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c_qvgxt4015hw2 | 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 ... | Fekete problem |
c_m3exvol1rsp4 | 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 i... | Fekete–Szegő inequality |
c_wqap7emj236q | 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... | Feller–Tornier constant |
c_dx25ag8vm4uz | 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 in... | Fermat curve |
c_g92l9to7gzpv | 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. | Fermat curve |
c_j9n6i8js3dbq | It is isogenous to a product of simple abelian varieties with complex multiplication. The Fermat curve also has gonality: n − 1. {\displaystyle n-1.\ } | Fermat curve |
c_fwjqmbjq2za2 | 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 ... | Generalizations of Fibonacci numbers |
c_rjrfh6kxxd1z | 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. | Fibonacci polynomials |
c_whwcbsi614wp | 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 someti... | Fibonacci ratio |
c_nt4tu10xc6td | 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 Fibonacc... | Fibonacci ratio |
c_m5tapi3qfx57 | 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 ... | Fibonomial coefficient |
c_dob61vnewzpl | 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 ... | Fibonorial |
c_s3glbn1vzyqd | 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 {\dis... | Fictitious domain method |
c_kkw49alaisuy | 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 ... | Fitting lemma |
c_ln4rwazz425k | 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 pro... | Fontaine–Mazur conjecture |
c_r5oar3z4d44c | 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 ... | FKG inequality |
c_1joogbyakwqc | 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 ine... | FKG inequality |
c_cezang51tlbs | 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 the... | Fourier inversion |
c_gjovvpslo2dv | 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 .} | Fourier inversion |
c_rg1x2hfe1zlx | 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}}.} | Fourier inversion |
c_zco6adi0ax3v | 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 abov... | Fourier inversion |
c_jkg26oa9hiia | 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. | Sine transform |
c_zz1zxq3hzi7u | In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. | Fourier transform on finite groups |
c_b09c4awxy0qx | 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 = ... | Fox H-function |
c_9k42j162nmm4 | 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 deriv... | Fox derivative |
c_xgdajd158s7l | 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 A... | Fox–Wright Psi function |
c_vxhxz0gi6yjx | 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. | Fraňková–Helly selection theorem |
c_6ggtacvc04vu | 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 co... | Fredholm alternative |
c_nrxvyx478uji | 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 Iv... | Fredholm determinant |
c_2zfkot2atljy | 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 Adomi... | Fredholm equation |
c_onqc1xy05bz3 | 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 fro... | Freidlin–Wentzell theorem |
c_jnut2sq392q1 | 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. T... | Rosenfeld projective plane |
c_65vnxnigwsy0 | 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-kn... | Freudenthal spectral theorem |
c_wkquwhz675e5 | 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 replac... | Frobenius determinant theorem |
c_u4huavljgqvg | 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 fin... | Frobenius mapping |
c_ddhge7qpthdm | 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... | Frobenius mapping |
c_i8mrs21rh4by | 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 satis... | Frobenius inner product |
c_s3wjw7l97bkj | 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 u... | Fréchet derivative |
c_sd7891u94720 | 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. | Fréchet distance |
c_barwtna3nvzc | 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 ... | Cofinite filter |
c_if37vrlds80s | 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 u... | Frölicher spectral sequence |
c_miomcr25jrcy | 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 ... | Frölicher–Nijenhuis bracket |
c_zn3g7vecgk9c | 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 ... | Fubini–Study metric |
c_awc5ci0v9372 | 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. | Fubini–Study metric |
c_mdfpme5h675j | 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. | Fuchs relation |
c_w8k2tx3kr1gt | 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 prov... | Fulton–Hansen connectedness theorem |
c_z9gvd1la903a | 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 Meij... | Meijer G-function |
c_1ldqpibkjamn | 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 fun... | Meijer G-function |
c_qbxj3jghus3w | 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 th... | Gabriel-Popescu theorem |
c_1uc0yo7iio8z | 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 a... | Gan–Gross–Prasad conjecture |
c_w3iehkz8l728 | In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules Vλ in terms of standard polytabloids. | Garnir relations |
c_r7dqusz5p4c7 | 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 B... | Gateaux derivative |
c_lvsfrfhz44u4 | 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 m... | Gateaux derivative |
c_jc156t5apxvq | 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 nu... | Gauss's circle problem |
c_ct1tv5mhzf4l | 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... | Class number problem |
c_7i1ocvj7jzkd | 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 ... | Gauss iterated map |
c_vyppui7ir7io | 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{b... | Q-binomial theorem |
c_qv6j7z7cmh8q | 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. | Gaussian isoperimetric inequality |
c_yyrohxs6yjvu | 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 linea... | Hypergeometric series |
c_zbn79duo01zk | 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... | Gauss-Kuzmin distribution |
c_4bhweikwl46k | 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 ... | Gauss–Kuzmin–Wirsing operator |
c_wb8womqnj8l3 | 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_... | Gauss–Manin connection |
c_eimr8v9pcyy1 | 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... | Gelfand representation |
c_d7e6tu87raqc | 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 ... | Gelfand–Naimark theorem |
c_swv9ijkeishg | 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 canoni... | Gelfand–Tsetlin integrable system |
c_rpnf43jw9h76 | In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. | Gelfond's theorem |
c_br5drwcwwfys | 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 (seque... | Genocchi number |
c_pafmc8oyj6dw | 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 Hi... | Gershgorin circle theorem |
c_fr9acrnipkb8 | 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. | Gibbons–Hawking Ansatz |
c_3sy73e01wojo | 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 stat... | Gibbs random field |
c_lyetlx8fhxdb | 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, th... | Gibbs random field |
c_75j2i2e7r5js | 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... | Gibbs random field |
c_186s7lezdyh2 | 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 measu... | Gibbs random field |
c_npbcqu83gt1u | 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... | Gibbs random field |
c_gg0hd99urd0j | 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 Four... | Gibbs phenomenon |
c_sns8uds7d98f | 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). | 3-manifold |
c_w2jpl97xnc4y | 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. | 3-manifold |
c_gzvgssde25lr | 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 tetr... | 3-manifold |
c_y3u4wref7s6t | 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). | Gieseking manifold |
c_75ruwxd67zg2 | 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. | Gieseking manifold |
c_mk3x84yyye3s | 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. | Gieseking manifold |
c_es7x4f7g0ayk | 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 squar... | Gieseking manifold |
c_8uwnko270j2l | 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. | Gilbert–Pollack conjecture |
c_0fc584mo1cq2 | 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 mathematic... | Glaisher–Kinkelin constant |
c_td4162ixxif4 | 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! | Glaisher–Kinkelin constant |
c_mkj7blfcwuvm | }{n^{n+{\frac {1}{2}}}\,e^{-n}}}} which shows that just as π is obtained from approximation of the factorials, A can also be obtained from a similar approximation to the hyperfactorials. An equivalent definition for A involving the Barnes G-function, given by G(n) = Πn−2k=1 k! = n−1/K(n) where Γ(n) is the gamma functio... | Glaisher–Kinkelin constant |
c_wzmvhpw5fgz8 | 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}}=... | Goldbach–Euler theorem |
c_3ltr6poqbis5 | 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. | Golod–Shafarevich theorem |
c_owexrrg1mre3 | 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... | Golomb sequence |
c_ljf86moxjpci | In mathematics, the Golomb–Dickman constant arises in the theory of random permutations and in number theory. Its value is λ = 0.62432998854355087099293638310083724 … {\displaystyle \lambda =0.62432998854355087099293638310083724\dots } (sequence A084945 in the OEIS)It is not known whether this constant is rational or i... | Golomb–Dickman constant |
c_fomraczvze0g | 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 ... | Euler–Gompertz constant |
c_5wybdqruum6g | 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.} ... | Euler–Gompertz constant |
c_iv8c68wsbmqb | The numerical value of δ {\displaystyle \delta } is about δ = 0.596347362323194074341078499369279376074 … {\displaystyle \delta =0.596347362323194074341078499369279376074\dots } When Euler studied divergent infinite series, he encountered δ {\displaystyle \delta } via, for example, the above integral representations. L... | Euler–Gompertz constant |
c_de6a51cz7jc4 | 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). | Goncharov conjecture |
c_zdxs7tovyyfm | 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... | Goormaghtigh conjecture |
c_03811qu0g26c | In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian. The theorem is usually stated as "knots are determined by their co... | Gordon–Luecke theorem |
c_6s7l460p06zo | Often two knots are considered equivalent if they are isotopic. The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic. These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a n... | Gordon–Luecke theorem |
c_16wrxgsszw95 | 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... | Gordon–Luecke theorem |
c_6ofopxbgm7tc | 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... | Gordon–Luecke theorem |
c_5a6olqktnsy0 | In mathematics, the Gorenstein–Walter theorem, proved by Gorenstein and Walter (1965a, 1965b, 1965c), states that if a finite group G has a dihedral Sylow 2-subgroup, and O(G) is the maximal normal subgroup of odd order, then G/O(G) is isomorphic to a 2-group, or the alternating group A7, or a subgroup of PΓL2(q) conta... | Gorenstein–Walter theorem |
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