{s}=\sum _{n=0}^{\infty }{s \choose n}z^{n}=1+{s \choose 1}z+{s \choose 2}z^{2}+\cdots .} + + +A proof for this identity can be obtained by showing that it satisfies the differential equation + + + + + ( + 1 + + + z + ) + + + + d + ( + 1 + + + z + + ) + + s + + + + + d + z + + + + = + s + ( + 1 + + + z + + ) + + s + + + . + + + {\displaystyle (1+z){\frac {d(1+z)^{s}}{dz}}=s(1+z)^{s}.} + + +The + + + + log + + + {\displaystyle \log } + + of the gamma function, and its derivative the digamma function, can both have Newtonian series found by taking their binomial transform as sequences over the integers: + + + + + + + + + log + ⁡ + ( + Γ + ( + s + + + 1 + ) + ) + + + + = + + ∑ + + n + = + 1 + + + ∞ + + + + + + ( + + + s + n + + + ) + + + + + ∑ + + k + = + 1 + + + n + + + ( + − + 1 + + ) + + n + − + k + + + + + + ( + + + + n + − + 1 + + + k + − + 1 + + + + ) + + + + log + ⁡ + ( + k + ) + + + + + ψ + ( + s + + + 1 + ) + + + γ + = + + H + + s + + + + + + = + + ∑ + + n + = + 1 + + + ∞ + + + + + + ( + + + s + n + + + ) + + + + + + + ( + − + 1 + + ) + + n + − + 1 + + + + n + + + + + + + + + {\displaystyle {\begin{aligned}\log(\Gamma (s+1))&=\sum _{n=1}^{\infty }{s \choose n}\sum _{k=1}^{n}(-1)^{n-k}{n-1 \choose k-1}\log(k)\\\psi (s+1)+\gamma =H_{s}&=\sum _{n=1}^{\infty }{s \choose n}{\frac {(-1)^{n-1}}{n}}\end{aligned}}} + + +These are both valid in the right half-plane + + + + ℜ + ( + s + ) + > + 0 + + + {\displaystyle \Re (s)>0} + +, as proven by Charles Hermite in 1900 and Moritz Abraham Stern in 1847 (see Digamma function#Newton series) respectively. +The Stirling numbers of the second kind are given by the finite sum + + + + + + { + + + + + n + + + + + k + + + + + } + + = + + + 1 + + k + ! + + + + + ∑ + + j + = + 0 + + + k + + + ( + − + 1 + + ) + + k + − + j + + + + + + ( + + + k + j + + + ) + + + + + j + + n + + + . + + + {\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}={\frac {1}{k!}}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}j^{n}.} + + +This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0: + + + + + + Δ + + k + + + + x + + n + + + = + + ∑ + + j + = + 0 + + + k + + + ( + − + 1 + + ) + + k + − + j + + + + + + ( + + + k + j + + + ) + + + + ( + x + + + j + + ) + + n + + + . + + + {\displaystyle \Delta ^{k}x^{n}=\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}(x+j)^{n}.} + + +A related identity forms the basis of the Nörlund–Rice integral: + + + + + + ∑ + + k + = + 0 + + + n + + + + + + ( + + + n + k + + + ) + + + + + + + ( + − + 1 + + ) + + n + − + k + + + + + s + − + k + + + + = + + + + n + ! + + + s + ( + s + − + 1 + ) + ( + s + − + 2 + ) + ⋯ + ( + s + − + n + ) + + + + = + + + + Γ + ( + n + + + 1 + ) + Γ + ( + s + − + n + ) + + + Γ + ( + s + + + 1 + ) + + + + = + B + ( + n + + + 1 + , + s + − + n + ) + , + s + ∉ + { + 0 + , + … + , + n + } + + + {\displaystyle \sum _{k=0}^{n}{n \choose k}{\frac {(-1)^{n-k}}{s-k}}={\frac {n!}{s(s-1)(s-2)\cdots (s-n)}}={\frac {\Gamma (n+1)\Gamma (s-n)}{\Gamma (s+1)}}=B(n+1,s-n),s\notin \{0,\ldots ,n\}} + + +where + + + + Γ + ( + x + ) + + + {\displaystyle \Gamma (x)} + + is the Gamma function and + + + + B + ( + x + , + y + ) + + + {\displaystyle B(x,y)} + + is the Beta function. +The trigonometric functions have umbral identities: + + + + + + ∑ + + n + = + 0 + + + ∞ + + + ( + − + 1 + + ) + + n + + + + + + ( + + + s + + 2 + n + + + + ) + + + + = + + 2 + + s + + / + + 2 + + + cos + ⁡ + + + + π + s + + 4 + + + + + {\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n}=2^{s/2}\cos {\frac {\pi s}{4}}} + + +and + + + + + + ∑ + + n + = + 0 + + + ∞ + + + ( + − + 1 + + ) + + n + + + + + + ( + + + s + + 2 + n + + + 1 + + + + ) + + + + = + + 2 + + s + + / + + 2 + + + sin + ⁡ + + + + π + s + + 4 + + + + + {\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n+1}=2^{s/2}\sin {\frac {\pi s}{4}}} + + +The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial + + + + ( + s + + ) + + n + + + + + {\displaystyle (s)_{n}} + +. The first few terms of the sin series are + + + + + s + − + + + + ( + s + + ) + + 3 + + + + + 3 + ! + + + + + + + + + ( + s + + ) + + 5 + + + + + 5 + ! + + + + − + + + + ( + s + + ) + + 7 + + + + + 7 + ! + + + + + + ⋯ + + + {\displaystyle s-{\frac {(s)_{3}}{3!}}+{\frac {(s)_{5}}{5!}}-{\frac {(s)_{7}}{7!}}+\cdots } + + +which can be recognized as resembling the Taylor series for sin x, with (s)n standing in the place of xn. +In analytic number theory it is of interest to sum + + + + + + + ∑ + + k + = + 0 + + + + B + + k + + + + z + + k + + + , + + + {\displaystyle \!\sum _{k=0}B_{k}z^{k},} + + +where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as + + + + + + ∑ + + k + = + 0 + + + + B + + k + + + + z + + k + + + = + + ∫ + + 0 + + + ∞ + + + + e + + − + t + + + + + + t + z + + + + e + + t + z + + + − + 1 + + + + + d + t + = + + ∑ + + k + = + 1 + + + + + z + + ( + k + z + + + 1 + + ) + + 2 + + + + + + . + + + {\displaystyle \sum _{k=0}B_{k}z^{k}=\int _{0}^{\infty }e^{-t}{\frac {tz}{e^{tz}-1}}\,dt=\sum _{k=1}{\frac {z}{(kz+1)^{2}}}.} + + +The general relation gives the Newton series + + + + + + ∑ + + k + = + 0 + + + + + + + B + + k + + + ( + x + ) + + + z + + k + + + + + + + + + ( + + + + 1 + − + s + + k + + + ) + + + + s + − + 1 + + + + = + + z + + s + − + 1 + + + ζ + ( + s + , + x + + + z + ) + , + + + {\displaystyle \sum _{k=0}{\frac {B_{k}(x)}{z^{k}}}{\frac {1-s \choose k}{s-1}}=z^{s-1}\zeta (s,x+z),} + + +where + + + + ζ + + + {\displaystyle \zeta } + + is the Hurwitz zeta function and + + + + + B + + k + + + ( + x + ) + + + {\displaystyle B_{k}(x)} + + the Bernoulli polynomial. The series does not converge, the identity holds formally. +Another identity is + + + + + + + 1 + + Γ + ( + x + ) + + + + = + + ∑ + + k + = + 0 + + + ∞ + + + + + + ( + + + + x + − + a + + k + + + ) + + + + + ∑ + + j + = + 0 + + + k + + + + + + ( + − + 1 + + ) + + k + − + j + + + + + Γ + ( + a + + + j + ) + + + + + + + ( + + + k + j + + + ) + + + + , + + + {\displaystyle {\frac {1}{\Gamma (x)}}=\sum _{k=0}^{\infty }{x-a \choose k}\sum _{j=0}^{k}{\frac {(-1)^{k-j}}{\Gamma (a+j)}}{k \choose j},} + + +which converges for + + + + x + > + a + + + {\displaystyle x>a} + +. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent) + + + + + f + ( + x + ) + = + + ∑ + + k + = + 0 + + + + + + ( + + + + + + x + − + a + + h + + + k + + + ) + + + + + ∑ + + j + = + 0 + + + k + + + ( + − + 1 + + ) + + k + − + j + + + + + + ( + + + k + j + + + ) + + + + f + ( + a + + + j + h + ) + . + + + {\displaystyle f(x)=\sum _{k=0}{{\frac {x-a}{h}} \choose k}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}f(a+jh).} + + + +== See also == +Binomial transform +List of factorial and binomial topics +Nörlund–Rice integral +Carlson's theorem + + +== References == + +Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals", Theoretical Computer Science 144 (1995) pp 101–124. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Table_of_prime_factors-0.md b/data/en.wikipedia.org/wiki/Table_of_prime_factors-0.md new file mode 100644 index 000000000..ef50da534 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Table_of_prime_factors-0.md @@ -0,0 +1,537 @@ +--- +title: "Table of prime factors" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/Table_of_prime_factors" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:21.457299+00:00" +instance: "kb-cron" +--- + +The tables contain the prime factorization of the natural numbers from 1 to 1000. +When n is a prime number, the prime factorization is just n itself, written in bold below. +The number 1 is called a unit. It has no prime factors and is neither prime nor composite. + +== Properties == +Many properties of a natural number + + + + n + + + {\displaystyle n} + + can be seen or directly computed from the prime factorization of + + + + n + + + {\displaystyle n} + +. + +The multiplicity of a prime factor + + + + p + + + {\displaystyle p} + + of + + + + n + + + {\displaystyle n} + + is the largest exponent + + + + m + + + {\displaystyle m} + + for which + + + + + p + + m + + + + + {\displaystyle p^{m}} + + divides + + + + n + + + {\displaystyle n} + +. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is + + + + 1 + + + {\displaystyle 1} + + (since + + + + p + = + + p + + 1 + + + + + {\displaystyle p=p^{1}} + +). The multiplicity of a prime which does not divide + + + + n + + + {\displaystyle n} + + may be called + + + + 0 + + + {\displaystyle 0} + + or may be considered undefined. + + + + + ω + ( + n + ) + + + {\displaystyle \omega (n)} + + and + + + + Ω + ( + n + ) + + + {\displaystyle \Omega (n)} + +, the prime omega functions, count the number of prime factors of a natural number + + + + n + + + {\displaystyle n} + +. + + + + + ω + ( + n + ) + + + {\displaystyle \omega (n)} + + (little omega) is the number of distinct prime factors of + + + + n + + + {\displaystyle n} + +. + + + + + Ω + ( + n + ) + + + {\displaystyle \Omega (n)} + + (big omega) is the number of prime factors of + + + + n + + + {\displaystyle n} + + counted with multiplicity (so it is the sum of all prime factor multiplicities). +A prime number has + + + + Ω + ( + n + ) + = + ω + ( + n + ) + = + 1 + + + {\displaystyle \Omega (n)=\omega (n)=1} + +. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers. +A composite number has + + + + Ω + ( + n + ) + ≥ + ω + ( + n + ) + > + 1 + + + {\displaystyle \Omega (n)\geq \omega (n)>1} + +. The first: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 (sequence A002808 in the OEIS). All numbers above 1 are either prime or composite. 1 is neither. +A semiprime has + + + + Ω + ( + n + ) + = + 2 + + + {\displaystyle \Omega (n)=2} + + (so it is composite). The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34 (sequence A001358 in the OEIS). +A + + + + k + + + {\displaystyle k} + +-almost prime (for a natural number + + + + k + + + {\displaystyle k} + +) has + + + + Ω + ( + n + ) + = + k + + + {\displaystyle \Omega (n)=k} + + (so it is composite if + + + + k + > + 1 + + + {\displaystyle k>1} + +). +An even number has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 (sequence A005843 in the OEIS). +An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 (sequence A005408 in the OEIS). All integers are either even or odd. +A square has even multiplicity for all prime factors (it is of the form + + + + + a + + 2 + + + + + {\displaystyle a^{2}} + + for some + + + + a + + + {\displaystyle a} + +). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS). +A cube has all multiplicities divisible by 3 (it is of the form + + + + + a + + 3 + + + + + {\displaystyle a^{3}} + + for some + + + + a + + + {\displaystyle a} + +). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 (sequence A000578 in the OEIS). +A perfect power has a common divisor + + + + m + > + 1 + + + {\displaystyle m>1} + + for all multiplicities (it is of the form + + + + + a + + m + + + + + {\displaystyle a^{m}} + + for some + + + + a + > + 1 + + + {\displaystyle a>1} + + and + + + + m + > + 1 + + + {\displaystyle m>1} + +). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 (sequence A001597 in the OEIS). 1 is sometimes included. +A powerful number (also called squarefull) has multiplicity greater than 1 for all its prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72 (sequence A001694 in the OEIS). +A prime power has only one prime factor, i.e. + + + + ω + ( + n + ) + = + 1 + + + {\displaystyle \omega (n)=1} + +. The first: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 (sequence A000961 in the OEIS). 1 is sometimes included. +An Achilles number is powerful but not a perfect power. The first: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968 (sequence A052486 in the OEIS). +A square-free integer has no prime factor with multiplicity greater than 1. The first: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17 (sequence A005117 in the OEIS). A number where some but not all prime factors have multiplicity greater than 1 is neither square-free nor squarefull, but squareful. +The Liouville function + + + + λ + ( + n + ) + + + {\displaystyle \lambda (n)} + + is 1 if + + + + Ω + ( + n + ) + + + {\displaystyle \Omega (n)} + + is even, and is -1 if + + + + Ω + ( + n + ) + + + {\displaystyle \Omega (n)} + + is odd. +The Möbius function + + + + μ + ( + n + ) + + + {\displaystyle \mu (n)} + + is 0 if + + + + n + + + {\displaystyle n} + + is not square-free. Otherwise + + + + μ + ( + n + ) + + + {\displaystyle \mu (n)} + + is 1 if + + + + Ω + ( + n + ) + + + {\displaystyle \Omega (n)} + + is even, and is −1 if + + + + Ω + ( + n + ) + + + {\displaystyle \Omega (n)} + + is odd. +A sphenic number is square-free and the product of 3 distinct primes, i.e. it has + + + + ω + ( + n + ) + = + Ω + ( + n + ) + = + 3 + + + {\displaystyle \omega (n)=\Omega (n)=3} + +. The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154 (sequence A007304 in the OEIS). \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Table_of_prime_factors-1.md b/data/en.wikipedia.org/wiki/Table_of_prime_factors-1.md new file mode 100644 index 000000000..45360e7d9 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Table_of_prime_factors-1.md @@ -0,0 +1,657 @@ +--- +title: "Table of prime factors" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/Table_of_prime_factors" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:16:21.457299+00:00" +instance: "kb-cron" +--- + + + + + + a + + 0 + + + ( + n + ) + + + {\displaystyle a_{0}(n)} + +, sometimes called the integer logarithm, is the sum of primes dividing + + + + n + + + {\displaystyle n} + +, counted with multiplicity. It is an additive function. +A Ruth-Aaron pair is a pair of two consecutive numbers + + + + ( + n + , + n + + + 1 + ) + + + {\displaystyle (n,n+1)} + + with + + + + + a + + 0 + + + ( + n + ) + = + + a + + 0 + + + ( + n + + + 1 + ) + + + {\displaystyle a_{0}(n)=a_{0}(n+1)} + +. The first (by + + + + n + + + {\displaystyle n} + + value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 (sequence A039752 in the OEIS). Another definition is where the same prime is only counted once; if so, the first (by + + + + n + + + {\displaystyle n} + + value): 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299 (sequence A006145 in the OEIS). +A primorial + + + + + p + + n + + + # + + + {\displaystyle p_{n}\#} + + is the product of all primes from 2 to + + + + + p + + n + + + + + {\displaystyle p_{n}} + +. The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810 (sequence A002110 in the OEIS). + + + + 1 + # + = + 1 + + + {\displaystyle 1\#=1} + + is sometimes included. +A factorial + + + + n + ! + + + {\displaystyle n!} + + is the product of all numbers from 1 to + + + + n + + + {\displaystyle n} + +. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). + + + + 0 + ! + = + 1 + + + {\displaystyle 0!=1} + + is sometimes included. +A + + + + k + + + {\displaystyle k} + +-smooth number (for a natural number + + + + k + + + {\displaystyle k} + +) has its prime factors + + + + ≤ + k + + + {\displaystyle \leq k} + + (so it is also + + + + j + + + {\displaystyle j} + +-smooth for any + + + + j + > + k + + + {\displaystyle j>k} + +). + + + + + m + + + {\displaystyle m} + + is smoother than + + + + n + + + {\displaystyle n} + + if the largest prime factor of + + + + m + + + {\displaystyle m} + + is less than the largest of + + + + n + + + {\displaystyle n} + +. +A regular number has no prime factor greater than 5 (so it is 5-smooth). The first: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 (sequence A051037 in the OEIS). +A + + + + k + + + {\displaystyle k} + +-powersmooth number has all + + + + + p + + m + + + ≤ + k + + + {\displaystyle p^{m}\leq k} + + where + + + + p + + + {\displaystyle p} + + is a prime factor with multiplicity + + + + m + + + {\displaystyle m} + +. +A frugal number has more digits than the number of digits in its prime factorization (when written like the tables below with multiplicities above 1 as exponents). The first in decimal: 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250 (sequence A046759 in the OEIS). +An equidigital number has the same number of digits as its prime factorization. The first in decimal: 1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17 (sequence A046758 in the OEIS). +An extravagant number has fewer digits than its prime factorization. The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 (sequence A046760 in the OEIS). +An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital. + + + + + g + c + d + ( + m + , + n + ) + + + {\displaystyle gcd(m,n)} + + (greatest common divisor of + + + + m + + + {\displaystyle m} + + and + + + + n + + + {\displaystyle n} + +) is the product of all prime factors which are both in + + + + m + + + {\displaystyle m} + + and + + + + n + + + {\displaystyle n} + + (with the smallest multiplicity for + + + + m + + + {\displaystyle m} + + and + + + + n + + + {\displaystyle n} + +). + + + + + m + + + {\displaystyle m} + + and + + + + n + + + {\displaystyle n} + + are coprime (also called relatively prime) if they have no common prime factors, which implies + + + + g + c + d + ( + m + , + n + ) + = + 1 + + + {\displaystyle gcd(m,n)=1} + +. + + + + + l + c + m + ( + m + , + n + ) + + + {\displaystyle lcm(m,n)} + + (least common multiple of + + + + m + + + {\displaystyle m} + + and + + + + n + + + {\displaystyle n} + +) is the product of all prime factors of + + + + m + + + {\displaystyle m} + + or + + + + n + + + {\displaystyle n} + + (with the largest multiplicity for + + + + m + + + {\displaystyle m} + + or + + + + n + + + {\displaystyle n} + +). + + + + + g + c + d + ( + m + , + n + ) + × + l + c + m + ( + m + , + n + ) + = + m + × + n + + + {\displaystyle gcd(m,n)\times lcm(m,n)=m\times n} + +. Finding the prime factors is often harder than computing + + + + g + c + d + + + {\displaystyle gcd} + + and + + + + l + c + m + + + {\displaystyle lcm} + + using other algorithms which do not require known prime factorization. + + + + + m + + + {\displaystyle m} + + is a divisor of + + + + n + + + {\displaystyle n} + + (also called + + + + m + + + {\displaystyle m} + + divides + + + + n + + + {\displaystyle n} + +, or + + + + n + + + {\displaystyle n} + + is divisible by + + + + m + + + {\displaystyle m} + +) if all prime factors of + + + + m + + + {\displaystyle m} + + have at least the same multiplicity in + + + + n + + + {\displaystyle n} + +. +The divisors of + + + + n + + + {\displaystyle n} + + are all products of some or all prime factors of + + + + n + + + {\displaystyle n} + + (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. +Divisors and properties related to divisors are shown in table of divisors. + +== 1 to 100 == + +== 101 to 200 == + +== 201 to 300 == + +== 301 to 400 == + +== 401 to 500 == + +== 501 to 600 == + +== 601 to 700 == + +== 701 to 800 == + +== 801 to 900 == + +== 901 to 1000 == + +== See also == +Fundamental theorem of arithmetic – Integers have unique prime factorizations +List of prime numbers +Table of divisors \ No newline at end of file