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wiki_1500_chunk_0 | Theory of equivalence | In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism. For example, if M and N are two Riemannian manifolds with metrics g and h, respectively, when is there a diffeomorphism ϕ: M → N {\displaystyle \phi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1501_chunk_0 | Cartan's lemma | In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan: In exterior algebra: Suppose that v1, ..., vp are linearly independent elements of a vector space V and w1, ..., wp are such that v 1 ∧ w 1 + ⋯ + v p ∧ w p = 0 {\displaystyle v_{1}\wedge w_{1}+\cdots +v_... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1502_chunk_0 | Cartan's theorems A and B | In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology. Theorem B is stated in cohomological terms (a formulat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1503_chunk_0 | Cartan's theorems A and B | The analogue of Theorem B in this context is as follows (Hartshorne 1977, Theorem III.3.7): These theorems have many important applications. For instance, they imply that a holomorphic function on a closed complex submanifold, Z, of a Stein manifold X can be extended to a holomorphic function on all of X. At a deeper l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1504_chunk_0 | Cartan's theorems A and B | quasi-coherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see (Serre 1956) (resp. (Serre 1957) and (Hartshorne 1977, Theorem III.3.7)). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1505_chunk_0 | Casey's theorem | In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1506_chunk_0 | Castelnuovo's contraction theorem | In mathematics, Castelnuovo's contraction theorem is used in the classification theory of algebraic surfaces to construct the minimal model of a given smooth algebraic surface. More precisely, let X {\displaystyle X} be a smooth projective surface over C {\displaystyle \mathbb {C} } and C {\displaystyle C} a (−1)-curve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1507_chunk_0 | Catalan's constant | In mathematics, Catalan's constant G, is defined by G = β ( 2 ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 = 1 1 2 − 1 3 2 + 1 5 2 − 1 7 2 + 1 9 2 − ⋯ , {\displaystyle G=\beta (2)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1508_chunk_0 | Cauchy integral formula | In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a hol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1509_chunk_0 | Cayley's Ω process | In mathematics, Cayley's Ω process, introduced by Arthur Cayley (1846), is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action. As a partial differential operator acting on functions of n2 variables xij, the omega operator is given by the dete... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1510_chunk_0 | Charlier polynomials | In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by C n ( x ; μ ) = 2 F 0 ( − n , − x ; − ; − 1 / μ ) = ( − 1 ) n n ! L n ( − 1 − x ) ( − 1 μ ) , {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1511_chunk_0 | Chebyshev distance | In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. It is named after Pafnuty Chebyshev. It is also known as chessboard distance, since ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1512_chunk_0 | Chebyshev's sum inequality | In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if a 1 ≥ a 2 ≥ ⋯ ≥ a n {\displaystyle a_{1}\geq a_{2}\geq \cdots \geq a_{n}\quad } and b 1 ≥ b 2 ≥ ⋯ ≥ b n , {\displaystyle \quad b_{1}\geq b_{2}\geq \cdots \geq b_{n},} then 1 n ∑ k = 1 n a k b k ≥ ( 1 n ∑ k = 1 n a k ) ( 1 n ∑ k = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1513_chunk_0 | Choi's theorem on completely positive maps | In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon–Nikodym" theorem for completely positive maps. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1514_chunk_0 | Choquet theory | In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1515_chunk_0 | Choquet theory | The two ends of a line segment determine the points in between: in vector terms the segment from v to w consists of the λv + (1 − λ)w with 0 ≤ λ ≤ 1. The classical result of Hermann Minkowski says that in Euclidean space, a bounded, closed convex set C is the convex hull of its extreme point set E, so that any c in C i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1516_chunk_0 | Chow's theorem | In mathematics, Chow's theorem may refer to a number of theorems due to Wei-Liang Chow: Chow's theorem: The theorem that asserts that any analytic subvariety in projective space is actually algebraic. Chow–Rashevskii theorem: In sub-Riemannian geometry, the theorem that asserts that any two points are connected by a ho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1517_chunk_0 | Chrystal's equation | In mathematics, Chrystal's equation is a first order nonlinear ordinary differential equation, named after the mathematician George Chrystal, who discussed the singular solution of this equation in 1896. The equation reads as ( d y d x ) 2 + A x d y d x + B y + C x 2 = 0 {\displaystyle \left({\frac {dy}{dx}}\right)^{2}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1518_chunk_0 | Church numerals | Terms that are usually considered primitive in other notations (such as integers, booleans, pairs, lists, and tagged unions) are mapped to higher-order functions under Church encoding. The Church-Turing thesis asserts that any computable operator (and its operands) can be represented under Church encoding. In the untyp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1519_chunk_0 | Clarke's generalized Jacobian | In mathematics, Clarke's generalized Jacobian is a generalization of the Jacobian matrix of a smooth function to non-smooth functions. It was introduced by Clarke (1983). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1520_chunk_0 | Clausen's formula | In mathematics, Clausen's formula, found by Thomas Clausen (1828), expresses the square of a Gaussian hypergeometric series as a generalized hypergeometric series. It states 2 F 1 2 = 3 F 2 {\displaystyle \;_{2}F_{1}\left^{2}=\;_{3}F_{2}\left} In particular it gives conditions for a hypergeometric series to be positi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1521_chunk_0 | Green's conjecture | In mathematics, Clifford's theorem on special divisors is a result of William K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve C. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1522_chunk_0 | Cohn's theorem | In mathematics, Cohn's theorem states that a nth-degree self-inversive polynomial p ( z ) {\displaystyle p(z)} has as many roots in the open unit disk D = { z ∈ C: | z | < 1 } {\displaystyle D=\{z\in \mathbb {C} :|z|<1\}} as the reciprocal polynomial of its derivative. Cohn's theorem is useful for studying the distribu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1523_chunk_0 | Cohn's theorem | The coefficients of self-inversive polynomials satisfy the relations. p k = ω p ¯ n − k , 0 ⩽ k ⩽ n . {\displaystyle p_{k}=\omega {\bar {p}}_{n-k},\qquad 0\leqslant k\leqslant n.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1524_chunk_0 | Cohn's theorem | In the case where ω = 1 , {\displaystyle \omega =1,} a self-inversive polynomial becomes a complex-reciprocal polynomial (also known as a self-conjugate polynomial). If its coefficients are real then it becomes a real self-reciprocal polynomial. The formal derivative of p ( z ) {\displaystyle p(z)} is a (n − 1)th-degre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1525_chunk_0 | Cohn's theorem | {\displaystyle q(z)=p'(z)=p_{1}+2p_{2}z+\cdots +np_{n}z^{n-1}.} Therefore, Cohn's theorem states that both p ( z ) {\displaystyle p(z)} and the polynomial q ∗ ( z ) = z n − 1 q ¯ n − 1 ( 1 / z ¯ ) = z n − 1 p ¯ ′ ( 1 / z ¯ ) = n p ¯ n + ( n − 1 ) p ¯ n − 1 z + ⋯ + p ¯ 1 z n − 1 {\displaystyle q^{*}(z)=z^{n-1}{\bar {q}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1526_chunk_0 | Cramer's paradox | It is the result of a naive understanding or a misapplication of two theorems: Bézout's theorem states that the number of points of intersection of two algebraic curves is equal to the product of their degrees, provided that certain necessary conditions are met. In particular, two curves of degree n {\displaystyle n} g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1527_chunk_0 | Danzer's configuration | Every middle layer graph is Hamiltonian.Danzer's configuration DCD(4) is the fourth term of an infinite series of ( ( 2 n − 1 n ) n ) {\displaystyle ({\tbinom {2n-1}{n}}_{n})} configurations DCD(n), where DCD(1) is the trivial configuration (11), DCD(2) is the trilateral (32) and DCD(3) is the Desargues configuration (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1528_chunk_0 | Darboux function | In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval. When ƒ is continuously differentiable (ƒ in C1()), ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1529_chunk_0 | De Gua's theorem | In mathematics, De Gua's theorem is a three-dimensional analog of the Pythagorean theorem named after Jean Paul de Gua de Malves. It states that if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1530_chunk_0 | Dedekind sum | In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have subsequently been much studied in number theory, and have occurred in some prob... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1531_chunk_0 | Degen's eight-square identity | However, in the 1960s, H. Zassenhaus, W. Eichhorn, and A. Pfister (independently) showed there can be a non-bilinear identity for 16 squares. Note that each quadrant reduces to a version of Euler's four-square identity: and similarly for the other three quadrants. Comment: The proof of the eight-square identity is by a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1532_chunk_0 | Degen's eight-square identity | The eight-square identity can be written in the form of a product of two inner products of 8-dimensional vectors, yielding again an inner product of 8-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b). This defines the octonion multiplication rule a×b, which reflects Degen's 8-square identity and the mathematics of octonio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1533_chunk_0 | Dehn's lemma | In mathematics, Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disk's interior, implies the existence of another piecewise-linear map of the disk which is an embedding and is identical to the original on the boundary of the disk. This theorem was thou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1534_chunk_0 | Denisyuk polynomials | In mathematics, Denisyuk polynomials Den(x) or Mn(x) are generalizations of the Laguerre polynomials introduced by Denisyuk (1954) given by the generating function | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1535_chunk_0 | Descartes' rule of signs | By a linear fractional transformation of the variable, one may use Descartes' rule of signs for getting a similar information on the number of roots in any interval. This is the basic idea of Budan's theorem and Budan–Fourier theorem. By repeating the division of an interval into two intervals, one gets eventually a li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1536_chunk_0 | Dickson's lemma | In mathematics, Dickson's lemma states that every set of n {\displaystyle n} -tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it to prove a result in number theory about perfect numbers. However, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1537_chunk_0 | Dieudonné's theorem | In mathematics, Dieudonné's theorem, named after Jean Dieudonné, is a theorem on when the Minkowski sum of closed sets is closed. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1538_chunk_0 | Dini criterion | In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by Ulisse Dini (1880). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1539_chunk_0 | Diophantine geometry | In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Diophantine geometry is part of the broader field of ari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1540_chunk_0 | Lobachevsky integral formula | In mathematics, Dirichlet integrals play an important role in distribution theory. We can see the Dirichlet integral in terms of distributions. One of those is the improper integral of the sinc function over the positive real line, ∫ 0 ∞ sin x x d x = ∫ 0 ∞ sin 2 x x 2 d x = π 2 . {\displaystyle \int _{0}^{\infty }... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1541_chunk_0 | Dirichlet's test | In mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1542_chunk_0 | Regulator of an algebraic number field | In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are. The state... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1543_chunk_0 | Regulator of an algebraic number field | The theory for real quadratic fields is essentially the theory of Pell's equation. The rank is positive for all number fields besides Q {\displaystyle \mathbb {Q} } and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a determinant called the regulator. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1544_chunk_0 | Regulator of an algebraic number field | The theorem not only applies to the maximal order OK but to any order O ⊂ OK. There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structure of the group of S-units, determining the rank of the unit group in localizations of rings of integers. Also, the Galois modul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1545_chunk_0 | Dixon's identity | In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon M... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1546_chunk_0 | Dodgson condensation | In mathematics, Dodgson condensation or method of contractants is a method of computing the determinants of square matrices. It is named for its inventor, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author). The method in the case of an n × n matrix is to construct an (n − 1) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1547_chunk_0 | Doob's martingale inequality | In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time. As the name suggests, the result is usually given in the cas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1548_chunk_0 | Dvoretzky's theorem | In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are ap... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1549_chunk_0 | E-function | In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendental number theory, and are more special than G-functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1550_chunk_0 | E6 (mathematics) | In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras e 6 {\displaystyle {\mathfrak {e}}_{6}} , all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. The designation E6 comes from the Cartan–Killing c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1551_chunk_0 | E6 (mathematics) | The fundamental group of the complex form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. Its fundamental representation is 27-dimensional (complex), and a basis is given by the 27 lines on a cubic surface. The dual representation, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1552_chunk_0 | E7 (mathematics) | In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simpl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1553_chunk_0 | E8 Lie algebra | In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1554_chunk_0 | Eisenstein criterion | In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1555_chunk_0 | Eisenstein's theorem | In mathematics, Eisenstein's theorem, named after the German mathematician Gotthold Eisenstein, applies to the coefficients of any power series which is an algebraic function with rational number coefficients. Through the theorem, it is readily demonstrable, for example, that the exponential function must be a transcen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1556_chunk_0 | Eisenstein–Kronecker number | In mathematics, Eisenstein–Kronecker numbers are an analogue for imaginary quadratic fields of generalized Bernoulli numbers. They are defined in terms of classical Eisenstein–Kronecker series, which were studied by Kenichi Bannai and Shinichi Kobayashi using the Poincaré bundle.Eisenstein–Kronecker numbers are algebra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1557_chunk_0 | Erdős space | If the set of all homeomorphisms of the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (for n ≥ 2 {\displaystyle n\geq 2} ) that leave invariant the set Q n {\displaystyle \mathbb {Q} ^{n}} of rational vectors is endowed with the compact-open topology, it becomes homeomorphic to the Erdős space.Erdős space also s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1558_chunk_0 | Esakia duality | Let H be a Heyting algebra, X denote the set of prime filters of H, and ≤ denote set-theoretic inclusion on the prime filters of H. Also, for each a ∈ H, let φ(a) = {x ∈ X: a ∈ x}, and let τ denote the topology on X generated by {φ(a), X − φ(a): a ∈ H}. Theorem: (X, τ, ≤) is an Esakia space, called the Esakia dual of H... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1559_chunk_0 | Esenin-Volpin's theorem | In mathematics, Esenin-Volpin's theorem states that weight of an infinite compact dyadic space is the supremum of the weights of its points. It was introduced by Alexander Esenin-Volpin (1949). It was generalized by (Efimov 1965) and (Turzański 1992). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1560_chunk_0 | Euclid number | In mathematics, Euclid numbers are integers of the form En = pn # + 1, where pn # is the nth primorial, i.e. the product of the first n prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1561_chunk_0 | Euler's differential equation | In mathematics, Euler's differential equation is a first-order non-linear ordinary differential equation, named after Leonhard Euler. It is given by: d y d x + a 0 + a 1 y + a 2 y 2 + a 3 y 3 + a 4 y 4 a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 = 0 {\displaystyle {\frac {dy}{dx}}+{\frac {\sqrt {a_{0}+a_{1}y+a_{2}y^{2}+a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1562_chunk_0 | Four squares formula | In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1563_chunk_0 | Euler's identity | In mathematics, Euler's identity (also known as Euler's equation) is the equality where e is Euler's number, the base of natural logarithms, i is the imaginary unit, which by definition satisfies i2 = −1, and π is pi, the ratio of the circumference of a circle to its diameter.Euler's identity is named after the Swiss m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1564_chunk_0 | F4 (mathematics) | In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1565_chunk_0 | F4 (mathematics) | There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras. The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E8. In older books and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1566_chunk_0 | Fatou's lemma | In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1567_chunk_0 | Bernoulli's formula | In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers as a polynomial in n. In modern notation, Faulhaber's formula is Here, ( p + 1 k ) {\textstyle {\binom {p+1}{k}}} is the binomial coefficient "p +... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1568_chunk_0 | Favard theorem | In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable 3-term recurrence relation is a sequence of orthogonal polynomials. The theorem was introduced in the theory of orthogonal polynomials by Favard (1935) and Shohat (1938), though essential... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1569_chunk_0 | Fejér's theorem | In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following: | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1570_chunk_0 | Klein J-invariant | In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that j ( e 2 π i / 3... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1571_chunk_0 | Fenchel duality | In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel. Let ƒ be a proper convex function on Rn and let g be a proper concave function on Rn. Then, if regularity conditions are satisfied, inf x ( f ( x ) − g ( x ) ) = sup p ( g ∗ ( p ) − f ∗ ( p ) ) . {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1572_chunk_0 | Fermat's theorem (stationary points) | In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is a theorem in r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1573_chunk_0 | Ferrers function | In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions. They are named after Norman Macleod Ferrers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1574_chunk_0 | Fischer's inequality | In mathematics, Fischer's inequality gives an upper bound for the determinant of a positive-semidefinite matrix whose entries are complex numbers in terms of the determinants of its principal diagonal blocks. Suppose A, C are respectively p×p, q×q positive-semidefinite complex matrices and B is a p×q complex matrix. Le... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1575_chunk_0 | Fischer's inequality | {\displaystyle \det(M)\leq \det(A)\det(C).} If M is positive-definite, equality is achieved in Fischer's inequality if and only if all the entries of B are 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1576_chunk_0 | Fischer's inequality | Inductively one may conclude that a similar inequality holds for a block decomposition of M with multiple principal diagonal blocks. Considering 1×1 blocks, a corollary is Hadamard's inequality. On the other hand, Fischer's inequality can also be proved by using Hadamard's inequality, see the proof of Theorem 7.8.5 in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1577_chunk_0 | KPP equation | In mathematics, Fisher's equation (named after statistician and biologist Ronald Fisher) also known as the Kolmogorov–Petrovsky–Piskunov equation (named after Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov), KPP equation or Fisher–KPP equation is the partial differential equation:It is a kind of reaction–diffu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1578_chunk_0 | Contact homology | In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1579_chunk_0 | Contact homology | Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current inve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1580_chunk_0 | Contact homology | Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action functional. For the (instanton) version for three-manifolds, it is th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1581_chunk_0 | Contact homology | Loosely speaking, Floer homology is the Morse homology of the function on the infinite-dimensional manifold. A Floer chain complex is formed from the abelian group spanned by the critical points of the function (or possibly certain collections of critical points). The differential of the chain complex is defined by cou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1582_chunk_0 | Contact homology | Floer homology is the homology of this chain complex. The gradient flow line equation, in a situation where Floer's ideas can be successfully applied, is typically a geometrically meaningful and analytically tractable equation. For symplectic Floer homology, the gradient flow equation for a path in the loopspace is (a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1583_chunk_0 | Fourier synthesis | In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1584_chunk_0 | Fourier synthesis | The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1585_chunk_0 | Fourier–Bessel series | In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions. Fourier–Bessel series are used in the solution to partial differential equations, particularly in cylindrical coordinate systems. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1586_chunk_0 | Fredholm index | In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator T: X → Y between two Banach spaces with finite-dimensional kernel ker T {\displaystyle \ker ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1587_chunk_0 | Fredholm theory | In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kern... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1588_chunk_0 | Fredholm's theorem | In mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear algebra, or in terms of the Fredholm operator on Banach spaces. The Fredho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1589_chunk_0 | Wave surface | In mathematics, Fresnel's wave surface, found by Augustin-Jean Fresnel in 1822, is a quartic surface describing the propagation of light in an optically biaxial crystal. Wave surfaces are special cases of tetrahedroids which are in turn special cases of Kummer surfaces. In projective coordinates (w:x:y:z) the wave surf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1590_chunk_0 | Friedrichs' inequality | In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent. Friedri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1591_chunk_0 | Frobenius integration theorem | In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and suffic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1592_chunk_0 | Frobenius integration theorem | The theorem is foundational in differential topology and calculus on manifolds. Contact geometry studies 1-forms that maximally violates the assumptions of Frobenius' theorem. An example is shown on the right. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1593_chunk_0 | Fubini's theorem on differentiation | In mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null sets. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1594_chunk_0 | Fuchs's theorem | In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form has a solution expressible by a generalised Frobenius series when p ( x ) {\displaystyle p(x)} , q ( x ) {\displaystyle q(x)} and g ( x ) {\displaystyle g(x)} are analytic at x = a {\displaystyle x=a}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1595_chunk_0 | Fuglede's theorem | In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1596_chunk_0 | Fujita conjecture | In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved as of 2017. It is named after Takao Fujita, who formulated it in 1985. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1597_chunk_0 | Per Enflo | In mathematics, Functional analysis is concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. In fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1598_chunk_0 | Per Enflo | An important example of a Banach space is a Hilbert space, where the norm arises from an inner product. Hilbert spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics, stochastic processes, and time-series analysis. Besides studying spaces of functions, functiona... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1599_chunk_0 | G2 (mathematics) | In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak {g}}_{2},} as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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