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In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups. They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated homogeneous spaces. Since the use of Lie group methods became one of the standard techniques in twentieth century mathematics, many phenomena can now be referred back to decompositions. The same ideas are often applied to Lie groups, Lie algebras, algebraic groups and p-adic number analogues, making it harder to summarise the facts into a unified theory.
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https://en.wikipedia.org/wiki/Lie_group_decomposition
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In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is R n {\displaystyle \mathbb {R} ^{n}} and T n {\displaystyle \mathbb {T} ^{n}} (see real coordinate space and the circle group respectively) which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other.
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https://en.wikipedia.org/wiki/Left_invariant
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However, by restricting our attention to the simply connected Lie groups, the Lie group-Lie algebra correspondence will be one-to-one.In this article, a Lie group refers to a real Lie group. For the complex and p-adic cases, see complex Lie group and p-adic Lie group. In this article, manifolds (in particular Lie groups) are assumed to be second countable; in particular, they have at most countably many connected components.
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https://en.wikipedia.org/wiki/Left_invariant
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In mathematics, Light's associativity test is a procedure invented by F. W. Light for testing whether a binary operation defined in a finite set by a Cayley multiplication table is associative. The naive procedure for verification of the associativity of a binary operation specified by a Cayley table, which compares the two products that can be formed from each triple of elements, is cumbersome. Light's associativity test simplifies the task in some instances (although it does not improve the worst-case runtime of the naive algorithm, namely O ( n 3 ) {\displaystyle {\mathcal {O}}\left(n^{3}\right)} for sets of size n {\displaystyle n} ).
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https://en.wikipedia.org/wiki/Light's_associativity_test
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In mathematics, Lill's method is a visual method of finding the real roots of a univariate polynomial of any degree. It was developed by Austrian engineer Eduard Lill in 1867. A later paper by Lill dealt with the problem of complex roots.Lill's method involves drawing a path of straight line segments making right angles, with lengths equal to the coefficients of the polynomial. The roots of the polynomial can then be found as the slopes of other right-angle paths, also connecting the start to the terminus, but with vertices on the lines of the first path.
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https://en.wikipedia.org/wiki/Lill's_method
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In mathematics, Lindelöf's lemma is a simple but useful lemma in topology on the real line, named for the Finnish mathematician Ernst Leonard Lindelöf.
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https://en.wikipedia.org/wiki/Lindelöf's_lemma
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In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.
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https://en.wikipedia.org/wiki/Lindelöf's_theorem
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In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system. The formula is named after the French mathematician Joseph Liouville. Jacobi's formula provides another representation of the same mathematical relationship. Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other(s), see the example application below.
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https://en.wikipedia.org/wiki/Liouville's_formula
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In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. These are called nonelementary antiderivatives. A standard example of such a function is e − x 2 , {\displaystyle e^{-x^{2}},} whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics.
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https://en.wikipedia.org/wiki/Liouville's_theorem_(differential_algebra)
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Other examples include the functions sin ( x ) x {\displaystyle {\frac {\sin(x)}{x}}} and x x . {\displaystyle x^{x}.} Liouville's theorem states that elementary antiderivatives, if they exist, are in the same differential field as the function, plus possibly a finite number of applications of the logarithm function.
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https://en.wikipedia.org/wiki/Liouville's_theorem_(differential_algebra)
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In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that any smooth conformal mapping on a domain of Rn, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions). This theorem severely limits the variety of possible conformal mappings in R3 and higher-dimensional spaces.
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https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)
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By contrast, conformal mappings in R2 can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem. Generalizations of the theorem hold for transformations that are only weakly differentiable (Iwaniec & Martin 2001, Chapter 5). The focus of such a study is the non-linear Cauchy–Riemann system that is a necessary and sufficient condition for a smooth mapping f: Ω → Rn to be conformal: D f T D f = | det D f | 2 / n I {\displaystyle Df^{\mathrm {T} }Df=\left|\det Df\right|^{2/n}I} where Df is the Jacobian derivative, T is the matrix transpose, and I is the identity matrix.
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https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)
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A weak solution of this system is defined to be an element f of the Sobolev space W1,nloc(Ω, Rn) with non-negative Jacobian determinant almost everywhere, such that the Cauchy–Riemann system holds at almost every point of Ω. Liouville's theorem is then that every weak solution (in this sense) is a Möbius transformation, meaning that it has the form f ( x ) = b + α A ( x − a ) | x − a | ε , D f = α A | x − a | ε ( I − ε x − a | x − a | ( x − a ) T | x − a | ) , {\displaystyle f(x)=b+{\frac {\alpha A(x-a)}{|x-a|^{\varepsilon }}},\qquad Df={\frac {\alpha A}{|x-a|^{\varepsilon }}}\left(I-\varepsilon {\frac {x-a}{|x-a|}}{\frac {(x-a)^{\mathrm {T} }}{|x-a|}}\right),} where a, b are vectors in Rn, α is a scalar, A is a rotation matrix, ε = 0 or 2, and the matrix in parentheses is I or a Householder matrix (so, orthogonal). Equivalently stated, any quasiconformal map of a domain in Euclidean space that is also conformal is a Möbius transformation. This equivalent statement justifies using the Sobolev space W1,n, since f ∈ W1,nloc(Ω, Rn) then follows from the geometrical condition of conformality and the ACL characterization of Sobolev space.
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https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)
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The result is not optimal however: in even dimensions n = 2k, the theorem also holds for solutions that are only assumed to be in the space W1,kloc, and this result is sharp in the sense that there are weak solutions of the Cauchy–Riemann system in W1,p for any p < k that are not Möbius transformations. In odd dimensions, it is known that W1,n is not optimal, but a sharp result is not known. Similar rigidity results (in the smooth case) hold on any conformal manifold.
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https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)
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The group of conformal isometries of an n-dimensional conformal Riemannian manifold always has dimension that cannot exceed that of the full conformal group SO(n + 1, 1). Equality of the two dimensions holds exactly when the conformal manifold is isometric with the n-sphere or projective space. Local versions of the result also hold: The Lie algebra of conformal Killing fields in an open set has dimension less than or equal to that of the conformal group, with equality holding if and only if the open set is locally conformally flat.
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https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)
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In mathematics, Littlewood's Tauberian theorem is a strengthening of Tauber's theorem introduced by John Edensor Littlewood (1911).
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https://en.wikipedia.org/wiki/Littlewood's_Tauberian_theorem
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In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering.
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https://en.wikipedia.org/wiki/Loewner_order
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In mathematics, Luna's slice theorem, introduced by Luna (1973), describes the local behavior of an action of a reductive algebraic group on an affine variety. It is an analogue in algebraic geometry of the theorem that a compact Lie group acting on a smooth manifold X has a slice at each point x, in other words a subvariety W such that X looks locally like G×Gx W. (see slice theorem (differential geometry).)
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https://en.wikipedia.org/wiki/Luna's_slice_theorem
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In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent λ {\displaystyle \lambda } ) in the a−b plane for given periodic sequences of a and b. In the images, yellow corresponds to λ < 0 {\displaystyle \lambda <0} (stability), and blue corresponds to λ > 0 {\displaystyle \lambda >0} (chaos). Lyapunov fractals were discovered in the late 1980s by the Germano-Chilean physicist Mario Markus from the Max Planck Institute of Molecular Physiology. They were introduced to a large public by a science popularization article on recreational mathematics published in Scientific American in 1991.
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https://en.wikipedia.org/wiki/Lyapunov_fractal
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In mathematics, Lüroth's theorem asserts that every field that lies between two other fields K and K(X) must be generated as an extension of K by a single element of K(X). This result is named after Jacob Lüroth, who proved it in 1876.
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https://en.wikipedia.org/wiki/Lüroth's_theorem
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In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup Γ {\displaystyle \Gamma } of S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )} as modular forms. They are eigenforms of the hyperbolic Laplace operator Δ {\displaystyle \Delta } defined on H {\displaystyle \mathbb {H} } and satisfy certain growth conditions at the cusps of a fundamental domain of Γ {\displaystyle \Gamma } . In contrast to modular forms, Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.
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https://en.wikipedia.org/wiki/Maass_wave_forms
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In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis (1916). It is often used to derive binomial identities, most notably Dixon's identity.
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https://en.wikipedia.org/wiki/MacMahon_master_theorem
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In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable t, but later realized that it is more natural to associate them with affine root systems rather than finite root systems, in which case the variable t can be replaced by several different variables t=(t1,...,tk), one for each of the k orbits of roots in the affine root system.
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https://en.wikipedia.org/wiki/Macdonald's_constant_term_conjecture
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The Macdonald polynomials are polynomials in n variables x=(x1,...,xn), where n is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-variable orthogonal polynomials as special cases. Koornwinder polynomials are Macdonald polynomials of certain non-reduced root systems. They have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.
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https://en.wikipedia.org/wiki/Macdonald's_constant_term_conjecture
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In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by Koornwinder (1992) and I. G. Macdonald (1987, important special cases), that generalize the Askey–Wilson polynomials. They are the Macdonald polynomials attached to the non-reduced affine root system of type (C∨n, Cn), and in particular satisfy (van Diejen 1996, Sahi 1999) analogues of Macdonald's conjectures (Macdonald 2003, Chapter 5.3). In addition Jan Felipe van Diejen showed that the Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Macdonald-Koornwinder polynomials and found complete sets of concrete commuting difference operators diagonalized by them (van Diejen 1995).
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https://en.wikipedia.org/wiki/Koornwinder_polynomials
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Furthermore, there is a large class of interesting families of multivariable orthogonal polynomials associated with classical root systems which are degenerate cases of the Macdonald-Koornwinder polynomials (van Diejen 1999). The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras (Noumi 1995, Sahi 1999, Macdonald 2003). The Macdonald-Koornwinder polynomial in n variables associated to the partition λ is the unique Laurent polynomial invariant under permutation and inversion of variables, with leading monomial xλ, and orthogonal with respect to the density ∏ 1 ≤ i < j ≤ n ( x i x j , x i / x j , x j / x i , 1 / x i x j ; q ) ∞ ( t x i x j , t x i / x j , t x j / x i , t / x i x j ; q ) ∞ ∏ 1 ≤ i ≤ n ( x i 2 , 1 / x i 2 ; q ) ∞ ( a x i , a / x i , b x i , b / x i , c x i , c / x i , d x i , d / x i ; q ) ∞ {\displaystyle \prod _{1\leq i
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https://en.wikipedia.org/wiki/Koornwinder_polynomials
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In mathematics, Machin-like formulae are a popular technique for computing π (the ratio of the circumference to the diameter of a circle) to a large number of digits. They are generalizations of John Machin's formula from 1706: π 4 = 4 arctan 1 5 − arctan 1 239 {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}} which he used to compute π to 100 decimal places.Machin-like formulas have the form where c 0 {\displaystyle c_{0}} is a positive integer, c n {\displaystyle c_{n}} are signed non-zero integers, and a n {\displaystyle a_{n}} and b n {\displaystyle b_{n}} are positive integers such that a n < b n {\displaystyle a_{n}
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https://en.wikipedia.org/wiki/Machin-like_formulas
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In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a1, a2, ..., an be positive real numbers, and for k = 1, 2, ..., n define the averages Sk as follows: S k = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n a i 1 a i 2 ⋯ a i k ( n k ) . {\displaystyle S_{k}={\frac {\displaystyle \sum _{1\leq i_{1}<\cdots
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https://en.wikipedia.org/wiki/Maclaurin's_inequality
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In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval on the reals), and "purely atomic parts", using the counting measure on some discrete space. The theorem is due to Dorothy Maharam. It was extended to localizable measure spaces by Irving Segal.The result is important to classical Banach space theory, in that, when considering the Banach space given as an Lp space of measurable functions over a general measurable space, it is sufficient to understand it in terms of its decomposition into non-atomic and atomic parts.
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https://en.wikipedia.org/wiki/Maharam's_theorem
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Maharam's theorem can also be translated into the language of abelian von Neumann algebras. Every abelian von Neumann algebra is isomorphic to a product of σ-finite abelian von Neumann algebras, and every σ-finite abelian von Neumann algebra is isomorphic to a spatial tensor product of discrete abelian von Neumann algebras; that is, algebras of bounded functions on a discrete set. A similar theorem was given by Kazimierz Kuratowski for Polish spaces, stating that they are isomorphic, as Borel spaces, to either the reals, the integers, or a finite set. == References ==
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https://en.wikipedia.org/wiki/Maharam's_theorem
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In mathematics, Mahler's 3/2 problem concerns the existence of "Z-numbers". A Z-number is a real number x such that the fractional parts of x ( 3 2 ) n {\displaystyle x\left({\frac {3}{2}}\right)^{n}} are less than 1/2 for all positive integers n. Kurt Mahler conjectured in 1968 that there are no Z-numbers. More generally, for a real number α, define Ω(α) as Ω ( α ) = inf θ > 0 ( lim sup n → ∞ { θ α n } − lim inf n → ∞ { θ α n } ) . {\displaystyle \Omega (\alpha )=\inf _{\theta >0}\left({\limsup _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace -\liminf _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace }\right).} Mahler's conjecture would thus imply that Ω(3/2) exceeds 1/2. Flatto, Lagarias, and Pollington showed that Ω ( p q ) > 1 p {\displaystyle \Omega \left({\frac {p}{q}}\right)>{\frac {1}{p}}} for rational p/q > 1 in lowest terms.
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https://en.wikipedia.org/wiki/Mahler's_3/2_problem
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In mathematics, Mahler's compactness theorem, proved by Kurt Mahler (1946), is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in which a lattice could degenerate (go off to infinity) in a sequence of lattices. In intuitive terms it says that this is possible in just two ways: becoming coarse-grained with a fundamental domain that has ever larger volume; or containing shorter and shorter vectors.
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https://en.wikipedia.org/wiki/Mahler's_compactness_theorem
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It is also called his selection theorem, following an older convention used in naming compactness theorems, because they were formulated in terms of sequential compactness (the possibility of selecting a convergent subsequence). Let X be the space G L n ( R ) / G L n ( Z ) {\displaystyle \mathrm {GL} _{n}(\mathbb {R} )/\mathrm {GL} _{n}(\mathbb {Z} )} that parametrises lattices in R n {\displaystyle \mathbb {R} ^{n}} , with its quotient topology. There is a well-defined function Δ on X, which is the absolute value of the determinant of a matrix – this is constant on the cosets, since an invertible integer matrix has determinant 1 or −1.
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https://en.wikipedia.org/wiki/Mahler's_compactness_theorem
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Mahler's compactness theorem states that a subset Y of X is relatively compact if and only if Δ is bounded on Y, and there is a neighbourhood N of 0 in R n {\displaystyle \mathbb {R} ^{n}} such that for all Λ in Y, the only lattice point of Λ in N is 0 itself. The assertion of Mahler's theorem is equivalent to the compactness of the space of unit-covolume lattices in R n {\displaystyle \mathbb {R} ^{n}} whose systole is larger or equal than any fixed ε > 0 {\displaystyle \varepsilon >0} . Mahler's compactness theorem was generalized to semisimple Lie groups by David Mumford; see Mumford's compactness theorem.
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https://en.wikipedia.org/wiki/Mahler's_compactness_theorem
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In mathematics, Mahler's inequality, named after Kurt Mahler, states that the geometric mean of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means: ∏ k = 1 n ( x k + y k ) 1 / n ≥ ∏ k = 1 n x k 1 / n + ∏ k = 1 n y k 1 / n {\displaystyle \prod _{k=1}^{n}(x_{k}+y_{k})^{1/n}\geq \prod _{k=1}^{n}x_{k}^{1/n}+\prod _{k=1}^{n}y_{k}^{1/n}} when xk, yk > 0 for all k.
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https://en.wikipedia.org/wiki/Mahler's_inequality
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In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. It is the p-adic counterpart to the Stone-Weierstrass theorem for continuous real-valued functions on a closed interval.
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https://en.wikipedia.org/wiki/Mahler's_theorem
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In mathematics, Maillet's determinant Dp is the determinant of the matrix introduced by Maillet (1913) whose entries are R(s/r) for s,r = 1, 2, ..., (p – 1)/2 ∈ Z/pZ for an odd prime p, where and R(a) is the least positive residue of a mod p (Muir 1930, pages 340–342). Malo (1914) calculated the determinant Dp for p = 3, 5, 7, 11, 13 and found that in these cases it is given by (–p)(p – 3)/2, and conjectured that it is given by this formula in general. Carlitz & Olson (1955) showed that this conjecture is incorrect; the determinant in general is given by Dp = (–p)(p – 3)/2h−, where h− is the first factor of the class number of the cyclotomic field generated by pth roots of 1, which happens to be 1 for p less than 23.
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https://en.wikipedia.org/wiki/Maillet's_determinant
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In particular this verifies Maillet's conjecture that the determinant is always non-zero. Chowla and Weil had previously found the same formula but did not publish it. Their results have been extended to all non-prime odd numbers by K. Wang(1982).
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https://en.wikipedia.org/wiki/Maillet's_determinant
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In mathematics, Malmquist's theorem, is the name of any of the three theorems proved by Axel Johannes Malmquist (1913, 1920, 1941). These theorems restrict the forms of first order algebraic differential equations which have transcendental meromorphic or algebroid solutions.
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https://en.wikipedia.org/wiki/Malmquist's_theorem
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In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the determinant for them and most linear algebra theorems like Cramer's rule, Cayley–Hamilton theorem, etc. hold true for them. Any matrix with commuting elements is a Manin matrix. These matrices have applications in representation theory in particular to Capelli's identity, Yangian and quantum integrable systems.
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https://en.wikipedia.org/wiki/Manin_matrices
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Manin matrices are particular examples of Manin's general construction of "non-commutative symmetries" which can be applied to any algebra. From this point of view they are "non-commutative endomorphisms" of polynomial algebra C. Taking (q)-(super)-commuting variables one will get (q)-(super)-analogs of Manin matrices, which are closely related to quantum groups. Manin works were influenced by the quantum group theory. He discovered that quantized algebra of functions Funq(GL) can be defined by the requirement that T and Tt are simultaneously q-Manin matrices. In that sense it should be stressed that (q)-Manin matrices are defined only by half of the relations of related quantum group Funq(GL), and these relations are enough for many linear algebra theorems.
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https://en.wikipedia.org/wiki/Manin_matrices
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In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative. See also geometrical properties of polynomial roots.
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https://en.wikipedia.org/wiki/Marden's_theorem
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In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.
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https://en.wikipedia.org/wiki/Maschke's_theorem
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In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation d 2 y d x 2 + ( a − 2 q cos ( 2 x ) ) y = 0 , {\displaystyle {\frac {d^{2}y}{dx^{2}}}+(a-2q\cos(2x))y=0,} where a, q are real-valued parameters. Since we may add π/2 to x to change the sign of q, it is a usual convention to set q ≥ 0. They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads. They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation (PDE) boundary value problems possessing elliptic symmetry.
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https://en.wikipedia.org/wiki/Mathieu_differential_equation
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In mathematics, Matsumoto zeta functions are a type of zeta function introduced by Kohji Matsumoto in 1990. They are functions of the form ϕ ( s ) = ∏ p 1 A p ( p − s ) {\displaystyle \phi (s)=\prod _{p}{\frac {1}{A_{p}(p^{-s})}}} where p is a prime and Ap is a polynomial.
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https://en.wikipedia.org/wiki/Matsumoto_zeta_function
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In mathematics, Matsushima's formula, introduced by Matsushima (1967), is a formula for the Betti numbers of a quotient of a symmetric space G/H by a discrete group, in terms of unitary representations of the group G. The Matsushima–Murakami formula is a generalization giving dimensions of spaces of automorphic forms, introduced by Matsushima & Murakami (1968). == References ==
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https://en.wikipedia.org/wiki/Matsushima's_formula
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In mathematics, Mazur's lemma is a result in the theory of normed vector spaces. It shows that any weakly convergent sequence in a normed space has a sequence of convex combinations of its members that converges strongly to the same limit, and is used in the proof of Tonelli's theorem.
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https://en.wikipedia.org/wiki/Mazur's_lemma
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In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner (1934). They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by M n ( x , β , γ ) = ∑ k = 0 n ( − 1 ) k ( n k ) ( x k ) k ! ( x + β ) n − k γ − k {\displaystyle M_{n}(x,\beta ,\gamma )=\sum _{k=0}^{n}(-1)^{k}{n \choose k}{x \choose k}k! (x+\beta )_{n-k}\gamma ^{-k}}
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https://en.wikipedia.org/wiki/Meixner_polynomials
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In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period (sequence A028416 in the OEIS). If the period of the decimal representation of a/p is 2n, so that a p = 0. a 1 a 2 a 3 … a n a n + 1 … a 2 n ¯ {\displaystyle {\frac {a}{p}}=0. {\overline {a_{1}a_{2}a_{3}\dots a_{n}a_{n+1}\dots a_{2n}}}} then the digits in the second half of the repeating decimal period are the 9s complement of the corresponding digits in its first half. In other words, a i + a i + n = 9 {\displaystyle a_{i}+a_{i+n}=9} a 1 … a n + a n + 1 … a 2 n = 10 n − 1.
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https://en.wikipedia.org/wiki/Midy's_Theorem
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{\displaystyle a_{1}\dots a_{n}+a_{n+1}\dots a_{2n}=10^{n}-1.} For example, 1 13 = 0. 076923 ¯ and 076 + 923 = 999.
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https://en.wikipedia.org/wiki/Midy's_Theorem
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{\displaystyle {\frac {1}{13}}=0. {\overline {076923}}{\text{ and }}076+923=999.} 1 17 = 0.
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https://en.wikipedia.org/wiki/Midy's_Theorem
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0588235294117647 ¯ and 05882352 + 94117647 = 99999999. {\displaystyle {\frac {1}{17}}=0. {\overline {0588235294117647}}{\text{ and }}05882352+94117647=99999999.}
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https://en.wikipedia.org/wiki/Midy's_Theorem
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In mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more structure than sets. Let T be a finitely splitting rooted tree of height ω, n a positive integer, and S T n {\displaystyle \mathbb {S} _{T}^{n}} the collection of all strongly embedded subtrees of T of height n. In one of its simple forms, Milliken's tree theorem states that if S T n = C 1 ∪ . . . ∪ C r {\displaystyle \mathbb {S} _{T}^{n}=C_{1}\cup ...\cup C_{r}} then for some strongly embedded infinite subtree R of T, S R n ⊂ C i {\displaystyle \mathbb {S} _{R}^{n}\subset C_{i}} for some i ≤ r. This immediately implies Ramsey's theorem; take the tree T to be a linear ordering on ω vertices. Define S n = ⋃ T S T n {\displaystyle \mathbb {S} ^{n}=\bigcup _{T}\mathbb {S} _{T}^{n}} where T ranges over finitely splitting rooted trees of height ω. Milliken's tree theorem says that not only is S n {\displaystyle \mathbb {S} ^{n}} partition regular for each n < ω, but that the homogeneous subtree R guaranteed by the theorem is strongly embedded in T.
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https://en.wikipedia.org/wiki/Milliken's_tree_theorem
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In mathematics, Milnor K-theory is an algebraic invariant (denoted K ∗ ( F ) {\displaystyle K_{*}(F)} for a field F {\displaystyle F} ) defined by John Milnor (1970) as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic K-theory and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for K 1 {\displaystyle K_{1}} and K 2 {\displaystyle K_{2}} . Fortunately, it can be shown Milnor K-theory is a part of algebraic K-theory, which in general is the easiest part to compute.
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https://en.wikipedia.org/wiki/Milnor_ring
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In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space.
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https://en.wikipedia.org/wiki/Milnor_fiber
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In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality.
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https://en.wikipedia.org/wiki/Minkowski's_first_inequality_for_convex_bodies
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In mathematics, Minkowski's question-mark function, denoted ? (x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. It also maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.
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https://en.wikipedia.org/wiki/Minkowski's_question-mark_function
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In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell.
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https://en.wikipedia.org/wiki/Minkowski's_second_theorem
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In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the origin and which has volume greater than 2 n {\displaystyle 2^{n}} contains a non-zero integer point (meaning a point in Z n {\displaystyle \mathbb {Z} ^{n}} that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers. It can be extended from the integers to any lattice L {\displaystyle L} and to any symmetric convex set with volume greater than 2 n d ( L ) {\displaystyle 2^{n}\,d(L)} , where d ( L ) {\displaystyle d(L)} denotes the covolume of the lattice (the absolute value of the determinant of any of its bases).
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https://en.wikipedia.org/wiki/Minkowski's_theorem
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In mathematics, Mitchell's group is a complex reflection group in 6 complex dimensions of order 108 × 9!, introduced by Mitchell (1914). It has the structure 6.PSU4(F3).2. As a complex reflection group it has 126 reflections of order 2, and its ring of invariants is a polynomial algebra with generators of degrees 6, 12, 18, 24, 30, 42. Coxeter gives it group symbol 3 and Coxeter-Dynkin diagram .Mitchell's group is an index 2 subgroup of the automorphism group of the Coxeter–Todd lattice.
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https://en.wikipedia.org/wiki/Mitchell's_group
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In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Mittag-Leffler (1908)
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https://en.wikipedia.org/wiki/Mittag-Leffler_summation
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In mathematics, Molien's formula computes the generating function attached to a linear representation of a group G on a finite-dimensional vector space, that counts the homogeneous polynomials of a given total degree that are invariants for G. It is named for Theodor Molien. Precisely, it says: given a finite-dimensional complex representation V of G and R n = C n = Sym n ( V ∗ ) {\displaystyle R_{n}=\mathbb {C} _{n}=\operatorname {Sym} ^{n}(V^{*})} , the space of homogeneous polynomial functions on V of degree n (degree-one homogeneous polynomials are precisely linear functionals), if G is a finite group, the series (called Molien series) can be computed as: ∑ n = 0 ∞ dim ( R n G ) t n = ( # G ) − 1 ∑ g ∈ G det ( 1 − t g | V ∗ ) − 1 . {\displaystyle \sum _{n=0}^{\infty }\dim(R_{n}^{G})t^{n}=(\#G)^{-1}\sum _{g\in G}\det(1-tg|V^{*})^{-1}.} Here, R n G {\displaystyle R_{n}^{G}} is the subspace of R n {\displaystyle R_{n}} that consists of all vectors fixed by all elements of G; i.e., invariant forms of degree n. Thus, the dimension of it is the number of invariants of degree n. If G is a compact group, the similar formula holds in terms of Haar measure.
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https://en.wikipedia.org/wiki/Molien's_formula
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In mathematics, Monk's formula, found by Monk (1959), is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold. Write tij for the transposition (i j), and si = ti,i+1. Then 𝔖sr = x1 + ⋯ + xr, and Monk's formula states that for a permutation w, S s r S w = ∑ i ≤ r < j ℓ ( w t i j ) = ℓ ( w ) + 1 S w t i j , {\displaystyle {\mathfrak {S}}_{s_{r}}{\mathfrak {S}}_{w}=\sum _{{i\leq r
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https://en.wikipedia.org/wiki/Monk's_formula
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In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, sequential Monte Carlo (also known as a particle filter), and mean-field particle methods.
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https://en.wikipedia.org/wiki/MISER_algorithm
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In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery (1973) that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) is 1 − ( sin ( π u ) π u ) 2 + δ ( u ) , {\displaystyle 1-\left({\frac {\sin(\pi u)}{\pi u}}\right)^{\!2}+\delta (u),} which, as Freeman Dyson pointed out to him, is the same as the pair correlation function of random Hermitian matrices.
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https://en.wikipedia.org/wiki/Pair_correlation_conjecture
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In mathematics, Moreau's theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms of the resolvent operator.
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https://en.wikipedia.org/wiki/Moreau's_theorem
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In mathematics, Moss E. Sweedler (1969, p. 89–90) introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative.
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https://en.wikipedia.org/wiki/Sweedler's_Hopf_algebra
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In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by Mostow (1968) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad (1973) in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm.
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https://en.wikipedia.org/wiki/Mostow's_rigidity_theorem
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Besson, Courtois & Gallot (1996) gave the simplest available proof. While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic n {\displaystyle n} -manifold (for n > 2 {\displaystyle n>2} ) is a point, for a hyperbolic surface of genus g > 1 {\displaystyle g>1} there is a moduli space of dimension 6 g − 6 {\displaystyle 6g-6} that parameterizes all metrics of constant curvature (up to diffeomorphism), a fact essential for Teichmüller theory. There is also a rich theory of deformation spaces of hyperbolic structures on infinite volume manifolds in three dimensions.
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https://en.wikipedia.org/wiki/Mostow's_rigidity_theorem
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In mathematics, Motz's problem is a problem which is widely employed as a benchmark for singularity problems to compare the effectiveness of numerical methods. The problem was first presented in 1947 by H. Motz in the paper "The treatment of singularities of partial differential equations by relaxation methods".
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https://en.wikipedia.org/wiki/Motz's_problem
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In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of root groups. In a book on the topic, Tits and Richard Weiss classify them all. An earlier theorem, proved independently by Tits and Weiss, showed that a Moufang polygon must be a generalized 3-gon, 4-gon, 6-gon, or 8-gon, so the purpose of the aforementioned book was to analyze these four cases.
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https://en.wikipedia.org/wiki/Moufang_polygon
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In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means.
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https://en.wikipedia.org/wiki/Muirhead's_Inequality
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In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metric is compact. It was proved by David Mumford (1971) as a consequence of a theorem about the compactness of sets of discrete subgroups of semisimple Lie groups generalizing Mahler's compactness theorem.
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https://en.wikipedia.org/wiki/Mumford's_compactness_theorem
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In mathematics, Nakayama's conjecture is a conjecture about Artinian rings, introduced by Nakayama (1958). The generalized Nakayama conjecture is an extension to more general rings, introduced by Auslander and Reiten (1975). Leuschke & Huneke (2004) proved some cases of the generalized Nakayama conjecture. Nakayama's conjecture states that if all the modules of a minimal injective resolution of an Artin algebra R are injective and projective, then R is self-injective.
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https://en.wikipedia.org/wiki/Nakayama's_conjecture
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In mathematics, Nambooripad order (also called Nambooripad's partial order) is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig, some authors refer to it as Hartwig–Nambooripad order. "Natural" here means that the order is defined in terms of the operation on the semigroup. In general Nambooripad's order in a regular semigroup is not compatible with multiplication. It is compatible with multiplication only if the semigroup is pseudo-inverse (locally inverse).
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https://en.wikipedia.org/wiki/Natural_partial_order
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In mathematics, Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem. This was soon generalized to flows generated by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu–Poisson manifolds with more than one Hamiltonian.
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https://en.wikipedia.org/wiki/Nambu_dynamics
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In mathematics, Nesbitt's inequality states that for positive real numbers a, b and c, a b + c + b a + c + c a + b ≥ 3 2 . {\displaystyle {\frac {a}{b+c}}+{\frac {b}{a+c}}+{\frac {c}{a+b}}\geq {\frac {3}{2}}.} It is an elementary special case (N = 3) of the difficult and much studied Shapiro inequality, and was published at least 50 years earlier. There is no corresponding upper bound as any of the 3 fractions in the inequality can be made arbitrarily large.
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https://en.wikipedia.org/wiki/Nesbitt's_inequality
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In mathematics, Nevanlinna's criterion in complex analysis, proved in 1920 by the Finnish mathematician Rolf Nevanlinna, characterizes holomorphic univalent functions on the unit disk which are starlike. Nevanlinna used this criterion to prove the Bieberbach conjecture for starlike univalent functions.
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https://en.wikipedia.org/wiki/Carathéodory's_lemma
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In mathematics, Neville's algorithm is an algorithm used for polynomial interpolation that was derived by the mathematician Eric Harold Neville in 1934. Given n + 1 points, there is a unique polynomial of degree ≤ n which goes through the given points. Neville's algorithm evaluates this polynomial. Neville's algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences. It is similar to Aitken's algorithm (named after Alexander Aitken), which is nowadays not used.
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https://en.wikipedia.org/wiki/Neville's_schema
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In mathematics, Newick tree format (or Newick notation or New Hampshire tree format) is a way of representing graph-theoretical trees with edge lengths using parentheses and commas. It was adopted by James Archie, William H. E. Day, Joseph Felsenstein, Wayne Maddison, Christopher Meacham, F. James Rohlf, and David Swofford, at two meetings in 1986, the second of which was at Newick's restaurant in Dover, New Hampshire, US. The adopted format is a generalization of the format developed by Meacham in 1984 for the first tree-drawing programs in Felsenstein's PHYLIP package.
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https://en.wikipedia.org/wiki/Newick_format
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In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.
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https://en.wikipedia.org/wiki/Newton's_identities
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In mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant. Isaac Newton stated it as lemma 28 of section VI of book 1 of Newton's Principia, and used it to show that the position of a planet moving in an orbit is not an algebraic function of time. There has been some controversy about whether or not this theorem is correct because Newton did not state exactly what he meant by an oval, and for some interpretations of the word oval the theorem is correct, while for others it is false. If "oval" means merely a continuous closed convex curve, then there are counterexamples, such as triangles or one of the lobes of Huygens lemniscate y2 = x2 − x4, while Arnold (1989) pointed that if "oval" an infinitely differentiable convex curve then Newton's claim is correct and his argument has the essential steps of a rigorous proof. Vassiliev (2002) generalized Newton's theorem to higher dimensions.
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https://en.wikipedia.org/wiki/Newton's_theorem_about_ovals
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In mathematics, Nikiel's conjecture in general topology was a conjectural characterization of the continuous image of a compact total order. The conjecture was first formulated by Jacek Nikiel in 1986. The conjecture was proven by Mary Ellen Rudin in 1999.The conjecture states that a compact topological space is the continuous image of a total order if and only if it is a monotonically normal space. == Notes ==
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https://en.wikipedia.org/wiki/Nikiel's_conjecture
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In mathematics, Nirenberg's conjecture, now Osserman's theorem, states that if a neighborhood of the sphere is omitted by the Gauss map of a complete minimal surface, then the surface in question is a plane. It was proved by Robert Osserman in 1959.
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https://en.wikipedia.org/wiki/Nirenberg's_conjecture
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In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0° ≤ θ ≤ 90° for which the sine of θ degrees is also a rational number are: sin 0 ∘ = 0 , sin 30 ∘ = 1 2 , sin 90 ∘ = 1. {\displaystyle {\begin{aligned}\sin 0^{\circ }&=0,\\\sin 30^{\circ }&={\frac {1}{2}},\\\sin 90^{\circ }&=1.\end{aligned}}} In radians, one would require that 0 ≤ x ≤ π/2, that x/π be rational, and that sin x be rational. The conclusion is then that the only such values are sin 0 = 0, sin π/6 = 1/2, and sin π/2 = 1. The theorem appears as Corollary 3.12 in Niven's book on irrational numbers.The theorem extends to the other trigonometric functions as well. For rational values of θ, the only rational values of the sine or cosine are 0, ±1/2, and ±1; the only rational values of the secant or cosecant are ±1 and ±2; and the only rational values of the tangent or cotangent are 0 and ±1.
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https://en.wikipedia.org/wiki/Niven's_theorem
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In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian L, Noether identities can be defined as a differential operator whose kernel contains a range of the Euler–Lagrange operator of L. Any Euler–Lagrange operator obeys Noether identities which therefore are separated into the trivial and non-trivial ones. A Lagrangian L is called degenerate if the Euler–Lagrange operator of L satisfies non-trivial Noether identities. In this case Euler–Lagrange equations are not independent.
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https://en.wikipedia.org/wiki/Noether_identities
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Noether identities need not be independent, but satisfy first-stage Noether identities, which are subject to the second-stage Noether identities and so on. Higher-stage Noether identities also are separated into the trivial and non-trivial once. A degenerate Lagrangian is called reducible if there exist non-trivial higher-stage Noether identities.
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https://en.wikipedia.org/wiki/Noether_identities
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Yang–Mills gauge theory and gauge gravitation theory exemplify irreducible Lagrangian field theories. Different variants of second Noether’s theorem state the one-to-one correspondence between the non-trivial reducible Noether identities and the non-trivial reducible gauge symmetries. Formulated in a very general setting, second Noether’s theorem associates to the Koszul–Tate complex of reducible Noether identities, parameterized by antifields, the BRST complex of reducible gauge symmetries parameterized by ghosts. This is the case of covariant classical field theory and Lagrangian BRST theory.
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https://en.wikipedia.org/wiki/Noether_identities
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In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre also a projective line. Then the theorem states that S is rational.
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https://en.wikipedia.org/wiki/Noether's_theorem_on_rationality_for_surfaces
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In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that A codimension-one foliation of a compact 3-manifold whose universal covering space is not contractible must have a compact leaf.
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https://en.wikipedia.org/wiki/Novikov's_compact_leaf_theorem
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In mathematics, Oka's lemma, proved by Kiyoshi Oka, states that in a domain of holomorphy in C n {\displaystyle \mathbb {C} ^{n}} , the function − log d ( z ) {\displaystyle -\log d(z)} is plurisubharmonic, where d {\displaystyle d} is the distance to the boundary. This property shows that the domain is pseudoconvex. Historically, this lemma was first shown in the Hartogs domain in the case of two variables, also Oka's lemma is the inverse of the Levi's problem (unramified Riemann domain over C n {\displaystyle \mathbb {C} ^{n}} ). So maybe that's why Oka called Levi's problem as "problème inverse de Hartogs", and the Levi's problem is occasionally called Hartogs' Inverse Problem.
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https://en.wikipedia.org/wiki/Oka's_lemma
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In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles and right triangles, as shown by F. Balitrand in 1916.
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https://en.wikipedia.org/wiki/Ono's_inequality
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In mathematics, Osgood's lemma, introduced by William Fogg Osgood (1899), is a proposition in complex analysis. It states that a continuous function of several complex variables that is holomorphic in each variable separately is holomorphic. The assumption that the function is continuous can be dropped, but that form of the lemma is much harder to prove and is known as Hartogs' theorem. There is no analogue of this result for real variables.
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https://en.wikipedia.org/wiki/Osgood's_lemma
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If we assume that a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is globally continuous and separately differentiable on each variable (all partial derivatives exist everywhere), it is not true that f {\displaystyle f} will necessarily be differentiable. A counterexample in two dimensions is given by f ( x , y ) = 2 x 2 y + y 3 x 2 + y 2 . {\displaystyle f(x,y)={\dfrac {2x^{2}y+y^{3}}{x^{2}+y^{2}}}.} If in addition we define f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} , this function is everywhere continuous and has well-defined partial derivatives in x {\displaystyle x} and y {\displaystyle y} everywhere (also at the origin), but is not differentiable at the origin.
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https://en.wikipedia.org/wiki/Osgood's_lemma
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In mathematics, Ostrowski numeration, named after Alexander Ostrowski, is either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers and a non-integer representation of real numbers. Fix a positive irrational number α with continued fraction expansion . Let (qn) be the sequence of denominators of the convergents pn/qn to α: so qn = anqn−1 + qn−2. Let αn denote Tn(α) where T is the Gauss map T(x) = {1/x}, and write βn = (−1)n+1 α0 α1 ... αn: we have βn = anβn−1 + βn−2.
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https://en.wikipedia.org/wiki/Ostrowski_numeration
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In mathematics, Owen's T function T(h, a), named after statistician Donald Bruce Owen, is defined by T ( h , a ) = 1 2 π ∫ 0 a e − 1 2 h 2 ( 1 + x 2 ) 1 + x 2 d x ( − ∞ < h , a < + ∞ ) . {\displaystyle T(h,a)={\frac {1}{2\pi }}\int _{0}^{a}{\frac {e^{-{\frac {1}{2}}h^{2}(1+x^{2})}}{1+x^{2}}}dx\quad \left(-\infty
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https://en.wikipedia.org/wiki/Owen's_T_function
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In mathematics, Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by: P n ( x ) = { 1 , if n = 1 0 , if n = 2 x , if n = 3 x P n − 2 ( x ) + P n − 3 ( x ) , if n ≥ 4. {\displaystyle P_{n}(x)={\begin{cases}1,&{\mbox{if }}n=1\\0,&{\mbox{if }}n=2\\x,&{\mbox{if }}n=3\\xP_{n-2}(x)+P_{n-3}(x),&{\mbox{if }}n\geq 4.\end{cases}}} The first few Padovan polynomials are: P 1 ( x ) = 1 {\displaystyle P_{1}(x)=1\,} P 2 ( x ) = 0 {\displaystyle P_{2}(x)=0\,} P 3 ( x ) = x {\displaystyle P_{3}(x)=x\,} P 4 ( x ) = 1 {\displaystyle P_{4}(x)=1\,} P 5 ( x ) = x 2 {\displaystyle P_{5}(x)=x^{2}\,} P 6 ( x ) = 2 x {\displaystyle P_{6}(x)=2x\,} P 7 ( x ) = x 3 + 1 {\displaystyle P_{7}(x)=x^{3}+1\,} P 8 ( x ) = 3 x 2 {\displaystyle P_{8}(x)=3x^{2}\,} P 9 ( x ) = x 4 + 3 x {\displaystyle P_{9}(x)=x^{4}+3x\,} P 10 ( x ) = 4 x 3 + 1 {\displaystyle P_{10}(x)=4x^{3}+1\,} P 11 ( x ) = x 5 + 6 x 2 .
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https://en.wikipedia.org/wiki/Padovan_polynomials
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{\displaystyle P_{11}(x)=x^{5}+6x^{2}.\,} The Padovan numbers are recovered by evaluating the polynomials Pn−3(x) at x = 1. Evaluating Pn−3(x) at x = 2 gives the nth Fibonacci number plus (−1)n. (sequence A008346 in the OEIS) The ordinary generating function for the sequence is ∑ n = 1 ∞ P n ( x ) t n = t 1 − x t 2 − t 3 . {\displaystyle \sum _{n=1}^{\infty }P_{n}(x)t^{n}={\frac {t}{1-xt^{2}-t^{3}}}.}
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https://en.wikipedia.org/wiki/Padovan_polynomials
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In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. They were discovered by Émile Picard (1889), Paul Painlevé (1900, 1902), Richard Fuchs (1905), and Bertrand Gambier (1910).
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https://en.wikipedia.org/wiki/Painleve_equations
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In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally.
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https://en.wikipedia.org/wiki/Paley_digraph
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