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Fekete problem Summary Fekete_problem In mathematics, the Fekete problem is, given a natural number N and a real s ≥ 0, to find the points x1,...,xN on the 2-sphere for which the s-energy, defined by ∑ 1 ≤ i < j ≤ N ‖ x i − x j ‖ − s {\displaystyle \sum _{1\leq i 0 and by ∑ 1 ≤ i < j ≤ N log ‖ x i − x j ‖ − 1 {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fekete–Szegő inequality Summary Fekete–Szegő_inequality In mathematics, the Fekete–Szegő inequality is an inequality for the coefficients of univalent analytic functions found by Fekete and Szegő (1933), related to the Bieberbach conjecture. Finding similar estimates for other classes of functions is called the Fekete–... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Feller–Tornier constant Summary Feller–Tornier_constant In mathematics, the Feller–Tornier constant CFT is the density of the set of all positive integers that have an even number of distinct prime factors raised to a power larger than one (ignoring any prime factors which appear only to the first power). It is named a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fermat curve Summary Fermat_curve In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation: X n + Y n = Z n . {\displaystyle X^{n}+Y^{n}=Z^{n}.\ } Therefore, in terms of the affine plane its equation is: x n + y n = 1. {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fermat curve Summary Fermat_curve The Fermat curve is non-singular and has genus: ( n − 1 ) ( n − 2 ) / 2. {\displaystyle (n-1)(n-2)/2.\ } This means genus 0 for the case n = 2 (a conic) and genus 1 only for n = 3 (an elliptic curve). The Jacobian variety of the Fermat curve has been studied in depth. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fermat curve Summary Fermat_curve It is isogenous to a product of simple abelian varieties with complex multiplication. The Fermat curve also has gonality: n − 1. {\displaystyle n-1.\ } | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalizations of Fibonacci numbers Summary Tribonacci_constant In mathematics, the Fibonacci numbers form a sequence defined recursively by: F n = { 0 n = 0 1 n = 1 F n − 1 + F n − 2 n > 1 {\displaystyle F_{n}={\begin{cases}0&n=0\\1&n=1\\F_{n-1}+F_{n-2}&n>1\end{cases}}} That is, after two starting values, each number... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fibonacci polynomials Summary Lucas_polynomial In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fibonacci ratio Summary Fibonacci_series In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fibonacci ratio Summary Fibonacci_series They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fibonomial coefficient Summary Fibonomial_coefficient In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as ( n k ) F = F n F n − 1 ⋯ F n − k + 1 F k F k − 1 ⋯ F 1 = n ! F k ! F ( n − k ) ! F {\displaystyle {\binom {n}{k}}_{F}={\frac {F_{n}F_{n-1}\cdots F_{n-k+1}}{F_{k}F_{k-1}\cd... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fibonorial Summary Fibonorial In mathematics, the Fibonorial n!F, also called the Fibonacci factorial, where n is a nonnegative integer, is defined as the product of the first n positive Fibonacci numbers, i.e. n ! F := ∏ i = 1 n F i , n ≥ 0 , {\displaystyle {n! }_{F}:=\prod _{i=1}^{n}F_{i},\quad n\geq 0,} where Fi is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fictitious domain method Summary Fictitious_domain_method In mathematics, the Fictitious domain method is a method to find the solution of a partial differential equations on a complicated domain D {\displaystyle D} , by substituting a given problem posed on a domain D {\displaystyle D} , with a new problem posed on a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fitting lemma Summary Fitting_lemma In mathematics, the Fitting lemma – named after the mathematician Hans Fitting – is a basic statement in abstract algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M is either an automorphism or nilpotent.As an imm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fontaine–Mazur conjecture Summary Fontaine–Mazur_conjecture In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by Fontaine and Mazur (1995) about when p-adic representations of Galois groups of number fields can be constructed from representations on étale cohomology groups of a varieties. S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
FKG inequality Summary FKG_inequality In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), due to Cees M. Fortuin, Pieter W. Kasteleyn, and Jean Gin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
FKG inequality Summary FKG_inequality variables, called Harris inequality, is due to Theodore Edward Harris (1960), see below. One generalization of the FKG inequality is the Holley inequality (1974) below, and an even further generalization is the Ahlswede–Daykin "four functions" theorem (1978). Furthermore, it has th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fourier inversion Summary Fourier_inversion_theorem In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fourier inversion Summary Fourier_inversion_theorem In other words, the theorem says that f ( x ) = ∬ R 2 e 2 π i ( x − y ) ⋅ ξ f ( y ) d y d ξ . {\displaystyle f(x)=\iint _{\mathbb {R} ^{2}}e^{2\pi i(x-y)\cdot \xi }\,f(y)\,dy\,d\xi .} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fourier inversion Summary Fourier_inversion_theorem This last equation is called the Fourier integral theorem. Another way to state the theorem is that if R {\displaystyle R} is the flip operator i.e. ( R f ) ( x ) := f ( − x ) {\displaystyle (Rf)(x):=f(-x)} , then F − 1 = F R = R F . {\displaystyle {\mathcal {F}}^{-1}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fourier inversion Summary Fourier_inversion_theorem The theorem holds if both f {\displaystyle f} and its Fourier transform are absolutely integrable (in the Lebesgue sense) and f {\displaystyle f} is continuous at the point x {\displaystyle x} . However, even under more general conditions versions of the Fourier inver... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sine transform Summary Sine_transform In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fourier transform on finite groups Summary Fourier_transform_on_finite_groups In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fox H-function Summary H_function In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral H p , q m , n = 1 2 π i ∫ L ∏ j = 1 m Γ ( b j + B j s ) ∏ j = 1 n Γ ( 1 − a j − A j s ) ∏ j = m ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fox derivative Summary Fox_derivative In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the fre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fox–Wright Psi function Summary Fox–Wright_Psi_function In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935):... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fraňková–Helly selection theorem Summary Fraňková–Helly_selection_theorem In mathematics, the Fraňková–Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana Fraňková. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fredholm alternative Summary Fredholm_alternative In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm opera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fredholm determinant Summary Fredholm_determinant In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fredholm equation Summary Fredholm_equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Freidlin–Wentzell theorem Summary Freidlin–Wentzell_theorem In mathematics, the Freidlin–Wentzell theorem (due to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentzell theorem gives an estimate for the probability that a (sca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rosenfeld projective plane Summary Rosenfeld_projective_plane In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Freudenthal spectral theorem Summary Freudenthal_spectral_theorem In mathematics, the Freudenthal spectral theorem is a result in Riesz space theory proved by Hans Freudenthal in 1936. It roughly states that any element dominated by a positive element in a Riesz space with the principal projection property can in a sen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frobenius determinant theorem Summary Frobenius_determinant_theorem In mathematics, the Frobenius determinant theorem was a conjecture made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it (reproduced in (Dedekind 1968), with an English translation in (Curtis 2003, p. 51)). ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frobenius mapping Summary Frobenius_mapping In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping φ that takes r in R to rp is a ring endomorphism of R. The image of φ is then Rp, the subring of R consisting of p-th powe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frobenius mapping Summary Frobenius_mapping The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping φ*: Spec(Rp) → Spec(R)of affine schemes. Even in cases where Rp = R this is not the identity, unless R is the prime field. Mappings created by fibre produc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frobenius inner product Summary Frobenius_inner_product In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted ⟨ A , B ⟩ F {\displaystyle \langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }} . The operation is a component-wise inner produ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fréchet derivative Summary Fréchet_derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fréchet distance Summary Fréchet_distance In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cofinite filter Summary Fréchet_filter In mathematics, the Fréchet filter, also called the cofinite filter, on a set X {\displaystyle X} is a certain collection of subsets of X {\displaystyle X} (that is, it is a particular subset of the power set of X {\displaystyle X} ). A subset F {\displaystyle F} of X {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frölicher spectral sequence Summary Frölicher_spectral_sequence In mathematics, the Frölicher spectral sequence (often misspelled as Fröhlicher) is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory that are valid in general only for Kähler manifolds. It ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frölicher–Nijenhuis bracket Summary Frölicher–Nijenhuis_bracket In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of connections, notably the Ehresmann connection, as well as in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fubini–Study metric Summary Fubini-Study_metric In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.A Hermitian form in (th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fubini–Study metric Summary Fubini-Study_metric The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2n+1)-sphere. In algebraic geometry, one uses a normalization making CPn a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fuchs relation Summary Fuchs_relation In mathematics, the Fuchs relation is a relation between the starting exponents of formal series solutions of certain linear differential equations, so called Fuchsian equations. It is named after Lazarus Immanuel Fuchs. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fulton–Hansen connectedness theorem Summary Fulton–Hansen_connectedness_theorem In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Meijer G-function Summary Meijer_G-function In mathematics, the G-function was introduced by Cornelis Simon Meijer (1936) as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the Mac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Meijer G-function Summary Meijer_G-function A notable property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a functional equation that allows to liberate from a G-function G(z) any factor zρ that is a constant power of its argumen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gabriel-Popescu theorem Summary Gabriel-Popescu_theorem In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by Pierre Gabriel and Nicolae Popescu (1964). It characterizes certain abelian categories (the Grothendieck categories) as quotients of module categories... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gan–Gross–Prasad conjecture Summary Gan–Gross–Prasad_conjecture In mathematics, the Gan–Gross–Prasad conjecture is a restriction problem in the representation theory of real or p-adic Lie groups posed by Gan Wee Teck, Benedict Gross, and Dipendra Prasad. The problem originated from a conjecture of Gross and Prasad for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Garnir relations Summary Garnir_relations In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules Vλ in terms of standard polytabloids. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gateaux derivative Summary Gateaux_derivative In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died at age 25 in World War I, it is defined for functions between loca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gateaux derivative Summary Gateaux_derivative Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as Tikhomirov (2001), draw a further distinc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gauss's circle problem Summary Gauss_circle_problem In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r {\displaystyle r} . This number is approximated by the area of the circle, so the real problem is to a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Class number problem Summary Class_number_problem_for_imaginary_quadratic_fields In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields Q ( d ) {\displaystyle \mathbb {Q} ({\sqrt {d}})} (for negat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gauss iterated map Summary Gauss_iterated_map In mathematics, the Gauss map (also known as Gaussian map or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function: x n + 1 = exp ( − α x n 2 ) + β , {\displaystyle x_{n+1}=\exp(-\alpha x_{n}^{2})+\beta ,\,} where α and β... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-binomial theorem Summary Gaussian_coefficient In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as ( n k ) q {\displaystyle {\binom {n}{k}}_{q}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gaussian isoperimetric inequality Summary Gaussian_isoperimetric_inequality In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hypergeometric series Summary Hypergeometric_series In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gauss-Kuzmin distribution Summary Gauss–Kuzmin_distribution In mathematics, the Gauss–Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1). The distributio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gauss–Kuzmin–Wirsing operator Summary Gauss–Kuzmin–Wirsing_operator In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in differential geometry.) It is named after Carl ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gauss–Manin connection Summary Gauss–Manin_connection In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties V s {\displaystyle V_{s}} . The fibers of the vector bundle are the de Rham cohomology groups H D R k ( V s ) {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gelfand representation Summary Gelfand_duality In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: a way of representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gelfand–Naimark theorem Summary Gelfand–Naimark_theorem In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant poin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gelfand–Tsetlin integrable system Summary Gelfand–Zeitlin_integrable_system In mathematics, the Gelfand–Zeitlin system (also written Gelfand–Zetlin system, Gelfand–Cetlin system, Gelfand–Tsetlin system) is an integrable system on conjugacy classes of Hermitian matrices. It was introduced by Guillemin and Sternberg (198... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gelfond's theorem Summary Gelfond's_theorem In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Genocchi number Summary Genocchi_number In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation − 2 t 1 + e − t = ∑ n = 0 ∞ G n t n n ! {\displaystyle {\frac {-2t}{1+e^{-t}}}=\sum _{n=0}^{\infty }G_{n}{\frac {t^{n}}{n!}}} The first few Genocchi numbers ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gershgorin circle theorem Summary Gershgorin_circle_theorem In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's name has been transliterated in several different ways, inc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gibbons–Hawking Ansatz Summary Gibbons–Hawking_Ansatz In mathematics, the Gibbons–Hawking ansatz is a method of constructing gravitational instantons introduced by Gary Gibbons and Stephen Hawking (1978, 1979). It gives examples of hyperkähler manifolds in dimension 4 that are invariant under a circle action. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gibbs random field Summary Gibbs_random_field In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gibbs random field Summary Gibbs_random_field Here, E is a function from the space of states to the real numbers; in physics applications, E(x) is interpreted as the energy of the configuration x. The parameter β is a free parameter; in physics, it is the inverse temperature. The normalizing constant Z(β) is the partit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gibbs random field Summary Gibbs_random_field Traditional approaches in statistical physics studied the limit of intensive properties as the size of a finite system approaches infinity (the thermodynamic limit). When the energy function can be written as a sum of terms that each involve only variables from a finite sub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gibbs random field Summary Gibbs_random_field A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary condition... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gibbs random field Summary Gibbs_random_field A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. The Gibbs measure of an infinite system is not necessarily unique, in contrast to th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gibbs phenomenon Summary Gibbs_phenomenon In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The N {\textstyle N} th partial Fourier series of the function (formed by summing the N {\textstyle N... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
3-manifold Gieseking manifold 3-manifold > Important examples of 3-manifolds > Gieseking manifold In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
3-manifold Gieseking manifold 3-manifold > Important examples of 3-manifolds > Gieseking manifold The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
3-manifold Gieseking manifold 3-manifold > Important examples of 3-manifolds > Gieseking manifold Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner. More... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gieseking manifold Summary Gieseking_manifold In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately V ≈ 1.0149416 {\displaystyle V\approx 1.0149416} . It was discover... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gieseking manifold Summary Gieseking_manifold Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epste... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gieseking manifold Summary Gieseking_manifold Moreover, the angle made by the faces is π / 3 {\displaystyle \pi /3} . The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together. The Gieseking manifold has a double cover homeomorphic to the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gieseking manifold Summary Gieseking_manifold The underlying compact manifold has a Klein bottle boundary, and the first homology group of the Gieseking manifold is the integers. The Gieseking manifold is a fiber bundle over the circle with fiber the once-punctured torus and monodromy given by ( x , y ) → ( x + y , x )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gilbert–Pollack conjecture Summary Gilbert–Pollack_conjecture In mathematics, the Gilbert–Pollak conjecture is an unproven conjecture on the ratio of lengths of Steiner trees and Euclidean minimum spanning trees for the same point sets in the Euclidean plane. It was proposed by Edgar Gilbert and Henry O. Pollak in 1968... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Glaisher–Kinkelin constant Summary Glaisher–Kinkelin_constant In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function. The constant appears in a number of sums and integrals, especially those involving ga... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Glaisher–Kinkelin constant Summary 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 |
Glaisher–Kinkelin constant Summary 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 b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goldbach–Euler theorem Summary 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 + ⋯ = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Golod–Shafarevich theorem Summary Class_field_tower 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 |
Golomb sequence Summary Golomb_sequence In mathematics, the Golomb sequence, named after Solomon W. Golomb (but also called Silverman's sequence), is a monotonically increasing integer sequence where an is the number of times that n occurs in the sequence, starting with a1 = 1, and with the property that for n > 1 each... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Golomb–Dickman constant Summary 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 OEI... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Euler–Gompertz constant Summary 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 −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Euler–Gompertz constant Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Euler–Gompertz constant Summary 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goncharov conjecture Summary Goncharov_conjecture In mathematics, the Goncharov conjecture is a conjecture introduced by Goncharov (1995) suggesting that the cohomology of certain motivic complexes coincides with pieces of K-groups. It extends a conjecture due to Zagier (1991). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goormaghtigh conjecture Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gordon–Luecke theorem Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gordon–Luecke theorem Summary 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 Gordo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gordon–Luecke theorem Summary 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 cycle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gordon–Luecke theorem Summary 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). ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gorenstein–Walter theorem Summary 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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