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wiki_5800_chunk_0 | Glaisher–Kinkelin constant | This formula displays a similarity between A and π which is perhaps best illustrated by noting Stirling's formula: 2 π = lim n → ∞ n ! n n + 1 2 e − n {\displaystyle {\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5801_chunk_0 | Glaisher–Kinkelin constant | }{n^{n+{\frac {1}{2}}}\,e^{-n}}}} which shows that just as π is obtained from approximation of the factorials, A can also be obtained from a similar approximation to the hyperfactorials. An equivalent definition for A involving the Barnes G-function, given by G(n) = Πn−2k=1 k! = n−1/K(n) where Γ(n) is the gamma functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5802_chunk_0 | Goldbach–Euler theorem | In mathematics, the Goldbach–Euler theorem (also known as Goldbach's theorem), states that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 and omitting repetitions, converges to 1: ∑ p ∞ 1 p − 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + ⋯ = 1. {\displaystyle \sum _{p}^{\infty }{\frac {1}{p-1}}=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5803_chunk_0 | Golod–Shafarevich theorem | In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5804_chunk_0 | Golomb–Dickman constant | In mathematics, the Golomb–Dickman constant arises in the theory of random permutations and in number theory. Its value is λ = 0.62432998854355087099293638310083724 … {\displaystyle \lambda =0.62432998854355087099293638310083724\dots } (sequence A084945 in the OEIS)It is not known whether this constant is rational or i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5805_chunk_0 | Euler–Gompertz constant | In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by δ {\displaystyle \delta } , appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz. It can be defined by the continued fraction δ = 1 2 − 1 4 − 4 6 − 9 8 − ⋱ − n 2 2 n + 2 − … , {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5806_chunk_0 | Euler–Gompertz constant | The most frequent appearance of δ {\displaystyle \delta } is in the following integrals: δ = ∫ 0 ∞ ln ( 1 + x ) e − x d x = ∫ 0 ∞ e − x 1 + x d x = ∫ 0 1 1 1 − ln ( x ) d x . {\displaystyle \delta =\int _{0}^{\infty }\ln(1+x)e^{-x}dx=\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}dx=\int _{0}^{1}{\frac {1}{1-\ln(x)}}dx.} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5807_chunk_0 | Euler–Gompertz constant | The numerical value of δ {\displaystyle \delta } is about δ = 0.596347362323194074341078499369279376074 … {\displaystyle \delta =0.596347362323194074341078499369279376074\dots } When Euler studied divergent infinite series, he encountered δ {\displaystyle \delta } via, for example, the above integral representations. L... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5808_chunk_0 | Goormaghtigh conjecture | In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation x m − 1 x − 1 = y n − 1 y − 1 {\displaystyle {\frac {x^{m}-1}{x-1}}={\frac {y^{n}-1}{y... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5809_chunk_0 | Gordon–Luecke theorem | In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian. The theorem is usually stated as "knots are determined by their co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5810_chunk_0 | Gordon–Luecke theorem | Often two knots are considered equivalent if they are isotopic. The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic. These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5811_chunk_0 | Gordon–Luecke theorem | The theorem was proved by Cameron Gordon and John Luecke. Essential ingredients of the proof are their joint work with Marc Culler and Peter Shalen on the cyclic surgery theorem, combinatorial techniques in the style of Litherland, thin position, and Scharlemann cycles. For link complements, it is not in fact true that... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5812_chunk_0 | Gordon–Luecke theorem | For example, JHC Whitehead proved that there are infinitely many links whose complements are all homeomorphic to the Whitehead link. His construction is to twist along a disc spanning an unknotted component (as is the case for either component of the Whitehead link). Another method is to twist along an annulus spanning... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5813_chunk_0 | Gorenstein–Walter theorem | In mathematics, the Gorenstein–Walter theorem, proved by Gorenstein and Walter (1965a, 1965b, 1965c), states that if a finite group G has a dihedral Sylow 2-subgroup, and O(G) is the maximal normal subgroup of odd order, then G/O(G) is isomorphic to a 2-group, or the alternating group A7, or a subgroup of PΓL2(q) conta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5814_chunk_0 | Grace–Walsh–Szegő theorem | In mathematics, the Grace–Walsh–Szegő coincidence theorem is a result named after John Hilton Grace, Joseph L. Walsh, and Gábor Szegő. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5815_chunk_0 | Graham–Rothschild theorem | In mathematics, the Graham–Rothschild theorem is a theorem that applies Ramsey theory to combinatorics on words and combinatorial cubes. It is named after Ronald Graham and Bruce Lee Rothschild, who published its proof in 1971. Through the work of Graham, Rothschild, and Klaus Leeb in 1972, it became part of the founda... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5816_chunk_0 | Grassmannian variety | In mathematics, the Grassmannian Gr(k, V ) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V. For example, the Grassmannian Gr(1, V ) is the space of lines through the origin in V, so it is the same as the projective space of one dimension lower than V.When V is a real... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5817_chunk_0 | Grassmannian variety | The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to Gr(2, R4) and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general. Notations for the Grassmannian va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5818_chunk_0 | Grauert-Riemenschneider conjecture | In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold, due to Grauert and Riemenschneider (1970). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5819_chunk_0 | Griess algebra | In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct M. The Monster fixes (vect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5820_chunk_0 | Griewank function | In mathematics, the Griewank function is often used in testing of optimization. It is defined as follows: 1 + 1 4000 ∑ i = 1 n x i 2 − ∏ i = 1 n cos ( x i i ) {\displaystyle 1+{\frac {1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}\cos \left({\frac {x_{i}}{\sqrt {i}}}\right)} The following paragraphs display the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5821_chunk_0 | Taubes's Gromov invariant | In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold, where the curves are holomorphic with respect to an auxiliary compatible almost complex structure. (Multiple covers of 2-tori with self-intersection 0 are also counted.) T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5822_chunk_0 | Taubes's Gromov invariant | Much of the analytical complexity connected to this invariant comes from properly counting multiply covered pseudoholomorphic curves so that the result is invariant of the choice of almost complex structure. The crux is a topologically defined index for pseudoholomorphic curves which controls embeddedness and bounds th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5823_chunk_0 | Gross–Koblitz formula | In mathematics, the Gross–Koblitz formula, introduced by Gross and Koblitz (1979) expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation and generalizes the Stickelberger theorem. B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5824_chunk_0 | Grothendieck existence theorem | In mathematics, the Grothendieck existence theorem, introduced by Grothendieck (1961, section 5), gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes over infinitesimal neighborhoods over a subscheme of a scheme S to schemes over S. The theorem can be vi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5825_chunk_0 | Grothendieck group | In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5826_chunk_0 | Grothendieck constant | In mathematics, the Grothendieck inequality states that there is a universal constant K G {\displaystyle K_{G}} with the following property. If Mij is an n × n (real or complex) matrix with | ∑ i , j M i j s i t j | ≤ 1 {\displaystyle {\Big |}\sum _{i,j}M_{ij}s_{i}t_{j}{\Big |}\leq 1} for all (real or complex) numbers ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5827_chunk_0 | Grothendieck conjecture | In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case. It is a conjecture of Alexander Gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5828_chunk_0 | Grünwald–Letnikov derivative | In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 186... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5829_chunk_0 | Gudermannian function | In mathematics, the Gudermannian function relates a hyperbolic angle measure ψ {\textstyle \psi } to a circular angle measure ϕ {\textstyle \phi } called the gudermannian of ψ {\textstyle \psi } and denoted gd ψ {\textstyle \operatorname {gd} \psi } . The Gudermannian function reveals a close relationship between the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5830_chunk_0 | Gudermannian function | {\textstyle m=1.} The real Gudermannian function is typically defined for − ∞ < ψ < ∞ {\textstyle -\infty <\psi <\infty } to be the integral of the hyperbolic secant ϕ = gd ψ ≡ ∫ 0 ψ sech t d t = arctan ( sinh ψ ) . {\displaystyle \phi =\operatorname {gd} \psi \equiv \int _{0}^{\psi }\operatorname {sech} t\,\ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5831_chunk_0 | Gudermannian function | The real inverse Gudermannian function can be defined for − 1 2 π < ϕ < 1 2 π {\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi } as the integral of the secant ψ = gd − 1 ϕ = ∫ 0 ϕ sec t d t = arsinh ( tan ϕ ) . {\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\operatorname {sec} t\,\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5832_chunk_0 | Gudermannian function | {\displaystyle \psi =\operatorname {lam} \phi .} In the context of geodesy and navigation for latitude ϕ {\textstyle \phi } , k gd − 1 ϕ {\displaystyle k\operatorname {gd} ^{-1}\phi } (scaled by arbitrary constant k {\textstyle k} ) was historically called the meridional part of ϕ {\displaystyle \phi } (French: latit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5833_chunk_0 | H-derivative | In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5834_chunk_0 | HM-GM-AM-QM inequalities | In mathematics, the HM-GM-AM-QM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} are positive real numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5835_chunk_0 | Haar transform | In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5836_chunk_0 | Haar transform | This property can, however, be an advantage for the analysis of signals with sudden transitions (discrete signals), such as monitoring of tool failure in machines.The Haar wavelet's mother wavelet function ψ ( t ) {\displaystyle \psi (t)} can be described as ψ ( t ) = { 1 0 ≤ t < 1 2 , − 1 1 2 ≤ t < 1 , 0 otherwise. {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5837_chunk_0 | Hadamard derivative | In mathematics, the Hadamard derivative is a concept of directional derivative for maps between Banach spaces. It is particularly suited for applications in stochastic programming and asymptotic statistics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5838_chunk_0 | Elementwise division | In mathematics, the Hadamard product (also known as the element-wise product, entrywise product: ch. 5 or Schur product) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication"... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5839_chunk_0 | Hadwiger–Finsler inequality | In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths a, b and c and area T, then | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5840_chunk_0 | Jordan decomposition theorem | In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} and any signed measure μ {\displaystyle \mu } defined on the σ {\displaystyle \sigma } -algebra Σ {\displaystyle \Sigma } , there exist two Σ {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5841_chunk_0 | Hahn–Exton q-Bessel function | In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, and satisfies the Hahn-Exton q-difference equation (Swarttouw (1992)). This function was introduced by Hahn (1953) in a special case and by Exton (1983) in general. The Hahn–Exton q-Bessel funct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5842_chunk_0 | Hales–Jewett theorem | In mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory named after Alfred W. Hales and Robert I. Jewett, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure; it is impossible for such objects to be "completely random".An ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5843_chunk_0 | Hall algebra | In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by Steinitz (1901) but forgotten until it was rediscovered by Philip Hall (1959), both of whom published no more than brief summaries of their work. The Hall po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5844_chunk_0 | Hall-Littlewood polynomials | In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials. They were first defined indirectly by Philip Hall using the Hall algebra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5845_chunk_0 | Halpern–Läuchli theorem | In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern–Läuchli theorem, but the proper attribution... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5846_chunk_0 | Hamburger moment problem | In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (m0, m1, m2, ...), does there exist a positive Borel measure μ (for instance, the measure determined by the cumulative distribution function of a random variable) on the real line such that m n = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5847_chunk_0 | Hamiltonian cycle polynomial | In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as ham ( A ) = ∑ σ ∈ H n ∏ i = 1 n a i , σ ( i ) {\displaystyle \operatorname {ham} (A)=\sum _{\sigma \in H_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}} where H n {\displaystyle H_{n}} is the set of n-permutations havin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5848_chunk_0 | Hamiltonian cycle polynomial | Hence if it's possible to polynomial-time assign weights from a field of characteristic 2 to a digraph's arcs that make its weighted adjacency matrix unitary and having a non-zero Hamiltonian cycle polynomial then the digraph is Hamiltonian. Therefore the Hamiltonian cycle problem is computable on such graphs in polyno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5849_chunk_0 | Hamiltonian cycle polynomial | For k = 1 {\displaystyle k=1} the latter statement can be re-formulated as the # 2 {\displaystyle _{2}} P-completeness of computing, for a given unitary n×n-matrix U {\displaystyle U} over a field of characteristic 2, the n×n-matrix H ( U ) {\displaystyle H(U)} whose i,j-th entry is the Hamiltonian cycle polynomial of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5850_chunk_0 | Hamiltonian cycle polynomial | . + a n 2 ) ham ( U ) {\displaystyle \operatorname {ham} \left({\begin{matrix}U&{Ua}\\a^{T}&1\end{matrix}}\right)=(a_{1}^{2}+...+a_{n}^{2})\operatorname {ham} (U)} where a {\displaystyle a} is an arbitrary n-vector (what can be interpreted as the polynomial-time computability of the Hamiltonian cycle polynomial of an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5851_chunk_0 | Hamiltonian cycle polynomial | Besides, in characteristic 2 for square matrices X, Y ham ( X Y Y X ) {\displaystyle \operatorname {ham} \left({\begin{matrix}X&Y\\Y&X\end{matrix}}\right)} is the square of the sum, over all the pairs of non-equal indexes i,j, of the i,j-th entry of Y multiplied by the Hamiltonian cycle polynomial of the matrix recei... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5852_chunk_0 | Hamiltonian cycle polynomial | These two types of transformation don't compress the matrix, but keep its size unchanged. However, in a number of cases their application allows to reduce the matrix's size by some of the above-mentioned compression operators. Hence there is a variety of matrix compression operators performed in polynomial time and pre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5853_chunk_0 | Fourier–Bessel transform | In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind Jν(kr). The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r axis. The necessary coefficient Fν of each Bessel funct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5854_chunk_0 | Fourier–Bessel transform | The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5855_chunk_0 | Haran's diamond theorem | In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5856_chunk_0 | Hardy–Littlewood zeta-function conjectures | In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5857_chunk_0 | Hardy–Ramanujan theorem | In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy states that the normal order of the number ω(n) of distinct prime factors of a number n is log(log(n)). Roughly speaking, this means that most numbers have about this number of distinct prime factors. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5858_chunk_0 | Hardy–Littlewood circle method | In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy, S. Ramanujan, and J. E. Littlewood, who developed it in a series of papers on Waring's problem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5859_chunk_0 | Distributional character | In mathematics, the Harish-Chandra character, named after Harish-Chandra, of a representation of a semisimple Lie group G on a Hilbert space H is a distribution on the group G that is analogous to the character of a finite-dimensional representation of a compact group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5860_chunk_0 | Harish-Chandra isomorphism | In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} of the universal enveloping algebra U ( g ) {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5861_chunk_0 | Cas (mathematics) | In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942, and is one of many known Fourier-related tran... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5862_chunk_0 | Hartogs–Rosenthal theorem | In mathematics, the Hartogs–Rosenthal theorem is a classical result in complex analysis on the uniform approximation of continuous functions on compact subsets of the complex plane by rational functions. The theorem was proved in 1931 by the German mathematicians Friedrich Hartogs and Arthur Rosenthal and has been wide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5863_chunk_0 | Hartree equation | In mathematics, the Hartree equation, named after Douglas Hartree, is i ∂ t u + ∇ 2 u = V ( u ) u {\displaystyle i\,\partial _{t}u+\nabla ^{2}u=V(u)u} in R d + 1 {\displaystyle \mathbb {R} ^{d+1}} where V ( u ) = ± | x | − n ∗ | u | 2 {\displaystyle V(u)=\pm |x|^{-n}*|u|^{2}} and 0 < n < d {\displaystyle 0 | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5864_chunk_0 | Hasse derivative | In mathematics, the Hasse derivative is a generalisation of the derivative which allows the formulation of Taylor's theorem in coordinate rings of algebraic varieties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5865_chunk_0 | Hasse invariant of a quadratic form | In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt. The quadratic form Q may be taken as a diagonal form Σ aixi2.Its invariant is then defined as the product of the classe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5866_chunk_0 | Hasse invariant of an algebra | In mathematics, the Hasse invariant of an algebra is an invariant attached to a Brauer class of algebras over a field. The concept is named after Helmut Hasse. The invariant plays a role in local class field theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5867_chunk_0 | Hasse–Weil zeta function | In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reducing modulo each prime number p. It is a global L-function defined as an Euler product... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5868_chunk_0 | Hasse-Witt matrix | In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the differentials of the first kind. It is a g × g matrix where C has genus g.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5869_chunk_0 | Hausdorff maximal principle | In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. The Hausdorff maximal principle is one o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5870_chunk_0 | Hausdorff moment problem | In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence (m0, m1, m2, ...) be the sequence of moments m n = ∫ 0 1 x n d μ ( x ) {\displaystyle m_{n}=\int _{0}^{1}x^{n}\,d\mu (x)} of some Borel measure μ supported on the closed unit int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5871_chunk_0 | Haynsworth inertia additivity formula | In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partitioned.The inertia of a Hermitian matrix H is defined as the ordered triple I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5872_chunk_0 | Hecke algebra | In mathematics, the Hecke algebra is the algebra generated by Hecke operators. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5873_chunk_0 | Heine-Cantor theorem | In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if f: M → N {\displaystyle f\colon M\to N} is a continuous function between two metric spaces M {\displaystyle M} and N {\displaystyle N} , and M {\displaystyle M} is compact, then f {\displaystyle f} is uniformly continuou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5874_chunk_0 | Heine–Stieltjes polynomials | In mathematics, the Heine–Stieltjes polynomials or Stieltjes polynomials, introduced by T. J. Stieltjes (1885), are polynomial solutions of a second-order Fuchsian equation, a differential equation all of whose singularities are regular. The Fuchsian equation has the form d 2 S d z 2 + ( ∑ j = 1 N γ j z − a j ) d S d z... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5875_chunk_0 | Heinz mean | In mathematics, the Heinz mean (named after E. Heinz) of two non-negative real numbers A and B, was defined by Bhatia as: H x ( A , B ) = A x B 1 − x + A 1 − x B x 2 , {\displaystyle \operatorname {H} _{x}(A,B)={\frac {A^{x}B^{1-x}+A^{1-x}B^{x}}{2}},} with 0 ≤ x ≤ 1/2. For different values of x, this Heinz mean inter... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5876_chunk_0 | Heisenberg group | In mathematics, the Heisenberg group H {\displaystyle H} , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ( 1 a c 0 1 b 0 0 1 ) {\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}} under the operation of matrix multiplication. Elements a, b and c can be taken fro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5877_chunk_0 | Hellinger integral | In mathematics, the Hellinger integral is an integral introduced by Hellinger (1909) that is a special case of the Kolmogorov integral. It is used to define the Hellinger distance in probability theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5878_chunk_0 | Paraxial Helmholtz equation | In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation where ∇2 is the Laplace operator, k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5879_chunk_0 | Henstock integral | In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of inequivalent definitions of the integr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5880_chunk_0 | Henstock integral | Denjoy was interested in a definition that would allow one to integrate functions like f ( x ) = 1 x sin ( 1 x 3 ) . {\displaystyle f(x)={\frac {1}{x}}\sin \left({\frac {1}{x^{3}}}\right).} This function has a singularity at 0, and is not Lebesgue integrable. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5881_chunk_0 | Henstock integral | However, it seems natural to calculate its integral except over the interval and then let ε, δ → 0. Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition quite complicated. Other definitions were given by Nikolai Luzin (using variations... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5882_chunk_0 | Henstock integral | It took a while to understand that the Perron and Denjoy integrals are actually identical. Later, in 1957, the Czech mathematician Jaroslav Kurzweil discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the gauge integral. Ralph Henstock independently i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5883_chunk_0 | Herbrand–Ribet theorem | In mathematics, the Herbrand–Ribet theorem is a result on the class group of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the n-th Bernoulli number B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5884_chunk_0 | Herglotz–Zagier function | In mathematics, the Herglotz–Zagier function, named after Gustav Herglotz and Don Zagier, is the function F ( x ) = ∑ n = 1 ∞ { Γ ′ ( n x ) Γ ( n x ) − log ( n x ) } 1 n . {\displaystyle F(x)=\sum _{n=1}^{\infty }\left\{{\frac {\Gamma ^{\prime }(nx)}{\Gamma (nx)}}-\log(nx)\right\}{\frac {1}{n}}.} introduced by Zagier... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5885_chunk_0 | Hermite polynomials | In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets for wavelet transform analysis probability, such as the Edgeworth series, as well as in connection with Brownian motion; combinatorics, as an example of an Appell seq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5886_chunk_0 | Hermite–Hadamard inequality | In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ: → R is convex, then the following chain of inequalities hold: f ( a + b 2 ) ≤ 1 b − a ∫ a b f ( x ) d x ≤ f ( a ) + f ( b ) 2 . {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5887_chunk_0 | Heronian mean | In mathematics, the Heronian mean H of two non-negative real numbers A and B is given by the formula It is named after Hero of Alexandria. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5888_chunk_0 | Hessian determinant | In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5889_chunk_0 | Hilbert metric | In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space Rn. It was introduced by David Hilbert (1895) as a generalization of Cayley's formula for the distance in the Cayley–Klein model of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5890_chunk_0 | Hilbert projection theorem | In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x {\displaystyle x} in a Hilbert space H {\displaystyle H} and every nonempty closed convex C ⊆ H , {\displaystyle C\subseteq H,} there exists a unique vector m ∈ C {\displaystyle m\in C} for which ‖ c −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5891_chunk_0 | Hilbert's reciprocity law | In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert sym... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5892_chunk_0 | Hilbert–Burch theorem | In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. Hilbert (1890) proved a version of this theorem for polynomial rings, and Burch (1968, p. 944) proved a more general version. Sever... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5893_chunk_0 | Hilbert–Mumford criterion | In mathematics, the Hilbert–Mumford criterion, introduced by David Hilbert and David Mumford, characterizes the semistable and stable points of a group action on a vector space in terms of eigenvalues of 1-parameter subgroups (Dieudonné & Carrell 1970, 1971, p.58). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5894_chunk_0 | Hilbert-Pólya conjecture | In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5895_chunk_0 | Hilbert–Speiser theorem | In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension of Q, which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields. Hilbert–Speiser Theorem. A finite abeli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5896_chunk_0 | Hilbert–Speiser theorem | This result was introduced by Hilbert (1897, Satz 132, 1998, theorem 132) in his Zahlbericht and by Speiser (1916, corollary to proposition 8.1). In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of Gaussian periods. For example if we take n a prime numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5897_chunk_0 | Hilbert–Speiser theorem | For a field K contained in it, the field trace can be used to construct such a basis in K also (see the article on Gaussian periods). Then in the case of n squarefree and odd, Q(ζn) is a compositum of subfields of this type for the primes p dividing n (this follows from a simple argument on ramification). This decompos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5898_chunk_0 | Hill differential equation | In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation d 2 y d t 2 + f ( t ) y = 0 , {\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0,} where f ( t ) {\displaystyle f(t)} is a periodic function by minimal period π {\displaystyle \pi } . By these we mean ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5899_chunk_0 | Hirzebruch–Riemann–Roch theorem | In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions. The result paved the way for the Grothendieck... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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