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wiki_1400_chunk_0 | Map coloring | In mathematics there is a close link between map coloring and graph coloring, since every map showing different areas has a corresponding graph. By far the most famous result in this area is the four color theorem, which states that any planar map can be colored with at most four colors. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1401_chunk_0 | No-go theorem | In mathematics there is the concept of proof of impossibility referring to problems impossible to solve. The difference between this impossibility and that of the no-go theorems is: a proof of impossibility states a category of logical proposition that may never be true; a no-go theorem instead presents a sequence of e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1402_chunk_0 | Parthasarathy's theorem | In mathematics – and in particular the study of games on the unit square – Parthasarathy's theorem is a generalization of Von Neumann's minimax theorem. It states that a particular class of games has a mixed value, provided that at least one of the players has a strategy that is restricted to absolutely continuous dist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1403_chunk_0 | Stechkin's lemma | In mathematics – more specifically, in functional analysis and numerical analysis – Stechkin's lemma is a result about the ℓq norm of the tail of a sequence, when the whole sequence is known to have finite ℓp norm. Here, the term "tail" means those terms in the sequence that are not among the N largest terms, for an ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1404_chunk_0 | Bochner measurable function | In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e., f ( t ) = lim n → ∞ f n ( t ) for almost every t , {\displaystyle f(t)=\lim _{n\righta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1405_chunk_0 | Densely defined | In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1406_chunk_0 | Itō diffusion | In mathematics – specifically, in stochastic analysis – an Itô diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1407_chunk_0 | Doob's martingale convergence theorems | In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any super... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1408_chunk_0 | Bochner identity | In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician Salomon Bochner. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1409_chunk_0 | Geodesically convex | In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convexity" of a set or function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1410_chunk_0 | Berezin transform | In mathematics — specifically, in complex analysis — the Berezin transform is an integral operator acting on functions defined on the open unit disk D of the complex plane C. Formally, for a function ƒ: D → C, the Berezin transform of ƒ is a new function Bƒ: D → C defined at a point z ∈ D by ( B f ) ( z ) = ∫ D ( 1 − |... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1411_chunk_0 | Maximising measure | In mathematics — specifically, in ergodic theory — a maximising measure is a particular kind of probability measure. Informally, a probability measure μ is a maximising measure for some function f if the integral of f with respect to μ is "as big as it can be". The theory of maximising measures is relatively young and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1412_chunk_0 | Assouad dimension | In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by Patrice Assouad in his 1977 PhD thesis and later published in 1979, although the same notion had been studied in 1928 by Georges Bouligand. As well as being... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1413_chunk_0 | Rate function | In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. Such functions are used to formulate large deviation principle. A large deviation principle quantifies the asymptotic probability of rare events for a sequence of probabilities. A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1414_chunk_0 | Contraction principle (large deviations theory) | In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a probability measure) to a large deviation principle on another space via a continuous function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1415_chunk_0 | Cylindrical σ-algebra | In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures or probability measures of random variables on Banach spaces. For a product space, the cylinder σ-algebra is the one that is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1416_chunk_0 | Concentration dimension | In mathematics — specifically, in probability theory — the concentration dimension of a Banach space-valued random variable is a numerical measure of how "spread out" the random variable is compared to the norm on the space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1417_chunk_0 | Dynkin's formula | In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion at a stopping time. It may be seen as a stochastic generalization of the (second) fundamental theorem of calculus. It is named after the Russian mathematic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1418_chunk_0 | Green measure | In mathematics — specifically, in stochastic analysis — the Green measure is a measure associated to an Itō diffusion. There is an associated Green formula representing suitably smooth functions in terms of the Green measure and first exit times of the diffusion. The concepts are named after the British mathematician G... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1419_chunk_0 | Infinitesimal generator (stochastic processes) | In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes a great deal of information about the process. The generator is used in evolution equatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1420_chunk_0 | Besov measure | In mathematics — specifically, in the fields of probability theory and inverse problems — Besov measures and associated Besov-distributed random variables are generalisations of the notions of Gaussian measures and random variables, Laplace distributions, and other classical distributions. They are particularly useful ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1421_chunk_0 | Question Mark | In mathematics, "?" commonly denotes Minkowski's question mark function. In equations, it can mean "questioned" as opposed to "defined". U+225F ≟ QUESTIONED EQUAL TO U+2A7B ⩻ LESS-THAN WITH QUESTION MARK ABOVE U+2A7C ⩼ GREATER-THAN WITH QUESTION MARK ABOVEIn linear logic, the question mark denotes one of the exponentia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1422_chunk_0 | 0.999... | More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1423_chunk_0 | 1 + 1 + 1 + 1 + ⋯ | In mathematics, 1 + 1 + 1 + 1 + ⋯, also written ∑ n = 1 ∞ n 0 {\displaystyle \sum _{n=1}^{\infty }n^{0}} , ∑ n = 1 ∞ 1 n {\displaystyle \sum _{n=1}^{\infty }1^{n}} , or simply ∑ n = 1 ∞ 1 {\displaystyle \sum _{n=1}^{\infty }1} , is a divergent series, meaning that its sequence of partial sums does not converge to a lim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1424_chunk_0 | 1 + 1 + 1 + 1 + ⋯ | = − 1 2 {\displaystyle \zeta (0)={\frac {1}{\pi }}\lim _{s\rightarrow 0}\ \sin \left({\frac {\pi s}{2}}\right)\ \zeta (1-s)={\frac {1}{\pi }}\lim _{s\rightarrow 0}\ \left({\frac {\pi s}{2}}-{\frac {\pi ^{3}s^{3}}{48}}+...\right)\ \left(-{\frac {1}{s}}+...\right)=-{\frac {1}{2}}} where the power series expansion for ζ(s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1425_chunk_0 | 1 − 2 + 3 − 4 + ⋯ | In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ..... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1426_chunk_0 | 1 − 2 + 3 − 4 + ⋯ | Many of these summability methods easily assign to 1 − 2 + 3 − 4 + ... a "value" of 1/4. Cesàro summation is one of the few methods that do not sum 1 − 2 + 3 − 4 + ..., so the series is an example where a slightly stronger method, such as Abel summation, is required. The series 1 − 2 + 3 − 4 + ... is closely related to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1427_chunk_0 | 6-sphere coordinates | The three coordinates are u = x x 2 + y 2 + z 2 , v = y x 2 + y 2 + z 2 , w = z x 2 + y 2 + z 2 . {\displaystyle u={\frac {x}{x^{2}+y^{2}+z^{2}}},\quad v={\frac {y}{x^{2}+y^{2}+z^{2}}},\quad w={\frac {z}{x^{2}+y^{2}+z^{2}}}.} Since inversion is its own inverse, the equations for x, y, and z in terms of u, v, and w are ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1428_chunk_0 | A-equivalence | Since Ω ( M x , N y ) {\displaystyle \Omega (M_{x},N_{y})} is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in questio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1429_chunk_0 | A-equivalence | Open sets in the topology are then unions of these base sets. Consider the orbit of some map germ o r b G ( f ) . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1430_chunk_0 | Abel's formula | In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation. The relation can be gene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1431_chunk_0 | Abel's formula | It provides useful identities relating the solutions, and is also useful as a part of other techniques such as the method of variation of parameters. It is especially useful for equations such as Bessel's equation where the solutions do not have a simple analytical form, because in such cases the Wronskian is difficult... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1432_chunk_0 | Abel's inequality | In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1433_chunk_0 | Abel's irreducibility theorem | In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1434_chunk_0 | Abel's summation formula | In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1435_chunk_0 | Abel's test | In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power ser... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1436_chunk_0 | Abel's Theorem | In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1437_chunk_0 | Tauberian theorem | In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1438_chunk_0 | Tauberian theorem | There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1439_chunk_0 | Abel–Goncharov interpolation | In mathematics, Abel–Goncharov interpolation determines a polynomial such that various higher derivatives are the same as those of a given function at given points. It was introduced by Whittaker (1935) and rediscovered by Goncharov (1954). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1440_chunk_0 | Abhyankar's lemma | In mathematics, Abhyankar's lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar's lemma states that if A, B, C are local fields such that A and B are finite extensions of C, with ramification indices a and b, and B is tame... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1441_chunk_0 | Alcuin's sequence | In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: x 3 ( 1 − x 2 ) ( 1 − x 3 ) ( 1 − x 4 ) = x 3 + x 5 + x 6 + 2 x 7 + x 8 + 3 x 9 + ⋯ . {\displaystyle {\frac {x^{3}}{(1-x^{2})(1-x^{3})(1-x^{4})}}=x^{3}+x^{5}+x^{6}+2x^{7}+x^{8}+3x^{9}+\cdots ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1442_chunk_0 | Alcuin's sequence | This is the generalization of problem 12 appearing in Propositiones ad Acuendos Juvenes ("Problems to Sharpen the Young") usually attributed to Alcuin. That problem is given as, Problem 12: A certain father died and left as an inheritance to his three sons 30 glass flasks, of which 10 were full of oil, another 10 were ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1443_chunk_0 | Ibn al-Haytham | In mathematics, Alhazen built on the mathematical works of Euclid and Thabit ibn Qurra and worked on "the beginnings of the link between algebra and geometry".He developed a formula for summing the first 100 natural numbers, using a geometric proof to prove the formula. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1444_chunk_0 | Anderson acceleration | In mathematics, Anderson acceleration, also called Anderson mixing, is a method for the acceleration of the convergence rate of fixed-point iterations. Introduced by Donald G. Anderson, this technique can be used to find the solution to fixed point equations f ( x ) = x {\displaystyle f(x)=x} often arising in the field... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1445_chunk_0 | Anderson's theorem | In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative function f over an n-dimensional convex body K does not decrease if K is translated inwards towards the origin. This is a natural statement, since the graph of f ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1446_chunk_0 | Appell series | In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series 2F1 of one variable. Appell established the set of partial differential equations of which these functions are solutions, and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1447_chunk_0 | Apéry's constant | In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number ζ ( 3 ) = ∑ n = 1 ∞ 1 n 3 = lim n → ∞ ( 1 1 3 + 1 2 3 + ⋯ + 1 n 3 ) , {\displaystyle {\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1448_chunk_0 | Apéry's theorem | In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number ζ ( 3 ) = ∑ n = 1 ∞ 1 n 3 = 1 1 3 + 1 2 3 + 1 3 3 + ⋯ = 1.2020569 … {\displaystyle \zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1449_chunk_0 | Arithmetic scheme | In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1450_chunk_0 | Arakelyan's theorem | In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1451_chunk_0 | Arnold's cat map | In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name.Thinking of the torus T 2 {\displaystyle \mathbb {T} ^{2}} as the quotient space R 2 / Z 2 {\displaystyle \mathbb {R} ^{2}/\mathbb ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1452_chunk_0 | Arnold's spectral sequence | In mathematics, Arnold's spectral sequence (also spelled Arnol'd) is a spectral sequence used in singularity theory and normal form theory as an efficient computational tool for reducing a function to canonical form near critical points. It was introduced by Vladimir Arnold in 1975. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1453_chunk_0 | Artin-Verdier duality | In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by Michael Artin and Jean-Louis Verdier (1964), that generalizes Tate duality. It shows that, as far as etale (or flat) cohomology is concerned, the ring of integers ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1454_chunk_0 | Atkin–Lehner involution | The newforms are defined as a vector subspace of the modular forms of level N, complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the Petersson inner product. The Hecke operators, which act on the space of all cusp forms, preserve the subspace of newforms and are self-adjoint ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1455_chunk_0 | Auerbach's lemma | In mathematics, Auerbach's lemma, named after Herman Auerbach, is a theorem in functional analysis which asserts that a certain property of Euclidean spaces holds for general finite-dimensional normed vector spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1456_chunk_0 | BCK algebra | In mathematics, BCI and BCK algebras are algebraic structures in universal algebra, which were introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1457_chunk_0 | BF-algebra | In mathematics, BF algebras are a class of algebraic structures arising out of a symmetric "Yin Yang" concept for Bipolar Fuzzy logic, the name was introduced by Andrzej Walendziak in 2007. The name covers discrete versions, but a canonical example arises in the BF space x of pairs of (false-ness, truth-ness). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1458_chunk_0 | Baire function | In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose characteristic function is a Baire function. (There are other sim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1459_chunk_0 | Banach algebra cohomology | In mathematics, Banach algebra cohomology of a Banach algebra with coefficients in a bimodule is a cohomology theory defined in a similar way to Hochschild cohomology of an abstract algebra, except that one takes the topology into account so that all cochains and so on are continuous. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1460_chunk_0 | Beez's theorem | In mathematics, Beez's theorem, introduced by Richard Beez in 1875, implies that if n > 3 then in general an (n – 1)-dimensional hypersurface immersed in Rn cannot be deformed. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1461_chunk_0 | Belyi's theorem | In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1462_chunk_0 | Bender–Dunne polynomials | In mathematics, Bender–Dunne polynomials are a two-parameter family of sequences of orthogonal polynomials studied by Carl M. Bender and Gerald Dunne (1988, 1996). They may be defined by the recursion: P 0 ( x ) = 1 {\displaystyle P_{0}(x)=1} , P 1 ( x ) = x {\displaystyle P_{1}(x)=x} ,and for n > 1 {\displaystyle n>1}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1463_chunk_0 | Bendixson's inequality | In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902. The inequality puts limits on the imaginary and real parts of characteristic roots (eigenvalues) of real matrices. A special case of this inequality leads to the result that characteristic roots o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1464_chunk_0 | Bendixson's inequality | If λ {\displaystyle \lambda } is any characteristic root of A {\displaystyle A} , then | Im ( λ ) | ≤ α n ( n − 1 ) 2 . {\displaystyle \left|\operatorname {Im} (\lambda )\right|\leq \alpha {\sqrt {\frac {n(n-1)}{2}}}.\,{}} If A {\displaystyle A} is symmetric then α = 0 {\displaystyle \alpha =0} and consequently the i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1465_chunk_0 | Berger's isoembolic inequality | In mathematics, Berger's isoembolic inequality is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the m-dimensional sphere with its usual "round" metric. The theorem is named after the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1466_chunk_0 | Bernoulli's inequality | In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x {\displaystyle 1+x} . It is often employed in real analysis. It has several useful variants: | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1467_chunk_0 | Proof of Bertrand's postulate | In mathematics, Bertrand's postulate (actually now a theorem) states that for each n ≥ 2 {\displaystyle n\geq 2} there is a prime p {\displaystyle p} such that n < p < 2 n {\displaystyle n | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1468_chunk_0 | Bhargava factorial | In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava as part of his thesis in Harvard University in 1996. The Bhargava factorial has the property that many number-theoretic r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1469_chunk_0 | Bhāskara I's sine approximation formula | In mathematics, Bhāskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhāskara I (c. 600 – c. 680), a seventh-century Indian mathematician. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1470_chunk_0 | Bhāskara I's sine approximation formula | This formula is given in his treatise titled Mahabhaskariya. It is not known how Bhāskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhāskara might have used to arrive at his formula. The formula is elegant and simple, and it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1471_chunk_0 | Birch's theorem | In mathematics, Birch's theorem, named for Bryan John Birch, is a statement about the representability of zero by odd degree forms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1472_chunk_0 | Birkhoff factorization | In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by George David Birkhoff (1909), is the factorization of an invertible matrix M with coefficients that are Laurent polynomials in z into a product M = M+M0M−, where M+ has entries that are polynomials in z, M0 is diagonal, and M− has entries t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1473_chunk_0 | Bochner formula | In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold ( M , g ) {\displaystyle (M,g)} to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1474_chunk_0 | Bochner theorem | In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1475_chunk_0 | Bochner's tube theorem | In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in C n {\displaystyle \mathbb {C} ^{n}} can be extended to the convex hull of this domain. Theorem Let ω ⊂ R n {\displaystyle \omega \subset \mathbb {R} ^{n}} be a connected open set. Then every fun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1476_chunk_0 | Edge-of-the-wedge theorem | In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum field theory to construct the analytic continuation of Wight... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1477_chunk_0 | Bondy's theorem | In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1478_chunk_0 | Borel summation | In mathematics, Borel summation is a summation method for divergent series, introduced by Émile Borel (1899). It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1479_chunk_0 | Borwein's algorithm | In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. They devised several other algorithms. They published the book Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1480_chunk_0 | Brandt matrix | In mathematics, Brandt matrices are matrices, introduced by Brandt (1943), that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and that give a representation of the Hecke algebra. Eichler (1955) calculated the traces of the Brandt matrices. Let O... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1481_chunk_0 | Brandt matrix | Fix an integer m. Let ej denote the number of units in the right order of Ij and let Bij denote the number of α in Ij−1Ii with reduced norm N(α) equal to mN(Ii)/N(Ij). The Brandt matrix B(m) is the H×H matrix with entries Bij. Up to conjugation by a permutation matrix it is independent of the choice of representatives ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1482_chunk_0 | Brandt semigroup | Define a matrix P {\displaystyle P} of dimension | I | × | J | {\displaystyle |I|\times |J|} with entries in G 0 = G ∪ { 0 } . {\displaystyle G^{0}=G\cup \{0\}.} Then, it can be shown that every 0-simple semigroup is of the form S = ( I × G 0 × J ) {\displaystyle S=(I\times G^{0}\times J)} with the operation ( i , a , ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1483_chunk_0 | Brandt semigroup | As Brandt semigroups are also inverse semigroups, the construction is more specialized and in fact, I = J (Howie 1995). Thus, a Brandt semigroup has the form S = ( I × G 0 × I ) {\displaystyle S=(I\times G^{0}\times I)} with the operation ( i , a , j ) ∗ ( k , b , n ) = ( i , a p j k b , n ) {\displaystyle (i,a,j)*(k,b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1484_chunk_0 | Brenke polynomials | In mathematics, Brenke polynomials are special cases of generalized Appell polynomials, and Brenke–Chihara polynomials are the Brenke polynomials that are also orthogonal polynomials. Brenke (1945) introduced sequences of Brenke polynomials Pn, which are special cases of generalized Appell polynomials with generating f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1485_chunk_0 | Brown's representability theorem | In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor. More specifically, we are given F: Hotcop → Set,and the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1486_chunk_0 | Brownian Motion | In mathematics, Brownian motion is described by the Wiener process, a continuous-time stochastic process named in honor of Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1487_chunk_0 | Budan's Theorem | In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number. It was published in 1807 by François Budan de Boislaurent. A similar theorem was published independently by Joseph Fourier in 1820. Each of these theorems is a coro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1488_chunk_0 | Burnside theorem | In mathematics, Burnside's theorem in group theory states that if G is a finite group of order p a q b {\displaystyle p^{a}q^{b}} where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1489_chunk_0 | Busemann's theorem | In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was motivated by his theory of area in Finsler spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1490_chunk_0 | Bäcklund transform | A Bäcklund transform which relates solutions of the same equation is called an invariant Bäcklund transform or auto-Bäcklund transform. If such a transform can be found, much can be deduced about the solutions of the equation especially if the Bäcklund transform contains a parameter. However, no systematic way of findi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1491_chunk_0 | Bézout's lemma | In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem: Here the greatest common divisor of 0 and 0 is taken to be 0. The integers x and y are called Bézout coefficients for (a, b); they are not unique. A pair of Bézout coefficients can be computed by the ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1492_chunk_0 | Bézout's lemma | As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as 3 = 15 × (−9) + 69 × 2, with Bézout coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1493_chunk_0 | Bôcher's theorem | In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1494_chunk_0 | Capelli's identity | In mathematics, Capelli's identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} . It can be used to relate an invariant ƒ to t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1495_chunk_0 | Carathéodory existence theorem | In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem sh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1496_chunk_0 | Carathéodory's theorem (conformal mapping) | In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that any conformal mapping sending the unit disk to some region in the complex plane bounded by a Jordan curve extends contin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1497_chunk_0 | Carleman's equation | In mathematics, Carleman's equation is a Fredholm integral equation of the first kind with a logarithmic kernel. Its solution was first given by Torsten Carleman in 1922. The equation is ∫ a b ln | x − t | y ( t ) d t = f ( x ) {\displaystyle \int _{a}^{b}\ln |x-t|\,y(t)\,dt=f(x)} The solution for b − a ≠ 4 is y ( x ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1498_chunk_0 | Carmichael's totient function conjecture | In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states that, for every n there is at least one other integer m ≠ n such that φ(m) = φ(n). Robert Carmichael first stated this ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1499_chunk_0 | Cartan criterion | In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on g {\displaystyle {\mathfrak {g}}} defined by the formula B ( u , v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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