# kb

id: list-of-formulae-involving-20260505-1
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-1.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b2Nm, when struck by the other object. (This gives the digits of π in base b up to N digits past the radix point.) == Formulae yielding π == === Integrals === 2 ∫ − 1 1 1 − x 2 d x = π {\displaystyle 2\int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx=\pi } (integrating two halves y ( x ) = 1 − x 2 {\displaystyle y(x)={\sqrt {1-x^{2}}}} to obtain the area of the unit circle) ∫ 0 2 4 − x 2 d x = π {\displaystyle \int _{0}^{2}{\sqrt {4-x^{2}}}\,dx=\pi } (integrating a quarter of a circle with a radius of two x 2 + y 2 = 4 {\displaystyle x^{2}+y^{2}=4} to obtain 4 π / 4 {\displaystyle {4\pi }/4} ) ∫ − ∞ ∞ sech ⁡ x d x = π {\displaystyle \int _{-\infty }^{\infty }\operatorname {sech} x\,dx=\pi } ∫ − ∞ ∞ ∫ t ∞ e − 1 / 2 t 2 − x 2 + x t d x d t = ∫ − ∞ ∞ ∫


id: list-of-formulae-involving-20260505-2
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-2.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
∑ k = 0 ∞ ( − 1 ) k 2 10 k ( − 2 5 4 k + 1 − 1 4 k + 3 + 2 8 10 k + 1 − 2 6 10 k + 3 − 2 2 10 k + 5 − 2 2 10 k + 7 + 1 10 k + 9 ) = 2 6 π {\displaystyle \sum _{k=0}^{\infty }{\frac {{(-1)}^{k}}{2^{10k}}}\left(-{\frac {2^{5}}{4k+1}}-{\frac {1}{4k+3}}+{\frac {2^{8}}{10k+1}}-{\frac {2^{6}}{10k+3}}-{\frac {2^{2}}{10k+5}}-{\frac {2^{2}}{10k+7}}+{\frac {1}{10k+9}}\right)=2^{6}\pi } Plouffe's series for calculating arbitrary decimal digits of π: ∑ k = 1 ∞ k 2 k k ! 2 ( 2 k ) ! = π + 3 {\displaystyle \sum _{k=1}^{\infty }k{\frac {2^{k}k!^{2}}{(2k)!}}=\pi +3} === Other infinite series === ζ ( 2 ) = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ⋯ = π 2 6 {\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}} (see also Basel problem and Riemann zeta function) ζ ( 4 ) = 1 1 4 + 1 2 4 + 1 3 4 + 1 4 4 + ⋯ = π 4 90 {\displaystyle \zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}} ζ


id: list-of-formulae-involving-20260505-3
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-3.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 k + 1 = ( − 1 ) k E 2 k 2 ( 2 k ) ! ( π 2 ) 2 k + 1 , k ∈ N 0 {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2k+1}}}=(-1)^{k}{\frac {E_{2k}}{2(2k)!}}\left({\frac {\pi }{2}}\right)^{2k+1},\quad k\in \mathbb {N} _{0}} where E 2 k {\displaystyle E_{2k}} is the 2 k {\displaystyle 2k} th Euler number. ∑ n = 0 ∞ ( 1 2 n ) ( − 1 ) n 2 n + 1 = 1 − 1 6 − 1 40 − ⋯ = π 4 {\displaystyle \sum _{n=0}^{\infty }{\binom {\frac {1}{2}}{n}}{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{6}}-{\frac {1}{40}}-\cdots ={\frac {\pi }{4}}} ∑ n = 0 ∞ 1 ( 4 n + 1 ) ( 4 n + 3 ) = 1 1 ⋅ 3 + 1 5 ⋅ 7 + 1 9 ⋅ 11 + ⋯ = π 8 {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{(4n+1)(4n+3)}}={\frac {1}{1\cdot 3}}+{\frac {1}{5\cdot 7}}+{\frac {1}{9\cdot 11}}+\cdots ={\frac {\pi }{8}}} ∑ n = 1 ∞ ( − 1 ) ( n 2 + n ) / 2 + 1 | G ( ( − 1 ) n + 1


id: list-of-formulae-involving-20260505-4
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-4.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
π 4 = 5 arctan ⁡ 1 7 + 2 arctan ⁡ 3 79 {\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}} π 4 = 6 arctan ⁡ 1 8 + 2 arctan ⁡ 1 57 + arctan ⁡ 1 239 {\displaystyle {\frac {\pi }{4}}=6\arctan {\frac {1}{8}}+2\arctan {\frac {1}{57}}+\arctan {\frac {1}{239}}} π 4 = 12 arctan ⁡ 1 49 + 32 arctan ⁡ 1 57 − 5 arctan ⁡ 1 239 + 12 arctan ⁡ 1 110443 {\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}} π 4 = 44 arctan ⁡ 1 57 + 7 arctan ⁡ 1 239 − 12 arctan ⁡ 1 682 + 24 arctan ⁡ 1 12943 {\displaystyle {\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}} === Infinite products === π 4 = ( ∏ p ≡ 1 ( mod 4 ) p p − 1 ) ⋅ ( ∏ p ≡ 3 ( mod 4 ) p p + 1 ) = 3 4 ⋅ 5 4 ⋅ 7 8 ⋅ 11 12 ⋅ 13 12 ⋯ , {\displaystyle {\frac {\pi }{4}}=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p-1}}\right)\cdot \left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p+1}}\right)={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot


id: list-of-formulae-involving-20260505-5
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-5.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
whenever a + b + c = a b c {\displaystyle a+b+c=abc} and a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} are positive real numbers (see List of trigonometric identities). A special case is π = arctan ⁡ 1 + arctan ⁡ 2 + arctan ⁡ 3. {\displaystyle \pi =\arctan 1+\arctan 2+\arctan 3.} === Complex functions === e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} (Euler's identity) The following equivalences are true for any complex z {\displaystyle z} : e z ∈ R ↔ ℑ z ∈ π Z {\displaystyle e^{z}\in \mathbb {R} \leftrightarrow \Im z\in \pi \mathbb {Z} } e z = 1 ↔ z ∈ 2 π i Z {\displaystyle e^{z}=1\leftrightarrow z\in 2\pi i\mathbb {Z} } Also 1 e z − 1 = lim N → ∞ ∑ n = − N N 1 z − 2 π i n − 1 2 , z ∈ C . {\displaystyle {\frac {1}{e^{z}-1}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z-2\pi in}}-{\frac {1}{2}},\quad z\in \mathbb {C} .} Suppose a lattice Ω {\displaystyle \Omega } is generated by two periods ω 1 , ω 2 {\displaystyle \omega _{1},\omega _{2}} . We define the quasi-periods of this lattice by η


id: list-of-formulae-involving-20260505-6
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-6.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
( 2 n n ) ∼ 4 n π n {\displaystyle {\binom {2n}{n}}\sim {\frac {4^{n}}{\sqrt {\pi n}}}} (asymptotic growth rate of the central binomial coefficients) C n ∼ 4 n π n 3 {\displaystyle C_{n}\sim {\frac {4^{n}}{\sqrt {\pi n^{3}}}}} (asymptotic growth rate of the Catalan numbers) n ! ∼ 2 π n ( n e ) n {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} (Stirling's approximation) log ⁡ n ! ≃ ( n + 1 2 ) log ⁡ n − n + log ⁡ 2 π 2 {\displaystyle \log n!\simeq \left(n+{\frac {1}{2}}\right)\log n-n+{\frac {\log 2\pi }{2}}} ∑ k = 1 n φ ( k ) ∼ 3 n 2 π 2 {\displaystyle \sum _{k=1}^{n}\varphi (k)\sim {\frac {3n^{2}}{\pi ^{2}}}} (where φ {\displaystyle \varphi } is Euler's totient function) ∑ k = 1 n φ ( k ) k ∼ 6 n π 2 {\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}\sim {\frac {6n}{\pi ^{2}}}} The symbol ∼ {\displaystyle \sim } means that the ratio of the left-hand side and the right-hand side tends to one as n → ∞ {\displaystyle n\to \infty } . The symbol ≃ {\displaystyle \simeq } means that the difference between the left-hand side and the right-hand side tends to


id: list-of-formulae-involving-20260505-7
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-7.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
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


id: list-of-topics-related-to-20260505-0
file: data/en.wikipedia.org/wiki/List_of_topics_related_to_π-0.md
category: reference
title: List of topics related to π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_topics_related_to_π
date_saved: 2026-05-05T08:16:11.270763+00:00
instance: kb-cron
parent: list-of-topics-related-to-20260505-0
summary:
This is a list of topics related to pi (π), the fundamental mathematical constant. 2π theorem Approximations of π Arithmetic–geometric mean Bailey–Borwein–Plouffe formula Basel problem Borwein's algorithm Buffon's needle Cadaeic Cadenza Chronology of computation of π Circle Euler's identity Six nines in pi Gauss–Legendre algorithm Gaussian function History of π A History of Pi Indiana Pi Bill Leibniz formula for pi Lindemann–Weierstrass theorem (Proof that π is transcendental) List of circle topics List of formulae involving π Liu Hui's π algorithm Mathematical constant (sorted by continued fraction representation) Mathematical constants and functions Method of exhaustion Milü Pi Pi (art project) Pi (letter) Pi Day PiFast PiHex Pi in the Sky Pilish Pimania Piphilology Proof that π is irrational Proof that 22/7 exceeds π Proof of Wallis product Rabbi Nehemiah Radian Ramanujan–Sato series Rhind Mathematical Papyrus Salamin–Brent algorithm Software for calculating π Squaring the circle Turn (geometry) Viète's formula


id: list-of-planar-symmetry-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_planar_symmetry_groups-0.md
category: reference
title: List of planar symmetry groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_planar_symmetry_groups
date_saved: 2026-05-05T08:16:12.595642+00:00
instance: kb-cron
parent: list-of-planar-symmetry-groups-20260505-0
summary:
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: 2 families of rosette groups – 2D point groups 7 frieze groups – 2D line groups 17 wallpaper groups – 2D space groups. == Rosette groups == There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group. == Frieze groups == The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each. == Wallpaper groups == The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular). The p1 and p2 groups, with no reflectional symmetry, are repeated in all classes. The


id: list-of-mathematical-properties-of-points-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_properties_of_points-0.md
category: reference
title: List of mathematical properties of points
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_properties_of_points
date_saved: 2026-05-05T08:16:13.832116+00:00
instance: kb-cron
parent: list-of-mathematical-properties-of-points-20260505-0
summary:
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


id: list-of-polygons-20260505-0
file: data/en.wikipedia.org/wiki/List_of_polygons-0.md
category: reference
title: List of polygons
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polygons
date_saved: 2026-05-05T08:16:15.076521+00:00
instance: kb-cron
parent: list-of-polygons-20260505-0
summary:
In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides, and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners. The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well. == Greek numbers == Polygons are primarily named by prefixes from Ancient Greek numbers. == Systematic polygon names == To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" connector is not included by some authors. Extending the system up to 999 is expressed with these prefixes. == List of n-gons by Greek numerical prefixes == == See also == Platonic solid


id: list-of-polyhedral-stellations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-0.md
category: reference
title: List of polyhedral stellations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyhedral_stellations
date_saved: 2026-05-05T08:16:16.272155+00:00
instance: kb-cron
parent: list-of-polyhedral-stellations-20260505-0
summary:
In three-dimensional space, applying the operation of stellation to a polyhedron extends its faces (or edges and planes) until they generate new vertices that bound a newly formed figure. Stellation represents the dual action to faceting a polyhedron. Originating from studies of star polyhedra in 14th century Europe, a proper mathematical account of polyhedral stellations was given by Johannes Kepler in his 1619 classic work, Harmonices Mundi. Progress later ensued on detailing and enumerating stellations of prominent stars, such as the regular Kepler-Poinsot polyhedra, with developments on different stellation methods occurring in the 1900s, principally from Coxeter et al. (1938) and soon afterward, Pawley (1975). A short generalized table of the most notable polyhedral stellations belonging to convex uniform polyhedra is provided, with complete sets of stellations for the Platonic solids (including the fifty-nine icosahedral stellations), as well as for select Catalan solids (e.g., the rhombic dodecahedron and the rhombic triacontahedron). Stellations of non-convex uniform polyhedra of structure with facial planes passing through their centers (i.e., hemipolyhedral), which render unbounded vertices, are also included; these are stellations to infinity — per Wenninger (1983) — conforming to extensions on traditional definitions of polyhedra. == Background == === Star polytopes === Experimentation


id: list-of-polyhedral-stellations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-1.md
category: reference
title: List of polyhedral stellations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyhedral_stellations
date_saved: 2026-05-05T08:16:16.272155+00:00
instance: kb-cron
parent: list-of-polyhedral-stellations-20260505-0
summary:
== Stellations of Platonic solids == Only three of the five Platonic solids produce stellations: the regular octahedron, regular dodecahedron, and regular icosahedron. The regular tetrahedron and cube are unable to generate stellations when extending their faces, since extending their vertices only form one possible convex hull. === Stellations of the octahedron === The stella octangula (or stellated octahedron) is the only stellation of the regular octahedron. This stellation is made of self-dual tetrahedra, as the simplest regular polyhedral compound: === Stellations of the dodecahedron === All stellations of the regular dodecahedron are Kepler-Poinsot polyhedra: === Stellations of the icosahedron === Coxeter et al. (1938) detailed the stellations of the regular icosahedron with rules proposed by J. C. P. Miller. As found in Coxeter et al. (1999), the following table lists all stellations of the icosahedron per the Crennells' indexing (in it, the regular icosahedron (or snub octahedron) stellation core is indexed as "1"): "Cells" (du Val notation) correspond to the internal congruent spaces formed by extending face-planes of the regular icosahedron. A subset of these are illustrated in Wenninger (1989), alongside constructions for physical models (W19–W66). == Stellations of Catalan solids == === Stellations of the rhombic dodecahedron ===


id: list-of-polynomial-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_polynomial_topics-0.md
category: reference
title: List of polynomial topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polynomial_topics
date_saved: 2026-05-05T08:16:17.484027+00:00
instance: kb-cron
parent: list-of-polynomial-topics-20260505-0
summary:
This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics. == Terminology == Degree: The maximum exponents among the monomials. Factor: An expression being multiplied. Linear factor: A factor of degree one. Coefficient: An expression multiplying one of the monomials of the polynomial. Root (or zero) of a polynomial: Given a polynomial p(x), the x values that satisfy p(x) = 0 are called roots (or zeroes) of the polynomial p. Graphing End behaviour – Concavity – Orientation – Tangency point – Inflection point – Point where concavity changes. == Basics == Polynomial Coefficient Monomial Polynomial long division Synthetic division Polynomial factorization Rational function Partial fraction Partial fraction decomposition over R Vieta's formulas Integer-valued polynomial Algebraic equation Factor theorem Polynomial remainder theorem === Elementary abstract algebra === See also Theory of equations below. Polynomial ring Greatest common divisior of two polynomials Symmetric function Homogeneous polynomial Polynomial SOS (sum of squares) == Theory of equations == Polynomial family Quadratic function Cubic function Quartic function Quintic function Sextic function Septic function Octic function Completing the square Abel–Ruffini theorem Bring radical Binomial theorem Blossom (functional) Root of a function nth root (radical) Surd Square root


id: list-of-regular-polytopes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-0.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. == Overview == This table shows a summary of regular polytope counts by rank. == 1-polytopes == There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol { }, or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion and gives it the Schläfli symbol { }. Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. == 2-polytopes (polygons) == The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than


id: list-of-regular-polytopes-20260505-1
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-1.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
==== Spherical ==== In spherical geometry, regular spherical polyhedra (tilings of the sphere) exist that would otherwise be degenerate as polytopes. These are the hosohedra {2,n} and their dual dihedra {n,2}. Coxeter calls these cases "improper" tessellations. The first few cases (n from 2 to 6) are listed below. Star-dihedra and hosohedra {p/q, 2} and {2, p/q} also exist for any star polygon {p/q}. === Stars === The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}: As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow. There are infinitely many failed star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are {5/2,4}, {5/2,9}, {7/2,3}, {5/2,5/2}, {7/2,7/3}, {4,5/2}, and {3,7/3}. === Skew polyhedra === Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures. For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n}


id: list-of-regular-polytopes-20260505-2
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-2.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
A vertex figure (of a 5-polytope) is a 4-polytope, seen by the arrangement of neighboring vertices to each vertex. An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge. A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face. A regular 5-polytope { p , q , r , s } {\displaystyle \{p,q,r,s\}} exists only if { p , q , r } {\displaystyle \{p,q,r\}} and { q , r , s } {\displaystyle \{q,r,s\}} are regular 4-polytopes. The space it fits in is based on the expression: cos 2 ⁡ ( π q ) sin 2 ⁡ ( π p ) + cos 2 ⁡ ( π r ) sin 2 ⁡ ( π s ) {\displaystyle {\frac {\cos ^{2}\left({\frac {\pi }{q}}\right)}{\sin ^{2}\left({\frac {\pi }{p}}\right)}}+{\frac {\cos ^{2}\left({\frac {\pi }{r}}\right)}{\sin ^{2}\left({\frac {\pi }{s}}\right)}}} < 1 {\displaystyle <1} : Spherical 4-space tessellation or 5-space polytope = 1 {\displaystyle =1} : Euclidean 4-space tessellation > 1 {\displaystyle >1} : hyperbolic 4-space tessellation Enumeration of these constraints produce 3 convex polytopes, no star polytopes, 3 tessellations of Euclidean 4-space, and 5 tessellations of paracompact hyperbolic 4-space. The


id: list-of-regular-polytopes-20260505-3
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-3.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
==== Hyperbolic tilings ==== Tessellations of hyperbolic 2-space are hyperbolic tilings. There are infinitely many regular tilings in H2. As stated above, every positive integer pair {p,q} such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle (p, q, r) the same holds true for 1/p + 1/q + 1/r < 1. There are a number of different ways to display the hyperbolic plane, including the Poincaré disk model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens. There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q<pq/2. {3,7}, {3,8}, {3,9} ... {3,∞} {4,5}, {4,6}, {4,7} ... {4,∞} {5,4}, {5,5}, {5,6} ... {5,∞} {6,4}, {6,5}, {6,6} ... {6,∞} {7,3}, {7,4}, {7,5} ... {7,∞} {8,3}, {8,4}, {8,5} ... {8,∞} {9,3}, {9,4}, {9,5} ... {9,∞} ... {∞,3}, {∞,4}, {∞,5} ... {∞,∞} A sampling: The tilings {p, ∞} have ideal vertices, on the


id: list-of-regular-polytopes-20260505-4
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-4.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
There are no regular compact or paracompact hyperbolic star-honeycombs in H3: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical. Ideal vertices now appear when the vertex figure is a Euclidean tiling, becoming inscribable in a horosphere rather than a sphere. They are dual to ideal cells (Euclidean tilings rather than finite polyhedra). As the last number in the Schläfli symbol rises further, the vertex figure becomes hyperbolic, and vertices become ultra-ideal (so the edges do not meet within hyperbolic space). In honeycombs {p, q, ∞} the edges intersect the Poincaré ball only in one ideal point; the rest of the edge has become ultra-ideal. Continuing further would lead to edges that are completely ultra-ideal, both for the honeycomb and for the fundamental simplex (though still infinitely many {p, q} would meet at such edges). In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point. === 5-apeirotopes === ==== Tessellations of Euclidean 4-space ==== There are three kinds of infinite regular tessellations (honeycombs) that can tessellate Euclidean four-dimensional space: There are also the two improper cases {4,3,4,2}


id: list-of-ppad-complete-problems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_PPAD-complete_problems-0.md
category: reference
title: List of PPAD-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_PPAD-complete_problems
date_saved: 2026-05-05T08:16:19.979284+00:00
instance: kb-cron
parent: list-of-ppad-complete-problems-20260505-0
summary:
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.


id: list-of-prime-knots-20260505-0
file: data/en.wikipedia.org/wiki/List_of_prime_knots-0.md
category: reference
title: List of prime knots
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_prime_knots
date_saved: 2026-05-05T08:16:22.748133+00:00
instance: kb-cron
parent: list-of-prime-knots-20260505-0
summary:
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes. == Table of prime knots == === Six or fewer crossings === === Seven crossings === === Eight crossings === === Nine crossings === === Ten crossings === === Higher === Conway knot 11n34 Kinoshita–Terasaka knot 11n42 == Table of prime links == === Eight or fewer crossings === === Higher === == See also == List of knots List of mathematical knots and links Knot tabulation (−2,3,7) pretzel knot == Notes == == External links == "The Rolfsen Knot Table", The Knot Atlas. "KnotInfo", Indiana.edu.


id: list-of-mersenne-primes-and-perfect-numbers-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers-0.md
category: reference
title: List of Mersenne primes and perfect numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers
date_saved: 2026-05-05T08:16:25.190446+00:00
instance: kb-cron
parent: list-of-mersenne-primes-and-perfect-numbers-20260505-0
summary:
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.


id: list-of-repunit-primes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_repunit_primes-0.md
category: reference
title: List of repunit primes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_repunit_primes
date_saved: 2026-05-05T08:16:26.460363+00:00
instance: kb-cron
parent: list-of-repunit-primes-20260505-0
summary:
This is a list of repunit primes in various bases. == Base 2 repunit primes == Base-2 repunit primes are called Mersenne primes. == Base 3 repunit primes == The first few base-3 repunit primes are 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (sequence A076481 in the OEIS), corresponding to n {\displaystyle n} of 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117, ... (sequence A028491 in the OEIS). == Base 4 repunit primes == The only base-4 repunit prime is 5 ( 11 4 {\displaystyle 11_{4}} ). 4 n − 1 = ( 2 n + 1 ) ( 2 n − 1 ) {\displaystyle 4^{n}-1=\left(2^{n}+1\right)\left(2^{n}-1\right)} , and 3 always divides 2 n + 1 {\displaystyle 2^{n}+1} when n is odd and 2 n − 1 {\displaystyle 2^{n}-1} when n is even. For n greater than 2, both 2 n + 1 {\displaystyle 2^{n}+1} and 2 n − 1 {\displaystyle 2^{n}-1} are greater than 3, so removing the factor of 3 still leaves two factors greater than 1. Therefore, the number cannot be prime. == Base 5 repunit primes == The first few base-5 repunit primes are


id: list-of-repunit-primes-20260505-1
file: data/en.wikipedia.org/wiki/List_of_repunit_primes-1.md
category: reference
title: List of repunit primes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_repunit_primes
date_saved: 2026-05-05T08:16:26.460363+00:00
instance: kb-cron
parent: list-of-repunit-primes-20260505-0
summary:
2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, 13, 136, 220, 162, 35, 10, 218, 19, 26, 39, 12, 22, 67, 120, 195, 48, 54, 463, 38, 41, 17, 808, 404, 46, 76, 793, 38, 28, 215, 37, 236, 59, 15, 514, 260, 498, 6, 2, 95, 3, ... (sequence A066180 in the OEIS) Smallest base b {\displaystyle b} such that R p ( − b ) {\displaystyle R_{p}(-b)} is prime (where p {\displaystyle p} is the n {\displaystyle n} th prime) are 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, 159, 10, 16, 209, 2,


id: probability-and-statistics-20260505-0
file: data/en.wikipedia.org/wiki/Probability_and_statistics-0.md
category: reference
title: Probability and statistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Probability_and_statistics
date_saved: 2026-05-05T08:16:27.641847+00:00
instance: kb-cron
parent: probability-and-statistics-20260505-0
summary:
Probability and statistics are two closely related fields in mathematics that are sometimes combined for academic purposes. They are covered in multiple articles and lists: Probability Statistics Glossary of probability and statistics Notation in probability and statistics Timeline of probability and statistics Publications named for both fields include the following: Brazilian Journal of Probability and Statistics Counterexamples in Probability and Statistics Probability and Mathematical Statistics Theory of Probability and Mathematical Statistics == References ==


id: list-of-mathematical-proofs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_proofs-0.md
category: reference
title: List of mathematical proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_proofs
date_saved: 2026-05-05T08:16:28.793352+00:00
instance: kb-cron
parent: list-of-mathematical-proofs-20260505-0
summary:
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


id: list-of-long-mathematical-proofs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-0.md
category: reference
title: List of long mathematical proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs
date_saved: 2026-05-05T08:16:29.984359+00:00
instance: kb-cron
parent: list-of-long-mathematical-proofs-20260505-0
summary:
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


id: list-of-long-mathematical-proofs-20260505-1
file: data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-1.md
category: reference
title: List of long mathematical proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs
date_saved: 2026-05-05T08:16:29.984359+00:00
instance: kb-cron
parent: list-of-long-mathematical-proofs-20260505-0
summary:
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


id: list-of-pspace-complete-problems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_PSPACE-complete_problems-0.md
category: reference
title: List of PSPACE-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_PSPACE-complete_problems
date_saved: 2026-05-05T08:16:31.189075+00:00
instance: kb-cron
parent: list-of-pspace-complete-problems-20260505-0
summary:
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


id: list-of-q-analogs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_q-analogs-0.md
category: reference
title: List of q-analogs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_q-analogs
date_saved: 2026-05-05T08:16:33.618494+00:00
instance: kb-cron
parent: list-of-q-analogs-20260505-0
summary:
This is a list of q-analogs in mathematics and related fields. == Algebra == Iwahori–Hecke algebra Quantum affine algebra Quantum enveloping algebra Quantum group == Analysis == Jackson integral q-derivative q-difference polynomial Quantum calculus == Combinatorics == LLT polynomial q-binomial coefficient q-Pochhammer symbol q-Vandermonde identity == Orthogonal polynomials == q-Bessel polynomials q-Charlier polynomials q-Hahn polynomials q-Jacobi polynomials: Big q-Jacobi polynomials Continuous q-Jacobi polynomials Little q-Jacobi polynomials q-Krawtchouk polynomials q-Laguerre polynomials Continuous q-Legendre polynomials q-Meixner polynomials q-Meixner–Pollaczek polynomials q-Racah polynomials == Probability and statistics == Gaussian q-distribution q-exponential distribution q-Weibull distribution Tsallis q-Gaussian Tsallis entropy == Special functions == Basic hypergeometric series Elliptic gamma function Hahn–Exton q-Bessel function Jackson q-Bessel function q-exponential q-gamma function q-theta function == See also == Lists of mathematics topics


id: list-of-mathematical-topics-in-quantum-theory-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory-0.md
category: reference
title: List of mathematical topics in quantum theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory
date_saved: 2026-05-05T08:16:34.798624+00:00
instance: kb-cron
parent: list-of-mathematical-topics-in-quantum-theory-20260505-0
summary:
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


id: list-of-random-number-generators-20260505-0
file: data/en.wikipedia.org/wiki/List_of_random_number_generators-0.md
category: reference
title: List of random number generators
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_random_number_generators
date_saved: 2026-05-05T08:16:35.976218+00:00
instance: kb-cron
parent: list-of-random-number-generators-20260505-0
summary:
Random number generators are important in many kinds of technical applications, including physics, engineering or mathematical computer studies (e.g., Monte Carlo simulations), cryptography and gambling (on game servers). This list includes many common types, regardless of quality or applicability to a given use case. == Pseudorandom number generators (PRNGs) == The following algorithms are pseudorandom number generators. == Cryptographic algorithms == Cipher algorithms and cryptographic hashes can be used as very high-quality pseudorandom number generators. However, generally they are considerably slower (typically by a factor 2–10) than fast, non-cryptographic random number generators. These include: Stream ciphers. Popular choices are Salsa20 or ChaCha (often with the number of rounds reduced to 8 for speed), ISAAC, HC-128 and RC4. Block ciphers in counter mode. Common choices are AES (which is very fast on systems supporting it in hardware), TwoFish, Serpent and Camellia. Cryptographic hash functions A few cryptographically secure pseudorandom number generators do not rely on cipher algorithms but try to link mathematically the difficulty of distinguishing their output from a `true' random stream to a computationally difficult problem. These approaches are theoretically important but are too slow to be practical in most applications. They include: Blum–Micali algorithm (1984) Blum Blum Shub


id: list-of-real-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_real_analysis_topics-0.md
category: reference
title: List of real analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_real_analysis_topics
date_saved: 2026-05-05T08:16:37.250027+00:00
instance: kb-cron
parent: list-of-real-analysis-topics-20260505-0
summary:
This is a list of articles that are considered real analysis topics. See also: glossary of real and complex analysis. == General topics == === Limits === Limit of a sequence Subsequential limit – the limit of some subsequence Limit of a function (see List of limits for a list of limits of common functions) One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below Squeeze theorem – confirms the limit of a function via comparison with two other functions Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions === Sequences and series === (see also list of mathematical series) Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number Harmonic progression – a


id: list-of-real-analysis-topics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_real_analysis_topics-1.md
category: reference
title: List of real analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_real_analysis_topics
date_saved: 2026-05-05T08:16:37.250027+00:00
instance: kb-cron
parent: list-of-real-analysis-topics-20260505-0
summary:
== Fundamental theorems == Monotone convergence theorem – relates monotonicity with convergence Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc Taylor's theorem – gives an approximation of a k {\displaystyle k} times differentiable function around a given point by a k {\displaystyle k} -th order Taylor-polynomial. L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms Abel's theorem – relates the limit of a power series to the sum of its coefficients Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function Darboux's theorem – states that all functions that result from the


id: list-of-properties-of-sets-of-reals-20260505-0
file: data/en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals-0.md
category: reference
title: List of properties of sets of reals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals
date_saved: 2026-05-05T08:16:38.430648+00:00
instance: kb-cron
parent: list-of-properties-of-sets-of-reals-20260505-0
summary:
This article lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory, which has a rather different emphasis from general topology. == Definability properties == Borel set Analytic set C-measurable set Projective set Inductive set Infinity-Borel set Suslin set Homogeneously Suslin set Weakly homogeneously Suslin set Set of uniqueness == Regularity properties == Property of Baire Lebesgue measurable Universally measurable set Perfect set property Universally Baire set == Largeness and smallness properties == Meager set Comeager set - A comeager set is one whose complement is meager. Null set Conull set Dense set Nowhere dense set


id: list-of-recreational-number-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_recreational_number_theory_topics-0.md
category: reference
title: List of recreational number theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_recreational_number_theory_topics
date_saved: 2026-05-05T08:16:39.664179+00:00
instance: kb-cron
parent: list-of-recreational-number-theory-topics-20260505-0
summary:
This is a list of recreational number theory topics (see number theory, recreational mathematics). Listing here is not pejorative: many famous topics in number theory have origins in challenging problems posed purely for their own sake. See list of number theory topics for pages dealing with aspects of number theory with more consolidated theories. == Number sequences == Integer sequence Fibonacci sequence Golden mean base Fibonacci coding Lucas sequence Padovan sequence Figurate numbers Polygonal number Triangular number Square number Pentagonal number Hexagonal number Heptagonal number Octagonal number Nonagonal number Decagonal number Centered polygonal number Centered square number Centered pentagonal number Centered hexagonal number Tetrahedral number Pyramidal number Triangular pyramidal number Square pyramidal number Pentagonal pyramidal number Hexagonal pyramidal number Heptagonal pyramidal number Octahedral number Star number Perfect number Quasiperfect number Almost perfect number Multiply perfect number Hyperperfect number Semiperfect number Primitive semiperfect number Unitary perfect number Weird number Untouchable number Amicable number Sociable number Abundant number Deficient number Amenable number Aliquot sequence Super-Poulet number Lucky number Powerful number Primeval number Palindromic number Telephone number Triangular square number Harmonic divisor number Sphenic number Smith number Double Mersenne number Zeisel number Heteromecic number Niven numbers Superparticular number Highly composite number Highly totient


id: list-of-mathematics-reference-tables-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics_reference_tables-0.md
category: reference
title: List of mathematics reference tables
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics_reference_tables
date_saved: 2026-05-05T08:16:41.067246+00:00
instance: kb-cron
parent: list-of-mathematics-reference-tables-20260505-0
summary:
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


id: list-of-mathematical-topics-in-relativity-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity-0.md
category: reference
title: List of mathematical topics in relativity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity
date_saved: 2026-05-05T08:16:42.212375+00:00
instance: kb-cron
parent: list-of-mathematical-topics-in-relativity-20260505-0
summary:
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


id: list-of-representation-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_representation_theory_topics-0.md
category: reference
title: List of representation theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_representation_theory_topics
date_saved: 2026-05-05T08:16:43.462653+00:00
instance: kb-cron
parent: list-of-representation-theory-topics-20260505-0
summary:
This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics, which is more directed towards the mathematical analysis aspects of representation theory. See also: Glossary of representation theory == General representation theory == Linear representation Unitary representation Trivial representation Irreducible representation Semisimple Complex representation Real representation Quaternionic representation Pseudo-real representation Symplectic representation Schur's lemma Restricted representation == Representation theory of groups == Group representation Group ring Maschke's theorem Regular representation Character (mathematics) Character theory Class function Representation theory of finite groups Modular representation theory Frobenius reciprocity Restricted representation Induced representation Peter–Weyl theorem Young tableau Spherical harmonic Hecke operator Representation theory of the symmetric group Representation theory of diffeomorphism groups Permutation representation Affine representation Projective representation Central extension == Representation theory of Lie groups and Lie algebras == Representation of a Lie group Lie algebra representation, Representation of a Lie superalgebra Universal enveloping algebra Casimir element Infinitesimal character Harish-Chandra homomorphism Fundamental representation Antifundamental representation Bifundamental representation Adjoint representation Weight (representation theory) Cartan's theorem Spinor Wigner's classification, Representation theory of the Poincaré group Wigner–Eckart theorem Stone–von Neumann theorem Orbit method Kirillov character formula Weyl character formula Discrete series representation Principal series representation Borel–Weil–Bott theorem


id: list-of-formulas-in-riemannian-geometry-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-0.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise. == Christoffel symbols, covariant derivative == In a smooth coordinate chart, the Christoffel symbols of the first kind are given by Γ k i j = 1 2 ( ∂ ∂ x j g k i + ∂ ∂ x i g k j − ∂ ∂ x k g i j ) = 1 2 ( g k i , j + g k j , i − g i j , k ) , {\displaystyle \Gamma _{kij}={\frac {1}{2}}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,,} and the Christoffel symbols of the second kind by Γ m i j = g m k Γ k i j = 1 2 g m k ( ∂ ∂ x j g k i + ∂ ∂ x i g k j − ∂ ∂ x k g i j ) = 1 2 g m k ( g k i , j + g k j , i − g i j , k ) . {\displaystyle {\begin{aligned}\Gamma


id: list-of-formulas-in-riemannian-geometry-20260505-1
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-1.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
R i j k l = ∂ Γ j k l ∂ x i − ∂ Γ i k l ∂ x j + ( Γ j k p Γ i p l − Γ i k p Γ j p l ) {\displaystyle {R_{ijk}}^{l}={\frac {\partial \Gamma _{jk}^{l}}{\partial x^{i}}}-{\frac {\partial \Gamma _{ik}^{l}}{\partial x^{j}}}+{\big (}\Gamma _{jk}^{p}\Gamma _{ip}^{l}-\Gamma _{ik}^{p}\Gamma _{jp}^{l}{\big )}} R ( u , v ) w = ∇ u ∇ v w − ∇ v ∇ u w − ∇ [ u , v ] w {\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w} ==== (3,1) Riemann curvature tensor ==== R j k l i = ∂ Γ l j i ∂ x k − ∂ Γ k j i ∂ x l + ( Γ k p i Γ l j p − Γ l p i Γ k j p ) {\displaystyle {R_{jkl}^{i}}={\frac {\partial \Gamma _{lj}^{i}}{\partial x^{k}}}-{\frac {\partial \Gamma _{kj}^{i}}{\partial x^{l}}}+{\big (}\Gamma _{kp}^{i}\Gamma _{lj}^{p}-\Gamma _{lp}^{i}\Gamma _{kj}^{p}{\big )}} ==== Ricci curvature ==== R i k = R i j k j {\displaystyle R_{ik}={R_{ijk}}^{j}} Ric ⁡ ( v , w ) = tr ⁡ ( u ↦ R ( u , v ) w ) {\displaystyle \operatorname {Ric} (v,w)=\operatorname {tr} (u\mapsto


id: list-of-formulas-in-riemannian-geometry-20260505-2
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-2.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
div g ⁡ Ric = 1 2 d R {\displaystyle \operatorname {div} _{g}\operatorname {Ric} ={\frac {1}{2}}dR} Equivalently: g p q ∇ p G q k = 0 {\displaystyle g^{pq}\nabla _{p}G_{qk}=0} div g ⁡ G = 0 {\displaystyle \operatorname {div} _{g}G=0} ==== Ricci identity ==== If X {\displaystyle X} is a vector field then ∇ i ∇ j X k − ∇ j ∇ i X k = − R i j p k X p , {\displaystyle \nabla _{i}\nabla _{j}X^{k}-\nabla _{j}\nabla _{i}X^{k}=-{R_{ijp}}^{k}X^{p},} which is just the definition of the Riemann tensor. If ω {\displaystyle \omega } is a one-form then ∇ i ∇ j ω k − ∇ j ∇ i ω k = R i j k p ω p . {\displaystyle \nabla _{i}\nabla _{j}\omega _{k}-\nabla _{j}\nabla _{i}\omega _{k}={R_{ijk}}^{p}\omega _{p}.} More generally, if T {\displaystyle T} is a (0,k)-tensor field then ∇ i ∇ j T l 1 ⋯ l k − ∇ j ∇ i T l 1 ⋯ l k = R i j l 1 p T p l 2 ⋯ l k + ⋯ + R i j l k p T l 1 ⋯ l k − 1 p . {\displaystyle \nabla _{i}\nabla _{j}T_{l_{1}\cdots


id: list-of-formulas-in-riemannian-geometry-20260505-3
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-3.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
R i k ℓ m = 1 2 ( ∂ 2 g i m ∂ x k ∂ x ℓ + ∂ 2 g k ℓ ∂ x i ∂ x m − ∂ 2 g i ℓ ∂ x k ∂ x m − ∂ 2 g k m ∂ x i ∂ x ℓ ) {\displaystyle R_{ik\ell m}={\frac {1}{2}}\left({\frac {\partial ^{2}g_{im}}{\partial x^{k}\partial x^{\ell }}}+{\frac {\partial ^{2}g_{k\ell }}{\partial x^{i}\partial x^{m}}}-{\frac {\partial ^{2}g_{i\ell }}{\partial x^{k}\partial x^{m}}}-{\frac {\partial ^{2}g_{km}}{\partial x^{i}\partial x^{\ell }}}\right)} R ℓ i j k = ∂ ∂ x j Γ ℓ i k − ∂ ∂ x k Γ ℓ i j {\displaystyle R^{\ell }{}_{ijk}={\frac {\partial }{\partial x^{j}}}\Gamma ^{\ell }{}_{ik}-{\frac {\partial }{\partial x^{k}}}\Gamma ^{\ell }{}_{ij}} == Conformal change == Let g {\displaystyle g} be a Riemannian or pseudo-Riemannian metric on a smooth manifold M {\displaystyle M} , and φ {\displaystyle \varphi } a smooth real-valued function on M {\displaystyle M} . Then g ~ = e 2 φ g {\displaystyle {\tilde {g}}=e^{2\varphi }g} is also a Riemannian metric on M {\displaystyle M} . We say that g ~ {\displaystyle {\tilde {g}}} is (pointwise) conformal to g {\displaystyle g} . Evidently, conformality of metrics is an equivalence relation. Here


id: list-of-formulas-in-riemannian-geometry-20260505-4
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-4.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
=== Hodge operator on p-forms === ∗ ~ i 1 ⋯ i n − p j 1 ⋯ j p = e ( n − 2 p ) φ ∗ i 1 ⋯ i n − p j 1 ⋯ j p {\displaystyle {\widetilde {\ast }}_{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}=e^{(n-2p)\varphi }\ast _{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}} ∗ ~ = e ( n − 2 p ) φ ∗ {\displaystyle {\widetilde {\ast }}=e^{(n-2p)\varphi }\ast } === Codifferential on p-forms === d ∗ ~ j 1 ⋯ j p − 1 i 1 ⋯ i p = e − 2 φ ( d ∗ ) j 1 ⋯ j p − 1 i 1 ⋯ i p − ( n − 2 p ) e − 2 φ ∇ i 1 φ δ j 1 i 2 ⋯ δ j p − 1 i p {\displaystyle {\widetilde {d^{\ast }}}_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}=e^{-2\varphi }(d^{\ast })_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}-(n-2p)e^{-2\varphi }\nabla ^{i_{1}}\varphi \delta _{j_{1}}^{i_{2}}\cdots \delta _{j_{p-1}}^{i_{p}}} d ∗ ~ = e − 2 φ d ∗ − ( n − 2 p ) e − 2 φ ι ∇ φ {\displaystyle {\widetilde {d^{\ast }}}=e^{-2\varphi }d^{\ast }-(n-2p)e^{-2\varphi }\iota _{\nabla \varphi }} === Laplacian on functions === Δ ~ Φ = e −


id: list-of-formulas-in-riemannian-geometry-20260505-5
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-5.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
∂ ∂ 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


id: list-of-rules-of-inference-20260505-0
file: data/en.wikipedia.org/wiki/List_of_rules_of_inference-0.md
category: reference
title: List of rules of inference
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rules_of_inference
date_saved: 2026-05-05T08:16:46.303073+00:00
instance: kb-cron
parent: list-of-rules-of-inference-20260505-0
summary:
This is a list of rules of inference, logical laws that relate to mathematical formulae. == Introduction == Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules. Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation φ ⊢ ψ {\displaystyle \varphi \vdash \psi } indicates such a subderivation from the temporary assumption φ {\displaystyle \varphi } to ψ {\displaystyle \psi } . == Rules for propositional calculus == === Rules for negations === Reductio ad absurdum (or Negation Introduction) φ ⊢ ψ {\displaystyle \varphi \vdash \psi } φ ⊢ ¬ ψ _ {\displaystyle {\underline {\varphi \vdash \lnot \psi }}} ¬ φ {\displaystyle \lnot \varphi } Reductio ad absurdum (related to the law of excluded middle) ¬ φ ⊢ ψ {\displaystyle \lnot \varphi \vdash \psi


id: list-of-rules-of-inference-20260505-1
file: data/en.wikipedia.org/wiki/List_of_rules_of_inference-1.md
category: reference
title: List of rules of inference
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rules_of_inference
date_saved: 2026-05-05T08:16:46.303073+00:00
instance: kb-cron
parent: list-of-rules-of-inference-20260505-0
summary:
ψ {\displaystyle \psi } Restriction 1: β {\displaystyle \beta } is a variable which does not occur in φ {\displaystyle \varphi } . Restriction 2: There is no occurrence, free or bound, of β {\displaystyle \beta } in ψ {\displaystyle \psi } . Restriction 3: β {\displaystyle \beta } is not mentioned in any hypothesis or undischarged assumptions. == Rules of substructural logic == The following are special cases of universal generalization and existential elimination; these occur in substructural logics, such as linear logic. Rule of weakening (or monotonicity of entailment) (aka no-cloning theorem) α ⊢ β {\displaystyle \alpha \vdash \beta } α , α ⊢ β ¯ {\displaystyle {\overline {\alpha ,\alpha \vdash \beta }}} Rule of contraction (or idempotency of entailment) (aka no-deleting theorem) α , α , γ ⊢ β _ {\displaystyle {\underline {\alpha ,\alpha ,\gamma \vdash \beta }}} α , γ ⊢ β {\displaystyle \alpha ,\gamma \vdash \beta } == Table: Rules of Inference == The rules above can be summed up in the following table. The "Tautology" column shows how to interpret the notation of a given rule. All rules use the basic logic operators. A complete table of "logic operators" is shown by a truth


id: list-of-runge-kutta-methods-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-0.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation d y d t = f ( t , y ) . {\displaystyle {\frac {dy}{dt}}=f(t,y).} Explicit Runge–Kutta methods take the form y n + 1 = y n + h ∑ i = 1 s b i k i k 1 = f ( t n , y n ) , k 2 = f ( t n + c 2 h , y n + h ( a 21 k 1 ) ) , k 3 = f ( t n + c 3 h , y n + h ( a 31 k 1 + a 32 k 2 ) ) , ⋮ k i = f ( t n + c i h , y n + h ∑ j = 1 i − 1 a i j k j ) . {\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\\k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\\&\;\;\vdots \\k_{i}&=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{i-1}a_{ij}k_{j}\right).\end{aligned}}} Stages for implicit methods of s stages take the more general form, with the solution to be found over all s k i = f ( t n + c i h , y n + h ∑ j = 1 s a i j k j


id: list-of-runge-kutta-methods-20260505-1
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-1.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
0 0 0 0 0 1 / 3 1 / 3 0 0 0 2 / 3 − 1 / 3 1 0 0 1 1 − 1 1 0 1 / 8 3 / 8 3 / 8 1 / 8 {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\1/3&1/3&0&0&0\\2/3&-1/3&1&0&0\\1&1&-1&1&0\\\hline &1/8&3/8&3/8&1/8\\\end{array}}} ==== Ralston's fourth-order method ==== This fourth order method has minimum truncation error. 0 0 0 0 0 2 5 2 5 0 0 0 14 − 3 5 16 − 2 889 + 1 428 5 1 024 3 785 − 1 620 5 1 024 0 0 1 − 3 365 + 2 094 5 6 040 − 975 − 3 046 5 2 552 467 040 + 203 968 5 240 845 0 263 + 24 5 1 812 125 − 1000 5 3 828 3 426 304 + 1 661 952 5 5 924 787 30 − 4 5 123 {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\{\frac {2}{5}}&{\frac {2}{5}}&0&0&0\\{\frac {14-3{\sqrt {5}}}{16}}&{\frac {-2\,889+1\,428{\sqrt {5}}}{1\,024}}&{\frac {3\,785-1\,620{\sqrt {5}}}{1\,024}}&0&0\\1&{\frac {-3\,365+2\,094{\sqrt {5}}}{6\,040}}&{\frac {-975-3\,046{\sqrt {5}}}{2\,552}}&{\frac {467\,040+203\,968{\sqrt {5}}}{240\,845}}&0\\\hline &{\frac {263+24{\sqrt {5}}}{1\,812}}&{\frac {125-1000{\sqrt {5}}}{3\,828}}&{\frac {3\,426\,304+1\,661\,952{\sqrt {5}}}{5\,924\,787}}&{\frac {30-4{\sqrt {5}}}{123}}\\\end{array}}} === Fifth-order methods === ==== Nyström's fifth-order method ==== This fifth-order method was a correction of the one proposed originally by Kutta's work. 0 0 0


id: list-of-runge-kutta-methods-20260505-2
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-2.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
=== Gauss–Legendre methods === These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau: 1 2 − 3 6 1 4 1 4 − 3 6 1 2 + 3 6 1 4 + 3 6 1 4 1 2 1 2 1 2 + 3 2 1 2 − 3 2 {\displaystyle {\begin{array}{c|cc}{\frac {1}{2}}-{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}&{\frac {1}{4}}-{\frac {\sqrt {3}}{6}}\\{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\&{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}\\\end{array}}} The Gauss–Legendre method of order six has Butcher tableau: 1 2 − 15 10 5 36 2 9 − 15 15 5 36 − 15 30 1 2 5 36 + 15 24 2 9 5 36 − 15 24 1 2 + 15 10 5 36 + 15 30 2 9 + 15 15 5 36 5 18 4 9 5 18 − 5 6 8 3 − 5 6 {\displaystyle {\begin{array}{c|ccc}{\frac {1}{2}}-{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}&{\frac {2}{9}}-{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{30}}\\{\frac {1}{2}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{24}}&{\frac {2}{9}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{24}}\\{\frac {1}{2}}+{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{30}}&{\frac {2}{9}}+{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}\\\hline &{\frac {5}{18}}&{\frac {4}{9}}&{\frac {5}{18}}\\&-{\frac {5}{6}}&{\frac {8}{3}}&-{\frac {5}{6}}\end{array}}} === Diagonally Implicit Runge–Kutta methods ===


id: list-of-runge-kutta-methods-20260505-3
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-3.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
1 / 2 1 / 2 0 0 0 2 / 3 1 / 6 1 / 2 0 0 1 / 2 − 1 / 2 1 / 2 1 / 2 0 1 3 / 2 − 3 / 2 1 / 2 1 / 2 3 / 2 − 3 / 2 1 / 2 1 / 2 {\displaystyle {\begin{array}{c|cccc}1/2&1/2&0&0&0\\2/3&1/6&1/2&0&0\\1/2&-1/2&1/2&1/2&0\\1&3/2&-3/2&1/2&1/2\\\hline &3/2&-3/2&1/2&1/2\\\end{array}}} === Lobatto methods === There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis. All are implicit methods, have order 2s − 2 and they all have c1 = 0 and cs = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta. ==== Lobatto IIIA methods ==== The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule: 0 0 0 1 1


id: list-of-runge-kutta-methods-20260505-4
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-4.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
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


id: list-of-self-intersecting-polygons-20260505-0
file: data/en.wikipedia.org/wiki/List_of_self-intersecting_polygons-0.md
category: reference
title: List of self-intersecting polygons
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_self-intersecting_polygons
date_saved: 2026-05-05T08:16:48.847957+00:00
instance: kb-cron
parent: list-of-self-intersecting-polygons-20260505-0
summary:
Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross. Some types of self-intersecting polygons are: the crossed quadrilateral, with four edges the antiparallelogram, a crossed quadrilateral with alternate edges of equal length the crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle, hence having two edges parallel Star polygons pentagram, with five edges hexagram, with six edges heptagram, with seven edges octagram, with eight edges enneagram or nonagram, with nine edges decagram, with ten edges hendecagram, with eleven edges dodecagram, with twelve edges icositetragram, with twenty four edges 257-gram, with two hundred and fifty seven edges == See also == List of regular polytopes and compounds § Stars Complex polygon


id: list-of-mathematical-series-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-0.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, 0 0 {\displaystyle 0^{0}} is taken to have the value 1 {\displaystyle 1} { x } {\displaystyle \{x\}} denotes the fractional part of x {\displaystyle x} B n ( x ) {\displaystyle B_{n}(x)} is a Bernoulli polynomial. B n {\displaystyle B_{n}} is a Bernoulli number, and here, B 1 = − 1 2 . {\displaystyle B_{1}=-{\frac {1}{2}}.} E n {\displaystyle E_{n}} is an Euler number. ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. ψ n ( z ) {\displaystyle \psi _{n}(z)} is a polygamma function. Li s ⁡ ( z ) {\displaystyle \operatorname {Li} _{s}(z)} is a polylogarithm. ( n k ) {\displaystyle n \choose k} is binomial coefficient exp ⁡ ( x ) {\displaystyle \exp(x)} denotes exponential of x {\displaystyle x} == Sums of powers == See Faulhaber's formula. ∑ k = 0 m k n − 1 = B n ( m + 1 ) − B n n {\displaystyle \sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}} The first few values are:


id: list-of-mathematical-series-20260505-1
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-1.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ k = 0 ∞ k 2 z k k ! = ( z + z 2 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{2}{\frac {z^{k}}{k!}}=(z+z^{2})e^{z}} (cf. second moment of Poisson distribution) ∑ k = 0 ∞ k 3 z k k ! = ( z + 3 z 2 + z 3 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{3}{\frac {z^{k}}{k!}}=(z+3z^{2}+z^{3})e^{z}} ∑ k = 0 ∞ k 4 z k k ! = ( z + 7 z 2 + 6 z 3 + z 4 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{4}{\frac {z^{k}}{k!}}=(z+7z^{2}+6z^{3}+z^{4})e^{z}} ∑ k = 0 ∞ k n z k k ! = z d d z ∑ k = 0 ∞ k n − 1 z k k ! = e z T n ( z ) {\displaystyle \sum _{k=0}^{\infty }k^{n}{\frac {z^{k}}{k!}}=z{\frac {d}{dz}}\sum _{k=0}^{\infty }k^{n-1}{\frac {z^{k}}{k!}}\,\!=e^{z}T_{n}(z)} where T n ( z ) {\displaystyle T_{n}(z)} is the Touchard polynomials. === Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship === ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! = sin ⁡ z {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\sin z} ∑ k = 0 ∞ z


id: list-of-mathematical-series-20260505-2
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-2.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ k = 0 ∞ ( 4 k ) ! 2 4 k 2 ( 2 k ) ! ( 2 k + 1 ) ! z k = 1 − 1 − z z , | z | < 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {(4k)!}{2^{4k}{\sqrt {2}}(2k)!(2k+1)!}}z^{k}={\sqrt {\frac {1-{\sqrt {1-z}}}{z}}},|z|<1} ∑ k = 0 ∞ 2 2 k ( k ! ) 2 ( k + 1 ) ( 2 k + 1 ) ! z 2 k + 2 = ( arcsin ⁡ z ) 2 , | z | ≤ 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k}(k!)^{2}}{(k+1)(2k+1)!}}z^{2k+2}=\left(\arcsin {z}\right)^{2},|z|\leq 1} ∑ n = 0 ∞ ∏ k = 0 n − 1 ( 4 k 2 + α 2 ) ( 2 n ) ! z 2 n + ∑ n = 0 ∞ α ∏ k = 0 n − 1 [ ( 2 k + 1 ) 2 + α 2 ] ( 2 n + 1 ) ! z 2 n + 1 = e α arcsin ⁡ z , | z | ≤ 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(4k^{2}+\alpha ^{2})}{(2n)!}}z^{2n}+\sum _{n=0}^{\infty }{\frac {\alpha \prod _{k=0}^{n-1}[(2k+1)^{2}+\alpha ^{2}]}{(2n+1)!}}z^{2n+1}=e^{\alpha \arcsin {z}},|z|\leq 1} === Binomial coefficients === ( 1


id: list-of-mathematical-series-20260505-3
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-3.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ A ∈ P ( E ) c a r d ( A ) = n 2 n − 1 , where E is a finite set, and card( E ) = n {\displaystyle \sum _{A\ \in \ {\mathcal {P}}(E)}card(A)=n2^{n-1}{\text{, where }}E{\text{ is a finite set, and card(}}E{\text{) = n}}} == Trigonometric functions == Sums of sines and cosines arise in Fourier series. ∑ k = 1 ∞ cos ⁡ ( k θ ) k = − 1 2 ln ⁡ ( 2 − 2 cos ⁡ θ ) = − ln ⁡ ( 2 sin ⁡ θ 2 ) , 0 < θ < 2 π {\displaystyle \sum _{k=1}^{\infty }{\frac {\cos(k\theta )}{k}}=-{\frac {1}{2}}\ln(2-2\cos \theta )=-\ln \left(2\sin {\frac {\theta }{2}}\right),0<\theta <2\pi } ∑ k = 1 ∞ sin ⁡ ( k θ ) k = π − θ 2 , 0 < θ < 2 π {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi } ∑ k = 1 ∞ ( − 1 ) k − 1 k cos ⁡ ( k θ ) = 1 2 ln ⁡ ( 2 + 2 cos ⁡ θ ) = ln ⁡ ( 2 cos ⁡ θ 2 ) ,


id: list-of-mathematical-series-20260505-4
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-4.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ m ≠ 0 n − 1 1 1 − z m = n − 1 2 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {1}{1-z_{m}}}={\frac {n-1}{2}}} ∑ m ≠ 0 n − 1 1 ( 1 − z m ) 2 = ( n − 1 ) ( 5 − n ) 12 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {1}{(1-z_{m})^{2}}}={\frac {(n-1)(5-n)}{12}}} Let k {\displaystyle k} be an integer 0 < k < n {\displaystyle 0<k<n} then we also got: ∑ m = 0 n − 1 z m k = 0 {\displaystyle \sum _{m=0}^{n-1}z_{m}^{k}=0} ∑ m = 0 n − 1 ( x − z m ) k = n ⋅ x k {\displaystyle \sum _{m=0}^{n-1}(x-z_{m})^{k}=n\cdot x^{k}} ∑ m ≠ 0 n − 1 ( z m ) k 1 − z m = 2 k − n − 1 2 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {(z_{m})^{k}}{1-z_{m}}}={\frac {2k-n-1}{2}}} ∑ m ≠ 0 n − 1 ( z m ) k ( 1 − z m ) 2 = 6 k ( n + 2 − k ) − ( n + 1 ) ( n + 5 ) 12 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {(z_{m})^{k}}{(1-z_{m})^{2}}}={\frac {6k(n+2-k)-(n+1)(n+5)}{12}}} == Rational functions == ∑ n = a + 1 ∞


id: list-of-mathematical-series-20260505-5
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-5.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
S ( a , b ) = ( − 1 ) c ( ∑ k = 0 ∞ ( − 1 ) k a k + d − ∑ k = 0 c − 1 ( − 1 ) k a k + d ) = ( − 1 ) c ( S ( a , d ) − ∑ k = 0 c − 1 ( − 1 ) k a k + d ) {\displaystyle S(a,b)=(-1)^{c}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{ak+d}}-\sum _{k=0}^{c-1}{\frac {(-1)^{k}}{ak+d}}\right)=(-1)^{c}\left(S(a,d)-\sum _{k=0}^{c-1}{\frac {(-1)^{k}}{ak+d}}\right)} what that means is that all the infinite choices of a {\displaystyle a} and b {\displaystyle b} can essentially be boiled down to the cases where g c d ( a , b ) = 1 {\displaystyle gcd(a,b)=1} and a > b > 0 {\displaystyle a>b>0} . If we assume those two things we can then write: S ( a , b ) = 1 a ( π 2 sin ⁡ ( π b a ) − 2 ∑ m = 0 ⌊ a 2 ⌋ cos ⁡ ( π ( 2 m + 1 ) b a ) ln ⁡ ( sin ⁡ ( π 2 m + 1 2 a ) ) )


id: list-of-mathematical-series-20260505-6
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-6.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ 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


id: list-of-set-identities-and-relations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-0.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. The binary operations of set union ( ∪ {\displaystyle \cup } ) and intersection ( ∩ {\displaystyle \cap } ) satisfy many identities. Several of these identities or "laws" have well established names. == Notation == Throughout this article, capital letters (such as A , B , C , L , M , R , S , {\displaystyle A,B,C,L,M,R,S,} and X {\displaystyle X} ) will denote sets. On the left hand side of an identity, typically, L {\displaystyle L} will be the leftmost set, M {\displaystyle M} will be the middle set, and R {\displaystyle R} will be the rightmost set. This is to facilitate applying identities to expressions that are complicated or use the same symbols as the identity. For example, the identity ( L ∖ M ) ∖ R = ( L ∖ R ) ∖ ( M ∖ R ) {\displaystyle (L\,\setminus \,M)\,\setminus \,R~=~(L\,\setminus \,R)\,\setminus \,(M\,\setminus \,R)} may be read as: ( Left set


id: list-of-set-identities-and-relations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-1.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∪ L = L (Idempotence) L ∩ L = L (Idempotence) L △ L = ∅ (Nilpotence of index 2) L ∖ L = ∅ (Nilpotence of index 2) {\displaystyle {\begin{alignedat}{10}L\cup L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\cap L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\,\triangle \,L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]L\setminus L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]\end{alignedat}}} Domination/Absorbing element: Definition: z {\displaystyle z} is called a left absorbing element of a binary operator ∗ {\displaystyle \,\ast \,} if z ∗ R = z {\displaystyle z\,\ast \,R=z} for all R {\displaystyle R} and it is called a right absorbing element of ∗ {\displaystyle \,\ast \,} if L ∗ z = z {\displaystyle L\,\ast \,z=z} for all L . {\displaystyle L.} A left absorbing element that is also a right absorbing element if called an absorbing element. Absorbing elements are also sometime called annihilating elements or zero elements. A universe set is an absorbing element of binary union ∪ . {\displaystyle \cup .} The empty set ∅ {\displaystyle \varnothing } is an absorbing element of binary intersection ∩ {\displaystyle \cap } and binary Cartesian product × , {\displaystyle \times ,} and it is also a left absorbing element of set subtraction ∖ : {\displaystyle \,\setminus :} X


id: list-of-set-identities-and-relations-20260505-2
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-2.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
X ∖ ( L ∩ R ) = ( X ∖ L ) ∪ ( X ∖ R ) Also written ( L ∩ R ) ∁ = L ∁ ∪ R ∁ (De Morgan's law) X ∖ ( L ∪ R ) = ( X ∖ L ) ∩ ( X ∖ R ) Also written ( L ∪ R ) ∁ = L ∁ ∩ R ∁ (De Morgan's law) {\displaystyle {\begin{alignedat}{10}X\setminus (L\cap R)&=(X\setminus L)\cup (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cap R)^{\complement }=L^{\complement }\cup R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]X\setminus (L\cup R)&=(X\setminus L)\cap (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cup R)^{\complement }=L^{\complement }\cap R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]\end{alignedat}}} === Commutativity === Unions, intersection, and symmetric difference are commutative operations: L ∪ R = R ∪ L (Commutativity) L ∩ R = R ∩ L (Commutativity) L △ R = R △ L (Commutativity) {\displaystyle {\begin{alignedat}{10}L\cup R&\;=\;&&R\cup L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\cap R&\;=\;&&R\cap L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\,\triangle R&\;=\;&&R\,\triangle L&&\quad {\text{ (Commutativity)}}\\[1.4ex]\end{alignedat}}} Set subtraction is not commutative. However, the commutativity of set subtraction can be characterized: from ( L ∖ R ) ∩ ( R ∖ L ) = ∅ {\displaystyle (L\,\setminus \,R)\cap (R\,\setminus \,L)=\varnothing } it follows that: L ∖


id: list-of-set-identities-and-relations-20260505-3
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-3.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
===== Empty set ===== A set L {\displaystyle L} is empty if the sentence ∀ x ( x ∉ L ) {\displaystyle \forall x(x\not \in L)} is true, where the notation x ∉ L {\displaystyle x\not \in L} is shorthand for ¬ ( x ∈ L ) . {\displaystyle \lnot (x\in L).} If L {\displaystyle L} is any set then the following are equivalent: L {\displaystyle L} is not empty, meaning that the sentence ¬ [ ∀ x ( x ∉ L ) ] {\displaystyle \lnot [\forall x(x\not \in L)]} is true (literally, the logical negation of " L {\displaystyle L} is empty" holds true). (In classical mathematics) L {\displaystyle L} is inhabited, meaning: ∃ x ( x ∈ L ) {\displaystyle \exists x(x\in L)} In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set L {\displaystyle L} that is not empty (where by definition, " L {\displaystyle L} is empty" means that the statement ∀ x ( x ∉ L ) {\displaystyle \forall x(x\not \in L)} is true) might not have an inhabitant (which is an x {\displaystyle x}


id: list-of-set-identities-and-relations-20260505-4
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-4.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
=== Precedence rules === There is no universal agreement on the order of precedence of the basic set operators. Nevertheless, many authors use precedence rules for set operators, although these rules vary with the author. One common convention is to associate intersection L ∩ R = { x : ( x ∈ L ) ∧ ( x ∈ R ) } {\displaystyle L\cap R=\{x:(x\in L)\land (x\in R)\}} with logical conjunction (and) L ∧ R {\displaystyle L\land R} and associate union L ∪ R = { x : ( x ∈ L ) ∨ ( x ∈ R ) } {\displaystyle L\cup R=\{x:(x\in L)\lor (x\in R)\}} with logical disjunction (or) L ∨ R , {\displaystyle L\lor R,} and then transfer the precedence of these logical operators (where ∧ {\displaystyle \,\land \,} has precedence over ∨ {\displaystyle \,\lor \,} ) to these set operators, thereby giving ∩ {\displaystyle \,\cap \,} precedence over ∪ . {\displaystyle \,\cup .\,} So for example, L ∪ M ∩ R {\displaystyle L\cup M\cap R} would mean L ∪ ( M ∩ R ) {\displaystyle L\cup (M\cap R)} since it would be associated with the logical statement L ∨ M ∧ R = L ∨ ( M ∧


id: list-of-set-identities-and-relations-20260505-5
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-5.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) ∖ R = L ∖ ( M ∖ R ) {\textstyle \;(L\setminus M)\setminus R=L\setminus (M\setminus R)\;} if and only if ( R ∖ M ) ∖ L = R ∖ ( M ∖ L ) , {\displaystyle \;(R\setminus M)\setminus L=R\setminus (M\setminus L),\;} where the only difference between the left and right hand side set equalities is that the locations of L and R {\displaystyle L{\text{ and }}R} have been swapped. === Distributivity === Definition: If ∗ and ∙ {\displaystyle \ast {\text{ and }}\bullet } are binary operators then ∗ {\displaystyle \,\ast \,} left distributes over ∙ {\displaystyle \,\bullet \,} if L ∗ ( M ∙ R ) = ( L ∗ M ) ∙ ( L ∗ R ) for all L , M , R {\displaystyle L\,\ast \,(M\,\bullet \,R)~=~(L\,\ast \,M)\,\bullet \,(L\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R} while ∗ {\displaystyle \,\ast \,} right distributes over ∙ {\displaystyle \,\bullet \,} if ( L ∙ M ) ∗ R = ( L ∗ R ) ∙ ( M ∗ R ) for all L , M , R . {\displaystyle (L\,\bullet \,M)\,\ast \,R~=~(L\,\ast \,R)\,\bullet \,(M\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R.} The operator ∗ {\displaystyle \,\ast


id: list-of-set-identities-and-relations-20260505-6
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-6.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∪ ( M ∩ R ) = ( L ∪ M ) ∩ ( L ∪ R ) (Left-distributivity of ∪ over ∩ ) L ∪ ( M ∪ R ) = ( L ∪ M ) ∪ ( L ∪ R ) (Left-distributivity of ∪ over ∪ ) L ∩ ( M ∪ R ) = ( L ∩ M ) ∪ ( L ∩ R ) (Left-distributivity of ∩ over ∪ ) L ∩ ( M ∩ R ) = ( L ∩ M ) ∩ ( L ∩ R ) (Left-distributivity of ∩ over ∩ ) L ∩ ( M △ R ) = ( L ∩ M ) △ ( L ∩ R ) (Left-distributivity of ∩ over △ ) L × ( M ∩ R ) = ( L × M ) ∩ ( L × R ) (Left-distributivity of × over ∩ ) L × ( M ∪ R ) = ( L × M ) ∪ ( L × R ) (Left-distributivity of × over ∪ ) L × ( M ∖ R ) = ( L × M ) ∖ ( L × R ) (Left-distributivity of × over ∖ )


id: list-of-set-identities-and-relations-20260505-7
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-7.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∖ ( M ∪ R ) ⊆ ( L ∖ M ) ∪ ( L ∖ R ) = L ∖ ( M ∩ R ) {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cap \,R)\\[1.4ex]\end{alignedat}}} L ∖ ( M ∩ R ) ⊇ ( L ∖ M ) ∩ ( L ∖ R ) = L ∖ ( M ∪ R ) {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cap \,R)~&~~{\color {red}{\supseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)\\[1.4ex]\end{alignedat}}} where equality holds for one (or equivalently, for both) of the above two inclusion formulas if and only if L ∩ M = L ∩ R . {\displaystyle L\,\cap \,M=L\,\cap \,R.} The following statements are equivalent: L ∩ M = L ∩ R {\displaystyle L\cap M\,=\,L\cap R} L ∖ M = L ∖ R {\displaystyle L\,\setminus \,M\,=\,L\,\setminus \,R} L ∖ ( M ∩ R ) = ( L ∖ M ) ∩ ( L ∖ R ) ; {\displaystyle L\,\setminus \,(M\,\cap \,R)=(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R);} that is, ∖ {\displaystyle \,\setminus \,} left distributes over ∩ {\displaystyle \,\cap \,} for these three particular sets L ∖ ( M ∪ R ) = ( L ∖ M ) ∪ ( L ∖ R )


id: list-of-set-identities-and-relations-20260505-8
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-8.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) △ R = ( L ∖ ( M ∪ R ) ) ∪ ( R ∖ L ) ∪ ( L ∩ M ∩ R ) (the three outermost sets are pairwise disjoint) {\displaystyle {\begin{alignedat}{4}(L\setminus M)~\triangle ~R&=(L\setminus (M\cup R))\cup (R\setminus L)\cup (L\cap M\cap R)~~~{\text{ (the three outermost sets are pairwise disjoint) }}\\\end{alignedat}}} ( L ∖ M ) × R = ( L × R ) ∖ ( M × R ) (Distributivity) {\displaystyle (L\,\setminus M)\times R=(L\times R)\,\setminus (M\times R)~~~~~{\text{ (Distributivity)}}} ==== L\(M ⁎ R) ==== Set subtraction on the left, and parentheses on the right L ∖ ( M ∪ R ) = ( L ∖ M ) ∩ ( L ∖ R ) (De Morgan's law) = ( L ∖ M ) ∖ R = ( L ∖ R ) ∖ M {\displaystyle {\begin{alignedat}{3}L\setminus (M\cup R)&=(L\setminus M)&&\,\cap \,(&&L\setminus R)~~~~{\text{ (De Morgan's law) }}\\&=(L\setminus M)&&\,\,\setminus &&R\\&=(L\setminus R)&&\,\,\setminus &&M\\\end{alignedat}}} L ∖ ( M ∩ R ) = ( L ∖ M ) ∪ ( L ∖ R ) (De Morgan's law) {\displaystyle {\begin{alignedat}{4}L\setminus (M\cap R)&=(L\setminus M)\cup (L\setminus R)~~~~{\text{ (De Morgan's law) }}\\\end{alignedat}}} where the above two sets that are the subjects of De Morgan's laws always


id: list-of-set-identities-and-relations-20260505-9
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-9.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∪ M ) ∪ ( M ∪ R ) = L ∪ M ∪ R ( L ∪ M ) ∩ ( M ∪ R ) = M ∪ ( L ∩ R ) ( L ∪ M ) ∖ ( M ∪ R ) = L ∖ ( M ∪ R ) ( L ∪ M ) △ ( M ∪ R ) = ( L ∖ ( M ∪ R ) ) ∪ ( R ∖ ( L ∪ M ) ) = ( L △ R ) ∖ M ( L ∩ M ) ∪ ( M ∩ R ) = M ∩ ( L ∪ R ) ( L ∩ M ) ∩ ( M ∩ R ) = L ∩ M ∩ R ( L ∩ M ) ∖ ( M ∩ R ) = ( L ∩ M ) ∖ R ( L ∩ M ) △ ( M ∩ R ) = [ ( L ∩ M ) ∪ ( M ∩ R ) ] ∖ ( L ∩ M ∩ R ) ( L ∖ M ) ∪ ( M ∖ R ) = ( L ∪ M ) ∖


id: list-of-set-identities-and-relations-20260505-10
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-10.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) ∪ ( L ∖ R ) = L ∖ ( M ∩ R ) ( L ∖ M ) ∩ ( L ∖ R ) = L ∖ ( M ∪ R ) ( L ∖ M ) ∖ ( L ∖ R ) = ( L ∩ R ) ∖ M ( L ∖ M ) △ ( L ∖ R ) = L ∩ ( M △ R ) = ( L ∩ M ) △ ( L ∩ R ) {\displaystyle {\begin{alignedat}{9}(L\,\setminus M)&\,\cup \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cap \,R)\\[1.4ex](L\,\setminus M)&\,\cap \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cup \,R)\\[1.4ex](L\,\setminus M)&\,\setminus \,&&(&&L\,\setminus R)&&\;=\;&&(L\,\cap \,R)\,\setminus \,M\\[1.4ex](L\,\setminus M)&\,\triangle \,&&(&&L\,\setminus R)&&\;=\;&&L\,\cap \,(M\,\triangle \,R)\\[1.4ex]&\,&&\,&&\,&&\;=\;&&(L\cap M)\,\triangle \,(L\cap R)\\[1.4ex]\end{alignedat}}} === Other simplifications === Other properties: L ∩ M = R and L ∩ R = M if and only if M = R ⊆ L . {\displaystyle L\cap M=R\;{\text{ and }}\;L\cap R=M\qquad {\text{ if and only if }}\qquad M=R\subseteq L.} If L ⊆ M {\displaystyle L\subseteq M} then L ∖ R = L ∩ ( M ∖ R ) . {\displaystyle L\setminus R=L\cap (M\setminus R).} L × ( M ∖ R ) = ( L × M ) ∖ ( L × R ) {\displaystyle L\times


id: list-of-set-identities-and-relations-20260505-11
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-11.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L × M × R ) ∩ ( L 2 × M 2 × R 2 ) = ( L ∩ L 2 ) × ( M ∩ M 2 ) × ( R ∩ R 2 ) {\displaystyle (L\times M\times R)\cap \left(L_{2}\times M_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)} === Binary ⋃ of finite ⨯ === ( L × R ) ∪ ( L 2 × R 2 ) = [ ( L ∖ L 2 ) × R ] ∪ [ ( L 2 ∖ L ) × R 2 ] ∪ [ ( L ∩ L 2 ) × ( R ∪ R 2 ) ] = [ L × ( R ∖ R 2 ) ] ∪ [ L 2 × ( R 2 ∖ R ) ] ∪ [ ( L ∪ L 2 ) × ( R ∩ R 2 ) ] {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\cup ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\setminus L_{2}\right)\times R\right]~\cup ~\left[\left(L_{2}\setminus L\right)\times R_{2}\right]~\cup ~\left[\left(L\cap L_{2}\right)\times \left(R\cup R_{2}\right)\right]\\[0.5ex]~&=~\left[L\times \left(R\setminus R_{2}\right)\right]~\cup ~\left[L_{2}\times \left(R_{2}\setminus R\right)\right]~\cup ~\left[\left(L\cup L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\\end{alignedat}}} === Difference \ of finite ⨯ === ( L × R ) ∖ ( L 2 × R 2 ) = [ ( L ∖ L 2 )


id: list-of-set-identities-and-relations-20260505-12
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-12.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
== Arbitrary families of sets == Let ( L i ) i ∈ I , {\displaystyle \left(L_{i}\right)_{i\in I},} ( R j ) j ∈ J , {\displaystyle \left(R_{j}\right)_{j\in J},} and ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} be indexed families of sets. Whenever the assumption is needed, then all indexing sets, such as I {\displaystyle I} and J , {\displaystyle J,} are assumed to be non-empty. === Definitions === A family of sets or (more briefly) a family refers to a set whose elements are sets. An indexed family of sets is a function from some set, called its indexing set, into some family of sets. An indexed family of sets will be denoted by ( L i ) i ∈ I , {\displaystyle \left(L_{i}\right)_{i\in I},} where this notation assigns the symbol I {\displaystyle I} for the indexing set and for every index i ∈ I , {\displaystyle i\in I,} assigns the symbol L i {\displaystyle L_{i}} to the value of the function at i . {\displaystyle i.} The function itself may then be denoted by the symbol L ∙ , {\displaystyle L_{\bullet },} which is obtained from


id: list-of-set-identities-and-relations-20260505-13
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-13.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
otherwise, if you are working in a model in which "the class of all things x {\displaystyle x} " is not a set (by far the most common situation) then ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} is undefined because ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} consists of everything, which makes ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} a proper class and not a set. Assumption: Henceforth, whenever a formula requires some indexing set to be non-empty in order for an arbitrary intersection to be well-defined, then this will automatically be assumed without mention. A consequence of this is the following assumption/definition: A finite intersection of sets or an intersection of finitely many sets refers to the intersection of a finite collection of one or more sets. Some authors adopt the so called nullary intersection convention, which is the convention that an empty intersection of sets is equal to some canonical set. In particular, if all sets are subsets of some set X {\displaystyle X} then some author may declare that the empty intersection of these sets be equal to X


id: list-of-set-identities-and-relations-20260505-14
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-14.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
⋂ i ∈ I ( L i ∪ R i ) {\displaystyle ~\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~~\color {Red}{\neq }\color {Black}{}~~\bigcap _{i\in I}\left(L_{i}\cup R_{i}\right)~} (an example of this is given above). The single intersection on the right hand side must be over all pairs ( i , j ) ∈ I × I : {\displaystyle (i,j)\in I\times I:} ( ⋂ i ∈ I L i ) ∪ ( ⋂ i ∈ I R i ) = ⋂ j ∈ I i ∈ I , ( L i ∪ R j ) . {\displaystyle ~\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~~=~~\bigcap _{\stackrel {i\in I,}{j\in I}}\left(L_{i}\cup R_{j}\right).~} ==== Arbitrary ⋂'s and arbitrary ⋃'s ==== ===== Incorrectly distributing by swapping ⋂ and ⋃ ===== Naively swapping ⋃ i ∈ I {\displaystyle \;{\textstyle \bigcup \limits _{i\in I}}\;} and ⋂ j ∈ J {\displaystyle \;{\textstyle \bigcap \limits _{j\in J}}\;} may produce a different set The following inclusion always holds: In general, equality need not hold and moreover, the right hand side depends on how for each fixed i ∈ I , {\displaystyle i\in I,} the sets ( S i , j ) j ∈ J {\displaystyle \left(S_{i,j}\right)_{j\in J}} are labelled; and analogously, the left hand side


id: list-of-set-identities-and-relations-20260505-15
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-15.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Equality in Inclusion 1 ∪∩ is a subset of ∩∪ can hold under certain circumstances, such as in 7e, which is the special case where ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is ( L i ∖ R j ) ( i , j ) ∈ I × J {\displaystyle \left(L_{i}\setminus R_{j}\right)_{(i,j)\in I\times J}} (that is, S i , j : = L i ∖ R j {\displaystyle S_{i,j}\colon =L_{i}\setminus R_{j}} with the same indexing sets I {\displaystyle I} and J {\displaystyle J} ), or such as in 7f, which is the special case where ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is ( L i ∖ R j ) ( j , i ) ∈ J × I {\displaystyle \left(L_{i}\setminus R_{j}\right)_{(j,i)\in J\times I}} (that is, S ^ j , i : = L i ∖ R j {\displaystyle {\hat {S}}_{j,i}\colon =L_{i}\setminus R_{j}} with the indexing sets I {\displaystyle I} and J {\displaystyle J} swapped). For a correct formula that extends the distributive laws, an approach other than just switching ∪ {\displaystyle \cup } and ∩


id: list-of-set-identities-and-relations-20260505-16
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-16.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Example application: To apply the general formula to the case of ( C k ) k ∈ K {\displaystyle \left(C_{k}\right)_{k\in K}} and ( D l ) l ∈ L , {\displaystyle \left(D_{l}\right)_{l\in L},} use I : = { 1 , 2 } , {\displaystyle I\colon =\{1,2\},} J 1 : = K , {\displaystyle J_{1}\colon =K,} J 2 : = L , {\displaystyle J_{2}\colon =L,} and let T 1 , k : = C k {\displaystyle T_{1,k}\colon =C_{k}} for all k ∈ J 1 {\displaystyle k\in J_{1}} and let T 2 , l : = D l {\displaystyle T_{2,l}\colon =D_{l}} for all l ∈ J 2 . {\displaystyle l\in J_{2}.} Every map f ∈ ∏ J ∙ = def ∏ i ∈ I J i = J 1 × J 2 = K × L {\displaystyle f\in {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}=J_{1}\times J_{2}=K\times L} can be bijectively identified with the pair ( f ( 1 ) , f ( 2 ) ) ∈ K × L {\displaystyle \left(f(1),f(2)\right)\in K\times L} (the inverse sends ( k , l ) ∈ K × L {\displaystyle (k,l)\in K\times L} to the map f ( k , l ) ∈


id: list-of-set-identities-and-relations-20260505-17
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-17.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
and if I = J {\displaystyle I=J} then also: === Cartesian products Π of arbitrarily many sets === ==== Intersections ⋂ of Π ==== If ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is a family of sets then Moreover, a tuple ( x i ) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} belongs to the set in Eq. 8 above if and only if x i ∈ S i , j {\displaystyle x_{i}\in S_{i,j}} for all i ∈ I {\displaystyle i\in I} and all j ∈ J . {\displaystyle j\in J.} In particular, if ( L i ) i ∈ I {\displaystyle \left(L_{i}\right)_{i\in I}} and ( R i ) i ∈ I {\displaystyle \left(R_{i}\right)_{i\in I}} are two families indexed by the same set then ( ∏ i ∈ I L i ) ∩ ∏ i ∈ I R i = ∏ i ∈ I ( L i ∩ R i ) {\displaystyle \left(\prod _{i\in I}L_{i}\right)\cap \prod _{i\in I}R_{i}~=~\prod _{i\in I}\left(L_{i}\cap R_{i}\right)} So for instance, ( L × R ) ∩ ( L 2 × R 2 ) = ( L ∩ L 2 ) × ( R ∩ R


id: list-of-set-identities-and-relations-20260505-18
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-18.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L × ( ⋃ i ∈ I R i ) = ⋃ i ∈ I ( L × R i ) (Left-distributivity of × over ∪ ) L × ( ⋂ i ∈ I R i ) = ⋂ i ∈ I ( L × R i ) (Left-distributivity of × over ⋂ i ∈ I when I ≠ ∅ ) ( ⋃ i ∈ I L i ) × R = ⋃ i ∈ I ( L i × R ) (Right-distributivity of × over ∪ ) ( ⋂ i ∈ I L i ) × R = ⋂ i ∈ I ( L i × R ) (Right-distributivity of × over ⋂ i ∈ I when I ≠ ∅ ) {\displaystyle {\begin{alignedat}{5}L\times \left(\bigcup _{i\in I}R_{i}\right)&\;=\;\;&&\bigcup _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times \left(\bigcap _{i\in I}R_{i}\right)&\;=\;\;&&\bigcap _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\left(\bigcup _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcup _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]\left(\bigcap _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcap _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\end{alignedat}}} ==== Distributing arbitrary Π over arbitrary ⋃ ==== Suppose that


id: list-of-set-identities-and-relations-20260505-19
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-19.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( ∏ i ∈ I L i ) ∖ ∏ i ∈ I R i = ⋃ j ∈ I ∏ i ∈ I { L j ∖ R j if i = j L i if i ≠ j = ⋃ j ∈ I [ ( L j ∖ R j ) × ∏ j ≠ i i ∈ I , L i ] = ⋃ L j ⊈ R j j ∈ I , [ ( L j ∖ R j ) × ∏ j ≠ i i ∈ I , L i ] {\displaystyle {\begin{alignedat}{9}\left(\prod _{i\in I}L_{i}\right)~\setminus ~\prod _{i\in I}R_{i}~&=~\;~\bigcup _{j\in I}\;~\prod _{i\in I}{\begin{cases}L_{j}\,\setminus \,R_{j}&{\text{ if }}i=j\\L_{i}&{\text{ if }}i\neq j\\\end{cases}}\\[0.5ex]~&=~\;~\bigcup _{j\in I}\;~{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.5ex]~&=~\bigcup _{\stackrel {j\in I,}{L_{j}\not \subseteq R_{j}}}{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.3ex]\end{alignedat}}} so for instance, ( L × R ) ∖ ( L 2 × R 2 ) = [ ( L ∖ L 2 ) × R ] ∪ [ L × ( R ∖ R 2 ) ] {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\setminus ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]~\cup ~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}} and ( L × M × R ) ∖ ( L 2 × M 2 ×


id: list-of-set-identities-and-relations-20260505-20
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-20.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Any such set R {\displaystyle R} necessarily contains f ( A ) {\displaystyle f(A)} as a subset. Any set not entirely contained in the domain of f {\displaystyle f} cannot be f {\displaystyle f} -saturated. A = f − 1 ( f ( A ) ) . {\displaystyle A=f^{-1}(f(A)).} A ⊇ f − 1 ( f ( A ) ) {\displaystyle A\supseteq f^{-1}(f(A))} and A ⊆ domain ⁡ f . {\displaystyle A\subseteq \operatorname {domain} f.} The inclusion L ∩ domain ⁡ f ⊆ f − 1 ( f ( L ) ) {\displaystyle L\cap \operatorname {domain} f\subseteq f^{-1}(f(L))} always holds, where if A ⊆ domain ⁡ f {\displaystyle A\subseteq \operatorname {domain} f} then this becomes A ⊆ f − 1 ( f ( A ) ) . {\displaystyle A\subseteq f^{-1}(f(A)).} A ⊆ domain ⁡ f {\displaystyle A\subseteq \operatorname {domain} f} and if a ∈ A {\displaystyle a\in A} and x ∈ domain ⁡ f {\displaystyle x\in \operatorname {domain} f} satisfy f ( x ) = f ( a ) , {\displaystyle f(x)=f(a),} then x ∈ A . {\displaystyle x\in A.} Whenever a fiber of f {\displaystyle f} intersects A , {\displaystyle A,} then A {\displaystyle A} contains the entire fiber.


id: list-of-set-identities-and-relations-20260505-21
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-21.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
If ( A i ) i ∈ I {\displaystyle \left(A_{i}\right)_{i\in I}} is a family of arbitrary subsets of X = domain ⁡ f , {\displaystyle X=\operatorname {domain} f,} which means that A i ⊆ X {\displaystyle A_{i}\subseteq X} for all i , {\displaystyle i,} then Conditional Equality 10a becomes: === (Pre)Images of binary set operations === Throughout, let L {\displaystyle L} and R {\displaystyle R} be any sets and let f : X → Y {\displaystyle f:X\to Y} be any function. Summary As the table below shows, set equality is not guaranteed only for images of: intersections, set subtractions, and symmetric differences. Preimages preserve set operations Preimages of sets are well-behaved with respect to all basic set operations: f − 1 ( L ∪ R ) = f − 1 ( L ) ∪ f − 1 ( R ) f − 1 ( L ∩ R ) = f − 1 ( L ) ∩ f − 1 ( R ) f − 1 ( L ∖ R ) = f − 1 ( L ) ∖ f − 1 ( R ) f − 1 ( L △ R ) = f − 1 ( L ) △


id: list-of-set-identities-and-relations-20260505-22
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-22.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
The containment f ( L △ R ) ⊋ f ( L ) △ f ( R ) {\displaystyle ~f(L~\triangle ~R)~\supsetneq ~f(L)~\triangle ~f(R)~} is strict: { y } = f ( L △ R ) ≠ f ( L ) △ f ( R ) = { y } △ { y } = ∅ {\displaystyle \{y\}~=~f\left(L~\triangle ~R\right)~\neq ~f(L)~\triangle ~f(R)~=~\{y\}\triangle \{y\}~=~\varnothing } In words: functions might not distribute over symmetric difference △ {\displaystyle \,\triangle \,} (which can be defined as the set subtraction of two sets: L △ R = ( L ∪ R ) ∖ ( L ∩ R ) {\displaystyle L\triangle R=(L\cup R)\setminus (L\cap R)} ). The containment f ( L ∩ R ) ⊊ f ( L ) ∩ f ( R ) {\displaystyle ~f(L\cap R)~\subsetneq ~f(L)\cap f(R)~} is strict: ∅ = f ( ∅ ) = f ( L ∩ R ) ≠ f ( L ) ∩ f ( R ) = { y } ∩ { y } = { y } {\displaystyle \varnothing ~=~f(\varnothing )~=~f(L\cap R)~\neq ~f(L)\cap f(R)~=~\{y\}\cap \{y\}~=~\{y\}} In words: functions might not distribute over set intersection ∩ {\displaystyle \,\cap \,} (which can be defined as the set subtraction of two sets:


id: list-of-set-identities-and-relations-20260505-23
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-23.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
f ( L △ R ) = f ( L ) △ f ( R ) for all L , R ⊆ X . {\displaystyle f(L\,\triangle \,R)=f(L)\,\triangle \,f(R)\;{\text{ for all }}\,L,R\subseteq X.} (The equals sign = {\displaystyle \,=\,} can be replaced with ⊆ {\displaystyle \,\subseteq \,} ). Any one of the four statements (b) - (e) but with the words "for all" replaced with any one of the following: "for all singleton subsets" In particular, the statement that results from (d) gives a characterization of injectivity that explicitly involves only one point (rather than two): f {\displaystyle f} is injective if and only if f ( x ) ∉ f ( X ∖ { x } ) for every x ∈ X . {\displaystyle f(x)\not \in f(X\setminus \{x\})\;{\text{ for every }}\,x\in X.} "for all disjoint singleton subsets" For statement (d), this is the same as: "for all singleton subsets" (because the definition of "pairwise disjoint" is satisfies vacuously by any family that consists of exactly 1 set). "for all disjoint subsets" In particular, if a map is not known to be injective then barring additional information, there is no guarantee that any of the equalities in statements (b) - (e)


id: list-of-set-identities-and-relations-20260505-24
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-24.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∩ domain ⁡ f ⊆ R {\displaystyle L\cap \operatorname {domain} f\,\subseteq R} R ⊆ L {\displaystyle R\subseteq L} L ⊆ R {\displaystyle L\subseteq R} In addition, the following always hold: f ( f − 1 ( L ) ∩ R ) = L ∩ f ( R ) {\displaystyle f\left(f^{-1}(L)\cap R\right)~=~L\cap f(R)} f ( f − 1 ( L ) ∪ R ) = ( L ∩ Im ⁡ f ) ∪ f ( R ) {\displaystyle f\left(f^{-1}(L)\cup R\right)~=~(L\cap \operatorname {Im} f)\cup f(R)} ===== Conditions for f(L\R) = f(L)\f(R) ===== f ( L ∖ R ) ⊇ f ( L ) ∖ f ( R ) always holds {\displaystyle f(L\setminus R)~\supseteq ~f(L)\setminus f(R)\qquad \qquad {\text{ always holds}}} Characterizations of equality: The following statements are equivalent: f ( L ∖ R ) = f ( L ) ∖ f ( R ) {\displaystyle f(L\setminus R)~=~f(L)\setminus f(R)} f ( L ∖ R ) ⊆ f ( L ) ∖ f ( R ) {\displaystyle f(L\setminus R)~\subseteq ~f(L)\setminus f(R)} L ∩ f − 1 ( f ( R ) ) ⊆ R {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~R} L ∩ f − 1 ( f ( R ) ) = L ∩ R ∩


id: list-of-set-identities-and-relations-20260505-25
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-25.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Characterizations of equality: The following statements are equivalent: f ( L △ R ) = f ( L ) △ f ( R ) {\displaystyle f\left(L~\triangle ~R\right)=f(L)~\triangle ~f(R)} f ( L △ R ) ⊆ f ( L ) △ f ( R ) {\displaystyle f\left(L~\triangle ~R\right)\subseteq f(L)~\triangle ~f(R)} f ( L ∖ R ) = f ( L ) ∖ f ( R ) {\displaystyle f(L\,\setminus \,R)=f(L)\,\setminus \,f(R)} and f ( R ∖ L ) = f ( R ) ∖ f ( L ) {\displaystyle f(R\,\setminus \,L)=f(R)\,\setminus \,f(L)} f ( L ∖ R ) ⊆ f ( L ) ∖ f ( R ) {\displaystyle f(L\,\setminus \,R)\subseteq f(L)\,\setminus \,f(R)} and f ( R ∖ L ) ⊆ f ( R ) ∖ f ( L ) {\displaystyle f(R\,\setminus \,L)\subseteq f(R)\,\setminus \,f(L)} L ∩ f − 1 ( f ( R ) ) ⊆ R {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~R} and R ∩ f − 1 ( f ( L ) ) ⊆ L {\displaystyle R\cap f^{-1}(f(L))~\subseteq ~L} The inclusions L ∩ f − 1 ( f ( R ) ) ⊆ f − 1 ( f ( L ) ) {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~f^{-1}(f(L))} and R ∩ f −


id: list-of-set-identities-and-relations-20260505-26
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-26.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
f ( X ∖ R ) = f ( X ) ∖ { y ∈ f ( R ) : f − 1 ( y ) ⊆ R } . {\displaystyle f(X\setminus R)\,=\,f(X)\setminus \left\{y\in f(R):f^{-1}(y)\subseteq R\right\}.} From this it follows that: f ( X ∖ R ) = f ( X ) ∖ f ( R ) if and only if f ( R ) = { y ∈ f ( R ) : f − 1 ( y ) ⊆ R } if and only if f − 1 ( f ( R ) ) ⊆ R . {\displaystyle f(X\setminus R)=f(X)\setminus f(R)\quad {\text{ if and only if }}\quad f(R)=\left\{y\in f(R):f^{-1}(y)\subseteq R\right\}\quad {\text{ if and only if }}\quad f^{-1}(f(R))\subseteq R.} If f R := { y ∈ f ( X ) : f − 1 ( y ) ⊆ R } {\displaystyle f_{R}:=\left\{y\in f(X):f^{-1}(y)\subseteq R\right\}} then f ( X ∖ R ) = f ( X ) ∖ f R , {\displaystyle f(X\setminus R)=f(X)\setminus f_{R},} which can be written more symmetrically as f ( X ∖ R ) = f X ∖ f R {\displaystyle f(X\setminus R)=f_{X}\setminus f_{R}} (since f X = f ( X ) {\displaystyle f_{X}=f(X)} ). =====


id: list-of-set-identities-and-relations-20260505-27
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-27.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∩ f − 1 ( y ) ⊆ L ∖ R {\displaystyle L\cap f^{-1}(y)\subseteq L\setminus R~} if and only if L ∩ R ∩ f − 1 ( y ) = ∅ {\displaystyle ~L\cap R\cap f^{-1}(y)=\varnothing ~} if and only if R ∩ f − 1 ( y ) ⊆ R ∖ L {\displaystyle ~R\cap f^{-1}(y)\subseteq R\setminus L~} if and only if y ∉ f ( L ∩ R ) . {\displaystyle ~y\not \in f(L\cap R).} The sets U {\displaystyle U} and V {\displaystyle V} mentioned above could, in particular, be any of the sets f ( L ∪ R ) , Im ⁡ f , {\displaystyle f(L\cup R),\;\operatorname {Im} f,} or Y , {\displaystyle Y,} for example. === (Pre)Images of set operations on (pre)images === Let L {\displaystyle L} and R {\displaystyle R} be arbitrary sets, f : X → Y {\displaystyle f:X\to Y} be any map, and let A ⊆ X {\displaystyle A\subseteq X} and C ⊆ Y . {\displaystyle C\subseteq Y.} (Pre)Images of operations on images Since f ( L ) ∖ f ( L ∖ R ) = { y ∈ f ( L ∩ R ) : L ∩ f − 1 ( y


id: list-of-set-identities-and-relations-20260505-28
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-28.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
The map F ∙ = ( F j ) j ∈ J : X → ∏ j ∈ J Y j {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}} should not be confused with the Cartesian product ∏ j ∈ J F j {\displaystyle \prod _{j\in J}F_{j}} of these maps, which is by definition is the map ∏ j ∈ J F j : ∏ j ∈ J X → ∏ j ∈ J Y j ( x j ) j ∈ J ↦ ( F j ( x j ) ) j ∈ J {\displaystyle {\begin{alignedat}{4}\prod _{j\in J}F_{j}:\;&&\prod _{j\in J}X&&~\;\to \;~&\prod _{j\in J}Y_{j}\\[0.3ex]&&\left(x_{j}\right)_{j\in J}&&~\;\mapsto \;~&\left(F_{j}\left(x_{j}\right)\right)_{j\in J}\\\end{alignedat}}} with domain ∏ j ∈ J X = X J {\displaystyle \prod _{j\in J}X=X^{J}} rather than X . {\displaystyle X.} Preimage and images of a Cartesian product Suppose F ∙ = ( F j ) j ∈ J : X → ∏ j ∈ J Y j . {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}.} If A ⊆ X {\displaystyle A~\subseteq ~X} then F ∙ ( A ) ⊆ ∏ j ∈ J F j ( A ) . {\displaystyle F_{\bullet }(A)~~\;\color {Red}{\subseteq }\color {Black}{}\;~~\prod _{j\in J}F_{j}(A).} If B ⊆ ∏ j ∈ J


id: list-of-set-identities-and-relations-20260505-29
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-29.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
where N {\displaystyle \mathbb {N} } denotes the natural numbers. A notation S ∙ = ( S i ) i ∈ I {\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}} indicates that S ∙ {\displaystyle S_{\bullet }} is a net directed by ( I , ≤ ) , {\displaystyle (I,\leq ),} which (by definition) is a sequence if the set I , {\displaystyle I,} which is called the net's indexing set, is the natural numbers (that is, if I = N {\displaystyle I=\mathbb {N} } ) and ≤ {\displaystyle \,\leq \,} is the natural order on N . {\displaystyle \mathbb {N} .} Disjoint and monotone sequences of sets If S i ∩ S j = ∅ {\displaystyle S_{i}\cap S_{j}=\varnothing } for all distinct indices i ≠ j {\displaystyle i\neq j} then S ∙ {\displaystyle S_{\bullet }} is called a pairwise disjoint or simply a disjoint. A sequence or net S ∙ {\displaystyle S_{\bullet }} of set is called increasing or non-decreasing if (resp. decreasing or non-increasing) if for all indices i ≤ j , {\displaystyle i\leq j,} S i ⊆ S j {\displaystyle S_{i}\subseteq S_{j}} (resp. S i ⊇ S j {\displaystyle S_{i}\supseteq S_{j}} ). A sequence or net S ∙ {\displaystyle S_{\bullet }}


id: list-of-set-identities-and-relations-20260505-30
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-30.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
closed under finite intersections (resp. closed under finite unions) if whenever L , R ∈ L {\displaystyle L,R\in {\mathcal {L}}} then L ∩ R ∈ L {\displaystyle L\cap R\in {\mathcal {L}}} (respectively, L ∪ R ∈ L {\displaystyle L\cup R\in {\mathcal {L}}} ). closed under countable intersections (resp. closed under countable unions) if whenever L 1 , L 2 , L 3 , … {\displaystyle L_{1},L_{2},L_{3},\ldots } are elements of L {\displaystyle {\mathcal {L}}} then so is their intersections ⋂ i = 1 ∞ L i := L 1 ∩ L 2 ∩ L 3 ∩ ⋯ {\displaystyle \bigcap _{i=1}^{\infty }L_{i}:=L_{1}\cap L_{2}\cap L_{3}\cap \cdots } (resp. so is their union ⋃ i = 1 ∞ L i := L 1 ∪ L 2 ∪ L 3 ∪ ⋯ {\displaystyle \bigcup _{i=1}^{\infty }L_{i}:=L_{1}\cup L_{2}\cup L_{3}\cup \cdots } ). closed under complementation in (or with respect to) X {\displaystyle X} if whenever L ∈ L {\displaystyle L\in {\mathcal {L}}} then X ∖ L ∈ L . {\displaystyle X\setminus L\in {\mathcal {L}}.} A family L {\displaystyle {\mathcal {L}}} of sets is called a/an: π−system if L ≠ ∅ {\displaystyle {\mathcal {L}}\neq \varnothing } and L {\displaystyle {\mathcal {L}}} is closed under finite-intersections. Every


id: list-of-set-identities-and-relations-20260505-31
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-31.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ( ∪ ) { X } = { X } if L ≠ ∅ {\displaystyle {\mathcal {L}}\;(\cup )\;\{X\}=\{X\}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing } L ( ∩ ) { ∅ } = { ∅ } if L ≠ ∅ {\displaystyle {\mathcal {L}}\;(\cap )\;\{\varnothing \}=\{\varnothing \}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing } L ( ∪ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\cup )\;\varnothing =\varnothing } L ( ∩ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\cap )\;\varnothing =\varnothing } L ( ∖ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\setminus )\;\varnothing =\varnothing } ∅ ( ∖ ) R = ∅ {\displaystyle \varnothing \;(\setminus )\;{\mathcal {R}}=\varnothing } === Power set === P ( L ∩ R ) = P ( L ) ∩ P ( R ) {\displaystyle {\mathcal {P}}(L\cap R)~=~{\mathcal {P}}(L)\cap {\mathcal {P}}(R)} P ( L ∪ R ) = P ( L ) ( ∪ ) P ( R ) ⊇ P ( L ) ∪ P ( R ) . {\displaystyle {\mathcal {P}}(L\cup R)~=~{\mathcal {P}}(L)\ (\cup )\ {\mathcal {P}}(R)~\supseteq ~{\mathcal {P}}(L)\cup {\mathcal {P}}(R).} If L {\displaystyle L} and R {\displaystyle R} are subsets of a vector space X {\displaystyle X} and if s {\displaystyle s} is a scalar then P ( s


id: list-of-set-identities-and-relations-20260505-32
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-32.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
== References == Artin, Michael (1991). Algebra. Prentice Hall. ISBN 81-203-0871-9. Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures. Springer. ISBN 1-85233-905-5.. Bylinski, Czeslaw (2004). "Some Basic Properties of Sets". Journal of Formalized Mathematics. 1. Retrieved 5 October 2021. Courant, Richard, Herbert Robbins, Ian Stewart, What is mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. "SUPPLEMENT TO CHAPTER II THE ALGEBRA OF SETS". Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011. Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303. Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020. Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. {{cite book}}: ISBN / Date incompatibility


id: list-of-alternative-set-theories-20260505-0
file: data/en.wikipedia.org/wiki/List_of_alternative_set_theories-0.md
category: reference
title: List of alternative set theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_alternative_set_theories
date_saved: 2026-05-05T08:16:54.419564+00:00
instance: kb-cron
parent: list-of-alternative-set-theories-20260505-0
summary:
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 ==


id: list-of-set-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_set_theory_topics-0.md
category: reference
title: List of set theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_theory_topics
date_saved: 2026-05-05T08:16:55.603082+00:00
instance: kb-cron
parent: list-of-set-theory-topics-20260505-0
summary:
This page is a list of articles related to set theory. == Articles on individual set theory topics == == Lists related to set theory == Glossary of set theory List of large cardinal properties List of properties of sets of reals List of set identities and relations == Set theorists == == Societies and organizations == Association for Symbolic Logic The Cabal


id: list-of-types-of-sets-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_sets-0.md
category: reference
title: List of types of sets
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_sets
date_saved: 2026-05-05T08:16:56.774657+00:00
instance: kb-cron
parent: list-of-types-of-sets-20260505-0
summary:
Sets can be classified according to the properties they have. == Relative to set theory == Empty set Finite set, Infinite set Countable set, Uncountable set Power set == Relative to a topology == Closed set Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere dense set == Relative to a metric == Bounded set Totally bounded set == Relative to measurability == Borel set Baire set Measurable set, Non-measurable set Universally measurable set == Relative to a measure == Negligible set Null set Haar null set == In a linear space == Convex set Balanced set, Absolutely convex set == Relative to the real/complex numbers == Fractal set == Ways of defining sets/Relation to descriptive set theory == Recursive set Recursively enumerable set Arithmetical set Diophantine set Hyperarithmetical set Analytical set Analytic set, Coanalytic set Suslin set Projective set Inhabited set == More general objects still called sets == Multiset == See also == List of set identities and relations – Equalities for combinations of sets List of types of functions


id: list-of-shapes-with-known-packing-constant-20260505-0
file: data/en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant-0.md
category: reference
title: List of shapes with known packing constant
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant
date_saved: 2026-05-05T08:16:58.046518+00:00
instance: kb-cron
parent: list-of-shapes-with-known-packing-constant-20260505-0
summary:
The packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown. The following is a list of bodies in Euclidean spaces whose packing constant is known. Fejes Tóth proved that in the plane, a point symmetric body has a packing constant that is equal to its translative packing constant and its lattice packing constant. Therefore, any such body for which the lattice packing constant was previously known, such as any ellipse, consequently has a known packing constant. In addition to these bodies, the packing constants of hyperspheres in 8 and 24 dimensions are almost exactly known. == References ==


id: list-of-solids-derived-from-the-sphere-20260505-0
file: data/en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere-0.md
category: reference
title: List of solids derived from the sphere
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere
date_saved: 2026-05-05T08:16:59.236190+00:00
instance: kb-cron
parent: list-of-solids-derived-from-the-sphere-20260505-0
summary:
This page lists solids derived from a sphere. == Solids from cutting a sphere with one or more planes == Dome Spherical cap Spherical sector Spherical segment Spherical shell Spherical wedge == Solids from deforming a sphere == Ellipsoid Spheroid Solid bounded by Morin surface Any Genus 0 surface == Solids from intersecting a sphere with other solids or curved planes == Reuleaux tetrahedron Spherical lens == Notes ==


id: list-of-spherical-symmetry-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_spherical_symmetry_groups-0.md
category: reference
title: List of spherical symmetry groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups
date_saved: 2026-05-05T08:17:01.639486+00:00
instance: kb-cron
parent: list-of-spherical-symmetry-groups-20260505-0
summary:
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway used a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6. == Involutional symmetry == There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and central point symmetry (Ci). == Cyclic symmetry == There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry) == Dihedral symmetry == There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case).


id: sum-of-squares-20260505-0
file: data/en.wikipedia.org/wiki/Sum_of_squares-0.md
category: reference
title: Sum of squares
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sum_of_squares
date_saved: 2026-05-05T08:17:21.476492+00:00
instance: kb-cron
parent: sum-of-squares-20260505-0
summary:
In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts: == Statistics == For partitioning of variance, see Partition of sums of squares For the "sum of squared deviations", see Least squares For the "sum of squared differences", see Mean squared error For the "sum of squared error", see Residual sum of squares For the "sum of squares due to lack of fit", see Lack-of-fit sum of squares For sums of squares relating to model predictions, see Explained sum of squares For sums of squares relating to observations, see Total sum of squares For sums of squared deviations, see Squared deviations from the mean For modelling involving sums of squares, see Analysis of variance For modelling involving the multivariate generalisation of sums of squares, see Multivariate analysis of variance == Number theory == Fermat's theorem on sums of two squares says which primes are sums of two squares. The sum of two squares theorem generalizes Fermat's theorem to specify which composite numbers are the sums of two squares. Legendre's three-square theorem states which numbers can be expressed as the sum of three squares For representing an integer as a sum of squares of 4 integers, see


id: sums-of-powers-20260505-0
file: data/en.wikipedia.org/wiki/Sums_of_powers-0.md
category: reference
title: Sums of powers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sums_of_powers
date_saved: 2026-05-05T08:17:23.640704+00:00
instance: kb-cron
parent: sums-of-powers-20260505-0
summary:
In mathematics and statistics, sums of powers occur in a number of contexts: Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities. There are only finitely many positive integers that are not sums of distinct squares. The largest one is 128. The same applies for sums of distinct cubes (largest one is 12,758), distinct fourth powers (largest is 5,134,240), etc. See for a generalization to sums of polynomials. Faulhaber's formula expresses 1 k + 2 k + 3 k + ⋯ + n k {\displaystyle 1^{k}+2^{k}+3^{k}+\cdots +n^{k}} as a polynomial in n, or alternatively in terms of a Bernoulli polynomial. Fermat's right triangle theorem states that there is no solution in positive integers for a 2 = b 4 + c 4 {\displaystyle a^{2}=b^{4}+c^{4}} and a 4 = b 4 + c 2 {\displaystyle a^{4}=b^{4}+c^{2}} . Fermat's Last Theorem states that x k + y k = z k {\displaystyle x^{k}+y^{k}=z^{k}} is impossible in positive integers with k > 2. The equation of a


id: list-of-sums-of-reciprocals-20260505-0
file: data/en.wikipedia.org/wiki/List_of_sums_of_reciprocals-0.md
category: reference
title: List of sums of reciprocals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_sums_of_reciprocals
date_saved: 2026-05-05T08:17:27.040927+00:00
instance: kb-cron
parent: list-of-sums-of-reciprocals-20260505-0
summary:
In mathematics and number theory, the sum of reciprocals (or sum of inverses) is defined as the sum of reciprocals of some series of positive integers (counting numbers). It is a sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given as a sequence whose nth term is defined as the sum of the first n reciprocals. For sums of reciprocals over a series of finitely many numbers, key questions include whether there is a simple expression for the value of the sum, whether the sum must be less than a certain value, and whether the sum is ever an integer. When the sum of reciprocals is over an infinite series, questions include: First, does the sequence of sums diverge—that is, does it eventually exceed any given number—or does it converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positive integers is said to be large if the sum of its reciprocals diverges, and small if it converges.) Second, if it converges, what is a simple expression for the value it converges to? Is that value rational or irrational, and is that value


id: list-of-sums-of-reciprocals-20260505-1
file: data/en.wikipedia.org/wiki/List_of_sums_of_reciprocals-1.md
category: reference
title: List of sums of reciprocals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_sums_of_reciprocals
date_saved: 2026-05-05T08:17:27.040927+00:00
instance: kb-cron
parent: list-of-sums-of-reciprocals-20260505-0
summary:
A sum-free sequence of increasing positive integers is one for which no number is the sum of any subset of the previous ones. The sum of the reciprocals of the numbers in any sum-free sequence is less than 2.8570 . The sum of the reciprocals of the heptagonal numbers converges to a known value that is not only irrational but also transcendental, and for which there exists a complicated formula. The sum of the reciprocals of the twin primes, of which there may be finitely many or infinitely many, is known to be finite and is called Brun's constant, approximately 1.90216 . (By convention, the reciprocal of 5 {\displaystyle 5} appears twice in the sum.) The sum of the reciprocals of the Proth primes, of which there may be finitely many or infinitely many, is known to be finite, approximately 0.747392479 . The prime quadruplets are pairs of twin primes with only one odd number between them. The sum of the reciprocals of the numbers in prime quadruplets is approximately 0.8706 . The sum of the reciprocals of the perfect powers (including duplicates) is 1 . The sum of the reciprocals of the perfect powers (excluding duplicates) is approximately 0.8745


id: list-of-sums-of-reciprocals-20260505-2
file: data/en.wikipedia.org/wiki/List_of_sums_of_reciprocals-2.md
category: reference
title: List of sums of reciprocals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_sums_of_reciprocals
date_saved: 2026-05-05T08:17:27.040927+00:00
instance: kb-cron
parent: list-of-sums-of-reciprocals-20260505-0
summary:
=== Divergent series === The n-th partial sum of the harmonic series, which is the sum of the reciprocals of the first n positive integers, diverges as n goes to infinity, albeit extremely slowly: The sum of the first 1043 terms is less than 100 . The difference between the cumulative sum and the natural logarithm of n converges to the Euler–Mascheroni constant, commonly denoted as γ , {\displaystyle \gamma ,} which is approximately 0.5772. The sum of the reciprocals of the primes diverges. Given coprime positive integers a and b, the sum of the reciprocal of the primes of the form an + b diverges. This result is used to prove the Dirichlet's theorem, which states that there are infinitely many primes p {\displaystyle p} satisfying p ≡ a ( m o d b ) {\displaystyle p\equiv a{\text{ }}(\mathrm {mod} {\text{ }}b)} . The sum of the reciprocals of the primes of the form 4n + 1 is divergent. By Fermat's theorem on sums of two squares, it follows that the sum of reciprocals of numbers of the form a 2 + b 2 , {\displaystyle a^{2}+b^{2},} where a and b are non-negative integers, not both equal to 0,


id: table-of-mathematical-symbols-by-introduction-date-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_mathematical_symbols_by_introduction_date-0.md
category: reference
title: Table of mathematical symbols by introduction date
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_mathematical_symbols_by_introduction_date
date_saved: 2026-05-05T08:17:28.247985+00:00
instance: kb-cron
parent: table-of-mathematical-symbols-by-introduction-date-20260505-0
summary:
The following table lists many specialized symbols commonly used in modern mathematics, ordered by their introduction date. == See also == History of mathematical notation History of the Hindu–Arabic numeral system Glossary of mathematical symbols List of mathematical symbols by subject Mathematical notation Mathematical operators and symbols in Unicode == Sources == == External links == RapidTables: Math Symbols List Jeff Miller: Earliest Uses of Various Mathematical Symbols


id: list-of-types-of-systems-theory-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_systems_theory-0.md
category: reference
title: List of types of systems theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_systems_theory
date_saved: 2026-05-05T08:17:29.461574+00:00
instance: kb-cron
parent: list-of-types-of-systems-theory-20260505-0
summary:
This list of types of systems theory gives an overview of different types of systems theory, which are mentioned in scientific book titles or articles. The following more than 40 types of systems theory are all explicitly named systems theory and represent a unique conceptual framework in a specific field of science. Systems theory has been formalized since the 1950s, and a long set of specialized systems theories and cybernetics exist. In the beginnings, general systems theory was developed by Ludwig von Bertalanffy to overcome the over-specialisation of the modern times and as a worldview using holism. The systems theories nowadays are closer to the traditional specialisation than to holism, by interdependencies and mutual division by mutually-different specialists. == A == Abstract systems theory (also see: formal system) Action Theory Adaptive systems theory (also see: complex adaptive system) Applied general systems theory (also see: general systems theory) Applied multidimensional systems theory Archaeological systems theory (also see: Systems theory in archaeology) Systems theory in anthropology Associated systems theory == B == Behavioral systems theory Biochemical systems theory Biomatrix systems theory Body system == C == Complex adaptive systems theory (also see: complex adaptive system) Complex systems theory (also see: complex systems)


id: tarjan-s-algorithm-20260505-0
file: data/en.wikipedia.org/wiki/Tarjan's_algorithm-0.md
category: reference
title: Tarjan's algorithm
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tarjan's_algorithm
date_saved: 2026-05-05T08:17:30.674857+00:00
instance: kb-cron
parent: tarjan-s-algorithm-20260505-0
summary:
Tarjan's algorithm may refer to one of several algorithms attributed to Robert Tarjan, including: Tarjan's strongly connected components algorithm Tarjan's off-line lowest common ancestors algorithm Tarjan's algorithm for finding bridges in an undirected graph Tarjan's algorithm for finding simple circuits in a directed graph == See also == List of algorithms == References ==


id: list-of-tessellations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_tessellations-0.md
category: reference
title: List of tessellations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_tessellations
date_saved: 2026-05-05T08:17:31.940951+00:00
instance: kb-cron
parent: list-of-tessellations-20260505-0
summary:
This is a list of tessellations. == Spherical == == Planar == == Hyperbolic == == See also == Tessellation Uniform tiling Convex uniform honeycombs List of k-uniform tilings List of Euclidean uniform tilings Uniform tilings in hyperbolic plane == References ==


id: list-of-theorems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_theorems-0.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures List of derivatives and integrals in alternative calculi List of equations List of fundamental theorems List of hypotheses List of inequalities Lists of integrals List of laws List of lemmas List of limits List of logarithmic identities List of mathematical functions List of mathematical identities List of mathematical proofs List of misnamed theorems List of scientific laws List of theories Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields. == Logics and foundations == Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical logic) Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory) Church–Rosser theorem (lambda calculus) Compactness theorem (mathematical logic) Conservativity theorem (mathematical logic) Craig's theorem (mathematical logic) Craig's interpolation theorem (mathematical logic) Cut-elimination theorem (proof theory) Deduction theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set theory) Feferman–Vaught theorem (model theory) Friedberg–Muchnik theorem (mathematical logic) Fundamental theorem of equivalence relations


id: list-of-theorems-20260505-1
file: data/en.wikipedia.org/wiki/List_of_theorems-1.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Number theory == 15 and 290 theorems (number theory) Albert–Brauer–Hasse–Noether theorem (algebras) Ankeny–Artin–Chowla theorem (number theory) Apéry's theorem (number theory) Artin–Verdier duality theorem (number theory) ATS theorem (number theory) Auxiliary polynomial theorem (Diophantine approximation) Ax–Kochen theorem (number theory) Baker's theorem (number theory) Barban–Davenport–Halberstam theorem (analytic number theory) Basel problem (mathematical analysis) Beatty's theorem (Diophantine approximation) Behrend's theorem (number theory) Bertrand's postulate (number theory) Birch's theorem (algebraic number theory) Bombieri's theorem (number theory) Bombieri–Friedlander–Iwaniec theorem (number theory) Brauer–Siegel theorem (number theory) Brun's theorem (number theory) Brun–Titchmarsh theorem (number theory) Carmichael's theorem (Fibonacci numbers) Chebotarev's density theorem (number theory) Chen's theorem (number theory) Chowla–Mordell theorem (number theory) Cohn's irreducibility criterion (polynomials) Critical line theorem (number theory) Davenport–Schmidt theorem (number theory, Diophantine approximations) Dirichlet's approximation theorem (Diophantine approximations) Dirichlet's theorem on arithmetic progressions (number theory) Dirichlet's unit theorem (algebraic number theory) Equidistribution theorem (ergodic theory) Erdős–Kac theorem (number theory) Euclid's theorem (number theory) Euclid–Euler theorem (number theory) Euler's theorem (number theory) Fermat's Last Theorem (number theory) Fermat's little theorem (number theory) Fermat's theorem on sums of two squares (number theory) Ferrero–Washington theorem (algebraic number theory) Ford's theorem (number theory) Franel–Landau theorem (number theory) Gelfond–Schneider theorem (transcendental number theory) Glaisher's theorem (number theory)


id: list-of-theorems-20260505-2
file: data/en.wikipedia.org/wiki/List_of_theorems-2.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Linear and multilinear algebra; matrix theory == Amitsur–Levitzki theorem (linear algebra) Binomial inverse theorem (linear algebra) Birkhoff–Von Neumann theorem (linear algebra) Bregman–Minc inequality (discrete mathematics) Cauchy-Binet formula (linear algebra) Cayley–Hamilton theorem (Linear algebra) Dimension theorem for vector spaces (vector spaces, linear algebra) Euler's rotation theorem (geometry) Exchange theorem (linear algebra) Gamas's Theorem (multilinear algebra) Gershgorin circle theorem (matrix theory) Inverse eigenvalues theorem (linear algebra) Perron–Frobenius theorem (matrix theory) Principal axis theorem (linear algebra) Rank–nullity theorem (linear algebra) Rouché–Capelli theorem (Linear algebra) Sinkhorn's theorem (matrix theory) Specht's theorem (matrix theory) Spectral theorem (linear algebra, functional analysis) Sylvester's determinant theorem (determinants) Sylvester's law of inertia (quadratic forms) Witt's theorem (quadratic forms) == Associative rings and algebras == Artin–Wedderburn theorem (abstract algebra) Artin–Zorn theorem (algebra) Brauer–Cartan–Hua theorem (ring theory) Frobenius theorem (abstract algebras) Goldie's theorem (ring theory) Jacobson density theorem (ring theory) Jacobson–Bourbaki theorem (algebra) Levitzky's theorem (ring theory) Regev's theorem (ring theory) Skolem–Noether theorem (simple algebras) Wedderburn's little theorem (ring theory) Wedderburn's theorem (abstract algebra) == Nonassociative rings and algebras == Ado's theorem (Lie algebra) Goddard–Thorn theorem (vertex algebras) Hurwitz's theorem (normed division algebras) Jacobson–Morozov theorem (Lie algebra) Levi's theorem (Lie groups) Lie's theorem (Lie algebra) Poincaré–Birkhoff–Witt theorem (universal enveloping algebras)


id: list-of-theorems-20260505-3
file: data/en.wikipedia.org/wiki/List_of_theorems-3.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Functions of a complex variable == Akhiezer's theorem (complex analysis) Arakelyan's theorem (complex analysis) Area theorem (conformal mapping) (complex analysis) Beurling–Lax theorem (Hardy spaces) Bloch's theorem (complex analysis) Bôcher's theorem (complex analysis) Borel–Carathéodory theorem (complex analysis) Branching theorem (complex manifold) Carathéodory's theorem (complex analysis) Carleson–Jacobs theorem (complex analysis) Carlson's theorem (complex analysis) Cauchy integral theorem (complex analysis) Cauchy–Hadamard theorem (complex analysis) Clifford's theorem on special divisors (algebraic curves) Corona theorem (complex analysis) de Branges's theorem (complex analysis) De Franchis theorem (Riemann surfaces) Edge-of-the-wedge theorem (complex analysis) Farrell–Markushevich theorem (complex analysis) Fatou's theorem (complex analysis) Fundamental theorem of algebra (complex analysis) Gauss–Lucas theorem (complex analysis) Grunsky's theorem (complex analysis) Hadamard three-circle theorem (complex analysis) Hadamard three-lines theorem (complex analysis) Hardy's theorem (complex analysis) Hartogs–Rosenthal theorem (complex analysis) Harnack's theorem (complex analysis) Hurwitz's automorphisms theorem (algebraic curves) Hurwitz's theorem (complex analysis) Identity theorem (complex analysis) Identity theorem for Riemann surfaces (Riemann surfaces) Koebe 1/4 theorem (complex analysis) Lagrange inversion theorem (mathematical analysis, combinatorics) Lagrange reversion theorem (mathematical analysis, combinatorics) Laurent expansion theorem (complex analysis) Lindelöf's theorem (complex analysis) Liouville's theorem (complex analysis, entire functions) Liouville's theorem (conformal mappings) Looman–Menchoff theorem (complex analysis) Marden's theorem (polynomials) Mergelyan's theorem (complex analysis) Measurable Riemann


id: list-of-theorems-20260505-4
file: data/en.wikipedia.org/wiki/List_of_theorems-4.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Operator theory == Aronszajn–Smith theorem (functional analysis) Atiyah–Singer index theorem (elliptic differential operators, harmonic analysis) Atkinson's theorem (operator theory) Babuška–Lax–Milgram theorem (partial differential equations) Banach–Stone theorem (operator theory) Bauer–Fike theorem (spectral theory) Bounded inverse theorem (operator theory) Browder–Minty theorem (operator theory) Choi's theorem on completely positive maps (operator theory) Commutation theorem (von Neumann algebra) Fuglede's theorem (functional analysis) Gelfand–Mazur theorem (Banach algebra) Gelfand–Naimark theorem (functional analysis) Hardy–Littlewood maximal theorem (real analysis) Hellinger–Toeplitz theorem (functional analysis) Hilbert–Schmidt theorem (functional analysis) Hille–Yosida theorem (functional analysis) Kaplansky density theorem (von Neumann algebra) Kuiper's theorem (operator theory, topology) Lax–Milgram theorem (partial differential equations) Lions–Lax–Milgram theorem (partial differential equations) Lumer–Phillips theorem (semigroup theory) Marcinkiewicz theorem (functional analysis) Mazur–Ulam theorem (normed spaces) Mercer's theorem (functional analysis) Min-max theorem (functional analysis) Moreau's theorem (convex analysis) Nash–Moser theorem (mathematical analysis) Open mapping theorem (functional analysis) Peetre theorem (functional analysis) Riesz–Thorin theorem (functional analysis) Ryll-Nardzewski fixed-point theorem (functional analysis) Sazonov's theorem (functional analysis) Schröder–Bernstein theorems for operator algebras (operator algebras) Stinespring factorization theorem (operator theory) Stone's theorem on one-parameter unitary groups (functional analysis) Sz.-Nagy's dilation theorem (operator theory) Tomita's theorem (operator algebras) Von Neumann bicommutant theorem (functional analysis) Von Neumann's theorem (operator theory) == Calculus of variations and


id: list-of-theorems-20260505-5
file: data/en.wikipedia.org/wiki/List_of_theorems-5.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== General topology == Arzelà–Ascoli theorem (functional analysis) Baire category theorem (topology, metric spaces) Bing metrization theorem (general topology) Bolzano–Weierstrass theorem (real analysis, calculus) Borsuk–Ulam theorem (topology) Brouwer fixed-point theorem (topology) Cantor's intersection theorem (real analysis) Closed graph theorem (functional analysis) Extreme value theorem (calculus) Fixed-point theorems in infinite-dimensional spaces Hairy ball theorem (algebraic topology) Hahn–Mazurkiewicz theorem (continuum theory) Heine–Borel theorem (real analysis) Heine–Cantor theorem (metric geometry) Jordan curve theorem (topology) Kuratowski's closure-complement problem (topology) Lebesgue covering dimension (dimension theory) Metrization theorems (topological spaces) Nagata–Smirnov metrization theorem(general topology) Netto's theorem (topology) Parovicenko's theorem (topology) Tietze extension theorem (general topology) Tychonoff's theorem (general topology) == Algebraic topology == Acyclic models theorem (algebraic topology) Blakers–Massey theorem (homotopy theory) Bott periodicity theorem (homotopy theory) Brown's representability theorem (homotopy theory) Cellular approximation theorem (algebraic topology) Dold–Thom theorem (algebraic topology) Eilenberg–Ganea theorem (homological algebra, algebraic topology) Eilenberg–Zilber theorem (algebraic topology) Euler's polyhedron theorem (polyhedra) Excision theorem (homology theory) Freudenthal suspension theorem (homotopy theory) Hilton–Milnor theorem (algebraic topology) Homotopy excision theorem (algebraic topology) Hopf theorem (differential topology) Hurewicz theorem (algebraic topology) Künneth theorem (algebraic topology) Invariance of domain (algebraic topology) Landweber exact functor theorem (algebraic topology) Lefschetz fixed-point theorem (fixed points, algebraic topology) Lefschetz–Hopf theorem (topology)


id: list-of-theorems-20260505-6
file: data/en.wikipedia.org/wiki/List_of_theorems-6.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Mechanics of deformable solids == Betti's theorem (physics) Castigliano's first and second theorems (structural analysis) Clapeyron's theorem (physics) Saint-Venant's theorem (physics) Theorem of three moments (physics) == Fluid mechanics == Buckingham π theorem (dimensional analysis) Helmholtz's theorems (physics) Kelvin's circulation theorem (physics) Kutta–Joukowski theorem (physics) Reynolds transport theorem (fluid dynamics) Taylor–Proudman theorem (physics) == Optics, electromagnetic theory == Blondel's theorem (electric power) (physics) Earnshaw's theorem (electrostatics) Maximum power theorem (electrical circuits) Norton's theorem (electrical networks) Optical theorem (physics) Poynting's theorem (physics) Thévenin's theorem (electrical circuits) == Classical thermodynamics, heat transfer == Carnot's theorem (thermodynamics) Clausius theorem (physics) == Quantum theory == Adiabatic theorem (physics) Bell's theorem (quantum mechanics) Bogoliubov–Parasyuk theorem (quantum field theory) Byers–Yang theorem (quantum mechanics) C-theorem (physics) Cluster decomposition theorem (quantum field theory) Coleman–Mandula theorem (quantum field theory) Elitzur's theorem (quantum field theory, statistical field theory) Furry's theorem (quantum field theory) Gell-Mann and Low theorem (quantum field theory) Gleason's theorem (Hilbert space) Goldstone's theorem (physics) Haag's theorem (quantum field theory) Haag–Łopuszański–Sohnius theorem (physics) Hellmann–Feynman theorem (physics) Kinoshita–Lee–Nauenberg theorem (quantum field theory) Kochen–Specker theorem (physics) Kramers' theorem (physics) Nielsen–Ninomiya theorem (quantum field theory) No-broadcasting theorem (quantum information theory) No-cloning theorem (quantum computation) No-communication theorem (quantum information theory) No-deleting


id: list-of-theorems-called-fundamental-20260505-0
file: data/en.wikipedia.org/wiki/List_of_theorems_called_fundamental-0.md
category: reference
title: List of theorems called fundamental
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems_called_fundamental
date_saved: 2026-05-05T08:17:34.373025+00:00
instance: kb-cron
parent: list-of-theorems-called-fundamental-20260505-0
summary:
In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory. Some of these are classification theorems of objects which are mainly dealt with in the field. For instance, the fundamental theorem of curves describes classification of regular curves in space up to translation and rotation. Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself. == Fundamental theorems of mathematical topics == Fundamental theorem of algebra Fundamental theorem of algebraic K-theory Fundamental theorem of arithmetic Fundamental theorem of Boolean algebra Fundamental theorem of calculus Fundamental theorem of calculus for line integrals Fundamental theorem of curves Fundamental theorem of cyclic groups Fundamental theorem of dynamical systems Fundamental theorem of equivalence relations Fundamental theorem of exterior calculus Fundamental theorem


id: list-of-misnamed-theorems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_misnamed_theorems-0.md
category: reference
title: List of misnamed theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_misnamed_theorems
date_saved: 2026-05-05T08:17:35.596362+00:00
instance: kb-cron
parent: list-of-misnamed-theorems-20260505-0
summary:
This is a list of misnamed theorems in mathematics. It includes theorems (and lemmas, corollaries, conjectures, laws, and perhaps even the odd object) that are well known in mathematics, but which are not named for the originator. That is, the items on this list illustrate Stigler's law of eponymy (which is not, of course, due to Stephen Stigler, who credits Robert K Merton). == Applied mathematics == Benford's law. This was first stated in 1881 by Simon Newcomb, and rediscovered in 1938 by Frank Benford. The first rigorous formulation and proof seems to be due to Ted Hill in 1988.; see also the contribution by Persi Diaconis. Bertrand's ballot theorem. This result concerning the probability that the winner of an election was ahead at each step of ballot counting was first published by W. A. Whitworth in 1878, but named after Joseph Louis François Bertrand who rediscovered it in 1887. A common proof uses André's reflection method, though the proof by Désiré André did not use any reflections. == Algebra == Burnside's lemma. This was stated and proved without attribution in Burnside's 1897 textbook, but it had previously been discussed by Augustin Cauchy, in 1845, and by Georg Frobenius in


id: list-of-misnamed-theorems-20260505-1
file: data/en.wikipedia.org/wiki/List_of_misnamed_theorems-1.md
category: reference
title: List of misnamed theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_misnamed_theorems
date_saved: 2026-05-05T08:17:35.596362+00:00
instance: kb-cron
parent: list-of-misnamed-theorems-20260505-0
summary:
== Number theory == Bézout's theorem. The statement may have been made first by Isaac Newton in 1665. The matter of a proof was taken up by Colin MacLaurin (c. 1720) and Leonhard Euler as well as Étienne Bézout (c. 1750). However, Bézout's "proof" was incorrect. The first correct proof seems to be due mostly to Georges-Henri Halphen in the 1870s. Cramer's rule. It is named after Gabriel Cramer (1704–1752), who published the rule in his 1750 Introduction à l'analyse des lignes courbes algébriques, although Colin Maclaurin also published the method in his 1748 Treatise of Algebra (and probably knew of the method as early as 1729). Pell's equation. The solution of the equation x2 − dy2 = 1, where x and y are unknown positive integers and where d is a known positive integer which is not a perfect square, is ascribed to John Pell. It seems to have been discovered by Fermat, who set it as a challenge problem in 1657. The first European solution is found in a joint work in 1658 by John Wallis and Lord Brouncker; in 1668, a shorter solution was given in an edition of a third mathematician's work by Pell; see ref.


id: list-of-topologies-20260505-0
file: data/en.wikipedia.org/wiki/List_of_topologies-0.md
category: reference
title: List of topologies
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_topologies
date_saved: 2026-05-05T08:17:38.117299+00:00
instance: kb-cron
parent: list-of-topologies-20260505-0
summary:
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property. == Discrete and indiscrete == Discrete topology − All subsets are open. Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open. == Cardinality and ordinals == Cocountable topology Given a topological space ( X , τ ) , {\displaystyle (X,\tau ),} the cocountable extension topology on X {\displaystyle X} is the topology having as a subbasis the union of τ and the family of all subsets of X {\displaystyle X} whose complements in X {\displaystyle X} are countable. Cofinite topology Double-pointed cofinite topology Ordinal number topology Pseudo-arc Ran space Tychonoff plank === Finite spaces === Discrete two-point space − The simplest example of a totally disconnected discrete space. Finite topological space Pseudocircle − A finite topological space on 4 elements that fails to satisfy any separation axiom besides T0. However, from the viewpoint of algebraic topology, it has the remarkable property that it


id: list-of-topologies-20260505-1
file: data/en.wikipedia.org/wiki/List_of_topologies-1.md
category: reference
title: List of topologies
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_topologies
date_saved: 2026-05-05T08:17:38.117299+00:00
instance: kb-cron
parent: list-of-topologies-20260505-0
summary:
Alexandroff plank Appert topology − A Hausdorff, perfectly normal (T6), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact. Arens square Bullet-riddled square - The space [ 0 , 1 ] 2 ∖ Q 2 , {\displaystyle [0,1]^{2}\setminus \mathbb {Q} ^{2},} where [ 0 , 1 ] 2 ∩ Q 2 {\displaystyle [0,1]^{2}\cap \mathbb {Q} ^{2}} is the set of bullets. Neither of these sets is Jordan measurable although both are Lebesgue measurable. Cantor tree Comb space Dieudonné plank Double origin topology Dunce hat (topology) Either–or topology Excluded point topology − A topological space where the open sets are defined in terms of the exclusion of a particular point. Fort space Half-disk topology Hilbert cube − [ 0 , 1 / 1 ] × [ 0 , 1 / 2 ] × [ 0 , 1 / 3 ] × ⋯ {\displaystyle [0,1/1]\times [0,1/2]\times [0,1/3]\times \cdots } with the product topology. Infinite broom Integer broom topology K-topology Knaster–Kuratowski fan Long line (topology) Moore plane, also called the Niemytzki plane − A first countable, separable, completely regular, Hausdorff, Moore space that is not normal, Lindelöf, metrizable, second countable, nor locally compact. It also an uncountable closed


id: list-of-topology-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_topology_topics-0.md
category: reference
title: List of topology topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_topology_topics
date_saved: 2026-05-05T08:17:39.346041+00:00
instance: kb-cron
parent: list-of-topology-topics-20260505-0
summary:
In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it


id: list-of-algebraic-topology-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_algebraic_topology_topics-0.md
category: reference
title: List of algebraic topology topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algebraic_topology_topics
date_saved: 2026-05-05T08:17:40.558645+00:00
instance: kb-cron
parent: list-of-algebraic-topology-topics-20260505-0
summary:
This is a list of algebraic topology topics. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. == Homology (mathematics) == Simplex Simplicial complex Polytope Triangulation Barycentric subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial category Chain (algebraic topology) Betti number Euler characteristic Genus Riemann–Hurwitz formula Singular homology Cellular homology Relative homology Mayer–Vietoris sequence Excision theorem Universal coefficient theorem Cohomology List of cohomology theories Cocycle class Cup product Cohomology ring De Rham cohomology Čech cohomology Alexander–Spanier cohomology Intersection cohomology Lusternik–Schnirelmann category Poincaré duality Fundamental class Applications Jordan curve theorem Brouwer fixed point theorem Invariance of domain Lefschetz fixed-point theorem Hairy ball theorem Degree of a continuous mapping Borsuk–Ulam theorem Ham sandwich theorem Homology sphere == Homotopy theory == Homotopy Path (topology) Fundamental group Homotopy group Seifert–van


id: list-of-general-topology-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_general_topology_topics-0.md
category: reference
title: List of general topology topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_general_topology_topics
date_saved: 2026-05-05T08:17:41.721700+00:00
instance: kb-cron
parent: list-of-general-topology-topics-20260505-0
summary:
This is a list of general topology topics. == Basic concepts == Topological space Topological property Open set, closed set Clopen set Closure Boundary Interior Density G-delta set, F-sigma set Closeness Neighborhood Continuity (topology) Homeomorphism Local homeomorphism Open and closed maps Embedding Germ Basis Subbasis Open cover Locally finite space Covering space Atlas == Limits == Limit point Net Filter Ultrafilter == Topological properties == Baire category theorem Nowhere dense Baire space Banach–Mazur game Meagre set Comeagre set === Compactness and countability === Compact space Relatively compact subspace Heine–Borel theorem Tychonoff's theorem Finite intersection property Compactification Measure of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable space Lindelöf space Sigma-compact space === Connectedness === Connected space Simply connected space Path connected space === Separation axioms === T0 space T1 space Hausdorff space Completely Hausdorff space Regular space Tychonoff space Normal space Urysohn's lemma Tietze extension theorem Paracompact Separated sets == Topological constructions == Direct sum and the dual construction product Subspace and the dual construction quotient Topological tensor product == Examples == Discrete space Locally constant function Trivial topology Cofinite topology Cocountable topology Finer topology Product topology Restricted product


id: list-of-geometric-topology-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_geometric_topology_topics-0.md
category: reference
title: List of geometric topology topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_geometric_topology_topics
date_saved: 2026-05-05T08:17:42.922521+00:00
instance: kb-cron
parent: list-of-geometric-topology-topics-20260505-0
summary:
This is a list of geometric topology topics. == Low-dimensional topology == === Knot theory === Knot (mathematics) Link (knot theory) Wild knots Examples of knots (and links) Unknot Trefoil knot Figure-eight knot (mathematics) Borromean rings Types of knots (and links) Torus knot Prime knot Alternating knot Hyperbolic link Knot invariants Crossing number Linking number Skein relation Knot polynomials Alexander polynomial Jones polynomial Knot group Writhe Quandle Seifert surface Braids Braid theory Braid group Kirby calculus === Surfaces === Genus (mathematics) Examples Positive Euler characteristic 2-disk Sphere Real projective plane Zero Euler characteristic Annulus Möbius strip Torus Klein bottle Negative Euler characteristic The boundary of the pretzel is a genus three surface Embedded/Immersed in Euclidean space Cross-cap Boy's surface Roman surface Steiner surface Alexander horned sphere Klein bottle Mapping class group Dehn twist Nielsen–Thurston classification === Three-manifolds === Moise's Theorem (see also Hauptvermutung) Poincaré conjecture Thurston elliptization conjecture Thurston's geometrization conjecture Hyperbolic 3-manifolds Spherical 3-manifolds Euclidean 3-manifolds, Bieberbach Theorem, Flat manifolds, Crystallographic groups Seifert fiber space Heegaard splitting Waldhausen conjecture Compression body Handlebody Incompressible surface Dehn's lemma Loop theorem (aka the Disk theorem) Sphere theorem Haken manifold JSJ decomposition Branched surface Lamination Examples 3-sphere Torus bundles Surface bundles over the


id: list-of-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_transforms-0.md
category: reference
title: List of transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_transforms
date_saved: 2026-05-05T08:17:44.168043+00:00
instance: kb-cron
parent: list-of-transforms-20260505-0
summary:
This is a list of transforms in mathematics. == Integral transforms == Abel transform Aboodh transform Bateman transform Fourier transform Fourier cosine transform Fourier sine transform Fractional Fourier transform Short-time Fourier transform Rectangular mask short-time Fourier transform Fourier–Bros–Iagolnitzer transform Gabor transform Hankel transform Hartley transform Hermite transform Hilbert transform Hilbert–Schmidt integral operator Jacobi transform Laguerre transform Laplace transform Inverse Laplace transform Two-sided Laplace transform Inverse two-sided Laplace transform Laplace–Carson transform Laplace–Stieltjes transform Legendre transform Associated Legendre transform Linear canonical transform Mellin transform Inverse Mellin transform Poisson–Mellin–Newton cycle N-transform Radon transform X-ray transform Shehu transform Stieltjes transformation Sumudu transform Wavelet transform (integral) Weierstrass transform == Discrete transforms == Binomial transform Discrete Fourier transform, DFT Fast Fourier transform, a popular implementation of the DFT Discrete cosine transform Modified discrete cosine transform Discrete Hartley transform Discrete sine transform Discrete wavelet transform Hadamard transform (or, Walsh–Hadamard transform) Fast wavelet transform Hankel transform, the determinant of the Hankel matrix Discrete Chebyshev transform Equivalent, up to a diagonal scaling, to a discrete cosine transform Finite Legendre transform Irrational base discrete weighted transform Mojette transform Number-theoretic transform Spherical Harmonic transform Stirling transform == Discrete-time transforms == These transforms have a continuous frequency domain: Discrete-time Fourier transform Z-transform


id: list-of-triangle-inequalities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-0.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of the form "less than", "less than or equal to", "greater than", or "greater than or equal to". The parameters in a triangle inequality can be the side lengths, the semiperimeter, the angle measures, the values of trigonometric functions of those angles, the area of the triangle, the medians of the sides, the altitudes, the lengths of the internal angle bisectors from each angle to the opposite side, the perpendicular bisectors of the sides, the distance from an arbitrary point to another point, the inradius, the exradii, the circumradius, and/or other quantities. Unless otherwise specified, this article deals with triangles in the Euclidean plane. == Main parameters and notation == The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c; the semiperimeter s = ⁠a + b + c/2⁠ (half the perimeter p); the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the


id: list-of-triangle-inequalities-20260505-1
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-1.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
each holding with equality only when a = b = c. This says that in the non-equilateral case the harmonic mean of the sides is less than their geometric mean, which in turn is less than their arithmetic mean, and which in turn is less than their quadratic mean. == Angles == cos ⁡ A + cos ⁡ B + cos ⁡ C ≤ 3 2 . {\displaystyle \cos A+\cos B+\cos C\leq {\frac {3}{2}}.} ( 1 − cos ⁡ A ) ( 1 − cos ⁡ B ) ( 1 − cos ⁡ C ) ≥ cos ⁡ A ⋅ cos ⁡ B ⋅ cos ⁡ C . {\displaystyle (1-\cos A)(1-\cos B)(1-\cos C)\geq \cos A\cdot \cos B\cdot \cos C.} cos 4 ⁡ A 2 + cos 4 ⁡ B 2 + cos 4 ⁡ C 2 ≤ s 3 2 a b c {\displaystyle \cos ^{4}{\frac {A}{2}}+\cos ^{4}{\frac {B}{2}}+\cos ^{4}{\frac {C}{2}}\leq {\frac {s^{3}}{2abc}}} for semi-perimeter s, with equality only in the equilateral case. a + b + c ≥ 2 b c cos ⁡ A + 2 c a cos ⁡ B + 2 a b cos ⁡ C . {\displaystyle a+b+c\geq 2{\sqrt {bc}}\cos A+2{\sqrt {ca}}\cos B+2{\sqrt {ab}}\cos C.} sin


id: list-of-triangle-inequalities-20260505-2
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-2.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
a b + b c + c a ≥ 4 3 ⋅ T {\displaystyle ab+bc+ca\geq 4{\sqrt {3}}\cdot T} and T ≤ a b c 2 a + b + c a 3 + b 3 + c 3 + a b c ≤ 1 4 3 ( a + b + c ) 3 ( a b c ) 4 a 3 + b 3 + c 3 6 ≤ 3 4 ( a b c ) 2 3 . {\displaystyle T\leq {\frac {abc}{2}}{\sqrt {\frac {a+b+c}{a^{3}+b^{3}+c^{3}+abc}}}\leq {\frac {1}{4}}{\sqrt[{6}]{\frac {3(a+b+c)^{3}(abc)^{4}}{a^{3}+b^{3}+c^{3}}}}\leq {\frac {\sqrt {3}}{4}}(abc)^{\frac {2}{3}}.} From the rightmost upper bound on T, using the arithmetic-geometric mean inequality, is obtained the isoperimetric inequality for triangles: T ≤ 3 36 ( a + b + c ) 2 = 3 9 s 2 {\displaystyle T\leq {\frac {\sqrt {3}}{36}}(a+b+c)^{2}={\frac {\sqrt {3}}{9}}s^{2}} for semiperimeter s. This is sometimes stated in terms of perimeter p as p 2 ≥ 12 3 ⋅ T , {\displaystyle p^{2}\geq 12{\sqrt {3}}\cdot T,} with equality for the equilateral triangle. This is strengthened by T ≤ 3 4 ( a b c ) 2 3 . {\displaystyle T\leq {\frac {\sqrt {3}}{4}}\left(abc\right)^{\frac {2}{3}}.} Bonnesen's inequality also strengthens the isoperimetric inequality: π 2 (


id: list-of-triangle-inequalities-20260505-3
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-3.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
== Altitudes == The altitudes ha, etc. each connect a vertex to the opposite side and are perpendicular to that side. They satisfy both h a + h b + h c ≤ 3 2 ( a + b + c ) {\displaystyle h_{a}+h_{b}+h_{c}\leq {\frac {\sqrt {3}}{2}}(a+b+c)} and h a 2 + h b 2 + h c 2 ≤ 3 4 ( a 2 + b 2 + c 2 ) . {\displaystyle h_{a}^{2}+h_{b}^{2}+h_{c}^{2}\leq {\frac {3}{4}}(a^{2}+b^{2}+c^{2}).} In addition, if a ≥ b ≥ c , {\displaystyle a\geq b\geq c,} then a + h a ≥ b + h b ≥ c + h c . {\displaystyle a+h_{a}\geq b+h_{b}\geq c+h_{c}.} We also have h a 2 ( b 2 + c 2 ) ⋅ h b 2 ( c 2 + a 2 ) ⋅ h c 2 ( a 2 + b 2 ) ≤ ( 3 8 ) 3 . {\displaystyle {\frac {h_{a}^{2}}{(b^{2}+c^{2})}}\cdot {\frac {h_{b}^{2}}{(c^{2}+a^{2})}}\cdot {\frac {h_{c}^{2}}{(a^{2}+b^{2})}}\leq \left({\frac {3}{8}}\right)^{3}.} For internal angle bisectors ta, tb, tc from vertices A, B, C and circumcenter R and incenter r, we have h a t a + h b t b + h c t c ≥ R + 4 r


id: list-of-triangle-inequalities-20260505-4
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-4.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
2 ( P A + P B + P C ) > A B + B C + C A > P A + P B + P C , {\displaystyle 2(PA+PB+PC)>AB+BC+CA>PA+PB+PC,} and more strongly than the second of these inequalities is: If A B {\displaystyle AB} is the shortest side of the triangle, then P A + P B + P C ≤ A C + B C . {\displaystyle PA+PB+PC\leq AC+BC.} We also have Ptolemy's inequality P A ⋅ B C + P B ⋅ C A > P C ⋅ A B {\displaystyle PA\cdot BC+PB\cdot CA>PC\cdot AB} for interior point P and likewise for cyclic permutations of the vertices. If we draw perpendiculars from interior point P to the sides of the triangle, intersecting the sides at D, E, and F, we have P A ⋅ P B ⋅ P C ≥ ( P D + P E ) ( P E + P F ) ( P F + P D ) . {\displaystyle PA\cdot PB\cdot PC\geq (PD+PE)(PE+PF)(PF+PD).} Further, the Erdős–Mordell inequality states that P A + P B + P C P D + P E + P F ≥ 2 {\displaystyle {\frac {PA+PB+PC}{PD+PE+PF}}\geq 2}


id: list-of-triangle-inequalities-20260505-5
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-5.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
P A 4 + P B 4 + P C 4 + k ( P A ⋅ P B ⋅ P C ) 4 3 ≥ 16 ( k + 3 ) r 4 {\displaystyle PA^{4}+PB^{4}+PC^{4}+k(PA\cdot PB\cdot PC)^{\frac {4}{3}}\geq 16(k+3)r^{4}} for k = 0, 1, ..., 9. Furthermore, for circumradius R, ( P A ⋅ P B ) 3 2 + ( P B ⋅ P C ) 3 2 + ( P C ⋅ P A ) 3 2 ≥ 12 R r 2 ; {\displaystyle \left(PA\cdot PB\right)^{\frac {3}{2}}+\left(PB\cdot PC\right)^{\frac {3}{2}}+\left(PC\cdot PA\right)^{\frac {3}{2}}\geq 12Rr^{2};} ( P A ⋅ P B ) 2 + ( P B ⋅ P C ) 2 + ( P C ⋅ P A ) 2 ≥ 8 ( R + r ) R r 2 ; {\displaystyle (PA\cdot PB)^{2}+(PB\cdot PC)^{2}+(PC\cdot PA)^{2}\geq 8(R+r)Rr^{2};} ( P A ⋅ P B ) 2 + ( P B ⋅ P C ) 2 + ( P C ⋅ P A ) 2 ≥ 48 r 4 ; {\displaystyle (PA\cdot PB)^{2}+(PB\cdot PC)^{2}+(PC\cdot PA)^{2}\geq 48r^{4};} ( P A ⋅ P B ) 2 + ( P B ⋅ P C ) 2 + ( P C ⋅ P A )


id: list-of-triangle-inequalities-20260505-6
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-6.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
s ≤ ( 3 3 − 4 ) r + 2 R . {\displaystyle s\leq (3{\sqrt {3}}-4)r+2R.} We also have, for all acute triangles, s > 2 R + r . {\displaystyle s>2R+r.} For incircle center I, let AI, BI, and CI extend beyond I to intersect the circumcircle at D, E, and F respectively. Then A I I D + B I I E + C I I F ≥ 3. {\displaystyle {\frac {AI}{ID}}+{\frac {BI}{IE}}+{\frac {CI}{IF}}\geq 3.} In terms of the vertex angles we have cos ⁡ A ⋅ cos ⁡ B ⋅ cos ⁡ C ≤ ( r R 2 ) 2 . {\displaystyle \cos A\cdot \cos B\cdot \cos C\leq \left({\frac {r}{R{\sqrt {2}}}}\right)^{2}.} Denote as R A , R B , R C {\displaystyle R_{A},R_{B},R_{C}} the tanradii of the triangle. Then 4 R ≤ 1 R A + 1 R B + 1 R C ≤ 2 r {\displaystyle {\frac {4}{R}}\leq {\frac {1}{R_{A}}}+{\frac {1}{R_{B}}}+{\frac {1}{R_{C}}}\leq {\frac {2}{r}}} with equality only in the equilateral case, and 9 2 r ≤ R A + R B + R C ≤ 2 R + 1 2 r {\displaystyle {\frac {9}{2}}r\leq R_{A}+R_{B}+R_{C}\leq 2R+{\frac {1}{2}}r} with equality only in the equilateral case. ===


id: list-of-triangle-inequalities-20260505-7
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-7.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
=== Inscribed triangle === If three points D, E, F on the respective sides AB, BC, and CA of a reference triangle ABC are the vertices of an inscribed triangle, which thereby partitions the reference triangle into four triangles, then the area of the inscribed triangle is greater than the area of at least one of the other interior triangles, unless the vertices of the inscribed triangle are at the midpoints of the sides of the reference triangle (in which case the inscribed triangle is the medial triangle and all four interior triangles have equal areas): Area(DEF) ≥ min ( Area(BED), Area(CFE), Area(ADF) ) . {\displaystyle {\text{Area(DEF)}}\geq \min({\text{Area(BED), Area(CFE), Area(ADF)}}).} === Inscribed squares === An acute triangle has three inscribed squares, each with one side coinciding with part of a side of the triangle and with the square's other two vertices on the remaining two sides of the triangle. (A right triangle has only two distinct inscribed squares.) If one of these squares has side length xa and another has side length xb with xa < xb, then 1 ≥ x a x b ≥ 2 2 3 ≈ 0.94. {\displaystyle 1\geq {\frac {x_{a}}{x_{b}}}\geq {\frac {2{\sqrt {2}}}{3}}\approx 0.94.} Moreover, for


id: list-of-trigonometric-identities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-0.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. == Pythagorean identities == The basic relationship between the sine and cosine is given by the Pythagorean identity: sin 2 ⁡ θ + cos 2 ⁡ θ = 1 , {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,} where sin 2 ⁡ θ {\displaystyle \sin ^{2}\theta } means ( sin ⁡ θ ) 2 {\displaystyle {(\sin \theta )}^{2}} and cos 2 ⁡ θ {\displaystyle \cos ^{2}\theta } means ( cos ⁡ θ ) 2 . {\displaystyle {(\cos \theta )}^{2}.} This can be viewed as a version of the Pythagorean theorem, and follows from


id: list-of-trigonometric-identities-20260505-1
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-1.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
== Angle sum and difference identities == These are also known as the angle addition and subtraction theorems (or formulae). sin ⁡ ( α + β ) = sin ⁡ α cos ⁡ β + cos ⁡ α sin ⁡ β sin ⁡ ( α − β ) = sin ⁡ α cos ⁡ β − cos ⁡ α sin ⁡ β cos ⁡ ( α + β ) = cos ⁡ α cos ⁡ β − sin ⁡ α sin ⁡ β cos ⁡ ( α − β ) = cos ⁡ α cos ⁡ β + sin ⁡ α sin ⁡ β {\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}} The angle difference identities for sin ⁡ ( α − β ) {\displaystyle \sin(\alpha -\beta )} and cos ⁡ ( α − β ) {\displaystyle \cos(\alpha -\beta )} can be derived from the angle sum versions (and vice versa) by substituting − β {\displaystyle -\beta } for β {\displaystyle \beta } and using the


id: list-of-trigonometric-identities-20260505-2
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-2.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
tan ⁡ ( ∑ i θ i ) = sin ( ∑ i θ i ) / ∏ i cos ⁡ θ i cos ( ∑ i θ i ) / ∏ i cos ⁡ θ i = ∑ odd k ≥ 1 ( − 1 ) k − 1 2 ∑ A ⊆ { 1 , 2 , 3 , … } | A | = k ∏ i ∈ A tan ⁡ θ i ∑ even k ≥ 0 ( − 1 ) k 2 ∑ A ⊆ { 1 , 2 , 3 , … } | A | = k ∏ i ∈ A tan ⁡ θ i = e 1 − e 3 + e 5 − ⋯ e 0 − e 2 + e 4 − ⋯ cot ⁡ ( ∑ i θ i ) = e 0 − e 2 + e 4 − ⋯ e 1 − e 3 + e 5 − ⋯ {\displaystyle {\begin{aligned}\tan {\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {{\sin }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}{{\cos }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}}\\[10pt]&={\frac {\displaystyle \sum _{{\text{odd}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}{\displaystyle \sum


id: list-of-trigonometric-identities-20260505-3
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-3.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
sec ⁡ ( α + β + γ ) = sec ⁡ α sec ⁡ β sec ⁡ γ 1 − tan ⁡ α tan ⁡ β − tan ⁡ α tan ⁡ γ − tan ⁡ β tan ⁡ γ csc ⁡ ( α + β + γ ) = sec ⁡ α sec ⁡ β sec ⁡ γ tan ⁡ α + tan ⁡ β + tan ⁡ γ − tan ⁡ α tan ⁡ β tan ⁡ γ . {\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma }}.\end{aligned}}} === Ptolemy's theorem === Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a cyclic quadrilateral A B C D {\displaystyle ABCD} , as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths


id: list-of-trigonometric-identities-20260505-4
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-4.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
sin ⁡ ( 3 θ ) = 3 sin ⁡ θ − 4 sin 3 ⁡ θ = 4 sin ⁡ θ sin ⁡ ( π 3 − θ ) sin ⁡ ( π 3 + θ ) cos ⁡ ( 3 θ ) = 4 cos 3 ⁡ θ − 3 cos ⁡ θ = 4 cos ⁡ θ cos ⁡ ( π 3 − θ ) cos ⁡ ( π 3 + θ ) tan ⁡ ( 3 θ ) = 3 tan ⁡ θ − tan 3 ⁡ θ 1 − 3 tan 2 ⁡ θ = tan ⁡ θ tan ⁡ ( π 3 − θ ) tan ⁡ ( π 3 + θ ) cot ⁡ ( 3 θ ) = 3 cot ⁡ θ − cot 3 ⁡ θ 1 − 3 cot 2 ⁡ θ sec ⁡ ( 3 θ ) = sec 3 ⁡ θ 4 − 3 sec 2 ⁡ θ csc ⁡ ( 3 θ ) = csc 3 ⁡ θ 3 csc 2 ⁡ θ − 4 {\displaystyle {\begin{aligned}\sin(3\theta )&=3\sin \theta -4\sin ^{3}\theta &&=4\sin \theta \sin \left({\tfrac {\pi }{3}}-\theta \right)\sin \left({\tfrac {\pi }{3}}+\theta \right)\\\cos(3\theta )&=4\cos ^{3}\theta -3\cos \theta &&=4\cos


id: list-of-trigonometric-identities-20260505-5
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-5.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
cos ⁡ ( ( n − 1 ) x + x ) = cos ⁡ ( ( n − 1 ) x ) cos ⁡ x − sin ⁡ ( ( n − 1 ) x ) sin ⁡ x cos ⁡ ( ( n − 1 ) x − x ) = cos ⁡ ( ( n − 1 ) x ) cos ⁡ x + sin ⁡ ( ( n − 1 ) x ) sin ⁡ x {\displaystyle {\begin{aligned}\cos {\bigl (}(n-1)x+x{\bigr )}&=\cos {\bigl (}(n-1)x{\bigr )}\cos x-\sin {\bigl (}(n-1)x{\bigr )}\sin x\\\cos {\bigl (}(n-1)x-x{\bigr )}&=\cos {\bigl (}(n-1)x{\bigr )}\cos x+\sin {\bigl (}(n-1)x{\bigr )}\sin x\end{aligned}}} It follows by induction that cos ⁡ ( n x ) {\displaystyle \cos(nx)} is a polynomial of cos ⁡ x , {\displaystyle \cos x,} the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition. Similarly, sin ⁡ ( n x ) {\displaystyle \sin(nx)} can be computed from sin ⁡ ( ( n − 1 ) x ) , {\displaystyle \sin((n-1)x),} sin ⁡ ( ( n − 2 ) x ) , {\displaystyle \sin((n-2)x),} and cos ⁡ x {\displaystyle \cos x} with sin ⁡ ( n x ) = 2 cos ⁡ x sin ⁡


id: list-of-trigonometric-identities-20260505-6
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-6.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
The product-to-sum identities or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations. See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae. === Product-to-sum identities === The product of two sines or cosines of different angles can be converted to a sum of trigonometric functions of a sum and difference of those angles: cos ⁡ θ cos ⁡ φ = 1 2 ( cos ⁡ ( θ − φ ) + cos ⁡ ( θ + φ ) ) , sin ⁡ θ sin ⁡ φ = 1 2 ( cos ⁡ ( θ − φ ) − cos ⁡ ( θ + φ ) ) , sin ⁡ θ cos ⁡ φ = 1 2 ( sin ⁡ ( θ + φ ) + sin ⁡ ( θ − φ ) ) , cos ⁡ θ sin ⁡ φ = 1 2 ( sin ⁡ ( θ + φ ) − sin ⁡ ( θ − φ ) ) .


id: list-of-trigonometric-identities-20260505-7
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-7.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
cot ⁡ ( z − a 1 ) cot ⁡ ( z − a 2 ) = − 1 + cot ⁡ ( a 1 − a 2 ) cot ⁡ ( z − a 1 ) + cot ⁡ ( a 2 − a 1 ) cot ⁡ ( z − a 2 ) . {\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).} === Finite products of trigonometric functions === For coprime integers n, m ∏ k = 1 n ( 2 a + 2 cos ⁡ ( 2 π k m n + x ) ) = 2 ( T n ( a ) + ( − 1 ) n + m cos ⁡ ( n x ) ) {\displaystyle \prod _{k=1}^{n}\left(2a+2\cos \left({\frac {2\pi km}{n}}+x\right)\right)=2\left(T_{n}(a)+{(-1)}^{n+m}\cos(nx)\right)} where Tn is the Chebyshev polynomial. The following relationship holds for the sine function ∏ k = 1 n − 1 sin ⁡ ( k π n ) = n 2 n − 1 . {\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}}.} More generally for an integer n > 0 sin ⁡ ( n x ) = 2 n − 1 ∏ k = 0 n − 1 sin ⁡ ( k π n + x ) = 2


id: list-of-trigonometric-identities-20260505-8
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-8.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
sin ⁡ ( A + B ) − sin ⁡ ( A − B ) = 2 cos ⁡ A sin ⁡ B . {\displaystyle \sin(A+B)-\sin(A-B)=2\cos A\sin B.} Then let's examine the following formula, 2 sin ⁡ α ∑ k = 1 n cos ⁡ ( 2 k − 1 ) α = 2 sin ⁡ α cos ⁡ α + 2 sin ⁡ α cos ⁡ 3 α + 2 sin ⁡ α cos ⁡ 5 α + ⋯ + 2 sin ⁡ α cos ⁡ ( 2 n − 1 ) α {\displaystyle 2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha =2\sin \alpha \cos \alpha +2\sin \alpha \cos 3\alpha +2\sin \alpha \cos 5\alpha +\cdots +2\sin \alpha \cos(2n-1)\alpha } and this formula can be written by using the above identity, 2 sin ⁡ α ∑ k = 1 n cos ⁡ ( 2 k − 1 ) α = ∑ k = 1 n ( sin ⁡ ( 2 k α ) − sin ⁡ ( 2 ( k − 1 ) α ) ) = ( sin ⁡ 2 α − sin ⁡ 0 ) + ( sin ⁡ 4 α − sin ⁡ 2 α ) + ( sin ⁡ 6


id: list-of-trigonometric-identities-20260505-9
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-9.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
sin ⁡ x = x ∏ n = 1 ∞ ( 1 − x 2 π 2 n 2 ) , cos ⁡ x = ∏ n = 1 ∞ ( 1 − x 2 π 2 ( n − 1 2 ) ) 2 ) , sinh ⁡ x = x ∏ n = 1 ∞ ( 1 + x 2 π 2 n 2 ) , cosh ⁡ x = ∏ n = 1 ∞ ( 1 + x 2 π 2 ( n − 1 2 ) ) 2 ) . {\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right),\\[10mu]\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right).\end{aligned}}} == Inverse trigonometric functions == The following identities give the result of composing a trigonometric function with an inverse trigonometric function. sin ⁡ ( arcsin ⁡ x ) = x cos ⁡ ( arcsin ⁡ x ) = 1 − x 2 tan ⁡ ( arcsin ⁡ x ) = x 1 − x 2 sin ⁡ ( arccos ⁡ x ) = 1 − x 2 cos ⁡ ( arccos ⁡ x ) = x tan ⁡ ( arccos


id: list-of-trigonometric-identities-20260505-10
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-10.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
arctan ⁡ 1 p = arctan ⁡ 1 p + q + arctan ⁡ q p 2 + p q + 1 . {\displaystyle \arctan {\frac {1}{p}}=\arctan {\frac {1}{p+q}}+\arctan {\frac {q}{p^{2}+pq+1}}.} The curious identity known as Morrie's law, cos ⁡ 20 ∘ ⋅ cos ⁡ 40 ∘ ⋅ cos ⁡ 80 ∘ = 1 8 , {\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}},} is a special case of an identity that contains one variable: ∏ j = 0 k − 1 cos ⁡ ( 2 j x ) = sin ⁡ ( 2 k x ) 2 k sin ⁡ x . {\displaystyle \prod _{j=0}^{k-1}\cos \left(2^{j}x\right)={\frac {\sin \left(2^{k}x\right)}{2^{k}\sin x}}.} Similarly, sin ⁡ 20 ∘ ⋅ sin ⁡ 40 ∘ ⋅ sin ⁡ 80 ∘ = 3 8 {\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}} is a special case of an identity with x = 20 ∘ {\displaystyle x=20^{\circ }} : sin ⁡ x ⋅ sin ⁡ ( 60 ∘ − x ) ⋅ sin ⁡ ( 60 ∘ + x ) = sin ⁡ 3 x 4 . {\displaystyle \sin x\cdot \sin \left(60^{\circ }-x\right)\cdot \sin \left(60^{\circ }+x\right)={\frac {\sin 3x}{4}}.} For the


id: list-of-trigonometric-identities-20260505-11
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-11.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
If n is an odd number ( n = 2 m + 1 {\displaystyle n=2m+1} ) we can make use of the symmetries to get ∏ k = 1 m tan ⁡ k π 2 m + 1 = 2 m + 1 {\displaystyle \prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}} The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved: ∏ k = 1 n sin ⁡ ( 2 k − 1 ) π 4 n = ∏ k = 1 n cos ⁡ ( 2 k − 1 ) π 4 n = 2 2 n {\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}} === Computing π === An efficient way to compute π to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula: π 4 = 4 arctan ⁡ 1 5 − arctan ⁡ 1 239 {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}} or, alternatively, by using an identity of Leonhard Euler: π 4 = 5 arctan


id: list-of-trigonometric-identities-20260505-12
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-12.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
== Further "conditional" identities for the case α + β + γ = 180° == A conditional trigonometric identity is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied. The following formulae apply to arbitrary plane triangles and follow from α + β + γ = 180 ∘ , {\displaystyle \alpha +\beta +\gamma =180^{\circ },} as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur). tan ⁡ α + tan ⁡ β + tan ⁡ γ = tan ⁡ α tan ⁡ β tan ⁡ γ 1 = cot ⁡ β cot ⁡ γ + cot ⁡ γ cot ⁡ α + cot ⁡ α cot ⁡ β cot ⁡ ( α 2 ) + cot ⁡ ( β 2 ) + cot ⁡ ( γ 2 ) = cot ⁡ ( α 2 ) cot ⁡ ( β 2 ) cot ⁡ ( γ 2 ) 1 = tan ⁡ ( β 2 ) tan ⁡ ( γ 2 ) + tan ⁡ ( γ 2 ) tan ⁡ ( α 2 ) + tan ⁡ (


id: list-of-trigonometric-identities-20260505-13
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-13.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
=== Dirichlet kernel === The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity: 1 + 2 cos ⁡ x + 2 cos ⁡ ( 2 x ) + 2 cos ⁡ ( 3 x ) + ⋯ + 2 cos ⁡ ( n x ) = sin ⁡ ( ( n + 1 2 ) x ) sin ⁡ ( 1 2 x ) . {\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin \left({\frac {1}{2}}x\right)}}.} The convolution of any integrable function of period 2 π {\displaystyle 2\pi } with the Dirichlet kernel coincides with the function's n {\displaystyle n} th-degree Fourier approximation. The same holds for any measure or generalized function. === Tangent half-angle substitution === If we set t = tan ⁡ x 2 , {\displaystyle t=\tan {\frac {x}{2}},} then sin ⁡ x = 2 t 1 + t 2 ; cos ⁡ x = 1 − t 2 1 + t 2 ; e i x = 1 + i t 1 − i t ; d x = 2 d t 1 + t 2 , {\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}};\qquad dx={\frac {2\,dt}{1+t^{2}}},} where e i x =


id: outline-of-trigonometry-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_trigonometry-0.md
category: reference
title: Outline of trigonometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_trigonometry
date_saved: 2026-05-05T08:17:48.718303+00:00
instance: kb-cron
parent: outline-of-trigonometry-20260505-0
summary:
The following outline is provided as an overview of and topical guide to trigonometry: Trigonometry – branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. == Basics == Geometry – mathematics concerned with questions of shape, size, the relative position of figures, and the properties of space. Geometry is used extensively in trigonometry. Angle – the angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Ratio – a ratio indicates how many times one number contains another == Content of trigonometry == Trigonometry Trigonometric functions Trigonometric identities Euler's formula == Scholars == Archimedes Aristarchus Aryabhata Bhaskara I Claudius Ptolemy Euclid Hipparchus Madhava of Sangamagrama Ptolemy Pythagoras Regiomontanus == History == Aristarchus's inequality Bhaskara I's sine approximation formula Greek astronomy Indian astronomy Jyā, koti-jyā and utkrama-jyā Madhava's sine table Ptolemy's table of chords Rule of marteloio Āryabhaṭa's sine table == Fields ==


id: list-of-undecidable-problems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_undecidable_problems-0.md
category: reference
title: List of undecidable problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_undecidable_problems
date_saved: 2026-05-05T08:17:51.782326+00:00
instance: kb-cron
parent: list-of-undecidable-problems-20260505-0
summary:
In computability theory, an undecidable problem is a decision problem for which an effective method (algorithm) to derive the correct answer does not exist. More formally, an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language. There are uncountably many undecidable problems, so the list below is necessarily incomplete. Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e., such undecidable languages may be recursively enumerable. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not. For undecidability in axiomatic mathematics, see List of statements undecidable in ZFC. == Problems about abstract machines == The halting problem (determining whether a Turing machine halts on a given input) and the mortality problem (determining whether it halts for every starting configuration). Determining whether a Turing machine is a busy beaver champion (i.e., is the longest-running among halting Turing machines with the same number of states and symbols). Rice's theorem states that for all nontrivial properties of partial functions, it is undecidable whether a given machine


id: list-of-undecidable-problems-20260505-1
file: data/en.wikipedia.org/wiki/List_of_undecidable_problems-1.md
category: reference
title: List of undecidable problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_undecidable_problems
date_saved: 2026-05-05T08:17:51.782326+00:00
instance: kb-cron
parent: list-of-undecidable-problems-20260505-0
summary:
== Other problems == The problem of determining if a given set of Wang tiles can tile the plane. The problem of determining if a given set of polyominoes can tile the plane. The problem of determining the Kolmogorov complexity of a string. Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions, or in a piecewise-linear flow in three dimensions. Finding the capacity of an information-stable finite state machine channel. In network coding, determining whether a network is solvable. Determining whether a player has a winning strategy in a game of Magic: The Gathering. Planning in a partially observable Markov decision process. Planning air travel from one destination to another, when fares are taken into account. == See also == Lists of problems List of unsolved problems Reduction (complexity) Unknowability == Notes == == Bibliography == Brookshear, J. Glenn (1989). Theory of Computation: Formal Languages, Automata, and Complexity. Redwood City, California: Benjamin/Cummings Publishing Company, Inc. Appendix C includes impossibility of algorithms deciding if a grammar contains ambiguities, and impossibility of verifying program correctness by an algorithm


id: list-of-statements-independent-of-zfc-20260505-0
file: data/en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC-0.md
category: reference
title: List of statements independent of ZFC
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC
date_saved: 2026-05-05T08:17:53.109673+00:00
instance: kb-cron
parent: list-of-statements-independent-of-zfc-20260505-0
summary:
The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC. == Axiomatic set theory == In 1931, Kurt Gödel proved his incompleteness theorems, establishing that many mathematical theories, including ZFC, cannot prove their own consistency. Assuming ω-consistency of such a theory, the consistency statement can also not be disproven, meaning it is independent. A few years later, other arithmetic statements were defined that are independent of any such theory, see for example Rosser's trick. The following set theoretic statements are independent of ZFC, among others: the continuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The following four independence results are also due to Gödel/Cohen.); the generalized continuum hypothesis (GCH); a related independent


id: list-of-statements-independent-of-zfc-20260505-1
file: data/en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC-1.md
category: reference
title: List of statements independent of ZFC
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC
date_saved: 2026-05-05T08:17:53.109673+00:00
instance: kb-cron
parent: list-of-statements-independent-of-zfc-20260505-0
summary:
One can write down a concrete polynomial p ∈ Z[x1, ..., x9] such that the statement "there are integers m1, ..., m9 with p(m1, ..., m9) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). This follows from Yuri Matiyasevich's resolution of Hilbert's tenth problem; the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent. == Measure theory == A stronger version of Fubini's theorem for positive functions, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, is independent of ZFC. On the one hand, CH implies that there exists a function on the unit square whose iterated integrals are not equal — the function is simply the indicator function of an ordering of [0, 1] equivalent to a well ordering of the cardinal ω1. A similar example can be constructed using MA. On the other hand, the consistency of the strong Fubini theorem was first shown by Friedman. It can also be deduced from a variant of Freiling's axiom of symmetry. == Topology == The Normal Moore Space conjecture, namely that every normal


id: list-of-euclidean-uniform-tilings-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Euclidean_uniform_tilings-0.md
category: reference
title: List of Euclidean uniform tilings
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Euclidean_uniform_tilings
date_saved: 2026-05-05T08:17:54.404656+00:00
instance: kb-cron
parent: list-of-euclidean-uniform-tilings-20260505-0
summary:
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. John Conway called these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex. These 11 uniform tilings have 32 different uniform colorings. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are not color-uniform.) In addition to the 11 convex uniform tilings, there are also 14 known nonconvex tilings, using star polygons, and reverse orientation vertex configurations. A further 28 uniform tilings are known using apeirogons. If zigzags are also allowed, there are 23 more known uniform tilings and 10 more known families depending on a parameter: in 8 cases the parameter is continuous, and in the other 2 it


id: list-of-uniform-polyhedra-20260505-0
file: data/en.wikipedia.org/wiki/List_of_uniform_polyhedra-0.md
category: reference
title: List of uniform polyhedra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_uniform_polyhedra
date_saved: 2026-05-05T08:17:55.737868+00:00
instance: kb-cron
parent: list-of-uniform-polyhedra-20260505-0
summary:
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both. This list includes these: all 75 nonprismatic uniform polyhedra; a few representatives of the infinite sets of prisms and antiprisms; one degenerate polyhedron, Skilling's figure with overlapping edges. It was proven in Sopov (1970) that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide. Not included are: The uniform polyhedron compounds. 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter); The uniform tilings (infinite polyhedra) 11 Euclidean convex uniform


id: uniform-tilings-in-hyperbolic-plane-20260505-0
file: data/en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane-0.md
category: reference
title: Uniform tilings in hyperbolic plane
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane
date_saved: 2026-05-05T08:17:57.549662+00:00
instance: kb-cron
parent: uniform-tilings-in-hyperbolic-plane-20260505-0
summary:
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry. Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given the Schläfli symbol {7,3}. Uniform tilings may be regular (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) or semi-regular (if neither edge- nor face-transitive). For right triangles (p q 2), there are two regular tilings, represented by Schläfli symbol {p,q} and {q,p}. == Wythoff construction == There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where ⁠1/p⁠ + ⁠1/q⁠ + ⁠1/r⁠ < 1, where p, q, r are


id: uniform-tilings-in-hyperbolic-plane-20260505-1
file: data/en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane-1.md
category: reference
title: Uniform tilings in hyperbolic plane
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane
date_saved: 2026-05-05T08:17:57.549662+00:00
instance: kb-cron
parent: uniform-tilings-in-hyperbolic-plane-20260505-0
summary:
=== (4 3 3) === The (4 3 3) triangle group, Coxeter group [(4,3,3)], orbifold (*433) contains these uniform tilings. Without right angles in the fundamental triangle, the Wythoff constructions are slightly different. For instance in the (4,3,3) triangle family, the snub form has six polygons around a vertex and its dual has hexagons rather than pentagons. In general the vertex figure of a snub tiling in a triangle (p,q,r) is p. 3.q.3.r.3, being 4.3.3.3.3.3 in this case below. === (4 4 3) === The (4 4 3) triangle group, Coxeter group [(4,4,3)], orbifold (*443) contains these uniform tilings. === (4 4 4) === The (4 4 4) triangle group, Coxeter group [(4,4,4)], orbifold (*444) contains these uniform tilings. === (5 3 3) === The (5 3 3) triangle group, Coxeter group [(5,3,3)], orbifold (*533) contains these uniform tilings. === (5 4 3) === The (5 4 3) triangle group, Coxeter group [(5,4,3)], orbifold (*543) contains these uniform tilings. === (5 4 4) === The (5 4 4) triangle group, Coxeter group [(5,4,4)], orbifold (*544) contains these uniform tilings. === (6 3 3) === The (6 3 3) triangle group, Coxeter group [(6,3,3)], orbifold (*633) contains these uniform tilings. ===


id: list-of-united-states-regional-mathematics-competitions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_United_States_regional_mathematics_competitions-0.md
category: reference
title: List of United States regional mathematics competitions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_United_States_regional_mathematics_competitions
date_saved: 2026-05-05T08:17:58.773121+00:00
instance: kb-cron
parent: list-of-united-states-regional-mathematics-competitions-20260505-0
summary:
Many math competitions in the United States have regional restrictions. Of these, most are statewide. For a more complete list, please visit here . The contests include: == Alabama == Alabama Statewide High School Mathematics Contest Virgil Grissom High School Math Tournament Vestavia Hills High School Math Tournament == Arizona == Great Plains Math League AATM State High School Contest == California == Bay Area Math Olympiad Berkeley Math Tournament Lawrence Livermore National Laboratories Annual High School Math Challenge Cal Poly Math Contest and Trimathlon [1] Polya Competition Bay Area Math Meet College of Creative Studies Math Competition LA Math Cup Math Day at the Beach hosted by CSULB[2] Math Field Day for San Diego Middle Schools Mesa Day Math Contest at UC Berkeley Santa Barbara County Math Superbowl Pomona College Mathematical Talent Search Redwood Empire Mathematics Tournament hosted by Humboldt State (middle and high school) San Diego Math League and San Diego Math Olympiad hosted by the San Diego Math Circle Santa Clara University High School Mathematics Contest Southern California Mathematics Competition, hosted at USC Stanford Mathematics Tournament UCSD/GSDMC High School Honors Mathematics Contest == Colorado == Colorado Mathematics Olympiad == District of Columbia == Moody's Mega Math ==


id: list-of-variational-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_variational_topics-0.md
category: reference
title: List of variational topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_variational_topics
date_saved: 2026-05-05T08:17:59.915350+00:00
instance: kb-cron
parent: list-of-variational-topics-20260505-0
summary:
This is a list of variational topics in from mathematics and physics. See calculus of variations for a general introduction. Action (physics) Averaged Lagrangian Brachistochrone curve Calculus of variations Catenoid Cycloid Dirichlet principle Euler–Lagrange equation cf. Action (physics) Fermat's principle Functional (mathematics) Functional derivative Functional integral Geodesic Isoperimetry Lagrangian Lagrangian mechanics Legendre transformation Luke's variational principle Minimal surface Morse theory Noether's theorem Path integral formulation Plateau's problem Prime geodesic Principle of least action Soap bubble Soap film Tautochrone curve


id: vector-algebra-relations-20260505-0
file: data/en.wikipedia.org/wiki/Vector_algebra_relations-0.md
category: reference
title: Vector algebra relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_algebra_relations
date_saved: 2026-05-05T08:18:01.266487+00:00
instance: kb-cron
parent: vector-algebra-relations-20260505-0
summary:
The following are important identities in vector algebra. Identities that only involve the magnitude of a vector ‖ A ‖ {\displaystyle \|\mathbf {A} \|} and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there. Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs, if not earlier. == Magnitudes == The magnitude of a vector A can be expressed using the dot product: ‖ A ‖ 2 = A ⋅ A {\displaystyle \|\mathbf {A} \|^{2}=\mathbf {A\cdot A} } In three-dimensional Euclidean space, the magnitude of a vector is determined from its three components using Pythagoras' theorem: ‖ A ‖ 2 = A 1 2 + A 2 2 + A 3 2 {\displaystyle \|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}} == Inequalities == The Cauchy–Schwarz inequality: A ⋅ B ≤ ‖ A ‖ ‖ B ‖ {\displaystyle \mathbf {A} \cdot \mathbf {B} \leq \left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|} The triangle inequality: ‖ A + B ‖ ≤ ‖ A ‖ + ‖ B ‖ {\displaystyle \|\mathbf {A+B} \|\leq \|\mathbf {A}


id: vector-algebra-relations-20260505-1
file: data/en.wikipedia.org/wiki/Vector_algebra_relations-1.md
category: reference
title: Vector algebra relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_algebra_relations
date_saved: 2026-05-05T08:18:01.266487+00:00
instance: kb-cron
parent: vector-algebra-relations-20260505-0
summary:
== Addition and multiplication of vectors == Commutativity of addition: A + B = B + A {\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} } . Commutativity of scalar product: A ⋅ B = B ⋅ A {\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A} } . Anticommutativity of cross product: A × B = − ( B × A ) {\displaystyle \mathbf {A} \times \mathbf {B} =\mathbf {-} (\mathbf {B} \times \mathbf {A} )} . Distributivity of multiplication by a scalar over addition: c ( A + B ) = c A + c B {\displaystyle c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B} } . Distributivity of scalar product over addition: ( A + B ) ⋅ C = A ⋅ C + B ⋅ C {\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C} } . Distributivity of vector product over addition: ( A + B ) × C = A × C + B × C {\displaystyle (\mathbf {A} +\mathbf {B} )\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C} } . Scalar triple product: A ⋅ (


id: vector-algebra-relations-20260505-2
file: data/en.wikipedia.org/wiki/Vector_algebra_relations-2.md
category: reference
title: Vector algebra relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_algebra_relations
date_saved: 2026-05-05T08:18:01.266487+00:00
instance: kb-cron
parent: vector-algebra-relations-20260505-0
summary:
A consequence of the previous equation: | A B C | D = ( A ⋅ D ) ( B × C ) + ( B ⋅ D ) ( C × A ) + ( C ⋅ D ) ( A × B ) . {\displaystyle |\mathbf {A} \,\mathbf {B} \,\mathbf {C} |\,\mathbf {D} =(\mathbf {A} \cdot \mathbf {D} )\left(\mathbf {B} \times \mathbf {C} \right)+\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)+\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right).} In 3 dimensions, a vector D can be expressed in terms of basis vectors {A,B,C} as: D = D ⋅ ( B × C ) | A B C | A + D ⋅ ( C × A ) | A B C | B + D ⋅ ( A × B ) | A B C | C . {\displaystyle \mathbf {D} \ =\ {\frac {\mathbf {D} \cdot (\mathbf {B} \times \mathbf {C} )}{|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |}}\ \mathbf {A} +{\frac {\mathbf {D} \cdot (\mathbf {C} \times \mathbf {A} )}{|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |}}\ \mathbf {B} +{\frac {\mathbf {D} \cdot (\mathbf {A} \times \mathbf {B} )}{|\mathbf {A} \,\mathbf {B}


id: vector-calculus-identities-20260505-0
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-0.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
The following are important identities involving derivatives and integrals in vector calculus. == Operator notation == === Gradient === For a function f ( x , y , z ) {\displaystyle f(x,y,z)} in three-dimensional Cartesian coordinate variables, the gradient is the vector field: grad ⁡ ( f ) = ∇ f = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k {\displaystyle \operatorname {grad} (f)=\nabla f={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} } where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables ψ ( x 1 , … , x n ) {\displaystyle \psi (x_{1},\ldots ,x_{n})} , also called a scalar field, the gradient is the vector field: ∇ ψ = ( ∂ ∂ x 1 , … , ∂ ∂ x n ) ψ = ∂ ψ ∂ x 1 e 1 + ⋯ + ∂ ψ ∂ x n e n {\displaystyle


id: vector-calculus-identities-20260505-1
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-1.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
curl ⁡ F = ∇ × F = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) × ( F x , F y , F z ) = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z | = ( ∂ F z ∂ y − ∂ F y ∂ z ) i + ( ∂ F x ∂ z − ∂ F z ∂ x ) j + ( ∂ F y ∂ x − ∂ F x ∂ y ) k {\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {F} &=\nabla \times \mathbf {F} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}\\[1em]&={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}\\[1em]&=\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {k} \end{aligned}}} where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. As the name implies the curl is a measure of how much nearby vectors tend in a circular direction. In Einstein notation,


id: vector-calculus-identities-20260505-2
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-2.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
∇ ⋅ ( A × B ) = ∇ A ⋅ ( A × B ) + ∇ B ⋅ ( A × B ) = ( ∇ A × A ) ⋅ B + ( ∇ B × A ) ⋅ B = ( ∇ A × A ) ⋅ B − ( A × ∇ B ) ⋅ B = ( ∇ A × A ) ⋅ B − A ⋅ ( ∇ B × B ) = ( ∇ × A ) ⋅ B − A ⋅ ( ∇ × B ) {\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {A} \times \mathbf {B} )&=\nabla _{\mathbf {A} }\cdot (\mathbf {A} \times \mathbf {B} )+\nabla _{\mathbf {B} }\cdot (\mathbf {A} \times \mathbf {B} )\\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} +(\nabla _{\mathbf {B} }\times \mathbf {A} )\cdot \mathbf {B} \\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} -(\mathbf {A} \times \nabla _{\mathbf {B} })\cdot \mathbf {B} \\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla _{\mathbf {B} }\times \mathbf {B} )\\[2pt]&=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla \times \mathbf {B} )\end{aligned}}} An alternative method is to use the Cartesian components of


id: vector-calculus-identities-20260505-3
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-3.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
For the remainder of this article, Feynman subscript notation will be used where appropriate. == First derivative identities == For scalar fields ψ {\displaystyle \psi } , ϕ {\displaystyle \phi } and vector fields A {\displaystyle \mathbf {A} } , B {\displaystyle \mathbf {B} } , we have the following derivative identities. === Distributive properties === === First derivative associative properties === === Product rule for multiplication by a scalar === We have the following generalizations of the product rule in single-variable calculus. === Quotient rule for division by a scalar === === Chain rule === Let f ( x ) {\displaystyle f(x)} be a one-variable function from scalars to scalars, r ( t ) = ( x 1 ( t ) , … , x n ( t ) ) {\displaystyle \mathbf {r} (t)=(x_{1}(t),\ldots ,x_{n}(t))} a parametrized curve, ϕ : R n → R {\displaystyle \phi \!:\mathbb {R} ^{n}\to \mathbb {R} } a function from vectors to scalars, and A : R n → R n {\displaystyle \mathbf {A} \!:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} a vector field. We have the following special cases of the multi-variable chain rule. For a vector transformation u : R n → R


id: vector-calculus-identities-20260505-4
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-4.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
=== Curl of divergence is not defined === The divergence of a vector field A is a scalar, and the curl of a scalar quantity is undefined. Therefore, ∇ × ( ∇ ⋅ A ) is undefined. {\displaystyle \nabla \times (\nabla \cdot \mathbf {A} ){\text{ is undefined.}}} === Second derivative associative properties === === A mnemonic === The figure to the right is a mnemonic for some of these identities. The abbreviations used are: D: divergence, C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist. == Summary of important identities == === Differentiation === ==== Gradient ==== ==== Divergence ==== ==== Curl ==== ==== Vector-dot-Del Operator ==== ==== Second derivatives ==== ==== Third derivatives ==== === Integration === Below, the curly symbol ∂ means "boundary of" a surface or solid. ==== Surface–volume integrals ==== In the following surface–volume integral theorems, V denotes a three-dimensional volume with


id: vector-calculus-identities-20260505-5
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-5.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
∮ ∂ S A × ( B × d ℓ ) = ∬ S ( ∇ × ( A B T ) ) ⋅ d S + ∬ S ( ∇ ⋅ ( B A T ) ) × d S {\displaystyle \oint _{\partial S}\mathbf {A} \times (\mathbf {B} \times d{\boldsymbol {\ell }})\ =\ \iint _{S}\left(\nabla \times \left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)\right)\cdot d\mathbf {S} +\iint _{S}\left(\nabla \cdot \left(\mathbf {B} \mathbf {A} ^{\textsf {T}}\right)\right)\times d\mathbf {S} } ∮ ∂ S ( B ⋅ d ℓ ) A = ∬ S ( d S ⋅ [ ∇ × B − B × ∇ ] ) A {\displaystyle \oint _{\partial S}(\mathbf {B} \cdot d{\boldsymbol {\ell }})\mathbf {A} =\iint _{S}(d\mathbf {S} \cdot \left[\nabla \times \mathbf {B} -\mathbf {B} \times \nabla \right])\mathbf {A} } Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a definite integral): ==== Endpoint-curve integrals ==== In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points q − p = ∂ P {\displaystyle \mathbf {q} -\mathbf {p} =\partial P} and integration along P is from


id: list-of-vector-spaces-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_vector_spaces_in_mathematics-0.md
category: reference
title: List of vector spaces in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_vector_spaces_in_mathematics
date_saved: 2026-05-05T08:18:04.096795+00:00
instance: kb-cron
parent: list-of-vector-spaces-in-mathematics-20260505-0
summary:
This is a list of vector spaces in abstract mathematics, by Wikipedia page. Banach space Besov space Bochner space Dual space Euclidean space Fock space Fréchet space Hardy space Hilbert space Hölder space LF-space Lp space Minkowski space Montel space Morrey–Campanato space Orlicz space Riesz space Schwartz space Sobolev space Tsirelson space


id: list-of-wavelet-related-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_wavelet-related_transforms-0.md
category: reference
title: List of wavelet-related transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_wavelet-related_transforms
date_saved: 2026-05-05T08:18:06.741131+00:00
instance: kb-cron
parent: list-of-wavelet-related-transforms-20260505-0
summary:
A list of wavelet related transforms: Continuous wavelet transform (CWT) Discrete wavelet transform (DWT) Multiresolution analysis (MRA) Lifting scheme Binomial QMF (BQMF) Fast wavelet transform (FWT) Complex wavelet transform Non or undecimated wavelet transform, the downsampling is omitted Newland transform, an orthonormal basis of wavelets is formed from appropriately constructed top-hat filters in frequency space Wavelet packet decomposition (WPD), detail coefficients are decomposed and a variable tree can be formed Stationary wavelet transform (SWT), no downsampling and the filters at each level are different e-decimated discrete wavelet transform, depends on if the even or odd coefficients are selected in the downsampling Second generation wavelet transform (SGWT), filters and wavelets are not created in the frequency domain Dual-tree complex wavelet transform (DTCWT), two trees are used for decomposion to produce the real and complex coefficients WITS: Where Is The Starlet, a collection of a hundredth of wavelet names in -let and associated multiscale, directional, geometric, representations, from activelets to x-lets through bandelets, chirplets, contourlets, curvelets, noiselets, wedgelets ...


id: list-of-wenninger-polyhedron-models-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Wenninger_polyhedron_models-0.md
category: reference
title: List of Wenninger polyhedron models
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Wenninger_polyhedron_models
date_saved: 2026-05-05T08:18:08.055382+00:00
instance: kb-cron
parent: list-of-wenninger-polyhedron-models-20260505-0
summary:
This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes. It contains the 75 nonprismatic uniform polyhedra, as well as 44 stellated forms of the convex regular and quasiregular polyhedra. Models listed here can be cited as "Wenninger Model Number N", or WN for brevity. The polyhedra are grouped in 5 tables: Regular (1–5), Semiregular (6–18), regular star polyhedra (20–22,41), Stellations and compounds (19–66), and uniform star polyhedra (67–119). The four regular star polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings. == Platonic solids (regular convex polyhedra) W1 to W5 == == Archimedean solids (Semiregular) W6 to W18 == == Kepler–Poinsot polyhedra (Regular star polyhedra) W20, W21, W22 and W41 == == Stellations: models W19 to W66 == === Stellations of octahedron === === Stellations of dodecahedron === === Stellations of icosahedron === === Stellations of cuboctahedron === === Stellations of icosidodecahedron === == Uniform nonconvex solids


id: list-of-zeta-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_zeta_functions-0.md
category: reference
title: List of zeta functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_zeta_functions
date_saved: 2026-05-05T08:18:09.230852+00:00
instance: kb-cron
parent: list-of-zeta-functions-20260505-0
summary:
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.} Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta function Arithmetic zeta function Artin–Mazur zeta function of a dynamical system Barnes zeta function or double zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta function of a number field Duursma zeta function of error-correcting codes Epstein zeta function of a quadratic form Goss zeta function of a function field Hasse–Weil zeta function of a variety Height zeta function of a variety Hurwitz zeta function, a generalization of the Riemann zeta function Igusa zeta function Ihara zeta function of a graph L-function, a "twisted" zeta function Lefschetz zeta function of a morphism Lerch zeta function, a generalization of the Riemann zeta function Local zeta function of a characteristic-p variety Matsumoto zeta function Minakshisundaram–Pleijel zeta function of a Laplacian Motivic zeta function of a motive Multiple zeta function, or Mordell–Tornheim zeta function of several variables p-adic zeta function of a p-adic number Prime zeta function,


id: index-of-meteorology-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_meteorology_articles-0.md
category: reference
title: Index of meteorology articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_meteorology_articles
date_saved: 2026-05-05T08:18:10.971529+00:00
instance: kb-cron
parent: index-of-meteorology-articles-20260505-0
summary:
This is a list of meteorology topics. The terms relate to meteorology, the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. (see also: List of meteorological phenomena) == A == advection aeroacoustics aerobiology aerology air parcel (in meteorology) air quality index (AQI) airshed (in meteorology) American Geophysical Union (AGU) American Meteorological Society (AMS) anabatic wind anemometer annular hurricane anticyclone (in meteorology) apparent wind Atlantic Oceanographic and Meteorological Laboratory (AOML) Atlantic hurricane season atmometer atmosphere Atmospheric Model Intercomparison Project (AMIP) Atmospheric Radiation Measurement (ARM) (atmospheric boundary layer [ABL]) planetary boundary layer (PBL) atmospheric chemistry atmospheric circulation atmospheric convection atmospheric dispersion modeling atmospheric electricity atmospheric icing atmospheric physics atmospheric pressure atmospheric sciences atmospheric stratification atmospheric thermodynamics atmospheric window (see under Threats) == B == ball lightning balloon (aircraft) baroclinity barotropicity barometer ("to measure atmospheric pressure") berg wind biometeorology blizzard bomb (meteorology) buoyancy Bureau of Meteorology (in Australia) == C == Canada Weather Extremes Canadian Hurricane Centre (CHC) Cape Verde-type hurricane capping inversion (in meteorology) (see "severe thunderstorms" in paragraph 5) carbon cycle carbon fixation carbon flux carbon monoxide (see under Atmospheric presence) ceiling balloon ("to determine the height of the base of clouds above ground level") ceilometer


id: index-of-meteorology-articles-20260505-1
file: data/en.wikipedia.org/wiki/Index_of_meteorology_articles-1.md
category: reference
title: Index of meteorology articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_meteorology_articles
date_saved: 2026-05-05T08:18:10.971529+00:00
instance: kb-cron
parent: index-of-meteorology-articles-20260505-0
summary:
== E == Earth's atmosphere Earth's magnetic field Earth System Research Laboratories (ESRL) economics of global warming Emagram effect of Hurricane Katrina on New Orleans (see also Hurricane Katrina effects by region) effect of sun angle on climate Energy meteorology Enhanced Fujita scale (EF scale) eolian processes equator (see under Equatorial seasons and climate) equilibrium level (EL) equivalent potential temperature equivalent temperature European Centre for Medium-Range Weather Forecasts (ECMWF) European Climate Change Programme (ECCP) European emission standards (for motor vehicles) European Severe Storms Laboratory (ESSL) European windstorm evaporation evaporative cooler evaporative cooling evaporite (a mineral sediment resulting from evaporation of saline water) evapotranspiration (ET) (sum of evaporation and plant transpiration) exhaust gas recirculation (EGR) (exhaust gas recycling) exosphere (layer of atmosphere) extratropical cyclone (mid-latitude cyclone) extreme weather extremes on Earth == F == fire whirl firestorm fog forensic meteorology free convective layer (FCL) freezing rain (front [meteorology]: see) surface weather analysis frontogenesis frontolysis frost frost creep (frost heave) frost flowers (frost castles) (ice castles) (ice ribbons) (ice blossoms) frost heaving (frost heave) frost law frost line (frost point: see) dew point (dewpoint) frostbite Fujita scale (F scale) (for measuring tornadoes) fulgurite (full lunar eclipse: see) lunar eclipse full-spectrum light funnel


id: index-of-meteorology-articles-20260505-2
file: data/en.wikipedia.org/wiki/Index_of_meteorology_articles-2.md
category: reference
title: Index of meteorology articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_meteorology_articles
date_saved: 2026-05-05T08:18:10.971529+00:00
instance: kb-cron
parent: index-of-meteorology-articles-20260505-0
summary:
== N == NASA Clean Air Study NASA Earth Observatory NASA World Wind (virtual globe) National Ambient Air Quality Standards (NAAQS) (in the US) National Center for Atmospheric Research (NCAR) (in the US) National Centers for Environmental Prediction (NCEP) (in the US) National Climatic Data Center (NCDC) (in the US) National Emissions Standards for Hazardous Air Pollutants (NESHAPS) (in the US) (National Environmental Satellite, Data and Information Service: see) National Oceanic and Atmospheric Administration (NOAA) (in the US) National Geomagnetism Program (in the US) National Hurricane Center (NHC) (in the US) National Map (in the US) National Oceanic and Atmospheric Administration (NOAA) (in the US) (National Severe Storms Forecast Center [NSSFC]: renamed) Storm Prediction Center (SPC) (in the US) National Severe Storms Laboratory (NSSL) (in the US) National Snow and Ice Data Center (NSIDC) (in the US) National Solar Observatory (in the US) National Weather Association (NWA) (in the US) National Weather Center (NWC) (in the US) National Weather Service bulletin for New Orleans region (at 10:11 a.m., August 28, 2005) National Weather Service (NWS) nautical almanac nephology nephoscope night sky nimbus cloud nitrogen cycle (nitrogen pollution: see) eutrophication (see under Atmospheric deposition) NOAA Weather Radio All Hazards (NWR) (of


id: outline-of-meteorology-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_meteorology-0.md
category: reference
title: Outline of meteorology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_meteorology
date_saved: 2026-05-05T08:18:12.231405+00:00
instance: kb-cron
parent: outline-of-meteorology-20260505-0
summary:
The following outline is provided as an overview of and topical guide to the field of Meteorology. Meteorology The interdisciplinary, scientific study of the Earth's atmosphere with the primary focus being to understand, explain, and forecast weather events. Meteorology, is applied to and employed by a wide variety of diverse fields, including the military, energy production, transport, agriculture, and construction. == Essence of meteorology == Meteorology Climate – the average and variations of weather in a region over long periods of time. Meteorology – the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting (in contrast with climatology). Weather – the set of all the phenomena in a given atmosphere at a given time. == Branches of meteorology == Microscale meteorology – the study of atmospheric phenomena about 1 km or less, smaller than mesoscale, including small and generally fleeting cloud "puffs" and other small cloud features Mesoscale meteorology – the study of weather systems about 5 kilometers to several hundred kilometers, smaller than synoptic scale systems but larger than microscale and storm-scale cumulus systems, such as sea breezes, squall lines, and mesoscale convective complexes Synoptic scale meteorology – is a horizontal length scale of the


id: outline-of-meteorology-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_meteorology-1.md
category: reference
title: Outline of meteorology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_meteorology
date_saved: 2026-05-05T08:18:12.231405+00:00
instance: kb-cron
parent: outline-of-meteorology-20260505-0
summary:
History of weather forecasting – prior to the invention of meteorological instruments, weather analysis and prediction relied on pattern recognition, which was not always reliable History of surface weather analysis – initially used to study storm behavior, now used to explain current weather and as an aid in short term weather forecasting == Meteorological phenomena == Atmospheric pressure – the pressure at any given point in the Earth's atmosphere Cloud – a visible mass of droplets or frozen crystals floating in the atmosphere above the surface of a planet Rain – precipitation in which separate drops of water fall to the Earth from clouds, a product of the condensation of atmospheric water vapor Snow – precipitation in the form of crystalline water ice, consisting of a multitude of snowflakes that fall from clouds Freezing rain – precipitation that falls from a cloud as snow, melts completely on its way down, then passes through a layer of below-freezing air becoming supercooled, at which point it will freeze upon impact with any object encountered Sleet – term used in the United States and Canada for precipitation consisting of small, translucent ice balls, usually smaller than hailstones Tropical cyclone – a storm system


id: list-of-atlantic-hurricane-records-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records-0.md
category: reference
title: List of Atlantic hurricane records
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records
date_saved: 2026-05-05T08:18:13.570754+00:00
instance: kb-cron
parent: list-of-atlantic-hurricane-records-20260505-0
summary:
As of November 2025, there have been 1,753 tropical cyclones of at least tropical storm intensity, 972 at hurricane intensity, and 342 at major hurricane intensity within the Atlantic Ocean since 1851, the first Atlantic hurricane season to be included in the official Atlantic tropical cyclone record. Though a majority of these cyclones have fallen within climatological averages, prevailing atmospheric conditions occasionally lead to anomalous tropical systems which at times reach extremes in statistical record-keeping including in duration and intensity. The scope of this list is limited to tropical cyclone records solely within the North Atlantic Ocean and is subdivided by their reason for notability. == Tropical cyclogenesis == === Most active / least active Atlantic hurricane seasons === Most Atlantic hurricane seasons prior to the weather satellite era include seven or fewer recorded tropical storms or hurricanes. As the usage of satellite data was not available until the mid-1960s, early storm counts are less reliable. Before the advent of the airplane or means of tracking storms, the ones recorded were storms that affected mainly populated areas. An undercount bias of zero to six tropical cyclones per year between 1851 and 1885 and zero to four per year between 1886


id: list-of-atlantic-hurricane-records-20260505-1
file: data/en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records-1.md
category: reference
title: List of Atlantic hurricane records
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records
date_saved: 2026-05-05T08:18:13.570754+00:00
instance: kb-cron
parent: list-of-atlantic-hurricane-records-20260505-0
summary:
Generally speaking, the intensity of a tropical cyclone is determined by either the storm's maximum sustained winds or lowest barometric pressure. The following table lists the most intense Atlantic hurricanes in terms of their lowest barometric pressure. In terms of wind speed, Allen from 1980 and Melissa from 2025 are tied for the strongest Atlantic tropical cyclones on record, with maximum sustained winds of 190 mph (305 km/h). For many years, it was thought that Hurricane Camille also attained this intensity, but this conclusion was changed in 2014. The original measurements of Camille are suspect since wind speed instrumentation used at the time would likely be damaged by winds of such intensity. Nonetheless, their central pressures are low enough to rank them among the strongest recorded Atlantic hurricanes. This is also why despite there being 2 hurricanes in the month of January, neither of them are ranked here as both of them had a higher pressure than Unnamed Subtropical Storm (2023). Owing to their intensity, the strongest Atlantic hurricanes have all attained Category 5 classification. Hurricane Opal, the most intense Category 4 hurricane recorded, intensified to reach a minimum pressure of 916 mbar (hPa; 27.05 inHg), a pressure typical of


id: list-of-atlantic-hurricane-records-20260505-2
file: data/en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records-2.md
category: reference
title: List of Atlantic hurricane records
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records
date_saved: 2026-05-05T08:18:13.570754+00:00
instance: kb-cron
parent: list-of-atlantic-hurricane-records-20260505-0
summary:
=== Fastest intensification === Fastest intensification from a tropical depression to a hurricane (1-minute sustained surface winds) – 12 hoursHarvey 1981 – 35 mph (55 km/h) to 80 mph (130 km/h) – from 1200 UTC September 12 to 0000 UTC September 13 Fastest intensification from a tropical depression to a Category 5 hurricane (1-minute sustained surface winds) – 54 hoursMilton 2024 – 35 mph (55 km/h) to 175 mph (280 km/h) – from 1200 UTC October 5 to 1800 UTC October 7 Fastest intensification from a tropical storm to a Category 5 hurricane (1-minute sustained surface winds) – 24 hoursWilma 2005 – 70 mph (110 km/h) to 175 mph (275 km/h) – from 0600 UTC October 18 to 0600 UTC October 19 Maximum pressure drop in 12 hours – 83 mbarWilma 2005 – 975 millibars (28.8 inHg) to 892 millibars (26.3 inHg) – from 1800 UTC October 18 to 0600 UTC October 19 Maximum pressure drop in 24 hours – 97 mbarWilma 2005 – 979 millibars (28.9 inHg) to 882 millibars (26.0 inHg) – from 1200 UTC October 18 to 1200 UTC October 19 == Landfall == === Earliest and latest Atlantic landfalls === === Southern and northernmost Atlantic landfalls


id: list-of-countries-by-average-yearly-temperature-20260505-0
file: data/en.wikipedia.org/wiki/List_of_countries_by_average_yearly_temperature-0.md
category: reference
title: List of countries by average yearly temperature
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_countries_by_average_yearly_temperature
date_saved: 2026-05-05T08:18:14.880921+00:00
instance: kb-cron
parent: list-of-countries-by-average-yearly-temperature-20260505-0
summary:
This is a list of countries and sovereign states by temperature. Average yearly temperature is calculated by averaging the minimum and maximum daily temperatures in the country, averaged for the years 1991 – 2020, from World Bank Group, derived from raw gridded climatologies from the Climatic Research Unit. == See also == List of countries by average annual precipitation == References ==


id: list-of-rogue-waves-incidents-20260505-0
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-0.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
This list of rogue waves compiles incidents of known and likely rogue waves – also known as freak waves, monster waves, killer waves, and extreme waves. These are dangerous and rare ocean surface waves that unexpectedly reach at least twice the height of the tallest waves around them, and are often described by witnesses as "walls of water". They occur in deep water, usually far out at sea, and are a threat even to capital ships, ocean liners and land structures such as lighthouses. == Background == Anecdotal evidence from mariners' testimonies and incidents of wave damage to ships has long suggested the existence of rogue waves; however, their scientific measurement was positively confirmed only following measurements of the Draupner wave, a rogue wave at the Draupner platform, in the North Sea on 1 January 1995. In this event, minor damage was inflicted on the platform, confirming that the reading was valid. In modern oceanography, rogue waves are defined not as the biggest possible waves at sea, but instead as extreme sized waves for a given sea state. Many of these encounters are only reported in the media, and are not examples of open ocean rogue waves. Often a huge


id: list-of-rogue-waves-incidents-20260505-1
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-1.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
On 4 August 1498, during the third voyage of Christopher Columbus, Columbus's ships were in the Gulf of Paria off the southern tip of Trinidad when Columbus heard a tremendous roar from behind his flagship. He observed a wave approaching from astern that was as tall as his ships' masts. Columbus described the wave as lifting his ships higher than anything he ever had experienced before, after which the ships dropped into a huge trough. On 28 September 1853 a huge wave struck the emigrant ship Annie Jane while she was off Barra in the Outer Hebrides. The wave caused the poop deck to collapse, instantly crushing 200 people to death. Annie Jane soon sank, and her 102 survivors were washed ashore on a piece of her deck. On 11 March 1861 at midday the lighthouse on Eagle Island off the west coast of Ireland was struck by a large wave that smashed 23 panes, washing some of the lamps down the stairs, and broken glass damaged the reflectors beyond repair. In order to damage the uppermost portion of the lighthouse, water had to surmount a seaside cliff measuring 40 m (130 ft) in height and a further 26 m


id: list-of-rogue-waves-incidents-20260505-2
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-2.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
Water rose as high as the bottom of her lower foretop, filling it with water, sweeping her forward deck clear, smashing her forebridge – much of which was found in pieces on her upper deck – wrecking her chart house, shifting the roof of her conning tower, and flooding her forward main gun turret, mess decks, and flats. Five of her crew died, and 17 others suffered serious injuries. At midnight on 5–6 May 1916 the British polar explorer Ernest Shackleton was at the tiller of the small sailboat James Caird in the Southern Ocean during a storm when he thought he saw the bad weather clearing in the west, astern. He then realized that what he thought was a line of white clouds above a clear dark sky was actually the crest of a single enormous wave that struck and nearly swamped the boat. Shackleton reported that the wave was larger than any he had ever seen before in his 26 years of seafaring. On 29 August 1916 at about 4:40 p.m., the United States Navy armored cruiser USS Memphis was wrecked in Santo Domingo harbor in the Dominican Republic when struck in rapid succession by three waves of


id: list-of-rogue-waves-incidents-20260505-3
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-3.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
=== Second half of the 20th century === On 5 February 1963, the French Navy light cruiser Jeanne d'Arc encountered a rogue wave while serving as the training ship of the French Naval Academy. On 12 April 1966, the Italian liner Michelangelo was in the Atlantic Ocean steaming toward New York City in a storm, frequently encountering 20-metre (66 ft) waves, when a giant wave struck her, breaking 18 metres (59 ft) over her bow. It traveled 70 metres (230 ft) aft along her deck and struck her superstructure, smashing 2-centimetre (0.8 in) thick windows on her bridge over 25 metres (82 ft) above the waterline. It tore a hole in the superstructure, killed two passengers instantly, fatally injured a crewman who died a few hours later, and injured over 50 other people, 10 of them gravely. In 1968 the sailing yacht Suhaili was in the Southern Ocean in high winds and a 40-to-50-foot (12 to 15 m) swell when her captain came on deck and sighted a breaking wave approaching from astern which stretched from horizon to horizon and was twice the height of the swell, indicating a height of at least 80 feet (24 m). He climbed into


id: list-of-rogue-waves-incidents-20260505-4
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-4.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
=== 21st century === The Bahamian-registered cruise ships MS Bremen and MS Caledonian Star encountered 30-meter (98 ft) freak waves in the South Atlantic in 2001. Bridge windows on both ships were smashed, and all power and instrumentation lost. Naval Research Laboratory ocean-floor pressure sensors detected a freak wave caused by Hurricane Ivan in the Gulf of Mexico in 2004. The wave was around 27.7 meters (91 ft) high from peak to trough, and around 200 meters (660 ft) long. Norwegian Dawn, (three waves in succession, off the coast of Georgia, 16 April 2005): "The sea had actually calmed down when the 21-metre (69 ft) wave seemed to come out of thin air… Our captain, who has 20 years on the job, said he never saw anything like it." "The water exerted enough force to shear off the welds for the aluminum rail supports on the [ninth- and tenth-level] balconies of two cabins, allowing the teak balcony rails to break loose and crash into the cabin windows. The broken glass filling the drains compounded the water damage by allowing a large amount of water to enter the two cabins and damage the carpets in 61 other cabins. The ship's operating


id: list-of-meteorologists-20260505-0
file: data/en.wikipedia.org/wiki/List_of_meteorologists-0.md
category: reference
title: List of meteorologists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_meteorologists
date_saved: 2026-05-05T08:18:18.835134+00:00
instance: kb-cron
parent: list-of-meteorologists-20260505-0
summary:
== A–B == Cleveland Abbe Aristotle June Bacon-Bercey David Bates Francis Beaufort Tor Bergeron Jacob Bjerknes Vilhelm Bjerknes Howard B. Bluestein Bert Bolin Léon Teisserenc de Bort Harold E. Brooks Keith Browning David Brunt C.H.D. Buys Ballot Horace R. Byers == C–D == Matthew Cappucci DaNa Carlis Jule Gregory Charney Coching Chu Isaac Cline Balfour Currie John Dalton Robert Davies-Jones William Morris Davis Igor Delijanić Jean-André Deluc Charles A. Doswell III Heinrich Wilhelm Dove == E–F == V. Walfrid Ekman Kerry Emanuel John Farrah William Ferrel Rich Fields John Park Finley Michael Fish Robert FitzRoy Bert Foord Benjamin Franklin Tetsuya Theodore Fujita == G–H == Francis Galton Audronė Galvonaitė Allison Göhler Ernest Gold William M. Gray Thomas P. Grazulis Willis Ray Gregg George Hadley Edmond Halley Julius von Hann Kenneth Hare Ingrid Holford Brian Hoskins Luke Howard == J–K == Robert Johns Jan-Erik Johnsen John Knox Wladimir Köppen Shen Kuo == L–O == Hubert Lamb Christopher Landsea Leslie R. Lemon Johannes Letzmann Richard Lindzen Mònica López Edward Norton Lorenz David M. Ludlum Paul Markowski Charles F. Marvin Cliff Mass Patrick McTaggart-Cowan Pomponius Mela Milutin Milanković James T. Moore Joel Myers Aili Nurminen Isidoro Orlanski == P–R == Erik Palmén Allen


id: list-of-meteorological-photos-and-videos-20260505-0
file: data/en.wikipedia.org/wiki/List_of_meteorological_photos_and_videos-0.md
category: reference
title: List of meteorological photos and videos
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_meteorological_photos_and_videos
date_saved: 2026-05-05T08:18:20.153817+00:00
instance: kb-cron
parent: list-of-meteorological-photos-and-videos-20260505-0
summary:
There are several notable photographs or videos in the field of meteorology, the study of the Earth's atmosphere, climate, and weather. These images or videos may be referred to as historic, most important, most iconic, or most influential weather-related photographs or weather-related videos. == List == === 19th century === === 20th century === === 21st century === == See also == List of photographs considered the most important == References ==


id: list-of-cat-documentaries-television-series-and-cartoons-20260505-0
file: data/en.wikipedia.org/wiki/List_of_cat_documentaries,_television_series_and_cartoons-0.md
category: reference
title: List of cat documentaries, television series and cartoons
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_cat_documentaries,_television_series_and_cartoons
date_saved: 2026-05-05T08:18:34.681061+00:00
instance: kb-cron
parent: list-of-cat-documentaries-television-series-and-cartoons-20260505-0
summary:
List of cat documentaries, television series and cartoons includes serious documentaries, television series and cartoons, in alphabetical order, related to cats . == Documentaries == Cat Empire: Docuseries (2021), Ubique Film Cats: Caressing the Tiger (1991), National Geographic Cats: Choosing, Caring and Training (2008), Revolution LLC Kedi (2016), Termite Films The Lion in Your Living Room (2015), Canadian Broadcasting Company Science of Cats (2014), National Geographic Secret Life of Cats (2014), National Geographic The Standard of Perfection: Show Cats (2009), Public Broadcasting Service Understanding Cats (2009), Public Broadcasting Service The World of Cats (2008), Columbia River Entertainment === Big cats documentaries === Big Cats (2018, 3 episodes), BBC Lions of Darkness (1993) National Geographic Lions of the African Night (1987) National Geographic Tiger King: Murder, Mayhem and Madness (2020, 7 episodes), Netflix === Documentary and reality series === Cats 101 (2009–2012), Animal Planet My Cat from Hell (2011–2016), Animal Planet Must Love Cats (2011–2012), Animal Planet == Cartoons == Garfield (1978–present; TV series 1988–1994, 2009–2016, 2019–2020), Universal Press Syndicate Heathcliff (1980–1981, 1984–1985), Ruby-Spears Productions, et al. The Itchy & Scratchy Show (1988–1997, as an element of The Simpsons), 20th Television Krazy Kat (1913–1944), King Features Syndicate The Ren and Stimpy


id: list-of-nature-conservation-organizations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_nature_conservation_organizations-0.md
category: reference
title: List of nature conservation organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_nature_conservation_organizations
date_saved: 2026-05-05T08:18:36.001086+00:00
instance: kb-cron
parent: list-of-nature-conservation-organizations-20260505-0
summary:
This is a list of nature conservation organisations that primarily aim to protect species, their habitats, and ecosystems. == List of international conservation organisations == African Wildlife Foundation - ensuring that wildlife and wild lands thrive in modern Africa Bat Conservation International - working to conserve the world's bats and their habitats BirdLife International - a global partnership of organisations that strive to conserve birds Center for Biological Diversity - protecting endangered species through legal action, petitions, media and activism Conservation International - secure the critical benefits that nature provides to humanity Durrell Wildlife Conservation Trust - mission to save species from extinction EcoHealth Alliance - protecting the health of people, animals, and the environment from emerging infectious diseases Fauna and Flora International - safeguard the future of southern Africa's large mammal populations Greenpeace - raising environmental issues to public knowledge International Anti-Poaching Foundation - created a structured military-like approach to conservation in Africa International Union for Conservation of Nature - a partnership of 1400 organisations working in the field of nature conservation MarAlliance - non-profit that supports the effective management and conservation of large marine wildlife and their critical habitats with the support of dependent communities. National Audubon Society


id: index-of-conservation-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_conservation_articles-0.md
category: reference
title: Index of conservation articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_conservation_articles
date_saved: 2026-05-05T08:18:37.189423+00:00
instance: kb-cron
parent: index-of-conservation-articles-20260505-0
summary:
This is an index of conservation topics. It is an alphabetical index of articles relating to conservation biology and conservation of the natural environment. == A == Abiotic stress - Adaptive management - Adventive plant - Aerial-seeding - Agreed Measures for the Conservation of Antarctic Fauna and Flora - Agroecology - American Prairie Foundation - Anti-whaling - Assisted migration - Assisted migration of forests in North America == B == Biodegradation - Biodiversity - Biodiversity action plan - Biodiversity hotspot - Biogenic - Biodiversity Outcomes Framework - Biogeographic realm - Biogeography - Bioinformatics - Biological integrity - Biomagnification - Biomaterial - Biome - Biomimicry - Biomonitoring - Biophilia hypothesis - Biophilic design - Biopiracy - Bioregion - Bioregionalism - Biosafety - Biosalinity - Biosecurity - Biosphere - Biosphere reserve - Biostatistics - Biosurvey - Biotechnology - Bioterrorism - Biotransference - Bird conservation - Blue-listed - Bottom trawling - Buffer zone == C == Captive breeding - Cave conservation - Charismatic megafauna - CITES - Common species - Community-based conservation - Compassionate conservation - Conference of Governors - Conservation agriculture - Conservation area - Conservation authority - Conservation biology - Conservation Commons - Conservation community - Conservation dependent - Conservation designation


id: natural-regions-of-germany-20260505-0
file: data/en.wikipedia.org/wiki/Natural_regions_of_Germany-0.md
category: reference
title: Natural regions of Germany
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Natural_regions_of_Germany
date_saved: 2026-05-05T08:18:38.455696+00:00
instance: kb-cron
parent: natural-regions-of-germany-20260505-0
summary:
This division of Germany into major natural regions takes account primarily of geomorphological, geological, hydrological, and pedological criteria in order to divide the country into large, physical units with a common geographical basis. Political boundaries play no part in this, apart from defining the national border. In addition to a division of Germany by natural regions, the federal authorities have also produced a division by so-called landscape areas (Landschaftsräume) that is based more on human utilisation of various regions and so has clearly different boundaries. == Groundwork by the Federal Institute of Regional Studies (BfL) == The natural region classification of Germany, as used today by the Federal Office for Nature Conservation (Bundesamt für Naturschutz or BfN) and by most state institutions, is largely based on the work in producing the Handbook of Natural Region Divisions of Germany between the years 1953 to 1962. This divided the present federal territory (then West and East Germany) into 86 so-called major landscape unit groups (Haupteinheitgruppen) each with a two-digit number between 01 and 90. These, in turn, were subdivided into up to ten, in some cases more, major landscape units (Haupteinheiten), each with a three-digit number. The handbook was accompanied by 1:100,000


id: natural-regions-of-germany-20260505-1
file: data/en.wikipedia.org/wiki/Natural_regions_of_Germany-1.md
category: reference
title: Natural regions of Germany
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Natural_regions_of_Germany
date_saved: 2026-05-05T08:18:38.455696+00:00
instance: kb-cron
parent: natural-regions-of-germany-20260505-0
summary:
The first version by Heinrich Müller-Miny appeared in the updated handbook map, the overview maps of the individual map sheets from 1960 and in the ninth edition of the handbook (1962) In the map sheets from publication year 1964 this version was slightly changed: The Upper Palatine-Upper Main Hills (Oberpfälzisch-Obermainische Hügelland) was raised from the third to the second level. The secondary landscape region of Lössbörden was expanded, especially in the west. There was another change in the map sheets from 1979 (Sheet 182 Burghausen) until their last publication (Sheet München and Tegernsee 1994): The primary landscape region, Central Uplands (Mittelgebirge), was divided in two into Mittelgebirgsschwelle and Schichtstufenland The classification of the Natural regions in Saxony, published after that of the Institute for Regional Studies and continually revised, allocates the larger northern part of the major unit group of Upper Lusatia (Oberlausitz) to the Lössbörden, so that only the Lusatian Mountains (Lausitzer Bergland) in the south remains within the Central Uplands. For clarity the first and second levels of the major landscape regions are organised from north to south and from west to east. Within a secondary or tertiary landscape region the list follows the numerical order in the


id: natural-regions-of-germany-20260505-2
file: data/en.wikipedia.org/wiki/Natural_regions_of_Germany-2.md
category: reference
title: Natural regions of Germany
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Natural_regions_of_Germany
date_saved: 2026-05-05T08:18:38.455696+00:00
instance: kb-cron
parent: natural-regions-of-germany-20260505-0
summary:
Upper Rhine Plain (D53) 23 Rhine-Main Basin 22 Northern Upper Rhine Plain 21 Middle Upper Rhine Plain 20 Southern Upper Rhine Plain ==== South German Scarplands ==== Source: (arranged by escarpments Grundgebirge/Buntsandstein, Muschelkalk, Keuper-Lias and Malm from north to south, internally from west to east. 14 Odenwald-Spessart-Rhön (D55) 15 Black Forest (D54) Swabian-Franconian Gäue 12 Neckar and Tauber Gäu Plateaus (D57) 13 Main Franconia Plateau (D56) Swabian-Franconian Keuper-Lias Lands 10 Swabian Keuper-Lias Lands (D58) 11 Franconian Keuper-Lias Lands (D59) Swabian-Franconian Jura 09 Swabian Jura (D60) 08 Franconian Jura (D61) 16 Upper Rhine Region (Dinkelberg and Upper Rhine Valley) (D69) ==== Upper Palatine-Upper Main Hills ==== 07 Upper Palatine-Upper Main Hills (D62) === Alpine Foreland === Source: arranged from north to south and from west to east) ==== Northern Alpine Foreland ==== 04 Iller-Lech Plateau (D64) Lower Bavarian Upland and Isar-Inn Gravel Plateau (D65) 06 Lower Bavarian Upland 05 Isar-Inn Gravel Plateau ==== Southern Alpine Foreland ==== Pre-Alpine Hills and Moorland (= Southern Alpine Foreland; D66) 03 Subalpine New Moraine Land 02* Nagelfluh Hills and Basins between Lake Constance and Wertach === Alps === The following 3 groups were still counted as part of (ex-)group 02 in the Handbook; the German


id: list-of-glaciers-20260505-0
file: data/en.wikipedia.org/wiki/List_of_glaciers-0.md
category: reference
title: List of glaciers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_glaciers
date_saved: 2026-05-05T08:18:39.696555+00:00
instance: kb-cron
parent: list-of-glaciers-20260505-0
summary:
A glacier (US: GLAY-shər) or (UK: ) is a persistent body of dense ice that is constantly moving under its own weight; it forms where the accumulation of snow exceeds its ablation (melting and sublimation) over many years, often centuries. Glaciers slowly deform and flow due to stresses induced by their weight, creating crevasses, seracs, and other distinguishing features. Because glacial mass is affected by long-term climate changes, e.g., precipitation, mean temperature, and cloud cover, glacial mass changes are considered among the most sensitive indicators of climate change. There are about 198,000 to 200,000 glaciers in the world. Catalogs of glaciers include: World Glacier Inventory World Glacier Monitoring Service Global Land Ice Measurements from Space (GLIMS) Glacier Database Randolph Glacier Inventory (RGI) == Glaciers by continent == === Africa === Africa, specifically East Africa, has contained glacial regions, possibly as far back as the last glacier maximum 10 to 15 thousand years ago. Seasonal snow does exist on the highest peaks of East Africa as well as in the Drakensberg Range of South Africa, the Stormberg Mountains, and the Atlas Mountains in Morocco. Currently, the only remaining glaciers on the continent exist on Mount Kilimanjaro, Mount Kenya, and the Rwenzori.


id: list-of-habitats-of-principal-importance-in-wales-20260505-0
file: data/en.wikipedia.org/wiki/List_of_habitats_of_principal_importance_in_Wales-0.md
category: reference
title: List of habitats of principal importance in Wales
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_habitats_of_principal_importance_in_Wales
date_saved: 2026-05-05T08:18:40.968477+00:00
instance: kb-cron
parent: list-of-habitats-of-principal-importance-in-wales-20260505-0
summary:
Wales is obliged by law to maintain lists of species and habitats of principal importance for biodiversity conservation; the other countries within the UK: Scotland, England and Northern Ireland, have their own laws for this purpose. Public bodies, including local authorities now have a legal duty to have regard to conserving biodiversity in the exercise of their normal functions. In Wales, that obligation originally derived from section 42 of the Natural Environment and Rural Communities (NERC) Act 2006. However, this requirement for Wales has since been superseded by an almost identical requirement enshrined within the Environment (Wales) Act 2016. == Selection == The habitats that have been designated to be of "principal importance for the purpose of conserving biodiversity" derive from lists originally drawn up for the UK Biodiversity Action Plan (UK BAP). These lists were reviewed in 2007, and the total number of UK BAP habitats increased from 45 to 65, and the number of UK BAP species increased from under 600 to 1,150. From these, the formal list just for Wales (and laid out below) now contains 53 of those 65 habitats. === Legal obligations === Section 6 of the Environment (Wales) Act 2016 places a legal obligation


id: list-of-herbaria-in-europe-20260505-0
file: data/en.wikipedia.org/wiki/List_of_herbaria_in_Europe-0.md
category: reference
title: List of herbaria in Europe
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_herbaria_in_Europe
date_saved: 2026-05-05T08:18:42.184594+00:00
instance: kb-cron
parent: list-of-herbaria-in-europe-20260505-0
summary:
This is a list of herbaria in Europe, organized first by region where the herbarium is located (using the United Nations geoscheme for Europe), then within each region by size of the collection. For other continents, see List of herbaria. == Eastern Europe == The tables below list herbaria located in Eastern Europe as defined by the United Nations geoscheme for Europe. === Black Sea === The following table includes herbaria located in countries on the Black Sea, including Bulgaria, Moldova, Romania, and Ukraine. === Czech Republic === The following table includes herbaria located in the Czech Republic. === Hungary === The following table includes herbaria located in Hungary. === Poland === The following table includes herbaria located in Poland. === Russia === The following table includes herbaria located in European Russia. === Ukraine === National Herbarium of Ukraine (KW) == Northern Europe == The tables below list herbaria located in Northern Europe as defined by the United Nations geoscheme for Europe: Ireland, the British Isles, Baltic states, Scandinavia, and Iceland. === Baltic states === The following table includes herbaria located in the Baltic states: Estonia, Latvia, and Lithuania. === Britain & Ireland === The following table includes herbaria located


id: list-of-herbaria-in-north-america-20260505-0
file: data/en.wikipedia.org/wiki/List_of_herbaria_in_North_America-0.md
category: reference
title: List of herbaria in North America
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_herbaria_in_North_America
date_saved: 2026-05-05T08:18:43.376729+00:00
instance: kb-cron
parent: list-of-herbaria-in-north-america-20260505-0
summary:
This is a list of herbaria in North America, organized first by country or region where the herbarium is located, then within each region by size of the collection. All herbarium codes follow the Index Herbariorum. For other continents, see List of herbaria. == Canada == == Central America and the Caribbean == The table below lists herbaria located in Central America and the Caribbean. == Mexico == == United States == == References ==


id: list-of-herbaria-in-turkey-20260505-0
file: data/en.wikipedia.org/wiki/List_of_herbaria_in_Turkey-0.md
category: reference
title: List of herbaria in Turkey
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_herbaria_in_Turkey
date_saved: 2026-05-05T08:18:44.619963+00:00
instance: kb-cron
parent: list-of-herbaria-in-turkey-20260505-0
summary:
The following is a list of herbaria in Turkey. Herbaria are established within faculties and institutes of credible universities. Those created in departments that are concerned with natural sciences such as botany, ecology, biogeography and climatology are mostly used to do research on the genetics of the plants, to examine their distribution on specific geographical locations and to protect them for future generations, while those affiliated with more practice-based departments such as pharmacy are used as a resource for drug production techniques. The first herbarium in Turkey was opened in 1933 within the Faculty of Science at Ankara University. Founded by Kurt Krause and Hikmet Birand, this herbarium is the oldest among its counterparts in the country and has the most number of species within its 200,000-piece collection. Therefore, it is also called Herbarium Turcicum (Turkish Herbarium). The Sugar Institute Herbarium, Atatürk University Faculty of Science Herbarium and Ege University Faculty of Pharmacy Herbarium are closed to visitors and scientific researchers. However, thanks to a specially created website, it is possible to access all the data related to the Ege University Faculty of Pharmacy Herbarium remotely. Altınbaş University Faculty of Pharmacy Herbarium, which opened in 2018, is the first herbarium


id: list-of-herbaria-20260505-0
file: data/en.wikipedia.org/wiki/List_of_herbaria-0.md
category: reference
title: List of herbaria
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_herbaria
date_saved: 2026-05-05T08:18:45.907753+00:00
instance: kb-cron
parent: list-of-herbaria-20260505-0
summary:
This is a list of active herbaria, organized first by continent where the herbarium is located, then within each continent by size of the collection. The list is based on the Index Herbariorum, a global directory of herbaria and their associated staff. == Background == A herbarium (plural "herbaria") is a collection of preserved plant specimens. These specimens may be whole plants or plant parts: these will usually be in a dried form, mounted on a sheet, but depending upon the material may also be kept in alcohol or other preservative. The same term is often used in mycology to describe an equivalent collection of preserved fungi and in phycology to describe a collection of algae. To preserve their form and color, plants collected in the field are spread flat on sheets of newsprint and dried, usually in a plant press, between blotters or absorbent paper. The specimens, which are then mounted on sheets of stiff white paper, are labeled with all essential data, such as collector, date and place found, description of the plant, elevation, and special habitat conditions. The sheet is then placed in a protective case. As a precaution against insect damage, the pressed plant is frozen


id: iucn-protected-area-categories-20260505-0
file: data/en.wikipedia.org/wiki/IUCN_protected_area_categories-0.md
category: reference
title: IUCN protected area categories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/IUCN_protected_area_categories
date_saved: 2026-05-05T08:18:47.158801+00:00
instance: kb-cron
parent: iucn-protected-area-categories-20260505-0
summary:
IUCN protected area categories, or IUCN protected area management categories, are categories used to classify protected areas in a system developed by the International Union for Conservation of Nature (IUCN). The enlisting of such areas is part of a strategy being used toward the conservation of the world's natural environment and biodiversity. The IUCN has developed the protected area management categories system to define, record and classify the wide variety of specific aims and concerns when categorising protected areas and their objectives. Further supplementary guidelines have been developed specific to marine protected areas (MPAs). This categorisation method is recognised on a global scale by national governments and international bodies such as the United Nations and the Convention on Biological Diversity. == Categories == === Category I === ==== Category Ia – strict nature reserve ==== A strict nature reserve (IUCN Category Ia) is an area which is protected from all but light human use in order to protect its biodiversity and also possibly its geological/geomorphological features. These areas are often home to dense native ecosystems where all human disturbance except scientific study, environmental monitoring and education is prohibited. Because these areas are so strictly protected, they provide ideal pristine environments


id: iucn-protected-area-categories-20260505-1
file: data/en.wikipedia.org/wiki/IUCN_protected_area_categories-1.md
category: reference
title: IUCN protected area categories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/IUCN_protected_area_categories
date_saved: 2026-05-05T08:18:47.158801+00:00
instance: kb-cron
parent: iucn-protected-area-categories-20260505-0
summary:
=== Category V – protected landscape or seascape === A protected landscape or protected seascape (IUCN Category V) covers a body of land or ocean with an explicit natural conservation plan, but usually also accommodates a range of for-profit activities. The main objective is to safeguard regions that have built up a distinct and valuable ecological, biological, cultural, or scenic character. In contrast with previous categories, Category V permits surrounding communities to interact more with the area, contributing to the area's sustainable management and engaging with its natural and cultural heritage. Landscapes and seascapes that fall into this category should represent an integral balance between people and nature and can sustain activities such as traditional agricultural and forestry systems on conditions that ensure the continued protection or ecological restoration of the area. Category V is one of the more flexible classifications of protected areas. As a result, protected landscapes and seascapes may be able to accommodate contemporary developments, such as ecotourism, at the same time as maintaining the historical management practices that may procure the sustainability of agrobiodiversity and aquatic biodiversity. === Category VI – protected area with sustainable use of natural resources === Though human involvement is a large


id: list-of-animals-by-number-of-legs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_animals_by_number_of_legs-0.md
category: reference
title: List of animals by number of legs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_animals_by_number_of_legs
date_saved: 2026-05-05T08:18:48.428358+00:00
instance: kb-cron
parent: list-of-animals-by-number-of-legs-20260505-0
summary:
The following is a list of selected animals in order of increasing number of legs, from 0 legs to 653 pairs of legs, the maximum recorded in the animal kingdom. Each entry provides the relevant taxa up to the rank of phylum. Each entry also provides the common name of the animal. If the relevant taxon includes different animals with different common names, then the entry provides the common name of a familiar example. If juveniles have fewer legs than adults, then the animal is listed by the number of legs recorded in mature adults. If this number varies among adults within the taxon, then this variation is noted in a comment. In counting legs, this list follows conventions adopted in the relevant literature. For example, millipedes with gonopods are listed by numbers that exclude leg pairs that become gonopods, but numbers for millipedes with telopods include leg pairs that become telopods. Animals have been selected so that each number from 0 to 55 leg pairs has one example listed. Each of these examples is listed by a number closely associated with the relevant taxon, either because that number is the one most commonly observed in that taxon or because


id: list-of-long-distance-footpaths-20260505-0
file: data/en.wikipedia.org/wiki/List_of_long-distance_footpaths-0.md
category: reference
title: List of long-distance footpaths
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long-distance_footpaths
date_saved: 2026-05-05T08:18:49.672015+00:00
instance: kb-cron
parent: list-of-long-distance-footpaths-20260505-0
summary:
This is a list of some long-distance footpaths used for walking and hiking. == Africa == === Lesotho === 3 Ranges Trail: 288.3 km (179.1 mi) a hiking trail that passes three mountain ranges, namely Thaba Putsoa Range, Central Range and ends at Drakensburg Range. It starts at Lesotho lowest point, at the confluence of Makhaleng and Senqu and ends at the country's highest point, Thabana Ntlenyane (3482m). === Namibia === Fish River Canyon: 90 km (56 mi) route in the ǀAi-ǀAis/Richtersveld Transfrontier Park === South Africa === Otter Trail: 44 km (27 mi) section of the Garden Route along the Cape coast Drakensberg Grand Traverse: rugged 205 km (127 mi) trek in KwaZulu-Natal === Egypt === Sinai Trail: 250 km (160 mi) thru-hike from Nuweiba to Mount Catherine === Uganda === Sir Samuel and Lady Florence Baker Trail: 575 km (357 mi) trek through northern Uganda == Asia == === Bangladesh === Jhiri Path: ascent of Keokradong, one of the highest mountains in Bangladesh. === Bhutan === Snowman Trek: 328 km (204 mi) trek through high passes near the border with Tibet [1] === Hong Kong === Hong Kong Trail: 50 km (31 mi) across Hong Kong Island Lantau


id: list-of-long-distance-footpaths-20260505-1
file: data/en.wikipedia.org/wiki/List_of_long-distance_footpaths-1.md
category: reference
title: List of long-distance footpaths
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long-distance_footpaths
date_saved: 2026-05-05T08:18:49.672015+00:00
instance: kb-cron
parent: list-of-long-distance-footpaths-20260505-0
summary:
=== Türkiye === Sufi Trail – 801 km (498 mi) trail from Istanbul to Konya, following the Ottoman Sultans Hajj route Lycian Way – 540 km (340 mi) trail around the south coast of Turkey Saint Paul Trail – 500 km (310 mi) trail following St Paul's 1st journey Sultans Trail runs 2,200 km (1,400 mi) from Vienna through the Balkans, terminating in Istanbul Kaçkar Trails – 8 multi-day trails in Northeast Turkey Evliya Çelebi Way – the first part of Evliya's route to Mecca in the 17th century Phrygian Way – three separate trekking and cycling trails of total 506 km (314 mi) in the Phrygian Valleys, western Turkey,[2] Carian Trail – 800 km (500 mi) hugging the south-west coast of Turkey == Europe == Some of the best known footpaths in Europe are joined by 12 designated European long-distance paths over some 70,000 km (43,000 mi). Some other popular international routes include: GR (Grande Randonnée) footpaths in Belgium, France, Spain and the Netherlands Tour du Mont Blanc, circling the Mont Blanc massif in Italy, Switzerland, and France. It covers a distance of roughly 170 km (110 mi) with 10 km (6.2 mi) of elevation change. Haute Randonnée Pyrénéenne


id: list-of-long-distance-footpaths-20260505-2
file: data/en.wikipedia.org/wiki/List_of_long-distance_footpaths-2.md
category: reference
title: List of long-distance footpaths
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long-distance_footpaths
date_saved: 2026-05-05T08:18:49.672015+00:00
instance: kb-cron
parent: list-of-long-distance-footpaths-20260505-0
summary:
Barbarossaweg: Korbach to Kyffhäuserdenkmal (332 km (206 mi)) Bergischer Weg: Essen to Königswinter (137 km (85 mi)) Birkenhainer Strasse: Hanau to Gemünden am Main (71 km (44 mi)) Bonifatiusweg: Mainz to Fulda (180 km (110 mi)) Christine-Koch-Weg: Menden to Bad Laasphe (123 km (76 mi)) Eselsweg: Schlüchtern to Großheubach (111 km (69 mi)) König-Ludwig-Weg: Berg am Starnberger See to Füssen (120 km (75 mi)) Maximiliansweg: Lindau to Berchtesgaden (350 km (220 mi)) Moselhöhenweg: Trier to Koblenz Pandurensteig: Waldmuenchen to Passau Rennsteig: Hörsel to Blankenstein/Saale through the Thuringian Forest (169 km (105 mi)) Rheinburgenweg Trail: Bingen to Koblenz to Rüdesheim am Rhein (c.200 km (120 mi)) Rheinhöhenweg Trail: Bonn to Alsheim/Wiesbaden (530 km (330 mi)) Rheinsteig: Bonn to Wiesbaden (320 km (200 mi)) Westweg (Black Forest western ridgeway): Pforzheim to Basel (279 km (173 mi)) Mittelweg (Black Forest central ridgeway): Pforzheim to Waldshut (233 km (145 mi)) Ostweg (Black Forest eastern ridgeway): Pforzheim to Schaffhausen (238 km (148 mi)) Black Forest North Perimeter Way: Mühlacker to Karlsruhe (71 km (44 mi)) Two Valleys Trail: round hiking trail from/to Waldkirch (106 km (66 mi)) Freiburg-Lake Constance Black Forest Trail: Freiburg to Konstanz (183 km (114 mi)) Saar-Hunsrück-Steig: Trier to Saarschleife in Mettlach


id: list-of-long-distance-footpaths-20260505-3
file: data/en.wikipedia.org/wiki/List_of_long-distance_footpaths-3.md
category: reference
title: List of long-distance footpaths
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long-distance_footpaths
date_saved: 2026-05-05T08:18:49.672015+00:00
instance: kb-cron
parent: list-of-long-distance-footpaths-20260505-0
summary:
=== Serbia === The Fruška Gora Transversal (Serbian: Fruškogorska Transverzala) – circular trail around Fruška Gora mountain – 160 km (100 mi) The Small Toplica Transversal (Serbian: Mala Toplička Transverzala) – 122 km (76 mi) === Slovakia === The SNP Heroes Trail, 770 km (480 mi) from Dukla Pass to Devín, across mountainous regions of Slovakia === Slovenia === Slovenian Mountain Hiking Trail 599 km (372 mi) from east (Maribor) to west (Ankaran) === Spain === Way of St. James (Spanish: El Camino de Santiago) Route of the Monasteries of Valencia (Spanish: Ruta de los Monasterios de Valencia) Grande Randonnée 7 (GR 7): From Tarifa to the mountain cabin of Fontferrera, part of the E4 route. Grande Randonnée 11 (GR 11): Pyrenees Trail, staying within Spain. Haute Randonnée Pyrénéenne (HRP): High Pyrenees Trail, along the border with France. Way of the Lighthouses: 200 km (120 mi) along the coastline from Malpica to Fisterra, in Galicia (Spain) === Sweden === Kungsleden, 425 km (264 mi) from the northern extremity of the country Sörmlandsleden 1,000 km (620 mi) of winding paths through the county of Sörmland. Bohusleden 360 km (220 mi) past at Göteborg, Uddevalla, Munkedal and ends finally in Strömstad Ingegerdsleden,


id: list-of-long-distance-footpaths-20260505-4
file: data/en.wikipedia.org/wiki/List_of_long-distance_footpaths-4.md
category: reference
title: List of long-distance footpaths
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long-distance_footpaths
date_saved: 2026-05-05T08:18:49.672015+00:00
instance: kb-cron
parent: list-of-long-distance-footpaths-20260505-0
summary:
=== Brazil === Atlantic Forest Trail 4,270 km (2,400 mi) follows the Serra do Mar mountain range, crossing five Brazilian States and connecting 10 national parks, and many historical trails and traditional communities. (very hard trail) Transmantiqueira Trail 1,200 km (750 mi) crosses the Mantiqueira range in a west–east direction (very hard trail) Petrópolis Teresópolis Traverse 36 km (22 mi) between Petrópolis and Teresópolis (medium to hard trail) Transcarioca Trail 183 km (114 mi) in Rio de Janeiro city (medium to hard trail) Caminho da Fé 538 km (334 mi). There are 3 different starting points: from Sao Carlos to Aparecida, São Paulo or from Cravinhos to Aparecida or from Mococa to Aparecida === Costa Rica === 280 km from Atlantic to Pacific coasts of Costa Rica El Camino de Costa Rica === Chile === Central Chile and Patagonia 3,000 km (1,900 mi) Greater Patagonian Trail from the capital Santiago de Chile to the Southern Patagonian Ice Field. Magallanes 118 km (73 mi). Full circuit of Torres del Paine, including Valle del Francés and access to the base of Torres del Paine. Dientes de Navarino Circuit ≈53 km (33 mi). 3‑7 days. Starts out of Puerto Williams. Access from mainland


id: list-of-national-wildlife-refuges-established-for-endangered-species-20260505-0
category: reference
title: List of National Wildlife Refuges established for endangered species
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_National_Wildlife_Refuges_established_for_endangered_species
date_saved: 2026-05-05T08:18:52.021938+00:00
instance: kb-cron
parent: list-of-national-wildlife-refuges-established-for-endangered-species-20260505-0
file:
data/en.wikipedia.org/wiki/List_of_National_Wildlife_Refuges_established_for_endangered_species-0.md

summary:
This is a list of National Wildlife Refuges established specifically for the protection of one or more endangered species on the United States Fish and Wildlife Service list of endangered mammals and birds. == References ==


id: list-of-natural-phenomena-20260505-0
file: data/en.wikipedia.org/wiki/List_of_natural_phenomena-0.md
category: reference
title: List of natural phenomena
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_natural_phenomena
date_saved: 2026-05-05T08:18:54.736426+00:00
instance: kb-cron
parent: list-of-natural-phenomena-20260505-0
summary:
A natural phenomenon is an observable event which is not man-made. Examples include: sunrise, weather, fog, thunder, tornadoes; biological processes, decomposition, germination; physical processes, wave propagation, erosion; tidal flow, and natural disasters such as electromagnetic pulses, volcanic eruptions, hurricanes and earthquakes. == History == Over many intervals of time, natural phenomena have been observed by a series of countless events as a feature created by nature. == Physical phenomena == The act of: Freezing Boiling Gravity Magnetism === Gallery === == Chemical phenomena == Oxidation Fire Rusting == Biological phenomena == Metabolism Catabolism Anabolism Decomposition – by which organic substances are broken down into a much simpler form of matter Fermentation – converts sugar to acids, gases and/or alcohol. Growth Birth Death Population decrease === Gallery === == Astronomical phenomena == Supernova Gamma ray bursts Quasars Blazars Pulsars Cosmic microwave background radiation == Geological phenomena == Mineralogic phenomena Lithologic phenomena Rock types Igneous rock Igneous formation processes Sedimentary rock Sedimentary formation processes (sedimentation) Quicksand Metamorphic rock Endogenic phenomena Plate tectonics Continental drift Earthquake Oceanic trench Phenomena associated with igneous activity Geysers and hot springs Bradyseism Volcanic eruption Earth's magnetic field Exogenic phenomena Slope phenomena Slump Landslide Weathering phenomena Erosion Glacial


id: list-of-common-household-pests-20260505-0
file: data/en.wikipedia.org/wiki/List_of_common_household_pests-0.md
category: reference
title: List of common household pests
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_household_pests
date_saved: 2026-05-05T08:18:56.020269+00:00
instance: kb-cron
parent: list-of-common-household-pests-20260505-0
summary:
This is a list of common household pests – undesired animals that have a history of living, invading, causing damage, eating human foods, acting as disease vectors or causing other harms in human habitation. == Mammals == Mice Field mice House mice Possums Brushtail possum Ringtail possum Rats Black rats Brown rats Wood rats Cotton rats == Invertebrates == Ants Argentine ants Carpenter ants Fire ants Odorous house ants Pharaoh ants Thief ants Bed bugs Beetles Woodworms Death watch beetles Furniture beetles Weevils Maize weevil Rice weevil Carpet beetles Fur beetles Varied carpet beetles Spider beetles Mealworm beetles Centipedes House centipedes Cockroaches Brown-banded cockroaches German cockroaches American cockroaches Oriental cockroaches Dust mites Earwigs Crickets House crickets Firebrats Flies Bottle flies Blue bottle flies Green bottle flies House flies Fruit flies Drain flies Phorid flies Gnats (Fungus gnats) Mosquitoes Moths Almond moths Indianmeal moths Clothes moths Common clothes moths Brown house moths Paper Lice Red spiders Silverfish Spiders Termites Dampwood termites Subterranean termites Woodlouse == See also == Home-stored product entomology List of notifiable diseases Noxious weed Pest (organism) == References ==


id: outline-of-the-cascade-range-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_the_Cascade_Range-0.md
category: reference
title: Outline of the Cascade Range
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_the_Cascade_Range
date_saved: 2026-05-05T08:18:57.324967+00:00
instance: kb-cron
parent: outline-of-the-cascade-range-20260505-0
summary:
The following outline is provided as an overview of and topical guide to the Cascade Range: The Cascade Range, or Cascades, is a major mountain range of western North America, extending from southern British Columbia through Washington and Oregon to Northern California. It includes both non-volcanic mountains, such as many of those in the North Cascades, and the notable volcanoes known as the High Cascades. The small part of the range in British Columbia is referred to as the Canadian Cascades or, locally, as the Cascade Mountains. The highest peak in the range is Mount Rainier in Washington at 14,411 feet (4,392 m). The Cascades are part of the Pacific Ocean's Ring of Fire, the ring of volcanoes and associated mountains around the Pacific Ocean. All of the eruptions in the contiguous United States over the last 200 years have been from the Cascade Volcanoes. The two most recent were Lassen Peak from 1914 to 1921 and a major eruption of Mount St. Helens in 1980. Minor eruptions of Mount St. Helens have also occurred since, most recently from 2004 to 2008. The Cascade Range is a part of the American Cordillera, a nearly continuous chain of mountain ranges (cordillera)


id: outline-of-the-cascade-range-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_the_Cascade_Range-1.md
category: reference
title: Outline of the Cascade Range
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_the_Cascade_Range
date_saved: 2026-05-05T08:18:57.324967+00:00
instance: kb-cron
parent: outline-of-the-cascade-range-20260505-0
summary:
=== National parks === Lassen Volcanic National Park was established in 1916 while its namesake peak was erupting. The park includes the most extensive and active thermal areas in the United States outside Yellowstone National Park. Crater Lake National Park preserves the remains of Mount Mazama, a large volcano that imploded thousands of years ago, forming a caldera that was later filled with rain and ground water, later to be known as Crater Lake. Mount Rainier National Park surrounds the Cascades' highest volcano, Mount Rainier, which in turn is covered by the largest glacier system in the United States south of Alaska. North Cascades National Park was carved out of a primitive part of the range composed of ancient metamorphic and sedimentary rock. Mount Baker and Glacier Peak are nearby. === National Scenic Areas === Columbia River Gorge National Scenic Area === National monuments === Mount St. Helens National Volcanic Monument was formed following the 1980 eruption of Mount St. Helens in order to preserve the devastated area and give scientists a chance to study its recovery. Newberry National Volcanic Monument includes the area around Newberry Volcano in central Oregon. Cascade–Siskiyou National Monument is located in southern Oregon at the


id: new-7-wonders-of-nature-20260505-0
file: data/en.wikipedia.org/wiki/New_7_Wonders_of_Nature-0.md
category: reference
title: New 7 Wonders of Nature
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/New_7_Wonders_of_Nature
date_saved: 2026-05-05T08:18:58.591022+00:00
instance: kb-cron
parent: new-7-wonders-of-nature-20260505-0
summary:
New 7 Wonders of Nature (2007–2011) was an initiative started in 2007 to create a list of seven natural wonders chosen by people through a global poll. It was the second in a series of Internet-based polls led by Swiss-born Canadian Bernard Weber and organized by the New 7 Wonders Foundation a Swiss-based foundation which Weber founded. The initiative followed an earlier New 7 Wonders of the World campaign and attracted 100 million votes from around the world before voting finished on November 11, 2011. == Stages of the campaign == The New 7 Wonders of Nature campaign started in 2007, immediately after the campaign to elect the man-made New 7 Wonders of the World, in which more than 100 million votes were cast. From over 440 participants representing over 220 countries and through a national qualification and race to become one of the Top 77, as well as the recommendations of the Panel of Experts led by Prof. Federico Mayor, the list of 28 "Official Finalist Candidates" was determined. Voting until November 2011, during which time the New 7 Wonders World Tour planned to visit each of the finalists to allow them to present themselves to the voters across


id: list-of-parasitic-organisms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_parasitic_organisms-0.md
category: reference
title: List of parasitic organisms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_parasitic_organisms
date_saved: 2026-05-05T08:18:59.799352+00:00
instance: kb-cron
parent: list-of-parasitic-organisms-20260505-0
summary:
This is an incomplete list of organisms that are true parasites upon other organisms. == Endoparasites == (endo = within; parasites that live inside their hosts) === Plants === Rafflesia Cuscuta Mistletoe === Parasitic worms === These can be categorized into three groups; cestodes, nematodes and trematodes. Examples include: Acanthocephala Ascariasis (roundworms) Cestoda (tapeworms) including: Taenia saginata (human beef tapeworm), Taenia solium (human pork tapeworm), Diphyllobothrium latum (fish tapeworm) and Echinococcosis (hydatid tapeworm) Clonorchis sinensis (the Chinese liver fluke) Dracunculus medinensis (Guinea worm) Enterobius vermicularis (pinworm) Filariasis Hookworm Loa loa Onchocerciasis (river blindness) Schistosomiasis Strongyloides stercoralis Tapeworm Toxocara canis (dog roundworm) Trichinella Whipworm === Protozoans === Entamoeba histolytica and Entamoeba coli - can cause Amoebiasis Acanthamoeba Balamuthia mandrillaris Giardia Cyclospora cayetanensis Cryptosporidium Toxoplasma gondii Leishmania - L. tropica, L. donovani, and L. mexicana are known to cause Leishmaniasis. Plasmodium - causes the fatal disease, Malaria. P. falciparum, P. vivax, and P. malariae are pathogenic to humans. Babesia === Fungi === Gymnosporangium and other rusts Pyrenophora teres Cordyceps === Arthropods === Pentastomida == Ectoparasites == (ecto = outside; parasites that live on but not within their hosts, for example, attached to their skin) === Arthropoda === Acari Varroa destructor Cymothoa exigua


id: list-of-pine-barrens-20260505-0
file: data/en.wikipedia.org/wiki/List_of_pine_barrens-0.md
category: reference
title: List of pine barrens
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_pine_barrens
date_saved: 2026-05-05T08:19:01.262537+00:00
instance: kb-cron
parent: list-of-pine-barrens-20260505-0
summary:
The following is a list of pine barrens. Canada Kaladar Jack Pine Barrens in Ontario United States Kentucky Hi Lewis Pine Barrens State Nature Preserve Maine Waterboro Barrens Preserve Hollis Plains Massachusetts Plymouth Pinelands Michigan Huron National Forest New Hampshire Ossipee Pine Barrens Concord Pine Barrens New Jersey Pine Barrens New York Albany Pine Bush Altona Flat Rock Jack Pine Barrens Long Island Central Pine Barrens Rome Sand Plains Shawangunk Ridge Pennsylvania Long Pond Barrens Scotia Barrens Rhode Island Kingston Pine Barrens West Virginia North Fork Mountain Wisconsin Great Lakes Barrens Moquah Barrens Research Natural Area Northwest Wisconsin Pine Barrens Gotham Jack Pine Barrens Spread Eagle Barrens Virginia Zuni Pine Barrens


id: seven-natural-wonders-20260505-0
file: data/en.wikipedia.org/wiki/Seven_Natural_Wonders-0.md
category: reference
title: Seven Natural Wonders
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Seven_Natural_Wonders
date_saved: 2026-05-05T08:19:03.022217+00:00
instance: kb-cron
parent: seven-natural-wonders-20260505-0
summary:
Seven Natural Wonders is a television series that was broadcast on BBC Two from 3 May to 20 June 2005. The programme takes an area of England each week and, from votes by the people living in that area, shows the 'seven natural wonders' of that area in a programme. == Episodes and locations == The series covered eight regions of England, having originated as a 'local' television project. There was also a series, looking at a similar selection of 'man-made' wonders for each of eleven regions of England. == References == == External links == Seven Natural Wonders at BBC Online


id: seven-natural-wonders-of-africa-20260505-0
file: data/en.wikipedia.org/wiki/Seven_Natural_Wonders_of_Africa-0.md
category: reference
title: Seven Natural Wonders of Africa
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Seven_Natural_Wonders_of_Africa
date_saved: 2026-05-05T08:19:04.178354+00:00
instance: kb-cron
parent: seven-natural-wonders-of-africa-20260505-0
summary:
The Seven Natural Wonders of Africa was a competition where the seven were selected by voting on February 11, 2013. == Seven Wonders of Africa == == See also == Wonders of the World (disambiguation) Wonders of the World == References ==


id: list-of-the-seven-natural-wonders-of-georgia-u-s-state-20260505-0
file: data/en.wikipedia.org/wiki/List_of_the_seven_natural_wonders_of_Georgia_(U.S._state)-0.md
category: reference
title: List of the seven natural wonders of Georgia (U.S. state)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_the_seven_natural_wonders_of_Georgia_(U.S._state)
date_saved: 2026-05-05T08:19:05.357691+00:00
instance: kb-cron
parent: list-of-the-seven-natural-wonders-of-georgia-u-s-state-20260505-0
summary:
The Seven Natural Wonders of Georgia are considered to be: Amicalola Falls Okefenokee Swamp Providence Canyon Radium Springs Stone Mountain Tallulah Gorge Warm Springs The first list of natural wonders was compiled by state librarian Ella May Thornton and published in the Atlanta Georgian magazine on December 26, 1926. That first list included: Amicalola Falls Jekyll Island Forest Marble vein in Longswamp Valley in Pickens County Okefenokee Swamp Stone Mountain Tallulah Gorge Warm Springs == References == == External links == Georgia Encyclopedia


id: seven-wonders-of-canada-20260505-0
file: data/en.wikipedia.org/wiki/Seven_Wonders_of_Canada-0.md
category: reference
title: Seven Wonders of Canada
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Seven_Wonders_of_Canada
date_saved: 2026-05-05T08:19:06.576406+00:00
instance: kb-cron
parent: seven-wonders-of-canada-20260505-0
summary:
The Seven Wonders of Canada was a 2007 competition sponsored by CBC Television's The National and CBC Radio One's Sounds Like Canada. They sought to determine Canada's "seven wonders" by receiving nominations from viewers, and then from on-line voting of the short list. After the vote, a panel of judges, Ra McGuire, Roy MacGregor and Roberta L. Jamieson, picked the winners based on geographic and poetic criteria. Their seven picks were revealed on The National on June 7, 2007, making the official Seven Wonders of Canada, the Canoe, the Igloo, Niagara Falls, Old Quebec City, Pier 21 Halifax, Prairie Skies, and the Rockies. CBC anchor Peter Mansbridge commented on the top winner, “it’s hard to imagine Canada being Canada without the canoe. Explorers, missionaries, fur traders and First Nations—they’re all linked by this subtle and simple craft. To many, the quintessential Canadian experience begins by picking up a paddle. That’s why the canoe is one of the seven wonders” (Osler 2014). There were over 25,000 nominations and 1 million votes cast, according to the CBC website. The top audience votes were the Sleeping Giant, Niagara Falls, the Bay of Fundy, Nahanni National Park Reserve, the Northern Lights, the Rockies, and


id: list-of-numbers-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numbers-0.md
category: reference
title: List of numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numbers
date_saved: 2026-05-05T08:19:08.279774+00:00
instance: kb-cron
parent: list-of-numbers-20260505-0
summary:
This is a list of notable numbers and articles about them. The list does not contain all numbers in existence as most of the number sets are infinite. Numbers may be included in the list based on their mathematical, historical or cultural notability, but all numbers have qualities that could arguably make them notable. Even the smallest "uninteresting" number is paradoxically interesting for that very property. This is known as the interesting number paradox. The definition of what is classed as a number is rather diffuse and based on historical distinctions. For example, the pair of numbers (3,4) is commonly regarded as a number when it is in the form of a complex number (3+4i), but not when it is in the form of a vector (3,4). This list will also be categorized with the standard convention of types of numbers. This list focuses on numbers as mathematical objects and is not a list of numerals, which are linguistic devices: nouns, adjectives, or adverbs that designate numbers. The distinction is drawn between the number five (an abstract object equal to 2+3), and the numeral five (the noun referring to the number). == Natural numbers == Natural numbers are a subset


id: list-of-numbers-20260505-1
file: data/en.wikipedia.org/wiki/List_of_numbers-1.md
category: reference
title: List of numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numbers
date_saved: 2026-05-05T08:19:08.279774+00:00
instance: kb-cron
parent: list-of-numbers-20260505-0
summary:
A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is trivially a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold Q {\displaystyle \mathbb {Q} } , Unicode U+211A ℚ DOUBLE-STRUCK CAPITAL Q); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient". Rational numbers such as 0.12 can be represented in infinitely many ways, e.g. zero-point-one-two (0.12), three twenty-fifths (⁠3/25⁠), nine seventy-fifths (⁠9/75⁠), etc. This can be mitigated by representing rational numbers in a canonical form as an irreducible fraction. A list of rational numbers is shown below. The names of fractions can be found at numeral (linguistics). == Real numbers == Real numbers are least upper bounds of sets of rational numbers that are bounded above, or greatest lower bounds of sets of rational numbers that are bounded below, or limits of convergent sequences of rational numbers. Real numbers that are not


id: list-of-group-0-isbn-publisher-codes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_group-0_ISBN_publisher_codes-0.md
category: reference
title: List of group-0 ISBN publisher codes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_group-0_ISBN_publisher_codes
date_saved: 2026-05-05T08:19:09.633484+00:00
instance: kb-cron
parent: list-of-group-0-isbn-publisher-codes-20260505-0
summary:
A list of publisher codes for (978) International Standard Book Numbers with a group code of zero. == Assignment == The group-0 publisher codes are assigned as follows: == 2-digit publisher codes == == 3-digit publisher codes == (Note: the status of codes not listed in this table is unclear; please help fill the gaps.) == 4-digit publisher codes == (Note: many codes are not yet listed in this table; please help fill the gaps.) == 5-digit publisher codes == (Note: many codes are not yet listed in this table; please help fill the gaps.) == 6-digit publisher codes == (Note: many codes are not yet listed in this table; please help fill the gaps.) == 7-digit publisher codes == (Note: many codes are not yet listed in this table; please help fill the gaps.) == See also == List of group-1 ISBN publisher codes List of ISBN identifier groups == References == == External links == https://www.isbn-international.org/range_file_generation http://www.books-by-isbn.com/ List of 2 and 3-digit publisher codes for ISBNs that start with a 0 from http://blog.openlibrary.org/2009/07/20/isbn-publisher-codes/ where there is also a complete list of publisher codes for ISBNs that start with a 0 or 1.


id: list-of-group-1-isbn-publisher-codes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_group-1_ISBN_publisher_codes-0.md
category: reference
title: List of group-1 ISBN publisher codes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_group-1_ISBN_publisher_codes
date_saved: 2026-05-05T08:19:10.811646+00:00
instance: kb-cron
parent: list-of-group-1-isbn-publisher-codes-20260505-0
summary:
A list of publisher codes for (978) International Standard Book Numbers with a group code of one. (Data from published items by these publishers.) == Assignment == The group-1 publisher codes are assigned as follows: == 2-digit publisher codes == == 3-digit publisher codes == == 4-digit publisher codes == == 5-digit publisher codes == == 6-digit publisher codes == == 7-digit publisher codes == == See also == List of group-0 ISBN publisher codes List of ISBN identifier groups == References == == External links == https://www.isbn-international.org/range_file_generation http://www.books-by-isbn.com/ http://www.books-by-isbn.com/cg-english_speaking_area_1.html List of 2 and 3-digit publisher codes for ISBNs that start with a 0 from http://blog.openlibrary.org/2009/07/20/isbn-publisher-codes/ where there is also a complete list of publisher codes for ISBNs that start with a 0 or 1.


id: list-of-british-bingo-nicknames-20260505-0
file: data/en.wikipedia.org/wiki/List_of_British_bingo_nicknames-0.md
category: reference
title: List of British bingo nicknames
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_British_bingo_nicknames
date_saved: 2026-05-05T08:19:12.021324+00:00
instance: kb-cron
parent: list-of-british-bingo-nicknames-20260505-0
summary:
In the game of bingo in the United Kingdom, callers announcing the numbers have traditionally used some nicknames to refer to particular numbers if they are drawn. The nicknames are sometimes known by the rhyming phrase 'bingo lingo', and there are rhymes for each number from 1 to 90, some of which date back to 1900. Some traditional games went up to 100. In some clubs, the 'bingo caller' will say the number, with the assembled players intoning the rhyme in a call and response manner, in others, the caller will say the rhyme and the players chant the number. One purpose of the nicknames is to allow called numbers to be clearly understood in a noisy environment. In 2003, Butlins holiday camps introduced some more modern calls devised by a professor of popular culture in an attempt to bring fresh interest to bingo. == Calls == == References == === Citations === === Sources ===


id: list-of-mathematical-constants-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_constants-0.md
category: reference
title: List of mathematical constants
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_constants
date_saved: 2026-05-05T08:19:13.299249+00:00
instance: kb-cron
parent: list-of-mathematical-constants-20260505-0
summary:
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


id: list-of-dates-predicted-for-apocalyptic-events-20260505-0
file: data/en.wikipedia.org/wiki/List_of_dates_predicted_for_apocalyptic_events-0.md
category: reference
title: List of dates predicted for apocalyptic events
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_dates_predicted_for_apocalyptic_events
date_saved: 2026-05-05T08:19:14.717462+00:00
instance: kb-cron
parent: list-of-dates-predicted-for-apocalyptic-events-20260505-0
summary:
Predictions of apocalyptic events that will result in the extinction of humanity, a collapse of civilization, or the destruction of the planet have been made since at least the beginning of the Common Era. Most predictions are related to Abrahamic religions, often standing for or similar to the eschatological events described in their scriptures. Christian predictions typically refer to events like the Rapture, Great Tribulation, Last Judgment, and the Second Coming of Christ. End-time events are normally predicted to occur within the lifetime of the person making the prediction and are usually made using the Bible—in particular the New Testament—as either the primary or exclusive source for the predictions. This often takes the form of mathematical calculations, such as trying to calculate the point in time where it will have been 6,000 years since the supposed creation of the Earth by the Abrahamic God, which according to the Talmud marks the deadline for the Messiah to appear. Predictions of the end from natural events have also been theorised by various scientists and scientific groups. While these predictions are generally accepted as plausible within the scientific community, the events and phenomena are not expected to occur for hundreds of thousands, or


id: table-of-divisors-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_divisors-0.md
category: reference
title: Table of divisors
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_divisors
date_saved: 2026-05-05T08:19:15.893082+00:00
instance: kb-cron
parent: table-of-divisors-20260505-0
summary:
The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which ⁠n/m⁠ is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since ⁠21/7⁠ = 3 (and therefore 7 is also a divisor of 21). If m is a divisor of n, then so is −m. The tables below only list positive divisors. == Key to the tables == d(n) is the number of the positive divisors of n, including 1 and n itself σ(n) is the sum of the positive divisors of n, including 1 and n itself s(n) is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n a deficient number is greater than the sum of its proper divisors; that is, s(n) < n a perfect number equals the sum of its proper divisors; that is, s(n) = n an abundant number is lesser than the sum of its proper divisors; that is, s(n) > n a highly abundant number has a sum of positive divisors that is greater than


id: e-number-20260505-0
file: data/en.wikipedia.org/wiki/E_number-0.md
category: reference
title: E number
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/E_number
date_saved: 2026-05-05T08:19:17.143071+00:00
instance: kb-cron
parent: e-number-20260505-0
summary:
E numbers, short for Europe numbers, are codes for substances used as food additives, including those found naturally in many foods, such as vitamin C, for use within the European Union (EU) and European Free Trade Association (EFTA). Commonly found on food labels, their safety assessment and approval are the responsibility of the European Food Safety Authority (EFSA). The fact that an additive has an E number implies that its use was at one time permitted in products for sale in the European Single Market; some of these additives are no longer allowed today. Having a single unified list for food additives was first agreed upon in 1962 with food colouring. In 1964, the directives for preservatives were added; In 1970 antioxidants were added; In 1974 emulsifiers, stabilisers, thickeners and gelling agents were added as well. == Numbering schemes == The numbering scheme follows that of the International Numbering System (INS) as determined by the Codex Alimentarius committee, though only a subset of the INS additives are approved for use in the European Union as food additives. Outside the European continent plus Russia, E numbers are also encountered on food labelling in other jurisdictions, including the Gulf Cooperation Council, South


id: list-of-integer-sequences-20260505-0
file: data/en.wikipedia.org/wiki/List_of_integer_sequences-0.md
category: reference
title: List of integer sequences
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_integer_sequences
date_saved: 2026-05-05T08:19:18.511115+00:00
instance: kb-cron
parent: list-of-integer-sequences-20260505-0
summary:
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. == General == == Figurate numbers == == Types of primes == == Base-dependent == == References == OEIS core sequences == External links == Index to OEIS


id: list-of-isbn-registration-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_ISBN_registration_groups-0.md
category: reference
title: List of ISBN registration groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_ISBN_registration_groups
date_saved: 2026-05-05T08:19:19.769731+00:00
instance: kb-cron
parent: list-of-isbn-registration-groups-20260505-0
summary:
The registration group or identifier group is the second element in a 13-digit ISBN (first element in a 10-digit ISBN) and indicates the country, geographic region, or language area where a book was published. The element ranges from one to five numerical digits. In 2007, the length of an ISBN changed from 10 to 13 digits, and a new 3-digit prefix (978 or 979) was added in front of 10-digit ISBNs. The following registration groups are compatible with or without a 978- prefix: 0–5 600–639 65–66 69990–69999 7 80–94 950–989 9900–9989 99900–99999 The following must have a 979- prefix: (979-0 is reserved for International Standard Music Numbers for sheet music) 979-10 through 979-13 979-8 Shorter registration group numbers are generally used for countries or regions with greater publishing volume. Because a longer number leaves room for fewer publishers and ISBNs, several countries have more than one number assigned. On the other hand, some countries (Australia, Switzerland, Fiji) have no unique number because they fall in a broader geographic region or language area. == With 10-digit ISBN or 978- prefix == === Prefixes of length 1 === === Prefixes of length 2 === === Prefixes of length 3 === === Prefixes


id: significance-of-numbers-in-judaism-20260505-0
file: data/en.wikipedia.org/wiki/Significance_of_numbers_in_Judaism-0.md
category: reference
title: Significance of numbers in Judaism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Significance_of_numbers_in_Judaism
date_saved: 2026-05-05T08:19:21.096993+00:00
instance: kb-cron
parent: significance-of-numbers-in-judaism-20260505-0
summary:
Various numbers play a significant role in Jewish texts or practice. Some such numbers were used as mnemonics to help remember concepts, while other numbers were considered to have intrinsic significance or allusive meaning. The song Echad Mi Yodea ("who knows one?"), sung at the Passover Seder, is known for recounting a religious concept or practice associated with each of the first 13 numbers. == In Jewish History == In Jewish historical study, numbers were believed to be a means for understanding the divine. This marriage between the symbolic and the physical found its pinnacle in the creation of the Tabernacle. The numerical dimensions of the temple are a "microcosm of creation ... that God used to create the Olamot-Universes." In the thought system of Maharal, each number has a consistent philosophical meaning: unity. dualism and multiplicity. the unity between two extremes. multiplicity in two directions, like the cardinal directions. the center point which unifies those four extremes. multiplicity in three dimensions. the center point which unifies all of nature, as with Shabbat. the supernatural realm which feeds nature, and the striving of man for a connection with the supernatural. the most complete multiplicity, including division between the natural and


id: significance-of-numbers-in-judaism-20260505-1
file: data/en.wikipedia.org/wiki/Significance_of_numbers_in_Judaism-1.md
category: reference
title: Significance of numbers in Judaism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Significance_of_numbers_in_Judaism
date_saved: 2026-05-05T08:19:21.096993+00:00
instance: kb-cron
parent: significance-of-numbers-in-judaism-20260505-0
summary:
== 8 == Eight are the days of the circumcision - שמונה ימי מילה Total number of days of Yom Tov in a year in Israel Number of days of Chanukah 8 days of sukkos Number of days of Pesach (Diaspora) According to the Zohar, the number eight signifies new beginnings because the eighth day was the first day after creation when God returned to work; the week began again. == 9 == The first nine days of the Hebrew month of Av are collectively known as "The Nine Days" (Tisha HaYamim), and are a period of semi-mourning leading up to Tisha B'Av, the ninth day of Av on which both Temples in Jerusalem were destroyed == 10 == The Ten Commandments - עשרה דיבריא The ten Plagues of Egypt Ten Jewish people form a minyan There are ten Sefirot (human and Godly characteristics) depicted in Kabbalah According to the Mishna, the world was created by ten divine utterances; ten generations passed between Adam and Noah and between Noah and Abraham; Abraham received ten trials from God; the Israelites received ten trials in the desert; there were ten plagues in Egypt; ten miracles occurred in the Temple; ten apparently supernatural


id: significance-of-numbers-in-judaism-20260505-2
file: data/en.wikipedia.org/wiki/Significance_of_numbers_in_Judaism-2.md
category: reference
title: Significance of numbers in Judaism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Significance_of_numbers_in_Judaism
date_saved: 2026-05-05T08:19:21.096993+00:00
instance: kb-cron
parent: significance-of-numbers-in-judaism-20260505-0
summary:
== 40 == Moses stayed on mount Sinai for 40 days after the giving of Torah. After the golden calf, he spend there another 40 days (and nights) The number of days the spies were in the land of Canaan Years in the desert—a generation Just as Mozes, the reign of king David and Solomon was also 40 years Days and nights of rain during the flood that occurred at the time of Noah Isaac's age at marriage to Rebecca Esau's age at marriage to his first two wives Number of days Jonah prophesies will pass before Nineveh is destroyed. (They repent) A mikveh must contain at least 40 se'ah (volume measurement) of water Number of years of the reign of David, Solomon, and the most righteous judges in the book of Judges Number of lashes for one who transgresses a commandment Number of days which the Torah was given Number of weeks a person is formed in their mother's womb Number of curses on Adam Minimum age at which a man could join the Sanhedrin == 42 == Letters in one of God's Divine Names 42 cities that refugees (See Cities of Refuge) can go to when they kill


id: list-of-prime-numbers-20260505-0
file: data/en.wikipedia.org/wiki/List_of_prime_numbers-0.md
category: reference
title: List of prime numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_prime_numbers
date_saved: 2026-05-05T08:19:22.349003+00:00
instance: kb-cron
parent: list-of-prime-numbers-20260505-0
summary:
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. The first 1,000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. The number 1 is neither prime nor composite. == The first 1,000 prime numbers == The following table lists the first 1,000 primes, with 20 columns of consecutive primes in each of the 50 rows. (sequence A000040 in the OEIS). The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×1018. That means 95,676,260,903,887,607 primes (nearly 1017), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2×1021) smaller than 1023. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2×1022) smaller than 1024, if the Riemann


id: list-of-prime-numbers-20260505-1
file: data/en.wikipedia.org/wiki/List_of_prime_numbers-1.md
category: reference
title: List of prime numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_prime_numbers
date_saved: 2026-05-05T08:19:22.349003+00:00
instance: kb-cron
parent: list-of-prime-numbers-20260505-0
summary:
Real Eisenstein primes are real Eisenstein integers that are irreducible. Equivalently, they are primes of the form 3k − 1, for a positive integer k. 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401 (OEIS: A003627) === Emirps === Emirps are primes that become a different prime after their base 10 digits have been reversed. The name "emirp" is the reverse of the word "prime". 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991 (OEIS: A006567) === Euclid primes === Euclid primes are primes p such that p−1 is a primorial. 3, 7, 31, 211, 2311, 200560490131 (OEIS: A018239) === Euler irregular primes === Euler irregular primes are primes p {\displaystyle p} that divide an Euler number E 2 n , {\displaystyle E_{2n},} for some 0 ≤ 2 n ≤ p − 3. {\displaystyle 0\leq 2n\leq p-3.} 19, 31, 43, 47,


id: list-of-prime-numbers-20260505-2
file: data/en.wikipedia.org/wiki/List_of_prime_numbers-2.md
category: reference
title: List of prime numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_prime_numbers
date_saved: 2026-05-05T08:19:22.349003+00:00
instance: kb-cron
parent: list-of-prime-numbers-20260505-0
summary:
The (p, p - 3) irregular primes are primes p such that (p, p − 3) is an irregular pair. 16843, 2124679 (OEIS: A088164) ==== (p, p − 5) irregular primes ==== The (p, p - 5) irregular primes are primes p such that (p, p − 5) is an irregular pair. 37 ==== (p, p − 9) irregular primes ==== The (p, p - 9) irregular primes are primes p such that (p, p − 9) is an irregular pair. 67, 877 (OEIS: A212557) === Isolated primes === Isolated primes are primes p such that both p − 2 and p + 2 are both composite. 2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 839, 853, 863, 877, 887, 907, 911, 919, 929, 937, 941, 947, 953,


id: list-of-prime-numbers-20260505-3
file: data/en.wikipedia.org/wiki/List_of_prime_numbers-3.md
category: reference
title: List of prime numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_prime_numbers
date_saved: 2026-05-05T08:19:22.349003+00:00
instance: kb-cron
parent: list-of-prime-numbers-20260505-0
summary:
Primes that remain the same when their decimal digits are read backwards. 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 (OEIS: A002385) === Palindromic wing primes === Primes of the form a ( 10 m − 1 ) 9 ± b × 10 m − 1 2 {\displaystyle {\frac {a{\big (}10^{m}-1{\big )}}{9}}\pm b\times 10^{\frac {m-1}{2}}} with 0 ≤ a ± b < 10 {\displaystyle 0\leq a\pm b<10} . This means all digits except the middle digit are equal. 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 (OEIS: A077798) === Partition primes === Partition function values that are prime. 2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 (OEIS: A049575) === Pell primes === Primes in the Pell number sequence P0 = 0, P1 = 1, Pn = 2Pn−1 + Pn−2. 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 (OEIS:


id: list-of-prime-numbers-20260505-4
file: data/en.wikipedia.org/wiki/List_of_prime_numbers-4.md
category: reference
title: List of prime numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_prime_numbers
date_saved: 2026-05-05T08:19:22.349003+00:00
instance: kb-cron
parent: list-of-prime-numbers-20260505-0
summary:
Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a. The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are the Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d. If a and d are relatively prime, the arithmetic progression contains infinitely many primes. 2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 (OEIS: A065091) 4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 (OEIS: A002144) 4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 (OEIS: A002145) 6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 (OEIS: A002476) 6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (OEIS: A007528) 8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 (OEIS: A007519)


id: list-of-prime-numbers-20260505-5
file: data/en.wikipedia.org/wiki/List_of_prime_numbers-5.md
category: reference
title: List of prime numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_prime_numbers
date_saved: 2026-05-05T08:19:22.349003+00:00
instance: kb-cron
parent: list-of-prime-numbers-20260505-0
summary:
==== Left-truncatable ==== Primes that remain prime when the leading decimal digit is successively removed. 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683 (OEIS: A024785) ==== Right-truncatable ==== Primes that remain prime when the least significant decimal digit is successively removed. 2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797 (OEIS: A024770) ==== Two-sided ==== Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes: 2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (OEIS: A020994) === Twin primes === Where (p, p+2) are both prime. (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349),


id: table-of-prime-factors-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_prime_factors-0.md
category: reference
title: Table of prime factors
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_prime_factors
date_saved: 2026-05-05T08:19:23.562059+00:00
instance: kb-cron
parent: table-of-prime-factors-20260505-0
summary:
The tables contain the prime factorization of the natural numbers from 1 to 1000. When n is a prime number, the prime factorization is just n itself, written in bold below. The number 1 is called a unit. It has no prime factors and is neither prime nor composite. == Properties == Many properties of a natural number n {\displaystyle n} can be seen or directly computed from the prime factorization of n {\displaystyle n} . The multiplicity of a prime factor p {\displaystyle p} of n {\displaystyle n} is the largest exponent m {\displaystyle m} for which p m {\displaystyle p^{m}} divides n {\displaystyle n} . The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 {\displaystyle 1} (since p = p 1 {\displaystyle p=p^{1}} ). The multiplicity of a prime which does not divide n {\displaystyle n} may be called 0 {\displaystyle 0} or may be considered undefined. ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} , the prime omega functions, count the number of prime factors of a natural number n {\displaystyle n} . ω ( n ) {\displaystyle \omega (n)}


id: table-of-prime-factors-20260505-1
file: data/en.wikipedia.org/wiki/Table_of_prime_factors-1.md
category: reference
title: Table of prime factors
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_prime_factors
date_saved: 2026-05-05T08:19:23.562059+00:00
instance: kb-cron
parent: table-of-prime-factors-20260505-0
summary:
a 0 ( n ) {\displaystyle a_{0}(n)} , sometimes called the integer logarithm, is the sum of primes dividing n {\displaystyle n} , counted with multiplicity. It is an additive function. A Ruth-Aaron pair is a pair of two consecutive numbers ( n , n + 1 ) {\displaystyle (n,n+1)} with a 0 ( n ) = a 0 ( n + 1 ) {\displaystyle a_{0}(n)=a_{0}(n+1)} . The first (by n {\displaystyle n} value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 (sequence A039752 in the OEIS). Another definition is where the same prime is only counted once; if so, the first (by n {\displaystyle n} value): 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299 (sequence A006145 in the OEIS). A primorial p n # {\displaystyle p_{n}\#} is the product of all primes from 2 to p n {\displaystyle p_{n}} . The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810 (sequence A002110 in the OEIS). 1 # = 1 {\displaystyle 1\#=1} is sometimes included. A factorial n ! {\displaystyle n!} is the product of all numbers from 1 to n {\displaystyle n} . The first: 1, 2,


id: recycling-codes-20260505-0
file: data/en.wikipedia.org/wiki/Recycling_codes-0.md
category: reference
title: Recycling codes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Recycling_codes
date_saved: 2026-05-05T08:19:24.672496+00:00
instance: kb-cron
parent: recycling-codes-20260505-0
summary:
Recycling codes are used to identify the materials out of which the item is made, to facilitate easier recycling process. The presence on an item of a recycling code, a chasing arrows logo, or a resin code, is not an automatic indicator that a material is recyclable; it is an explanation of what the item is made of. Codes have been developed for batteries, biomatter/organic material, glass, metals, paper, and plastics. Various countries have adopted different codes. For example, the table below shows the polymer resin (plastic) codes. In the United States there are fewer, because ABS is placed with "others" in group 7. A number of countries have a finer-grained system with more recycling codes. For example, China's polymer identification system has seven different classifications of plastic, five different symbols for post-consumer paths, and 140 identification codes. The lack of a code system in some countries has encouraged those who fabricate their own plastic products, such as RepRap and other prosumer 3-D printer users, to adopt a voluntary recycling code based on the more comprehensive Chinese system. == Resin identification codes and codes defined by the European Commission == == Chinese codes for plastics products == The Standardization Administration


id: list-of-numeral-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numeral_systems-0.md
category: reference
title: List of numeral systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numeral_systems
date_saved: 2026-05-05T08:19:25.906324+00:00
instance: kb-cron
parent: list-of-numeral-systems-20260505-0
summary:
There are many different numeral systems, that is, writing systems for expressing numbers. == By culture / time period == "A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. Some systems have two bases, a smaller (subbase) and a larger (base); an example is Roman numerals, which are organized by fives (V=5, L=50, D=500, the subbase) and tens (X=10, C=100, M=1,000, the base). == By type of notation == Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base. === Standard positional numeral systems === The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. There have been some proposals for standardisation. === Non-standard positional numeral systems === ==== Bijective numeration ==== ==== Signed-digit representation ==== ==== Complex bases ==== ==== Non-integer bases ==== ==== n-adic number


id: list-of-types-of-numbers-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_numbers-0.md
category: reference
title: List of types of numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_numbers
date_saved: 2026-05-05T08:19:27.311797+00:00
instance: kb-cron
parent: list-of-types-of-numbers-20260505-0
summary:
Numbers can be classified according to how they are represented or according to the properties that they have. == Main types == Natural numbers ( N {\displaystyle \mathbb {N} } ): The counting numbers {1, 2, 3, ...} are commonly called natural numbers; however, other definitions include 0, so that the non-negative integers {0, 1, 2, 3, ...} are also called natural numbers. Natural numbers including 0 are also sometimes called whole numbers. Alternatively natural numbers not including 0 are also sometimes called whole numbers instead. Integers ( Z {\displaystyle \mathbb {Z} } ): Positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}. Rational numbers ( Q {\displaystyle \mathbb {Q} } ): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but there are rational numbers that are not integers, such as −2/9. Real numbers ( R {\displaystyle \mathbb {R} } ): Numbers that correspond to points along a line. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true. Irrational numbers ( R ∖ Q {\displaystyle \mathbb {R} \setminus \mathbb {Q} }


id: list-of-types-of-numbers-20260505-1
file: data/en.wikipedia.org/wiki/List_of_types_of_numbers-1.md
category: reference
title: List of types of numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_numbers
date_saved: 2026-05-05T08:19:27.311797+00:00
instance: kb-cron
parent: list-of-types-of-numbers-20260505-0
summary:
== Algebraic numbers == Algebraic number: Any number that is the root of a non-zero polynomial with rational coefficients. Transcendental number: Any real or complex number that is not algebraic. Examples include e and π. Trigonometric number: Any number that is the sine or cosine of a rational multiple of π. Quadratic surd: A root of a quadratic equation with rational coefficients. Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds. Algebraic integer: A root of a monic polynomial with integer coefficients. == Non-standard numbers == Transfinite numbers: Numbers that are greater than any natural number. Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets. Infinitesimals: These are smaller than any positive real number, but are nonetheless greater than zero. These were used in the initial development of calculus, and are used in synthetic


id: list-of-airborne-wind-energy-organizations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_airborne_wind_energy_organizations-0.md
category: reference
title: List of airborne wind energy organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_airborne_wind_energy_organizations
date_saved: 2026-05-05T08:19:29.973741+00:00
instance: kb-cron
parent: list-of-airborne-wind-energy-organizations-20260505-0
summary:
This is a list of airborne wind energy or kite-energy organizations that are advancing airborne wind energy systems (AWES). In 2011 there were over 40 organizations involved worldwide, but this number has increased to over 60 in 2017. Categories of kite-energy or airborne-wind-energy organizations that are forming the nascent industry: education, academic, non-profit, for-profit, communication, research, original kite-energy equipment manufacturer, kite-line manufacturer, industry-wide association, history, testing, forum entity, library, cooperative, consortium, group, club, school, training school. Generation by kite-energy systems may involve pumping, electricity generators flown in the upper flying system (flygen), electric generators situated on the land or sea or on board a vessel (groundgen), simple lifting of objects (lifting), pulling hulls or other objects (traction), or transportation; systems generate energy to do special tasks. Systems may be scaled from tiny to utility size. == Organizations == == References ==


id: list-of-arctic-research-programs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Arctic_research_programs-0.md
category: reference
title: List of Arctic research programs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Arctic_research_programs
date_saved: 2026-05-05T08:19:31.370560+00:00
instance: kb-cron
parent: list-of-arctic-research-programs-20260505-0
summary:
This is a list of Arctic research programs: Arctic and Antarctic Research Institute Arctic and Northern Studies at University of Alaska Fairbanks Arctic Centre, University of Lapland The Arctic Institute | Center for Circumpolar Security Studies Arctic Institute of North America Arctic Research Consortium of the United States Arctic Research Foundation Arctic Research Office Arctic Submarine Laboratory Arctic Technology Centre Association of Polar Early Career Scientists (APECS) Byrd Polar Research Center Canadian High Arctic Ionospheric Network Centre for Polar Observation & Modelling Circumarctic Environmental Observatories Network Cooperative Institute for Arctic Research EISCAT Flashline Mars Arctic Research Station Institute of Arctic and Alpine Research Institute of Arctic Biology International Arctic Buoy Program International Arctic Research Center International Arctic Science Committee Malaurie Institute of Arctic Research Monaco-UVSQ National Centre for Polar and Ocean Research North Greenland Ice Core Project National Snow and Ice Data Center QUEEN SCICEX Scott Polar Research Institute Surface Heat Budget of the Arctic Ocean United States Arctic Research Commission University of the Arctic == See also == List of research stations in the Arctic


id: atmospheric-chemistry-observational-databases-20260505-0
file: data/en.wikipedia.org/wiki/Atmospheric_chemistry_observational_databases-0.md
category: reference
title: Atmospheric chemistry observational databases
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Atmospheric_chemistry_observational_databases
date_saved: 2026-05-05T08:19:32.580210+00:00
instance: kb-cron
parent: atmospheric-chemistry-observational-databases-20260505-0
summary:
Over the last two centuries many environmental chemical observations have been made from a variety of ground-based, airborne, and orbital platforms and deposited in databases. Many of these databases are publicly available. All of the instruments mentioned in this article give online public access to their data. These observations are critical in developing our understanding of the Earth's atmosphere and issues such as climate change, ozone depletion and air quality. Some of the external links provide repositories of many of these datasets in one place. For example, the Cambridge Atmospheric Chemical Database, is a large database in a uniform ASCII format. Each observation is augmented with the meteorological conditions such as the temperature, potential temperature, geopotential height, and equivalent PV latitude. == Ground-based and balloon observations == NDSC observations. The Network for the Detection for Stratospheric Change (NDSC) is a set of high-quality remote-sounding research stations for observing and understanding the physical and chemical state of the stratosphere. Ozone and key ozone-related chemical compounds and parameters are targeted for measurement. The NDSC is a major component of the international upper atmosphere research effort and has been endorsed by national and international scientific agencies, including the International Ozone Commission, the United


id: atmospheric-chemistry-observational-databases-20260505-1
file: data/en.wikipedia.org/wiki/Atmospheric_chemistry_observational_databases-1.md
category: reference
title: Atmospheric chemistry observational databases
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Atmospheric_chemistry_observational_databases
date_saved: 2026-05-05T08:19:32.580210+00:00
instance: kb-cron
parent: atmospheric-chemistry-observational-databases-20260505-0
summary:
== Space shuttle observations == ATMOS observations. The Atmospheric Trace Molecule Spectroscopy experiment (ATMOS) is an infrared spectrometer (a Fourier transform interferometer) that is designed to study the chemical composition of the atmosphere. In this section you will be able to read both general and detailed information as to why and how the instrument works. The ATMOS instrument has flown four times on the Space Shuttle since 1985. The predecessor to ATMOS, flown on aircraft and high-altitude balloon platforms, was born in the early 1970s out of concern for the effects of Super Sonic Transport exhaust products on the ozone layer. The experiment was redesigned for the Space Shuttle when the potential for ozone destruction by man-made chlorofluorocarbons was discovered and the need for global measurements became crucial. CRISTA observations. CRISTA is short for CRyogenic Infrared Spectrometers and Telescopes for the Atmosphere. It is a limb-scanning satellite experiment, designed and developed by the University of Wuppertal to measure infrared emissions of the Earth's atmosphere. Equipped with three telescopes and four spectrometers and cooled with liquid helium, CRISTA acquires global maps of temperature and atmospheric trace gases with very high horizontal and vertical resolution. The design enables the observation of small


id: list-of-bioinformatics-institutions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_bioinformatics_institutions-0.md
category: reference
title: List of bioinformatics institutions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_bioinformatics_institutions
date_saved: 2026-05-05T08:19:33.801986+00:00
instance: kb-cron
parent: list-of-bioinformatics-institutions-20260505-0
summary:
This is a list of major bioinformatics institutions. National Center for Biotechnology Information (NCBI) European Bioinformatics Institute (EMBL-EBI) Australia Bioinformatics Resource (EMBL-ABR) Swiss Institute of Bioinformatics (SIB) Scripps Research Institute (TSRI) European Molecular Biology Laboratory (EMBL) Wellcome Trust Sanger Institute (WTSI) Computational Biology Department Broad Institute Whitehead Institute The Institute for Genomic Research Center for Biomolecular Science and Engineering Netherlands Bioinformatics Centre COSBI Institute of Bioinformatics (IOB) Max Planck Institute for Molecular Cell Biology and Genetics (MPI-CBG) Partner Institute for Computational Biology Flatiron Institute DDBJ Center (DDBJ) Bioinformatics Institute (Singapore) Database Center for Life Science (DBCLS) == References == Benue state university department of Anatomy


id: canadian-university-scientific-research-organizations-20260505-0
file: data/en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations-0.md
category: reference
title: Canadian university scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations
date_saved: 2026-05-05T08:19:35.071794+00:00
instance: kb-cron
parent: canadian-university-scientific-research-organizations-20260505-0
summary:
Expenditures by Canadian universities on scientific research and development accounted for about 40% of all spending on scientific research and development in Canada in 2006. Research in the natural and social sciences in Canada, with a few important exceptions, is almost exclusively funded by the Canadian taxpayer and is distributed to universities by five important federal funding agencies, the Natural Sciences and Engineering Research Council (NSERC), the Canadian Institutes of Health Research (CIHR), and the Social Sciences and Humanities Research Council of Canada (SSHRC). Additional monies are also provided by the Canada Research Chairs organization, which provides financing for the staffing of research personnel at Canadian universities and the Canada Foundation for Innovation, which supports the acquisition of scientific research infrastructure by Canadian universities, colleges, research hospitals, and non-profit research institutions. In 2006, total spending on scientific and industrial research in Canada amounted to C$28.067 billion or about 2 percent of GDP. In 2006, Canadian universities spent C$10.890 billion on research and development, representing about 40 percent of all R&D spending in Canada and about .66 percent of Canada's GDP. Below are the names of those university institutions that carry out both natural and social science research, although the emphasis


id: canadian-university-scientific-research-organizations-20260505-1
file: data/en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations-1.md
category: reference
title: Canadian university scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations
date_saved: 2026-05-05T08:19:35.071794+00:00
instance: kb-cron
parent: canadian-university-scientific-research-organizations-20260505-0
summary:
Alberta Poultry Research Centre Alberta Veterinary Research Institute (AVRI) Canada Science & Research Institute Inc (CSRI/Non Gov/Tr Labs Partner) Centre for Enhanced Forest Management (EFM) Dairy Research and Technology Centre (DRTC) Environmental Research and Studies Centre Material Culture Institute (MCI) Engineering Alberta Centre for Surface Engineering and Science Construction Research Institute for Canada (CIRC) Imperial Oil Centre for Oil Sands Innovation Health Sciences Council Alberta Centre on Aging Alberta Institute for Human Nutrition Community-University Partnership (CUP) The John Dossetor Health Ethics Centre Medicine and Dentistry Alberta Asthma Centre Alberta Centre for Prions and Protein Folding Diseases Alberta Diabetes Research Institute Alberta Institute for Viral Immunology (AIVI) Alberta Peptide Institute (API) Alberta Transplant Applied Genomics Centre (ATAGC) Canadian VIGOUR Centre Centre for Health Evidence Centre for Neuroscience Digestive Health Care Centre for Colon Cancer (DHCCC) Glaxo Wellcome Heritage Research Institute Magnetic Resonance Diagnostics Centre (MRDC) Muttart Diabetes Research & Training Centre Perinatal Research Group Women and Children's Health Research Institute (WCHRI) Nursing Institute for Philosophical Nursing Research International Institute for Qualitative Methodology International Nursing Centre Pharmacy and Pharmaceutical Sciences Centre for Community Pharmacy Research and Interdisciplinary Studies (c/COMPRIS) Noujaim Institute School of Public Health Alberta Centre for Injury Control and Research


id: canadian-university-scientific-research-organizations-20260505-2
file: data/en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations-2.md
category: reference
title: Canadian university scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations
date_saved: 2026-05-05T08:19:35.071794+00:00
instance: kb-cron
parent: canadian-university-scientific-research-organizations-20260505-0
summary:
==== Research centres and institutes ==== Canadian Centre of Arts & Technology (CCAT) Canadian Centre for Cultural Innovation (CCCI) Centre for Accounting Research & Education (CARE) Centre for Advanced Studies in Finance (CASF) Centre for Advancement of Trenchless Technologies at Waterloo (CATT) Centre for Applied Cryptographic Research (CACR) Centre for Business, Entrepreneurship & Technology (CBET) Centre for Computational Mathematics in Industry & Commerce (CCMIC) Centre for Contact Lens Research (CCLR) Centre for Cultural Management (CCM) Centre for Education in Mathematics & Computing (CEMC) Centre for Mental Health Research (CMHR) Centre for Molecular Beams & Laser Chemistry Centre for Theoretical Neuroscience Heritage Resource Centre Institute for Computer Research (ICR) Institute for Innovation Research (IIR) Institute for Polymer Research (IPR) Institute for Quantitative Finance & Insurance (IQFI) Institute for Quantum Computing (IQC) Institute for Risk Research (IRR) Institute for Vision Science & Technology (IVST) Institute of Biochemistry & Molecular Biology Institute of Insurance and Pension Research (IIPR) Integrated Centre for Visualization, Design & Manufacturing (ICVDM) Mid-Size City Research Centre (MCRC) Nortel Networks Institute for Advanced Information Technology (NNI) Schlegel – UW Research Institute for Aging (RIA) Survey Research Centre (SRC) Waterloo Centre for the Advancement of Co-operative Education (WatCACE) Waterloo Centre for


id: canadian-university-scientific-research-organizations-20260505-3
file: data/en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations-3.md
category: reference
title: Canadian university scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations
date_saved: 2026-05-05T08:19:35.071794+00:00
instance: kb-cron
parent: canadian-university-scientific-research-organizations-20260505-0
summary:
==== Centres de recherche ==== Agora Jules Dupuit (AJD) Centre d'excellence en Centre Canadien d'études allemandes et européennes (CCEAE) Centre d'excellence en Centre d'études et de recherches internationales (CERIUM) Centre d'excellence en Centre d'études ethniques des universités montréalaises (CEETUM) Centre d'excellence en Centre d'étude des religions de l'Université de Montréal (CERUM) Centre d'excellence en neuromique (CEN) Centre d'excellence en Centre d'excellence pour le développement des jeunes enfants (CEDJE) Centre d'excellence en Centre de droit des affaires et du commerce international (CDACI) Centre d'excellence en Centre de formation et d'expertise en recherche en administration des services infirmiers (FERASI) Centre d'excellence en Centre de recherche en droit public (CRDP) Centre d'excellence en Centre de recherche en éthique de l'Université de Montréal (CREUM) Centre d'excellence en Centre de recherche en neuropsychologie et cognition (CERNEC) Centre de recherche en reproduction animale (CRRA) Centre de recherche en sciences neurologiques (CRSN) Centre de recherche interdisciplinaire sur la violence familiale et la violence faite aux ** femmes (CRI-VIFF) Centre de recherche interdisciplinaire sur les problèmes conjugaux et les aggressions ** sexuelles (CRIPCAS) Centre de recherche interdisciplinaire sur les technologies émergentes (CITÉ) Centre de recherche interuniversitaire sur la formation et la profession enseignante (CRIFPE) Centre de recherche interuniversitaire


id: canadian-university-scientific-research-organizations-20260505-4
file: data/en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations-4.md
category: reference
title: Canadian university scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations
date_saved: 2026-05-05T08:19:35.071794+00:00
instance: kb-cron
parent: canadian-university-scientific-research-organizations-20260505-0
summary:
=== Queen's University === Kingston, Ontario NSERC Funding 2003: C$M 21,571 Number of Canada Research Chairs – 54 ==== University centres and institutes ==== Centre for Neuroscience Studies (formerly Centre for the Study of Molecular Neuroscience) Centre for Water and the Environment GeoEngineering Centre High Performance Computing Virtual Laboratory (HPCVL) Human Mobility Research Centre Sudbury Neutrino Observatory Institute Southern African Research Centre ==== Faculty centres and institutes ==== Cancer Research Institute Cancer Clinical Trials Division Cancer Biology & Genetics Division Cancer Care & Epidemiology Division Centre for Health Services and Policy Research Centre for International Relations Charles Pentland Centre for Manufacturing of Advanced Ceramics and Nanomaterials Vladimir Krstic Centre for Studies in Primary Care Richard Birtwhistle Centre for the Study of Democracy Fuel Cell Research Centre John Deutsch Institute for the Study of Economic Policy Industrial Relations Centre Institute for Intergovernmental Relations Thomas Courchene The Monieson Centre Yolande Chan Surveillance Studies Centre, David Murakami Wood === McMaster University === Hamilton, Ontario. NSERC Funding 2003: C$M 20,694 Number of Canada Research Chairs – 62 ==== Research institutes ==== Origins Institute McMaster Institute for Applied Radiation Sciences(McIARS) McMaster Institute for Energy Studies McMaster Institute of Environment & Health Institute on Globalization and


id: canadian-university-scientific-research-organizations-20260505-5
file: data/en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations-5.md
category: reference
title: Canadian university scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations
date_saved: 2026-05-05T08:19:35.071794+00:00
instance: kb-cron
parent: canadian-university-scientific-research-organizations-20260505-0
summary:
==== University research institutes and centres ==== Alberta Global Forum Calgary Centre for Financial Research – under development Calgary Centre for Innovative Technology (CCIT) Calgary Institute for the Humanities] Canadian Centre for the Study of Higher Education Centre for Advanced Technologies of Life Sciences (CAT) Centre for Bioengineering Research and Education (CBRE) Centre for Environmental Engineering Research and Education (CEERE) Centre for Gifted Education Centre for Health and Policy Studies (CHAPS) Centre for Information Security and Cryptography Centre for Innovation Studies (THECIS) Centre for Mathematics in Life Sciences Centre for Microsystems Engineering (CME) Centre for Military and Strategic Studies (CMSS) Centre for Public Interest Accounting (CPIA) Centre for Research in the Fine Arts (CRFA) Centre for Social Work Research and Development iNFORMATICS Research Centre Institute for Advanced Policy Research Institute for Biocomplexity and Informatics Institute for Gender Research Institute for Quantum Science and Technology (formerly the Institute for Quantum Information Science) Institute for Space Research Institute for Sustainable Energy, Environment and Economy Institute for United States Policy Research Institute of Professional Communication (IPC) International Institute for Resource Industries and Sustainable Studies (IRIS) Kananaskis Field Stations Language Research Centre Latin American Research Centre (LARC) Markin Institute for Public Health Pipeline Engineering


id: canadian-university-scientific-research-organizations-20260505-6
file: data/en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations-6.md
category: reference
title: Canadian university scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations
date_saved: 2026-05-05T08:19:35.071794+00:00
instance: kb-cron
parent: canadian-university-scientific-research-organizations-20260505-0
summary:
==== Research centres, institutes, and groups ==== Atlantic Centre of Excellence for Women's Health Atlantic Health Promotion Research Centre Atlantic Institute of Criminology Atlantic Research Data Centre (ARDC) Centre for African Studies Centre for Foreign Policy Studies Dalhousie Infectious Disease Research Alliance (DIDRA) Dalhousie Inflammation Group Dalhousie Multiple Sclerosis Research Unit Energy at Dalhousie Health Law Institute International Ocean Institute Law and Technology Institute Marine and Environmental Law Institute Neuroscience Institute Pediatric Pain Research Lab Population Health Research Unit (PHRU) Institute for Research in Materials ==== Technical research facilities ==== Aquatron Atlantic Region Magnetic Resonance Centre Canadian Institute of Fisheries Technology (CIFT) Canadian Residential Energy End-use Data and Analysis Center (CREEDAC) Centre for Water Resources Studies (CWRS) Cosmogenic Nuclide Exposure Dating Facility Minerals Engineering Centre (MEC) Nova Scotia CAD/CAM Centre (NSCCC) Slowpoke Facility Trace Analysis Research Centre (TARC) ==== Affiliated research organizations ==== IWK Health Centre Global Information Networking Institute The Nova Scotia Hospital, QEII Health Sciences Centre Saint John Regional Hospital === University of Ottawa === Ottawa, Ontario. NSERC Funding 2003: C$M 14,127 Number of Canada Research Chairs – 49 ==== uOttawa research centres and institutes ==== Centre for Advanced Research in Environmental Genomics (CAREG) Centre for Catalysis Research


id: canadian-university-scientific-research-organizations-20260505-7
file: data/en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations-7.md
category: reference
title: Canadian university scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_university_scientific_research_organizations
date_saved: 2026-05-05T08:19:35.071794+00:00
instance: kb-cron
parent: canadian-university-scientific-research-organizations-20260505-0
summary:
=== Health and life sciences === Accel-Rx Health Sciences Accelerator – Accel-Rx (2014–2019) – Vancouver, British Columbia, Aging Gracefully across Environments using Technology to Support Wellness, Engagement and Long Life – AGE-WELL (2014–2019) – Toronto Rehabilitation Institute – University Health Network, Toronto, Ontario, Allergy, Genes and Environment Network – AllerGen (2004–2019) – McMaster University, Hamilton, Ontario, Biotherapeutics for Cancer Treatment – BioCanRx (2014–2019) – Ottawa Hospital Research Institute, Ottawa, Ontario, Canadian Frailty Network – CFN (2012–2017) – Queen's University, Kingston, Ontario, Canadian Glycomics Network – GlycoNet (2014–2019) – University of Alberta, Edmonton, Alberta, Cardiac Arrhythmia Network of Canada – CANet (2014–2019) – Western University, London, Ontario, CellCAN Regenerative Medicine and Cell Therapy Network – CellCAN (2014–2018) – Hôpital Maisonneuve-Rosemont, Montréal, Quebec, Centre for Commercialization of Cancer Immunotherapy – C3i (2016–2021) – Montréal, Quebec, Centre for Commercialization of Regenerative Medicine – CCRM (2011–2017) – Toronto, Ontario, Centre for Drug Research and Development – CDRD (2008–2018) – Vancouver, British Columbia, Centre for Imaging Technology Commercialization – CIMTEC (2011–2018) – London, Ontario, Centre for Probe Development and Commercialization – CPDC (2008–2018) – Hamilton, Ontario, Centre for Surgical Invention and Innovation – CSII (2009–2017) – Hamilton, Ontario, Centre for the Commercialization of Antibodies and


id: canadian-government-scientific-research-organizations-20260505-0
file: data/en.wikipedia.org/wiki/Canadian_government_scientific_research_organizations-0.md
category: reference
title: Canadian government scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_government_scientific_research_organizations
date_saved: 2026-05-05T08:19:36.331481+00:00
instance: kb-cron
parent: canadian-government-scientific-research-organizations-20260505-0
summary:
Expenditures by federal and provincial organizations on scientific research and development accounted for about 10% of all such spending in Canada in 2006. These organizations are active in natural and social science research, engineering research, industrial research and medical research. Below is a list of Canadian Federal and Provincial Government scientific research organizations as of January 2008. In some cases the agency mentioned is dedicated exclusively to scientific research, a good example being the National Research Council Canada. In other cases the organization conducts scientific research within the framework of a much larger mandate, such as the transportation research undertaken by the Transportation Development Centre in Montreal which occurs as part on the general transportation regulatory function of Transport Canada. While most of the organizations mentioned here are "brick and mortar", some, such as the Canadian Institutes of Health Research, are "virtual" and consist of dedicated groups of researchers who are geographically dispersed but remain in close contact through electronic means. Total funding for the organizations listed below amounted to about C$2.5 billion in 2006, or about 10% of all scientific research and development spending in Canada. == Federal organizations == === Devoted exclusively to scientific research === Atomic Energy


id: canadian-government-scientific-research-organizations-20260505-1
file: data/en.wikipedia.org/wiki/Canadian_government_scientific_research_organizations-1.md
category: reference
title: Canadian government scientific research organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canadian_government_scientific_research_organizations
date_saved: 2026-05-05T08:19:36.331481+00:00
instance: kb-cron
parent: canadian-government-scientific-research-organizations-20260505-0
summary:
=== Conducting scientific research in support of a larger mandate === Agriculture and Agri-Food Canada - HQ, Ottawa, Ontario Research organizations Agassiz Research and Development Centre - Agassiz, British Columbia Summerland Research and Development Centre - Summerland, British Columbia Lacombe Research and Development Centre - Lacombe, Alberta Lethbridge Research and Development Centre - Lethbridge, Alberta Saskatoon Research and Development Centre - Saskatoon, Saskatchewan Swift Current Research and Development Centre - Swift Current, Saskatchewan Brandon Research and Development Centre - Brandon, Manitoba Morden Research and Development Centre - Morden, Manitoba London Research and Development Centre - London, Ontario Ottawa Research and Development Centre - Ottawa, Ontario Guelph Research and Development Centre - Guelph, Ontario Harrow Research and Development Centre - Harrow, Ontario Sherbrooke Research and Development Centre - Sherbrooke (Lennoxville Sector), Quebec Saint-Hyacinthe Research and Development Centre - Saint-Hyacinthe, Quebec Quebec Research and Development Centre - Quebec, Quebec Saint-Jean-sur-Richelieu Research and Development Centre - Saint-Jean-sur-Richelieu, Quebec Fredericton Research and Development Centre - Fredericton, New Brunswick Charlottetown Research and Development Centre - Charlottetown, Prince Edward Island Kentville Research and Development Centre - Kentville, Nova Scotia St John's Research and Development Centre - St. John's, Newfoundland and Labrador Canadian Food Inspection Agency -


id: list-of-chemical-engineering-societies-20260505-0
file: data/en.wikipedia.org/wiki/List_of_chemical_engineering_societies-0.md
category: reference
title: List of chemical engineering societies
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_chemical_engineering_societies
date_saved: 2026-05-05T08:19:37.549412+00:00
instance: kb-cron
parent: list-of-chemical-engineering-societies-20260505-0
summary:
This is the list of chemical engineering societies in the world. They are sorted by continent and alphabetically. They include national or international ones, but not student societies or those otherwise restricted to a particular university or institution. == Africa == Ethiopian Society of Chemical Engineers (ESChE), Ethiopia Nigerian Society of Chemical Engineers South African Institution of Chemical Engineers (SAIChE), South Africa == Asia == Asia Pacific Confederation of Chemical Engineering (APCChE) Iranian Association of Chemical Engineering (IAChE) [1], I.R. Iran Indian Institute of Chemical Engineers (IIChe), India [2] Israel Institute of Chemical Engineers (IIChe), Israel [3] Korean Chemical Society (KCS), Korea [4] Korean Institute of Chemical Engineers (ko) (KIChE), Korea [5] Pakistan Institute of Chemical Engineers (PICHE), Pakistan [6] Philippine Institute of Chemical Engineers (PIChE), Philippines [7] Society of Chemical Engineers, Japan (SCEJ) (ja), Japan [8] Society of Chemical Engineers of Nepal, Kathmandu, Nepal [9] Taiwan Institute of Chemical Engineering (TwIChE), Taiwan [10] Thai Institute of Chemical Engineering and Applied Chemistry (TIChE), Thailand [11] == Europe == European Federation of Chemical Engineering (EFCE) (umbrella organization) Associazione Italiana Di Ingegneria Chimica (AIDIC), Italy Chamber of Chemical Engineers, Turkey [12] DECHEMA, Germany Institution of Chemical Engineers (IChemE), UK Société Française de


