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wiki_1700_chunk_0 | Jacobi transform | In mathematics, Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials P n α , β ( x ) {\displaystyle P_{n}^{\alpha ,\beta }(x)} as kernels of the transform .The Jacobi transform of a function F ( x ) {\displaystyle F(x)} is J { F ( x ) } = f α , ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1701_chunk_0 | Janiszewski's theorem | In mathematics, Janiszewski's theorem, named after the Polish mathematician Zygmunt Janiszewski, is a result concerning the topology of the plane or extended plane. It states that if A and B are closed subsets of the extended plane with connected intersection, then any two points that can be connected by paths avoiding... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1702_chunk_0 | Serre's conjecture II (algebra) | In mathematics, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero. A convers... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1703_chunk_0 | Serre's conjecture II (algebra) | (Note that such fields do indeed have cohomological dimension at most 2.) The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2.The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Susl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1704_chunk_0 | Jensen inequality | In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1705_chunk_0 | Jensen inequality | In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.Jensen's inequality generalizes the statement that the secant line of a convex function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1706_chunk_0 | Jensen inequality | {\displaystyle f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).} In the context of probability theory, it is generally stated in the following form: if X is a random variable and φ is a convex function, then φ ( E ) ≤ E . {\displaystyle \varphi (\operatorname {E} )\leq \operatorname {E} \left.} The difference be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1707_chunk_0 | Jordan operator algebra | In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algeb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1708_chunk_0 | Jordan operator algebra | Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are cal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1709_chunk_0 | Jordan operator algebra | Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1710_chunk_0 | Jordan's inequality | In mathematics, Jordan's inequality, named after Camille Jordan, states that 2 π x ≤ sin ( x ) ≤ x for x ∈ . {\displaystyle {\frac {2}{\pi }}x\leq \sin(x)\leq x{\text{ for }}x\in \left.} It can be proven through the geometry of circles (see drawing). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1711_chunk_0 | K-homology | In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space. In terms of C ∗ {\displaystyle C^{*}} -algebras, it classifies the Fredholm modules over an algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1712_chunk_0 | Real K-theory | In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1713_chunk_0 | Real K-theory | It can be seen as the study of certain kinds of invariants of large matrices.K-theory involves the construction of families of K-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1714_chunk_0 | Real K-theory | In high energy physics, K-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branes, Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds. In condensed matter physics K-theory has been used to classify to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1715_chunk_0 | KK-theory | In mathematics, KK-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980. It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1716_chunk_0 | KR-theory | In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by Atiyah (1966), motivated by applications to the Atiyah–Singer index theorem for real elliptic operators. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1717_chunk_0 | Kachurovskii's theorem | In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1718_chunk_0 | Kaplansky's theorem on quadratic forms | In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Irving Kaplansky. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1719_chunk_0 | Karamata's inequality | In mathematics, Karamata's inequality, named after Jovan Karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality, and generalizes in turn to the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1720_chunk_0 | Kazamaki's condition | In mathematics, Kazamaki's condition gives a sufficient criterion ensuring that the Doléans-Dade exponential of a local martingale is a true martingale. This is particularly important if Girsanov's theorem is to be applied to perform a change of measure. Kazamaki's condition is more general than Novikov's condition. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1721_chunk_0 | Kingman's subadditive ergodic theorem | In mathematics, Kingman's subadditive ergodic theorem is one of several ergodic theorems. It can be seen as a generalization of Birkhoff's ergodic theorem. Intuitively, the subadditive ergodic theorem is a kind of random variable version of Fekete's lemma (hence the name ergodic). As a result, it can be rephrased in th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1722_chunk_0 | Knuth up-arrow notation | In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1723_chunk_0 | Kolmogorov's normability criterion | In mathematics, Kolmogorov's normability criterion is a theorem that provides a necessary and sufficient condition for a topological vector space to be normable; that is, for the existence of a norm on the space that generates the given topology. The normability criterion can be seen as a result in same vein as the Nag... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1724_chunk_0 | Kostant convexity theorem | In mathematics, Kostant's convexity theorem, introduced by Bertram Kostant (1973), states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a general... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1725_chunk_0 | Kostka polynomial | In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory. The two-variable Kostka polynomials Kλμ(q, t) are known by several names including Kostka–Foulke... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1726_chunk_0 | Kostka polynomial | Here the indices λ and μ are integer partitions and Kλμ(q, t) is polynomial in the variables q and t. Sometimes one considers single-variable versions of these polynomials that arise by setting q = 0, i.e., by considering the polynomial Kλμ(t) = Kλμ(0, t). There are two slightly different versions of them, one called t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1727_chunk_0 | Kostka polynomial | In fact, they show that K λ μ ( t ) = ∑ T ∈ S S Y T ( λ , μ ) t c h a r g e ( T ) {\displaystyle K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{charge(T)}} where the sum is taken over all semi-standard Young tableaux with shape λ and weight μ. Here, charge is a certain combinatorial statistic on semi-standard Y... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1728_chunk_0 | Koszul duality | In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototype example, due to Joseph Bernstein, Israel Gelfand,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1729_chunk_0 | Krawtchouk matrices | In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points. The Krawtchouk matrix K(N) is an (N + 1) × (N + 1) matrix. The first few Krawtchouk matrices are: K ( 0 ) = , K ( 1 ) = , K ( 2 ) = , K ( 3 ) = , {\displaystyle K^{(0)}={\begin{bmatrix}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1730_chunk_0 | Krener's theorem | In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1731_chunk_0 | Kronecker coefficient | In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. They play an important role algebraic combinatorics and geometric complexity theory. They were introduced by Murnag... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1732_chunk_0 | Kronecker's congruence | In mathematics, Kronecker's congruence, introduced by Kronecker, states that Φ p ( x , y ) ≡ ( x − y p ) ( x p − y ) mod p , {\displaystyle \Phi _{p}(x,y)\equiv (x-y^{p})(x^{p}-y){\bmod {p}},} where p is a prime and Φp(x,y) is the modular polynomial of order p, given by Φ n ( x , j ) = ∏ τ ( x − j ( τ ) ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1733_chunk_0 | Kronecker's theorem on diophantine approximation | In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by Leopold Kronecker (1884). Kronecker's approximation theorem had been firstly proved by L. Kronecker in the end of the 19th century. It has been now revealed to relate to the idea of n-torus and Mahler measure since the later... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1734_chunk_0 | Kruskal tree theorem | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1735_chunk_0 | Contractibility of unit sphere in Hilbert space | In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the nor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1736_chunk_0 | Kummer sum | In mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus p, with p congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers. These sums were known and used before Kummer, in the theory of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1737_chunk_0 | Kummer congruence | In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer (1851). Kubota & Leopoldt (1964) used Kummer's congruences to define the p-adic zeta function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1738_chunk_0 | Kummer's theorem | In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1739_chunk_0 | Kuratowski convergence | In mathematics, Kuratowski convergence or Painlevé-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlevé in lectures on mathematical analysis in 1902, the concept was popularized in texts by Felix Hausdorff and Kazimierz Kuratowski. Intuitively, the Kura... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1740_chunk_0 | Kuratowski's intersection theorem | In mathematics, Kuratowski's intersection theorem is a result in general topology that gives a sufficient condition for a nested sequence of sets to have a non-empty intersection. Kuratowski's result is a generalisation of Cantor's intersection theorem. Whereas Cantor's result requires that the sets involved be compact... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1741_chunk_0 | Algebraic de Rham cohomology | In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calcul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1742_chunk_0 | Kōmura's theorem | In mathematics, Kōmura's theorem is a result on the differentiability of absolutely continuous Banach space-valued functions, and is a substantial generalization of Lebesgue's theorem on the differentiability of the indefinite integral, which is that Φ: → R given by Φ ( t ) = ∫ 0 t φ ( s ) d s , {\displaystyle \Phi (t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1743_chunk_0 | Sides of an equation | In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.More generally, these terms may apply to an inequation or inequality; the right-hand side is everything on the rig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1744_chunk_0 | Lady Windermere's Fan (mathematics) | In mathematics, Lady Windermere's Fan is a telescopic identity employed to relate global and local error of a numerical algorithm. The name is derived from Oscar Wilde's 1892 play Lady Windermere's Fan, A Play About a Good Woman. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1745_chunk_0 | Lafforgue's theorem | In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups and representations of Galois groups. The Langlands conjectures were introduced by Langlands (1967... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1746_chunk_0 | Laguerre transform | In mathematics, Laguerre transform is an integral transform named after the mathematician Edmond Laguerre, which uses generalized Laguerre polynomials L n α ( x ) {\displaystyle L_{n}^{\alpha }(x)} as kernels of the transform.The Laguerre transform of a function f ( x ) {\displaystyle f(x)} is L { f ( x ) } = f ~ α ( n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1747_chunk_0 | Landau's function | In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn. Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1748_chunk_0 | Landau's function | An element of order 6 in the group S5 can be written in cycle notation as (1 2) (3 4 5). Note that the same argument applies to the number 6, that is, g(6) = 6. There are arbitrarily long sequences of consecutive numbers n, n + 1, …, n + m on which the function g is constant.The integer sequence g(0) = 1, g(1) = 1, g(2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1749_chunk_0 | Landau's function | Equivalently (using little-o notation), g ( n ) = e ( 1 + o ( 1 ) ) n ln n {\displaystyle g(n)=e^{(1+o(1)){\sqrt {n\ln n}}}} . The statement that ln g ( n ) < L i − 1 ( n ) {\displaystyle \ln g(n)<{\sqrt {\mathrm {Li} ^{-1}(n)}}} for all sufficiently large n, where Li−1 denotes the inverse of the logarithmic integr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1750_chunk_0 | Laplace's method | In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form ∫ a b e M f ( x ) d x , {\displaystyle \int _{a}^{b}e^{Mf(x)}\,dx,} where f ( x ) {\displaystyle f(x)} is a twice-differentiable function, M is a large number, and the endpoints a and b could pos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1751_chunk_0 | Laplace principle (large deviations theory) | In mathematics, Laplace's principle is a basic theorem in large deviations theory which is similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−θφ(x)) over a fixed set A as θ becomes large. Such expressions can be used, for example, in statistical mechanics to determining the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1752_chunk_0 | Laver table | In mathematics, Laver tables (named after Richard Laver, who discovered them towards the end of the 1980s in connection with his works on set theory) are tables of numbers that have certain properties of algebraic and combinatorial interest. They occur in the study of racks and quandles. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1753_chunk_0 | Hsiang–Lawson's conjecture | In mathematics, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S3. The conjecture was featured by the Australian Mathematical Society Gazette as part of the Millennium Problems series.In March 2012, Simon Brendle gave a proof of this conjecture, based on maximum ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1754_chunk_0 | Density point | In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} , the "density" of A is 0 or 1 at almost every point in R n {\displaystyle \mathbb {R} ^{n}} . Additionally, the "density" of A is 1 at almost every point in A. Intuitively, this mean... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1755_chunk_0 | Density point | The set of points in the plane at which the density is neither 0 nor 1 is non-empty (the square boundary), but it is negligible. The Lebesgue density theorem is a particular case of the Lebesgue differentiation theorem. Thus, this theorem is also true for every finite Borel measure on Rn instead of Lebesgue measure, se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1756_chunk_0 | Lefschetz duality | In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz (1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. There are now numerous form... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1757_chunk_0 | Legendre moment | In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image processing including: pattern and object recognition, image indexing, line fitting, feature extraction, edge detection, and texture analysis. Legendre moments have b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1758_chunk_0 | Legendre transform (integral transform) | In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials P n ( x ) {\displaystyle P_{n}(x)} as kernels of the transform. Legendre transform is a special case of Jacobi transform. The Legendre transform of a function f ( x ) {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1759_chunk_0 | Legendre's equation | In mathematics, Legendre's equation is the Diophantine equation The equation is named for Adrien-Marie Legendre who proved in 1785 that it is solvable in integers x, y, z, not all zero, if and only if −bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1760_chunk_0 | Legendre's formula | In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1761_chunk_0 | Legendre's three-square theorem | In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers n = x 2 + y 2 + z 2 {\displaystyle n=x^{2}+y^{2}+z^{2}} if and only if n is not of the form n = 4 a ( 8 b + 7 ) {\displaystyle n=4^{a}(8b+7)} for nonnegative integers a and b. The firs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1762_chunk_0 | Lehmer's totient problem | In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem. It is known that φ(n) = n − 1 if and only if n is prime. So for every prime number n, we have φ(n) = n − 1 and thus in particular φ(n) divides n − 1. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1763_chunk_0 | Lehrbuch der Topologie | In mathematics, Lehrbuch der Topologie (German for "textbook of topology") is a book by Herbert Seifert and William Threlfall, first published in 1934 and published in an English translation in 1980. It was one of the earliest textbooks on algebraic topology, and was the standard reference on this topic for many years.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1764_chunk_0 | Lemoine's problem | In mathematics, Lemoine's problem is a certain construction problem in elementary plane geometry posed by the French mathematician Émile Lemoine (1840–1912) in 1868. The problem was published as Question 864 in Nouvelles Annales de Mathématiques (Series 2, Volume 7 (1868), p 191). The chief interest in the problem is t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1765_chunk_0 | Levinson's inequality | In mathematics, Levinson's inequality is the following inequality, due to Norman Levinson, involving positive numbers. Let a > 0 {\displaystyle a>0} and let f {\displaystyle f} be a given function having a third derivative on the range ( 0 , 2 a ) {\displaystyle (0,2a)} , and such that f ‴ ( x ) ≥ 0 {\displaystyle f'''... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1766_chunk_0 | Lie algebra cohomology | In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by Claude Chevalley and S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1767_chunk_0 | Left invariant | In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1768_chunk_0 | Left invariant | However, by restricting our attention to the simply connected Lie groups, the Lie group-Lie algebra correspondence will be one-to-one.In this article, a Lie group refers to a real Lie group. For the complex and p-adic cases, see complex Lie group and p-adic Lie group. In this article, manifolds (in particular Lie group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1769_chunk_0 | Light's associativity test | In mathematics, Light's associativity test is a procedure invented by F. W. Light for testing whether a binary operation defined in a finite set by a Cayley multiplication table is associative. The naive procedure for verification of the associativity of a binary operation specified by a Cayley table, which compares th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1770_chunk_0 | Lindelöf's theorem | In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip mu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1771_chunk_0 | Liouville's formula | In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system. The formula is named after... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1772_chunk_0 | Liouville's theorem (differential algebra) | In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. These are called n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1773_chunk_0 | Liouville's theorem (differential algebra) | Other examples include the functions sin ( x ) x {\displaystyle {\frac {\sin(x)}{x}}} and x x . {\displaystyle x^{x}.} Liouville's theorem states that elementary antiderivatives, if they exist, are in the same differential field as the function, plus possibly a finite number of applications of the logarithm function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1774_chunk_0 | Liouville's theorem (conformal mappings) | In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that any smooth conformal mapping on a domain of Rn, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1775_chunk_0 | Liouville's theorem (conformal mappings) | By contrast, conformal mappings in R2 can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem. Generalizations of the theorem hold for transformations that are only weakly differentiable (Iwaniec & Martin 2001, Chapter 5). The focus of s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1776_chunk_0 | Liouville's theorem (conformal mappings) | A weak solution of this system is defined to be an element f of the Sobolev space W1,nloc(Ω, Rn) with non-negative Jacobian determinant almost everywhere, such that the Cauchy–Riemann system holds at almost every point of Ω. Liouville's theorem is then that every weak solution (in this sense) is a Möbius transformation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1777_chunk_0 | Liouville's theorem (conformal mappings) | The result is not optimal however: in even dimensions n = 2k, the theorem also holds for solutions that are only assumed to be in the space W1,kloc, and this result is sharp in the sense that there are weak solutions of the Cauchy–Riemann system in W1,p for any p < k that are not Möbius transformations. In odd dimensio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1778_chunk_0 | Liouville's theorem (conformal mappings) | The group of conformal isometries of an n-dimensional conformal Riemannian manifold always has dimension that cannot exceed that of the full conformal group SO(n + 1, 1). Equality of the two dimensions holds exactly when the conformal manifold is isometric with the n-sphere or projective space. Local versions of the re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1779_chunk_0 | Littlewood's Tauberian theorem | In mathematics, Littlewood's Tauberian theorem is a strengthening of Tauber's theorem introduced by John Edensor Littlewood (1911). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1780_chunk_0 | Loewner order | In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in mat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1781_chunk_0 | Luna's slice theorem | In mathematics, Luna's slice theorem, introduced by Luna (1973), describes the local behavior of an action of a reductive algebraic group on an affine variety. It is an analogue in algebraic geometry of the theorem that a compact Lie group acting on a smooth manifold X has a slice at each point x, in other words a subv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1782_chunk_0 | Lyapunov fractal | In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.A Lyapunov fractal is constructed by mapping the regions of stabili... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1783_chunk_0 | Lüroth's theorem | In mathematics, Lüroth's theorem asserts that every field that lies between two other fields K and K(X) must be generated as an extension of K by a single element of K(X). This result is named after Jacob Lüroth, who proved it in 1876. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1784_chunk_0 | Maass wave forms | In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup Γ {\displaystyle \Gamma } of S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1785_chunk_0 | MacMahon master theorem | In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis (1916). It is often used to derive binomial identities, most notably Dixon's identity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1786_chunk_0 | Macdonald's constant term conjecture | In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one var... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1787_chunk_0 | Macdonald's constant term conjecture | The Macdonald polynomials are polynomials in n variables x=(x1,...,xn), where n is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-variable ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1788_chunk_0 | Machin-like formulas | In mathematics, Machin-like formulae are a popular technique for computing π (the ratio of the circumference to the diameter of a circle) to a large number of digits. They are generalizations of John Machin's formula from 1706: π 4 = 4 arctan 1 5 − arctan 1 239 {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1789_chunk_0 | Maclaurin's inequality | In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a1, a2, ..., an be positive real numbers, and for k = 1, 2, ..., n define the averages Sk as follows: S k = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n a i 1 a i 2 ⋯ a i k ( n k ) . {\displaystyle S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1790_chunk_0 | Maharam's theorem | In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval on the reals), and "purely... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1791_chunk_0 | Maharam's theorem | Maharam's theorem can also be translated into the language of abelian von Neumann algebras. Every abelian von Neumann algebra is isomorphic to a product of σ-finite abelian von Neumann algebras, and every σ-finite abelian von Neumann algebra is isomorphic to a spatial tensor product of discrete abelian von Neumann alge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1792_chunk_0 | Mahler's compactness theorem | In mathematics, Mahler's compactness theorem, proved by Kurt Mahler (1946), is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in which a lattice could degenerate (go off to infinity) in a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1793_chunk_0 | Mahler's compactness theorem | It is also called his selection theorem, following an older convention used in naming compactness theorems, because they were formulated in terms of sequential compactness (the possibility of selecting a convergent subsequence). Let X be the space G L n ( R ) / G L n ( Z ) {\displaystyle \mathrm {GL} _{n}(\mathbb {R} )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1794_chunk_0 | Mahler's compactness theorem | Mahler's compactness theorem states that a subset Y of X is relatively compact if and only if Δ is bounded on Y, and there is a neighbourhood N of 0 in R n {\displaystyle \mathbb {R} ^{n}} such that for all Λ in Y, the only lattice point of Λ in N is 0 itself. The assertion of Mahler's theorem is equivalent to the comp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1795_chunk_0 | Mahler's inequality | In mathematics, Mahler's inequality, named after Kurt Mahler, states that the geometric mean of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means: ∏ k = 1 n ( x k + y k ) 1 / n ≥ ∏ k = 1 n x k 1 / n + ∏ k = 1 n y k 1 / n {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1796_chunk_0 | Mahler's theorem | In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. It is the p-adic counterpart to the Stone-Weierstrass theorem for continuous real-valued functions on a closed interval. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1797_chunk_0 | Maillet's determinant | In mathematics, Maillet's determinant Dp is the determinant of the matrix introduced by Maillet (1913) whose entries are R(s/r) for s,r = 1, 2, ..., (p – 1)/2 ∈ Z/pZ for an odd prime p, where and R(a) is the least positive residue of a mod p (Muir 1930, pages 340–342). Malo (1914) calculated the determinant Dp for p = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1798_chunk_0 | Maillet's determinant | In particular this verifies Maillet's conjecture that the determinant is always non-zero. Chowla and Weil had previously found the same formula but did not publish it. Their results have been extended to all non-prime odd numbers by K. Wang(1982). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1799_chunk_0 | Malmquist's theorem | In mathematics, Malmquist's theorem, is the name of any of the three theorems proved by Axel Johannes Malmquist (1913, 1920, 1941). These theorems restrict the forms of first order algebraic differential equations which have transcendental meromorphic or algebroid solutions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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