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wiki_6000_chunk_0 | Laguerre functions | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6001_chunk_0 | Laguerre functions | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6002_chunk_0 | Lambert W-function | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6003_chunk_0 | Lambert W-function | {\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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6004_chunk_0 | Lambert W-function | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6005_chunk_0 | Lanczos approximation | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6006_chunk_0 | Landau–Kolmogorov inequality | 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 ) ‖ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6007_chunk_0 | Landweber exact functor theorem | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6008_chunk_0 | Langlands–Deligne local constant | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6009_chunk_0 | Langlands–Shahidi method | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6010_chunk_0 | Laplace limit | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6011_chunk_0 | Laplace limit | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6012_chunk_0 | Laplace operator | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6013_chunk_0 | Laplace operator | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6014_chunk_0 | Laplace operator | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6015_chunk_0 | Laplace transform applied to differential equations | 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 {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6016_chunk_0 | Complex frequency | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6017_chunk_0 | Laplace–Carson transform | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6018_chunk_0 | Laplacian of the indicator | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6019_chunk_0 | Primary submodule | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6020_chunk_0 | Primary submodule | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6021_chunk_0 | Primary submodule | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6022_chunk_0 | Laurent power series | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6023_chunk_0 | Lebedev–Milin inequality | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6024_chunk_0 | Lebedev–Milin inequality | {\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 == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6025_chunk_0 | Lebesgue constant (interpolation) | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6026_chunk_0 | Lebesgue differentiation theorem | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6027_chunk_0 | Lefschetz–Hopf theorem | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6028_chunk_0 | Lefschetz zeta function | 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}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6029_chunk_0 | Legendre chi function | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6030_chunk_0 | Legendre transformation | In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity (s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6031_chunk_0 | Lehmer mean | In mathematics, the Lehmer mean of a tuple x {\displaystyle x} of positive real numbers, named after Derrick Henry Lehmer, is defined as: L p ( x ) = ∑ k = 1 n x k p ∑ k = 1 n x k p − 1 . {\displaystyle L_{p}(\mathbf {x} )={\frac {\sum _{k=1}^{n}x_{k}^{p}}{\sum _{k=1}^{n}x_{k}^{p-1}}}.} The weighted Lehmer mean with re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6032_chunk_0 | Lehmer–Schur algorithm | In mathematics, the Lehmer–Schur algorithm (named after Derrick Henry Lehmer and Issai Schur) is a root-finding algorithm for complex polynomials, extending the idea of enclosing roots like in the one-dimensional bisection method to the complex plane. It uses the Schur-Cohn test to test increasingly smaller disks for t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6033_chunk_0 | Leibniz series | In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that an alternating series. It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), and was la... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6034_chunk_0 | Leray spectral sequence | In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6035_chunk_0 | Leray–Hirsch theorem | In mathematics, the Leray–Hirsch theorem is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s. It can be thought of as a mild generalization of the Künneth formula, which computes the cohomology of a product space as a t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6036_chunk_0 | Leray–Schauder degree | In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres ( S n , ∗ ) → ( S n , ∗ ) {\displaystyle (S^{n},*)\to (S^{n},*)} or equivalently to a boundary sphere preserving continuous maps between balls ( B n , S n − 1 ) → ( B n , S n − 1 ) {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6037_chunk_0 | Lerch zeta function | In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6038_chunk_0 | Levi-Civita field | In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member a {\displaystyle a} can be constructed as a formal series of the form a = ∑ q ∈ Q a q ε q , {\displaystyle a=\sum _{q\in \mat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6039_chunk_0 | Lickorish–Wallace theorem | In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted. The theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6040_chunk_0 | Lickorish–Wallace theorem | Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable automorphism of a closed orientable surface is generated by Dehn twists along 3g − 1 specific simple closed curves in the surface, where g denotes the genus of the surface. Wallace's proof was more general and involved adding hand... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6041_chunk_0 | E7½ (Lie algebra) | In mathematics, the Lie algebra E7½ is a subalgebra of E8 containing E7 defined by Landsberg and Manivel in order to fill the "hole" in a dimension formula for the exceptional series En of simple Lie algebras. This hole was observed by Cvitanovic, Deligne, Cohen and de Man. E7½ has dimension 190, and is not simple: as ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6042_chunk_0 | Lie product formula | In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula, named after Hale Trotter, states that for arbitrary m × m real or complex matrices A and B, e A + B = lim n → ∞ ( e A / n e B / n ) n , {\displaystyle e^{A+B}=\lim _{n\rightarrow \infty }(e^{A/n}e^{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6043_chunk_0 | Lie product formula | The formula has applications, for example, in the path integral formulation of quantum mechanics. It allows one to separate the Schrödinger evolution operator (propagator) into alternating increments of kinetic and potential operators (the Suzuki–Trotter decomposition, after Trotter and Masuo Suzuki). The same idea is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6044_chunk_0 | Lie–Kolchin theorem | In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and ρ: G → G L ( V ) {\displaystyle \rho \c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6045_chunk_0 | Lie–Kolchin theorem | This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all ρ ( g ) , g ∈ G {\displaystyle \rho (g),\,\,g\in G} . It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6046_chunk_0 | Lindelöf hypothesis | In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see Lindelöf (1908)) about the rate of growth of the Riemann zeta function on the critical line. This hypothesis is implied by the Riemann hypothesis. It says that for any ε > 0, as t tends to infinity (see big O no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6047_chunk_0 | Lions–Lax–Milgram theorem | In mathematics, the Lions–Lax–Milgram theorem (or simply Lions's theorem) is a result in functional analysis with applications in the study of partial differential equations. It is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear function can be "inverted" to show the ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6048_chunk_0 | Lions–Magenes lemma | In mathematics, the Lions–Magenes lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a criterion for moving a time derivative of a function out of its action (as a functional) on the function itself. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6049_chunk_0 | Liouville–Neumann series | In mathematics, the Liouville–Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6050_chunk_0 | Liouvillian function | In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions. More explicitly, a Liouvillian function is a function of one variable which is the composit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6051_chunk_0 | Liouvillian function | The logarithm function does not need to be explicitly included since it is the integral of 1 / x {\displaystyle 1/x} . It follows directly from the definition that the set of Liouvillian functions is closed under arithmetic operations, composition, and integration. It is also closed under differentiation. It is not clo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6052_chunk_0 | Path model | In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody algebras. Its most important application is to complex semisimple Lie algebras or equivalently compact semisimple Lie grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6053_chunk_0 | Littlewood subordination theorem | In mathematics, the Littlewood subordination theorem, proved by J. E. Littlewood in 1925, is a theorem in operator theory and complex analysis. It states that any holomorphic univalent self-mapping of the unit disk in the complex numbers that fixes 0 induces a contractive composition operator on various function spaces... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6054_chunk_0 | Littlewood-Richardson coefficient | In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6055_chunk_0 | Loewner equation | In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (conformal mappings of the open disk onto the complex plane with a curve ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6056_chunk_0 | Loewner equation | The Loewner semigroup generalizes the notion of a univalent semigroup. The Loewner differential equation has led to inequalities for univalent functions that played an important role in the solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner himself used his techniques in 1923 for proving the con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6057_chunk_0 | Loomis-Whitney inequality | In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a d {\displaystyle d} -dimensional set by the sizes of its ( d − 1 ) {\displaystyle (d-1)} -dimensional projections. The inequality has applications in incidence geometry, the study of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6058_chunk_0 | L1 space | In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6059_chunk_0 | Lumer–Phillips theorem | In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6060_chunk_0 | Lusternik–Schnirelmann theorem | In mathematics, the Lusternik–Schnirelmann theorem, aka Lusternik–Schnirelmann–Borsuk theorem or LSB theorem, says as follows. If the sphere Sn is covered by n + 1 closed sets, then one of these sets contains a pair (x, −x) of antipodal points. It is named after Lazar Lyusternik and Lev Schnirelmann, who published it i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6061_chunk_0 | Lyapunov exponents | In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector δ Z 0 {\displaystyle \delta \mathbf {Z} _{0}} dive... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6062_chunk_0 | Lyapunov exponents | Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the sy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6063_chunk_0 | Lyapunov-Schmidt reduction | In mathematics, the Lyapunov–Schmidt reduction or Lyapunov–Schmidt construction is used to study solutions to nonlinear equations in the case when the implicit function theorem does not work. It permits the reduction of infinite-dimensional equations in Banach spaces to finite-dimensional equations. It is named after A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6064_chunk_0 | Lyusternik–Fet theorem | In mathematics, the Lyusternik–Fet theorem states that on every compact Riemannian manifold there exists a closed geodesic. It is named after Lazar Lyusternik and Abram Ilyich Fet. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6065_chunk_0 | Lusternik–Schnirelmann category | In the modern normalization, the cup-length is a lower bound for the LS-category. It was, as originally defined for the case of X {\displaystyle X} a manifold, the lower bound for the number of critical points that a real-valued function on X {\displaystyle X} could possess (this should be compared with the result in M... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6066_chunk_0 | Lévy metric | In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6067_chunk_0 | Lévy–Steinitz theorem | In mathematics, the Lévy–Steinitz theorem identifies the set of values to which rearrangements of an infinite series of vectors in Rn can converge. It was proved by Paul Lévy in his first published paper when he was 19 years old. In 1913 Ernst Steinitz filled in a gap in Lévy's proof and also proved the result by a dif... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6068_chunk_0 | Maass–Selberg relations | In mathematics, the Maass–Selberg relations are some relations describing the inner products of truncated real analytic Eisenstein series, that in some sense say that distinct Eisenstein series are orthogonal. Hans Maass introduced the Maass–Selberg relations for the case of real analytic Eisenstein series on the upper... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6069_chunk_0 | Maass–Selberg relations | Harish-Chandra generalized the Maass–Selberg relations to Eisenstein series of higher rank semisimple group (and named the relations after Maass and Selberg) and found some analogous relations between Eisenstein integrals, that he also called Maass–Selberg relations. Informally, the Maass–Selberg relations say that the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6070_chunk_0 | Macdonald identities | In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald (1972). They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by Dyson (1972), and a 10-fold product iden... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6071_chunk_0 | Mahler measure | In mathematics, the Mahler measure M ( p ) {\displaystyle M(p)} of a polynomial p ( z ) {\displaystyle p(z)} with complex coefficients is defined as where p ( z ) {\displaystyle p(z)} factorizes over the complex numbers C {\displaystyle \mathbb {C} } as The Mahler measure can be viewed as a kind of height function. Usi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6072_chunk_0 | Mahler polynomial | In mathematics, the Mahler polynomials gn(x) are polynomials introduced by Mahler (1930) in his work on the zeros of the incomplete gamma function. Mahler polynomials are given by the generating function ∑ g n ( x ) t n / n ! = exp ( x ( 1 + t − e t ) ) {\displaystyle \displaystyle \sum g_{n}(x)t^{n}/n!=\exp(x(1+t-e^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6073_chunk_0 | Malgrange preparation theorem | In mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by B. Malgrange (1962–1963, 1964, 1967). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6074_chunk_0 | Malgrange–Ehrenpreis theorem | In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis (1954, 1955) and Bernard Malgrange (1955–1956). This means that the differential equation P ( ∂ ∂ x 1 , … , ∂ ∂... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6075_chunk_0 | Malliavin derivative | In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6076_chunk_0 | Manin conjecture | In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6077_chunk_0 | Manin–Drinfeld theorem | In mathematics, the Manin–Drinfeld theorem, proved by Manin (1972) and Drinfeld (1973), states that the difference of two cusps of a modular curve has finite order in the Jacobian variety. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6078_chunk_0 | Marcinkiewicz interpolation theorem | In mathematics, the Marcinkiewicz interpolation theorem, discovered by Józef Marcinkiewicz (1939), is a result bounding the norms of non-linear operators acting on Lp spaces. Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6079_chunk_0 | Marcinkiewicz–Zygmund inequality | In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables. It is a generalization of the rule for the sum of variances of independent random variables to moments of arbitrary order. It is a sp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6080_chunk_0 | Markov brothers' inequality | In mathematics, the Markov brothers' inequality is an inequality proved in the 1890s by brothers Andrey Markov and Vladimir Markov, two Russian mathematicians. This inequality bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial. For k = 1 it was proved by Andre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6081_chunk_0 | Markov spectrum | In mathematics, the Markov spectrum devised by Andrey Markov is a complicated set of real numbers arising in Markov Diophantine equation and also in the theory of Diophantine approximation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6082_chunk_0 | Markov-Kakutani fixed-point theorem | In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem is a key tool in one of the quickest pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6083_chunk_0 | Markus–Yamabe conjecture | In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. If the Jacobian matrix of a dynamical system at a fixed point is Hurwitz, then the fixed point is asymptotically stable. Markus-Yamabe conjecture asks if a similar result holds globally. Precisely, the conjecture states that if... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6084_chunk_0 | Markus–Yamabe conjecture | The conjecture is true for the two-dimensional case. However, counterexamples have been constructed in higher dimensions. Hence, in the two-dimensional case only, it can also be referred to as the Markus–Yamabe theorem. Related mathematical results concerning global asymptotic stability, which are applicable in dimensi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6085_chunk_0 | Mathai–Quillen formalism | In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by Atiyah and Jeffrey (1990), based on the Mathai–Quillen form constructed in Mathai and Quillen (1986). In more detail, using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Ro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6086_chunk_0 | Maurer–Cartan form | In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G. It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6087_chunk_0 | Mazur–Ulam theorem | In mathematics, the Mazur–Ulam theorem states that if V {\displaystyle V} and W {\displaystyle W} are normed spaces over R and the mapping f: V → W {\displaystyle f\colon V\to W} is a surjective isometry, then f {\displaystyle f} is affine. It was proved by Stanisław Mazur and Stanisław Ulam in response to a question r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6088_chunk_0 | Mazur–Ulam theorem | In this case, for any u {\displaystyle u} and v {\displaystyle v} in V {\displaystyle V} , and for any t {\displaystyle t} in {\displaystyle } , write and denote the closed ball of radius R around v by B ¯ ( v , R ) {\displaystyle {\bar {B}}(v,R)} . Then t u + ( 1 − t ) v {\displaystyle tu+(1-t)v} is the unique elemen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6089_chunk_0 | McKay graph | In mathematics, the McKay graph of a finite-dimensional representation V of a finite group G is a weighted quiver encoding the structure of the representation theory of G. Each node represents an irreducible representation of G. If χ i, χ j are irreducible representations of G, then there is an arrow from χ i to χ j if... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6090_chunk_0 | Mehler–Fock transform | In mathematics, the Mehler–Fock transform is an integral transform introduced by Mehler (1881) and rediscovered by Fock (1943). It is given by F ( x ) = ∫ 0 ∞ P i t − 1 / 2 ( x ) f ( t ) d t , ( 1 ≤ x ≤ ∞ ) , {\displaystyle F(x)=\int _{0}^{\infty }P_{it-1/2}(x)f(t)dt,\quad (1\leq x\leq \infty ),} where P is a Legendre ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6091_chunk_0 | Mellin inversion theorem | In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6092_chunk_0 | Mellin transform | In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6093_chunk_0 | Mellin transform | The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part c need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem. The transform is named after the Finnish mathematician Hjalmar Mellin, who intro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6094_chunk_0 | Mercator series | In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: ln ( 1 + x ) = x − x 2 2 + x 3 3 − x 4 4 + ⋯ {\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots } In summation notation, ln ( 1 + x ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n x n .... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6095_chunk_0 | Mertens conjecture | In mathematics, the Mertens conjecture is the statement that the Mertens function M ( n ) {\displaystyle M(n)} is bounded by ± n {\displaystyle \pm {\sqrt {n}}} . Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6096_chunk_0 | Signature cocycle | In mathematics, the Meyer signature cocycle, introduced by Meyer (1973). is an integer-valued 2-cocyle on a symplectic group that describes the signature of a fiber bundle whose base and fiber are both Riemann surfaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6097_chunk_0 | Milliken–Taylor theorem | In mathematics, the Milliken–Taylor theorem in combinatorics is a generalization of both Ramsey's theorem and Hindman's theorem. It is named after Keith Milliken and Alan D. Taylor. Let P f ( N ) {\displaystyle {\mathcal {P}}_{f}(\mathbb {N} )} denote the set of finite subsets of N {\displaystyle \mathbb {N} } , and de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6098_chunk_0 | Milliken–Taylor theorem | Let k {\displaystyle ^{k}} denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition k = C 1 ∪ C 2 ∪ ⋯ ∪ C r {\displaystyle ^{k}=C_{1}\cup C_{2}\cup \cdots \cup C_{r}} , there exist some i ≤ r and a sequence ⟨ a n ⟩ n = 0 ∞ ⊂ N {\displaystyle \langle a_{n}\rangle _{n=0}^{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6099_chunk_0 | Milman–Pettis theorem | In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive. The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939). S. Kakutani gave a different proof in 1939, and John R. Ringrose published a shorter proof in 1959. Mahlon M. Day (1941) gave examples... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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