id stringlengths 14 19 | title stringlengths 1 124 | text stringlengths 12 2.83k | source stringclasses 1
value |
|---|---|---|---|
wiki_5700_chunk_0 | Erdős–Szekeres theorem | In mathematics, the Erdős–Szekeres theorem asserts that, given r, s, any sequence of distinct real numbers with length at least (r − 1)(s − 1) + 1 contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s. The proof appeared in the same 1935 paper that mentions th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5701_chunk_0 | Erdős–Turán inequality | In mathematics, the Erdős–Turán inequality bounds the distance between a probability measure on the circle and the Lebesgue measure, in terms of Fourier coefficients. It was proved by Paul Erdős and Pál Turán in 1948.Let μ be a probability measure on the unit circle R/Z. The Erdős–Turán inequality states that, for any ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5702_chunk_0 | Erlangen program | In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über neuere geometrische Forschungen. It is named after the University Erlangen-Nürnberg, where Klein worked. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5703_chunk_0 | Erlangen program | By 1872, non-Euclidean geometries had emerged, but without a way to determine their hierarchy and relationships. Klein's method was fundamentally innovative in three ways: Projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was more rest... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5704_chunk_0 | Ernst equation | In mathematics, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist Frederick J. Ernst. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5705_chunk_0 | Algorithm | In mathematics, the Euclidean algorithm or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5706_chunk_0 | Algorithm | It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number a multitude composed of units": a counting num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5707_chunk_0 | Algorithm | To "measure" is to place a shorter measuring length s successively (q times) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q×s, q being the quotient, or remainder r is the "modulus", the integer-fractional part left over after the division.For ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5708_chunk_0 | Euclid's algorithm | In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Element... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5709_chunk_0 | Euclid's algorithm | It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5710_chunk_0 | Euclid's algorithm | For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5711_chunk_0 | Euclid's algorithm | By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5712_chunk_0 | Euclid's algorithm | A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two (with this version, the algorithm stops when reaching a zero remainder). With this improvement, the algorithm never requires more steps than five times... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5713_chunk_0 | Euclid's algorithm | Additional methods for improving the algorithm's efficiency were developed in the 20th century. The Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5714_chunk_0 | Euclid's algorithm | Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5715_chunk_0 | Distance formula | In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5716_chunk_0 | Distance formula | The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5717_chunk_0 | Euler function | In mathematics, the Euler function is given by ϕ ( q ) = ∏ k = 1 ∞ ( 1 − q k ) , | q | < 1. {\displaystyle \phi (q)=\prod _{k=1}^{\infty }(1-q^{k}),\quad |q|<1.} Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics and complex analysis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5718_chunk_0 | Euler numbers | In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion 1 cosh t = 2 e t + e − t = ∑ n = 0 ∞ E n n ! ⋅ t n {\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n! }}\cdot t^{n}} ,where cosh ( t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5719_chunk_0 | Euler numbers | The Euler numbers are related to a special value of the Euler polynomials, namely: E n = 2 n E n ( 1 2 ) . {\displaystyle E_{n}=2^{n}E_{n}({\tfrac {1}{2}}).} The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also oc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5720_chunk_0 | Euler-Maclaurin formula | In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5721_chunk_0 | Euler-Maclaurin formula | The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. It was later generalized to Darboux's formula. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5722_chunk_0 | Euler–Poisson–Darboux equation | In mathematics, the Euler–Poisson–Darboux equation is the partial differential equation u x , y + N ( u x + u y ) x + y = 0. {\displaystyle u_{x,y}+{\frac {N(u_{x}+u_{y})}{x+y}}=0.} This equation is named for Siméon Poisson, Leonhard Euler, and Gaston Darboux. It plays an important role in solving the classical wave eq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5723_chunk_0 | Tricomi equation | In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi. u x x + x u y y = 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5724_chunk_0 | Tricomi equation | {\displaystyle u_{xx}+xu_{yy}=0.\,} It is elliptic in the half plane x > 0, parabolic at x = 0 and hyperbolic in the half plane x < 0. Its characteristics are x d x 2 + d y 2 = 0 , {\displaystyle x\,dx^{2}+dy^{2}=0,\,} which have the integral y ± 2 3 x 3 / 2 = C , {\displaystyle y\pm {\frac {2}{3}}x^{3/2}=C,} where C i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5725_chunk_0 | Ext functor | In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5726_chunk_0 | Ext functor | In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book Homolog... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5727_chunk_0 | F. and M. Riesz theorem | In mathematics, the F. and M. Riesz theorem is a result of the brothers Frigyes Riesz and Marcel Riesz, on analytic measures. It states that for a measure μ on the circle, any part of μ that is not absolutely continuous with respect to the Lebesgue measure dθ can be detected by means of Fourier coefficients. More preci... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5728_chunk_0 | F. and M. Riesz theorem | The formulation here is as in Walter Rudin, Real and Complex Analysis, p. 335. The proof given uses the Poisson kernel and the existence of boundary values for the Hardy space H1. Expansions to this theorem were made by James E. Weatherbee in his 1968 dissertation: Some Extensions Of The F. And M. Riesz Theorem On Abso... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5729_chunk_0 | Fourier–Bros–Iagolnitzer transform | In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicists Jacques Bros and Daniel Iagolnitzer in order to characterise the local analyticity of functions (or distributions) on Rn. The transform provides an alterna... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5730_chunk_0 | FEE method | In mathematics, the FEE method, or fast E-function evaluation method, is the method of fast summation of series of a special form. It was constructed in 1990 by Ekaterina Karatsuba and is so-named because it makes fast computations of the Siegel E-functions possible, in particular of e x {\displaystyle e^{x}} . A class... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5731_chunk_0 | FEE method | Using the FEE, it is possible to prove the following theorem: Theorem: Let y = f ( x ) {\displaystyle y=f(x)} be an elementary transcendental function, that is the exponential function, or a trigonometric function, or an elementary algebraic function, or their superposition, or their inverse, or a superposition of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5732_chunk_0 | FEE method | The algorithms based on the method FEE include the algorithms for fast calculation of any elementary transcendental function for any value of the argument, the classical constants e, π , {\displaystyle \pi ,} the Euler constant γ , {\displaystyle \gamma ,} the Catalan and the Apéry constants, such higher transcendental... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5733_chunk_0 | Faber polynomials | In mathematics, the Faber polynomials Pm of a Laurent series f ( z ) = z − 1 + a 0 + a 1 z + ⋯ {\displaystyle \displaystyle f(z)=z^{-1}+a_{0}+a_{1}z+\cdots } are the polynomials such that P m ( f ) − z − m {\displaystyle \displaystyle P_{m}(f)-z^{-m}} vanishes at z=0. They were introduced by Faber (1903, 1919) and stud... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5734_chunk_0 | Fabius function | In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). It was also written down as the Fourier transform of f ^ ( z ) = ∏ m = 1 ∞ ( cos π z 2 m ) m {\displaystyle {\hat {f}}(z)=\prod _{m=1}^{\infty }\left(\cos {\frac {\pi z}{2^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5735_chunk_0 | Fabius function | {\displaystyle 0\leq x\leq 1/2.} It follows that f ( x ) {\displaystyle f(x)} is monotone increasing for 0 ≤ x ≤ 1 , {\displaystyle 0\leq x\leq 1,} with f ( 1 / 2 ) = 1 / 2 {\displaystyle f(1/2)=1/2} and f ( 1 ) = 1. {\displaystyle f(1)=1.} There is a unique extension of f to the real numbers that satisfies the same di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5736_chunk_0 | Fabry gap theorem | In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5737_chunk_0 | Farey graph | In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. With the restricted definition, each Farey sequence starts with ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5738_chunk_0 | Farrell–Markushevich theorem | In mathematics, the Farrell–Markushevich theorem, proved independently by O. J. Farrell (1899–1981) and A. I. Markushevich (1908–1979) in 1934, is a result concerning the approximation in mean square of holomorphic functions on a bounded open set in the complex plane by complex polynomials. It states that complex polyn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5739_chunk_0 | Fatou–Lebesgue theorem | In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these functions. The theorem is named after Pierre Fatou a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5740_chunk_0 | Faxén integral | In mathematics, the Faxén integral (also named Faxén function) is the following integral Fi ( α , β ; x ) = ∫ 0 ∞ exp ( − t + x t α ) t β − 1 d t , ( 0 ≤ Re ( α ) < 1 , Re ( β ) > 0 ) . {\displaystyle \operatorname {Fi} (\alpha ,\beta ;x)=\int _{0}^{\infty }\exp(-t+xt^{\alpha })t^{\beta -1}\mathrm {d} t,\qquad ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5741_chunk_0 | Federer–Morse theorem | In mathematics, the Federer–Morse theorem, introduced by Federer and Morse (1943), states that if f is a surjective continuous map from a compact metric space X to a compact metric space Y, then there is a Borel subset Z of X such that f restricted to Z is a bijection from Z to Y. Moreover, the inverse of that restrict... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5742_chunk_0 | Feferman–Schütte ordinal | In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte, the former of whom suggested the name Γ0.There is no standard notation for or... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5743_chunk_0 | Feit–Thompson conjecture | In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by Walter Feit and John G. Thompson (1962). The conjecture states that there are no distinct prime numbers p and q such that p q − 1 p − 1 {\displaystyle {\frac {p^{q}-1}{p-1}}} divides q p − 1 q − 1 {\displaystyle {\frac {q^{p}-1}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5744_chunk_0 | Odd order theorem | In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5745_chunk_0 | Fejér kernel | In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5746_chunk_0 | Fekete problem | In mathematics, the Fekete problem is, given a natural number N and a real s ≥ 0, to find the points x1,...,xN on the 2-sphere for which the s-energy, defined by ∑ 1 ≤ i < j ≤ N ‖ x i − x j ‖ − s {\displaystyle \sum _{1\leq i 0 and by ∑ 1 ≤ i < j ≤ N log ‖ x i − x j ‖ − 1 {\displaystyle \sum _{1\leq i 0, such points ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5747_chunk_0 | Fekete–Szegő inequality | In mathematics, the Fekete–Szegő inequality is an inequality for the coefficients of univalent analytic functions found by Fekete and Szegő (1933), related to the Bieberbach conjecture. Finding similar estimates for other classes of functions is called the Fekete–Szegő problem. The Fekete–Szegő inequality states that i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5748_chunk_0 | Fermat curve | In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation: X n + Y n = Z n . {\displaystyle X^{n}+Y^{n}=Z^{n}.\ } Therefore, in terms of the affine plane its equation is: x n + y n = 1. {\displaystyle x^{n}+y^{n}=1.\ } An in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5749_chunk_0 | Fibonacci ratio | They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5750_chunk_0 | Fitting lemma | In mathematics, the Fitting lemma – named after the mathematician Hans Fitting – is a basic statement in abstract algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M is either an automorphism or nilpotent.As an immediate consequence, we see that the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5751_chunk_0 | FKG inequality | In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), due to Cees M. Fortuin, Pieter W. Kasteleyn, and Jean Ginibre (1971). Informally, it says that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5752_chunk_0 | FKG inequality | variables, called Harris inequality, is due to Theodore Edward Harris (1960), see below. One generalization of the FKG inequality is the Holley inequality (1974) below, and an even further generalization is the Ahlswede–Daykin "four functions" theorem (1978). Furthermore, it has the same conclusion as the Griffiths ine... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5753_chunk_0 | Fourier inversion | In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely. The the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5754_chunk_0 | Fourier inversion | In other words, the theorem says that f ( x ) = ∬ R 2 e 2 π i ( x − y ) ⋅ ξ f ( y ) d y d ξ . {\displaystyle f(x)=\iint _{\mathbb {R} ^{2}}e^{2\pi i(x-y)\cdot \xi }\,f(y)\,dy\,d\xi .} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5755_chunk_0 | Fourier inversion | This last equation is called the Fourier integral theorem. Another way to state the theorem is that if R {\displaystyle R} is the flip operator i.e. ( R f ) ( x ) := f ( − x ) {\displaystyle (Rf)(x):=f(-x)} , then F − 1 = F R = R F . {\displaystyle {\mathcal {F}}^{-1}={\mathcal {F}}R=R{\mathcal {F}}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5756_chunk_0 | Fourier inversion | The theorem holds if both f {\displaystyle f} and its Fourier transform are absolutely integrable (in the Lebesgue sense) and f {\displaystyle f} is continuous at the point x {\displaystyle x} . However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals abov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5757_chunk_0 | Sine transform | In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5758_chunk_0 | Fox H-function | In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral H p , q m , n = 1 2 π i ∫ L ∏ j = 1 m Γ ( b j + B j s ) ∏ j = 1 n Γ ( 1 − a j − A j s ) ∏ j = m + 1 q Γ ( 1 − b j − B j s ) ∏ j = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5759_chunk_0 | Fox derivative | In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus. The Fox deriv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5760_chunk_0 | Fox–Wright Psi function | In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935): Upon changing the normalisation it becomes pFq(z) for A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5761_chunk_0 | Fraňková–Helly selection theorem | In mathematics, the Fraňková–Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana Fraňková. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5762_chunk_0 | Fredholm alternative | In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5763_chunk_0 | Fredholm determinant | In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The function is named after the mathematician Erik Iv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5764_chunk_0 | Fredholm equation | In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5765_chunk_0 | Freidlin–Wentzell theorem | In mathematics, the Freidlin–Wentzell theorem (due to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentzell theorem gives an estimate for the probability that a (scaled-down) sample path of an Itō diffusion will stray far fro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5766_chunk_0 | Rosenfeld projective plane | In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5767_chunk_0 | Freudenthal spectral theorem | In mathematics, the Freudenthal spectral theorem is a result in Riesz space theory proved by Hans Freudenthal in 1936. It roughly states that any element dominated by a positive element in a Riesz space with the principal projection property can in a sense be approximated uniformly by simple functions. Numerous well-kn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5768_chunk_0 | Frobenius determinant theorem | In mathematics, the Frobenius determinant theorem was a conjecture made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it (reproduced in (Dedekind 1968), with an English translation in (Curtis 2003, p. 51)). If one takes the multiplication table of a finite group G and replac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5769_chunk_0 | Frobenius mapping | The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping φ*: Spec(Rp) → Spec(R)of affine schemes. Even in cases where Rp = R this is not the identity, unless R is the prime field. Mappings created by fibre product with φ*, i.e. base changes, tend in scheme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5770_chunk_0 | Fréchet derivative | In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5771_chunk_0 | Frölicher spectral sequence | In mathematics, the Frölicher spectral sequence (often misspelled as Fröhlicher) is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory that are valid in general only for Kähler manifolds. It was introduced by Frölicher (1955). A spectral sequence is set u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5772_chunk_0 | Frölicher–Nijenhuis bracket | In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of connections, notably the Ehresmann connection, as well as in the more general study of projections in the tangent bundle. It ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5773_chunk_0 | Fubini–Study metric | In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.A Hermitian form in (the vector space) Cn+1 defines a unitary subgroup ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5774_chunk_0 | Fubini–Study metric | The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2n+1)-sphere. In algebraic geometry, one uses a normalization making CPn a Hodge manifold. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5775_chunk_0 | Fuchs relation | In mathematics, the Fuchs relation is a relation between the starting exponents of formal series solutions of certain linear differential equations, so called Fuchsian equations. It is named after Lazarus Immanuel Fuchs. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5776_chunk_0 | Fulton–Hansen connectedness theorem | In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1. It is named after William Fulton and Johan Hansen, who prov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5777_chunk_0 | Meijer G-function | In mathematics, the G-function was introduced by Cornelis Simon Meijer (1936) as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meij... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5778_chunk_0 | Meijer G-function | A notable property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a functional equation that allows to liberate from a G-function G(z) any factor zρ that is a constant power of its argument z, the closure implies that whenever a fun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5779_chunk_0 | Gabriel-Popescu theorem | In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by Pierre Gabriel and Nicolae Popescu (1964). It characterizes certain abelian categories (the Grothendieck categories) as quotients of module categories. There are several generalizations and variations of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5780_chunk_0 | Gateaux derivative | In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died at age 25 in World War I, it is defined for functions between locally convex topological vector spaces such as B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5781_chunk_0 | Gateaux derivative | Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as Tikhomirov (2001), draw a further distinction between the Gateaux differential (which m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5782_chunk_0 | Gauss iterated map | In mathematics, the Gauss map (also known as Gaussian map or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function: x n + 1 = exp ( − α x n 2 ) + β , {\displaystyle x_{n+1}=\exp(-\alpha x_{n}^{2})+\beta ,\,} where α and β are real parameters. Named after Johann Carl ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5783_chunk_0 | Q-binomial theorem | In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as ( n k ) q {\displaystyle {\binom {n}{k}}_{q}} or q {\displaystyle {\begin{bmatrix}n\\k\end{b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5784_chunk_0 | Gaussian isoperimetric inequality | In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5785_chunk_0 | Hypergeometric series | In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5786_chunk_0 | Gauss-Kuzmin distribution | In mathematics, the Gauss–Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1). The distribution is named after Carl Friedrich Gauss, who derived it around... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5787_chunk_0 | Gauss–Kuzmin–Wirsing operator | In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in differential geometry.) It is named after Carl Gauss, Rodion Kuzmin, and Eduard Wirsing. It occurs in the study of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5788_chunk_0 | Gauss–Manin connection | In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties V s {\displaystyle V_{s}} . The fibers of the vector bundle are the de Rham cohomology groups H D R k ( V s ) {\displaystyle H_{DR}^{k}(V_{s})} of the fibers V s {\displaystyle V_... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5789_chunk_0 | Gelfand representation | In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: a way of representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is an isometric isomorphism.In the former case, one... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5790_chunk_0 | Gelfand–Naimark theorem | In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5791_chunk_0 | Gelfond's theorem | In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5792_chunk_0 | Gershgorin circle theorem | In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's name has been transliterated in several different ways, including Geršgorin, Gerschgorin, Gershgorin, Hershhorn, and Hi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5793_chunk_0 | Gibbs random field | In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system X being in stat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5794_chunk_0 | Gibbs random field | Here, E is a function from the space of states to the real numbers; in physics applications, E(x) is interpreted as the energy of the configuration x. The parameter β is a free parameter; in physics, it is the inverse temperature. The normalizing constant Z(β) is the partition function. However, in infinite systems, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5795_chunk_0 | Gibbs random field | Traditional approaches in statistical physics studied the limit of intensive properties as the size of a finite system approaches infinity (the thermodynamic limit). When the energy function can be written as a sum of terms that each involve only variables from a finite subsystem, the notion of a Gibbs measure provides... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5796_chunk_0 | Gibbs random field | A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5797_chunk_0 | Gibbs random field | A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5798_chunk_0 | Gibbs phenomenon | In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The N {\textstyle N} th partial Fourier series of the function (formed by summing the N {\textstyle N} lowest constituent sinusoids of the Four... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5799_chunk_0 | Glaisher–Kinkelin constant | In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function. The constant appears in a number of sums and integrals, especially those involving gamma functions and zeta functions. It is named after mathematic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.