# kb

id: list-of-continuity-related-mathematical-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_continuity-related_mathematical_topics-0.md
category: reference
title: List of continuity-related mathematical topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_continuity-related_mathematical_topics
date_saved: 2026-05-05T08:13:43.792858+00:00
instance: kb-cron
parent: list-of-continuity-related-mathematical-topics-20260505-0
summary:
In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways. == Continuity of functions and measures == Continuous function Absolutely continuous function Absolute continuity of a measure with respect to another measure Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous. Sometimes it has a less inclusive meaning: a distribution whose c.d.f. is absolutely continuous with respect to Lebesgue measure. This less inclusive sense is equivalent to the condition that every set whose Lebesgue measure is 0 has probability 0. Geometric continuity Parametric continuity Smoothness == Continuum == Continuum (set theory), the real line or the corresponding cardinal number Linear continuum, any ordered set that shares certain properties of the real line Continuum (topology), a nonempty compact connected metric space (sometimes a Hausdorff space) Continuum hypothesis, a conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets and the cardinality of the set of all real numbers. The latter cardinality is equal to the cardinality of the set of all subsets of a countably infinite set. Cardinality of the continuum, a cardinal number that represents the


id: list-of-convexity-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_convexity_topics-0.md
category: reference
title: List of convexity topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_convexity_topics
date_saved: 2026-05-05T08:13:45.110385+00:00
instance: kb-cron
parent: list-of-convexity-topics-20260505-0
summary:
This is a list of convexity topics, by Wikipedia page. Alpha blending - the process of combining a translucent foreground color with a background color, thereby producing a new blended color. This is a convex combination of two colors allowing for transparency effects in computer graphics. Barycentric coordinates - a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron, etc.) is specified as the center of mass, or barycenter, of masses placed at its vertices. The coordinates are non-negative for points in the convex hull. Borsuk's conjecture - a conjecture about the number of pieces required to cover a body with a larger diameter. Solved by Hadwiger for the case of smooth convex bodies. Bond convexity - a measure of the non-linear relationship between price and yield duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates. A basic form of convexity in finance. Carathéodory's theorem (convex hull) - If a point x of Rd lies in the convex hull of a set P, there is a subset of P with d+1 or fewer points such that x lies in its


id: list-of-coordinate-charts-20260505-0
file: data/en.wikipedia.org/wiki/List_of_coordinate_charts-0.md
category: reference
title: List of coordinate charts
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_coordinate_charts
date_saved: 2026-05-05T08:13:46.208272+00:00
instance: kb-cron
parent: list-of-coordinate-charts-20260505-0
summary:
This article contains a non-exhaustive list of coordinate charts for Riemannian manifolds and pseudo-Riemannian manifolds. Coordinate charts are mathematical objects of topological manifolds, and they have multiple applications in theoretical and applied mathematics. When a differentiable structure and a metric are defined, greater structure exists, and this allows the definition of constructs such as integration and geodesics. == Charts for Riemannian and pseudo-Riemannian surfaces == The following charts (with appropriate metric tensors) can be used in the stated classes of Riemannian and pseudo-Riemannian surfaces: == See also == Coordinate chart Coordinate system Metric tensor List of mathematics lists, particularly: List of multivariable calculus topics List of Fourier analysis topics List of differential geometry topics == References ==


id: list-of-common-coordinate-transformations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-0.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
This is a list of some of the most commonly used coordinate transformations. == 2-dimensional == Let ( x , y ) {\displaystyle (x,y)} be the standard Cartesian coordinates, and ( r , θ ) {\displaystyle (r,\theta )} the standard polar coordinates. === To Cartesian coordinates === ==== From polar coordinates ==== x = r cos ⁡ θ y = r sin ⁡ θ ∂ ( x , y ) ∂ ( r , θ ) = [ cos ⁡ θ − r sin ⁡ θ sin ⁡ θ − r cos ⁡ θ ] Jacobian = det ∂ ( x , y ) ∂ ( r , θ ) = r {\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\[5pt]{\frac {\partial (x,y)}{\partial (r,\theta )}}&={\begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &{\phantom {-}}r\cos \theta \end{bmatrix}}\\[5pt]{\text{Jacobian}}=\det {\frac {\partial (x,y)}{\partial (r,\theta )}}&=r\end{aligned}}} ==== From log-polar coordinates ==== x = e ρ cos ⁡ θ , y = e ρ sin ⁡ θ . {\displaystyle {\begin{aligned}x&=e^{\rho }\cos \theta ,\\y&=e^{\rho }\sin \theta .\end{aligned}}} By using complex numbers ( x , y ) = x + i y ′ {\displaystyle (x,y)=x+iy'} , the transformation can be written as x + i y = e ρ + i θ {\displaystyle


id: list-of-common-coordinate-transformations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-1.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as illustrated). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range of 180°, running from 0° to 180°, and does not pose any problem when calculated from an arccosine, but beware for an arctangent. If, in the alternative definition, θ is chosen to run from −90° to +90°, in opposite direction of the earlier definition, it can be found uniquely from an arcsine, but beware of an arccotangent. In this case in all formulas below all arguments in θ should have sine and cosine exchanged, and as derivative also a plus and minus exchanged. All divisions by zero result in special cases of being directions along one of the main axes and are in practice most easily solved by observation. === To Cartesian coordinates === ==== From spherical coordinates ==== x = ρ sin ⁡ θ cos ⁡ φ y = ρ sin ⁡ θ sin ⁡ φ z = ρ cos ⁡ θ ∂


id: list-of-common-coordinate-transformations-20260505-2
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-2.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
∂ ( r , θ , h ) ∂ ( x , y , z ) = ( x x 2 + y 2 y x 2 + y 2 0 − y x 2 + y 2 x x 2 + y 2 0 0 0 1 ) {\displaystyle {\frac {\partial (r,\theta ,h)}{\partial (x,y,z)}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\\0&0&1\end{pmatrix}}} ==== From spherical coordinates ==== r = ρ sin ⁡ φ h = ρ cos ⁡ φ θ = θ ∂ ( r , h , θ ) ∂ ( ρ , φ , θ ) = ( sin ⁡ φ ρ cos ⁡ φ 0 cos ⁡ φ − ρ sin ⁡ φ 0 0 0 1 ) det ∂ ( r , h , θ ) ∂ ( ρ , φ , θ ) = − ρ {\displaystyle {\begin{aligned}r&=\rho \sin \varphi \\h&=\rho \cos \varphi \\\theta &=\theta \\{\frac {\partial (r,h,\theta )}{\partial (\rho ,\varphi ,\theta )}}&={\begin{pmatrix}\sin \varphi &\rho \cos \varphi &0\\\cos \varphi &-\rho \sin \varphi &0\\0&0&1\\\end{pmatrix}}\\\det {\frac {\partial (r,h,\theta )}{\partial (\rho ,\varphi ,\theta )}}&=-\rho \end{aligned}}} === Arc-length, curvature and torsion from Cartesian coordinates === s = ∫ 0 t x ′ 2 + y ′ 2 + z ′


id: index-of-cryptography-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-0.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
Articles related to cryptography include: == A == A5/1 • A5/2 • ABA digital signature guidelines • ABC (stream cipher) • Abraham Sinkov • Acoustic cryptanalysis • Adaptive chosen-ciphertext attack • Adaptive chosen plaintext and chosen ciphertext attack • Advantage (cryptography) • ADFGVX cipher • Adi Shamir • Advanced Access Content System • Advanced Encryption Standard • Advanced Encryption Standard process • Adversary • AEAD block cipher modes of operation • Affine cipher • Agnes Meyer Driscoll • AKA (security) • Akelarre (cipher) • Alan Turing • Alastair Denniston • Al Bhed language • Alex Biryukov • Alfred Menezes • Algebraic Eraser • Algorithmically random sequence • Alice and Bob • All-or-nothing transform • Alphabetum Kaldeorum • Alternating step generator • American Cryptogram Association • AN/CYZ-10 • Anonymous publication • Anonymous remailer • Antoni Palluth • Anubis (cipher) • Argon2 • ARIA (cipher) • Arlington Hall • Arne Beurling • Arnold Cipher • Array controller based encryption • Arthur Scherbius • Arvid Gerhard Damm • Asiacrypt • Atbash • Attribute-based encryption • Attack model • Auguste Kerckhoffs • Authenticated encryption • Authentication • Authorization certificate • Autokey cipher • Avalanche effect == B == B-Dienst • Babington Plot • Baby-step


id: index-of-cryptography-articles-20260505-1
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-1.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== D == D'Agapeyeff cipher • Daniel J. Bernstein • Data Authentication Algorithm • Data Encryption Standard • Datagram Transport Layer Security • David Chaum • David Kahn • David Naccache • David Wagner • David Wheeler (computer scientist) • Davies attack • Davies–Meyer hash • DEAL • Decipherment • Decisional Diffie–Hellman assumption • Decorrelation theory • Decrypt • DeCSS • Defence Signals Directorate • Degree of anonymity • Delegated Path Discovery • Delegated Path Validation • Deniable encryption • Derek Taunt • Derived unique key per transaction • DES Challenges • DES supplementary material • DES-X • Deterministic encryption • DFC (cipher) • Dictionary attack • Differential cryptanalysis • Differential-linear attack • Differential power analysis • Diffie–Hellman key exchange • Diffie–Hellman problem • DigiCipher 2 • Digital Fortress • Digital rights management • Digital signature • Digital Signature Algorithm • Digital signature forgery • Digital timestamping • Digital watermarking • Dilly Knox • Dining cryptographers problem • Diplomatic bag • Direct Anonymous Attestation • Discrete logarithm • Disk encryption • Disk encryption hardware • Disk encryption software • Distance-bounding protocol • Distinguishing attack • Distributed.net • DMA attack • dm-crypt • Dmitry Sklyarov • DomainKeys • Don Coppersmith •


id: index-of-cryptography-articles-20260505-2
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-2.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== K == Kaisa Nyberg • Kalyna (cipher) • Kasiski examination • KASUMI • KCDSA • KeePass • Kerberos (protocol) • Kerckhoffs's principle • Kevin McCurley (cryptographer) • Key-agreement protocol • Key authentication • Key clustering • Key (cryptography) • Key derivation function • Key distribution center • Key escrow • Key exchange • Keyfile • Key generation • Key generator • Key management • Key-recovery attack • Key schedule • Key server (cryptographic) • Key signature (cryptography) • Keysigning • Key signing party • Key size • Key space (cryptography) • Keystream • Key stretching • Key whitening • KG-84 • KHAZAD • Khufu and Khafre • Kiss (cryptanalysis) • KL-43 • KL-51 • KL-7 • Kleptography • KN-Cipher • Knapsack problem • Known-key distinguishing attack • Known-plaintext attack • KOI-18 • KOV-14 • Kryha • Kryptos • KSD-64 • Kupyna • Kuznyechik • KW-26 • KW-37 • KY-3 • KY-58 • KY-68 • KYK-13 == L == Lacida • Ladder-DES • Lamport signature • Lars Knudsen • Lattice-based cryptography • Laurance Safford • Lawrie Brown • LCS35 • Leo Marks • Leonard Adleman • Leon Battista Alberti • Leo Rosen • Leslie Yoxall • LEVIATHAN (cipher) • LEX


id: index-of-cryptography-articles-20260505-3
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-3.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== R == Rabbit (cipher) • Rabin cryptosystem • Rabin signature • RadioGatún • Rail fence cipher • Rainbow table • Ralph Merkle • Rambutan (cryptography) • Random function • Randomness tests • Random number generator attack • Random oracle • RC2 • RC4 • RC5 • RC6 • Rebound attack • Reciprocal cipher • Red/black concept • REDOC • Red Pike (cipher) • Reflector (cipher machine) • Regulation of Investigatory Powers Act 2000 • Reihenschieber • Rekeying (cryptography) • Related-key attack • Replay attack • Reservehandverfahren • Residual block termination • Rijndael key schedule • Rijndael S-box • Ring signature • RIPEMD • Rip van Winkle cipher • Robert Morris (cryptographer) • Rockex • Rolf Noskwith • Ron Rivest • Room 40 • Root certificate • Ross J. Anderson • Rossignols • ROT13 • Rotor machine • RSA RSA • RSA-100 • RSA-1024 • RSA-110 • RSA-120 • RSA-129 • RSA-130 • RSA-140 • RSA-150 • RSA-1536 • RSA-155 • RSA-160 • RSA-170 • RSA-180 • RSA-190 • RSA-200 • RSA-2048 • RSA-210 • RSA-220 • RSA-230 • RSA-232 • RSA-240 • RSA-250 • RSA-260 • RSA-270 • RSA-280 • RSA-290 • RSA-300 • RSA-309 • RSA-310 • RSA-320


id: list-of-curves-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_curves_topics-0.md
category: reference
title: List of curves topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_curves_topics
date_saved: 2026-05-05T08:13:49.986081+00:00
instance: kb-cron
parent: list-of-curves-topics-20260505-0
summary:
This is an alphabetical index of articles related to curves used in mathematics. Acnode Algebraic curve Arc Asymptote Asymptotic curve Barbier's theorem Bézier curve Bézout's theorem Birch and Swinnerton-Dyer conjecture Bitangent Bitangents of a quartic Cartesian coordinate system Caustic Cesàro equation Chord (geometry) Cissoid Circumference Closed timelike curve concavity Conchoid (mathematics) Confocal Contact (mathematics) Contour line Crunode Cubic Hermite curve Curvature Curve orientation Curve fitting Curve-fitting compaction Curve of constant width Curve of pursuit Curves in differential geometry Cusp Cyclogon De Boor algorithm Differential geometry of curves Eccentricity (mathematics) Elliptic curve cryptography Envelope (mathematics) Fenchel's theorem Genus (mathematics) Geodesic Geometric genus Great-circle distance Harmonograph Hedgehog (curve) [1] Hilbert's sixteenth problem Hyperelliptic curve cryptography Inflection point Inscribed square problem intercept, y-intercept, x-intercept Intersection number Intrinsic equation Isoperimetric inequality Jordan curve Jordan curve theorem Knot Limit cycle Linking coefficient List of circle topics Loop (knot) M-curve Mannheim curve[2] Meander (mathematics) Mordell conjecture Natural representation Opisometer Orbital elements Osculating circle Osculating plane Osgood curve Parallel (curve) Parallel transport Parametric curve Bézier curve Spline (mathematics) Hermite spline Beta spline B-spline Higher-order spline NURBS Perimeter Pi Plane curve Pochhammer contour Polar coordinate system Prime geodesic Projective line Ray Regular parametric representation Reuleaux triangle Ribaucour curve[3][4]


id: differentiation-rules-20260505-0
file: data/en.wikipedia.org/wiki/Differentiation_rules-0.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
This article is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. == Elementary rules of differentiation == Unless otherwise stated, all functions are functions of real numbers ( R {\textstyle \mathbb {R} } ) that return real values, although, more generally, the formulas below apply wherever they are well defined, including the case of complex numbers ( C {\textstyle \mathbb {C} } ). === Constant term rule === For any value of c {\textstyle c} , where c ∈ R {\textstyle c\in \mathbb {R} } , if f ( x ) {\textstyle f(x)} is the constant function given by f ( x ) = c {\textstyle f(x)=c} , then d f d x = 0 {\textstyle {\frac {df}{dx}}=0} . ==== Proof ==== Let c ∈ R {\textstyle c\in \mathbb {R} } and f ( x ) = c {\textstyle f(x)=c} . By the definition of the derivative: f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h = lim h → 0 ( c ) − ( c ) h = lim h → 0 0 h = lim


id: differentiation-rules-20260505-1
file: data/en.wikipedia.org/wiki/Differentiation_rules-1.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
h ′ ( x ) = − f ′ ( x ) ( f ( x ) ) 2 , {\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}},} wherever f {\textstyle f} is nonzero. In Leibniz's notation, this formula is written: d ( 1 f ) d x = − 1 f 2 d f d x . {\displaystyle {\frac {d\left({\frac {1}{f}}\right)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.} The reciprocal rule can be derived either from the quotient rule or from the combination of power rule and chain rule. === Quotient rule === If f {\textstyle f} and g {\textstyle g} are functions, then: ( f g ) ′ = f ′ g − g ′ f g 2 , {\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}},} wherever g {\textstyle g} is nonzero. This can be derived from the product rule and the reciprocal rule. === Generalized power rule === The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f {\textstyle f} and g {\textstyle g} , ( f g ) ′ = ( e g ln ⁡ f ) ′ = f g ( f ′ g f + g ′ ln ⁡ f ) , {\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln


id: differentiation-rules-20260505-2
file: data/en.wikipedia.org/wiki/Differentiation_rules-2.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
Γ ′ ( x ) = ∫ 0 ∞ t x − 1 e − t ln ⁡ t d t = Γ ( x ) ( ∑ n = 1 ∞ ( ln ⁡ ( 1 + 1 n ) − 1 x + n ) − 1 x ) = Γ ( x ) ψ ( x ) , {\displaystyle {\begin{aligned}\Gamma '(x)&=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt\\&=\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)\\&=\Gamma (x)\psi (x),\end{aligned}}} with ψ ( x ) {\textstyle \psi (x)} being the digamma function, expressed by the parenthesized expression to the right of Γ ( x ) {\textstyle \Gamma (x)} in the line above. === Riemann zeta function === ζ ( x ) = ∑ n = 1 ∞ 1 n x {\displaystyle \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}} ζ ′ ( x ) = − ∑ n = 1 ∞ ln ⁡ n n x = − ln ⁡ 2 2 x − ln ⁡ 3 3 x − ln ⁡ 4 4 x − ⋯ = − ∑ p prime p − x ln ⁡ p ( 1 − p − x ) 2 ∏ q prime , q ≠ p 1 1 − q


id: list-of-derivatives-and-integrals-in-alternative-calculi-20260505-0
file: data/en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi-0.md
category: reference
title: List of derivatives and integrals in alternative calculi
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi
date_saved: 2026-05-05T08:13:52.505671+00:00
instance: kb-cron
parent: list-of-derivatives-and-integrals-in-alternative-calculi-20260505-0
summary:
There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea. The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). == Table == In the following table; ψ ( x ) = Γ ′ ( x ) Γ ( x ) {\displaystyle \psi (x)={\frac {\Gamma '(x)}{\Gamma (x)}}} is the digamma function, K ⁡ ( x ) = e ζ ′ ( − 1 , x ) − ζ ′ ( − 1 ) = e z − z 2 2 + z 2 ln ⁡ ( 2 π ) − ψ ( − 2 ) ( z ) {\displaystyle \operatorname {K} (x)=e^{\zeta ^{\prime }(-1,x)-\zeta ^{\prime }(-1)}=e^{{\frac {z-z^{2}}{2}}+{\frac {z}{2}}\ln(2\pi )-\psi ^{(-2)}(z)}} is the K-function, ( ! x ) = Γ ( x + 1 , − 1 ) e {\displaystyle (!x)={\frac {\Gamma (x+1,-1)}{e}}} is subfactorial, B a ( x ) = − a ζ ( − a + 1 , x ) {\displaystyle B_{a}(x)=-a\zeta (-a+1,x)} are the generalized to real


id: list-of-differential-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_differential_geometry_topics-0.md
category: reference
title: List of differential geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_differential_geometry_topics
date_saved: 2026-05-05T08:13:53.815515+00:00
instance: kb-cron
parent: list-of-differential-geometry-topics-20260505-0
summary:
This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. == Differential geometry of curves and surfaces == === Differential geometry of curves === List of curves topics Frenet–Serret formulas Curves in differential geometry Line element Curvature Radius of curvature Osculating circle Curve Fenchel's theorem === Differential geometry of surfaces === Theorema egregium Gauss–Bonnet theorem First fundamental form Second fundamental form Gauss–Codazzi–Mainardi equations Dupin indicatrix Asymptotic curve Curvature Principal curvatures Mean curvature Gauss curvature Elliptic point Types of surfaces Minimal surface Ruled surface Conical surface Developable surface Nadirashvili surface == Foundations == === Calculus on manifolds === See also multivariable calculus, list of multivariable calculus topics Manifold Differentiable manifold Smooth manifold Banach manifold Fréchet manifold Tensor analysis Tangent vector Tangent space Tangent bundle Cotangent space Cotangent bundle Tensor Tensor bundle Vector field Tensor field Differential form Exterior derivative Lie derivative pullback (differential geometry) pushforward (differential) jet (mathematics) Contact (mathematics) jet bundle Frobenius theorem (differential topology) Integral curve === Differential topology === Diffeomorphism Large diffeomorphism Orientability characteristic class Chern class Pontrjagin class spin structure differentiable map submersion immersion Embedding Whitney embedding theorem Critical value Sard's theorem Saddle point Morse


id: outline-of-discrete-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_discrete_mathematics-0.md
category: reference
title: Outline of discrete mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_discrete_mathematics
date_saved: 2026-05-05T08:13:55.045428+00:00
instance: kb-cron
parent: outline-of-discrete-mathematics-20260505-0
summary:
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis. Included below are many of the standard terms used routinely in university-level courses and in research papers. This is not, however, intended as a complete list of mathematical terms; just a selection of typical terms of art that may be encountered. Logic – Study of correct reasoning Modal logic – Type of formal logic Set theory – Branch of mathematics that studies sets Number theory – Branch of pure mathematics Combinatorics – Branch of discrete mathematics Finite mathematics – Syllabus in college and university mathematics Graph theory – Area of discrete mathematics Digital geometry – Deals with digitized models or images of objects of the 2D or 3D Euclidean space Digital topology – Properties of 2D or 3D digital images that correspond to classic topological properties Algorithmics – Sequence of operations for


id: outline-of-discrete-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_discrete_mathematics-1.md
category: reference
title: Outline of discrete mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_discrete_mathematics
date_saved: 2026-05-05T08:13:55.045428+00:00
instance: kb-cron
parent: outline-of-discrete-mathematics-20260505-0
summary:
Left-hand side and right-hand side of an equation – Mathematical nomenclaturePages displaying short descriptions of redirect targets Linear equation – Equation that does not involve powers or products of variables Quadratic equation – Polynomial equation of degree two Solution point – Mathematical formula expressing equalityPages displaying short descriptions of redirect targets Arithmetic progression – Sequence of equally spaced numbers Recurrence relation – Pattern defining an infinite sequence of numbers Finite difference – Discrete analog of a derivative Difference operator – Pattern defining an infinite sequence of numbersPages displaying short descriptions of redirect targets Groups – Set with associative invertible operation Group isomorphism – Bijective group homomorphism Subgroups – Subset of a group that forms a group itselfPages displaying short descriptions of redirect targets Fermat's little theorem – A prime p divides a^p–a for any integer a Cryptography – Practice and study of secure communication techniques Faulhaber's formula – Expression for sums of powers === Mathematical relations === Binary relation – Relationship between elements of two sets Heterogeneous relation – Relationship between elements of two setsPages displaying short descriptions of redirect targets Reflexive relation – Binary relation that relates every element to itself Reflexive property of equality – Basic notion of


id: divisibility-rule-20260505-0
file: data/en.wikipedia.org/wiki/Divisibility_rule-0.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American. == Divisibility rules for numbers 1−30 == The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last n digits) the result must be examined by other means. For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits. == Step-by-step examples == === Divisibility by 2 === First, take any number (for this example it will be 376) and note


id: divisibility-rule-20260505-1
file: data/en.wikipedia.org/wiki/Divisibility_rule-1.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Divisibility by 5 === Divisibility by 5 is easily determined by checking the last digit in the number (475), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5. If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero, so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8). If the last digit in the number is 5, then the result will be the remaining digits multiplied by two, plus one. For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125 divided by 5 (125/5=25). Example. If the last digit is 0 110 (The original number) 11 0 (Take the last digit of the number, and check if it is 0 or


id: divisibility-rule-20260505-2
file: data/en.wikipedia.org/wiki/Divisibility_rule-2.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Take for instance the number 371 Change all occurrences of 7, 8 or 9 into 0, 1 and 2, respectively. In this example, we get: 301. This second step may be skipped, except for the left most digit, but following it may facilitate calculations later on. Now convert the first digit (3) into the following digit in the sequence 13264513... In our example, 3 becomes 2. Add the result in the previous step (2) to the second digit of the number, and substitute the result for both digits, leaving all remaining digits unmodified: 2 + 0 = 2. So 301 becomes 21. Repeat the procedure until you have a recognizable multiple of 7, or to make sure, a number between 0 and 6. So, starting from 21 (which is a recognizable multiple of 7), take the first digit (2) and convert it into the following in the sequence above: 2 becomes 6. Then add this to the second digit: 6 + 1 = 7. If at any point the first digit is 8 or 9, these become 1 or 2, respectively. But if it is a 7 it should become 0, only if no other digits follow. Otherwise, it should


id: divisibility-rule-20260505-3
file: data/en.wikipedia.org/wiki/Divisibility_rule-3.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
For all of the above examples, subtracting the first three digits from the last three results in a multiple of seven. Notice that leading zeros are permitted to form a 6-digit pattern. This phenomenon forms the basis for Steps B and C. Step B: If the integer is between 1001 and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer (leading zeros are allowed and can help you visualize the pattern). If the positive difference is less than 1000, apply Step A. This can be done by subtracting the first three digits from the last three digits. For example: 341,355 − 341,341 = 14 -> 1 − (4×2) = 1 − 8 = −7 YES 67,326 − 067,067 = 259 -> 25 − (9×2) = 25 − 18 = 7 YES The fact that 999,999 is a multiple of 7 can be used for determining divisibility of integers larger than one million by reducing the integer to a 6-digit number that can be determined using Step B. This can be done easily by adding the digits left of the first six to the last six and


id: divisibility-rule-20260505-4
file: data/en.wikipedia.org/wiki/Divisibility_rule-4.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Finding remainder of a number when divided by 7 7 − (1, 3, 2, −1, −3, −2, cycle repeats for the next six digits) Period: 6 digits. Recurring numbers: 1, 3, 2, −1, −3, −2 Minimum magnitude sequence (1, 3, 2, 6, 4, 5, cycle repeats for the next six digits) Period: 6 digits. Recurring numbers: 1, 3, 2, 6, 4, 5 Positive sequence Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7. Example: What is the remainder when 1036125837 is divided by 7? Multiplication of the rightmost digit = 1 × 7 = 7 Multiplication of the second rightmost digit = 3 × 3 = 9 Third rightmost digit = 8 × 2 = 16 Fourth rightmost digit = 5 × −1 = −5 Fifth rightmost digit = 2 × −3 = −6 Sixth rightmost digit = 1 × −2 = −2 Seventh rightmost digit = 6 × 1 = 6 Eighth rightmost digit = 3 × 3 =


id: divisibility-rule-20260505-5
file: data/en.wikipedia.org/wiki/Divisibility_rule-5.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Another digit grouping method of divisibility by 7 Method This method determines whether an integer is divisible by 7 by grouping digits in sets of three: Starting from the right (units place), divide the number into groups of three digits. If necessary, prepend leading zeros to complete the leftmost group. Starting from the rightmost group, alternately add and subtract the groups moving left. That is, subtract the second group from the first, add the third, subtract the fourth, and so on. Evaluate the resulting sum. If the result is divisible by 7, then the original number is divisible by 7; otherwise, it is not. Example To test the number 10,517,815: Group the digits: 010 | 517 | 815 Apply alternating operations: 815 − 517 + 010 = 308 Since 308 is divisible by 7, the original number is also divisible by 7. Proof of correctness This method is based on modular arithmetic. Observe that: 1000 ≡ −1 (mod 7) since 1000 leaves a remainder of 6 when divided by 7, and 6 ≡ −1 (mod 7). Any integer can be expressed as a sum of multiples of powers of 1000: N = a₀ + a₁·1000 + a₂·1000² + a₃·1000³ +


id: divisibility-rule-20260505-6
file: data/en.wikipedia.org/wiki/Divisibility_rule-6.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Composite divisors === A number is divisible by a given divisor if it is divisible by the highest power of each of its prime factors. For example, to determine divisibility by 36, check divisibility by 4 and by 9. Note that checking 3 and 12, or 2 and 18, would not be sufficient. A table of prime factors may be useful. A composite divisor may also have a rule formed using the same procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present in the divisor. For instance, one cannot make a rule for 14 that involves multiplying the equation by 7. This is not an issue for prime divisors because they have no smaller factors. To test the divisibility of a number by a power of 2 or a power of 5 (2n or 5n, in which n is a positive integer), one only need to look at the last n digits of that number. To test divisibility by any number expressed as the product of prime factors p 1 n p 2 m p 3 q {\displaystyle p_{1}^{n}p_{2}^{m}p_{3}^{q}} , we can separately test for


id: divisibility-rule-20260505-7
file: data/en.wikipedia.org/wiki/Divisibility_rule-7.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Proof using basic algebra === Many of the simpler rules can be produced using only algebraic manipulation, creating binomials and rearranging them. By writing a number as the sum of each digit times a power of 10 each digit's power can be manipulated individually. ==== Case where all digits are summed ==== This method works for divisors that are factors of 10 − 1 = 9. Using 3 as an example, 3 divides 9 = 10 − 1. That means 10 ≡ 1 ( mod 3 ) {\displaystyle 10\equiv 1{\pmod {3}}} (see modular arithmetic). The same for all the higher powers of 10: 10 n ≡ 1 n ≡ 1 ( mod 3 ) . {\displaystyle 10^{n}\equiv 1^{n}\equiv 1{\pmod {3}}.} They are all congruent to 1 modulo 3. Since two things that are congruent modulo 3 are either both divisible by 3 or both not, we can interchange values that are congruent modulo 3. So, in a number such as the following, we can replace all the powers of 10 by 1: 100 ⋅ a + 10 ⋅ b + 1 ⋅ c ≡ ( 1 ) a + ( 1 ) b + ( 1 ) c (


id: list-of-dynamical-systems-and-differential-equations-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics-0.md
category: reference
title: List of dynamical systems and differential equations topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics
date_saved: 2026-05-05T08:14:00.321935+00:00
instance: kb-cron
parent: list-of-dynamical-systems-and-differential-equations-topics-20260505-0
summary:
This is a list of dynamical system and differential equation topics, by Wikipedia page. See also list of partial differential equation topics, list of equations. == Dynamical systems, in general == Deterministic system (mathematics) Linear system Partial differential equation Dynamical systems and chaos theory Chaos theory Chaos argument Butterfly effect 0-1 test for chaos Bifurcation diagram Feigenbaum constant Sharkovskii's theorem Attractor Strange nonchaotic attractor Stability theory Mechanical equilibrium Astable Monostable Bistability Metastability Feedback Negative feedback Positive feedback Homeostasis Damping ratio Dissipative system Spontaneous symmetry breaking Turbulence Perturbation theory Control theory Non-linear control Adaptive control Hierarchical control Intelligent control Optimal control Dynamic programming Robust control Stochastic control System dynamics, system analysis Takens' theorem Exponential dichotomy Liénard's theorem Krylov–Bogolyubov theorem Krylov-Bogoliubov averaging method == Abstract dynamical systems == Measure-preserving dynamical system Ergodic theory Mixing (mathematics) Almost periodic function Symbolic dynamics Time scale calculus Arithmetic dynamics Sequential dynamical system Graph dynamical system Topological dynamical system == Dynamical systems, examples == List of chaotic maps Logistic map Lorenz attractor Lorenz-96 Iterated function system Tetration Ackermann function Horseshoe map Hénon map Arnold's cat map Population dynamics == Complex dynamics == Fatou set Julia set Mandelbrot set == Difference equations == Recurrence relation Matrix difference equation


id: list-of-eponyms-of-special-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_eponyms_of_special_functions-0.md
category: reference
title: List of eponyms of special functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_eponyms_of_special_functions
date_saved: 2026-05-05T08:14:02.125990+00:00
instance: kb-cron
parent: list-of-eponyms-of-special-functions-20260505-0
summary:
This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intended to include every mathematical eponym). Named symmetric functions, and other special polynomials, are included. == A == Niels Abel: Abel polynomials - Abelian function - Abel–Gontscharoff interpolating polynomial Sir George Biddell Airy: Airy function Waleed Al-Salam (1926–1996): Al-Salam polynomial - Al Salam–Carlitz polynomial - Al Salam–Chihara polynomial C. T. Anger: Anger–Weber function Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral Paul Émile Appell (1855–1930): Appell hypergeometric series, Appell polynomial, Generalized Appell polynomials Richard Askey: Askey–Wilson polynomial, Askey–Wilson function (with James A. Wilson) == B == Ernest William Barnes: Barnes G-function E. T. Bell: Bell polynomials Bender–Dunne polynomial Jacob Bernoulli: Bernoulli polynomial Friedrich Bessel: Bessel function, Bessel–Clifford function H. Blasius: Blasius functions R. P. Boas, R. C. Buck: Boas–Buck polynomial Böhmer integral Erland Samuel Bring: Bring radical de Bruijn function Buchstab function Burchnall, Chaundy: Burchnall–Chaundy polynomial == C == Leonard Carlitz: Carlitz polynomial Arthur Cayley, Capelli: Cayley–Capelli operator Celine's polynomial Charlier polynomial Pafnuty Chebyshev: Chebyshev polynomials Elwin Bruno Christoffel, Darboux: Christoffel–Darboux relation Cyclotomic polynomials == D == H. G. Dawson:


id: list-of-mathematical-examples-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_examples-0.md
category: reference
title: List of mathematical examples
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_examples
date_saved: 2026-05-05T08:14:04.638441+00:00
instance: kb-cron
parent: list-of-mathematical-examples-20260505-0
summary:
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a general theorem, could perhaps be considered an "example"). The discussion page for list of mathematical topics has some comments on this. Eventually this page may have its own discussion page. This page links to itself in order that edits to this page will be included among related changes when the user clicks on that button. The concrete example within the article titled Rao–Blackwell theorem is perhaps one of the best ways for a probabilist ignorant of statistical inference to get a quick impression of the flavor of that subject. == Uncategorized examples, alphabetized == Alexander horned sphere All horses are the same color Cantor function Cantor set Checking if a coin is biased Concrete illustration of the central limit theorem Differential equations of mathematical physics Dirichlet function Discontinuous linear map Efron's non-transitive


id: list-of-exceptional-set-concepts-20260505-0
file: data/en.wikipedia.org/wiki/List_of_exceptional_set_concepts-0.md
category: reference
title: List of exceptional set concepts
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_exceptional_set_concepts
date_saved: 2026-05-05T08:14:06.643357+00:00
instance: kb-cron
parent: list-of-exceptional-set-concepts-20260505-0
summary:
This is a list of exceptional set concepts. In mathematics, and in particular in mathematical analysis, it is very useful to be able to characterise subsets of a given set X as 'small', in some definite sense, or 'large' if their complement in X is small. There are numerous concepts that have been introduced to study 'small' or 'exceptional' subsets. In the case of sets of natural numbers, it is possible to define more than one concept of 'density', for example. See also list of properties of sets of reals. Almost all Almost always Almost everywhere Almost never Almost surely Analytic capacity Closed unbounded set Cofinal (mathematics) Cofinite Dense set IP set 2-large Large set (Ramsey theory) Meagre set Measure zero Natural density Negligible set Nowhere dense set Null set, conull set Partition regular Piecewise syndetic set Schnirelmann density Small set (combinatorics) Stationary set Syndetic set Thick set Thin set (Serre)


id: list-of-exponential-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_exponential_topics-0.md
category: reference
title: List of exponential topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_exponential_topics
date_saved: 2026-05-05T08:14:08.848554+00:00
instance: kb-cron
parent: list-of-exponential-topics-20260505-0
summary:
This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics. Accelerating change Approximating natural exponents (log base e) Artin–Hasse exponential Bacterial growth Baker–Campbell–Hausdorff formula Carlitz exponential Cell growth Barometric formula Beer–Lambert law Characterizations of the exponential function Catenary Compound interest C0-semigroup De Moivre's formula Derivative of the exponential map Doléans-Dade exponential Double exponential function Doubling time e-folding Elimination half-life Error exponent Euler's formula Euler's identity e (mathematical constant) Exponent Exponent bias Exponential (disambiguation) Exponential backoff Exponential decay Exponential dichotomy Exponential discounting Exponential diophantine equation Exponential dispersion model Exponential distribution Exponential error Exponential factorial Exponential family Exponential field Ordered exponential field Exponential formula Exponential function Exponential generating function Exponential-Golomb coding Exponential growth Exponential hierarchy Exponential integral Exponential integrator Exponential map (Lie theory) Exponential map (Riemannian geometry) Exponential map (discrete dynamical systems) Exponential notation Exponential object (category theory) Exponential polynomials—see also Touchard polynomials (combinatorics) Exponential response formula Exponential sheaf sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential type Exponentially equivalent measures Exponentiating by squaring Exponentiation Forgetting curve Four exponentials conjecture Fourier analysis List of Fourier analysis topics Gaussian function Gudermannian function Half-exponential function Half-life Hyperbolic function Inflation, inflation rate Interest


id: exterior-calculus-identities-20260505-0
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-0.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
This article summarizes several identities in exterior calculus, a mathematical calculus used in differential geometry. == Notation == The following notation is used in this article. === Manifold === M {\displaystyle M} , N {\displaystyle N} are n {\displaystyle n} -dimensional smooth manifolds, where n ∈ N {\displaystyle n\in \mathbb {N} } . That is, differentiable manifolds that can be differentiated enough times for the purposes on this page. p ∈ M {\displaystyle p\in M} , q ∈ N {\displaystyle q\in N} denote one point on each of the manifolds. The boundary of a manifold M {\displaystyle M} is a manifold ∂ M {\displaystyle \partial M} , which has dimension n − 1 {\displaystyle n-1} . An orientation on M {\displaystyle M} induces an orientation on ∂ M {\displaystyle \partial M} . We usually denote a submanifold by Σ ⊂ M {\displaystyle \Sigma \subset M} . === Tangent and cotangent bundles === T M {\displaystyle TM} , T ∗ M {\displaystyle T^{*}M} denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold M {\displaystyle M} . T p M {\displaystyle T_{p}M} , T q N {\displaystyle T_{q}N} denote the tangent spaces of M {\displaystyle M} , N {\displaystyle


id: exterior-calculus-identities-20260505-1
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-1.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Exterior derivative === The exterior derivative d k : Ω k ( M ) → Ω k + 1 ( M ) {\displaystyle d_{k}:\Omega ^{k}(M)\rightarrow \Omega ^{k+1}(M)} is defined for all 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} . We generally omit the subscript when it is clear from the context. For a 0 {\displaystyle 0} -form f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} we have d 0 f ∈ Ω 1 ( M ) {\displaystyle d_{0}f\in \Omega ^{1}(M)} as the 1 {\displaystyle 1} -form that gives the directional derivative, i.e., for the section X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} we have ( d 0 f ) ( X ) = ∂ X f {\displaystyle (d_{0}f)(X)=\partial _{X}f} , the directional derivative of f {\displaystyle f} along X {\displaystyle X} . For 0 < k ≤ n {\displaystyle 0<k\leq n} , ( d k ω ) ( X 0 , … , X k ) = ∑ 0 ≤ j ≤ k ( − 1 ) j d 0 ( ω ( X 0 , … , X ^ j , … , X k ) ) ( X


id: exterior-calculus-identities-20260505-2
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-2.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
A one-form α ∈ Ω 1 ( M ) {\displaystyle \alpha \in \Omega ^{1}(M)} corresponds to the unique vector field α ♯ ∈ Γ ( T M ) {\displaystyle \alpha ^{\sharp }\in \Gamma (TM)} such that for all X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} , we have: α ( X ) = g ( α ♯ , X ) . {\displaystyle \alpha (X)=g(\alpha ^{\sharp },X).} These mappings extend via multilinearity to mappings from k {\displaystyle k} -vector fields to k {\displaystyle k} -forms and k {\displaystyle k} -forms to k {\displaystyle k} -vector fields through ( A 1 ∧ A 2 ∧ ⋯ ∧ A k ) ♭ = A 1 ♭ ∧ A 2 ♭ ∧ ⋯ ∧ A k ♭ {\displaystyle (A_{1}\wedge A_{2}\wedge \cdots \wedge A_{k})^{\flat }=A_{1}^{\flat }\wedge A_{2}^{\flat }\wedge \cdots \wedge A_{k}^{\flat }} ( α 1 ∧ α 2 ∧ ⋯ ∧ α k ) ♯ = α 1 ♯ ∧ α 2 ♯ ∧ ⋯ ∧ α k ♯ . {\displaystyle (\alpha _{1}\wedge \alpha _{2}\wedge \cdots \wedge \alpha _{k})^{\sharp }=\alpha _{1}^{\sharp }\wedge \alpha _{2}^{\sharp }\wedge \cdots \wedge \alpha _{k}^{\sharp }.} === Hodge star === For an n-manifold M, the Hodge star


id: exterior-calculus-identities-20260505-3
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-3.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Cohomology === The k {\displaystyle k} -th cohomology of a manifold M {\displaystyle M} and its exterior derivative operators d 0 , … , d n − 1 {\displaystyle d_{0},\ldots ,d_{n-1}} is given by H k ( M ) := ker ( d k ) im ( d k − 1 ) {\displaystyle H^{k}(M):={\frac {{\text{ker}}(d_{k})}{{\text{im}}(d_{k-1})}}} Two closed k {\displaystyle k} -forms α , β ∈ Ω k ( M ) {\displaystyle \alpha ,\beta \in \Omega ^{k}(M)} are in the same cohomology class if their difference is an exact form i.e. [ α ] = [ β ] ⟺ α − β = d η for some η ∈ Ω k − 1 ( M ) {\displaystyle [\alpha ]=[\beta ]\ \ \Longleftrightarrow \ \ \alpha {-}\beta =d\eta \ {\text{ for some }}\eta \in \Omega ^{k-1}(M)} A closed surface of genus g {\displaystyle g} will have 2 g {\displaystyle 2g} generators which are harmonic. === Dirichlet energy === Given α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} , its Dirichlet energy is E D ( α ) := 1 2 ⟨ ⟨ d α , d α ⟩ ⟩ + 1 2 ⟨ ⟨ δ α ,


id: exterior-calculus-identities-20260505-4
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-4.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
⋆ ( λ 1 α + λ 2 β ) = λ 1 ( ⋆ α ) + λ 2 ( ⋆ β ) {\displaystyle {\star }(\lambda _{1}\alpha +\lambda _{2}\beta )=\lambda _{1}({\star }\alpha )+\lambda _{2}({\star }\beta )} for λ 1 , λ 2 ∈ R {\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {R} } ( linearity ) ⋆ ⋆ α = s ( − 1 ) k ( n − k ) α {\displaystyle {\star }{\star }\alpha =s(-1)^{k(n-k)}\alpha } for α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} , n = dim ⁡ ( M ) {\displaystyle n=\dim(M)} , and s = sign ⁡ ( g ) {\displaystyle s=\operatorname {sign} (g)} the sign of the metric ⋆ ( − 1 ) = s ( − 1 ) k ( n − k ) ⋆ {\displaystyle {\star }^{(-1)}=s(-1)^{k(n-k)}{\star }} ( inversion ) ⋆ ( f α ) = f ( ⋆ α ) {\displaystyle {\star }(f\alpha )=f({\star }\alpha )} for f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} ( commutative with 0 {\displaystyle 0} -forms ) ⟨ ⟨ α , α ⟩ ⟩ = ⟨ ⟨ ⋆ α , ⋆ α ⟩ ⟩ {\displaystyle \langle \!\langle


id: exterior-calculus-identities-20260505-5
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-5.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
( α ∧ β ∧ γ ) ( X , Y , Z ) = det [ α ( X ) α ( Y ) α ( Z ) β ( X ) β ( Y ) β ( Z ) γ ( X ) γ ( Y ) γ ( Z ) ] {\displaystyle (\alpha \wedge \beta \wedge \gamma )(X,Y,Z)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)&\alpha (Z)\\\beta (X)&\beta (Y)&\beta (Z)\\\gamma (X)&\gamma (Y)&\gamma (Z)\end{bmatrix}}} ( α 1 ∧ … ∧ α l ) ( X 1 , … , X l ) = det [ α 1 ( X 1 ) α 1 ( X 2 ) … α 1 ( X l ) α 2 ( X 1 ) α 2 ( X 2 ) … α 2 ( X l ) ⋮ ⋮ ⋱ ⋮ α l ( X 1 ) α l ( X 2 ) … α l ( X l ) ] {\displaystyle (\alpha _{1}\wedge \ldots \wedge \alpha _{l})(X_{1},\ldots ,X_{l})=\det {\begin{bmatrix}\alpha _{1}(X_{1})&\alpha _{1}(X_{2})&\dots &\alpha _{1}(X_{l})\\\alpha _{2}(X_{1})&\alpha _{2}(X_{2})&\dots &\alpha _{2}(X_{l})\\\vdots &\vdots &\ddots &\vdots \\\alpha _{l}(X_{1})&\alpha _{l}(X_{2})&\dots &\alpha _{l}(X_{l})\end{bmatrix}}} === Projection and rejection === ( − 1 ) k ι X ⋆ α = ⋆ ( X ♭ ∧ α


id: exterior-calculus-identities-20260505-6
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-6.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Identities in Euclidean 3-space === Let Euclidean metric g ( X , Y ) := ⟨ X , Y ⟩ = X ⋅ Y {\displaystyle g(X,Y):=\langle X,Y\rangle =X\cdot Y} . We use ∇ = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) {\displaystyle \nabla =\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)} differential operator R 3 {\displaystyle \mathbb {R} ^{3}} ι X α = g ( X , α ♯ ) = X ⋅ α ♯ {\displaystyle \iota _{X}\alpha =g(X,\alpha ^{\sharp })=X\cdot \alpha ^{\sharp }} for α ∈ Ω 1 ( M ) {\displaystyle \alpha \in \Omega ^{1}(M)} . d e t ( X , Y , Z ) = ⟨ X , Y × Z ⟩ = ⟨ X × Y , Z ⟩ {\displaystyle \mathbf {det} (X,Y,Z)=\langle X,Y\times Z\rangle =\langle X\times Y,Z\rangle } ( scalar triple product ) X × Y = ( ⋆ ( X ♭ ∧ Y ♭ ) ) ♯ {\displaystyle X\times Y=({\star }(X^{\flat }\wedge Y^{\flat }))^{\sharp }} ( cross product ) ι X α = − ( X × A ) ♭ {\displaystyle \iota _{X}\alpha =-(X\times A)^{\flat }} if α ∈ Ω 2 ( M )


id: list-of-factorial-and-binomial-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_factorial_and_binomial_topics-0.md
category: reference
title: List of factorial and binomial topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_factorial_and_binomial_topics
date_saved: 2026-05-05T08:14:15.007034+00:00
instance: kb-cron
parent: list-of-factorial-and-binomial-topics-20260505-0
summary:
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem Alternating factorial Antichain Beta function Bhargava factorial Binomial coefficient Pascal's triangle Binomial distribution Binomial proportion confidence interval Binomial-QMF (Daubechies wavelet filters) Binomial series Binomial theorem Binomial transform Binomial type Carlson's theorem Catalan number Fuss–Catalan number Central binomial coefficient Combination Combinatorial number system De Polignac's formula Difference operator Difference polynomials Digamma function Egorychev method Erdős–Ko–Rado theorem Euler–Mascheroni constant Faà di Bruno's formula Factorial Factorial moment Factorial number system Factorial prime Factoriangular number Gamma distribution Gamma function Gaussian binomial coefficient Gould's sequence Hyperfactorial Hypergeometric distribution Hypergeometric function identities Hypergeometric series Incomplete beta function Incomplete gamma function Jordan–Pólya number Kempner function Lah number Lanczos approximation Lozanić's triangle Macaulay representation of an integer Mahler's theorem Multinomial distribution Multinomial coefficient, Multinomial formula, Multinomial theorem Multiplicities of entries in Pascal's triangle Multiset Multivariate gamma function Narayana numbers Negative binomial distribution Nörlund–Rice integral Pascal matrix Pascal's pyramid Pascal's simplex Pascal's triangle Permutation List of permutation topics Pochhammer symbol (also falling, lower, rising, upper factorials) Poisson distribution Polygamma function Primorial Proof of Bertrand's postulate Sierpinski triangle Star of David theorem Stirling number Stirling transform Stirling's approximation Subfactorial Table of Newtonian


id: list-of-finite-simple-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_finite_simple_groups-0.md
category: reference
title: List of finite simple groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite_simple_groups
date_saved: 2026-05-05T08:14:16.205745+00:00
instance: kb-cron
parent: list-of-finite-simple-groups-20260505-0
summary:
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates. == Summary == The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = A3(2) and A2(4) both have order 20160, and that the group Bn(q) has the same order as Cn(q) for q odd, n > 2. The smallest of the latter pairs of groups are B3(3) and C3(3) which both have order 4585351680.) There is an unfortunate conflict between the notations for the alternating groups An and the groups of Lie


id: list-of-finite-simple-groups-20260505-1
file: data/en.wikipedia.org/wiki/List_of_finite_simple_groups-1.md
category: reference
title: List of finite simple groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite_simple_groups
date_saved: 2026-05-05T08:14:16.205745+00:00
instance: kb-cron
parent: list-of-finite-simple-groups-20260505-0
summary:
=== Ree groups, 2G2(32n+1) === Simplicity: Simple for n ≥ 1. The group 2G2(3) is not simple, but its derived group 2G2(3)′ is a simple subgroup of index 3. Order: q3 (q3 + 1) (q − 1), where q = 32n+1 Schur multiplier: Trivial for n ≥ 1 and for 2G2(3)′. Outer automorphism group: 1⋅f⋅1, where f = 2n + 1. Other names: Ree(32n+1), R(32n+1), E2∗(32n+1) . Isomorphisms: The derived group 2G2(3)′ is isomorphic to A1(8). Remarks: 2G2(32n+1) has a doubly transitive permutation representation on 33(2n+1) + 1 points and acts on a 7-dimensional vector space over the field with 32n+1 elements. == Sporadic groups == === Mathieu groups, M11, M12, M22, M23, M24 === === Janko groups, J1, J2, J3, J4 === === Conway groups, Co1, Co2, Co3 === === Fischer groups, Fi22, Fi23, Fi24′ === === Higman–Sims group, HS === Order: 29 ⋅ 32 ⋅ 53 ⋅ 7 ⋅ 11 = 44352000 Schur multiplier: Order 2. Outer automorphism group: Order 2. Remarks: It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in Co2 and in Co3. === McLaughlin group, McL === Order: 27 ⋅ 36 ⋅ 53 ⋅


id: list-of-finite-dimensional-nichols-algebras-20260505-0
file: data/en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras-0.md
category: reference
title: List of finite-dimensional Nichols algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
date_saved: 2026-05-05T08:14:17.618854+00:00
instance: kb-cron
parent: list-of-finite-dimensional-nichols-algebras-20260505-0
summary:
In mathematics, a Nichols algebra is a Hopf algebra in a braided category assigned to an object V in this category (e.g. a braided vector space). The Nichols algebra is a quotient of the tensor algebra of V enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in any pointed Hopf algebra and enabled their classification in important cases. The most well known examples for Nichols algebras are the Borel parts U q ( g ) + {\displaystyle U_{q}({\mathfrak {g}})^{+}} of the infinite-dimensional quantum groups when q is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts u q ( g ) + {\displaystyle u_{q}({\mathfrak {g}})^{+}} of the Frobenius–Lusztig kernel (small quantum group) when q is a root of unity. The following article lists all known finite-dimensional Nichols algebras B ( V ) {\displaystyle {\mathfrak {B}}(V)} where V {\displaystyle V} is a Yetter–Drinfel'd module over a finite group G {\displaystyle G} , where the group is generated by the support of V {\displaystyle V} . For more details on Nichols algebras see Nichols algebra. There are two major cases: G {\displaystyle G} abelian, which implies V {\displaystyle V} is


id: list-of-finite-dimensional-nichols-algebras-20260505-1
file: data/en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras-1.md
category: reference
title: List of finite-dimensional Nichols algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
date_saved: 2026-05-05T08:14:17.618854+00:00
instance: kb-cron
parent: list-of-finite-dimensional-nichols-algebras-20260505-0
summary:
=== Nichols algebras from Coxeter groups === For every finite coxeter system ( W , S ) {\displaystyle (W,S)} the Nichols algebra over the conjugacy class(es) of reflections was studied in (reflections on roots of different length are not conjugate, see fourth example fellow). They discovered in this way the following first Nichols algebras over nonabelian groups : The case S 2 ≅ Z 2 {\displaystyle \mathbb {S} _{2}\cong \mathbb {Z} _{2}} is the rank 1 diagonal Nichols algebra u i ( A 1 ) + {\displaystyle u_{i}(A_{1})^{+}} of dimension 2. === Other Nichols algebras of rank 1 === === Nichols algebras of rank 2, type Gamma-3 === These Nichols algebras were discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 2 type Gamma-4 === This Nichols algebra was discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 2, type T === This Nichols algebra was discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 3 involving Gamma-3 === This Nichols algebra was the last Nichols algebra discovered during the classification of Heckenberger and Vendramin. === Nichols algebras from diagram folding === The following families


id: list-of-first-order-theories-20260505-0
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-0.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties. == Preliminaries == For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally a σ-structure. Given a signature σ there is a unique first-order language Lσ that can be used to capture the first-order expressible facts about the σ-structure. There are two common ways to specify theories: List or describe a set of sentences in the language Lσ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in Lσ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields. An Lσ theory may: be consistent: no proof of contradiction exists; be satisfiable: there exists a σ-structure for which the sentences of the theory are all true (by the completeness theorem, satisfiability is equivalent to consistency); be complete:


id: list-of-first-order-theories-20260505-1
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-1.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
~ has an infinite number of equivalence classes; ~ has exactly n equivalence classes (for any fixed positive integer n); All equivalence classes are infinite; All equivalence classes have size exactly n (for any fixed positive integer n). The theory of an equivalence relation with exactly 2 infinite equivalence classes is an easy example of a theory which is ω-categorical but not categorical for any larger cardinal. The equivalence relation ~ should not be confused with the identity symbol '=': if x=y then x~y, but the converse is not necessarily true. Theories of equivalence relations are not all that difficult or interesting, but often give easy examples or counterexamples for various statements. The following constructions are sometimes used to produce examples of theories with certain spectra; in fact by applying them to a small number of explicit theories T one gets examples of complete countable theories with all possible uncountable spectra. If T is a theory in some language, we define a new theory 2T by adding a new binary relation to the language, and adding axioms stating that it is an equivalence relation, such that there are an infinite number of equivalence classes all of which are models of


id: list-of-first-order-theories-20260505-2
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-2.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
== Boolean algebras == There are several different signatures and conventions used for Boolean algebras: The signature has two constants, 0 and 1, and two binary functions ∧ and ∨ ("and" and "or"), and one unary function ¬ ("not"). This can be confusing as the functions use the same symbols as the propositional functions of first-order logic. In set theory, a common convention is that the language has two constants, 0 and 1, and two binary functions · and +, and one unary function −. The three functions have the same interpretation as the functions in the first convention. Unfortunately, this convention clashes badly with the next convention: In algebra, the usual convention is that the language has two constants, 0 and 1, and two binary functions · and +. The function · has the same meaning as ∧, but a+b means a∨b∧¬(a∧b). The reason for this is that the axioms for a Boolean algebra are then just the axioms for a ring with 1 plus ∀x x2 = x. Unfortunately this clashes with the standard convention in set theory given above. The axioms are: The axioms for a distributive lattice (see above) ∀a a∧¬a = 0, ∀a a∨¬a =


id: list-of-first-order-theories-20260505-3
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-3.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
== Rings and fields == The signature of (unital) rings has two constants 0 and 1, two binary functions + and ×, and, optionally, one unary negation function −. Rings Axioms: Addition makes the ring into an abelian group, multiplication is associative and has an identity 1, and multiplication is left and right distributive. Commutative rings The axioms for rings plus ∀x ∀y xy = yx. Fields The axioms for commutative rings plus ∀x (¬ x = 0 → ∃y xy = 1) and ¬ 1 = 0. Many of the examples given here have only universal, or algebraic axioms. The class of structures satisfying such a theory has the property of being closed under substructure. For example, a subset of a group closed under the group actions of multiplication and inverse is again a group. Since the signature of fields does not usually include multiplicative and additive inverse, the axioms for inverses are not universal, and therefore a substructure of a field closed under addition and multiplication is not always a field. This can be remedied by adding unary inverse functions to the language. For any positive integer n the property that all equations of degree n have a


id: list-of-first-order-theories-20260505-4
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-4.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
∀ A ∃ b ∃ c ∃ d b A ∧ c A ∧ d A ∧ ¬ b = c ∧ ¬ b = d ∧ ¬ c = d {\displaystyle \forall A\exists b\exists c\exists d\;bA\land cA\land dA\land \lnot b=c\land \lnot b=d\land \lnot c=d} (Every line has at least 3 points) Euclid did not state all the axioms for Euclidean geometry explicitly, and the first complete list was given by Hilbert in Hilbert's axioms. This is not a first-order axiomatization as one of Hilbert's axioms is a second order completeness axiom. Tarski's axioms are a first-order axiomatization of Euclidean geometry. Tarski showed this axiom system is complete and decidable by relating it to the complete and decidable theory of real closed fields. == Differential algebra == The theory DF of differential fields. The signature is that of fields (0, 1, +, −, ×) together with a unary function ∂, the derivation. The axioms are those for fields together with ∀ u ∀ v ∂ ( u v ) = u ∂ v + v ∂ u {\displaystyle \forall u\forall v\,\partial (uv)=u\,\partial v+v\,\partial u} ∀ u ∀ v ∂ ( u + v ) = ∂ u + ∂ v


id: list-of-first-order-theories-20260505-5
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-5.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
∀x ¬ Sx = 0 ∀x ¬ x = 0 → ∃y Sy = x ∀x∀y Sx = Sy → x = y ∀x x + 0 = x ∀x∀y x + Sy = S(x + y) ∀x x × 0 = 0 ∀x∀y x × Sy = (x × y) + x. IΣn is first-order Peano arithmetic with induction restricted to Σn formulas (for n = 0, 1, 2, ...). The theory IΣ0 is often denoted by IΔ0. This is a series of more and more powerful fragments of Peano arithmetic. The case n = 1 has about the same strength as primitive recursive arithmetic (PRA). Exponential function arithmetic (EFA) is IΣ0 with an axiom stating that xy exists for all x and y (with the usual properties). First-order Peano arithmetic, PA. The "standard" theory of arithmetic. The axioms are the axioms of Robinson arithmetic above, together with the axiom scheme of induction: ϕ ( 0 ) ∧ ( ∀ x ϕ ( x ) → ϕ ( S x ) ) → ( ∀ x ϕ ( x ) ) {\displaystyle \phi (0)\wedge (\forall x\phi (x)\rightarrow \phi (Sx))\rightarrow (\forall x\phi (x))} for any formula φ in the language


id: list-of-formal-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formal_systems-0.md
category: reference
title: List of formal systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formal_systems
date_saved: 2026-05-05T08:14:21.897478+00:00
instance: kb-cron
parent: list-of-formal-systems-20260505-0
summary:
This is a list of formal systems, also known as logical calculi. == Mathematical == Functional calculus, a way to apply various types of functions to operators Matrix calculus, a specialized notation for multivariable calculus over spaces of matrices Umbral calculus, the combinatorics of certain operations on polynomials Vector calculus (also called vector analysis), comprising specialized notations for multivariable analysis of vectors in an inner-product space === Logical === Predicate calculus, specifies the rules of inference governing the logic of predicates Propositional calculus, specifies the rules of inference governing the logic of propositions === In theoretical computer science (Formal language) === Modal μ-calculus, a common temporal logic used by formal verification methods such as model checking Lambda calculus, a formulation of the theory of reflexive functions that has deep connections to computational theory Kappa calculus, a reformulation of the first-order fragment of typed lambda calculus Rho calculus, introduced as a general means to uniformly integrate rewriting into lambda calculus Process calculus, a set of approaches to formulating formal models of concurrent systems Ambient calculus, a family of models for concurrent systems based on the concept of agent mobility Join calculus, a theoretical model for the design of distributed programming languages


id: list-of-formulas-in-elementary-geometry-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formulas_in_elementary_geometry-0.md
category: reference
title: List of formulas in elementary geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_elementary_geometry
date_saved: 2026-05-05T08:14:23.334016+00:00
instance: kb-cron
parent: list-of-formulas-in-elementary-geometry-20260505-0
summary:
This is a short list of some common mathematical shapes and figures and the formulas that describe them. == Two-dimensional shapes == Sources: == Three-dimensional shapes == This is a list of volume formulas of basic shapes: Cone – 1 3 π r 2 h {\textstyle {\frac {1}{3}}\pi r^{2}h} , where r {\textstyle r} is the base's radius and h {\textstyle h} is the cone's height; Cube – a 3 {\textstyle a^{3}} , where a {\textstyle a} is the side's length; Cuboid – a b c {\textstyle abc} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the sides' length; Cylinder – π r 2 h {\textstyle \pi r^{2}h} , where r {\textstyle r} is the base's radius and h {\textstyle h} is the cylinder's height; Ellipsoid – 4 3 π a b c {\textstyle {\frac {4}{3}}\pi abc} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the semi-major and semi-minor axes' length; Sphere – 4 3 π r 3 {\textstyle {\frac {4}{3}}\pi r^{3}} , where r {\textstyle r} is the radius; Parallelepiped – a b c K {\textstyle abc{\sqrt {K}}} , where a {\textstyle a} , b


id: list-of-fourier-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Fourier_analysis_topics-0.md
category: reference
title: List of Fourier analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Fourier_analysis_topics
date_saved: 2026-05-05T08:14:24.703918+00:00
instance: kb-cron
parent: list-of-fourier-analysis-topics-20260505-0
summary:
This is a list of Fourier analysis topics. == Fourier analysis == Multiplier (Fourier analysis) Fourier shell correlation Pinsky phenomenon === Fourier series === Generalized Fourier series Regressive discrete Fourier series Gibbs phenomenon Sigma approximation Dini test Poisson summation formula Spectrum continuation analysis Convergence of Fourier series Half-range Fourier series === Fourier transforms === List of Fourier-related transforms Fourier transform on finite groups Fractional Fourier transform Continuous Fourier transform Fourier operator Fourier inversion theorem Sine and cosine transforms Parseval's theorem Paley–Wiener theorem Projection-slice theorem Frequency spectrum === Discrete Fourier transforms === Discrete Fourier series Non-uniform discrete Fourier transform DFT matrix Entropy influence conjecture Fast cosine transform Fast Fourier transform Cyclotomic fast Fourier transform Sparse Fourier transform === Applications of Fourier analysis === Fourier amplitude sensitivity testing Fourier optics Quantum mechanics, for Wavefunctions Uncertainty principle Quantum Fourier transform == Algebra and analysis == Periodic function Almost periodic function ATS theorem Modulus of continuity Banach algebra Compact group Haar measure Hardy space Sobolev space Topological group === Harmonic analysis === Set of uniqueness Pontryagin duality Plancherel theorem Peter–Weyl theorem Fourier integral operator Oscillatory integral operator === Harmonic functions === Laplace operator Laplace equation Dirichlet problem Unit circle Unit disc Spherical harmonic ===


id: list-of-fourier-related-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Fourier-related_transforms-0.md
category: reference
title: List of Fourier-related transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Fourier-related_transforms
date_saved: 2026-05-05T08:14:26.129378+00:00
instance: kb-cron
parent: list-of-fourier-related-transforms-20260505-0
summary:
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component. == Continuous transforms == Applied to functions of continuous arguments, Fourier-related transforms include: Two-sided Laplace transform Mellin transform, another closely related integral transform Laplace transform: the Fourier transform may be considered a special case of the imaginary axis of the bilateral Laplace transform Fourier transform, with special cases: Fourier series When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients. When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values


id: index-of-fractal-related-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_fractal-related_articles-0.md
category: reference
title: Index of fractal-related articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_fractal-related_articles
date_saved: 2026-05-05T08:14:27.328439+00:00
instance: kb-cron
parent: index-of-fractal-related-articles-20260505-0
summary:
This is a list of fractal topics, by Wikipedia page, See also list of dynamical systems and differential equations topics. 1/f noise Apollonian gasket Attractor Box-counting dimension Cantor distribution Cantor dust Cantor function Cantor set Cantor space Chaos theory Coastline Constructal theory Dimension Dimension theory Dragon curve Fatou set Fractal Fractal antenna Fractal art Fractal compression Fractal flame Fractal landscape Fractal transform Fractint Graftal Iterated function system Horseshoe map How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Julia set Koch snowflake L-system Lebesgue covering dimension Lévy C curve Lévy flight List of fractals by Hausdorff dimension Lorenz attractor Lyapunov fractal Mandelbrot set Menger sponge Minkowski–Bouligand dimension Multifractal analysis Olbers' paradox Perlin noise Power law Rectifiable curve Scale-free network Self-similarity Sierpinski carpet Sierpiński curve Sierpinski triangle Space-filling curve T-square (fractal) Topological dimension


id: list-of-fractals-by-hausdorff-dimension-20260505-0
file: data/en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension-0.md
category: reference
title: List of fractals by Hausdorff dimension
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
date_saved: 2026-05-05T08:14:28.578190+00:00
instance: kb-cron
parent: list-of-fractals-by-hausdorff-dimension-20260505-0
summary:
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension. == Deterministic fractals == == Random and natural fractals == == See also == Fractal dimension Hausdorff dimension Scale invariance == Notes and references == == Further reading == Mandelbrot, Benoît (1982). The Fractal Geometry of Nature. W.H. Freeman. ISBN 0-7167-1186-9. Peitgen, Heinz-Otto (1988). Saupe, Dietmar (ed.). The Science of Fractal Images. Springer Verlag. ISBN 0-387-96608-0. Barnsley, Michael F. (1 January 1993). Fractals Everywhere. Morgan Kaufmann. ISBN 0-12-079061-0. Sapoval, Bernard; Mandelbrot, Benoît B. (2001). Universalités et fractales: jeux d'enfant ou délits d'initié?. Flammarion-Champs. ISBN 2-08-081466-4. == External links == The fractals on Mathworld Other fractals on Paul Bourke's website Soler's Gallery Fractals on mathcurve.com 1000fractales.free.fr - Project gathering fractals created with various software Fractals unleashed IFStile - software that computes the dimension of the boundary of self-affine tiles


id: list-of-functional-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_functional_analysis_topics-0.md
category: reference
title: List of functional analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_functional_analysis_topics
date_saved: 2026-05-05T08:14:29.761801+00:00
instance: kb-cron
parent: list-of-functional-analysis-topics-20260505-0
summary:
This is a list of functional analysis topics. See also: Glossary of functional analysis. == Hilbert space == == Functional analysis, classic results == == Operator theory == == Banach space examples == Lp space Hardy space Sobolev space Tsirelson space ba space == Real and complex algebras == == Topological vector spaces == == Amenability == Amenable group Von Neumann conjecture == Wavelets == == Quantum theory == See also list of mathematical topics in quantum theory == Probability == Free probability Bernstein's theorem == Non-linear == Fixed-point theorems in infinite-dimensional spaces == History == Stefan Banach (1892–1945) Hugo Steinhaus (1887–1972) John von Neumann (1903-1957) Alain Connes (born 1947) Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis Earliest Known Uses of Some of the Words of Mathematics: Matrices and Linear Algebra


id: list-of-mathematical-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_functions-0.md
category: reference
title: List of mathematical functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_functions
date_saved: 2026-05-05T08:14:30.991878+00:00
instance: kb-cron
parent: list-of-mathematical-functions-20260505-0
summary:
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics. A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are "anonymous", with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations. See also List of types of functions == Elementary functions == Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...) === Algebraic functions === Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer (or equivalently rational) coefficients. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree polynomial. Quintic function: Fifth degree polynomial. Rational functions: A ratio


id: list-of-types-of-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_functions-0.md
category: reference
title: List of types of functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_functions
date_saved: 2026-05-05T08:14:32.235773+00:00
instance: kb-cron
parent: list-of-types-of-functions-20260505-0
summary:
In mathematics, functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. == Relative to set theory == These properties concern the domain, the codomain and the image of functions. Injective function: has a distinct value for each distinct input. Also called an injection or, sometimes, one-to-one function. In other words, every element of the function's codomain is the image of at most one element of its domain. Surjective function: has a preimage for every element of the codomain, that is, the codomain equals the image. Also called a surjection or onto function. Bijective function: is both an injection and a surjection, and thus invertible. Identity function: maps any given element to itself. Constant function: has a fixed value regardless of its input. Empty function: whose domain equals the empty set. Set function: whose input is a set. Set-valued function: whose values are sets. Choice function called also selector or uniformizing function: assigns to each set one of its elements. == Relative to an operator (c.q. a group or other structure) == These properties concern how the function is affected by arithmetic operations on its argument. The following are special examples


id: list-of-types-of-functions-20260505-1
file: data/en.wikipedia.org/wiki/List_of_types_of_functions-1.md
category: reference
title: List of types of functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_functions
date_saved: 2026-05-05T08:14:32.235773+00:00
instance: kb-cron
parent: list-of-types-of-functions-20260505-0
summary:
== Ways of defining functions/relation to type theory == Polynomial function: defined by evaluating a polynomial. Linear function; also affine function. Quadratic function Cubic function Quartic function Quintic function Rational function: ratio of two polynomial functions. In particular, Möbius transformation called also linear fractional function. Algebraic function: defined as the root of a polynomial equation. Transcendental function: analytic but not algebraic. Also hypertranscendental function. Composite function: is formed by the composition of two functions f and g, by mapping x to f (g(x)). Inverse function: is declared by "doing the reverse" of a given function (e.g. arcsine is the inverse of sine). Implicit function: defined implicitly by a relation between the argument(s) and the value. Piecewise function: is defined by different expressions on different intervals. Computable function: an algorithm can do the job of the function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying the name of a dependent variable, and a way of calculating what it should map to. For this purpose, the ↦ {\displaystyle \mapsto } symbol or Church's λ {\displaystyle \lambda } is often used. Also, sometimes mathematicians notate a function's domain and codomain by writing e.g. f


id: list-of-examples-in-general-topology-20260505-0
file: data/en.wikipedia.org/wiki/List_of_examples_in_general_topology-0.md
category: reference
title: List of examples in general topology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_examples_in_general_topology
date_saved: 2026-05-05T08:14:33.479770+00:00
instance: kb-cron
parent: list-of-examples-in-general-topology-20260505-0
summary:
This is a list of useful examples in general topology, a field of mathematics. Alexandrov topology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open topology Compactification Discrete topology Double-pointed cofinite topology Extended real number line Finite topological space Hawaiian earring Hilbert cube Irrational cable on a torus Lakes of Wada Long line Order topology Lexicographical/dictionary order Ordinal number topology Real line Split interval Overlapping interval topology Moore plane Sierpiński space Sorgenfrey line Sorgenfrey plane Space-filling curve Topologist's sine curve Trivial topology Unit interval Zariski topology == See also == Counterexamples in Topology π-Base: An Interactive Encyclopedia of Topological Spaces


id: list-of-genetic-algorithm-applications-20260505-0
file: data/en.wikipedia.org/wiki/List_of_genetic_algorithm_applications-0.md
category: reference
title: List of genetic algorithm applications
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_genetic_algorithm_applications
date_saved: 2026-05-05T08:14:34.797128+00:00
instance: kb-cron
parent: list-of-genetic-algorithm-applications-20260505-0
summary:
This is a list of genetic algorithm (GA) applications. == Natural Sciences, Mathematics and Computer Science == Bayesian inference links to particle methods in Bayesian statistics and hidden Markov chain models Artificial creativity Chemical kinetics (gas and solid phases) Calculation of bound states and local-density approximations Code-breaking, using the GA to search large solution spaces of ciphers for the one correct decryption. Computer architecture: using GA to find out weak links in approximate computing such as lookahead. Configuration applications, particularly physics applications of optimal molecule configurations for particular systems like C60 (buckyballs) Construction of facial composites of suspects by eyewitnesses in forensic science. Data Center/Server Farm. Distributed computer network topologies Electronic circuit design, known as evolvable hardware Evolutionary image processing Feature selection for Machine Learning Feynman-Kac models File allocation for a distributed system Filtering and signal processing Finding hardware bugs. Game theory equilibrium resolution Genetic Algorithm for Rule Set Production Scheduling applications, including job-shop scheduling and scheduling in printed circuit board assembly. The objective being to schedule jobs in a sequence-dependent or non-sequence-dependent setup environment in order to maximize the volume of production while minimizing penalties such as tardiness. Satellite communication scheduling for the NASA Deep Space Network was shown


id: list-of-geodesic-polyhedra-and-goldberg-polyhedra-20260505-0
file: data/en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra-0.md
category: reference
title: List of geodesic polyhedra and Goldberg polyhedra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra
date_saved: 2026-05-05T08:14:36.120154+00:00
instance: kb-cron
parent: list-of-geodesic-polyhedra-and-goldberg-polyhedra-20260505-0
summary:
This is a list of selected geodesic polyhedra and Goldberg polyhedra, two infinite classes of polyhedra. Geodesic polyhedra and Goldberg polyhedra are duals of each other. The geodesic and Goldberg polyhedra are parameterized by integers m and n, with m > 0 {\displaystyle m>0} and n ≥ 0 {\displaystyle n\geq 0} . T is the triangulation number, which is equal to T = m 2 + m n + n 2 {\displaystyle T=m^{2}+mn+n^{2}} . == Icosahedral == == Octahedral == == Tetrahedral == == References == Wenninger, Magnus (1979), Spherical Models, Cambridge University Press, ISBN 978-0-521-29432-4, MR 0552023, archived from the original on July 4, 2008 Reprinted by Dover 1999 ISBN 978-0-486-40921-4


id: outline-of-geometry-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_geometry-0.md
category: reference
title: Outline of geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_geometry
date_saved: 2026-05-05T08:14:37.296179+00:00
instance: kb-cron
parent: outline-of-geometry-20260505-0
summary:
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Modern geometry also extends into non-Euclidean spaces, topology, and fractal dimensions, bridging pure mathematics with applications in physics, computer science, and data visualization. == Types, methodologies, and terminologies of geometry. == Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive solid geometry Contact geometry Convex geometry Descriptive geometry Differential geometry Digital geometry Discrete geometry Distance geometry Elliptic geometry Enumerative geometry Epipolar geometry Euclidean geometry Finite geometry Fractal geometry Geometry of numbers Hyperbolic geometry Incidence geometry Information geometry Integral geometry Inversive geometry Inversive ring geometry Klein geometry Lie sphere geometry Non-Euclidean geometry Noncommutative algebraic geometry Noncommutative geometry Ordered geometry Parabolic geometry Plane geometry Projective geometry Quantum geometry Riemannian geometry Ruppeiner geometry Solid geometry Spherical geometry Symplectic geometry Synthetic geometry Systolic geometry Taxicab geometry Toric geometry Transformation geometry Tropical geometry == General geometry concepts == === General concepts === Chirality Handedness Relative direction Mirror image Coordinate-free treatment Four-dimensional space Infinitesimal transformation Geometric progression Geometric shape Pi Angular velocity Linear velocity De Moivre's theorem Similar triangles


id: graduate-studies-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Graduate_Studies_in_Mathematics-0.md
category: reference
title: Graduate Studies in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Studies_in_Mathematics
date_saved: 2026-05-05T08:14:38.533221+00:00
instance: kb-cron
parent: graduate-studies-in-mathematics-20260505-0
summary:
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published in hardcover and e-book formats. == List of books == == See also == Graduate Texts in Mathematics == Notes == == External links == Graduate Studies in Mathematics on AMS Bookstore ISSN 1065-7339


id: graduate-texts-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-0.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level.


id: graduate-texts-in-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-1.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
== List of books == Introduction to Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ISBN 978-1-4613-8170-9) Measure and Category – A Survey of the Analogies between Topological and Measure Spaces, John C. Oxtoby (1980, 2nd ed., ISBN 978-0-387-90508-2) Topological Vector Spaces, H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ISBN 978-0-387-98726-2) A Course in Homological Algebra, Peter Hilton, Urs Stammbach (1997, 2nd ed., ISBN 978-0-387-94823-2) Categories for the Working Mathematician, Saunders Mac Lane (1998, 2nd ed., ISBN 978-0-387-98403-2) Projective Planes, Daniel R. Hughes, Fred C. Piper, (1982, ISBN 978-3-540-90043-6) A Course in Arithmetic, Jean-Pierre Serre (1996, ISBN 978-0-387-90040-7) Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring, (1973, ISBN 978-3-540-90050-4) Introduction to Lie Algebras and Representation Theory, James E. Humphreys (1997, ISBN 978-0-387-90053-7) A Course in Simple-Homotopy Theory, Marshall. M. Cohen, (1973, ISBN 0-387-90056-X) Functions of One Complex Variable I, John B. Conway (1978, 2nd ed., ISBN 978-0-387-90328-6) Advanced Mathematical Analysis, Richard Beals (1973, ISBN 978-0-387-90065-0) Rings and Categories of Modules, Frank W. Anderson, Kent R. Fuller (1992, 2nd ed., ISBN 978-0-387-97845-1) Stable Mappings and Their Singularities, Martin Golubitsky, Victor Guillemin, (1974, ISBN 978-0-387-90072-8) Lectures in Functional Analysis and Operator Theory, Sterling K. Berberian, (1974, ISBN


id: graduate-texts-in-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-2.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Koblitz (1993, 2nd ed., ISBN 978-0-387-97966-3) Representations of Compact Lie Groups, Theodor Bröcker, Tammo tom Dieck (1985, ISBN 978-3-540-13678-1) Finite Reflection Groups, L.C. Grove, C.T. Benson (1985, 2nd ed., ISBN 978-0-387-96082-1) Harmonic Analysis on Semigroups – Theory of Positive Definite and Related Functions, Christian Berg, Jens Peter Reus Christensen, Paul Ressel (1984, ISBN 978-0-387-90925-7) Galois Theory, Harold M. Edwards (1984, ISBN 978-0-387-90980-6) Lie Groups, Lie Algebras, and Their Representations, V. S. Varadarajan (1984, ISBN 978-0-387-90969-1) Complex Analysis, Serge Lang (1999, 4th ed., ISBN 978-0-387-98592-3) Modern Geometry — Methods and Applications Part II: The Geometry and Topology of Manifolds, B. A. Dubrovin, Anatoly Timofeevich Fomenko, Sergei Novikov (1985, ISBN 978-0-387-96162-0) SL2(R), Serge Lang (1985, ISBN 978-0-387-96198-9) The Arithmetic of Elliptic Curves, Joseph H. Silverman (2009, 2nd ed., ISBN 978-0-387-09493-9) Applications of Lie Groups to Differential Equations, Peter J. Olver (2nd ed., 1993, ISBN 978-1-4684-0276-6) Holomorphic Functions and Integral Representations in Several Complex Variables, R. Michael Range (1986, ISBN 978-0-387-96259-7) Univalent Functions and Teichmüller Spaces, O. Lehto (1987, ISBN 978-1-4613-8654-4) Algebraic Number Theory, Serge Lang (1994, 2nd ed., ISBN 978-0-387-94225-4) Elliptic Curves, Dale Husemöller (2004, 2nd ed., ISBN 978-0-387-95490-5) Elliptic Functions, Serge Lang (1987, 2nd ed., ISBN 978-0-387-96508-6) Brownian Motion and Stochastic


id: graduate-texts-in-mathematics-20260505-3
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-3.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Robert (2000, ISBN 978-0-387-98669-2) Theory of Bergman Spaces, Hakan Hedenmalm, Boris Korenblum, Kehe Zhu (2000, ISBN 978-0-387-98791-0) An Introduction to Riemann–Finsler Geometry, David Bao, Shiing-Shen Chern, Zhongmin Shen (2000, ISBN 978-1-4612-7070-6) Diophantine Geometry, Marc Hindry, Joseph H. Silverman (2000, ISBN 978-0-387-98975-4) Introduction to Topological Manifolds, John M. Lee (2011, 2nd ed., ISBN 978-1-4419-7939-1) The Symmetric Group — Representations, Combinatorial Algorithms, and Symmetric Functions, Bruce E. Sagan (2001, 2nd ed., ISBN 978-0-387-95067-9) Galois Theory, Jean-Pierre Escofier (2001, ISBN 978-0-387-98765-1) Rational Homotopy Theory, Yves Félix, Stephen Halperin, Jean-Claude Thomas (2000, ISBN 978-0-387-95068-6) Problems in Analytic Number Theory, M. Ram Murty (2007, 2nd ed., ISBN 978-0-387-95143-0) Algebraic Graph Theory, Chris Godsil, Gordon Royle (2001, ISBN 978-0-387-95241-3) Analysis for Applied Mathematics, Ward Cheney (2001, ISBN 978-0-387-95279-6) A Short Course on Spectral Theory, William Arveson (2002, ISBN 978-0-387-95300-7) Number Theory in Function Fields, Michael Rosen (2002, ISBN 978-0-387-95335-9) Algebra, Serge Lang (2002, Revised 3rd ed., ISBN 978-0-387-95385-4) Lectures on Discrete Geometry, Jiří Matoušek (2002, ISBN 978-0-387-95374-8) From Holomorphic Functions to Complex Manifolds, Klaus Fritzsche, Hans Grauert (2002, ISBN 978-0-387-95395-3) Partial Differential Equations, Jürgen Jost, (2013, 3rd ed., ISBN 978-1-4614-4808-2) Algebraic Functions and Projective Curves, David M. Goldschmidt, (2003, ISBN 978-0-387-95432-5) Matrices — Theory and Applications,


id: graduate-texts-in-mathematics-20260505-4
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-4.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Getz, Heekyoung Hahn (2024, ISBN 978-3-031-41151-9) An Invitation to Mathematical Logic, David Marker (2024, ISBN 978-3-031-55367-7) Fundamentals of Fourier Analysis, Loukas Grafakos (2024, ISBN 978-3-031-56499-4) A Course in Real Algebraic Geometry, Claus Scheiderer (2024, ISBN 978-3-031-69212-3) An Introduction to Microlocal Analysis, Peter Hintz (2025, ISBN 978-3-031-90705-0) Complex Analysis in One Variable and Riemann Surfaces, Mei-Chi Shaw, Charles M. Stanton (2025, ISBN 978-3-031-93641-8) Metric Lie Groups: Carnot-Carathéodory Spaces from the Homogeneous Viewpoint, Enrico Le Donne (2025, ISBN 978-3-031-98831-8) From Classical L-Functions to Modern Reciprocity Laws, Claus Sorensen (2025, ISBN 978-3-032-03034-4) Unitary Representations and Unitary Duals, Manfred Einsiedler, Thomas Ward (2025, ISBN 978-3-032-03898-2) Commutative Algebra, Shiro Goto, Kei-ichi Watanabe (2025, ISBN 978-981-95-4226-0) == See also == Graduate Studies in Mathematics == Notes == == External links == Springer-Verlag's Summary of Graduate Texts in Mathematics


id: list-of-graph-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_graph_theory_topics-0.md
category: reference
title: List of graph theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_graph_theory_topics
date_saved: 2026-05-05T08:14:41.041813+00:00
instance: kb-cron
parent: list-of-graph-theory-topics-20260505-0
summary:
This is a list of graph theory topics, by Wikipedia page. See glossary of graph theory for basic terminology. == Examples and types of graphs == == Graph coloring == == Paths and cycles == == Trees == === Terminology === Node Child node Parent node Leaf node Root node Root (graph theory) === Operations === Tree structure Tree data structure Cayley's formula Kőnig's lemma Tree (set theory) (need not be a tree in the graph-theory sense, because there may not be a unique path between two vertices) Tree (descriptive set theory) Euler tour technique == Graph limits == Graphon == Graphs in logic == Conceptual graph Entitative graph Existential graph Laws of Form Logical graph == Mazes and labyrinths == Labyrinth Maze Maze generation algorithm == Algorithms == Ant colony algorithm Breadth-first search Depth-first search Depth-limited search FKT algorithm Flood fill Graph exploration algorithm Matching (graph theory) Max flow min cut theorem Maximum-cardinality search Shortest path Dijkstra's algorithm Bellman–Ford algorithm A* algorithm Floyd–Warshall algorithm Topological sorting Pre-topological order == Other topics == == Networks, network theory == See list of network theory topics == Hypergraphs == Helly family Intersection (Line) Graphs of hypergraphs


id: list-of-graphs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_graphs-0.md
category: reference
title: List of graphs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_graphs
date_saved: 2026-05-05T08:14:43.462282+00:00
instance: kb-cron
parent: list-of-graphs-20260505-0
summary:
This partial list of graphs contains definitions of graphs and graph families. For collected definitions of graph theory terms that do not refer to individual graph types, such as vertex and path, see Glossary of graph theory. For links to existing articles about particular kinds of graphs, see Category:Graphs. Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different contexts. == Individual graphs == == Highly symmetric graphs == === Strongly regular graphs === The strongly regular graph on v vertices and rank k is usually denoted srg(v,k,λ,μ). === Symmetric graphs === A symmetric graph is one in which there is a symmetry (graph automorphism) taking any ordered pair of adjacent vertices to any other ordered pair; the Foster census lists all small symmetric 3-regular graphs. Every strongly regular graph is symmetric, but not vice versa. === Semi-symmetric graphs === == Graph families == === Complete graphs === The complete graph on n {\displaystyle n} vertices is often called the n {\displaystyle n} -clique


id: list-of-group-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_group_theory_topics-0.md
category: reference
title: List of group theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_group_theory_topics
date_saved: 2026-05-05T08:14:44.910624+00:00
instance: kb-cron
parent: list-of-group-theory-topics-20260505-0
summary:
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. == Structures and operations == == Basic properties of groups == === Group homomorphisms === == Basic types of groups == === Simple groups and their classification === Classification of finite simple groups === Permutation and symmetry groups === == Concepts groups share with other mathematics == == Mathematical objects making use of a group operation == == Mathematical


id: list-of-small-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_small_groups-0.md
category: reference
title: List of small groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_small_groups
date_saved: 2026-05-05T08:14:46.116505+00:00
instance: kb-cron
parent: list-of-small-groups-20260505-0
summary:
The following list in mathematics contains the finite groups of small order up to group isomorphism. == Counts == For n = 1, 2, … the number of nonisomorphic groups of order n is 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... (sequence A000001 in the OEIS) For labeled groups, see (sequence A034383 in the OEIS). == Glossary == Each group is named by Small Groups Library as Goi, where o is the order of the group, and i is the index used to label the group within that order. Common group names: Zn: the cyclic group of order n (the notation Cn is also used; it is isomorphic to the additive group of Z/nZ) Dihn: the dihedral group of order 2n (often the notation Dn or D2n is used) Dn: the dihedral group of order 2n, the same as Dihn (notation used in section List of small non-abelian groups) K4: the Klein four-group of order 4, isomorphic to Z2 × Z2 and Dih2 Sn: the symmetric group of degree n, containing the n! permutations of n elements An: the alternating group of degree n, containing the


id: list-of-harmonic-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_harmonic_analysis_topics-0.md
category: reference
title: List of harmonic analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_harmonic_analysis_topics
date_saved: 2026-05-05T08:14:47.331318+00:00
instance: kb-cron
parent: list-of-harmonic-analysis-topics-20260505-0
summary:
This is a list of harmonic analysis topics. See also list of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards the classical Fourier series and Fourier transform of mathematical analysis, mathematical physics and engineering. == Fourier analysis == === Fourier series === Periodic function Trigonometric function Trigonometric polynomial Exponential sum Dirichlet kernel Fejér kernel Gibbs phenomenon Parseval's identity Parseval's theorem Weyl differintegral Generalized Fourier series Orthogonal functions Orthogonal polynomials Empirical orthogonal functions Set of uniqueness === Fourier transform === Continuous Fourier transform Fourier inversion theorem Plancherel's theorem Convolution Convolution theorem Positive-definite function Poisson summation formula Paley–Wiener theorem Sobolev space Time–frequency representation Quantum Fourier transform == Topological groups == Topological abelian group Haar measure Discrete Fourier transform Dirichlet character Amenable group Von Neumann's conjecture Pontryagin duality Kronecker's theorem on diophantine approximation Almost periodic function Bohr compactification Wiener's tauberian theorem == Representation theory == Representation of a Lie group Unitary representation Irreducible representation Harish-Chandra character Restricted representation Induced representation Peter–Weyl theorem Spherical harmonic Casimir operator Hecke operator Stone–von Neumann theorem Discrete series representation Tempered representation Langlands program == Fast Fourier transform == Bluestein's FFT algorithm Cooley–Tukey FFT algorithm Rader's FFT algorithm Number-theoretic transform Irrational base discrete weighted


id: highly-composite-number-20260505-0
file: data/en.wikipedia.org/wiki/Highly_composite_number-0.md
category: reference
title: Highly composite number
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Highly_composite_number
date_saved: 2026-05-05T08:15:00.548879+00:00
instance: kb-cron
parent: highly-composite-number-20260505-0
summary:
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N. For example, 6 is highly composite because d(6) = 4, and for n = 1,2,3,4,5, you get d(n) = 1,2,2,3,2, respectively, which are all less than 4. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. Ramanujan wrote a paper on highly composite numbers in 1915. The mathematician Jean-Pierre Kahane suggested that Plato must have known about highly composite numbers as he deliberately chose such a number, 5040 (= 7!), as the ideal number of citizens in a city. Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040. == Examples == The first 41 highly composite numbers are listed in the table below (sequence A002182 in the


id: highly-composite-number-20260505-1
file: data/en.wikipedia.org/wiki/Highly_composite_number-1.md
category: reference
title: Highly composite number
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Highly_composite_number
date_saved: 2026-05-05T08:15:00.548879+00:00
instance: kb-cron
parent: highly-composite-number-20260505-0
summary:
lim sup x → ∞ log ⁡ Q ( x ) log ⁡ log ⁡ x ≤ 1.71 . {\displaystyle \limsup _{x\,\to \,\infty }{\frac {\log Q(x)}{\log \log x}}\leq 1.71\ .} == Related sequences == Highly composite numbers greater than 6 are also abundant numbers. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first highly composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800. 10 of the first 38 highly composite numbers are superior highly composite numbers. The sequence of highly composite numbers (sequence A002182 in the OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS). Highly composite numbers whose number of divisors is also a highly composite number are 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200 (sequence A189394 in the OEIS). It is known that this sequence is complete. A positive integer n is a


id: list-of-mathematics-history-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics_history_topics-0.md
category: reference
title: List of mathematics history topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics_history_topics
date_saved: 2026-05-05T08:15:01.857681+00:00
instance: kb-cron
parent: list-of-mathematics-history-topics-20260505-0
summary:
This is a list of mathematics history topics, by Wikipedia page. See also list of mathematicians, timeline of mathematics, history of mathematics, list of publications in mathematics. 1729 (anecdote) Adequality Archimedes Palimpsest Archimedes' use of infinitesimals Arithmetization of analysis Brachistochrone curve Chinese mathematics Cours d'Analyse Edinburgh Mathematical Society Erlangen programme Fermat's Last Theorem Greek mathematics Thomas Little Heath Hilbert's problems History of topos theory Hyperbolic quaternion Indian mathematics Islamic mathematics Italian school of algebraic geometry Kraków School of Mathematics Law of Continuity Lwów School of Mathematics Nicolas Bourbaki Non-Euclidean geometry Scottish Café Seven bridges of Königsberg Spectral theory Synthetic geometry Tautochrone curve Unifying theories in mathematics Waring's problem Warsaw School of Mathematics == Academic positions == Lowndean Professor of Astronomy and Geometry Lucasian professor Rouse Ball Professor of Mathematics Sadleirian Chair == See also == Mathematics portal


id: list-of-homological-algebra-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_homological_algebra_topics-0.md
category: reference
title: List of homological algebra topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_homological_algebra_topics
date_saved: 2026-05-05T08:15:03.084460+00:00
instance: kb-cron
parent: list-of-homological-algebra-topics-20260505-0
summary:
This is a list of homological algebra topics, by Wikipedia page. Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology. == Basic techniques == Cokernel Exact sequence Chain complex Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma Projective module Injective module Projective resolution Injective resolution Koszul complex Exact functor Derived functor Ext functor Tor functor Filtration (abstract algebra) Spectral sequence Abelian category Triangulated category Derived category == Applications == Group cohomology Galois cohomology Lie algebra cohomology Sheaf cohomology Whitehead problem Homological conjectures in commutative algebra


id: list-of-mathematical-identities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_identities-0.md
category: reference
title: List of mathematical identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_identities
date_saved: 2026-05-05T08:15:04.285513+00:00
instance: kb-cron
parent: list-of-mathematical-identities-20260505-0
summary:
This article lists mathematical identities, that is, identically true relations holding in mathematics. Binet-cauchy identity Binomial inverse theorem Binomial identity Brahmagupta–Fibonacci two-square identity Candido's identity Cassini and Catalan identities Degen's eight-square identity Difference of two squares Euler's four-square identity Euler's identity Fibonacci's identity see Brahmagupta–Fibonacci identity or Cassini and Catalan identities Heine's identity Hermite's identity Lagrange's identity Lagrange's trigonometric identities List of logarithmic identities MacWilliams identity Matrix determinant lemma Newton's identity Parseval's identity Pfister's sixteen-square identity Sherman–Morrison formula Sophie Germain identity Sun's curious identity Sylvester's determinant identity Vandermonde's identity Woodbury matrix identity == Identities for classes of functions == Exterior calculus identities Fibonacci identities: Combinatorial Fibonacci identities and Other Fibonacci identities Hypergeometric function identities List of integrals of logarithmic functions List of topics related to π List of trigonometric identities Inverse trigonometric functions Logarithmic identities Summation identities Vector calculus identities == See also == List of inequalities List of set identities and relations – Equalities for combinations of sets == External links == A Collection of Algebraic Identities Matrix Identities


id: list-of-impossible-puzzles-20260505-0
file: data/en.wikipedia.org/wiki/List_of_impossible_puzzles-0.md
category: reference
title: List of impossible puzzles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_impossible_puzzles
date_saved: 2026-05-05T08:15:05.485545+00:00
instance: kb-cron
parent: list-of-impossible-puzzles-20260505-0
summary:
This is a list of puzzles that cannot be solved. An impossible puzzle is a puzzle that cannot be resolved, either due to lack of sufficient information, or any number of logical impossibilities. 15 Puzzle – Slide fifteen numbered tiles into numerical order. It is impossible to solve in half of the starting positions. Five room puzzle – Cross each wall of a diagram exactly once with a continuous line. MU puzzle – Transform the string MI to MU according to a set of rules. Mutilated chessboard problem – Place 31 dominoes of size 2×1 on a chessboard with two opposite corners removed. Coloring the edges of the Petersen graph with three colors. Seven Bridges of Königsberg – Walk through a city while crossing each of seven bridges exactly once. Squaring the circle, the impossible problem of constructing a square with the same area as a given circle, using only a compass and straightedge. Three cups problem – Turn three cups right-side up after starting with one wrong and turning two at a time. Three utilities problem – Connect three cottages to gas, water, and electricity without crossing lines. Thirty-six officers problem – Arrange six regiments consisting of six officers


id: list-of-incomplete-proofs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_incomplete_proofs-0.md
category: reference
title: List of incomplete proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_incomplete_proofs
date_saved: 2026-05-05T08:15:06.686663+00:00
instance: kb-cron
parent: list-of-incomplete-proofs-20260505-0
summary:
This page lists notable examples of incomplete or incorrect published mathematical proofs. Most of these were accepted as complete or correct for several years but later discovered to contain gaps or errors. There are both examples where a complete proof was later found, or where the alleged result turned out to be false. == Results later proved rigorously == Euclid's Elements. Euclid's proofs are essentially correct, but strictly speaking sometimes contain gaps because he tacitly uses some unstated assumptions, such as the existence of intersection points. In 1899 David Hilbert gave a complete set of (second order) axioms for Euclidean geometry, called Hilbert's axioms, and between 1926 and 1959 Tarski gave some complete sets of first order axioms, called Tarski's axioms. Isoperimetric inequality. For three dimensions it states that the shape enclosing the maximum volume for its surface area is the sphere. It was formulated by Archimedes but not proved rigorously until the 19th century, by Hermann Schwarz. Infinitesimals. In the 18th century there was widespread use of infinitesimals in calculus, though these were not really well defined. Calculus was put on firm foundations in the 19th century, and Robinson put infinitesimals in a rigorous basis with the introduction of


id: list-of-incomplete-proofs-20260505-1
file: data/en.wikipedia.org/wiki/List_of_incomplete_proofs-1.md
category: reference
title: List of incomplete proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_incomplete_proofs
date_saved: 2026-05-05T08:15:06.686663+00:00
instance: kb-cron
parent: list-of-incomplete-proofs-20260505-0
summary:
== Incorrect results == In 1759 Euler claimed that there were no closed knight tours on a chess board with 3 rows, but in 1917 Ernest Bergholt found tours on 3 × 10 and 3 × 12 boards. Euler's conjecture on Graeco-Latin squares. In the 1780s Euler conjectured that no such squares exist for any oddly even number n ≡ 2 (mod 4). In 1959, R. C. Bose and S. S. Shrikhande constructed counterexamples of order 22. Then E. T. Parker found a counterexample of order 10 using a one-hour computer search. Finally Parker, Bose, and Shrikhande showed this conjecture to be false for all n ≥ 10. In 1798 A. M. Legendre claimed that 6 is not the sum of 2 rational cubes, which as Lamé pointed out in 1865 is false as 6 = (37/21)3 + (17/21)3. In 1803, Gian Francesco Malfatti claimed to prove that a certain arrangement of three circles would cover the maximum possible area inside a right triangle. However, to do so he made certain unwarranted assumptions about the configuration of the circles. It was shown in 1930 that circles in a different configuration could cover a greater area, and in 1967 that Malfatti's


id: list-of-incomplete-proofs-20260505-2
file: data/en.wikipedia.org/wiki/List_of_incomplete_proofs-2.md
category: reference
title: List of incomplete proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_incomplete_proofs
date_saved: 2026-05-05T08:15:06.686663+00:00
instance: kb-cron
parent: list-of-incomplete-proofs-20260505-0
summary:
This caused an error in the title of Janko's paper A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroup on J4: it does not have the full covering group as a subgroup, as the full covering group is larger than was realized at the time. The original statement of the classification of N-groups by Thompson in 1968 accidentally omitted the Tits group, though he soon fixed this. In 1975, Leitzel, Madan, and Queen incorrectly claimed that there are only 7 function fields over finite fields with genus > 0 and class number 1, but in 2013 Stirpe found another; there are in fact exactly 8. Busemann–Petty problem. Zhang published two papers in the Annals of Mathematics in 1994 and 1999, in the first of which he proved that the Busemann–Petty problem in R4 has a negative solution, and in the second of which he proved that it has a positive solution. Algebraic stacks. The book Laumon & Moret-Bailly (2000) on algebraic stacks mistakenly claimed that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. The results depending on this were repaired by Olsson (2007).


id: list-of-incomplete-proofs-20260505-3
file: data/en.wikipedia.org/wiki/List_of_incomplete_proofs-3.md
category: reference
title: List of incomplete proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_incomplete_proofs
date_saved: 2026-05-05T08:15:06.686663+00:00
instance: kb-cron
parent: list-of-incomplete-proofs-20260505-0
summary:
== Theories proven inconsistent == Frege's foundations of mathematics in his 1879 book Begriffsschrift turned out to be inconsistent because of Russell's paradox, found in 1901. Church's original published attempt in 1932 to define a formal system was inconsistent, as was his correction in 1933. The consistent part of his system later became the lambda calculus. Quine published his original description of the system Mathematical Logic in 1940, but in 1942 Rosser showed it was inconsistent. Wang found a correction in 1950; the consistency of this revised system is still unclear. In 1967 Reinhardt proposed Reinhardt cardinals, which Kunen showed to be inconsistent with ZFC in 1971, though they are not known to be inconsistent with ZF. Per Martin-Löf's original version of intuitionistic type theory proposed in 1971 was shown to be inconsistent by Jean-Yves Girard in 1972, and was replaced by a corrected version. == Status unclear == Uniform convergence. In his Cours d'Analyse of 1821, Cauchy "proved" that if a sum of continuous functions converges pointwise, then its limit is also continuous. However, Abel observed in 1826 that this is not the case. For the conclusion to hold, "pointwise convergence" must be replaced with "uniform convergence". It is


id: indefinite-product-20260505-0
file: data/en.wikipedia.org/wiki/Indefinite_product-0.md
category: reference
title: Indefinite product
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Indefinite_product
date_saved: 2026-05-05T08:15:08.101960+00:00
instance: kb-cron
parent: indefinite-product-20260505-0
summary:
In mathematics, the indefinite product operator is the inverse operator of Q ( f ( x ) ) = f ( x + 1 ) f ( x ) {\textstyle Q(f(x))={\frac {f(x+1)}{f(x)}}} . It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Thus Q ( ∏ x f ( x ) ) = f ( x ) . {\displaystyle Q\left(\prod _{x}f(x)\right)=f(x)\,.} More explicitly, if ∏ x f ( x ) = F ( x ) {\textstyle \prod _{x}f(x)=F(x)} , then F ( x + 1 ) F ( x ) = f ( x ) . {\displaystyle {\frac {F(x+1)}{F(x)}}=f(x)\,.} If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. Therefore, each indefinite product actually represents a family of functions, differing by a multiplicative constant. == Period rule == If T {\displaystyle T} is a period of function f ( x ) {\displaystyle f(x)} then ∏ x f ( T x ) = C f ( T x ) x − 1 {\displaystyle \prod _{x}f(Tx)=Cf(Tx)^{x-1}} == Connection to indefinite sum == Indefinite product can be expressed in terms of indefinite sum: ∏


id: list-of-inequalities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_inequalities-0.md
category: reference
title: List of inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_inequalities
date_saved: 2026-05-05T08:15:09.363624+00:00
instance: kb-cron
parent: list-of-inequalities-20260505-0
summary:
This article lists Wikipedia articles about named mathematical inequalities. == Inequalities in pure mathematics == === Analysis === Agmon's inequality Askey–Gasper inequality Babenko–Beckner inequality Bernoulli's inequality Bernstein's inequality (mathematical analysis) Bessel's inequality Bihari–LaSalle inequality Bohnenblust–Hille inequality Borell–Brascamp–Lieb inequality Brezis–Gallouet inequality Carleman's inequality Carlson's inequality Chebyshev–Markov–Stieltjes inequalities Chebyshev's sum inequality Clarkson's inequalities Eilenberg's inequality Fekete–Szegő inequality Fenchel's inequality Friedrichs' inequality Gagliardo–Nirenberg interpolation inequality Gårding's inequality Grothendieck inequality Grunsky's inequalities Hanner's inequalities Hardy's inequality Hardy–Littlewood inequality Hardy–Littlewood–Sobolev inequality Harnack's inequality Hausdorff–Young inequality Hermite–Hadamard inequality Hilbert's inequality Hölder's inequality Jackson's inequality Jensen's inequality Khabibullin's conjecture on integral inequalities Kantorovich inequality Karamata's inequality Korn's inequality Ladyzhenskaya's inequality Landau–Kolmogorov inequality Lebedev–Milin inequality Lieb–Thirring inequality Littlewood's 4/3 inequality Markov brothers' inequality Mashreghi–Ransford inequality Max–min inequality Minkowski's inequality Poincaré inequality Popoviciu's inequality Prékopa–Leindler inequality Rayleigh–Faber–Krahn inequality Remez inequality Riesz rearrangement inequality Schur test Shapiro inequality Sobolev inequality Steffensen's inequality Szegő inequality Three spheres inequality Trace inequalities Trudinger's theorem Turán's inequalities Von Neumann's inequality Wirtinger's inequality for functions Young's convolution inequality Young's inequality for products ==== Inequalities relating to means ==== Hardy–Littlewood maximal inequality Inequality of arithmetic and geometric means Ky Fan inequality Levinson's inequality Maclaurin's inequality Mahler's inequality Muirhead's inequality Newton's inequalities Stein–Strömberg theorem === Combinatorics ===


id: index-of-information-theory-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_information_theory_articles-0.md
category: reference
title: Index of information theory articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_information_theory_articles
date_saved: 2026-05-05T08:15:10.528092+00:00
instance: kb-cron
parent: index-of-information-theory-articles-20260505-0
summary:
This is a list of information theory topics. A Mathematical Theory of Communication algorithmic information theory arithmetic coding channel capacity Communication Theory of Secrecy Systems conditional entropy conditional quantum entropy confusion and diffusion cross-entropy data compression entropic uncertainty (Hirchman uncertainty) entropy encoding entropy (information theory) Fisher information Hick's law Huffman coding information bottleneck method information theoretic security information theory joint entropy Kullback–Leibler divergence lossless compression negentropy noisy-channel coding theorem (Shannon's theorem) principle of maximum entropy quantum information science range encoding redundancy (information theory) Rényi entropy self-information Shannon–Hartley theorem


id: list-of-integration-and-measure-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics-0.md
category: reference
title: List of integration and measure theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics
date_saved: 2026-05-05T08:15:12.959879+00:00
instance: kb-cron
parent: list-of-integration-and-measure-theory-topics-20260505-0
summary:
This is a list of integration and measure theory topics, by Wikipedia page. == Intuitive foundations == Length Area Volume Probability Moving average == Riemann integral == Riemann sum Riemann–Stieltjes integral Bounded variation Jordan content == Improper integrals == Cauchy principal value == Measure theory and the Lebesgue integral == Measure (mathematics) Sigma algebra Separable sigma algebra Filtration (abstract algebra) Borel algebra Borel measure Indicator function Lebesgue measure Lebesgue integration Lebesgue's density theorem Counting measure Complete measure Haar measure Outer measure Borel regular measure Radon measure Measurable function Null set, negligible set Almost everywhere, conull set Lp space Borel–Cantelli lemma Lebesgue's monotone convergence theorem Fatou's lemma Absolutely continuous Uniform absolute continuity Total variation Radon–Nikodym theorem Fubini's theorem Double integral Vitali set, non-measurable set == Extensions == Henstock–Kurzweil integral Amenable group Banach–Tarski paradox Hausdorff paradox == Integral equations == Fredholm equation Fredholm operator Liouville–Neumann series == Integral transforms == See also list of transforms, list of Fourier-related transforms Kernel (integral operator) Convolution Radon transform == Integral geometry == Buffon's needle Hadwiger's theorem mean width intrinsic volumes == Other == Stokes theorem Differentiation under the integral sign Contour integration Examples of contour integration == See also == List of calculus topics List


id: list-of-international-mathematical-olympiads-20260505-0
file: data/en.wikipedia.org/wiki/List_of_International_Mathematical_Olympiads-0.md
category: reference
title: List of International Mathematical Olympiads
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_International_Mathematical_Olympiads
date_saved: 2026-05-05T08:15:14.252706+00:00
instance: kb-cron
parent: list-of-international-mathematical-olympiads-20260505-0
summary:
The first of the International Mathematical Olympiads (IMOs) was held in Romania in 1959. The oldest of the International Science Olympiads, the IMO has since been held annually, except in 1980. That year, the competition initially planned to be held in Mongolia was cancelled due to the Soviet invasion of Afghanistan. Because the competition was initially founded for Eastern European countries participating in the Warsaw Pact, under the influence of the Eastern Bloc, the earlier IMOs were hosted only in Eastern European countries, gradually spreading to other nations. The first IMO was held in Romania in 1959. Seven countries entered – Bulgaria, Czechoslovakia, East Germany, Hungary, Poland, Romania and the Soviet Union – with the hosts finishing as the top-ranked nation. The number of participating countries has since risen: 14 countries took part in 1969, 50 in 1989, and 104 in 2009. North Korea is the only country whose entire team has been caught cheating, resulting in its disqualification at the 32nd IMO in 1991 and the 51st IMO in 2010. (However, the 2010 case was controversial.) There have been other disqualifications of contestants due to cheating, but such cases are not officially made public. In January 2011, Google gave


id: list-of-countries-by-medal-count-at-international-mathematical-olympiad-20260505-0
category: reference
title: List of countries by medal count at International Mathematical Olympiad
tags: science, encyclopedia
date_saved: 2026-05-05T08:15:15.567788+00:00
instance: kb-cron
parent: list-of-countries-by-medal-count-at-international-mathematical-olympiad-20260505-0
file:
data/en.wikipedia.org/wiki/List_of_countries_by_medal_count_at_International_Mathematical_Olympiad-0.md

summary:
The following is the top 100 list of countries by medal count at the International Mathematical Olympiad: == Notes == ^ This team is now defunct. == References ==

source_url:
https://en.wikipedia.org/wiki/List_of_countries_by_medal_count_at_International_Mathematical_Olympiad


id: list-of-irreducible-tits-indices-20260505-0
file: data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-0.md
category: reference
title: List of irreducible Tits indices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices
date_saved: 2026-05-05T08:15:16.864015+00:00
instance: kb-cron
parent: list-of-irreducible-tits-indices-20260505-0
summary:
In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field k, not assumed to be algebraically closed. The possible irreducible indices were classified by Jacques Tits, and this classification is reproduced below. (Because every index is a direct sum of irreducible indices, classifying all indices amounts to classifying irreducible indices.) == Organization of the list == An index can be represented as a Dynkin diagram with certain vertices drawn close to each other (the orbit of the vertices under the *-action of the Galois group of k) and with certain sets of vertices circled (the orbits of the non-distinguished vertices under the *-action). This representation captures the full information of the index except when the underlying Dynkin diagram is D4, in which case one must distinguish between an action by the cyclic group C3 or the permutation group S3. Alternatively, an index can be represented using the name of the underlying Dynkin diagram together with additional superscripts and subscripts, to be explained momentarily. This representation, together with the labeled Dynkin diagram described in the previous paragraph, captures the full information of the index.


id: list-of-irreducible-tits-indices-20260505-1
file: data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-1.md
category: reference
title: List of irreducible Tits indices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices
date_saved: 2026-05-05T08:15:16.864015+00:00
instance: kb-cron
parent: list-of-irreducible-tits-indices-20260505-0
summary:
Conditions d = 2a | 2n, d ≥ 1, rd ≤ n, n ≠ rd + 1. Distinguished vertices d, 2d, ..., rd. Description Algebraic group : If k has characteristic ≠ 2, the special unitary group SU2n/d(D,h), where D is a division algebra of degree d over k and h is a hermitian form of discriminant 1 and index r, relative to a k-linear involution σ of D, an "involution of the first kind such that the subring Dσ has dimension d(d + 1)/2; or equivalently, when d > 1 and char k ≠ 2, the group SU2n/d where D and h are as above except that h is anti-hermitian form of discriminant 1 and index r, and Dσ has dimension d(d − 1)/2. Special fields Over a finite field, d = 1 and n = r; over the reals, d = 1 and n − r = 2m, or d = 2 and n = 2r; over a p-adic field or local function field, d = 1 and r = n or n − 2, or d = 2 and n = 2r or 2r + 3; over a number field, d = 1 and n − r


id: list-of-irreducible-tits-indices-20260505-2
file: data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-2.md
category: reference
title: List of irreducible Tits indices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices
date_saved: 2026-05-05T08:15:16.864015+00:00
instance: kb-cron
parent: list-of-irreducible-tits-indices-20260505-0
summary:
=== E1337,0 === Image: Special fields: This type exists only over the reals and over some number fields; does not exist over any finite field nor over any local non-archimedean nor global function field. === E787,1 === Image: Special fields: This type does not exist over any finite field nor any local nor global field. === E667,1 === Image: Special fields: This type does not exist over any finite field nor over any local nor global field. === E487,1 === Image: Special fields: This type does not exist over any finite field nor any local nor global fields. === E317,2 === Image: Special fields: This type exists only over some number fields; does not exists over any finite field, nor any local nor global field. === E287,3 === Image: Special fields: This type exists only over the reals and over some number fields; does not exists over any finite field, nor local non-archimedean nor global function fields. === E97,4 === Image: Special fields: This type does not exist over any finite field; it exists over any local and global field. === E07,7 === Image: Special fields: This type exists over any field. == E8 == === E2488,0 === Image: Special


id: karp-s-21-np-complete-problems-20260505-0
file: data/en.wikipedia.org/wiki/Karp's_21_NP-complete_problems-0.md
category: reference
title: Karp's 21 NP-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Karp's_21_NP-complete_problems
date_saved: 2026-05-05T08:15:18.156275+00:00
instance: kb-cron
parent: karp-s-21-np-complete-problems-20260505-0
summary:
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the Boolean satisfiability problem is NP-complete (also called the Cook–Levin theorem) to show that there is a polynomial time many-one reduction from the Boolean satisfiability problem to each of 21 combinatorial and graph theoretical computational problems, thereby showing that they are all NP-complete. This was one of the first demonstrations that many natural computational problems occurring throughout computer science are computationally intractable, and it drove interest in the study of NP-completeness and the P versus NP problem. == The problems == Karp's 21 problems are shown below, many with their original names. The nesting indicates the direction of the reductions used. For example, Knapsack was shown to be NP-complete by reducing Exact cover to Knapsack. Satisfiability: the Boolean satisfiability problem for formulas in conjunctive normal form (often referred to as SAT) 0–1 integer programming (A variation in which only the restrictions must be satisfied, with no optimization) Clique (see also independent set problem) Set packing Vertex cover Set covering Feedback node set Feedback arc set Directed Hamilton


id: list-of-knot-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_knot_theory_topics-0.md
category: reference
title: List of knot theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_knot_theory_topics
date_saved: 2026-05-05T08:15:19.329939+00:00
instance: kb-cron
parent: list-of-knot-theory-topics-20260505-0
summary:
Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. == History == == Knots, links, braids == Knot (mathematics) gives a general introduction to the concept of a knot. Two classes of knots: torus knots and pretzel knots Cinquefoil knot also known as a (5, 2) torus knot. Figure-eight knot (mathematics) the only 4-crossing knot Granny knot (mathematics) and Square knot (mathematics) are a connected sum of two Trefoil knots Perko pair, two entries in a knot table that were later shown to be identical. Stevedore knot (mathematics), a prime knot with crossing number 6 Three-twist knot is the twist knot with three-half twists, also known as the


id: list-of-laplace-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Laplace_transforms-0.md
category: reference
title: List of Laplace transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Laplace_transforms
date_saved: 2026-05-05T08:15:20.618665+00:00
instance: kb-cron
parent: list-of-laplace-transforms-20260505-0
summary:
The following is a list of Laplace transforms for many common functions of a single variable. The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency). == Properties == The Laplace transform of a function f ( t ) {\displaystyle f(t)} can be obtained using the formal definition of the Laplace transform. However, some properties of the Laplace transform can be used to obtain the Laplace transform of some functions more easily. === Linearity === For functions f {\displaystyle f} and g {\displaystyle g} and for scalar a {\displaystyle a} , the Laplace transform satisfies L { a f ( t ) + b g ( t ) } = a L { f ( t ) } + b L { g ( t ) } {\displaystyle {\mathcal {L}}\{af(t)+bg(t)\}=a{\mathcal {L}}\{f(t)\}+b{\mathcal {L}}\{g(t)\}} and is, therefore, regarded as a linear operator. === Time shifting === The Laplace transform of f ( t − a ) u ( t − a ) {\displaystyle f(t-a)u(t-a)} , where u {\displaystyle u} is the Heaviside step function, is e − a s F ( s


id: list-of-large-cardinal-properties-20260505-0
file: data/en.wikipedia.org/wiki/List_of_large_cardinal_properties-0.md
category: reference
title: List of large cardinal properties
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_large_cardinal_properties
date_saved: 2026-05-05T08:15:21.962706+00:00
instance: kb-cron
parent: list-of-large-cardinal-properties-20260505-0
summary:
This page includes a list of large cardinal properties in the mathematical field of set theory. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, Vκ satisfies "there is an unbounded class of cardinals satisfying φ". The following table usually arranges cardinals in order of consistency strength, with size of the cardinal used as a tiebreaker. In a few cases (such as strongly compact cardinals) the exact consistency strength is not known and the table uses the current best guess. "Small" cardinals: 0, 1, 2, ..., ℵ 0 , ℵ 1 {\displaystyle \aleph _{0},\aleph _{1}} ,..., κ = ℵ κ {\displaystyle \kappa =\aleph _{\kappa }} , ... (see Aleph number) the height of the minimal transitive model of ZFC worldly cardinals weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals. reflecting cardinals pseudo uplifting cardinals, uplifting cardinals weakly compact (= Π11-indescribable), Πmn-indescribable, totally


id: list-of-lemmas-20260505-0
file: data/en.wikipedia.org/wiki/List_of_lemmas-0.md
category: reference
title: List of lemmas
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_lemmas
date_saved: 2026-05-05T08:15:23.303385+00:00
instance: kb-cron
parent: list-of-lemmas-20260505-0
summary:
This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. == Algebra == Abhyankar's lemma Aubin–Lions lemma Bergman's diamond lemma Fitting lemma Injective test lemma Hua's lemma (exponential sums) Krull's separation lemma Schanuel's lemma (projective modules) Schwartz–Zippel lemma Shapiro's lemma Stewart–Walker lemma (tensors) Whitehead's lemma (Lie algebras) Zariski's lemma === Algebraic geometry === Abhyankar's lemma Fundamental lemma (Langlands program) === Category theory === Five lemma Horseshoe lemma Nine lemma Short five lemma Snake lemma Splitting lemma Yoneda lemma === Linear algebra === Matrix determinant lemma Matrix inversion lemma === Group theory === Burnside's lemma also known as the Cauchy–Frobenius lemma Frattini's lemma (finite groups) Goursat's lemma Mautner's lemma (representation theory) Ping-pong lemma (geometric group theory) Schreier's subgroup lemma Schur's lemma (representation theory) Zassenhaus lemma === Polynomials === Gauss's lemma (polynomials) Schwartz–Zippel lemma === Ring theory and commutative algebra === Artin–Rees lemma Hensel's lemma (commutative rings) Nakayama lemma Noether's normalization lemma Prime avoidance lemma === Universal algebra === Jónsson's lemma == Analysis == Fekete's lemma Fundamental lemma of the calculus of variations Hopf lemma Sard's lemma (singularity theory)


id: list-of-lie-groups-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Lie_groups_topics-0.md
category: reference
title: List of Lie groups topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Lie_groups_topics
date_saved: 2026-05-05T08:15:25.912263+00:00
instance: kb-cron
parent: list-of-lie-groups-topics-20260505-0
summary:
This is a list of Lie group topics, by Wikipedia page. == Examples == See Table of Lie groups for a list General linear group, special linear group SL2(R) SL2(C) Unitary group, special unitary group SU(2) SU(3) Orthogonal group, special orthogonal group Rotation group SO(3) SO(8) Generalized orthogonal group, generalized special orthogonal group The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions. It is the group of hyperbolic motions of the Poincaré disk model of the Hyperbolic plane. Lorentz group Spinor group Symplectic group Exceptional groups G2 F4 E6 E7 E8 Affine group Euclidean group Poincaré group Heisenberg group == Lie algebras == Commutator Jacobi identity Universal enveloping algebra Baker–Campbell–Hausdorff formula Casimir invariant Killing form Kac–Moody algebra Affine Lie algebra Loop algebra Graded Lie algebra == Foundational results == One-parameter group, One-parameter subgroup Matrix exponential Infinitesimal transformation Lie's third theorem Maurer–Cartan form Cartan's theorem Cartan's criterion Local Lie group Formal group law Hilbert's fifth problem Hilbert–Smith conjecture Lie group decompositions Real form (Lie theory) Complex Lie group Complexification (Lie group) == Semisimple theory == Simple Lie group Compact Lie group, Compact real form Semisimple Lie algebra Root system Simply laced group ADE classification Maximal torus


id: table-of-lie-groups-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_Lie_groups-0.md
category: reference
title: Table of Lie groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_Lie_groups
date_saved: 2026-05-05T08:15:27.102675+00:00
instance: kb-cron
parent: table-of-lie-groups-20260505-0
summary:
This article gives a table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; see classification of low-dimensional real Lie algebras for up to four dimensions; and the list of Lie group topics. == Real Lie groups and their algebras == Column legend Cpt: Is this group G compact? (Yes or No) π 0 {\displaystyle \pi _{0}} : Gives the group of components of G. The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0). π 1 {\displaystyle \pi _{1}} : Gives the fundamental group of G whenever G is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0). UC: If G is not simply connected, gives the


id: list-of-limits-20260505-0
file: data/en.wikipedia.org/wiki/List_of_limits-0.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x. == Limits for general functions == === Definitions of limits and related concepts === lim x → c f ( x ) = L {\displaystyle \lim _{x\to c}f(x)=L} if and only if ∀ ε > 0 ∃ δ > 0 : 0 < | x − c | < δ ⟹ | f ( x ) − L | < ε {\displaystyle \forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon } . This is the (ε, δ)-definition of limit. The limit superior and limit inferior of a sequence are defined as lim sup n → ∞ x n = lim n → ∞ ( sup m ≥ n x m ) {\displaystyle \limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)} and lim inf n → ∞ x n = lim n → ∞ ( inf m ≥ n x m ) {\displaystyle \liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)} . A function, f ( x ) {\displaystyle f(x)} , is said to be continuous at a point, c, if


id: list-of-limits-20260505-1
file: data/en.wikipedia.org/wiki/List_of_limits-1.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
If lim x → c f ( x ) = lim x → c h ( x ) = L {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L} and f ( x ) ≤ g ( x ) ≤ h ( x ) {\displaystyle f(x)\leq g(x)\leq h(x)} for all x in an open interval that contains c, except possibly c itself, lim x → c g ( x ) = L . {\displaystyle \lim _{x\to c}g(x)=L.} This is known as the squeeze theorem. This applies even in the cases that f(x) and g(x) take on different values at c, or are discontinuous at c. == Polynomials and functions of the form xa == lim x → c a = a {\displaystyle \lim _{x\to c}a=a} === Polynomials in x === lim x → c x = c {\displaystyle \lim _{x\to c}x=c} lim x → c ( a x + b ) = a c + b {\displaystyle \lim _{x\to c}(ax+b)=ac+b} lim x → c x n = c n {\displaystyle \lim _{x\to c}x^{n}=c^{n}} if n is a positive integer lim x → ∞ x / a = { ∞ , a > 0 does not exist , a = 0 − ∞ ,


id: list-of-limits-20260505-2
file: data/en.wikipedia.org/wiki/List_of_limits-2.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
lim x → 1 ln ⁡ ( x ) x − 1 = 1 {\displaystyle \lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1} lim x → 0 ln ⁡ ( x + 1 ) x = 1 {\displaystyle \lim _{x\to 0}{\frac {\ln(x+1)}{x}}=1} lim x → 0 − ln ⁡ ( 1 + a ( e − x − 1 ) ) x = a {\displaystyle \lim _{x\to 0}{\frac {-\ln \left(1+a\left({e^{-x}-1}\right)\right)}{x}}=a} . This limit follows from L'Hôpital's rule. lim x → 0 x ln ⁡ x = 0 {\displaystyle \lim _{x\to 0}x\ln x=0} , hence lim x → 0 x x = 1 {\displaystyle \lim _{x\to 0}x^{x}=1} lim x → ∞ ln ⁡ x x = 0 {\displaystyle \lim _{x\to \infty }{\frac {\ln x}{x}}=0} === Logarithms to arbitrary bases === For b > 1, lim x → 0 + log b ⁡ x = − ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-\infty } lim x → ∞ log b ⁡ x = ∞ {\displaystyle \lim _{x\to \infty }\log _{b}x=\infty } For b < 1, lim x → 0 + log b ⁡ x = ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=\infty } lim x → ∞ log b ⁡ x = − ∞ {\displaystyle \lim _{x\to \infty


id: list-of-books-on-history-of-number-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems-0.md
category: reference
title: List of books on history of number systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems
date_saved: 2026-05-05T08:15:29.597547+00:00
instance: kb-cron
parent: list-of-books-on-history-of-number-systems-20260505-0
summary:
This list compiles notable works that explore the history and development of number systems across various civilizations and time periods. These works cover topics ranging from ancient numeral systems and arithmetic methods to the evolution of mathematical notations and the impact of numerals on science, trade, and culture. == Overview == Number systems have been central to the development of human civilization, enabling record-keeping, commerce, astronomy, and scientific advancement. Early systems such as tally marks and Roman numerals gradually gave way to more abstract and efficient representations like the Babylonian base-60 system and the Hindu–Arabic numerals, now standard worldwide. The invention of zero, positional notation, and symbolic mathematics has had profound philosophical and technological implications. == Notable works on the history of number systems == == Works on the history of zero == == Children's books on the history of numbers == == Historical texts == == References ==


id: list-of-regular-polytope-compounds-20260505-0
file: data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-0.md
category: reference
title: List of regular polytope compounds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytope_compounds
date_saved: 2026-05-05T08:15:32.110805+00:00
instance: kb-cron
parent: list-of-regular-polytope-compounds-20260505-0
summary:
This article lists the regular polytope compounds in Euclidean, spherical and hyperbolic spaces. == Two dimensional compounds == For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon. In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m. A further


id: list-of-regular-polytope-compounds-20260505-1
file: data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-1.md
category: reference
title: List of regular polytope compounds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytope_compounds
date_saved: 2026-05-05T08:15:32.110805+00:00
instance: kb-cron
parent: list-of-regular-polytope-compounds-20260505-0
summary:
Although the 5-cell and 24-cell are both self-dual, their dual compounds (the compound of two 5-cells and compound of two 24-cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any regular 4-polytope. However, they are vertex-, edge-, face-, and cell-transitive. === Euclidean 3-space compounds === The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[d{4,3,4}]{4,3,4}. === Hyperbolic 3-space compounds === C. W. L. Garner described two dual pairs of regular hyperbolic compound honeycombs in 1970: the compact pair 2{5,3,4}[5{4,3,5}] and [5{5,3,4}]2{4,3,5}, and the paracompact pair {6,3,3}[5{6,3,4}] and [5{4,3,6}]{3,3,6}. He did not consider vertex-regular compounds where the vertices are at infinity, or (reciprocally) cell-regular compounds where the cells are centred at infinity. In 2019, Peter McMullen (who focused only on the compact case) pointed out and filled a gap in Garner's proof of completeness, so that it is now proven that 2{5,3,4}[5{4,3,5}] and [5{5,3,4}]2{4,3,5} are the only compact regular


id: list-of-things-named-after-felix-hausdorff-20260505-0
file: data/en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff-0.md
category: reference
title: List of things named after Felix Hausdorff
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff
date_saved: 2026-05-05T08:15:33.394906+00:00
instance: kb-cron
parent: list-of-things-named-after-felix-hausdorff-20260505-0
summary:
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician. He made significant contributions to mathematics, specifically in the fields of set theory and topology. == Mathematics == Baker–Campbell–Hausdorff formula Gromov–Hausdorff convergence Hausdorff–Young inequality Hausdorff completion Hausdorff dimension Hausdorff distance Hausdorff gap Hausdorff maximal principle Hausdorff measure Hausdorff Medal Hausdorff moment problem Hausdorff paradox Hausdorff space == Places == Felix Hausdorff International Meeting Centre at the University of Greifswald Hausdorff Center for Mathematics == Space == 24947 Hausdorff == References ==


id: index-of-logarithm-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_logarithm_articles-0.md
category: reference
title: Index of logarithm articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_logarithm_articles
date_saved: 2026-05-05T08:15:34.685156+00:00
instance: kb-cron
parent: index-of-logarithm-articles-20260505-0
summary:
This is a list of logarithm topics, by Wikipedia page. See also the list of exponential topics. Acoustic power Amoeba (mathematics) Antilogarithm Apparent magnitude Baker's theorem Bel Benford's law Binary logarithm Bode plot Henry Briggs Bygrave slide rule Cologarithm Common logarithm Complex logarithm Discrete logarithm Discrete logarithm records e Representations of e El Gamal discrete log cryptosystem Harmonic series History of logarithms Hyperbolic sector Iterated logarithm Otis King Law of the iterated logarithm Linear form in logarithms Linearithmic List of integrals of logarithmic functions Log canonical singularity Log-likelihood ratio Log-log graph Log-normal distribution Log-periodic antenna Log semiring Log structure Log-Weibull distribution Logarithmic algorithm Logarithmic convolution Logarithmic decrement Logarithmic derivative Logarithmic differential Logarithmic differentiation Logarithmic distribution Logarithmic form Logarithmic graph paper Logarithmic growth Logarithmic identities Logarithmic mean Logarithmic number system Logarithmic scale Logarithmic spiral Logit LogSumExp Mantissa is a disambiguation page; see common logarithm for the traditional concept of mantissa; see significand for the modern concept used in computing. Matrix logarithm Mel scale Mercator projection Mercator series Moment magnitude scale John Napier Napierian logarithm Natural logarithm Natural logarithm of 2 Neper Offset logarithmic integral pH Plethystic logarithm Pollard's kangaroo algorithm Pollard's rho algorithm for logarithms Polylogarithm Polylogarithmic function Prime number theorem


id: list-of-mathematical-logic-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_logic_topics-0.md
category: reference
title: List of mathematical logic topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_logic_topics
date_saved: 2026-05-05T08:15:35.913586+00:00
instance: kb-cron
parent: list-of-mathematical-logic-topics-20260505-0
summary:
This is a list of mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and complexity topics for more theory of algorithms. == Working foundations == Peano axioms Giuseppe Peano Mathematical induction Structural induction Recursive definition Naive set theory Element (mathematics) Ur-element Singleton (mathematics) Simple theorems in the algebra of sets Algebra of sets Power set Empty set Non-empty set Empty function Universe (mathematics) Axiomatization Axiomatic system Axiom schema Axiomatic method Formal system Mathematical proof Direct proof Reductio ad absurdum Proof by exhaustion Constructive proof Nonconstructive proof Tautology Consistency proof Arithmetization of analysis Foundations of mathematics Formal language Principia Mathematica Hilbert's program Impredicative Definable real number Algebraic logic Boolean algebra (logic) Dialectica space categorical logic == Model theory == Finite model theory Descriptive complexity theory Model checking Trakhtenbrot's theorem Computable model theory Tarski's exponential function problem Undecidable problem Institutional model theory Institution (computer science) Non-standard analysis Non-standard calculus Hyperinteger Hyperreal number Transfer principle Overspill Elementary Calculus: An Infinitesimal Approach Criticism of non-standard analysis Standard part function Set theory Forcing (mathematics) Boolean-valued model Kripke semantics General frame Predicate logic First-order logic Infinitary logic Many-sorted logic Higher-order logic Lindström quantifier Second-order


id: outline-of-logic-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_logic-0.md
category: reference
title: Outline of logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_logic
date_saved: 2026-05-05T08:15:37.128800+00:00
instance: kb-cron
parent: outline-of-logic-20260505-0
summary:
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct (or valid) and incorrect (or fallacious) inferences. Logicians study the criteria for the evaluation of arguments. == Foundations of logic == Philosophy of logic Analytic-synthetic distinction Antinomy A priori and a posteriori Definition Description Entailment Identity (philosophy) Inference Logical form Logical implication Logical truth Logical consequence Name Necessity Material conditional Meaning (linguistic) Meaning (non-linguistic) Paradox (list) Possible world Presupposition Probability Quantification Reason Reasoning Reference Semantics Strict conditional Syntax (logic) Truth Truth value Validity == Branches of logic == Affine logic Alethic logic Aristotelian logic Boolean logic Buddhist logic Bunched logic Categorical logic Classical logic Computability logic Deontic logic Dependence logic Description logic


id: outline-of-logic-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_logic-1.md
category: reference
title: Outline of logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_logic
date_saved: 2026-05-05T08:15:37.128800+00:00
instance: kb-cron
parent: outline-of-logic-20260505-0
summary:
Finitary relation Antisymmetric relation Asymmetric relation Bijection Bijection, injection and surjection Binary relation Composition of relations Congruence relation Connected relation Converse relation Coreflexive relation Covering relation Cyclic order Dense relation Dependence relation Dependency relation Directed set Equivalence relation Euclidean relation Homogeneous relation Idempotence Intransitivity Involutive relation Partial equivalence relation Partial function Partially ordered set Preorder Prewellordering Propositional function Quasitransitive relation Reflexive relation Serial relation Surjective function Symmetric relation Ternary relation Transitive relation Trichotomy (mathematics) Well-founded relation == Mathematical logic == Mathematical logic === Set theory === Set theory list Aleph null Bijection, injection and surjection Binary set Cantor's diagonal argument Cantor's first uncountability proof Cantor's theorem Cardinality of the continuum Cardinal number Codomain Complement (set theory) Constructible universe Continuum hypothesis Countable set Decidable set Denumerable set Disjoint sets Disjoint union Domain of a function Effective enumeration Element (mathematics) Empty function Empty set Enumeration Extensionality Finite set Forcing (mathematics) Function (set theory) Function composition Generalized continuum hypothesis Index set Infinite set Intension Intersection (set theory) Inverse function Large cardinal Löwenheim–Skolem theorem Map (mathematics) Multiset Morse–Kelley set theory Naïve set theory One-to-one correspondence Ordered pair Partition of a set Pointed set Power set Projection (set theory) Proper subset Proper superset Range of


id: outline-of-logic-20260505-2
file: data/en.wikipedia.org/wiki/Outline_of_logic-2.md
category: reference
title: Outline of logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_logic
date_saved: 2026-05-05T08:15:37.128800+00:00
instance: kb-cron
parent: outline-of-logic-20260505-0
summary:
=== Journals === Journal of Logic, Language and Information Journal of Philosophical Logic Linguistics and Philosophy === Books === A System of Logic Attacking Faulty Reasoning Begriffsschrift Categories (Aristotle) Charles Sanders Peirce bibliography De Interpretatione Gödel, Escher, Bach Introduction to Mathematical Philosophy Language, Truth, and Logic Laws of Form Novum Organum On Formally Undecidable Propositions of Principia Mathematica and Related Systems Organon Philosophy of Arithmetic Polish Logic Port-Royal Logic Posterior Analytics Principia Mathematica Principles of Mathematical Logic Prior Analytics Rhetoric (Aristotle) Sophistical Refutations Sum of Logic The Art of Being Right The Foundations of Arithmetic Topics (Aristotle) Tractatus Logico-Philosophicus == Logic organizations == Association for Symbolic Logic == Logicians == List of logicians List of philosophers of language == See also == Glossary of logic Index of logic articles Mathematics List of basic mathematics topics List of mathematics articles Philosophy List of basic philosophy topics List of philosophy topics Outline of philosophy Outline of discrete mathematics – for introductory set theory and other supporting material == References == == External links == Taxonomy of Logical Fallacies forall x: an introduction to formal logic, by P.D. Magnus, covers sentential and quantified logic Translation Tips, by Peter Suber, for translating from English


id: list-of-manifolds-20260505-0
file: data/en.wikipedia.org/wiki/List_of_manifolds-0.md
category: reference
title: List of manifolds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_manifolds
date_saved: 2026-05-05T08:15:38.253557+00:00
instance: kb-cron
parent: list-of-manifolds-20260505-0
summary:
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics. For categorical listings see Category:Manifolds and its subcategories. == Generic families of manifolds == Euclidean space, Rn n-sphere, Sn n-torus, Tn Real projective space, RPn Complex projective space, CPn Quaternionic projective space, HPn Flag manifold Grassmann manifold Stiefel manifold Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics. == Manifolds of a specific dimension == === 1-manifolds === Circle, S1 Long line Real line, R Real projective line, RP1 ≅ S1 === 2-manifolds === Cylinder, S1 × R Klein bottle, RP2 # RP2 Klein quartic (a genus 3 surface) Möbius strip Real projective plane, RP2 Sphere, S2 Surface of genus g Torus Double torus === 3-manifolds === 3-sphere, S3 3-torus, T3 Poincaré homology sphere SO(3) ≅ RP3 Solid Klein bottle Solid torus Whitehead manifold Meyerhoff manifold Weeks manifold For more examples see 3-manifold. === 4-manifolds === Complex projective plane Del Pezzo surface E8 manifold Enriques surface Exotic R4 Hirzebruch surface K3 surface For more examples see 4-manifold. == Special types of manifolds == === Manifolds related


id: list-of-martin-gardner-mathematical-games-columns-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns-0.md
category: reference
title: List of Martin Gardner Mathematical Games columns
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns
date_saved: 2026-05-05T08:15:39.569085+00:00
instance: kb-cron
parent: list-of-martin-gardner-mathematical-games-columns-20260505-0
summary:
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for Scientific American magazine. During the next 5.5 years, until June 1986, Gardner wrote 9 more columns, bringing his total to 297. During this period other authors wrote most of the columns. In 1981, Gardner's column alternated with a new column by Douglas Hofstadter called "Metamagical Themas" (an anagram of "Mathematical Games"). The table below lists Gardner's columns. Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by "[cover]" in the table with a hyperlink to the cover. == Other articles by Gardner == Gardner wrote 5 other articles for Scientific American. His flexagon article in December 1956 was in all but name the first article in the series of Mathematical Games columns and led directly to the series which began the following month. These five articles are listed below. == References == == External links == A Quarter Century of Recreational Mathematics, by Martin Gardner preserved at the Internet Archive A subject index for the fifteen books of Martin Gardner's Mathematical Games columns The Top 10 Martin Gardner Scientific American Articles


id: list-of-named-matrices-20260505-0
file: data/en.wikipedia.org/wiki/List_of_named_matrices-0.md
category: reference
title: List of named matrices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_named_matrices
date_saved: 2026-05-05T08:15:45.006476+00:00
instance: kb-cron
parent: list-of-named-matrices-20260505-0
summary:
This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include the identity matrix given by I n = [ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 ] . {\displaystyle I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.} and the zero matrix of dimension m × n {\displaystyle m\times n} . For example: O 2 × 3 = ( 0 0 0 0 0 0 ) {\displaystyle O_{2\times 3}={\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}}} . Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the product of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including physics and chemistry, have particular matrices that are applied chiefly in these areas. == Constant matrices == The list below comprises matrices whose elements are constant for any given dimension (size) of matrix. The matrix


id: list-of-named-matrices-20260505-1
file: data/en.wikipedia.org/wiki/List_of_named_matrices-1.md
category: reference
title: List of named matrices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_named_matrices
date_saved: 2026-05-05T08:15:45.006476+00:00
instance: kb-cron
parent: list-of-named-matrices-20260505-0
summary:
== Matrices used in science and engineering == Cabibbo–Kobayashi–Maskawa matrix — a unitary matrix used in particle physics to describe the strength of flavour-changing weak decays. Density matrix — a matrix describing the statistical state of a quantum system. Hermitian, non-negative and with trace 1. Fundamental matrix (computer vision) — a 3 × 3 matrix in computer vision that relates corresponding points in stereo images. Fuzzy associative matrix — a matrix in artificial intelligence, used in machine learning processes. Gamma matrices — 4 × 4 matrices in quantum field theory. Gell-Mann matrices — a generalization of the Pauli matrices; these matrices are one notable representation of the infinitesimal generators of the special unitary group SU(3). Hamiltonian matrix — a matrix used in a variety of fields, including quantum mechanics and linear-quadratic regulator (LQR) systems. Irregular matrix — a matrix used in computer science which has a varying number of elements in each row. Overlap matrix — a type of Gramian matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system. S matrix — a matrix in quantum mechanics that connects asymptotic (infinite past and future) particle states. Scattering matrix - a


id: list-of-mathematics-based-methods-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics-based_methods-0.md
category: reference
title: List of mathematics-based methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics-based_methods
date_saved: 2026-05-05T08:15:46.364138+00:00
instance: kb-cron
parent: list-of-mathematics-based-methods-20260505-0
summary:
This is a list of mathematics-based methods. Adams' method (differential equations) Akra–Bazzi method (asymptotic analysis) Bisection method (root finding) Brent's method (root finding) Condorcet method (voting systems) Coombs' method (voting systems) Copeland's method (voting systems) Crank–Nicolson method (numerical analysis) D'Hondt method (voting systems) D21 – Janeček method (voting system) Discrete element method (numerical analysis) Domain decomposition method (numerical analysis) Epidemiological methods Euler's forward method Explicit and implicit methods (numerical analysis) Finite difference method (numerical analysis) Finite element method (numerical analysis) Finite volume method (numerical analysis) Highest averages method (voting systems) Method of exhaustion Method of infinite descent (number theory) Information bottleneck method Inverse chain rule method (calculus) Inverse transform sampling method (probability) Iterative method (numerical analysis) Jacobi method (linear algebra) Largest remainder method (voting systems) Level-set method Linear combination of atomic orbitals molecular orbital method (molecular orbitals) Method of characteristics Least squares method (optimization, statistics) Maximum likelihood method (statistics) Method of complements (arithmetic) Method of moving frames (differential geometry) Method of successive substitution (number theory) Monte Carlo method (computational physics, simulation) Newton's method (numerical analysis) Pemdas method (order of operation) Perturbation methods (functional analysis, quantum theory) Probabilistic method (combinatorics) Romberg's method (numerical analysis) Runge–Kutta method (numerical analysis) Sainte-Laguë method (voting


id: list-of-works-by-nicolas-minorsky-20260505-0
file: data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-0.md
category: reference
title: List of works by Nicolas Minorsky
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky
date_saved: 2026-05-05T08:15:47.573957+00:00
instance: kb-cron
parent: list-of-works-by-nicolas-minorsky-20260505-0
summary:
List of works by Nicolas Minorsky. == Books == Minorsky, N. (1947). Introduction to non-linear mechanics: Topological methods, analytical methods, non-linear resonance, relaxation oscillations. J.W. Edwards. ASIN B0007DXRVY. Minorsky, Nicolai (1955). Influence d'Henri Poincaré sur l'évolution moderne de la théorie des oscillations non linéaires [Influence of Henri Poincaré on the modern development of the theory of nonlinear oscillations]. Le livre du centenaire de la naissance de Henri Poincaré, 1854-1954 (in French). Paris, Gauthier-Villars. pp. 120–126. OCLC 10571426. Minorsky, N. (1958). Dynamics and Nonlinear Mechanics: The Theory of Oscillations. Surveys in Applied Mathematics. Wiley. pp. 110–206. ASIN B0006AVKQW. Minorsky, N. (1962). G. Szego; et al. (eds.). On some aspects of non-linear oscillations. Studies in Mathematical Analysis and Related Topics. Stanford Univ Pr (April 1962). ISBN 978-0804701402. {{cite book}}: ISBN / Date incompatibility (help) Minorsky, N. (1967). Théorie des Oscillations [Theory of Oscillations]. Memorial des Sciences Mathematiques, Fascicule CLXIII (in French). ASIN B000LQD7KI. Minorsky, Nicolai; Hetnarski, Richard B.; Mróz, Zenon (1967). Drgania nieliniowe [Nonlinear oscillations] (in Polish). Warszawa : Państwowe Wydawnictwo Naukowe. OCLC 749191659.{{cite book}}: CS1 maint: publisher location (link)


id: list-of-works-by-nicolas-minorsky-20260505-1
file: data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-1.md
category: reference
title: List of works by Nicolas Minorsky
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky
date_saved: 2026-05-05T08:15:47.573957+00:00
instance: kb-cron
parent: list-of-works-by-nicolas-minorsky-20260505-0
summary:
== Papers == Minorsky, N. (May 1922). "Directional stability of automatically steered bodies". Journal of the American Society of Naval Engineers. 34 (2): 280–309. doi:10.1111/j.1559-3584.1922.tb04958.x. Minorsky, N. (February 1927). "Phenomenon of direct-current self excitation in vacuum tubes circuits and in applications". J. Franklin Inst. 203 (2): 181–209. doi:10.1016/S0016-0032(27)92437-5. Minorsky, N. (April 1928). "La rotation de l'arc électrique dans champ magnétique radial" [The rotation of an arc in a radial magnetic field] (PDF). J. Phys. Radium (in French). 9 (4): 127–136. doi:10.1051/jphysrad:0192800904012700. Minorsky, N. (1928). "Mesure de la vitesse d'un aéronef par rapport au sol en l'absence supposée de tout repère extérieur" [Measuring the speed of an aircraft relative to the ground in the absence of any assumed approximate]. L'Aéronautique (in French). 113. Minorsky, N. (May 1930). "Automatic steering test". Journal of the American Society for Naval Engineers. 42 (2): 285–310. doi:10.1111/j.1559-3584.1930.tb05036.x. Minorsky, N. (June 1930). "Electronic conduction and ionization in crossed electric and magnetic fields". Journal of the Franklin Institute. 209 (6): 757–775. doi:10.1016/S0016-0032(30)91472-X. ISSN 0016-0032. Minorsky, N. (August 1934). "Ship stabilization by activated tank: An experimental investigation". The Engineer. 158: 154. Minorsky, N. (27 January 1936). "Une méthode d'intégration de quelques équations différentielles par un procédé électrique" [A method


id: list-of-works-by-nicolas-minorsky-20260505-2
file: data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-2.md
category: reference
title: List of works by Nicolas Minorsky
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky
date_saved: 2026-05-05T08:15:47.573957+00:00
instance: kb-cron
parent: list-of-works-by-nicolas-minorsky-20260505-0
summary:
Acad. Sci. 259. Minorsky, N. (1965). "Sur les oscillations quasi discontinues". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur les phénomènes paramétriques". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur ll'interaction des oscillations non linéaires". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur les oscillations quasi discontinues". C. R. Acad. Sci. Mécanique. 261. Minorsky, N. (1966). "Sur la méthode stroboscopique". C. R. Acad. Sci. 263. Minorsky, N. (1967). "Comments "On asynchronous quenching"". IEEE Transactions on Automatic Control. 12 (2): 225–227. doi:10.1109/TAC.1967.1098559. Minorsky, N. (1968). "Sur les équations différentielles aux différences". C. R. Acad. Sci. 266.


id: list-of-works-by-nicolas-minorsky-20260505-3
file: data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-3.md
category: reference
title: List of works by Nicolas Minorsky
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky
date_saved: 2026-05-05T08:15:47.573957+00:00
instance: kb-cron
parent: list-of-works-by-nicolas-minorsky-20260505-0
summary:
== Conferences == Minorsky, N. (1948). Self-excited oscillations in systems possessing retarded actions. 7th Int. Congr. Appl. Math, London, England. Minorsky, N. (1951). Sur le phénomène Béthenod. Actes Colloq. Int. Vibrations linéaires, Porqurolles, France. Minorsky, N. (1952). Non-linear control systems. Conf. Automat. Contr., London:Butterworth. Minorsky, N. (1952). Sur la méthode stroboscopique. Mem. lett. Accad. Sci. Ist., Bologna. Minorsky, N. (1952). Sur quelques oscillatoires contenant les paramètres à inertie. Ann. Fac. Sci. Univ. d'Aix, Marseilles, France. Minorsky, N. (1957). Nouvelles méthodes de la théorie des oscillations. Conf. Semin. Mat. Univ. Bari. Minorsky, N. (1961). On synchronization. Int. Union Theor. Appl. Mech. Symp. Nonlinear Vibrations, Kiev, U.S.S.R. Minorsky, N. (1963). On synchronization. Tr. Mizh. Symp. po Nelin. Kolebelina, vol. 1, Kiev, U.S.S.R. Minorsky, N. (1964). Les vibrations forcées dans les systèmes non linéaires. Coll. Int. Centre Nat. Rech. Sci., no. 148, Marseilles, France. Minorsky, N. (1965). Sur les systèmes non autonomes. Colloques Inl. Centre Nat. Rech. Sci., vol 148. == Patents == US patent 1306552, N. Minorsky, "Gyrometer", published 1919-06-10, assigned to The Sperry Gyroscope company US patent 1372184, N. Minorsky, "Angular-velocity-indicating apparatus", published 1921-03-22, assigned to N. Minorsky US patent 1436280, N. Minorsky, "Automatic steering device", published 1922-11-21, assigned to N.


id: list-of-multivariable-calculus-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_multivariable_calculus_topics-0.md
category: reference
title: List of multivariable calculus topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multivariable_calculus_topics
date_saved: 2026-05-05T08:15:48.804210+00:00
instance: kb-cron
parent: list-of-multivariable-calculus-topics-20260505-0
summary:
This is a list of multivariable calculus topics. See also multivariable calculus, vector calculus, list of real analysis topics, list of calculus topics. Closed and exact differential forms Contact (mathematics) Contour integral Contour line Critical point (mathematics) Curl (mathematics) Current (mathematics) Curvature Curvilinear coordinates Del Differential form Differential operator Directional derivative Divergence Divergence theorem Double integral Equipotential surface Euler's theorem on homogeneous functions Exterior derivative Flux Frenet–Serret formulas Gauss's law Gradient Green's theorem Green's identities Harmonic function Helmholtz decomposition Hessian matrix Hodge star operator Inverse function theorem Irrotational vector field Isoperimetry Jacobian matrix Lagrange multiplier Lamellar vector field Laplacian Laplacian vector field Level set Line integral Matrix calculus Mixed derivatives Monkey saddle Multiple integral Newtonian potential Parametric equation Parametric surface Partial derivative Partial differential equation Potential Real coordinate space Saddle point Scalar field Solenoidal vector field Stokes' theorem Submersion Surface integral Symmetry of second derivatives Taylor's theorem Total derivative Vector field Vector operator Vector potential


id: table-of-newtonian-series-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_Newtonian_series-0.md
category: reference
title: Table of Newtonian series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_Newtonian_series
date_saved: 2026-05-05T08:15:50.124582+00:00
instance: kb-cron
parent: table-of-newtonian-series-20260505-0
summary:
In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a n {\displaystyle a_{n}} written in the form f ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( s n ) a n = ∑ n = 0 ∞ ( − s ) n n ! a n {\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n}}{n!}}a_{n}} where ( s n ) {\displaystyle {s \choose n}} is the binomial coefficient and ( s ) n {\displaystyle (s)_{n}} is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus. == List == The generalized binomial theorem gives ( 1 + z ) s = ∑ n = 0 ∞ ( s n ) z n = 1 + ( s 1 ) z + ( s 2 ) z 2 + ⋯ . {\displaystyle (1+z)^{s}=\sum _{n=0}^{\infty }{s \choose n}z^{n}=1+{s \choose 1}z+{s \choose 2}z^{2}+\cdots .} A proof for this identity can be obtained by showing that it satisfies the differential equation ( 1 + z ) d ( 1 + z ) s d z = s ( 1 + z ) s . {\displaystyle


id: list-of-np-complete-problems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_NP-complete_problems-0.md
category: reference
title: List of NP-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_NP-complete_problems
date_saved: 2026-05-05T08:15:51.446008+00:00
instance: kb-cron
parent: list-of-np-complete-problems-20260505-0
summary:
This is a list of some of the more commonly known problems that are NP-complete when expressed as decision problems. As there are thousands of such problems known, this list is in no way comprehensive. Many problems of this type can be found in Garey & Johnson (1979). == Graphs and hypergraphs == Graphs occur frequently in everyday applications. Examples include biological or social networks, which contain hundreds, thousands and even billions of nodes in some cases (e.g. Facebook or LinkedIn). 1-planarity 3-dimensional matching Bandwidth problem Bipartite dimension Capacitated minimum spanning tree Route inspection problem (also called Chinese postman problem) for mixed graphs (having both directed and undirected edges). The program is solvable in polynomial time if the graph has all undirected or all directed edges. Variants include the rural postman problem. Clique cover problem Clique problem Complete coloring, a.k.a. achromatic number Cycle rank Degree-constrained spanning tree Domatic number Dominating set, a.k.a. domination number NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. Feedback vertex set Feedback arc set Graph coloring Graph homomorphism problem Graph partition into


id: list-of-np-complete-problems-20260505-1
file: data/en.wikipedia.org/wiki/List_of_NP-complete_problems-1.md
category: reference
title: List of NP-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_NP-complete_problems
date_saved: 2026-05-05T08:15:51.446008+00:00
instance: kb-cron
parent: list-of-np-complete-problems-20260505-0
summary:
By the same article existence of bounded modular square roots with arbitrarily composite modulus. Given positive integers A , B , C ≥ 0 {\displaystyle \textstyle A,B,C\geq 0} , decide existence of an integer x ∈ [ 0 , C ] {\displaystyle x\in [0,C]} such that x 2 ≡ A mod B {\displaystyle x^{2}\equiv A{\bmod {B}}} . The problem remains NP-complete even if a prime factorization of B {\displaystyle B} is provided. Serializability of database histories Set cover (also called "minimum cover" problem). This is equivalent, by transposing the incidence matrix, to the hitting set problem. Set packing Set splitting problem Scheduling to minimize weighted completion time Block Sorting (Sorting by Block Moves) Sparse approximation Variations of the Steiner tree problem. Specifically, with the discretized Euclidean metric, rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. Three-dimensional Ising model == See also == Existential theory of the reals § Complete problems Karp's 21 NP-complete problems List of PSPACE-complete problems Reduction (complexity) == Notes == == References == General Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences (1st ed.). New


id: list-of-number-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_number_theory_topics-0.md
category: reference
title: List of number theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_number_theory_topics
date_saved: 2026-05-05T08:15:52.778467+00:00
instance: kb-cron
parent: list-of-number-theory-topics-20260505-0
summary:
This is a list of topics in number theory. See also: List of recreational number theory topics Topics in cryptography == Divisibility rule == Composite number Highly composite number Even and odd numbers Parity Divisor, aliquot part Greatest common divisor Least common multiple Euclidean algorithm Coprime Euclid's lemma Bézout's identity, Bézout's lemma Extended Euclidean algorithm Table of divisors Prime number, prime power Bonse's inequality Prime factor Table of prime factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial == Fractions == Rational number Unit fraction Irreducible fraction = in lowest terms Dyadic fraction Recurring decimal Cyclic number Farey sequence Ford circle Stern–Brocot tree Dedekind sum Egyptian fraction == Modular arithmetic == Montgomery reduction Modular exponentiation Linear congruence theorem Successive over-relaxation Chinese remainder theorem Fermat's little theorem Proofs of Fermat's little theorem Fermat quotient Euler's totient function Noncototient Nontotient Euler's theorem Wilson's theorem Primitive root modulo n Multiplicative order Discrete logarithm Quadratic residue Euler's criterion Legendre symbol Gauss's lemma (number theory) Congruence of squares Luhn formula Mod n cryptanalysis == Arithmetic functions == Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor


id: list-of-numeral-system-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numeral_system_topics-0.md
category: reference
title: List of numeral system topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numeral_system_topics
date_saved: 2026-05-05T08:15:54.075389+00:00
instance: kb-cron
parent: list-of-numeral-system-topics-20260505-0
summary:
This is a list of Wikipedia articles on topics of numeral system and "numeric representations" See also: computer numbering formats and number names. == Arranged by base == Radix, radix point, mixed radix, exponentiation Unary numeral system (base 1) Tally marks – Numeral form used for counting Binary numeral system (base 2) Negative base numeral system (base −2) Ternary numeral system numeral system (base 3) Balanced ternary numeral system (base 3) Negative base numeral system (base −3) Quaternary numeral system (base 4) Quater-imaginary base (base 2i) Quinary numeral system (base 5) Pentadic numerals – Scandinavian numeral system Senary numeral system (base 6) Septenary numeral system (base 7) Octal numeral system (base 8) Nonary (novenary) numeral system (base 9) Decimal (denary) numeral system (base 10) Bi-quinary coded decimal – Numeral encoding scheme Negative base numeral system (base −10) Duodecimal (dozenal) numeral system (base 12) Hexadecimal numeral system (base 16) Vigesimal numeral system (base 20) Sexagesimal numeral system (base 60) == Arranged by culture == Aegean numbers – Numeral system used by the Minoans and MycenaeansPages displaying short descriptions of redirect targets Australian Aboriginal enumeration – Counting system used by Australian Aboriginals Armenian numerals Babylonian numerals – Numeral systemPages displaying short descriptions


id: list-of-numerical-libraries-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numerical_libraries-0.md
category: reference
title: List of numerical libraries
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_libraries
date_saved: 2026-05-05T08:15:55.235107+00:00
instance: kb-cron
parent: list-of-numerical-libraries-20260505-0
summary:
This is a list of numerical libraries, which are libraries used in software development for performing numerical calculations. It is not a complete listing but is instead a list of numerical libraries with articles on Wikipedia, with few exceptions. The choice of a typical library depends on a range of requirements such as: desired features (e.g. large dimensional linear algebra, parallel computation, partial differential equations), licensing, readability of API, portability or platform/compiler dependence (e.g. Linux, Windows, Visual C++, GCC), performance, ease-of-use, continued support from developers, standard compliance, specialized optimization in code for specific application scenarios or even the size of the code-base to be installed. == Multi-language == == C == == C++ == == Delphi == ALGLIB - an open source numerical analysis library. == .NET Framework languages: C#, F#, VB.NET and PowerShell == == Fortran == == Java == == OCaml == OCaml programming language has support for array programming in the standard library, also with a specific module named bigarrays for multi-dimensional, numerical arrays, with both C and Fortran layout options. A comprehensive support of numerical computations is provided by the library Owl Scientific Computing which provides methods for statistics, linear algebra (using OpenBLAS), differential equations, algorithmic


id: list-of-numerical-analysis-software-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-0.md
category: reference
title: List of numerical-analysis software
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical-analysis_software
date_saved: 2026-05-05T08:15:56.369537+00:00
instance: kb-cron
parent: list-of-numerical-analysis-software-20260505-0
summary:
Listed here are notable end-user computer applications intended for use with numerical or data analysis: == Numerical-software packages == Analytica is a widely used proprietary software tool for building and analyzing numerical models. It is a declarative and visual programming language based on influence diagrams. FlexPro is a program to analyze and present measurement data. It has a rich Excel-like user interface and a built-in vector programming language FPScript has a syntax similar to MATLAB. FreeMat, an open-source MATLAB-like environment with a GNU General Public License (GPL). GNU Octave is a high-level programming language, intended for mainly numerical computing. It has a convenient command-line interface to solve linear and nonlinear problems numerically, and to perform other numerical experiments using a language that is compatible mostly with MATLAB. The 4.0 and newer releases of Octave include a GUI. Several independently developed Linux programs (Cantor, KAlgebra) also offer GUI front-ends to Octave. An active community provides technical support to users. GroovyLab (formerly jLab), a research platform to build an open-source MATLAB-like environment in pure Java and Groovy. Supports interpreted j-Scripts (MATLAB-like) and compiled GroovySci (extension to Groovy) scripts that give direct interfacing to Java code and scripting access to many popular Java


id: list-of-numerical-analysis-software-20260505-1
file: data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-1.md
category: reference
title: List of numerical-analysis software
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical-analysis_software
date_saved: 2026-05-05T08:15:56.369537+00:00
instance: kb-cron
parent: list-of-numerical-analysis-software-20260505-0
summary:
== Interface-oriented == Baudline is a time-frequency browser for numerical signals analysis and scientific visualization. COMSOL Multiphysics is a finite-element analysis, solver and simulation software / FEA Software package for various physics and engineering applications, especially coupled phenomena, or multiphysics. Dataplot is provided by NIST. DADiSP is a commercial program focused on digital signal processing (DSP) that combines the numerical ability of MATLAB with a spreadsheet-like interface. Easy Java Simulations (EJS) is an open-source software tool, written in Java, for generating simulations. Euler Mathematical Toolbox is a powerful numerical laboratory with a programming language that can handle real, complex and interval numbers, vectors and matrices. It can produce 2D/3D plots. FEATool Multiphysics is a MATLAB GUI toolbox for finite element FEM and PDE multiphysics simulations. FEniCS Project is a collection of project for automated solutions to partial differential equations (PDEs). Hermes is a C++ library of advanced adaptive finite element algorithms to solve PDEs and multiphysics coupled problems. Fityk is a curve fitting and data-analysis program. Primarily used for peak fitting and analyzing peak data. FlexPro is a commercial program for interactive and automated analysis and presentation of mainly measurement data. It supports many binary instrument data formats and has


id: list-of-numerical-analysis-software-20260505-2
file: data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-2.md
category: reference
title: List of numerical-analysis software
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical-analysis_software
date_saved: 2026-05-05T08:15:56.369537+00:00
instance: kb-cron
parent: list-of-numerical-analysis-software-20260505-0
summary:
acslX is a software application for modeling and evaluating the performance of continuous systems described by time-dependent, nonlinear differential equations. ADMB is a software suite for non-linear statistical modeling based on C++ which uses automatic differentiation. AMPL is a mathematical modeling language for describing and solving high complexity problems for large-scale optimization. Calcpad is an open-source numerical software for engineering calculations. Supports numerical differentiation and integration, root and extrema finding, vector and matrix calculations. Ch, a commercial C/C++-based interpreted language with computational array for scientific numerical computation and visualization. APMonitor: APMonitor is a mathematical modeling language for describing and solving representations of physical systems in the form of differential and algebraic equations. Armadillo is C++ template library for linear algebra; includes various decompositions, factorisations, and statistics functions; its syntax (application programming interface (API) is similar to MATLAB. Clojure with numeric libraries Neanderthal, ClojureCUDA, and ClojureCL to call optimized matrix and linear algebra functions on CPU and GPU. Julia is designed for cloud parallel scientific computing in mind on LLVM-based just-in-time compilation (JIT) as a backend. Lightweight green threading (coroutines). Direct calls of C functions from code (no wrappers or special APIs needed), support for Unicode. Powerful shell-like abilities to manage


id: list-of-numerical-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-0.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
This is a list of numerical analysis topics. == General == Validated numerics Iterative method Rate of convergence — the speed at which a convergent sequence approaches its limit Order of accuracy — rate at which numerical solution of differential equation converges to exact solution Series acceleration — methods to accelerate the speed of convergence of a series Aitken's delta-squared process — most useful for linearly converging sequences Minimum polynomial extrapolation — for vector sequences Richardson extrapolation Shanks transformation — similar to Aitken's delta-squared process, but applied to the partial sums Van Wijngaarden transformation — for accelerating the convergence of an alternating series Abramowitz and Stegun — book containing formulas and tables of many special functions Digital Library of Mathematical Functions — successor of book by Abramowitz and Stegun Curse of dimensionality Local convergence and global convergence — whether you need a good initial guess to get convergence Superconvergence Discretization Difference quotient Complexity: Computational complexity of mathematical operations Smoothed analysis — measuring the expected performance of algorithms under slight random perturbations of worst-case inputs Symbolic-numeric computation — combination of symbolic and numeric methods Cultural and historical aspects: History of numerical solution of differential equations using computers Hundred-dollar, Hundred-digit Challenge problems


id: list-of-numerical-analysis-topics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-1.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
=== Basic concepts === Types of matrices appearing in numerical analysis: Sparse matrix Band matrix Bidiagonal matrix Tridiagonal matrix Pentadiagonal matrix Skyline matrix Circulant matrix Triangular matrix Diagonally dominant matrix Block matrix — matrix composed of smaller matrices Stieltjes matrix — symmetric positive definite with non-positive off-diagonal entries Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle) Wilkinson matrix — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues Convergent matrix — square matrix whose successive powers approach the zero matrix Algorithms for matrix multiplication: Strassen algorithm Coppersmith–Winograd algorithm Cannon's algorithm — a distributed algorithm, especially suitable for processors laid out in a 2d grid Freivalds' algorithm — a randomized algorithm for checking the result of a multiplication Matrix decompositions: LU decomposition — lower triangular times upper triangular QR decomposition — orthogonal matrix times triangular matrix RRQR factorization — rank-revealing QR factorization, can be used to compute rank of a matrix Polar decomposition — unitary matrix times positive-semidefinite Hermitian matrix Decompositions by similarity: Eigendecomposition — decomposition in terms of eigenvectors and eigenvalues Jordan normal form — bidiagonal matrix of a certain form; generalizes the eigendecomposition Weyr canonical


id: list-of-numerical-analysis-topics-20260505-2
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-2.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Spline (mathematics) — the piecewise polynomials used as interpolants Perfect spline — polynomial spline of degree m whose mth derivate is ±1 Cubic Hermite spline Centripetal Catmull–Rom spline — special case of cubic Hermite splines without self-intersections or cusps Monotone cubic interpolation Hermite spline Bézier curve De Casteljau's algorithm composite Bézier curve Generalizations to more dimensions: Bézier triangle — maps a triangle to R3 Bézier surface — maps a square to R3 B-spline Box spline — multivariate generalization of B-splines Truncated power function De Boor's algorithm — generalizes De Casteljau's algorithm Non-uniform rational B-spline (NURBS) T-spline — can be thought of as a NURBS surface for which a row of control points is allowed to terminate Kochanek–Bartels spline Coons patch — type of manifold parametrization used to smoothly join other surfaces together M-spline — a non-negative spline I-spline — a monotone spline, defined in terms of M-splines Smoothing spline — a spline fitted smoothly to noisy data Blossom (functional) — a unique, affine, symmetric map associated to a polynomial or spline See also: List of numerical computational geometry topics === Trigonometric interpolation === Trigonometric interpolation — interpolation by trigonometric polynomials Discrete Fourier transform — can be viewed as trigonometric interpolation


id: list-of-numerical-analysis-topics-20260505-3
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-3.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Orders of approximation Lebesgue's lemma Curve fitting Vector field reconstruction Modulus of continuity — measures smoothness of a function Least squares (function approximation) — minimizes the error in the L2-norm Minimax approximation algorithm — minimizes the maximum error over an interval (the L∞-norm) Equioscillation theorem — characterizes the best approximation in the L∞-norm Unisolvent point set — function from given function space is determined uniquely by values on such a set of points Stone–Weierstrass theorem — continuous functions can be approximated uniformly by polynomials, or certain other function spaces Approximation by polynomials: Linear approximation Bernstein polynomial — basis of polynomials useful for approximating a function Bernstein's constant — error when approximating |x| by a polynomial Remez algorithm — for constructing the best polynomial approximation in the L∞-norm Bernstein's inequality (mathematical analysis) — bound on maximum of derivative of polynomial in unit disk Mergelyan's theorem — generalization of Stone–Weierstrass theorem for polynomials Müntz–Szász theorem — variant of Stone–Weierstrass theorem for polynomials if some coefficients have to be zero Bramble–Hilbert lemma — upper bound on Lp error of polynomial approximation in multiple dimensions Discrete Chebyshev polynomials — polynomials orthogonal with respect to a discrete measure Favard's theorem — polynomials satisfying suitable 3-term


id: list-of-numerical-analysis-topics-20260505-4
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-4.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Algorithms for linear programming: Simplex algorithm Bland's rule — rule to avoid cycling in the simplex method Klee–Minty cube — perturbed (hyper)cube; simplex method has exponential complexity on such a domain Criss-cross algorithm — similar to the simplex algorithm Big M method — variation of simplex algorithm for problems with both "less than" and "greater than" constraints Interior point method Ellipsoid method Karmarkar's algorithm Mehrotra predictor–corrector method Column generation k-approximation of k-hitting set — algorithm for specific LP problems (to find a weighted hitting set) Linear complementarity problem Decompositions: Benders' decomposition Dantzig–Wolfe decomposition Theory of two-level planning Variable splitting Basic solution (linear programming) — solution at vertex of feasible region Fourier–Motzkin elimination Hilbert basis (linear programming) — set of integer vectors in a convex cone which generate all integer vectors in the cone LP-type problem Linear inequality Vertex enumeration problem — list all vertices of the feasible set === Convex optimization === Convex optimization Quadratic programming Linear least squares (mathematics) Total least squares Frank–Wolfe algorithm Sequential minimal optimization — breaks up large QP problems into a series of smallest possible QP problems Bilinear program Basis pursuit — minimize L1-norm of vector subject to linear constraints Basis pursuit denoising (BPDN) —


id: list-of-numerical-analysis-topics-20260505-5
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-5.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Semi-infinite programming — infinite number of variables and finite number of constraints, or other way around Shape optimization, Topology optimization — optimization over a set of regions Topological derivative — derivative with respect to changing in the shape Generalized semi-infinite programming — finite number of variables, infinite number of constraints === Uncertainty and randomness === Approaches to deal with uncertainty: Markov decision process Partially observable Markov decision process Robust optimization Wald's maximin model Scenario optimization — constraints are uncertain Stochastic approximation Stochastic optimization Stochastic programming Stochastic gradient descent Random optimization algorithms: Random search — choose a point randomly in ball around current iterate Simulated annealing Adaptive simulated annealing — variant in which the algorithm parameters are adjusted during the computation. Great Deluge algorithm Mean field annealing — deterministic variant of simulated annealing Bayesian optimization — treats objective function as a random function and places a prior over it Evolutionary algorithm Differential evolution Evolutionary programming Genetic algorithm, Genetic programming Genetic algorithms in economics MCACEA (Multiple Coordinated Agents Coevolution Evolutionary Algorithm) — uses an evolutionary algorithm for every agent Simultaneous perturbation stochastic approximation (SPSA) Luus–Jaakola Particle swarm optimization Stochastic tunneling Harmony search — mimicks the improvisation process of musicians see also the


id: list-of-numerical-analysis-topics-20260505-6
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-6.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Rectangle method — first-order method, based on (piecewise) constant approximation Trapezoidal rule — second-order method, based on (piecewise) linear approximation Simpson's rule — fourth-order method, based on (piecewise) quadratic approximation Adaptive Simpson's method Boole's rule — sixth-order method, based on the values at five equidistant points Newton–Cotes formulas — generalizes the above methods Romberg's method — Richardson extrapolation applied to trapezium rule Gaussian quadrature — highest possible degree with given number of points Chebyshev–Gauss quadrature — extension of Gaussian quadrature for integrals with weight (1 − x2)±1/2 on [−1, 1] Gauss–Hermite quadrature — extension of Gaussian quadrature for integrals with weight exp(−x2) on [−∞, ∞] Gauss–Jacobi quadrature — extension of Gaussian quadrature for integrals with weight (1 − x)α (1 + x)β on [−1, 1] Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight exp(−x) on [0, ∞] Gauss–Kronrod quadrature formula — nested rule based on Gaussian quadrature Gauss–Kronrod rules Tanh-sinh quadrature — variant of Gaussian quadrature which works well with singularities at the end points Clenshaw–Curtis quadrature — based on expanding the integrand in terms of Chebyshev polynomials Adaptive quadrature — adapting the subintervals in which the integration interval is divided depending on the integrand Monte Carlo


id: list-of-numerical-analysis-topics-20260505-7
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-7.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Finite difference — the discrete analogue of a differential operator Finite difference coefficient — table of coefficients of finite-difference approximations to derivatives Discrete Laplace operator — finite-difference approximation of the Laplace operator Eigenvalues and eigenvectors of the second derivative — includes eigenvalues of discrete Laplace operator Kronecker sum of discrete Laplacians — used for Laplace operator in multiple dimensions Discrete Poisson equation — discrete analogue of the Poisson equation using the discrete Laplace operator Stencil (numerical analysis) — the geometric arrangements of grid points affected by a basic step of the algorithm Compact stencil — stencil which only uses a few grid points, usually only the immediate and diagonal neighbours Higher-order compact finite difference scheme Non-compact stencil — any stencil that is not compact Five-point stencil — two-dimensional stencil consisting of a point and its four immediate neighbours on a rectangular grid Finite difference methods for heat equation and related PDEs: FTCS scheme (forward-time central-space) — first-order explicit Crank–Nicolson method — second-order implicit Finite difference methods for hyperbolic PDEs like the wave equation: Lax–Friedrichs method — first-order explicit Lax–Wendroff method — second-order explicit MacCormack method — second-order explicit Upwind scheme Upwind differencing scheme for convection — first-order scheme for convection–diffusion


id: list-of-numerical-analysis-topics-20260505-8
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-8.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
=== Other methods === Spectral method — based on the Fourier transformation Pseudo-spectral method Method of lines — reduces the PDE to a large system of ordinary differential equations Boundary element method (BEM) — based on transforming the PDE to an integral equation on the boundary of the domain Interval boundary element method — a version using interval arithmetics Analytic element method — similar to the boundary element method, but the integral equation is evaluated analytically Finite volume method — based on dividing the domain in many small domains; popular in computational fluid dynamics Godunov's scheme — first-order conservative scheme for fluid flow, based on piecewise constant approximation MUSCL scheme — second-order variant of Godunov's scheme AUSM — advection upstream splitting method Flux limiter — limits spatial derivatives (fluxes) in order to avoid spurious oscillations Riemann solver — a solver for Riemann problems (a conservation law with piecewise constant data) Properties of discretization schemes — finite volume methods can be conservative, bounded, etc. Discrete element method — a method in which the elements can move freely relative to each other Extended discrete element method — adds properties such as strain to each particle Movable cellular automaton — combination of cellular


id: list-of-numerical-analysis-topics-20260505-9
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-9.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
=== Grids and meshes === Grid classification / Types of mesh: Polygon mesh — consists of polygons in 2D or 3D Triangle mesh — consists of triangles in 2D or 3D Triangulation (geometry) — subdivision of given region in triangles, or higher-dimensional analogue Nonobtuse mesh — mesh in which all angles are less than or equal to 90° Point-set triangulation — triangle mesh such that given set of point are all a vertex of a triangle Polygon triangulation — triangle mesh inside a polygon Delaunay triangulation — triangulation such that no vertex is inside the circumcentre of a triangle Constrained Delaunay triangulation — generalization of the Delaunay triangulation that forces certain required segments into the triangulation Pitteway triangulation — for any point, triangle containing it has nearest neighbour of the point as a vertex Minimum-weight triangulation — triangulation of minimum total edge length Kinetic triangulation — a triangulation that moves over time Triangulated irregular network Quasi-triangulation — subdivision into simplices, where vertices are not points but arbitrary sloped line segments Volume mesh — consists of three-dimensional shapes Regular grid — consists of congruent parallelograms, or higher-dimensional analogue Unstructured grid Geodesic grid — isotropic grid on a sphere Mesh generation Image-based


id: list-of-numerical-computational-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics-0.md
category: reference
title: List of numerical computational geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics
date_saved: 2026-05-05T08:15:58.970321+00:00
instance: kb-cron
parent: list-of-numerical-computational-geometry-topics-20260505-0
summary:
List of numerical computational geometry topics enumerates the topics of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis. This area is also called "machine geometry", computer-aided geometric design, and geometric modelling. See List of combinatorial computational geometry topics for another flavor of computational geometry that states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character. == Curves == In the list of curves topics, the following ones are fundamental to geometric modelling. Parametric curve Bézier curve Spline Hermite spline Beta spline B-spline Higher-order spline NURBS Contour line == Surfaces == Bézier surface Isosurface Parametric surface == Other == Level-set method Computational topology


id: list-of-operator-splitting-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_operator_splitting_topics-0.md
category: reference
title: List of operator splitting topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_operator_splitting_topics
date_saved: 2026-05-05T08:16:00.110845+00:00
instance: kb-cron
parent: list-of-operator-splitting-topics-20260505-0
summary:
This is a list of operator splitting topics. == General == Alternating direction implicit method — finite difference method for parabolic, hyperbolic, and elliptic partial differential equations GRADELA — simple gradient elasticity model Matrix splitting — general method of splitting a matrix operator into a sum or difference of matrices Paul Tseng — resolved question on convergence of matrix splitting algorithms PISO algorithm — pressure-velocity calculation for Navier-Stokes equations Projection method (fluid dynamics) — computational fluid dynamics method Reactive transport modeling in porous media — modeling of chemical reactions and fluid flow through the Earth's crust Richard S. Varga — developed matrix splitting Strang splitting — specific numerical method for solving differential equations using operator splitting


id: list-of-mathematic-operators-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematic_operators-0.md
category: reference
title: List of mathematic operators
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematic_operators
date_saved: 2026-05-05T08:16:01.392792+00:00
instance: kb-cron
parent: list-of-mathematic-operators-20260505-0
summary:
In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differential operators. In the following L is an operator L : F → G {\displaystyle L:{\mathcal {F}}\to {\mathcal {G}}} which takes a function y ∈ F {\displaystyle y\in {\mathcal {F}}} to another function L [ y ] ∈ G {\displaystyle L[y]\in {\mathcal {G}}} . Here, F {\displaystyle {\mathcal {F}}} and G {\displaystyle {\mathcal {G}}} are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions. == See also == List of transforms List of Fourier-related transforms Transfer operator Fredholm operator Borel transform Glossary of mathematical symbols


id: list-of-order-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_order_theory_topics-0.md
category: reference
title: List of order theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_order_theory_topics
date_saved: 2026-05-05T08:16:02.632494+00:00
instance: kb-cron
parent: list-of-order-theory-topics-20260505-0
summary:
Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value and mathematical optimization. == Overview == Partially ordered set Preorder Totally ordered set Total preorder Chain Trichotomy Extended real number line Antichain Strict order Hasse diagram Directed acyclic graph Duality (order theory) Product order == Distinguished elements of partial orders == Greatest element (maximum, top, unit), Least element (minimum, bottom, zero) Maximal element, minimal element Upper bound Least upper bound (supremum, join) Greatest lower bound (infimum, meet) Limit superior and limit inferior Irreducible element Prime element Compact element == Subsets of partial orders == Cofinal and coinitial set, sometimes also called dense Meet-dense set and join-dense set Linked set (upwards and downwards) Directed set (upwards and downwards) centered and σ-centered set Net (mathematics) Upper set and lower set Ideal and filter Ultrafilter == Special types of partial orders == Completeness (order theory) Dense order Distributivity (order theory) Modular lattice Distributive lattice


id: list-of-order-structures-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_order_structures_in_mathematics-0.md
category: reference
title: List of order structures in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_order_structures_in_mathematics
date_saved: 2026-05-05T08:16:03.849108+00:00
instance: kb-cron
parent: list-of-order-structures-in-mathematics-20260505-0
summary:
In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities) Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions Total orders, orderings that specify, for every two distinct elements, which one is less than the other Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities) Well-orders, total orders in which every non-empty subset has a least element Well-quasi-orderings, a class of preorders generalizing the well-orders == See also == Glossary of order theory List of order theory topics


id: list-of-partial-differential-equation-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_partial_differential_equation_topics-0.md
category: reference
title: List of partial differential equation topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_partial_differential_equation_topics
date_saved: 2026-05-05T08:16:05.112454+00:00
instance: kb-cron
parent: list-of-partial-differential-equation-topics-20260505-0
summary:
This is a list of partial differential equation topics. == General topics == Partial differential equation Nonlinear partial differential equation list of nonlinear partial differential equations Boundary condition Boundary value problem Dirichlet problem, Dirichlet boundary condition Neumann boundary condition Stefan problem Wiener–Hopf problem Separation of variables Green's function Elliptic partial differential equation Singular perturbation Cauchy–Kovalevskaya theorem H-principle Atiyah–Singer index theorem Bäcklund transform Viscosity solution Weak solution Loewy decomposition of linear differential equations == Specific partial differential equations == Broer–Kaup equations Burgers' equation Euler equations Fokker–Planck equation Hamilton–Jacobi equation, Hamilton–Jacobi–Bellman equation Heat equation Laplace's equation Laplace operator Harmonic function Spherical harmonic Poisson integral formula Klein–Gordon equation Korteweg–de Vries equation Modified KdV–Burgers equation Maxwell's equations Navier–Stokes equations Poisson's equation Primitive equations (hydrodynamics) Schrödinger equation Wave equation == Numerical methods for PDEs == Finite difference Finite element method Finite volume method Boundary element method Multigrid Spectral method Computational fluid dynamics Alternating direction implicit == Related areas of mathematics == Calculus of variations Harmonic analysis Ordinary differential equation Sobolev space


id: list-of-partition-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_partition_topics-0.md
category: reference
title: List of partition topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_partition_topics
date_saved: 2026-05-05T08:16:06.339911+00:00
instance: kb-cron
parent: list-of-partition-topics-20260505-0
summary:
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are partition of a set or an ordered partition of a set, partition of a graph, partition of an integer, partition of an interval, partition of unity, partition of a matrix; see block matrix, and partition of the sum of squares in statistics problems, especially in the analysis of variance, quotition and partition, two ways of viewing the operation of division of integers. == Integer partitions == Composition (combinatorics) Ewens's sampling formula Ferrers graph Glaisher's theorem Landau's function Partition function (number theory) Pentagonal number theorem Plane partition Quotition and partition Rank of a partition Crank of a partition Solid partition Young tableau Young's lattice == Set partitions == Bell number Bell polynomials Dobinski's formula Cumulant Data clustering Equivalence relation Exact cover Knuth's Algorithm X Dancing Links Exponential formula Faà di Bruno's formula Feshbach–Fano partitioning Foliation Frequency partition Graph partition Kernel of a function Lamination (topology) Matroid partitioning Multipartition Multiplicative partition Noncrossing partition Ordered partition of a set Partition calculus Partition function (quantum field theory) Partition function (statistical mechanics) Derivation of the partition function Partition of an interval Partition of a


id: list-of-periodic-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_periodic_functions-0.md
category: reference
title: List of periodic functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_periodic_functions
date_saved: 2026-05-05T08:16:07.555333+00:00
instance: kb-cron
parent: list-of-periodic-functions-20260505-0
summary:
This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions. == Smooth functions == All trigonometric functions listed have period 2 π {\displaystyle 2\pi } , unless otherwise stated. For the following trigonometric functions: Un is the nth up/down number, Bn is the nth Bernoulli number in Jacobi elliptic functions, q = e − π K ( 1 − m ) K ( m ) {\displaystyle q=e^{-\pi {\frac {K(1-m)}{K(m)}}}} == Non-smooth functions == The following functions have period p {\displaystyle p} and take x {\displaystyle x} as their argument. The symbol ⌊ n ⌋ {\displaystyle \lfloor n\rfloor } is the floor function of n {\displaystyle n} and sgn {\displaystyle \operatorname {sgn} } is the sign function. K means Elliptic integral K(m) == Vector-valued functions == Epitrochoid Epicycloid (special case of the epitrochoid) Limaçon (special case of the epitrochoid) Hypotrochoid Hypocycloid (special case of the hypotrochoid) Spirograph (special case of the hypotrochoid) == Doubly periodic functions == Jacobi's elliptic functions Weierstrass's


id: list-of-permutation-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_permutation_topics-0.md
category: reference
title: List of permutation topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_permutation_topics
date_saved: 2026-05-05T08:16:08.751910+00:00
instance: kb-cron
parent: list-of-permutation-topics-20260505-0
summary:
This is a list of topics on mathematical permutations. == Particular kinds of permutations == Alternating permutation Circular shift Cyclic permutation Derangement Even and odd permutations—see Parity of a permutation Josephus permutation Parity of a permutation Separable permutation Stirling permutation Superpattern Transposition (mathematics) Unpredictable permutation == Combinatorics of permutations == Bijection Combination Costas array Cycle index Cycle notation Cycles and fixed points Cyclic order Direct sum of permutations Enumerations of specific permutation classes Factorial Falling factorial Permutation matrix Generalized permutation matrix Inversion (discrete mathematics) Major index Ménage problem Permutation graph Permutation pattern Permutation polynomial Permutohedron Rencontres numbers Robinson–Schensted correspondence Sum of permutations: Direct sum of permutations Skew sum of permutations Stanley–Wilf conjecture Symmetric function Szymanski's conjecture Twelvefold way == Permutation groups and other algebraic structures == === Groups === Alternating group Automorphisms of the symmetric and alternating groups Block (permutation group theory) Cayley's theorem Cycle index Frobenius group Galois group of a polynomial Jucys–Murphy element Landau's function Oligomorphic group O'Nan–Scott theorem Parker vector Permutation group Place-permutation action Primitive permutation group Rank 3 permutation group Representation theory of the symmetric group Schreier vector Strong generating set Symmetric group Symmetric inverse semigroup Weak order of permutations Wreath product Young symmetrizer Zassenhaus group


id: list-of-formulae-involving-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-0.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:
The following is a list of significant formulae involving the mathematical constant π. Many of these formulae can be found in the article Pi, or the article Approximations of π. == Euclidean geometry == π = C d = C 2 r {\displaystyle \pi ={\frac {C}{d}}={\frac {C}{2r}}} where C is the circumference of a circle, d is the diameter, and r is the radius. More generally, π = L w {\displaystyle \pi ={\frac {L}{w}}} where L and w are, respectively, the perimeter and the width of any curve of constant width. A = π r 2 {\displaystyle A=\pi r^{2}} where A is the area of a circle. More generally, A = π a b {\displaystyle A=\pi ab} where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b. C = 2 π agm ⁡ ( a , b ) ( a 1 2 − ∑ n = 2 ∞ 2 n − 1 ( a n 2 − b n 2 ) ) {\displaystyle C={\frac {2\pi }{\operatorname {agm} (a,b)}}\left(a_{1}^{2}-\sum _{n=2}^{\infty }2^{n-1}(a_{n}^{2}-b_{n}^{2})\right)} where C is the circumference of an ellipse with semi-major axis a and semi-minor axis b and a n , b n {\displaystyle


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


