Scrape wikipedia-science: 2535 new, 2697 updated, 5357 total (kb-cron)

data/en.wikipedia.org/wiki/Highly_composite_number-0.md

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+---
+title: "Highly composite number"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Highly_composite_number"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:15:00.548879+00:00"
+instance: "kb-cron"
+---
+
+A highly composite number is a positive integer that has more divisors than all smaller positive integers. 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}<p_{2}<\cdots <p_{k}}
+  
+ are prime, and the exponents 
+  
+    
+      
+        
+          c
+          
+            i
+          
+        
+      
+    
+    {\displaystyle c_{i}}
+  
+ are positive integers.
+Any factor of n must have the same or lesser multiplicity in each prime:
+
+  
+    
+      
+        
+          p
+          
+            1
+          
+          
+            
+              d
+              
+                1
+              
+            
+          
+        
+        ×
+        
+          p
+          
+            2
+          
+          
+            
+              d
+              
+                2
+              
+            
+          
+        
+        ×
+        ⋯
+        ×
+        
+          p
+          
+            k
+          
+          
+            
+              d
+              
+                k
+              
+            
+          
+        
+        ,
+        0
+        ≤
+        
+          d
+          
+            i
+          
+        
+        ≤
+        
+          c
+          
+            i
+          
+        
+        ,
+        0
+        <
+        i
+        ≤
+        k
+      
+    
+    {\displaystyle p_{1}^{d_{1}}\times p_{2}^{d_{2}}\times \cdots \times p_{k}^{d_{k}},0\leq d_{i}\leq c_{i},0<i\leq k}
+  
+
+So the number of divisors of n is:
+
+  
+    
+      
+        d
+        (
+        n
+        )
+        =
+        (
+        
+          c
+          
+            1
+          
+        
+        +
+        1
+        )
+        ×
+        (
+        
+          c
+          
+            2
+          
+        
+        +
+        1
+        )
+        ×
+        ⋯
+        ×
+        (
+        
+          c
+          
+            k
+          
+        
+        +
+        1
+        )
+        .
+      
+    
+    {\displaystyle d(n)=(c_{1}+1)\times (c_{2}+1)\times \cdots \times (c_{k}+1).}
+  
+
+Hence, for a highly composite number n,
+
+the k given prime numbers pi must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four divisors);
+the sequence of exponents must be non-increasing, that is 
+  
+    
+      
+        
+          c
+          
+            1
+          
+        
+        ≥
+        
+          c
+          
+            2
+          
+        
+        ≥
+        ⋯
+        ≥
+        
+          c
+          
+            k
+          
+        
+      
+    
+    {\displaystyle c_{1}\geq c_{2}\geq \cdots \geq c_{k}}
+  
+; otherwise, by exchanging two exponents we would again get a smaller number than n with the same number of divisors (for instance 18 = 21 × 32 may be replaced with 12 = 22 × 31; both have six divisors).
+Also, except in two special cases n = 4 and n = 36, the last exponent ck must equal 1. It means that 1, 4, and 36 are the only square highly composite numbers. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials or, alternatively, the smallest number for its prime signature.
+Note that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example, 96 = 25 × 3 satisfies the above conditions and has 12 divisors but is not highly composite since there is a smaller number (60) which has the same number of divisors.
+
+== Asymptotic growth and density ==
+If Q(x) denotes the number of highly composite numbers less than or equal to x, then there are two constants a and b, both greater than 1, such that
+
+  
+    
+      
+        (
+        log
+        ⁡
+        x
+        
+          )
+          
+            a
+          
+        
+        ≤
+        Q
+        (
+        x
+        )
+        ≤
+        (
+        log
+        ⁡
+        x
+        
+          )
+          
+            b
+          
+        
+        
+        .
+      
+    
+    {\displaystyle (\log x)^{a}\leq Q(x)\leq (\log x)^{b}\,.}
+  
+
+The first part of the inequality was proved by Paul Erdős in 1944 and the second part by Jean-Louis Nicolas in 1988.  We have
+
+  
+    
+      
+        1.13682
+        <
+        
+          lim inf
+          
+            x
+            
+            →
+            
+            ∞
+          
+        
+        
+          
+            
+              log
+              ⁡
+              Q
+              (
+              x
+              )
+            
+            
+              log
+              ⁡
+              log
+              ⁡
+              x
+            
+          
+        
+        ≤
+        1.44
+         
+      
+    
+    {\displaystyle 1.13682<\liminf _{x\,\to \,\infty }{\frac {\log Q(x)}{\log \log x}}\leq 1.44\ }
+  
+
+and

data/en.wikipedia.org/wiki/Highly_composite_number-1.md

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+---
+title: "Highly composite number"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/Highly_composite_number"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:15:00.548879+00:00"
+instance: "kb-cron"
+---
+
+  
+    
+      
+        
+          lim sup
+          
+            x
+            
+            →
+            
+            ∞
+          
+        
+        
+          
+            
+              log
+              ⁡
+              Q
+              (
+              x
+              )
+            
+            
+              log
+              ⁡
+              log
+              ⁡
+              x
+            
+          
+        
+        ≤
+        1.71
+         
+        .
+      
+    
+    {\displaystyle \limsup _{x\,\to \,\infty }{\frac {\log Q(x)}{\log \log x}}\leq 1.71\ .}
+  
+
+== Related sequences ==
+
+Highly composite numbers greater than 6 are also abundant numbers. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first highly composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800.
+10 of the first 38 highly composite numbers are superior highly composite numbers.
+The sequence of highly composite numbers (sequence A002182 in the OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS).
+Highly composite numbers whose number of divisors is also a highly composite number are
+
+1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200 (sequence A189394 in the OEIS).
+It is known that this sequence is complete.
+A positive integer n is a largely composite number if d(n) ≥ d(m) for all m ≤ n.  The counting function QL(x) of largely composite numbers satisfies
+
+  
+    
+      
+        (
+        log
+        ⁡
+        x
+        
+          )
+          
+            c
+          
+        
+        ≤
+        log
+        ⁡
+        
+          Q
+          
+            L
+          
+        
+        (
+        x
+        )
+        ≤
+        (
+        log
+        ⁡
+        x
+        
+          )
+          
+            d
+          
+        
+         
+      
+    
+    {\displaystyle (\log x)^{c}\leq \log Q_{L}(x)\leq (\log x)^{d}\ }
+  
+
+for positive c and d with 
+  
+    
+      
+        0.2
+        ≤
+        c
+        ≤
+        d
+        ≤
+        0.5
+      
+    
+    {\displaystyle 0.2\leq c\leq d\leq 0.5}
+  
+.
+Because the prime factorization of a highly composite number uses all of the first k primes, every highly composite number must be a practical number. Due to their ease of use in calculations involving fractions, many of these numbers are used in traditional systems of measurement and engineering designs.
+
+== See also ==
+Superior highly composite number
+Highly totient number
+Table of divisors
+Euler's totient function
+Round number
+Smooth number
+
+== Notes ==
+
+== References ==
+Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 45–46. ISBN 1-4020-4215-9. Zbl 1151.11300.
+Erdös, P. (1944). "On highly composite numbers" (PDF). Journal of the London Mathematical Society. Second Series. 19 (75_Part_3): 130–133. doi:10.1112/jlms/19.75_part_3.130. MR 0013381.
+Alaoglu, L.; Erdös, P. (1944). "On highly composite and similar numbers" (PDF). Transactions of the American Mathematical Society. 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087.
+Ramanujan, Srinivasa (1997). "Highly composite numbers" (PDF). Ramanujan Journal. 1 (2): 119–153. doi:10.1023/A:1009764017495. MR 1606180. S2CID 115619659. Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.
+
+== External links ==
+Weisstein, Eric W. "Highly Composite Number". MathWorld.
+Algorithm for computing Highly Composite Numbers
+First 10000 Highly Composite Numbers as factors
+Achim Flammenkamp, First 779674 HCN with sigma, tau, factors
+Online Highly Composite Numbers Calculator
+5040 and other Anti-Prime Numbers - Dr. James Grime by Dr. James Grime for Numberphile

data/en.wikipedia.org/wiki/Index_of_information_theory_articles-0.md

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+---
+title: "Index of information theory articles"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Index_of_information_theory_articles"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:15:10.528092+00:00"
+instance: "kb-cron"
+---
+
+This is a list of information theory topics.
+
+A Mathematical Theory of Communication
+algorithmic information theory
+arithmetic coding
+channel capacity
+Communication Theory of Secrecy Systems
+conditional entropy
+conditional quantum entropy
+confusion and diffusion
+cross-entropy
+data compression
+entropic uncertainty (Hirchman uncertainty)
+entropy encoding
+entropy (information theory)
+Fisher information
+Hick's law
+Huffman coding
+information bottleneck method
+information theoretic security
+information theory
+joint entropy
+Kullback–Leibler divergence
+lossless compression
+negentropy
+noisy-channel coding theorem (Shannon's theorem)
+principle of maximum entropy
+quantum information science
+range encoding
+redundancy (information theory)
+Rényi entropy
+self-information
+Shannon–Hartley theorem

data/en.wikipedia.org/wiki/Index_of_logarithm_articles-0.md

@@ -0,0 +1,97 @@
+---
+title: "Index of logarithm articles"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Index_of_logarithm_articles"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:15:34.685156+00:00"
+instance: "kb-cron"
+---
+
+This is a list of logarithm topics, by Wikipedia page. See also the list of exponential topics.
+
+Acoustic power
+Amoeba (mathematics)
+Antilogarithm
+Apparent magnitude
+Baker's theorem
+Bel
+Benford's law
+Binary logarithm
+Bode plot
+Henry Briggs
+Bygrave slide rule
+Cologarithm
+Common logarithm
+Complex logarithm
+Discrete logarithm
+Discrete logarithm records
+e
+Representations of e
+El Gamal discrete log cryptosystem
+Harmonic series
+History of logarithms
+Hyperbolic sector
+Iterated logarithm
+Otis King
+Law of the iterated logarithm
+Linear form in logarithms
+Linearithmic
+List of integrals of logarithmic functions
+Log canonical singularity
+Log-likelihood ratio
+Log-log graph
+Log-normal distribution
+Log-periodic antenna
+Log semiring
+Log structure
+Log-Weibull distribution
+Logarithmic algorithm
+Logarithmic convolution
+Logarithmic decrement
+Logarithmic derivative
+Logarithmic differential
+Logarithmic differentiation
+Logarithmic distribution
+Logarithmic form
+Logarithmic graph paper
+Logarithmic growth
+Logarithmic identities
+Logarithmic mean
+Logarithmic number system
+Logarithmic scale
+Logarithmic spiral
+Logit
+LogSumExp
+Mantissa is a disambiguation page; see common logarithm for the traditional concept of mantissa; see significand for the modern concept used in computing.
+Matrix logarithm
+Mel scale
+Mercator projection
+Mercator series
+Moment magnitude scale
+John Napier
+Napierian logarithm
+Natural logarithm
+Natural logarithm of 2
+Neper
+Offset logarithmic integral
+pH
+Plethystic logarithm
+Pollard's kangaroo algorithm
+Pollard's rho algorithm for logarithms
+Polylogarithm
+Polylogarithmic function
+Prime number theorem
+Richter magnitude scale
+Grégoire de Saint-Vincent
+Alphonse Antonio de Sarasa
+Schnorr signature
+Semi-log graph
+Significand
+Slide rule
+Smearing retransformation
+Sound intensity level
+Stochastic logarithm
+Super-logarithm
+Table of logarithms
+Weber-Fechner law

data/en.wikipedia.org/wiki/Karp's_21_NP-complete_problems-0.md

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+---
+title: "Karp's 21 NP-complete problems"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Karp's_21_NP-complete_problems"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:15:18.156275+00:00"
+instance: "kb-cron"
+---
+
+In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the Boolean satisfiability problem is NP-complete (also called the Cook–Levin theorem) to show that there is a polynomial time many-one reduction from the Boolean satisfiability problem to each of 21 combinatorial and graph theoretical computational problems, thereby showing that they are all NP-complete. This was one of the first demonstrations that many natural computational problems occurring throughout computer science are computationally intractable, and it drove interest in the study of NP-completeness and the P versus NP problem.
+
+
+== The problems ==
+Karp's 21 problems are shown below, many with their original names. The nesting indicates the direction of the reductions used. For example, Knapsack was shown to be NP-complete by reducing Exact cover to Knapsack.
+
+Satisfiability: the Boolean satisfiability problem for formulas in conjunctive normal form (often referred to as SAT)
+0–1 integer programming (A variation in which only the restrictions must be satisfied, with no optimization)
+Clique (see also independent set problem)
+Set packing
+Vertex cover
+Set covering
+Feedback node set
+Feedback arc set
+Directed Hamilton circuit (Karp's name, now usually called Directed Hamiltonian cycle)
+Undirected Hamilton circuit (Karp's name, now usually called Undirected Hamiltonian cycle)
+Satisfiability with at most 3 literals per clause (equivalent to 3-SAT)
+Chromatic number (also called the Graph Coloring Problem)
+Clique cover
+Exact cover
+Hitting set
+Steiner tree
+3-dimensional matching
+Knapsack (Karp's definition of Knapsack is closer to Subset sum)
+Job sequencing
+Partition
+Max cut
+
+
+== Approximations ==
+As time went on it was discovered that many of the problems can be solved efficiently if restricted to special cases, or can be solved within any fixed percentage of the optimal result. However, David Zuckerman showed in 1996 that every one of these 21 problems has a constrained optimization version that is impossible to approximate within any constant factor unless P = NP, by showing that Karp's approach to reduction generalizes to a specific type of approximability reduction. However, these may be different from the standard optimization versions of the problems, which may have approximation algorithms (as in the case of maximum cut).
+
+
+== See also ==
+List of NP-complete problems
+
+
+== Notes ==
+
+
+== References ==
+Cook, Stephen (1971). "The Complexity of Theorem Proving Procedures". Proc. 3rd Annual ACM Symposium on Theory of Computing (STOC). pp. 151–158. doi:10.1145/800157.805047. ISBN 9781450374644. S2CID 7573663.
+Karp, Richard M. (1972). "Reducibility Among Combinatorial Problems" (PDF). In R. E. Miller; J. W. Thatcher; J.D. Bohlinger (eds.). Complexity of Computer Computations. New York: Plenum. pp. 85–103. doi:10.1007/978-1-4684-2001-2_9. ISBN 978-1-4684-2003-6.{{cite book}}:  CS1 maint: publisher location (link)
+Zuckerman, David (1996). "On Unapproximable Versions of NP-Complete Problems". SIAM Journal on Computing. 25 (6): 1293–1304. doi:10.1137/S0097539794266407. [1]

data/en.wikipedia.org/wiki/List_of_International_Mathematical_Olympiads-0.md

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+---
+title: "List of International Mathematical Olympiads"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/List_of_International_Mathematical_Olympiads"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:15:14.252706+00:00"
+instance: "kb-cron"
+---
+
+The first of the International Mathematical Olympiads (IMOs) was held in Romania in 1959. The oldest of the International Science Olympiads, the IMO has since been held annually, except in 1980. That year, the competition initially planned to be held in Mongolia was cancelled due to the Soviet invasion of Afghanistan. Because the competition was initially founded for Eastern European countries participating in the Warsaw Pact, under the influence of the Eastern Bloc, the earlier IMOs were hosted only in Eastern European countries, gradually spreading to other nations.
+The first IMO was held in Romania in 1959. Seven countries entered – Bulgaria, Czechoslovakia, East Germany, Hungary, Poland, Romania and the Soviet Union – with the hosts finishing as the top-ranked nation. The number of participating countries has since risen: 14 countries took part in 1969, 50 in 1989, and 104 in 2009.
+North Korea is the only country whose entire team has been caught cheating, resulting in its disqualification at the 32nd IMO in 1991 and the 51st IMO in 2010. (However, the 2010 case was controversial.) There have been other disqualifications of contestants due to cheating, but such cases are not officially made public. In January 2011, Google gave €1 million to the IMO organization to help cover the costs of the events from 2011 to 2015.
+
+
+== List of Olympiads ==
+
+ 
+
+
+== See also ==
+Asian Pacific Mathematics Olympiad
+Provincial Mathematical Olympiad
+List of mathematics competitions
+List of International Mathematical Olympiad participants
+List of countries by medal count at International Mathematical Olympiad
+
+
+== Notes ==
+
+
+== References ==
+
+
+== Bibliography ==
+
+
+== External links ==
+Official IMO site Archived 2016-06-17 at the Wayback Machine
+International Mathematical Olympiad Info Page Archived 2015-05-26 at the Wayback Machine at Mathematical Association of America

data/en.wikipedia.org/wiki/List_of_Laplace_transforms-0.md

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+---
+title: "List of Laplace transforms"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/List_of_Laplace_transforms"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:15:20.618665+00:00"
+instance: "kb-cron"
+---
+
+The following is a list of Laplace transforms for many common functions of a single variable. The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency).
+
+
+== Properties ==
+
+The Laplace transform of a function 
+  
+    
+      
+        f
+        (
+        t
+        )
+      
+    
+    {\displaystyle f(t)}
+  
+ can be obtained using the formal definition of the Laplace transform. However, some properties of the Laplace transform can be used to obtain the Laplace transform of some functions more easily.
+
+
+=== Linearity ===
+For functions 
+  
+    
+      
+        f
+      
+    
+    {\displaystyle f}
+  
+ and 
+  
+    
+      
+        g
+      
+    
+    {\displaystyle g}
+  
+ and for scalar 
+  
+    
+      
+        a
+      
+    
+    {\displaystyle a}
+  
+, the Laplace transform satisfies
+
+  
+    
+      
+        
+          
+            L
+          
+        
+        {
+        a
+        f
+        (
+        t
+        )
+        +
+        b
+        g
+        (
+        t
+        )
+        }
+        =
+        a
+        
+          
+            L
+          
+        
+        {
+        f
+        (
+        t
+        )
+        }
+        +
+        b
+        
+          
+            L
+          
+        
+        {
+        g
+        (
+        t
+        )
+        }
+      
+    
+    {\displaystyle {\mathcal {L}}\{af(t)+bg(t)\}=a{\mathcal {L}}\{f(t)\}+b{\mathcal {L}}\{g(t)\}}
+  
+
+and is, therefore, regarded as a linear operator.
+
+
+=== Time shifting ===
+The Laplace transform of 
+  
+    
+      
+        f
+        (
+        t
+        −
+        a
+        )
+        u
+        (
+        t
+        −
+        a
+        )
+      
+    
+    {\displaystyle f(t-a)u(t-a)}
+  
+, where 
+  
+    
+      
+        u
+      
+    
+    {\displaystyle u}
+  
+ is the Heaviside step function, is 
+  
+    
+      
+        
+          e
+          
+            −
+            a
+            s
+          
+        
+        F
+        (
+        s
+        )
+      
+    
+    {\displaystyle e^{-as}F(s)}
+  
+.
+
+
+=== Frequency shifting ===
+The Laplace transform of 
+  
+    
+      
+        
+          e
+          
+            a
+            t
+          
+        
+        f
+        (
+        t
+        )
+      
+    
+    {\displaystyle e^{at}f(t)}
+  
+ is 
+  
+    
+      
+        F
+        (
+        s
+        −
+        a
+        )
+      
+    
+    {\displaystyle F(s-a)}
+  
+.
+
+
+== Explanatory notes ==
+The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t).
+The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 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 ==

data/en.wikipedia.org/wiki/List_of_Lie_groups_topics-0.md

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+---
+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)

data/en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns-0.md

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+---
+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

data/en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers-0.md

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+---
+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

data/en.wikipedia.org/wiki/List_of_NP-complete_problems-0.md

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+---
+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}
+  

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

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.

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

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.

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 ==

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 ==

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 ==

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.

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

data/en.wikipedia.org/wiki/List_of_homological_algebra_topics-0.md

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+---
+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

data/en.wikipedia.org/wiki/List_of_impossible_puzzles-0.md

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+---
+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 ==

data/en.wikipedia.org/wiki/List_of_incomplete_proofs-0.md

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+---
+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.

data/en.wikipedia.org/wiki/List_of_incomplete_proofs-2.md

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+---
+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).

data/en.wikipedia.org/wiki/List_of_incomplete_proofs-3.md

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+---
+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?

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

data/en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics-0.md

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+---
+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

data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-0.md

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+---
+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

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 ==

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

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

data/en.wikipedia.org/wiki/List_of_large_cardinal_properties-0.md

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+---
+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

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)

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 ==

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:

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

data/en.wikipedia.org/wiki/List_of_manifolds-0.md

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+---
+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 ==

data/en.wikipedia.org/wiki/List_of_mathematic_operators-0.md

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+---
+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

data/en.wikipedia.org/wiki/List_of_mathematical_constants-0.md

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+---
+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

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

data/en.wikipedia.org/wiki/List_of_mathematical_logic_topics-0.md

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+---
+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

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

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

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.

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

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

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

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"
 ---
 

data/en.wikipedia.org/wiki/List_of_mathematics_books-0.md

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+---
+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

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

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

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

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.

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

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

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+---
+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

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+---
+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.

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+---
+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 ==

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+---
+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 ==

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+---
+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

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+---
+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

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+---
+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

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+---
+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

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+---
+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

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+---
+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

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+---
+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

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+---
+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

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+---
+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

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+---
+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 ==

data/en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics-0.md

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+---
+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

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+---
+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

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