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In mathematics, Hua's lemma, named for Hua Loo-keng, is an estimate for exponential sums. It states that if P is an integral-valued polynomial of degree k, ε {\displaystyle \varepsilon } is a positive real number, and f a real function defined by f ( α ) = ∑ x = 1 N exp ( 2 π i P ( x ) α ) , {\displaystyle f(\alpha )... | https://en.wikipedia.org/wiki/Hua's_lemma |
In mathematics, Hudde's rules are two properties of polynomial roots described by Johann Hudde. 1. If r is a double root of the polynomial equation a 0 x n + a 1 x n − 1 + ⋯ + a n − 1 x + a n = 0 {\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}=0} and if b 0 , b 1 , … , b n − 1 , b n {\displaystyle b_{0},b... | https://en.wikipedia.org/wiki/Hudde's_rules |
This definition is a form of the modern theorem that if r is a double root of ƒ(x) = 0, then r is a root of ƒ '(x) = 0.2. If for x = a the polynomial a 0 x n + a 1 x n − 1 + ⋯ + a n − 1 x + a n {\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}} takes on a relative maximum or minimum value, then a is a root ... | https://en.wikipedia.org/wiki/Hudde's_rules |
This definition is a modification of Fermat's theorem in the form that if ƒ(a) is a relative maximum or minimum value of a polynomial ƒ(x), then ƒ '(a) = 0, where ƒ ' is the derivative of ƒ.Hudde was working with Frans van Schooten on a Latin edition of La Géométrie of René Descartes. In the 1659 edition of the transla... | https://en.wikipedia.org/wiki/Hudde's_rules |
In mathematics, Hurwitz determinants were introduced by Adolf Hurwitz (1895), who used them to give a criterion for all roots of a polynomial to have negative real part. | https://en.wikipedia.org/wiki/Hurwitz_determinant |
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz... | https://en.wikipedia.org/wiki/Hurwitz's_automorphisms_theorem |
The theorem is named after Adolf Hurwitz, who proved it in (Hurwitz 1893). Hurwitz's bound also holds for algebraic curves over a field of characteristic 0, and over fields of positive characteristic p>0 for groups whose order is coprime to p, but can fail over fields of positive characteristic p>0 when p divides the g... | https://en.wikipedia.org/wiki/Hurwitz's_automorphisms_theorem |
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into ... | https://en.wikipedia.org/wiki/Euclidean_Hurwitz_algebra |
The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwit... | https://en.wikipedia.org/wiki/Euclidean_Hurwitz_algebra |
In mathematics, Huygens mastered the methods of ancient Greek geometry, particularly the work of Archimedes, and was an adept user of the analytic geometry and infinitesimal techniques of Descartes and Fermat. His mathematical style can be best described as geometrical infinitesimal analysis of curves and of motion. Dr... | https://en.wikipedia.org/wiki/Christiaan_Huygens |
Huygens brought this type of geometrical analysis to a close, as more mathematicians turned away from classical geometry to the calculus for handling infinitesimals, limit processes, and motion.Huygens was moreover able to fully employ mathematics to answer questions of physics. Often this entailed introducing a simple... | https://en.wikipedia.org/wiki/Christiaan_Huygens |
Huygens favoured axiomatic presentations of his results, which require rigorous methods of geometric demonstration: although he allowed levels of uncertainty in the selection of primary axioms and hypotheses, the proofs of theorems derived from these could never be in doubt. Huygens's style of publication exerted an in... | https://en.wikipedia.org/wiki/Christiaan_Huygens |
In demanding such mathematical tractability and precision, Huygens set an example for eighteenth-century scientists such as Johann Bernoulli, Jean le Rond d'Alembert, and Charles-Augustin de Coulomb.Although never intended for publication, Huygens made use of algebraic expressions to represent physical entities in a ha... | https://en.wikipedia.org/wiki/Christiaan_Huygens |
In mathematics, Hölder summation is a method for summing divergent series introduced by Hölder (1882). | https://en.wikipedia.org/wiki/Hölder_summation |
In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found.The theorem also generalizes to the q {\displaystyl... | https://en.wikipedia.org/wiki/Hölder's_theorem |
In mathematics, Ihara's lemma, introduced by Ihara (1975, lemma 3.2) and named by Ribet (1984), states that the kernel of the sum of the two p-degeneracy maps from J0(N)×J0(N) to J0(Np) is Eisenstein whenever the prime p does not divide N. Here J0(N) is the Jacobian of the compactification of the modular curve of Γ0(N)... | https://en.wikipedia.org/wiki/Ihara's_lemma |
In mathematics, Ingleton's inequality is an inequality that is satisfied by the rank function of any representable matroid. In this sense it is a necessary condition for representability of a matroid over a finite field. Let M be a matroid and let ρ be its rank function, Ingleton's inequality states that for any subset... | https://en.wikipedia.org/wiki/Ingleton's_inequality |
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in the French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heurist... | https://en.wikipedia.org/wiki/Itô's_lemma |
In mathematics, Jacob's ladder is a surface with infinite genus and two ends. It was named after Jacob's ladder by Étienne Ghys (1995, Théorème A), because the surface can be constructed as the boundary of a ladder that is infinitely long in both directions. | https://en.wikipedia.org/wiki/Jacob's_ladder_surface |
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ( 1 − x ) α ( 1 + x ) β {\displaystyle (1-x)^{\alpha }(1+x)^{\beta }} on the... | https://en.wikipedia.org/wiki/Jacobi_polynomial |
In mathematics, Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials P n α , β ( x ) {\displaystyle P_{n}^{\alpha ,\beta }(x)} as kernels of the transform .The Jacobi transform of a function F ( x ) {\displaystyle F(x)} is J { F ( x ) } = f α , ... | https://en.wikipedia.org/wiki/Jacobi_transform |
In mathematics, Jacobsthal sums are finite sums of Legendre symbols related to Gauss sums. They were introduced by Jacobsthal (1907). | https://en.wikipedia.org/wiki/Jacobsthal_sum |
In mathematics, Janiszewski's theorem, named after the Polish mathematician Zygmunt Janiszewski, is a result concerning the topology of the plane or extended plane. It states that if A and B are closed subsets of the extended plane with connected intersection, then any two points that can be connected by paths avoiding... | https://en.wikipedia.org/wiki/Janiszewski's_theorem |
In mathematics, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero. A convers... | https://en.wikipedia.org/wiki/Serre's_conjecture_II_(algebra) |
(Note that such fields do indeed have cohomological dimension at most 2.) The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2.The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Susl... | https://en.wikipedia.org/wiki/Serre's_conjecture_II_(algebra) |
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in... | https://en.wikipedia.org/wiki/Jensen_inequality |
In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.Jensen's inequality generalizes the statement that the secant line of a convex function... | https://en.wikipedia.org/wiki/Jensen_inequality |
{\displaystyle f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).} In the context of probability theory, it is generally stated in the following form: if X is a random variable and φ is a convex function, then φ ( E ) ≤ E . {\displaystyle \varphi (\operatorname {E} )\leq \operatorname {E} \left.} The difference be... | https://en.wikipedia.org/wiki/Jensen_inequality |
In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algeb... | https://en.wikipedia.org/wiki/Jordan_operator_algebra |
Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are cal... | https://en.wikipedia.org/wiki/Jordan_operator_algebra |
Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and... | https://en.wikipedia.org/wiki/Jordan_operator_algebra |
In mathematics, Jordan's inequality, named after Camille Jordan, states that 2 π x ≤ sin ( x ) ≤ x for x ∈ . {\displaystyle {\frac {2}{\pi }}x\leq \sin(x)\leq x{\text{ for }}x\in \left.} It can be proven through the geometry of circles (see drawing). | https://en.wikipedia.org/wiki/Jordan's_inequality |
In mathematics, K {\displaystyle {\mathcal {K}}} -equivalence, or contact equivalence, is an equivalence relation between map germs. It was introduced by John Mather in his seminal work in Singularity theory in the 1960s as a technical tool for studying stable maps. Since then it has proved important in its own right. ... | https://en.wikipedia.org/wiki/K-equivalence |
In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space. In terms of C ∗ {\displaystyle C^{*}} -algebras, it classifies the Fredholm modules over an algebra. | https://en.wikipedia.org/wiki/K-homology |
An operator homotopy between two Fredholm modules ( H , F 0 , Γ ) {\displaystyle ({\mathcal {H}},F_{0},\Gamma )} and ( H , F 1 , Γ ) {\displaystyle ({\mathcal {H}},F_{1},\Gamma )} is a norm continuous path of Fredholm modules, t ↦ ( H , F t , Γ ) {\displaystyle t\mapsto ({\mathcal {H}},F_{t},\Gamma )} , t ∈ . {\displa... | https://en.wikipedia.org/wiki/K-homology |
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in ... | https://en.wikipedia.org/wiki/Real_K-theory |
It can be seen as the study of certain kinds of invariants of large matrices.K-theory involves the construction of families of K-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in al... | https://en.wikipedia.org/wiki/Real_K-theory |
In high energy physics, K-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branes, Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds. In condensed matter physics K-theory has been used to classify to... | https://en.wikipedia.org/wiki/Real_K-theory |
In mathematics, KK-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980. It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem,... | https://en.wikipedia.org/wiki/KK-theory |
In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by Atiyah (1966), motivated by applications to the Atiyah–Singer index theorem for real elliptic operators. | https://en.wikipedia.org/wiki/KR-theory |
In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative. | https://en.wikipedia.org/wiki/Kachurovskii's_theorem |
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Dan... | https://en.wikipedia.org/wiki/Kan_fibration |
In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Irving Kaplansky. | https://en.wikipedia.org/wiki/Kaplansky's_theorem_on_quadratic_forms |
In mathematics, Karamata's inequality, named after Jovan Karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality, and generalizes in turn to the ... | https://en.wikipedia.org/wiki/Karamata's_inequality |
In mathematics, Kazamaki's condition gives a sufficient criterion ensuring that the Doléans-Dade exponential of a local martingale is a true martingale. This is particularly important if Girsanov's theorem is to be applied to perform a change of measure. Kazamaki's condition is more general than Novikov's condition. | https://en.wikipedia.org/wiki/Kazamaki's_condition |
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov, then at the University of California, Davis, now at Columbia University. | https://en.wikipedia.org/wiki/Khovanov_homology |
In mathematics, Kingman's subadditive ergodic theorem is one of several ergodic theorems. It can be seen as a generalization of Birkhoff's ergodic theorem. Intuitively, the subadditive ergodic theorem is a kind of random variable version of Fekete's lemma (hence the name ergodic). As a result, it can be rephrased in th... | https://en.wikipedia.org/wiki/Kingman's_subadditive_ergodic_theorem |
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for th... | https://en.wikipedia.org/wiki/Knuth_up-arrow_notation |
In mathematics, Kolmogorov's normability criterion is a theorem that provides a necessary and sufficient condition for a topological vector space to be normable; that is, for the existence of a norm on the space that generates the given topology. The normability criterion can be seen as a result in same vein as the Nag... | https://en.wikipedia.org/wiki/Kolmogorov's_normability_criterion |
In mathematics, Kostant's convexity theorem, introduced by Bertram Kostant (1973), states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a general... | https://en.wikipedia.org/wiki/Kostant_convexity_theorem |
In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory. The two-variable Kostka polynomials Kλμ(q, t) are known by several names including Kostka–Foulke... | https://en.wikipedia.org/wiki/Kostka_polynomial |
Here the indices λ and μ are integer partitions and Kλμ(q, t) is polynomial in the variables q and t. Sometimes one considers single-variable versions of these polynomials that arise by setting q = 0, i.e., by considering the polynomial Kλμ(t) = Kλμ(0, t). There are two slightly different versions of them, one called t... | https://en.wikipedia.org/wiki/Kostka_polynomial |
In fact, they show that K λ μ ( t ) = ∑ T ∈ S S Y T ( λ , μ ) t c h a r g e ( T ) {\displaystyle K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{charge(T)}} where the sum is taken over all semi-standard Young tableaux with shape λ and weight μ. Here, charge is a certain combinatorial statistic on semi-standard Y... | https://en.wikipedia.org/wiki/Kostka_polynomial |
In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototype example, due to Joseph Bernstein, Israel Gelfand,... | https://en.wikipedia.org/wiki/Koszul_duality |
In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points. The Krawtchouk matrix K(N) is an (N + 1) × (N + 1) matrix. The first few Krawtchouk matrices are: K ( 0 ) = , K ( 1 ) = , K ( 2 ) = , K ( 3 ) = , {\displaystyle K^{(0)}={\begin{bmatrix}... | https://en.wikipedia.org/wiki/Krawtchouk_matrices |
In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable ... | https://en.wikipedia.org/wiki/Krener's_theorem |
In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. They play an important role algebraic combinatorics and geometric complexity theory. They were introduced by Murnag... | https://en.wikipedia.org/wiki/Kronecker_coefficient |
In mathematics, Kronecker's congruence, introduced by Kronecker, states that Φ p ( x , y ) ≡ ( x − y p ) ( x p − y ) mod p , {\displaystyle \Phi _{p}(x,y)\equiv (x-y^{p})(x^{p}-y){\bmod {p}},} where p is a prime and Φp(x,y) is the modular polynomial of order p, given by Φ n ( x , j ) = ∏ τ ( x − j ( τ ) ) {\displaystyl... | https://en.wikipedia.org/wiki/Kronecker's_congruence |
In mathematics, Kronecker's lemma (see, e.g., Shiryaev (1996, Lemma IV.3.2)) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers. Th... | https://en.wikipedia.org/wiki/Kronecker's_lemma |
In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by Leopold Kronecker (1884). Kronecker's approximation theorem had been firstly proved by L. Kronecker in the end of the 19th century. It has been now revealed to relate to the idea of n-torus and Mahler measure since the later... | https://en.wikipedia.org/wiki/Kronecker's_theorem_on_diophantine_approximation |
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. | https://en.wikipedia.org/wiki/Kruskal_tree_theorem |
In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the nor... | https://en.wikipedia.org/wiki/Contractibility_of_unit_sphere_in_Hilbert_space |
In mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus p, with p congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers. These sums were known and used before Kummer, in the theory of... | https://en.wikipedia.org/wiki/Kummer_sum |
In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer (1851). Kubota & Leopoldt (1964) used Kummer's congruences to define the p-adic zeta function. | https://en.wikipedia.org/wiki/Kummer_congruence |
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852). | https://en.wikipedia.org/wiki/Kummer's_theorem |
In mathematics, Kuratowski convergence or Painlevé-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlevé in lectures on mathematical analysis in 1902, the concept was popularized in texts by Felix Hausdorff and Kazimierz Kuratowski. Intuitively, the Kura... | https://en.wikipedia.org/wiki/Kuratowski_convergence |
In mathematics, Kuratowski's intersection theorem is a result in general topology that gives a sufficient condition for a nested sequence of sets to have a non-empty intersection. Kuratowski's result is a generalisation of Cantor's intersection theorem. Whereas Cantor's result requires that the sets involved be compact... | https://en.wikipedia.org/wiki/Kuratowski's_intersection_theorem |
In mathematics, Ky Fan's lemma (KFL) is a combinatorial lemma about labellings of triangulations. It is a generalization of Tucker's lemma. It was proved by Ky Fan in 1952. | https://en.wikipedia.org/wiki/Ky_Fan_lemma |
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calcul... | https://en.wikipedia.org/wiki/Algebraic_de_Rham_cohomology |
In mathematics, Kōmura's theorem is a result on the differentiability of absolutely continuous Banach space-valued functions, and is a substantial generalization of Lebesgue's theorem on the differentiability of the indefinite integral, which is that Φ: → R given by Φ ( t ) = ∫ 0 t φ ( s ) d s , {\displaystyle \Phi (t... | https://en.wikipedia.org/wiki/Kōmura's_theorem |
In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on M gives rise to a norm on di... | https://en.wikipedia.org/wiki/L2_cohomology |
It is closely related to intersection cohomology; indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology. Another such result is the Zucker conjecture, which states that for a Hermitian locally symmetric variety the L2 cohomology is isomorphic to the intersection cohomolog... | https://en.wikipedia.org/wiki/L2_cohomology |
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.More generally, these terms may apply to an inequation or inequality; the right-hand side is everything on the rig... | https://en.wikipedia.org/wiki/Sides_of_an_equation |
In mathematics, Lady Windermere's Fan is a telescopic identity employed to relate global and local error of a numerical algorithm. The name is derived from Oscar Wilde's 1892 play Lady Windermere's Fan, A Play About a Good Woman. | https://en.wikipedia.org/wiki/Lady_Windermere's_Fan_(mathematics) |
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups and representations of Galois groups. The Langlands conjectures were introduced by Langlands (1967... | https://en.wikipedia.org/wiki/Lafforgue's_theorem |
In mathematics, Laguerre transform is an integral transform named after the mathematician Edmond Laguerre, which uses generalized Laguerre polynomials L n α ( x ) {\displaystyle L_{n}^{\alpha }(x)} as kernels of the transform.The Laguerre transform of a function f ( x ) {\displaystyle f(x)} is L { f ( x ) } = f ~ α ( n... | https://en.wikipedia.org/wiki/Laguerre_transform |
In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn. Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be... | https://en.wikipedia.org/wiki/Landau's_function |
An element of order 6 in the group S5 can be written in cycle notation as (1 2) (3 4 5). Note that the same argument applies to the number 6, that is, g(6) = 6. There are arbitrarily long sequences of consecutive numbers n, n + 1, …, n + m on which the function g is constant.The integer sequence g(0) = 1, g(1) = 1, g(2... | https://en.wikipedia.org/wiki/Landau's_function |
Equivalently (using little-o notation), g ( n ) = e ( 1 + o ( 1 ) ) n ln n {\displaystyle g(n)=e^{(1+o(1)){\sqrt {n\ln n}}}} . The statement that ln g ( n ) < L i − 1 ( n ) {\displaystyle \ln g(n)<{\sqrt {\mathrm {Li} ^{-1}(n)}}} for all sufficiently large n, where Li−1 denotes the inverse of the logarithmic integr... | https://en.wikipedia.org/wiki/Landau's_function |
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form ∫ a b e M f ( x ) d x , {\displaystyle \int _{a}^{b}e^{Mf(x)}\,dx,} where f ( x ) {\displaystyle f(x)} is a twice-differentiable function, M is a large number, and the endpoints a and b could pos... | https://en.wikipedia.org/wiki/Laplace's_method |
In mathematics, Laplace's principle is a basic theorem in large deviations theory which is similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−θφ(x)) over a fixed set A as θ becomes large. Such expressions can be used, for example, in statistical mechanics to determining the... | https://en.wikipedia.org/wiki/Laplace_principle_(large_deviations_theory) |
In mathematics, Laver tables (named after Richard Laver, who discovered them towards the end of the 1980s in connection with his works on set theory) are tables of numbers that have certain properties of algebraic and combinatorial interest. They occur in the study of racks and quandles. | https://en.wikipedia.org/wiki/Laver_table |
In mathematics, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S3. The conjecture was featured by the Australian Mathematical Society Gazette as part of the Millennium Problems series.In March 2012, Simon Brendle gave a proof of this conjecture, based on maximum ... | https://en.wikipedia.org/wiki/Hsiang–Lawson's_conjecture |
In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let F ( x , y ) {\di... | https://en.wikipedia.org/wiki/Lazard's_universal_ring |
More or less by definition, the ring R has the following universal property: For every commutative ring S, one-dimensional formal group laws over S correspond to ring homomorphisms from R to S.The commutative ring R constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complica... | https://en.wikipedia.org/wiki/Lazard's_universal_ring |
In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} , the "density" of A is 0 or 1 at almost every point in R n {\displaystyle \mathbb {R} ^{n}} . Additionally, the "density" of A is 1 at almost every point in A. Intuitively, this mean... | https://en.wikipedia.org/wiki/Density_point |
In other words, for every measurable set A, the density of A is 0 or 1 almost everywhere in Rn. However, if μ(A) > 0 and μ(Rn \ A) > 0, then there are always points of Rn where the density is neither 0 nor 1. For example, given a square in the plane, the density at every point inside the square is 1, on the edges is 1/... | https://en.wikipedia.org/wiki/Density_point |
The set of points in the plane at which the density is neither 0 nor 1 is non-empty (the square boundary), but it is negligible. The Lebesgue density theorem is a particular case of the Lebesgue differentiation theorem. Thus, this theorem is also true for every finite Borel measure on Rn instead of Lebesgue measure, se... | https://en.wikipedia.org/wiki/Density_point |
In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz (1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. There are now numerous form... | https://en.wikipedia.org/wiki/Lefschetz_duality |
In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image processing including: pattern and object recognition, image indexing, line fitting, feature extraction, edge detection, and texture analysis. Legendre moments have b... | https://en.wikipedia.org/wiki/Legendre_moment |
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest genera... | https://en.wikipedia.org/wiki/Legendre_Polynomials |
In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials P n ( x ) {\displaystyle P_{n}(x)} as kernels of the transform. Legendre transform is a special case of Jacobi transform. The Legendre transform of a function f ( x ) {\displa... | https://en.wikipedia.org/wiki/Legendre_transform_(integral_transform) |
In mathematics, Legendre's equation is the Diophantine equation The equation is named for Adrien-Marie Legendre who proved in 1785 that it is solvable in integers x, y, z, not all zero, if and only if −bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwi... | https://en.wikipedia.org/wiki/Legendre's_equation |
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac. | https://en.wikipedia.org/wiki/Legendre's_formula |
In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers n = x 2 + y 2 + z 2 {\displaystyle n=x^{2}+y^{2}+z^{2}} if and only if n is not of the form n = 4 a ( 8 b + 7 ) {\displaystyle n=4^{a}(8b+7)} for nonnegative integers a and b. The firs... | https://en.wikipedia.org/wiki/Legendre's_three-square_theorem |
In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem. It is known that φ(n) = n − 1 if and only if n is prime. So for every prime number n, we have φ(n) = n − 1 and thus in particular φ(n) divides n − 1. ... | https://en.wikipedia.org/wiki/Lehmer's_totient_problem |
In mathematics, Lehrbuch der Topologie (German for "textbook of topology") is a book by Herbert Seifert and William Threlfall, first published in 1934 and published in an English translation in 1980. It was one of the earliest textbooks on algebraic topology, and was the standard reference on this topic for many years.... | https://en.wikipedia.org/wiki/Lehrbuch_der_Topologie |
In mathematics, Lemoine's problem is a certain construction problem in elementary plane geometry posed by the French mathematician Émile Lemoine (1840–1912) in 1868. The problem was published as Question 864 in Nouvelles Annales de Mathématiques (Series 2, Volume 7 (1868), p 191). The chief interest in the problem is t... | https://en.wikipedia.org/wiki/Lemoine's_problem |
In mathematics, Levinson's inequality is the following inequality, due to Norman Levinson, involving positive numbers. Let a > 0 {\displaystyle a>0} and let f {\displaystyle f} be a given function having a third derivative on the range ( 0 , 2 a ) {\displaystyle (0,2a)} , and such that f ‴ ( x ) ≥ 0 {\displaystyle f'''... | https://en.wikipedia.org/wiki/Levinson's_inequality |
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by Claude Chevalley and S... | https://en.wikipedia.org/wiki/Lie_algebra_cohomology |
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