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In mathematics, the Kervaire invariant is an invariant of a framed ( 4 k + 2 ) {\displaystyle (4k+2)} -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf. The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group.
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https://en.wikipedia.org/wiki/Kervaire_invariant
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It can be thought of as the simply-connected quadratic L-group L 4 k + 2 {\displaystyle L_{4k+2}} , and thus analogous to the other invariants from L-theory: the signature, a 4 k {\displaystyle 4k} -dimensional invariant (either symmetric or quadratic, L 4 k ≅ L 4 k {\displaystyle L^{4k}\cong L_{4k}} ), and the De Rham invariant, a ( 4 k + 1 ) {\displaystyle (4k+1)} -dimensional symmetric invariant L 4 k + 1 {\displaystyle L^{4k+1}} . In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1.
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https://en.wikipedia.org/wiki/Kervaire_invariant
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The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open.
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https://en.wikipedia.org/wiki/Kervaire_invariant
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In mathematics, the Kervaire semi-characteristic, introduced by Michel Kervaire (1956), is an invariant of closed manifolds M of dimension 4 n + 1 {\displaystyle 4n+1} taking values in Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } , given by k ( M ) = ∑ i = 0 2 n dim H 2 i ( M , R ) mod 2 {\displaystyle k(M)=\sum _{i=0}^{2n}\dim H^{2i}(M,\mathbb {R} ){\bmod {2}}} .Michael Atiyah and Isadore Singer (1971) showed that the Kervaire semi-characteristic of a differentiable manifold is given by the index of a skew-adjoint elliptic operator. Assuming M is oriented, the Atiyah vanishing theorem states that if M has two linearly independent vector fields, then k ( M ) = 0 {\displaystyle k(M)=0} .
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https://en.wikipedia.org/wiki/Kervaire_semi-characteristic
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In mathematics, the Khatri–Rao product of matrices is defined as A ∗ B = ( A i j ⊗ B i j ) i j {\displaystyle \mathbf {A} \ast \mathbf {B} =\left(\mathbf {A} _{ij}\otimes \mathbf {B} _{ij}\right)_{ij}} in which the ij-th block is the mipi × njqj sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal. The size of the product is then (Σi mipi) × (Σj njqj). For example, if A and B both are 2 × 2 partitioned matrices e.g.: A = = , B = = , {\displaystyle \mathbf {A} =\left=\left,\quad \mathbf {B} =\left=\left,} we obtain: A ∗ B = = . {\displaystyle \mathbf {A} \ast \mathbf {B} =\left=\left.} This is a submatrix of the Tracy–Singh product of the two matrices (each partition in this example is a partition in a corner of the Tracy–Singh product) and also may be called the block Kronecker product.
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https://en.wikipedia.org/wiki/Face-splitting_product
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In mathematics, the Khinchin integral (sometimes spelled Khintchine integral), also known as the Denjoy–Khinchin integral, generalized Denjoy integral or wide Denjoy integral, is one of a number of definitions of the integral of a function. It is a generalization of the Riemann and Lebesgue integrals. It is named after Aleksandr Khinchin and Arnaud Denjoy, but is not to be confused with the (narrow) Denjoy integral.
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https://en.wikipedia.org/wiki/Khinchin_integral
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In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick N {\displaystyle N} complex numbers x 1 , … , x N ∈ C {\displaystyle x_{1},\dots ,x_{N}\in \mathbb {C} } , and add them together each multiplied by a random sign ± 1 {\displaystyle \pm 1} , then the expected value of the sum's modulus, or the modulus it will be closest to on average, will be not too far off from | x 1 | 2 + ⋯ + | x N | 2 {\displaystyle {\sqrt {|x_{1}|^{2}+\cdots +|x_{N}|^{2}}}} .
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https://en.wikipedia.org/wiki/Khintchine_inequality
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In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras.
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https://en.wikipedia.org/wiki/Killing_form
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In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if M and N are 3-manifolds, resulting from Dehn surgery on framed links L and J respectively, then they are homeomorphic if and only if L and J are related by a sequence of Kirby moves. According to the Lickorish–Wallace theorem any closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere. Some ambiguity exists in the literature on the precise use of the term "Kirby moves".
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https://en.wikipedia.org/wiki/Kirby_calculus
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Different presentations of "Kirby calculus" have a different set of moves and these are sometimes called Kirby moves. Kirby's original formulation involved two kinds of move, the "blow-up" and the "handle slide"; Roger Fenn and Colin Rourke exhibited an equivalent construction in terms of a single move, the Fenn–Rourke move, that appears in many expositions and extensions of the Kirby calculus. Dale Rolfsen's book, Knots and Links, from which many topologists have learned the Kirby calculus, describes a set of two moves: 1) delete or add a component with surgery coefficient infinity 2) twist along an unknotted component and modify surgery coefficients appropriately (this is called the Rolfsen twist).
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https://en.wikipedia.org/wiki/Kirby_calculus
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This allows an extension of the Kirby calculus to rational surgeries. There are also various tricks to modify surgery diagrams.
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https://en.wikipedia.org/wiki/Kirby_calculus
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One such useful move is the slam-dunk. An extended set of diagrams and moves are used for describing 4-manifolds. A framed link in the 3-sphere encodes instructions for attaching 2-handles to the 4-ball.
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https://en.wikipedia.org/wiki/Kirby_calculus
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(The 3-dimensional boundary of this manifold is the 3-manifold interpretation of the link diagram mentioned above.) 1-handles are denoted by either a pair of 3-balls (the attaching region of the 1-handle) or, more commonly, unknotted circles with dots. The dot indicates that a neighborhood of a standard 2-disk with boundary the dotted circle is to be excised from the interior of the 4-ball. Excising this 2-handle is equivalent to adding a 1-handle; 3-handles and 4-handles are usually not indicated in the diagram.
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https://en.wikipedia.org/wiki/Kirby_calculus
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In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are inconsistent, in particular the version of Haskell Curry's combinatory logic introduced in 1930, and Alonzo Church's original lambda calculus, introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibited by Stephen Kleene and J. B. Rosser in 1935.
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https://en.wikipedia.org/wiki/Kleene–Rosser_paradox
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In mathematics, the Klein bottle () is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold. Other related non-orientable surfaces include the Möbius strip and the real projective plane.
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https://en.wikipedia.org/wiki/Klein_bottle
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While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary. The Klein bottle was first described in 1882 by the mathematician Felix Klein.
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https://en.wikipedia.org/wiki/Klein_bottle
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In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4.
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https://en.wikipedia.org/wiki/Klein_4-group
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The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups.
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https://en.wikipedia.org/wiki/Klein_4-group
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In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations: the first one, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not; the other one, named after Hellmuth Kneser, is about the topology of the set of all solutions of an initial value problem with continuous right hand side.
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https://en.wikipedia.org/wiki/Kneser's_theorem_(differential_equations)
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In mathematics, the Kneser–Tits problem, introduced by Tits (1964) based on a suggestion by Martin Kneser, asks whether the Whitehead group W(G,K) of a semisimple simply connected isotropic algebraic group G over a field K is trivial. The Whitehead group is the quotient of the rational points of G by the normal subgroup generated by K-subgroups isomorphic to the additive group.
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https://en.wikipedia.org/wiki/Kneser–Tits_conjecture
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In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials. Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an integral cohomology class, there is a complex-analytic embedding of M into complex projective space of some high enough dimension N. The fact that M embeds as an algebraic variety follows from its compactness by Chow's theorem. A Kähler manifold with a Hodge metric is occasionally called a Hodge manifold (named after W. V. D. Hodge), so Kodaira's results states that Hodge manifolds are projective.
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https://en.wikipedia.org/wiki/Kodaira_embedding_theorem
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The converse that projective manifolds are Hodge manifolds is more elementary and was already known. Kodaira also proved (Kodaira 1963), by recourse to the classification of compact complex surfaces, that every compact Kähler surface is a deformation of a projective Kähler surface. This was later simplified by Buchdahl to remove reliance on the classification (Buchdahl 2008).
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https://en.wikipedia.org/wiki/Kodaira_embedding_theorem
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In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications for the group with index q = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem.
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https://en.wikipedia.org/wiki/Kodaira_vanishing_theorem
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In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X.
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https://en.wikipedia.org/wiki/Kodaira–Spencer_map
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In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representation as dilations of a univalent holomorphic mapping, or a semigroup of mappings, of the unit disk in the complex numbers into itself.
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https://en.wikipedia.org/wiki/Koenigs_function
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In mathematics, the Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols {1,2} that is the sequence of run lengths in its own run-length encoding. It is named after the recreational mathematician William Kolakoski (1944–97) who described it in 1965, but it was previously discussed by Rufus Oldenburger in 1939.
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https://en.wikipedia.org/wiki/Kolakoski_sequence
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In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.
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https://en.wikipedia.org/wiki/Kolmogorov_continuity_theorem
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In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov.
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https://en.wikipedia.org/wiki/Kolmogorov_extension_theorem
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In mathematics, the Konhauser polynomials, introduced by Konhauser (1967), are biorthogonal polynomials for the distribution function of the Laguerre polynomials.
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https://en.wikipedia.org/wiki/Konhauser_polynomials
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In mathematics, the Kontorovich–Lebedev transform is an integral transform which uses a Macdonald function (modified Bessel function of the second kind) with imaginary index as its kernel. Unlike other Bessel function transforms, such as the Hankel transform, this transform involves integrating over the index of the function rather than its argument. The transform of a function ƒ(x) and its inverse (provided they exist) are given below: g ( y ) = ∫ 0 ∞ f ( x ) K i y ( x ) d x {\displaystyle g(y)=\int _{0}^{\infty }f(x)K_{iy}(x)\,dx} f ( x ) = 2 π 2 x ∫ 0 ∞ g ( y ) K i y ( x ) sinh ( π y ) y d y . {\displaystyle f(x)={\frac {2}{\pi ^{2}x}}\int _{0}^{\infty }g(y)K_{iy}(x)\sinh(\pi y)y\,dy.}
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https://en.wikipedia.org/wiki/Kontorovich–Lebedev_transform
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Laguerre previously studied a similar transform regarding Laguerre function as: g ( y ) = ∫ 0 ∞ f ( x ) e − x L y ( x ) d x {\displaystyle g(y)=\int _{0}^{\infty }f(x)e^{-x}L_{y}(x)\,dx} f ( x ) = ∫ 0 ∞ g ( y ) Γ ( y ) L y ( x ) d y . {\displaystyle f(x)=\int _{0}^{\infty }{\frac {g(y)}{\Gamma (y)}}L_{y}(x)\,dy.} Erdélyi et al., for instance, contains a short list of Kontorovich–Lebedev transforms as well references to the original work of Kontorovich and Lebedev in the late 1930s. This transform is mostly used in solving the Laplace equation in cylindrical coordinates for wedge shaped domains by the method of separation of variables.
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https://en.wikipedia.org/wiki/Kontorovich–Lebedev_transform
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In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich.
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https://en.wikipedia.org/wiki/Kontsevich_quantization_formula
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In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviors for an integrable PDE, such as a large number of explicit solutions, in particular soliton solutions, and an infinite number of conserved quantities, despite the nonlinearity which typically renders PDEs intractable. The KdV can be solved by the inverse scattering method (ISM). In fact, Gardner, Greene, Kruskal and Miura developed the classical inverse scattering method to solve the KdV equation. The KdV equation was first introduced by Boussinesq (1877, footnote on page 360) and rediscovered by Diederik Korteweg and Gustav de Vries (1895), who found the simplest solution, the one-soliton solution. Understanding of the equation and behavior of solutions was greatly advanced by the computer simulations of Zabusky and Kruskal in 1965 and then the development of the inverse scattering transform in 1967.
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https://en.wikipedia.org/wiki/Korteweg_de_Vries_equation
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In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the finite reflection group of a root system.
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https://en.wikipedia.org/wiki/Kostant_polynomial
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In mathematics, the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} (depending on two integer partitions λ {\displaystyle \lambda } and μ {\displaystyle \mu } ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ {\displaystyle \lambda } and weight μ {\displaystyle \mu } . They were introduced by the mathematician Carl Kostka in his study of symmetric functions (Kostka (1882)).For example, if λ = ( 3 , 2 ) {\displaystyle \lambda =(3,2)} and μ = ( 1 , 1 , 2 , 1 ) {\displaystyle \mu =(1,1,2,1)} , the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows. The three such tableaux are shown at right, and K ( 3 , 2 ) ( 1 , 1 , 2 , 1 ) = 3 {\displaystyle K_{(3,2)(1,1,2,1)}=3} .
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https://en.wikipedia.org/wiki/Kostka_number
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In mathematics, the Koszul cohomology groups K p , q ( X , L ) {\displaystyle K_{p,q}(X,L)} are groups associated to a projective variety X with a line bundle L. They were introduced by Mark Green (1984, 1984b), and named after Jean-Louis Koszul as they are closely related to the Koszul complex. Green (1989) surveys early work on Koszul cohomology, Eisenbud (2005) gives an introduction to Koszul cohomology, and Aprodu & Nagel (2010) gives a more advanced survey.
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https://en.wikipedia.org/wiki/Koszul_cohomology
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In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an M-regular sequence, and hence it can be used to prove basic facts about the depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of Krull dimension. Moreover, in certain circumstances, the complex is the complex of syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth.
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https://en.wikipedia.org/wiki/Koszul_complex
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In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson brackets: For example, δ 12 = 0 {\displaystyle \delta _{12}=0} because 1 ≠ 2 {\displaystyle 1\neq 2} , whereas δ 33 = 1 {\displaystyle \delta _{33}=1} because 3 = 3 {\displaystyle 3=3} . The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above.
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https://en.wikipedia.org/wiki/Kronecker_delta_function
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In linear algebra, the n × n {\displaystyle n\times n} identity matrix I {\displaystyle \mathbf {I} } has entries equal to the Kronecker delta: where i {\displaystyle i} and j {\displaystyle j} take the values 1 , 2 , ⋯ , n {\displaystyle 1,2,\cdots ,n} , and the inner product of vectors can be written as Here the Euclidean vectors are defined as n-tuples: a = ( a 1 , a 2 , … , a n ) {\displaystyle \mathbf {a} =(a_{1},a_{2},\dots ,a_{n})} and b = ( b 1 , b 2 , . . . , b n ) {\displaystyle \mathbf {b} =(b_{1},b_{2},...,b_{n})} and the last step is obtained by using the values of the Kronecker delta to reduce the summation over j {\displaystyle j} . It is common for i and j to be restricted to a set of the form {1, 2, ..., n} or {0, 1, ..., n − 1}, but the Kronecker delta can be defined on an arbitrary set.
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https://en.wikipedia.org/wiki/Kronecker_delta_function
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In mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian in a rectangular cuboid domain.
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https://en.wikipedia.org/wiki/Kronecker_sum_of_discrete_Laplacians
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In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
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https://en.wikipedia.org/wiki/Krull–Schmidt_theorem
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In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and were named after Russian-Ukrainian mathematicians and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems.
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https://en.wikipedia.org/wiki/Krylov–Bogolyubov_theorem
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In mathematics, the Kubilius model relies on a clarification and extension of a finite probability space on which the behaviour of additive arithmetic functions can be modeled by sum of independent random variables.The method was introduced in Jonas Kubilius's monograph Tikimybiniai metodai skaičių teorijoje (published in Lithuanian in 1959) / Probabilistic Methods in the Theory of Numbers (published in English in 1964) .Eugenijus Manstavičius and Fritz Schweiger wrote about Kubilius's work in 1992, "the most impressive work has been done on the statistical theory of arithmetic functions which almost created a new research area called Probabilistic Number Theory. A monograph (Probabilistic Methods in the Theory of Numbers) devoted to this topic was translated into English in 1964 and became very influential. ": xi
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https://en.wikipedia.org/wiki/Kubilius_model
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In mathematics, the Kummer variety of an abelian variety is its quotient by the map taking any element to its inverse. The Kummer variety of a 2-dimensional abelian variety is called a Kummer surface.
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https://en.wikipedia.org/wiki/Kummer_variety
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In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number hK of the maximal real subfield K = Q ( ζ p ) + {\displaystyle K=\mathbb {Q} (\zeta _{p})^{+}} of the p-th cyclotomic field. The conjecture was first made by Ernst Kummer on 28 December 1849 and 24 April 1853 in letters to Leopold Kronecker, reprinted in (Kummer 1975, pages 84, 93, 123–124), and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver (1946, p. 576), As of 2011, there is no particularly strong evidence either for or against the conjecture and it is unclear whether it is true or false, though it is likely that counterexamples are very rare.
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https://en.wikipedia.org/wiki/Vandiver_conjecture
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In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation or flame equation) is a fourth-order nonlinear partial differential equation. It is named after Yoshiki Kuramoto and Gregory Sivashinsky, who derived the equation in the late 1970s to model the diffusive–thermal instabilities in a laminar flame front. The Kuramoto–Sivashinsky equation is known for its chaotic behavior.
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https://en.wikipedia.org/wiki/Kuramoto–Sivashinsky_equation
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In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.Many classical selection results follow from this theorem and it is widely used in mathematical economics and optimal control.
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https://en.wikipedia.org/wiki/Kuratowski_and_Ryll-Nardzewski_measurable_selection_theorem
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In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam (1932), called also the Fubini theorem for category, is an analog of Fubini's theorem for arbitrary second countable Baire spaces. Let X and Y be second countable Baire spaces (or, in particular, Polish spaces), and let A ⊂ X × Y {\displaystyle A\subset X\times Y} . Then the following are equivalent if A has the Baire property: A is meager (respectively comeager). The set { x ∈ X: A x is meager (resp.
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https://en.wikipedia.org/wiki/Kuratowski–Ulam_theorem
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comeager) in Y } {\displaystyle \{x\in X:A_{x}{\text{ is meager (resp. comeager) in }}Y\}} is comeager in X, where A x = π Y {\displaystyle A_{x}=\pi _{Y}} , where π Y {\displaystyle \pi _{Y}} is the projection onto Y.Even if A does not have the Baire property, 2. follows from 1. Note that the theorem still holds (perhaps vacuously) for X an arbitrary Hausdorff space and Y a Hausdorff space with countable π-base. The theorem is analogous to the regular Fubini's theorem for the case where the considered function is a characteristic function of a subset in a product space, with the usual correspondences, namely, meagre set with a set of measure zero, comeagre set with one of full measure, and a set with the Baire property with a measurable set.
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https://en.wikipedia.org/wiki/Kuratowski–Ulam_theorem
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In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexamples. These matters were brought up by Aleksandr Gennadievich Kurosh as analogues of the Burnside problem in group theory. Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen).
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https://en.wikipedia.org/wiki/Kurosh_problem
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A special case is whether or not every nil algebra is locally nilpotent. For PI-algebras the Kurosh problem has a positive solution. Golod showed a counterexample to that case, as an application of the Golod–Shafarevich theorem.
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https://en.wikipedia.org/wiki/Kurosh_problem
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The Kurosh problem on group algebras concerns the augmentation ideal I. If I is a nil ideal, is the group algebra locally nilpotent? There is an important problem which is often referred as the Kurosh's problem on division rings. The problem asks whether there exists an algebraic (over the center) division ring which is not locally finite. This problem has not been solved until now.
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https://en.wikipedia.org/wiki/Kurosh_problem
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In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2022. It is formulated in various ways. Suppose that R is a ring.
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https://en.wikipedia.org/wiki/Köthe_conjecture
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One way to state the conjecture is that if R has no nil ideal, other than {0}, then it has no nil one-sided ideal, other than {0}. This question was posed in 1930 by Gottfried Köthe (1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings and right Noetherian rings, but a general solution remains elusive.
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https://en.wikipedia.org/wiki/Köthe_conjecture
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In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural.
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https://en.wikipedia.org/wiki/Functional_equation_(L-function)
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In mathematics, the Labs septic surface is a degree-7 (septic) nodal surface with 99 nodes found by Labs (2006). As of 2015, it has the largest known number of nodes of a degree-7 surface, though this number is still less than the best known upper bound of 104 nodes given by Varchenko (1983).
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https://en.wikipedia.org/wiki/Labs_septic
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In mathematics, the Lagrange numbers are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem.
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https://en.wikipedia.org/wiki/Lagrange_number
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In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let v be a function of x and y in terms of another function f such that v = x + y f ( v ) {\displaystyle v=x+yf(v)} Then for any function g, for small enough y: g ( v ) = g ( x ) + ∑ k = 1 ∞ y k k ! ( ∂ ∂ x ) k − 1 ( f ( x ) k g ′ ( x ) ) .
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https://en.wikipedia.org/wiki/Lagrange_reversion_theorem
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{\displaystyle g(v)=g(x)+\sum _{k=1}^{\infty }{\frac {y^{k}}{k! }}\left({\frac {\partial }{\partial x}}\right)^{k-1}\left(f(x)^{k}g'(x)\right).} If g is the identity, this becomes v = x + ∑ k = 1 ∞ y k k !
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https://en.wikipedia.org/wiki/Lagrange_reversion_theorem
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( ∂ ∂ x ) k − 1 ( f ( x ) k ) {\displaystyle v=x+\sum _{k=1}^{\infty }{\frac {y^{k}}{k! }}\left({\frac {\partial }{\partial x}}\right)^{k-1}\left(f(x)^{k}\right)} In which case the equation can be derived using perturbation theory. In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation for v mentioned above. However, his solution used cumbersome series expansions of logarithms. In 1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, which was based on relations between partial derivatives with respect to the variable x and the parameter y. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration.Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation.
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https://en.wikipedia.org/wiki/Lagrange_reversion_theorem
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In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of V is 2n). It may be identified with the homogeneous space U(n)/O(n),where U(n) is the unitary group and O(n) the orthogonal group. Following Vladimir Arnold it is denoted by Λ(n). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V. A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space V of dimension 2n. It may be identified with the homogeneous space of complex dimension 1/2n(n + 1) Sp(n)/U(n),where Sp(n) is the compact symplectic group.
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https://en.wikipedia.org/wiki/Lagrangian_Grassmannian
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In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bundles (covariant classical field theory). The variational bicomplex is a cochain complex of the differential graded algebra of exterior forms on jet manifolds of sections of a fiber bundle. Lagrangians and Euler–Lagrange operators on a fiber bundle are defined as elements of this bicomplex. Cohomology of the variational bicomplex leads to the global first variational formula and first Noether's theorem. Extended to Lagrangian theory of even and odd fields on graded manifolds, the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian BRST theory.
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https://en.wikipedia.org/wiki/Variational_bicomplex
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In mathematics, the Laguerre form is generally given as a third degree tensor-valued form, that can be written as, L = ( w 1 ) 2 D a 11 + 2 w 1 w 2 D a 12 + ( w 2 ) 2 D a 22 {\displaystyle {\mathfrak {L}}=(w^{1})^{2}Da_{11}+2w^{1}w^{2}Da_{12}+(w^{2})^{2}Da_{22}} .
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https://en.wikipedia.org/wiki/Laguerre_form
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In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of where n is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin).
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https://en.wikipedia.org/wiki/Laguerre_functions
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More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form These polynomials, usually denoted L0, L1, …, are a polynomial sequence which may be defined by the Rodrigues formula, reducing to the closed form of a following section. They are orthogonal polynomials with respect to an inner product The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables.
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https://en.wikipedia.org/wiki/Laguerre_functions
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Further see the Tricomi–Carlitz polynomials. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space.
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https://en.wikipedia.org/wiki/Laguerre_functions
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They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator. Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)
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https://en.wikipedia.org/wiki/Laguerre_functions
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In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function f(w) = wew, where w is any complex number and ew is the exponential function. For each integer k there is one branch, denoted by Wk(z), which is a complex-valued function of one complex argument. W0 is known as the principal branch. These functions have the following property: if z and w are any complex numbers, then w e w = z {\displaystyle we^{w}=z} holds if and only if w = W k ( z ) for some integer k .
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https://en.wikipedia.org/wiki/Lambert_W-function
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{\displaystyle w=W_{k}(z)\ \ {\text{ for some integer }}k.} When dealing with real numbers only, the two branches W0 and W−1 suffice: for real numbers x and y the equation y e y = x {\displaystyle ye^{y}=x} can be solved for y only if x ≥ −1/e; we get y = W0(x) if x ≥ 0 and the two values y = W0(x) and y = W−1(x) if −1/e ≤ x < 0.
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https://en.wikipedia.org/wiki/Lambert_W-function
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The Lambert W relation cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance, in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y′(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, an opened-form solution for the time-course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
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https://en.wikipedia.org/wiki/Lambert_W-function
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In mathematics, the Lanczos approximation is a method for computing the gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical alternative to the more popular Stirling's approximation for calculating the gamma function with fixed precision.
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https://en.wikipedia.org/wiki/Lanczos_approximation
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In mathematics, the Landau–Kolmogorov inequality, named after Edmund Landau and Andrey Kolmogorov, is the following family of interpolation inequalities between different derivatives of a function f defined on a subset T of the real numbers: ‖ f ( k ) ‖ L ∞ ( T ) ≤ C ( n , k , T ) ‖ f ‖ L ∞ ( T ) 1 − k / n ‖ f ( n ) ‖ L ∞ ( T ) k / n for 1 ≤ k < n . {\displaystyle \|f^{(k)}\|_{L_{\infty }(T)}\leq C(n,k,T){\|f\|_{L_{\infty }(T)}}^{1-k/n}{\|f^{(n)}\|_{L_{\infty }(T)}}^{k/n}{\text{ for }}1\leq k
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https://en.wikipedia.org/wiki/Landau–Kolmogorov_inequality
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In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law. The Landweber exact functor theorem (or LEFT for short) can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.
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https://en.wikipedia.org/wiki/Landweber_exact_functor_theorem
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In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product P = M A N {\displaystyle P=MAN} of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.
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https://en.wikipedia.org/wiki/Langlands_decomposition
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In mathematics, the Langlands–Deligne local constant, also known as the local epsilon factor or local Artin root number (up to an elementary real function of s), is an elementary function associated with a representation of the Weil group of a local field. The functional equation L(ρ,s) = ε(ρ,s)L(ρ∨,1−s)of an Artin L-function has an elementary function ε(ρ,s) appearing in it, equal to a constant called the Artin root number times an elementary real function of s, and Langlands discovered that ε(ρ,s) can be written in a canonical way as a product ε(ρ,s) = Π ε(ρv, s, ψv)of local constants ε(ρv, s, ψv) associated to primes v. Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis. Dwork (1956) proved the existence of the local constant ε(ρv, s, ψv) up to sign. The original proof of the existence of the local constants by Langlands (1970) used local methods and was rather long and complicated, and never published. Deligne (1973) later discovered a simpler proof using global methods.
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https://en.wikipedia.org/wiki/Langlands–Deligne_local_constant
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In mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number field. This includes Rankin–Selberg products for cuspidal automorphic representations of general linear groups. The method develops the theory of the local coefficient, which links to the global theory via Eisenstein series. The resulting L-functions satisfy a number of analytic properties, including an important functional equation.
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https://en.wikipedia.org/wiki/Langlands–Shahidi_method
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In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges. It is approximately 0.66274 34193 49181 58097 47420 97109 25290.Kepler's equation M = E − ε sin E relates the mean anomaly M with the eccentric anomaly E for a body moving in an ellipse with eccentricity ε. This equation cannot be solved for E in terms of elementary functions, but the Lagrange reversion theorem gives the solution as a power series in ε: E = M + sin ( M ) ε + 1 2 sin ( 2 M ) ε 2 + ( 3 8 sin ( 3 M ) − 1 8 sin ( M ) ) ε 3 + ⋯ {\displaystyle E=M+\sin(M)\,\varepsilon +{\tfrac {1}{2}}\sin(2M)\,\varepsilon ^{2}+\left({\tfrac {3}{8}}\sin(3M)-{\tfrac {1}{8}}\sin(M)\right)\,\varepsilon ^{3}+\cdots } or in general E = M + ∑ n = 1 + ∞ ε n 2 n − 1 n ! ∑ k = 0 ⌊ n / 2 ⌋ ( − 1 ) k ( n k ) ( n − 2 k ) n − 1 sin ( ( n − 2 k ) M ) {\displaystyle E=M\;+\;\sum _{n=1}^{+\infty }{\frac {\varepsilon ^{n}}{2^{n-1}\,n! }}\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}\,{\binom {n}{k}}\,(n-2k)^{n-1}\,\sin((n-2k)\,M)} Laplace realized that this series converges for small values of the eccentricity, but diverges for any value of M other than a multiple of π if the eccentricity exceeds a certain value that does not depend on M. The Laplace limit is this value.
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https://en.wikipedia.org/wiki/Laplace_limit
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It is the radius of convergence of the power series. It is given by the solution to the transcendental equation x exp ( 1 + x 2 ) 1 + 1 + x 2 = 1. {\displaystyle {\frac {x\exp({\sqrt {1+x^{2}}})}{1+{\sqrt {1+x^{2}}}}}=1.} No closed-form expression or infinite series is known for the Laplace limit.
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https://en.wikipedia.org/wiki/Laplace_limit
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In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ∇ ⋅ ∇ {\displaystyle \nabla \cdot \nabla } , ∇ 2 {\displaystyle \nabla ^{2}} (where ∇ {\displaystyle \nabla } is the nabla operator), or Δ {\displaystyle \Delta } . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form.
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https://en.wikipedia.org/wiki/Laplace_operator
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Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p). The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.
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https://en.wikipedia.org/wiki/Laplace_operator
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The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and fluid flow; the wave equation describes wave propagation; and the Schrödinger equation describes the wave function in quantum mechanics. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology.
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https://en.wikipedia.org/wiki/Laplace_operator
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Note that if the initial conditions are all zero, i.e. f ( i ) ( 0 ) = c i = 0 ∀ i ∈ { 0 , 1 , 2 , . . . n } {\displaystyle f^{(i)}(0)=c_{i}=0\quad \forall i\in \{0,1,2,...\ n\}} then the formula simplifies to f ( t ) = L − 1 { L { ϕ ( t ) } ∑ i = 0 n a i s i } {\displaystyle f(t)={\mathcal {L}}^{-1}\left\{{{\mathcal {L}}\{\phi (t)\} \over \sum _{i=0}^{n}a_{i}s^{i}}\right\}}
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https://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations
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In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t {\displaystyle t} , in the time domain) to a function of a complex variable s {\displaystyle s} (in the complex frequency domain, also known as s-domain, or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions f, the Laplace transform is the integral
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https://en.wikipedia.org/wiki/Complex_frequency
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In mathematics, the Laplace–Carson transform, named after Pierre Simon Laplace and John Renshaw Carson, is an integral transform with significant applications in the field of physics and engineering, particularly in the field of railway engineering.
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https://en.wikipedia.org/wiki/Laplace–Carson_transform
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In mathematics, the Laplacian of the indicator of the domain D is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the surface of D. It can be viewed as the surface delta prime function. It is analogous to the second derivative of the Heaviside step function in one dimension. It can be obtained by letting the Laplace operator work on the indicator function of some domain D. The Laplacian of the indicator can be thought of as having infinitely positive and negative values when evaluated very near the boundary of the domain D. From a mathematical viewpoint, it is not strictly a function but a generalized function or measure. Similarly to the derivative of the Dirac delta function in one dimension, the Laplacian of the indicator only makes sense as a mathematical object when it appears under an integral sign; i.e. it is a distribution function. Just as in the formulation of distribution theory, it is in practice regarded as a limit of a sequence of smooth functions; one may meaningfully take the Laplacian of a bump function, which is smooth by definition, and let the bump function approach the indicator in the limit.
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https://en.wikipedia.org/wiki/Laplacian_of_the_indicator
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In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by Emanuel Lasker (1905) for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether (1921). The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings.
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https://en.wikipedia.org/wiki/Primary_submodule
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The theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case, considering the ring as a module over itself, so that ideals are submodules.
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https://en.wikipedia.org/wiki/Primary_submodule
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This also generalizes the primary decomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of (irreducible) varieties. The first algorithm for computing primary decompositions for polynomial rings over a field of characteristic 0 was published by Noether's student Grete Hermann (1926). The decomposition does not hold in general for non-commutative Noetherian rings. Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals.
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https://en.wikipedia.org/wiki/Primary_submodule
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In mathematics, the Laurent series of a complex function f ( z ) {\displaystyle f(z)} is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.
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https://en.wikipedia.org/wiki/Laurent_power_series
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In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by Lebedev and Milin (1965) and Isaak Moiseevich Milin (1977). It was used in the proof of the Bieberbach conjecture, as it shows that the Milin conjecture implies the Robertson conjecture. They state that if ∑ k ≥ 0 β k z k = exp ( ∑ k ≥ 1 α k z k ) {\displaystyle \sum _{k\geq 0}\beta _{k}z^{k}=\exp \left(\sum _{k\geq 1}\alpha _{k}z^{k}\right)} for complex numbers β k {\displaystyle \beta _{k}} and α k {\displaystyle \alpha _{k}} , and n {\displaystyle n} is a positive integer, then ∑ k = 0 ∞ | β k | 2 ≤ exp ( ∑ k = 1 ∞ k | α k | 2 ) , {\displaystyle \sum _{k=0}^{\infty }|\beta _{k}|^{2}\leq \exp \left(\sum _{k=1}^{\infty }k|\alpha _{k}|^{2}\right),} ∑ k = 0 n | β k | 2 ≤ ( n + 1 ) exp ( 1 n + 1 ∑ m = 1 n ∑ k = 1 m ( k | α k | 2 − 1 / k ) ) , {\displaystyle \sum _{k=0}^{n}|\beta _{k}|^{2}\leq (n+1)\exp \left({\frac {1}{n+1}}\sum _{m=1}^{n}\sum _{k=1}^{m}(k|\alpha _{k}|^{2}-1/k)\right),} | β n | 2 ≤ exp ( ∑ k = 1 n ( k | α k | 2 − 1 / k ) ) .
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https://en.wikipedia.org/wiki/Lebedev–Milin_inequality
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{\displaystyle |\beta _{n}|^{2}\leq \exp \left(\sum _{k=1}^{n}(k|\alpha _{k}|^{2}-1/k)\right).} See also exponential formula (on exponentiation of power series). == References ==
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https://en.wikipedia.org/wiki/Lebedev–Milin_inequality
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In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are fixed). The Lebesgue constant for polynomials of degree at most n and for the set of n + 1 nodes T is generally denoted by Λn(T ). These constants are named after Henri Lebesgue.
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https://en.wikipedia.org/wiki/Lebesgue_constant_(interpolation)
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In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.
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https://en.wikipedia.org/wiki/Topological_dimension
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In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for Henri Lebesgue.
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https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem
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In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by John Leech (1967). It may also have been discovered (but not published) by Ernst Witt in 1940.
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https://en.wikipedia.org/wiki/Leech_lattice
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In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X {\displaystyle X} to itself by means of traces of the induced mappings on the homology groups of X {\displaystyle X} . It is named after Solomon Lefschetz, who first stated it in 1926. The counting is subject to an imputed multiplicity at a fixed point called the fixed-point index. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).
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https://en.wikipedia.org/wiki/Lefschetz–Hopf_theorem
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In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a continuous map f: X → X {\displaystyle f\colon X\to X} , the zeta-function is defined as the formal series ζ f ( t ) = exp ( ∑ n = 1 ∞ L ( f n ) t n n ) , {\displaystyle \zeta _{f}(t)=\exp \left(\sum _{n=1}^{\infty }L(f^{n}){\frac {t^{n}}{n}}\right),} where L ( f n ) {\displaystyle L(f^{n})} is the Lefschetz number of the n {\displaystyle n} -th iterate of f {\displaystyle f} . This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of f {\displaystyle f} .
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https://en.wikipedia.org/wiki/Lefschetz_zeta_function
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In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by As such, it resembles the Dirichlet series for the polylogarithm, and, indeed, is trivially expressible in terms of the polylogarithm as The Legendre chi function appears as the discrete Fourier transform, with respect to the order ν, of the Hurwitz zeta function, and also of the Euler polynomials, with the explicit relationships given in those articles. The Legendre chi function is a special case of the Lerch transcendent, and is given by
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https://en.wikipedia.org/wiki/Legendre_chi_function
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In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity k {\displaystyle \scriptstyle {k}} (the ellipse being defined parametrically by x = 1 − k 2 cos ( t ) {\displaystyle \scriptstyle {x={\sqrt {1-k^{2}}}\cos(t)}} , y = sin ( t ) {\displaystyle \scriptstyle {y=\sin(t)}} ). In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.
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https://en.wikipedia.org/wiki/Legendre_form
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In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory. It applies the concept of the Sieve of Eratosthenes to find upper or lower bounds on the number of primes within a given set of integers. Because it is a simple extension of Eratosthenes' idea, it is sometimes called the Legendre–Eratosthenes sieve.
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https://en.wikipedia.org/wiki/Legendre_sieve
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