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Hua's lemma
Summary
Hua's_lemma
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 )=\sum _{x=1}^{N}\exp(2\pi iP(x)\alpha ),} then ∫ 0 1 | f ( α ) | λ d α ≪ P , ε N μ ( λ ) {\displaystyle \int _{0}^{1}|f(\alpha )|^{\lambda }d\alpha \ll _{P,\varepsilon }N^{\mu (\lambda )}} ,where ( λ , μ ( λ ) ) {\displaystyle (\lambda ,\mu (\lambda ))} lies on a polygonal line with vertices ( 2 ν , 2 ν − ν + ε ) , ν = 1 , … , k . {\displaystyle (2^{\nu },2^{\nu }-\nu +\varepsilon ),\quad \nu =1,\ldots ,k.} == References ==
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Hudde's rules
Summary
Hudde's_rules
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_{1},\dots ,b_{n-1},b_{n}} are numbers in arithmetic progression, then r is also a root of a 0 b 0 x n + a 1 b 1 x n − 1 + ⋯ + a n − 1 b n − 1 x + a n b n = 0. {\displaystyle a_{0}b_{0}x^{n}+a_{1}b_{1}x^{n-1}+\cdots +a_{n-1}b_{n-1}x+a_{n}b_{n}=0.}
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Hudde's rules
Summary
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 of the equation n a 0 x n + ( n − 1 ) a 1 x n − 1 + ⋯ + 2 a n − 2 x 2 + a n − 1 x = 0. {\displaystyle na_{0}x^{n}+(n-1)a_{1}x^{n-1}+\cdots +2a_{n-2}x^{2}+a_{n-1}x=0.}
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Hudde's rules
Summary
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 translation, Hudde contributed two letters: "Epistola prima de Redvctione Ǣqvationvm" (pages 406 to 506), and "Epistola secvnda de Maximus et Minimus" (pages 507 to 16). These letters may be read by the Internet Archive link below.
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Hurwitz determinant
Summary
Hurwitz_determinant
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.
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Hurwitz's automorphisms theorem
Summary
Hurwitz's_automorphisms_theorem
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 group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve.
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Hurwitz's automorphisms theorem
Summary
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 group order. For example, the double cover of the projective line y2 = xp −x branched at all points defined over the prime field has genus g=(p−1)/2 but is acted on by the group SL2(p) of order p3−p.
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Euclidean Hurwitz algebra
Summary
Hurwitz_algebra
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 the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.
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Euclidean Hurwitz algebra
Summary
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 Hurwitz problem, solved also in Radon (1922). Subsequent proofs of the restrictions on the dimension have been given by Eckmann (1943) using the representation theory of finite groups and by Lee (1948) and Chevalley (1954) using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups and in quantum mechanics to the classification of simple Jordan algebras.
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Christiaan Huygens
Mathematics and physics
Christiaan_Huygens > Legacy > Mathematics and physics
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. Drawing inspiration and imagery from mechanics, it remained pure mathematics in form.
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Christiaan Huygens
Mathematics and physics
Christiaan_Huygens > Legacy > Mathematics and physics
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 model for describing a complicated situation, then analyzing it starting from simple arguments to their logical consequences, developing the necessary mathematics along the way. As he wrote at the end of a draft of De vi Centrifuga: Whatever you will have supposed not impossible either concerning gravity or motion or any other matter, if then you prove something concerning the magnitude of a line, surface, or body, it will be true; as for instance, Archimedes on the quadrature of the parabola, where the tendency of heavy objects has been assumed to act through parallel lines.
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Christiaan Huygens
Mathematics and physics
Christiaan_Huygens > Legacy > Mathematics and physics
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 influence in Newton's presentation of his own major works.Besides the application of mathematics to physics and physics to mathematics, Huygens relied on mathematics as methodology, specifically its ability to generate new knowledge about the world. Unlike Galileo, who used mathematics primarily as rhetoric or synthesis, Huygens consistently employed mathematics as a method of discovery and analysis, and insisted that the reduction of the physical to the geometrical satisfy exacting standards of fit between the real and the ideal.
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Christiaan Huygens
Mathematics and physics
Christiaan_Huygens > Legacy > Mathematics and physics
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 handful of his manuscripts on collisions. This would make him one of the first to employ mathematical formulae to describe relationships in physics, as it is done today. Huygens also came close to the modern idea of limit while working on his Dioptrica, though he never used the notion outside geometrical optics.
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Hölder summation
Summary
Hölder_summation
In mathematics, Hölder summation is a method for summing divergent series introduced by Hölder (1882).
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Hölder's theorem
Summary
Hölder's_theorem
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 {\displaystyle q} -gamma function.
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Ihara's lemma
Summary
Ihara's_lemma
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).
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Ingleton's inequality
Summary
Ingleton's_inequality
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 subsets X1, X2, X3 and X4 in the support of M, the inequality ρ(X1)+ρ(X2)+ρ(X1∪X2∪X3)+ρ(X1∪X2∪X4)+ρ(X3∪X4) ≤ ρ(X1∪X2)+ρ(X1∪X3)+ρ(X1∪X4)+ρ(X2∪X3)+ρ(X2∪X4) is satisfied.Aubrey William Ingleton, an English mathematician, wrote an important paper in 1969 in which he surveyed the representability problem in matroids. Although the article is mainly expository, in this paper Ingleton stated and proved Ingleton's inequality, which has found interesting applications in information theory, matroid theory, and network coding.
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Itô's lemma
Summary
Itō's_lemma
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 heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values. Kiyoshi Itô published a proof of the formula in 1951.
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Jacob's ladder surface
Summary
Jacob's_ladder_(manifold)
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.
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Jacobi polynomial
Summary
Hypergeometric_polynomial
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 interval {\displaystyle } . The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
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Jacobi transform
Summary
Jacobi_transform
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 , β ( n ) = ∫ − 1 1 ( 1 − x ) α ( 1 + x ) β P n α , β ( x ) F ( x ) d x {\displaystyle J\{F(x)\}=f^{\alpha ,\beta }(n)=\int _{-1}^{1}(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)\ F(x)\ dx} The inverse Jacobi transform is given by J − 1 { f α , β ( n ) } = F ( x ) = ∑ n = 0 ∞ 1 δ n f α , β ( n ) P n α , β ( x ) , where δ n = 2 α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) n ! ( α + β + 2 n + 1 ) Γ ( n + α + β + 1 ) {\displaystyle J^{-1}\{f^{\alpha ,\beta }(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{\delta _{n}}}f^{\alpha ,\beta }(n)P_{n}^{\alpha ,\beta }(x),\quad {\text{where}}\quad \delta _{n}={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n! (\alpha +\beta +2n+1)\Gamma (n+\alpha +\beta +1)}}}
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Jacobsthal sum
Summary
Jacobsthal_sum
In mathematics, Jacobsthal sums are finite sums of Legendre symbols related to Gauss sums. They were introduced by Jacobsthal (1907).
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Janiszewski's theorem
Summary
Janiszewski's_theorem
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 either A or B can be connected by a path avoiding both of them. The theorem has been used as a tool for proving the Jordan curve theorem and in complex function theory.
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Serre's conjecture II (algebra)
Summary
Serre's_conjecture_II_(algebra)
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 converse of the conjecture holds: if the field F is perfect and if the cohomology set H1(F, G) is zero for every semisimple simply connected algebraic group G then the p-cohomological dimension of F is at most 2 for every prime p.The conjecture holds in the case where F is a local field (such as p-adic field) or a global field with no real embeddings (such as Q(√−1)). This is a special case of the Kneser–Harder–Chernousov Hasse principle for algebraic groups over global fields.
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Serre's conjecture II (algebra)
Summary
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–Suslin theorem. Building on this result, the conjecture holds if G is a classical group. The conjecture also holds if G is one of certain kinds of exceptional group.
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Jensen inequality
Summary
Jensen's_Inequality
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 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below.
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Jensen inequality
Summary
Jensen's_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 lies above the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for t ∈ ), t f ( x 1 ) + ( 1 − t ) f ( x 2 ) , {\displaystyle tf(x_{1})+(1-t)f(x_{2}),} while the graph of the function is the convex function of the weighted means, f ( t x 1 + ( 1 − t ) x 2 ) . {\displaystyle f(tx_{1}+(1-t)x_{2}).} Thus, Jensen's inequality is f ( t x 1 + ( 1 − t ) x 2 ) ≤ t f ( x 1 ) + ( 1 − t ) f ( x 2 ) .
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Jensen inequality
Summary
Jensen's_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 between the two sides of the inequality, E ⁡ − φ ( E ⁡ ) {\displaystyle \operatorname {E} \left-\varphi \left(\operatorname {E} \right)} , is called the Jensen gap.
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Jordan operator algebra
Summary
Jordan_operator_algebra
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 algebras were devised by Alfsen, Shultz & Størmer (1978).
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Jordan operator algebra
Summary
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 called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras.
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Jordan operator algebra
Summary
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 the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.
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Jordan's inequality
Summary
Jordan's_inequality
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).
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K-equivalence
Summary
K-equivalence
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. Roughly speaking, two map germs ƒ, g are K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalent if ƒ−1(0) and g−1(0) are diffeomorphic.
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K-homology
Summary
K-homology
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.
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K-homology
Summary
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 ∈ . {\displaystyle t\in .} Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The K 0 ( A ) {\displaystyle K^{0}(A)} group is the abelian group of equivalence classes of even Fredholm modules over A. The K 1 ( A ) {\displaystyle K^{1}(A)} group is the abelian group of equivalence classes of odd Fredholm modules over A. Addition is given by direct summation of Fredholm modules, and the inverse of ( H , F , Γ ) {\displaystyle ({\mathcal {H}},F,\Gamma )} is ( H , − F , − Γ ) . {\displaystyle ({\mathcal {H}},-F,-\Gamma ).}
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Real K-theory
Summary
Real_K-theory
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 the field of operator algebras.
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Real K-theory
Summary
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 algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyah–Singer index theorem, and the Adams operations.
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Real K-theory
Summary
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 topological insulators, superconductors and stable Fermi surfaces. For more details, see K-theory (physics).
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KK-theory
Summary
KK-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, and the classification of extensions of C*-algebras by Lawrence G. Brown, Ronald G. Douglas, and Peter Arthur Fillmore in 1977. In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of nuclear C*-algebras, as it was the key to the solutions of many problems in operator K-theory, such as, for instance, the mere calculation of K-groups. Furthermore, it was essential in the development of the Baum–Connes conjecture and plays a crucial role in noncommutative topology. KK-theory was followed by a series of similar bifunctor constructions such as the E-theory and the bivariant periodic cyclic theory, most of them having more category-theoretic flavors, or concerning another class of algebras rather than that of the separable C*-algebras, or incorporating group actions.
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KR-theory
Summary
KR-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.
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Kachurovskii's theorem
Summary
Kachurovskii's_theorem
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.
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Kan fibration
Summary
Kan_fibration
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 Daniel Kan.
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Kaplansky's theorem on quadratic forms
Summary
Kaplansky's_theorem_on_quadratic_forms
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.
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Karamata's inequality
Summary
Karamata's_inequality
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 concept of Schur-convex functions.
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Kazamaki's condition
Summary
Kazamaki's_condition
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.
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Khovanov homology
Summary
Khovanov_homology
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.
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Kingman's subadditive ergodic theorem
Summary
Kingman's_subadditive_ergodic_theorem
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 the language of probability, e.g. using a sequence of random variables and expected values. The theorem is named after John Kingman.
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Knuth up-arrow notation
Summary
Knuth's_up-arrow_notation
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 the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc. Various notations have been used to represent hyperoperations. One such notation is H n ( a , b ) {\displaystyle H_{n}(a,b)} . Knuth's up-arrow notation ↑ {\displaystyle \uparrow } is another. For example: the single arrow ↑ {\displaystyle \uparrow } represents exponentiation (iterated multiplication) the double arrow ↑↑ {\displaystyle \uparrow \uparrow } represents tetration (iterated exponentiation) the triple arrow ↑↑↑ {\displaystyle \uparrow \uparrow \uparrow } represents pentation (iterated tetration) The general definition of the up-arrow notation is as follows (for a ≥ 0 , n ≥ 1 , b ≥ 0 {\displaystyle a\geq 0,n\geq 1,b\geq 0} ): Here, ↑ n {\displaystyle \uparrow ^{n}} stands for n arrows, so for example The square brackets are another notation for hyperoperations.
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Kolmogorov's normability criterion
Summary
Kolmogorov's_normability_criterion
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 Nagata–Smirnov metrization theorem and Bing metrization theorem, which gives a necessary and sufficient condition for a topological space to be metrizable. The result was proved by the Russian mathematician Andrey Nikolayevich Kolmogorov in 1934.
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Kostant convexity theorem
Summary
Kostant's_convexity_theorem
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 generalization of a result of Schur (1923), Horn (1954) and Thompson (1972) for hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all n by n complex self-adjoint matrices with given eigenvalues Λ = (λ1, ..., λn) is the convex polytope with vertices all permutations of the coordinates of Λ. Kostant used this to generalize the Golden–Thompson inequality to all compact groups.
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Kostka polynomial
Summary
Kostka_polynomial
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–Foulkes polynomials, Macdonald–Kostka polynomials or q,t-Kostka polynomials.
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Kostka polynomial
Summary
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 transformed Kostka polynomials.The one-variable specializations of the Kostka polynomials can be used to relate Hall-Littlewood polynomials Pμ to Schur polynomials sλ: s λ ( x 1 , … , x n ) = ∑ μ K λ μ ( t ) P μ ( x 1 , … , x n ; t ) . {\displaystyle s_{\lambda }(x_{1},\ldots ,x_{n})=\sum _{\mu }K_{\lambda \mu }(t)P_{\mu }(x_{1},\ldots ,x_{n};t).\ } These polynomials were conjectured to have non-negative integer coefficients by Foulkes, and this was later proved in 1978 by Alain Lascoux and Marcel-Paul Schützenberger.
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Kostka polynomial
Summary
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 Young tableaux. The Macdonald–Kostka polynomials can be used to relate Macdonald polynomials (also denoted by Pμ) to Schur polynomials sλ: s λ ( x 1 , … , x n ) = ∑ μ K λ μ ( q , t ) J μ ( x 1 , … , x n ; q , t ) {\displaystyle s_{\lambda }(x_{1},\ldots ,x_{n})=\sum _{\mu }K_{\lambda \mu }(q,t)J_{\mu }(x_{1},\ldots ,x_{n};q,t)\ } where J μ ( x 1 , … , x n ; q , t ) = P μ ( x 1 , … , x n ; q , t ) ∏ s ∈ μ ( 1 − q a r m ( s ) t l e g ( s ) + 1 ) . {\displaystyle J_{\mu }(x_{1},\ldots ,x_{n};q,t)=P_{\mu }(x_{1},\ldots ,x_{n};q,t)\prod _{s\in \mu }(1-q^{arm(s)}t^{leg(s)+1}).\ } Kostka numbers are special values of the one- or two-variable Kostka polynomials: K λ μ = K λ μ ( 1 ) = K λ μ ( 0 , 1 ) . {\displaystyle K_{\lambda \mu }=K_{\lambda \mu }(1)=K_{\lambda \mu }(0,1).\ }
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Koszul duality
Summary
Koszul_duality
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, and Sergei Gelfand, is the rough duality between the derived category of a symmetric algebra and that of an exterior algebra. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature.
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Krawtchouk matrices
Summary
Krawtchouk_matrices
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}1\end{bmatrix}},\qquad K^{(1)}=\left,\qquad K^{(2)}=\left,\qquad K^{(3)}=\left,} K ( 4 ) = , K ( 5 ) = . {\displaystyle K^{(4)}=\left,\qquad K^{(5)}=\left.}
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Krener's theorem
Summary
Krener's_theorem
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 set has nonempty interior in the topology of the corresponding orbit. Heuristically, Krener's theorem prohibits attainable sets from being hairy.
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Kronecker coefficient
Summary
Kronecker_coefficient
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 Murnaghan in 1938.
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Kronecker's congruence
Summary
Kronecker's_congruence
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 ) ) {\displaystyle \Phi _{n}(x,j)=\prod _{\tau }(x-j(\tau ))} for j the elliptic modular function and τ running through classes of imaginary quadratic integers of discriminant n.
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Kronecker's lemma
Summary
Kronecker's_lemma
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. The lemma is named after the German mathematician Leopold Kronecker.
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Kronecker's theorem on diophantine approximation
Summary
Kronecker's_theorem
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 half of the 20th century. In terms of physical systems, it has the consequence that planets in circular orbits moving uniformly around a star will, over time, assume all alignments, unless there is an exact dependency between their orbital periods.
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Kruskal tree theorem
Summary
Kruskal's_tree_theorem
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.
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Contractibility of unit sphere in Hilbert space
Summary
Contractibility_of_unit_sphere_in_Hilbert_space
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 norm topology on operators. A significant corollary, also referred to as Kuiper's theorem, is that this group is weakly contractible, ie. all its homotopy groups are trivial. This result has important uses in topological K-theory.
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Kummer sum
Summary
Kummer_sum
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 cyclotomy.
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Kummer congruence
Summary
Kummer_congruence
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.
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Kummer's theorem
Summary
Kummer's_theorem
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).
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Kuratowski convergence
Summary
Kuratowski_convergence
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 Kuratowski limit of a sequence of sets is where the sets "accumulate".
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Kuratowski's intersection theorem
Summary
Kuratowski's_intersection_theorem
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, Kuratowski's result allows them to be non-compact, but insists that their non-compactness "tends to zero" in an appropriate sense. The theorem is named for the Polish mathematician Kazimierz Kuratowski, who proved it in 1930.
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Ky Fan lemma
Summary
Ky_Fan_lemma
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.
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Algebraic de Rham cohomology
Summary
Sheaf_of_relative_differentials
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 calculus and geometry over the complex numbers to contexts where such methods are not available.
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Kōmura's theorem
Summary
Kōmura's_theorem
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)=\int _{0}^{t}\varphi (s)\,\mathrm {d} s,} is differentiable at t for almost every 0 < t < T when φ: → R lies in the Lp space L1(; R).
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L2 cohomology
Summary
L2_cohomology
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 differential forms and a volume form. L2 cohomology, which grew in part out of L2 d-bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker (1978) and Jeff Cheeger (1979).
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L2 cohomology
Summary
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 cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker 1982). This was proved in different ways by Eduard Looijenga (1988) and by Leslie Saper and Mark Stern (1990).
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Sides of an equation
Summary
Inhomogeneous_equation
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 right side of a test operator in an expression, with LHS defined similarly.
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Lady Windermere's Fan (mathematics)
Summary
Lady_Windermere's_Fan_(mathematics)
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.
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Lafforgue's theorem
Summary
Lafforgue's_theorem
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, 1970) and describe a correspondence between representations of the Weil group of an algebraic function field and representations of algebraic groups over the function field, generalizing class field theory of function fields from abelian Galois groups to non-abelian Galois groups.
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Laguerre transform
Summary
Laguerre_transform
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 ) = ∫ 0 ∞ e − x x α L n α ( x ) f ( x ) d x {\displaystyle L\{f(x)\}={\tilde {f}}_{\alpha }(n)=\int _{0}^{\infty }e^{-x}x^{\alpha }\ L_{n}^{\alpha }(x)\ f(x)\ dx} The inverse Laguerre transform is given by L − 1 { f ~ α ( n ) } = f ( x ) = ∑ n = 0 ∞ ( n + α n ) − 1 1 Γ ( α + 1 ) f ~ α ( n ) L n α ( x ) {\displaystyle L^{-1}\{{\tilde {f}}_{\alpha }(n)\}=f(x)=\sum _{n=0}^{\infty }{\binom {n+\alpha }{n}}^{-1}{\frac {1}{\Gamma (\alpha +1)}}{\tilde {f}}_{\alpha }(n)L_{n}^{\alpha }(x)}
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Landau's function
Summary
Landau's_function
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 recursively applied to itself before it returns to its starting sequence. For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so g(5) = 6.
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Landau's function
Summary
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) = 2, g(3) = 3, g(4) = 4, g(5) = 6, g(6) = 6, g(7) = 12, g(8) = 15, ... (sequence A000793 in the OEIS) is named after Edmund Landau, who proved in 1902 that lim n → ∞ ln ⁡ ( g ( n ) ) n ln ⁡ ( n ) = 1 {\displaystyle \lim _{n\to \infty }{\frac {\ln(g(n))}{\sqrt {n\ln(n)}}}=1} (where ln denotes the natural logarithm).
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Landau's function
Summary
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 integral function, is equivalent to the Riemann hypothesis. It can be shown that g ( n ) ≤ e n / e {\displaystyle g(n)\leq e^{n/e}} with the only equality between the functions at n = 0, and indeed g ( n ) ≤ exp ⁡ ( 1.05314 n ln ⁡ n ) . {\displaystyle g(n)\leq \exp \left(1.05314{\sqrt {n\ln n}}\right).}
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Laplace's method
Summary
Laplace's_method
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 possibly be infinite. This technique was originally presented in Laplace (1774). In Bayesian statistics, Laplace's approximation can refer to either approximating the posterior normalizing constant with Laplace's method or approximating the posterior distribution with a Gaussian centered at the maximum a posteriori estimate. Laplace approximations play a central role in the integrated nested Laplace approximations method for fast approximate Bayesian inference.
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Laplace principle (large deviations theory)
Summary
Laplace_principle_(large_deviations_theory)
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 limiting behaviour of a system as the temperature tends to absolute zero.
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Laver table
Summary
Laver_table
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.
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Hsiang–Lawson's conjecture
Summary
Hsiang–Lawson's_conjecture
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 principle techniques. == References ==
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Lazard's universal ring
Summary
Lazard's_universal_ring
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 ) {\displaystyle F(x,y)} be x + y + ∑ i , j c i , j x i y j {\displaystyle x+y+\sum _{i,j}c_{i,j}x^{i}y^{j}} for indeterminates c i , j {\displaystyle c_{i,j}} , and we define the universal ring R to be the commutative ring generated by the elements c i , j {\displaystyle c_{i,j}} , with the relations that are forced by the associativity and commutativity laws for formal group laws.
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Lazard's universal ring
Summary
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 complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degree 1, 2, 3, ..., where c i , j {\displaystyle c_{i,j}} has degree ( i + j − 1 ) {\displaystyle (i+j-1)} . Daniel Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring. Hence, topologists commonly regrade the Lazard ring so that c i , j {\displaystyle c_{i,j}} has degree 2 ( i + j − 1 ) {\displaystyle 2(i+j-1)} , because the coefficient ring of complex cobordism is evenly graded.
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Density point
Summary
Lebesgue's_density_theorem
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 means that the "edge" of A, the set of points in A whose "neighborhood" is partially in A and partially outside of A, is negligible. Let μ be the Lebesgue measure on the Euclidean space Rn and A be a Lebesgue measurable subset of Rn. Define the approximate density of A in a ε-neighborhood of a point x in Rn as d ε ( x ) = μ ( A ∩ B ε ( x ) ) μ ( B ε ( x ) ) {\displaystyle d_{\varepsilon }(x)={\frac {\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}}} where Bε denotes the closed ball of radius ε centered at x. Lebesgue's density theorem asserts that for almost every point x of A the density d ( x ) = lim ε → 0 d ε ( x ) {\displaystyle d(x)=\lim _{\varepsilon \to 0}d_{\varepsilon }(x)} exists and is equal to 0 or 1.
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Density point
Summary
Lebesgue's_density_theorem
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 1/2, and at the corners is 1/4.
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Density point
Summary
Lebesgue's_density_theorem
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, see Discussion.
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Lefschetz duality
Summary
Poincaré–Lefschetz_duality_theorem
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 formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality.
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Legendre moment
Summary
Legendre_moment
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 been studied as a means to reduce image moment calculation complexity by limiting the amount of information redundancy through approximation.
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Legendre Polynomials
Summary
Legendre_polynomial
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 generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, Big q-Legendre polynomials, and associated Legendre functions.
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Legendre transform (integral transform)
Summary
Legendre_transform_(integral_transform)
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 ) {\displaystyle f(x)} is J n { f ( x ) } = f ~ ( n ) = ∫ − 1 1 P n ( x ) f ( x ) d x {\displaystyle {\mathcal {J}}_{n}\{f(x)\}={\tilde {f}}(n)=\int _{-1}^{1}P_{n}(x)\ f(x)\ dx} The inverse Legendre transform is given by J n − 1 { f ~ ( n ) } = f ( x ) = ∑ n = 0 ∞ 2 n + 1 2 f ~ ( n ) P n ( x ) {\displaystyle {\mathcal {J}}_{n}^{-1}\{{\tilde {f}}(n)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\tilde {f}}(n)P_{n}(x)}
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Legendre's equation
Summary
Legendre's_equation
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, pairwise relatively prime integers, not all positive or all negative .
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Legendre's formula
Summary
Legendre's_formula
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.
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Legendre's three-square theorem
Summary
Legendre's_three-square_theorem
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 first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as n = 4 a ( 8 b + 7 ) {\displaystyle n=4^{a}(8b+7)} ) are 7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... (sequence A004215 in the OEIS).
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Lehmer's totient problem
Summary
Lehmer's_totient_problem
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. D. H. Lehmer conjectured in 1932 that there are no composite numbers with this property.
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Lehrbuch der Topologie
Summary
Lehrbuch_der_Topologie
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. Albert W. Tucker wrote a review.
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Lemoine's problem
Summary
Lemoine's_problem
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 that a discussion of the solution of the problem by Ludwig Kiepert published in Nouvelles Annales de Mathématiques (series 2, Volume 8 (1869), pp 40–42) contained a description of a hyperbola which is now known as the Kiepert hyperbola.
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Levinson's inequality
Summary
Levinson's_inequality
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'''(x)\geq 0} for all x ∈ ( 0 , 2 a ) {\displaystyle x\in (0,2a)} . Suppose 0 < x i ≤ a {\displaystyle 0
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Lie algebra cohomology
Summary
Lie_algebra_homology
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 Samuel Eilenberg (1948) to coefficients in an arbitrary Lie module.