text
stringlengths 9
3.55k
| source
stringlengths 31
280
|
|---|---|
In mathematics, the Fekete problem is, given a natural number N and a real s ≥ 0, to find the points x1,...,xN on the 2-sphere for which the s-energy, defined by ∑ 1 ≤ i < j ≤ N ‖ x i − x j ‖ − s {\displaystyle \sum _{1\leq i 0 and by ∑ 1 ≤ i < j ≤ N log ‖ x i − x j ‖ − 1 {\displaystyle \sum _{1\leq i 0, such points are called s-Fekete points, and for s = 0, logarithmic Fekete points (see Saff & Kuijlaars (1997)). More generally, one can consider the same problem on the d-dimensional sphere, or on a Riemannian manifold (in which case ||xi −xj|| is replaced with the Riemannian distance between xi and xj). The problem originated in the paper by Michael Fekete (1923) who considered the one-dimensional, s = 0 case, answering a question of Issai Schur. An algorithmic version of the Fekete problem is number 7 on the list of problems discussed by Smale (1998).
|
https://en.wikipedia.org/wiki/Fekete_problem
|
In mathematics, the Fekete–Szegő inequality is an inequality for the coefficients of univalent analytic functions found by Fekete and Szegő (1933), related to the Bieberbach conjecture. Finding similar estimates for other classes of functions is called the Fekete–Szegő problem. The Fekete–Szegő inequality states that if f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ {\displaystyle f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots } is a univalent analytic function on the unit disk and 0 ≤ λ < 1 {\displaystyle 0\leq \lambda <1} , then | a 3 − λ a 2 2 | ≤ 1 + 2 exp ( − 2 λ / ( 1 − λ ) ) . {\displaystyle |a_{3}-\lambda a_{2}^{2}|\leq 1+2\exp(-2\lambda /(1-\lambda )).}
|
https://en.wikipedia.org/wiki/Fekete–Szegő_inequality
|
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 after William Feller (1906–1970) and Erhard Tornier (1894–1982) C FT = 1 2 + ( 1 2 ∏ n = 1 ∞ ( 1 − 2 p n 2 ) ) = 1 2 ( 1 + ∏ n = 1 ∞ ( 1 − 2 p n 2 ) ) = 1 2 ( 1 + 1 ζ ( 2 ) ∏ n = 1 ∞ ( 1 − 1 p n 2 − 1 ) ) = 1 2 + 3 π 2 ∏ n = 1 ∞ ( 1 − 1 p n 2 − 1 ) = 0.66131704946 … {\displaystyle {\begin{aligned}C_{\text{FT}}&={1 \over 2}+\left({1 \over 2}\prod _{n=1}^{\infty }\left(1-{2 \over p_{n}^{2}}\right)\right)\\&={{1} \over {2}}\left(1+\prod _{n=1}^{\infty }\left(1-{{2} \over {p_{n}^{2}}}\right)\right)\\&={1 \over 2}\left(1+{{1} \over {\zeta (2)}}\prod _{n=1}^{\infty }\left(1-{{1} \over {p_{n}^{2}-1}}\right)\right)\\&={1 \over 2}+{{3} \over {\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{{1} \over {p_{n}^{2}-1}}\right)=0.66131704946\ldots \end{aligned}}} (sequence A065493 in the OEIS)
|
https://en.wikipedia.org/wiki/Feller–Tornier_constant
|
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation: X n + Y n = Z n . {\displaystyle X^{n}+Y^{n}=Z^{n}.\ } Therefore, in terms of the affine plane its equation is: x n + y n = 1. {\displaystyle x^{n}+y^{n}=1.\ } An integer solution to the Fermat equation would correspond to a nonzero rational number solution to the affine equation, and vice versa. But by Fermat's Last Theorem it is now known that (for n > 2) there are no nontrivial integer solutions to the Fermat equation; therefore, the Fermat curve has no nontrivial rational points.
|
https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Fermat_curve
|
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 is the sum of the two preceding numbers. The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers.
|
https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers
|
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://en.wikipedia.org/wiki/Fibonacci_polynomials
|
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 start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.
|
https://en.wikipedia.org/wiki/Fibonacci_ratio
|
They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone's bracts, though they don't occur in all species. Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.
|
https://en.wikipedia.org/wiki/Fibonacci_ratio
|
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}\cdots F_{1}}}={\frac {n!_{F}}{k!_{F}(n-k)!_{F}}}} where n and k are non-negative integers, 0 ≤ k ≤ n, Fj is the j-th Fibonacci number and n!F is the nth Fibonorial, i.e. n ! F := ∏ i = 1 n F i , {\displaystyle {n! }_{F}:=\prod _{i=1}^{n}F_{i},} where 0!F, being the empty product, evaluates to 1.
|
https://en.wikipedia.org/wiki/Fibonomial_coefficient
|
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 the ith Fibonacci number, and 0!F gives the empty product (defined as the multiplicative identity, i.e. 1). The Fibonorial n!F is defined analogously to the factorial n!. The Fibonorial numbers are used in the definition of Fibonomial coefficients (or Fibonacci-binomial coefficients) similarly as the factorial numbers are used in the definition of binomial coefficients.
|
https://en.wikipedia.org/wiki/Fibonorial
|
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 simple domain Ω {\displaystyle \Omega } containing D {\displaystyle D} .
|
https://en.wikipedia.org/wiki/Fictitious_domain_method
|
In mathematics, the Fitting lemma – named after the mathematician Hans Fitting – is a basic statement in abstract algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M is either an automorphism or nilpotent.As an immediate consequence, we see that the endomorphism ring of every finite-length indecomposable module is local. A version of Fitting's lemma is often used in the representation theory of groups. This is in fact a special case of the version above, since every K-linear representation of a group G can be viewed as a module over the group algebra KG.
|
https://en.wikipedia.org/wiki/Fitting_lemma
|
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. Some cases of this conjecture in dimension 2 were already proved by Dieulefait (2004).
|
https://en.wikipedia.org/wiki/Fontaine–Mazur_conjecture
|
In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), due to Cees M. Fortuin, Pieter W. Kasteleyn, and Jean Ginibre (1971). Informally, it says that in many random systems, increasing events are positively correlated, while an increasing and a decreasing event are negatively correlated. It was obtained by studying the random cluster model. An earlier version, for the special case of i.i.d.
|
https://en.wikipedia.org/wiki/FKG_inequality
|
variables, called Harris inequality, is due to Theodore Edward Harris (1960), see below. One generalization of the FKG inequality is the Holley inequality (1974) below, and an even further generalization is the Ahlswede–Daykin "four functions" theorem (1978). Furthermore, it has the same conclusion as the Griffiths inequalities, but the hypotheses are different.
|
https://en.wikipedia.org/wiki/FKG_inequality
|
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function f: R → C {\displaystyle f:\mathbb {R} \to \mathbb {C} } satisfying certain conditions, and we use the convention for the Fourier transform that ( F f ) ( ξ ) := ∫ R e − 2 π i y ⋅ ξ f ( y ) d y , {\displaystyle ({\mathcal {F}}f)(\xi ):=\int _{\mathbb {R} }e^{-2\pi iy\cdot \xi }\,f(y)\,dy,} then f ( x ) = ∫ R e 2 π i x ⋅ ξ ( F f ) ( ξ ) d ξ . {\displaystyle f(x)=\int _{\mathbb {R} }e^{2\pi ix\cdot \xi }\,({\mathcal {F}}f)(\xi )\,d\xi .}
|
https://en.wikipedia.org/wiki/Fourier_inversion
|
In other words, the theorem says that f ( x ) = ∬ R 2 e 2 π i ( x − y ) ⋅ ξ f ( y ) d y d ξ . {\displaystyle f(x)=\iint _{\mathbb {R} ^{2}}e^{2\pi i(x-y)\cdot \xi }\,f(y)\,dy\,d\xi .}
|
https://en.wikipedia.org/wiki/Fourier_inversion
|
This last equation is called the Fourier integral theorem. Another way to state the theorem is that if R {\displaystyle R} is the flip operator i.e. ( R f ) ( x ) := f ( − x ) {\displaystyle (Rf)(x):=f(-x)} , then F − 1 = F R = R F . {\displaystyle {\mathcal {F}}^{-1}={\mathcal {F}}R=R{\mathcal {F}}.}
|
https://en.wikipedia.org/wiki/Fourier_inversion
|
The theorem holds if both f {\displaystyle f} and its Fourier transform are absolutely integrable (in the Lebesgue sense) and f {\displaystyle f} is continuous at the point x {\displaystyle x} . However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.
|
https://en.wikipedia.org/wiki/Fourier_inversion
|
In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics.
|
https://en.wikipedia.org/wiki/Sine_transform
|
In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.
|
https://en.wikipedia.org/wiki/Fourier_transform_on_finite_groups
|
In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral H p , q m , n = 1 2 π i ∫ L ∏ j = 1 m Γ ( b j + B j s ) ∏ j = 1 n Γ ( 1 − a j − A j s ) ∏ j = m + 1 q Γ ( 1 − b j − B j s ) ∏ j = n + 1 p Γ ( a j + A j s ) z − s d s , {\displaystyle H_{p,q}^{\,m,n}\!\left={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}+B_{j}s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}-A_{j}s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}-B_{j}s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}+A_{j}s)}}z^{-s}\,ds,} where L is a certain contour separating the poles of the two factors in the numerator.
|
https://en.wikipedia.org/wiki/Fox_H-function
|
In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus. The Fox derivative was developed in a series of five papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953.
|
https://en.wikipedia.org/wiki/Fox_derivative
|
In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935): Upon changing the normalisation it becomes pFq(z) for A1...p = B1...q = 1. The Fox–Wright function is a special case of the Fox H-function (Srivastava & Manocha 1984, p. 50): A special case of Fox–Wright function appears as a part of the normalizing constant of the Modified half-normal distribution with the pdf on ( 0 , ∞ ) {\displaystyle (0,\infty )} is given as f ( x ) = 2 β α 2 x α − 1 exp ( − β x 2 + γ x ) Ψ ( α 2 , γ β ) {\displaystyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}} , where Ψ ( α , z ) = 1 Ψ 1 ( ( α , 1 2 ) ( 1 , 0 ) ; z ) {\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)} denotes the Fox–Wright Psi function.
|
https://en.wikipedia.org/wiki/Fox–Wright_Psi_function
|
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://en.wikipedia.org/wiki/Fraňková–Helly_selection_theorem
|
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.
|
https://en.wikipedia.org/wiki/Fredholm_alternative
|
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The function is named after the mathematician Erik Ivar Fredholm. Fredholm determinants have had many applications in mathematical physics, the most celebrated example being Gábor Szegő's limit formula, proved in response to a question raised by Lars Onsager and C. N. Yang on the spontaneous magnetization of the Ising model.
|
https://en.wikipedia.org/wiki/Fredholm_determinant
|
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian.
|
https://en.wikipedia.org/wiki/Fredholm_equation
|
In mathematics, the Freidlin–Wentzell theorem (due to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentzell theorem gives an estimate for the probability that a (scaled-down) sample path of an Itō diffusion will stray far from the mean path. This statement is made precise using rate functions. The Freidlin–Wentzell theorem generalizes Schilder's theorem for standard Brownian motion.
|
https://en.wikipedia.org/wiki/Freidlin–Wentzell_theorem
|
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table at right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B, despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction. The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the octonions": G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space – see prehomogeneous vector space).
|
https://en.wikipedia.org/wiki/Rosenfeld_projective_plane
|
In mathematics, the Freudenthal spectral theorem is a result in Riesz space theory proved by Hans Freudenthal in 1936. It roughly states that any element dominated by a positive element in a Riesz space with the principal projection property can in a sense be approximated uniformly by simple functions. Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-known Radon–Nikodym theorem, the validity of the Poisson formula and the spectral theorem from the theory of normal operators can all be shown to follow as special cases of the Freudenthal spectral theorem.
|
https://en.wikipedia.org/wiki/Freudenthal_spectral_theorem
|
In mathematics, the Frobenius determinant theorem was a conjecture made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it (reproduced in (Dedekind 1968), with an English translation in (Curtis 2003, p. 51)). If one takes the multiplication table of a finite group G and replaces each entry g with the variable xg, and subsequently takes the determinant, then the determinant factors as a product of n irreducible polynomials, where n is the number of conjugacy classes. Moreover, each polynomial is raised to a power equal to its degree. Frobenius proved this surprising conjecture, and it became known as the Frobenius determinant theorem.
|
https://en.wikipedia.org/wiki/Frobenius_determinant_theorem
|
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 powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring automorphism.
|
https://en.wikipedia.org/wiki/Frobenius_mapping
|
The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping φ*: Spec(Rp) → Spec(R)of affine schemes. Even in cases where Rp = R this is not the identity, unless R is the prime field. Mappings created by fibre product with φ*, i.e. base changes, tend in scheme theory to be called geometric Frobenius. The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear.
|
https://en.wikipedia.org/wiki/Frobenius_mapping
|
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 product of two matrices as though they are vectors, and satisfies the axioms for an inner product. The two matrices must have the same dimension - same number of rows and columns, but are not restricted to be square matrices.
|
https://en.wikipedia.org/wiki/Frobenius_inner_product
|
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces. The Fréchet derivative should be contrasted to the more general Gateaux derivative which is a generalization of the classical directional derivative. The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis.
|
https://en.wikipedia.org/wiki/Fréchet_derivative
|
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://en.wikipedia.org/wiki/Fréchet_distance
|
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 {\displaystyle X} belongs to the Fréchet filter if and only if the complement of F {\displaystyle F} in X {\displaystyle X} is finite. Any such set F {\displaystyle F} is said to be cofinite in X {\displaystyle X} , which is why it is alternatively called the cofinite filter on X {\displaystyle X} . The Fréchet filter is of interest in topology, where filters originated, and relates to order and lattice theory because a set's power set is a partially ordered set under set inclusion (more specifically, it forms a lattice). The Fréchet filter is named after the French mathematician Maurice Fréchet (1878-1973), who worked in topology.
|
https://en.wikipedia.org/wiki/Cofinite_filter
|
In mathematics, the Frölicher spectral sequence (often misspelled as Fröhlicher) is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory that are valid in general only for Kähler manifolds. It was introduced by Frölicher (1955). A spectral sequence is set up, the degeneration of which would give the results of Hodge theory and Dolbeault's theorem.
|
https://en.wikipedia.org/wiki/Frölicher_spectral_sequence
|
In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of connections, notably the Ehresmann connection, as well as in the more general study of projections in the tangent bundle. It was introduced by Alfred Frölicher and Albert Nijenhuis (1956) and is related to the work of Schouten (1940). It is related to but not the same as the Nijenhuis–Richardson bracket and the Schouten–Nijenhuis bracket.
|
https://en.wikipedia.org/wiki/Frölicher–Nijenhuis_bracket
|
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.A Hermitian form in (the vector space) Cn+1 defines a unitary subgroup U(n+1) in GL(n+1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CPn is a symmetric space.
|
https://en.wikipedia.org/wiki/Fubini–Study_metric
|
The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2n+1)-sphere. In algebraic geometry, one uses a normalization making CPn a Hodge manifold.
|
https://en.wikipedia.org/wiki/Fubini–Study_metric
|
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://en.wikipedia.org/wiki/Fuchs_relation
|
In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1. It is named after William Fulton and Johan Hansen, who proved it in 1979. The formal statement is that if V and W are irreducible algebraic subvarieties of a projective space P, all over an algebraically closed field, and if dim ( V ) + dim ( W ) > dim ( P ) {\displaystyle \dim(V)+\dim(W)>\dim(P)} in terms of the dimension of an algebraic variety, then the intersection U of V and W is connected. More generally, the theorem states that if Z {\displaystyle Z} is a projective variety and f: Z → P n × P n {\displaystyle f\colon Z\to P^{n}\times P^{n}} is any morphism such that dim f ( Z ) > n {\displaystyle \dim f(Z)>n} , then f − 1 Δ {\displaystyle f^{-1}\Delta } is connected, where Δ {\displaystyle \Delta } is the diagonal in P n × P n {\displaystyle P^{n}\times P^{n}} . The special case of intersections is recovered by taking Z = V × W {\displaystyle Z=V\times W} , with f {\displaystyle f} the natural inclusion.
|
https://en.wikipedia.org/wiki/Fulton–Hansen_connectedness_theorem
|
In mathematics, the G-function was introduced by Cornelis Simon Meijer (1936) as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's G-function was able to include those as particular cases as well. The first definition was made by Meijer using a series; nowadays the accepted and more general definition is via a line integral in the complex plane, introduced in its full generality by Arthur Erdélyi in 1953. With the modern definition, the majority of the established special functions can be represented in terms of the Meijer G-function.
|
https://en.wikipedia.org/wiki/Meijer_G-function
|
A notable property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a functional equation that allows to liberate from a G-function G(z) any factor zρ that is a constant power of its argument z, the closure implies that whenever a function is expressible as a G-function of a constant multiple of some constant power of the function argument, f(x) = G(cxγ), the derivative and the antiderivative of this function are expressible so too. The wide coverage of special functions also lends power to uses of Meijer's G-function other than the representation and manipulation of derivatives and antiderivatives. For example, the definite integral over the positive real axis of any function g(x) that can be written as a product G1(cxγ)·G2(dxδ) of two G-functions with rational γ/δ equals just another G-function, and generalizations of integral transforms like the Hankel transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels. A still more general function, which introduces additional parameters into Meijer's G-function, is Fox's H-function and is used for Matrix transform by Ram Kishore SaxenaOne application of the Meijer G-function has been the particle spectrum of radiation from an inertial horizon in the moving mirror model of the dynamical Casimir effect (Good 2020).
|
https://en.wikipedia.org/wiki/Meijer_G-function
|
In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by Pierre Gabriel and Nicolae Popescu (1964). It characterizes certain abelian categories (the Grothendieck categories) as quotients of module categories. There are several generalizations and variations of the Gabriel–Popescu theorem, given by Kuhn (1994) (for an AB5 category with a set of generators), Lowen (2004), Porta (2010) (for triangulated categories).
|
https://en.wikipedia.org/wiki/Gabriel-Popescu_theorem
|
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 special orthogonal groups but was later generalized to include all four classical groups. In the cases considered, it is known that the multiplicity of the restrictions is at most one and the conjecture describes when the multiplicity is precisely one.
|
https://en.wikipedia.org/wiki/Gan–Gross–Prasad_conjecture
|
In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules Vλ in terms of standard polytabloids.
|
https://en.wikipedia.org/wiki/Garnir_relations
|
In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died at age 25 in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.
|
https://en.wikipedia.org/wiki/Gateaux_derivative
|
Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as Tikhomirov (2001), draw a further distinction between the Gateaux differential (which may be nonlinear) and the Gateaux derivative (which they take to be linear). In most applications, continuous linearity follows from some more primitive condition which is natural to the particular setting, such as imposing complex differentiability in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis.
|
https://en.wikipedia.org/wiki/Gateaux_derivative
|
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 accurately bound the error term describing how the number of points differs from the area. The first progress on a solution was made by Carl Friedrich Gauss, hence its name.
|
https://en.wikipedia.org/wiki/Gauss's_circle_problem
|
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 negative integers d) having class number n. It is named after Carl Friedrich Gauss. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as d → − ∞ {\displaystyle d\to -\infty } . The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.
|
https://en.wikipedia.org/wiki/Class_number_problem
|
In mathematics, the Gauss map (also known as Gaussian map or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function: x n + 1 = exp ( − α x n 2 ) + β , {\displaystyle x_{n+1}=\exp(-\alpha x_{n}^{2})+\beta ,\,} where α and β are real parameters. Named after Johann Carl Friedrich Gauss, the function maps the bell shaped Gaussian function similar to the logistic map.
|
https://en.wikipedia.org/wiki/Gauss_iterated_map
|
In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as ( n k ) q {\displaystyle {\binom {n}{k}}_{q}} or q {\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}_{q}} , is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over F q {\displaystyle \mathbb {F} _{q}} , a finite field with q elements; i.e. it is the number of points in the finite Grassmannian G r ( k , F q n ) {\displaystyle \mathrm {Gr} (k,\mathbb {F} _{q}^{n})} .
|
https://en.wikipedia.org/wiki/Q-binomial_theorem
|
In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure.
|
https://en.wikipedia.org/wiki/Gaussian_isoperimetric_inequality
|
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953) and Olde Daalhuis (2010). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.
|
https://en.wikipedia.org/wiki/Hypergeometric_series
|
In mathematics, the Gauss–Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1). The distribution is named after Carl Friedrich Gauss, who derived it around 1800, and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929. It is given by the probability mass function p ( k ) = − log 2 ( 1 − 1 ( 1 + k ) 2 ) . {\displaystyle p(k)=-\log _{2}\left(1-{\frac {1}{(1+k)^{2}}}\right)~.}
|
https://en.wikipedia.org/wiki/Gauss-Kuzmin_distribution
|
In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in differential geometry.) It is named after Carl Gauss, Rodion Kuzmin, and Eduard Wirsing. It occurs in the study of continued fractions; it is also related to the Riemann zeta function.
|
https://en.wikipedia.org/wiki/Gauss–Kuzmin–Wirsing_operator
|
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties V s {\displaystyle V_{s}} . The fibers of the vector bundle are the de Rham cohomology groups H D R k ( V s ) {\displaystyle H_{DR}^{k}(V_{s})} of the fibers V s {\displaystyle V_{s}} of the family. It was introduced by Yuri Manin (1958) for curves S and by Alexander Grothendieck (1966) in higher dimensions. Flat sections of the bundle are described by differential equations; the best-known of these is the Picard–Fuchs equation, which arises when the family of varieties is taken to be the family of elliptic curves. In intuitive terms, when the family is locally trivial, cohomology classes can be moved from one fiber in the family to nearby fibers, providing the 'flat section' concept in purely topological terms. The existence of the connection is to be inferred from the flat sections.
|
https://en.wikipedia.org/wiki/Gauss–Manin_connection
|
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: a way of representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is an isometric isomorphism.In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand–Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix.
|
https://en.wikipedia.org/wiki/Gelfand_representation
|
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra.
|
https://en.wikipedia.org/wiki/Gelfand–Naimark_theorem
|
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 (1983), who named it after the Gelfand–Zeitlin basis, an early example of canonical basis, introduced by I. M. Gelfand and M. L. Cetlin in 1950s. Kostant and Wallach (2006) introduced a complex version of this integrable system.
|
https://en.wikipedia.org/wiki/Gelfand–Tsetlin_integrable_system
|
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
|
https://en.wikipedia.org/wiki/Gelfond's_theorem
|
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 are 0, −1, −1, 0, 1, 0, −3, 0, 17 (sequence A226158 in the OEIS), see OEIS: A001469.
|
https://en.wikipedia.org/wiki/Genocchi_number
|
In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's name has been transliterated in several different ways, including Geršgorin, Gerschgorin, Gershgorin, Hershhorn, and Hirschhorn.
|
https://en.wikipedia.org/wiki/Gershgorin_circle_theorem
|
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://en.wikipedia.org/wiki/Gibbons–Hawking_Ansatz
|
In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system X being in state x (equivalently, of the random variable X having value x) as P ( X = x ) = 1 Z ( β ) exp ( − β E ( x ) ) . {\displaystyle P(X=x)={\frac {1}{Z(\beta )}}\exp(-\beta E(x)).}
|
https://en.wikipedia.org/wiki/Gibbs_random_field
|
Here, E is a function from the space of states to the real numbers; in physics applications, E(x) is interpreted as the energy of the configuration x. The parameter β is a free parameter; in physics, it is the inverse temperature. The normalizing constant Z(β) is the partition function. However, in infinite systems, the total energy is no longer a finite number and cannot be used in the traditional construction of the probability distribution of a canonical ensemble.
|
https://en.wikipedia.org/wiki/Gibbs_random_field
|
Traditional approaches in statistical physics studied the limit of intensive properties as the size of a finite system approaches infinity (the thermodynamic limit). When the energy function can be written as a sum of terms that each involve only variables from a finite subsystem, the notion of a Gibbs measure provides an alternative approach. Gibbs measures were proposed by probability theorists such as Dobrushin, Lanford, and Ruelle and provided a framework to directly study infinite systems, instead of taking the limit of finite systems.
|
https://en.wikipedia.org/wiki/Gibbs_random_field
|
A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measure conditional on the frozen degrees of freedom. The Hammersley–Clifford theorem implies that any probability measure that satisfies a Markov property is a Gibbs measure for an appropriate choice of (locally defined) energy function. Therefore, the Gibbs measure applies to widespread problems outside of physics, such as Hopfield networks, Markov networks, Markov logic networks, and boundedly rational potential games in game theory and economics.
|
https://en.wikipedia.org/wiki/Gibbs_random_field
|
A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique. The existence of more than one Gibbs measure is associated with statistical phenomena such as symmetry breaking and phase coexistence.
|
https://en.wikipedia.org/wiki/Gibbs_random_field
|
In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The N {\textstyle N} th partial Fourier series of the function (formed by summing the N {\textstyle N} lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere (pointwise convergence on continuous points) except points of discontinuity.The Gibbs phenomenon was observed by experimental physicists and was believed to be due to imperfections in the measuring apparatus, but it is in fact a mathematical result. It is one cause of ringing artifacts in signal processing.
|
https://en.wikipedia.org/wiki/Gibbs_phenomenon
|
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 was discovered by Hugo Gieseking (1912).
|
https://en.wikipedia.org/wiki/3-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 the face with vertices 3,1,0 in that order.
|
https://en.wikipedia.org/wiki/3-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. 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.
|
https://en.wikipedia.org/wiki/3-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 discovered by Hugo Gieseking (1912).
|
https://en.wikipedia.org/wiki/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. Epstein and Robert C. Penner.
|
https://en.wikipedia.org/wiki/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 figure-eight knot complement.
|
https://en.wikipedia.org/wiki/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 ) . {\displaystyle (x,y)\to (x+y,x).} The square of this map is Arnold's cat map and this gives another way to see that the Gieseking manifold is double covered by the complement of the figure-eight knot.
|
https://en.wikipedia.org/wiki/Gieseking_manifold
|
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://en.wikipedia.org/wiki/Gilbert–Pollack_conjecture
|
In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function. The constant appears in a number of sums and integrals, especially those involving gamma functions and zeta functions. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin. Its approximate value is: A = 1.28242712910062263687... (sequence A074962 in the OEIS).The Glaisher–Kinkelin constant A can be given by the limit: A = lim n → ∞ H ( n ) n n 2 2 + n 2 + 1 12 e − n 2 4 {\displaystyle A=\lim _{n\rightarrow \infty }{\frac {H(n)}{n^{{\frac {n^{2}}{2}}+{\frac {n}{2}}+{\frac {1}{12}}}\,e^{-{\frac {n^{2}}{4}}}}}} where H(n) = Πnk=1 kk is the hyperfactorial.
|
https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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 function is: A = lim n → ∞ ( 2 π ) n 2 n n 2 2 − 1 12 e − 3 n 2 4 + 1 12 G ( n + 1 ) {\displaystyle A=\lim _{n\rightarrow \infty }{\frac {\left(2\pi \right)^{\frac {n}{2}}n^{{\frac {n^{2}}{2}}-{\frac {1}{12}}}e^{-{\frac {3n^{2}}{4}}+{\frac {1}{12}}}}{G(n+1)}}} .The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as: ζ ′ ( − 1 ) = 1 12 − ln A {\displaystyle \zeta '(-1)={\tfrac {1}{12}}-\ln A} ∑ k = 2 ∞ ln k k 2 = − ζ ′ ( 2 ) = π 2 6 ( 12 ln A − γ − ln 2 π ) {\displaystyle \sum _{k=2}^{\infty }{\frac {\ln k}{k^{2}}}=-\zeta '(2)={\frac {\pi ^{2}}{6}}\left(12\ln A-\gamma -\ln 2\pi \right)} where γ is the Euler–Mascheroni constant.
|
https://en.wikipedia.org/wiki/Glaisher–Kinkelin_constant
|
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}}={{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{15}}+{\frac {1}{24}}+{\frac {1}{26}}+{\frac {1}{31}}}+\cdots =1.} This result was first published in Euler's 1737 paper "Variæ observationes circa series infinitas". Euler attributed the result to a letter (now lost) from Goldbach.
|
https://en.wikipedia.org/wiki/Goldbach–Euler_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://en.wikipedia.org/wiki/Golod–Shafarevich_theorem
|
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 an is the smallest unique integer which makes it possible to satisfy the condition. For example, a1 = 1 says that 1 only occurs once in the sequence, so a2 cannot be 1 too, but it can be 2, and therefore must be 2. The first few values are 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12 (sequence A001462 in the OEIS).
|
https://en.wikipedia.org/wiki/Golomb_sequence
|
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 irrational.
|
https://en.wikipedia.org/wiki/Golomb–Dickman_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 \delta ={\frac {1}{2-{\frac {1}{4-{\frac {4}{6-{\frac {9}{{8-\qquad \qquad } \atop {\qquad }{}^{\ddots }{-{\frac {n^{2}}{2n+2-\dots }}}}}}}}}}},} or, alternatively, by δ = 1 − 1 3 − 2 5 − 6 7 − 12 9 − ⋱ − n ( n + 1 ) 2 n + 3 − … {\displaystyle \delta =1-{\frac {1}{3-{\frac {2}{5-{\frac {6}{7-{\frac {12}{{9-\qquad \qquad } \atop {\qquad }{}^{\ddots }{-{\frac {n(n+1)}{2n+3-\dots }}}}}}}}}}}} or δ = 1 1 + 1 1 + 1 1 + 2 1 + 2 1 + 3 1 + 3 1 + 4 1 + … . {\displaystyle \delta ={\frac {1}{1+{\frac {1}{1+{\frac {1}{1+{\frac {2}{1+{\frac {2}{1+{\frac {3}{1+{\frac {3}{1+{\frac {4}{1+\dots }}}}}}}}}}}}}}}}.}
|
https://en.wikipedia.org/wiki/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.} The first integral defines δ {\displaystyle \delta } , and the second and third follow from an integration of parts and a variable substitution respectively.
|
https://en.wikipedia.org/wiki/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. Le Lionnais called δ {\displaystyle \delta } the Gompertz constant because of its role in survival analysis.In 2009 Alexander Aptekarev proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational. This result was improved in 2012 by Tanguy Rivoal where he proved that at least one of them is transcendental.
|
https://en.wikipedia.org/wiki/Euler–Gompertz_constant
|
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://en.wikipedia.org/wiki/Goncharov_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-1}}} satisfying x > y > 1 {\displaystyle x>y>1} and n , m > 2 {\displaystyle n,m>2} are 5 3 − 1 5 − 1 = 2 5 − 1 2 − 1 = 31 {\displaystyle {\frac {5^{3}-1}{5-1}}={\frac {2^{5}-1}{2-1}}=31} and 90 3 − 1 90 − 1 = 2 13 − 1 2 − 1 = 8191. {\displaystyle {\frac {90^{3}-1}{90-1}}={\frac {2^{13}-1}{2-1}}=8191.}
|
https://en.wikipedia.org/wiki/Goormaghtigh_conjecture
|
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 complements"; however this is slightly ambiguous as it considers two knots to be equivalent if there is a self-homeomorphism taking one knot to the other. Thus mirror images are neglected.
|
https://en.wikipedia.org/wiki/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 nontrivial knot in the 3-sphere can yield the 3-sphere.
|
https://en.wikipedia.org/wiki/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 links are determined by their complements.
|
https://en.wikipedia.org/wiki/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 two components. Gordon proved that for the class of links where these two constructions are not possible there are finitely many links in this class with a given complement.
|
https://en.wikipedia.org/wiki/Gordon–Luecke_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) containing PSL2(q) for q an odd prime power. Note that A5 ≈ PSL2(4) ≈ PSL2(5) and A6 ≈ PSL2(9).
|
https://en.wikipedia.org/wiki/Gorenstein–Walter_theorem
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.