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c_j9tv0np0hmxp | 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 name... | Kervaire invariant |
c_k08fnwao2ew3 | 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... | Kervaire invariant |
c_xdjy8zkv5dws | 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. | Kervaire invariant |
c_aolsincnalk7 | 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)=\su... | Kervaire semi-characteristic |
c_j8ek5rzw599t | 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 t... | Face-splitting product |
c_9uupxwbf1ic0 | 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... | Khinchin integral |
c_fjvfrqmvg3o6 | 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... | Khintchine inequality |
c_o241vm2v5s99 | 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 Li... | Killing form |
c_osteq39ckkzx | 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 resp... | Kirby calculus |
c_rvoip1qdm3qh | 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... | Kirby calculus |
c_5lc7109w1tsi | This allows an extension of the Kirby calculus to rational surgeries. There are also various tricks to modify surgery diagrams. | Kirby calculus |
c_bbo1uj3m9q9f | 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. | Kirby calculus |
c_faywu596nl2m | (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 bou... | Kirby calculus |
c_j5qhsre3z0tk | 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... | Kleene–Rosser paradox |
c_cmrtve8f9359 | 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... | Klein bottle |
c_cjv4qdmn1n7x | 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. | Klein bottle |
c_4ki9o7jr01jz | 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 rectangl... | Klein 4-group |
c_htmrgqcixjym | 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. | Klein 4-group |
c_qfidnzdiel4e | 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 ... | Kneser's theorem (differential equations) |
c_1fvca81exwaz | 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 subgrou... | Kneser–Tits conjecture |
c_h3geqreqlqr2 | 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... | Kodaira embedding theorem |
c_5qzyp15idufx | 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... | Kodaira embedding theorem |
c_i2pbczmsl2i9 | 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 numb... | Kodaira vanishing theorem |
c_nma406zqz2q5 | 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. | Kodaira–Spencer map |
c_4u0i1afgx38z | 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 i... | Koenigs function |
c_557y44bqup8a | 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 w... | Kolakoski sequence |
c_n9uto8783eqa | 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. | Kolmogorov continuity theorem |
c_ng773f4zivll | 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... | Kolmogorov extension theorem |
c_byzyjjyy1klh | In mathematics, the Konhauser polynomials, introduced by Konhauser (1967), are biorthogonal polynomials for the distribution function of the Laguerre polynomials. | Konhauser polynomials |
c_eek26lheifcr | 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 fu... | Kontorovich–Lebedev transform |
c_joi8ts5s18tx | 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.} Erdél... | Kontorovich–Lebedev transform |
c_8laqk5dqpo91 | 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. | Kontsevich quantization formula |
c_6ofjpd6wlqev | 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 ... | Korteweg de Vries equation |
c_w5vf293xpsyn | 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. | Kostant polynomial |
c_kex2c5ey5jz4 | 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 }... | Kostka number |
c_2a6vxen14cqi | 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 ea... | Koszul cohomology |
c_v9rv4yhc2c18 | 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 ... | Koszul complex |
c_iuicxyd43875 | 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 ... | Kronecker delta function |
c_j9uy426lfinw | 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 Eucl... | Kronecker delta function |
c_jtva8xjy51o3 | 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. | Kronecker sum of discrete Laplacians |
c_10ljanmzdq81 | 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. | Krull–Schmidt theorem |
c_m5rec2c6th7l | 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 we... | Krylov–Bogolyubov theorem |
c_tj5px01f4ce8 | 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... | Kubilius model |
c_nfohhwspjecy | 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. | Kummer variety |
c_ncpt4134a7n6 | 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 an... | Vandiver conjecture |
c_soi5r1h5gqrv | 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 fl... | Kuramoto–Sivashinsky equation |
c_47og4jhzk7zk | 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 sel... | Kuratowski and Ryll-Nardzewski measurable selection theorem |
c_7kmnd3x34645 | 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... | Kuratowski–Ulam theorem |
c_g04m7pavuscj | 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 (per... | Kuratowski–Ulam theorem |
c_54gvrs7fqvta | 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 t... | Kurosh problem |
c_vmatzkkdl8os | 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. | Kurosh problem |
c_k7wsd5v52msc | 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 i... | Kurosh problem |
c_mjaeuwf52m5h | 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. | Köthe conjecture |
c_98tgdi7l0oxg | 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 Noet... | Köthe conjecture |
c_wurqlsqza3ip | 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. | Functional equation (L-function) |
c_e9onoeyykgle | 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). | Labs septic |
c_eodnirvs4u2e | 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. | Lagrange number |
c_v13qym24e3fj | 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,... | Lagrange reversion theorem |
c_6rdj5hmrnlww | {\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 ! | Lagrange reversion theorem |
c_efs730qyipij | ( ∂ ∂ 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 im... | Lagrange reversion theorem |
c_e6t2oojwdqfg | 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. Followin... | Lagrangian Grassmannian |
c_yz2pe777pa6k | 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 coc... | Variational bicomplex |
c_33ux81o94t87 | 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}} . | Laguerre form |
c_9j7moqkivggd | 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 u... | Laguerre functions |
c_m5uoroj6wwtp | 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 fo... | Laguerre functions |
c_ybj7vbks1utx | 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. | Laguerre functions |
c_f2lkzb9cseiq | 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-c... | Laguerre functions |
c_r7xxftdcp4o0 | 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), whi... | Lambert W-function |
c_hxn5worhj5zr | {\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... | Lambert W-function |
c_oiiec3idk5c3 | 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 sol... | Lambert W-function |
c_8mrojfqxhrfi | 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. | Lanczos approximation |
c_sb9sr28bt0et | 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 ) ‖ ... | Landau–Kolmogorov inequality |
c_id2jywsc4cl1 | 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 const... | Landweber exact functor theorem |
c_syqkeeuvg3w5 | 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. | Langlands decomposition |
c_rgkizodorv4b | 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-f... | Langlands–Deligne local constant |
c_i73m0h4hh4il | 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 l... | Langlands–Shahidi method |
c_2wle9crq0agm | 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... | Laplace limit |
c_bd9hmhihrzr1 | 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. | Laplace limit |
c_sn308uy3rdae | 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 op... | Laplace operator |
c_vypubkpyk1d7 | 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 celest... | Laplace operator |
c_f0urwjzbg3s5 | 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 m... | Laplace operator |
c_vqa0e4jz25e4 | 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 {... | Laplace transform applied to differential equations |
c_o5p226t2vhjj | 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-dom... | Complex frequency |
c_rqtk25oezvsm | 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. | Laplace–Carson transform |
c_nnk025y54sr3 | 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 i... | Laplacian of the indicator |
c_0ropqs2nokws | 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... | Primary submodule |
c_rfeveg134e1i | 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 intersec... | Primary submodule |
c_29olr4fsxa3u | 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 ... | Primary submodule |
c_h2vfvgrtd5f5 | 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... | Laurent power series |
c_05yd3p14tkai | 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 c... | Lebedev–Milin inequality |
c_o7gnvoojxne7 | {\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 == | Lebedev–Milin inequality |
c_a9forvika65h | 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 deg... | Lebesgue constant (interpolation) |
c_qvvycb5vqvnr | 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. | Topological dimension |
c_4bluf5u1xh3u | 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. | Lebesgue differentiation theorem |
c_8vbq35r7waam | 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. | Leech lattice |
c_p81yya90l9h4 | 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 sta... | Lefschetz–Hopf theorem |
c_y4nvrz0kaceb | 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}... | Lefschetz zeta function |
c_i3n4q8vrlhrw | 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 tran... | Legendre chi function |
c_0dh81kzmdzbk | 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}} (... | Legendre form |
c_q9t9bp1fy34a | 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 so... | Legendre sieve |
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