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c_vs030jtjcw3s | Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem). For example, in contemporary geometry, point, line, and contains are some primitive notions. Instead of attempting to define them, their interplay is ruled (in Hilbert's axiom system) by axioms like "For every two points there exists a line that contains them both". | Undefined term |
c_lpmb4ee1owz6 | In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. | Loop algebra |
c_m3zldalk8foi | In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See also random walk for more general treatment of this topic. | Uniform spanning tree |
c_nlh9pqrf3l60 | In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory. | Low-dimensional topology |
c_o6blma9lgnoh | In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure. Low-rank approximation is closely related to numerous other techniques, including principal component analysis, factor analysis, total least squares, latent semantic analysis, orthogonal regression, and dynamic mode decomposition. | Low-rank approximation |
c_lps7fx4ju9vj | The order of the entries of the vectors x {\displaystyle \mathbf {x} } or y {\displaystyle \mathbf {y} } does not affect the majorization, e.g., the statement ( 1 , 2 ) ≺ ( 0 , 3 ) {\displaystyle (1,2)\prec (0,3)} is simply equivalent to ( 2 , 1 ) ≺ ( 3 , 0 ) {\displaystyle (2,1)\prec (3,0)} . As a consequence, majorization is not a partial order, since x ≻ y {\displaystyle \mathbf {x} \succ \mathbf {y} } and y ≻ x {\displaystyle \mathbf {y} \succ \mathbf {x} } do not imply x = y {\displaystyle \mathbf {x} =\mathbf {y} } , it only implies that the components of each vector are equal, but not necessarily in the same order. The majorization partial order on finite dimensional vectors, described here, can be generalized to the Lorenz ordering, a partial order on distribution functions. | Majorization |
c_cxmc87kb5fr9 | For example, a wealth distribution is Lorenz-greater than another if its Lorenz curve lies below the other. As such, a Lorenz-greater wealth distribution has a higher Gini coefficient, and has more income disparity. Various other generalizations of majorization are discussed in chapters 14 and 15 of. | Majorization |
c_m3p81u36xgfk | In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes. | List of logarithmic identities |
c_blxyffedsog5 | In mathematics, many sequences of numbers or of polynomials are indexed by nonnegative integers, for example, the Bernoulli numbers and the Bell numbers. In both mechanics and statistics, the zeroth moment is defined, representing total mass in the case of physical density, or total probability, i.e. one, for a probability distribution. The zeroth law of thermodynamics was formulated after the first, second, and third laws, but considered more fundamental, thus its name. In biology, an organism is said to have zero-order intentionality if it shows "no intention of anything at all". This would include a situation where the organism's genetically predetermined phenotype results in a fitness benefit to itself, because it did not "intend" to express its genes. In the similar sense, a computer may be considered from this perspective a zero-order intentional entity, as it does not "intend" to express the code of the programs it runs.In biological or medical experiments, initial measurements made before any experimental time has passed are said to be on the 0 day of the experiment.In genomics, both 0-based and 1-based systems are used for genome coordinates.Patient zero (or index case) is the initial patient in the population sample of an epidemiological investigation. | Zero-based numbering |
c_kqb9euuieubx | In mathematics, mathematical maturity is an informal term often used to refer to the quality of having a general understanding and mastery of the way mathematicians operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated exposure to mathematical concepts. It is a gauge of mathematics students' erudition in mathematical structures and methods, and can overlap with other related concepts such as mathematical intuition and mathematical competence. The topic is occasionally also addressed in literature in its own right. | Mathematical maturity |
c_it3mf3b0lhfl | In mathematics, mathematical optimization (or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem involves maximizing or minimizing a real function by selecting input values of the function and computing the corresponding values of the function. The solution process includes satisfying general necessary and sufficient conditions for optimality. For optimization problems, specialized notation may be used as to the function and its input(s). | Mathematical economics |
c_rwpvqq8yyh6n | More generally, optimization includes finding the best available element of some function given a defined domain and may use a variety of different computational optimization techniques.Economics is closely enough linked to optimization by agents in an economy that an influential definition relatedly describes economics qua science as the "study of human behavior as a relationship between ends and scarce means" with alternative uses. Optimization problems run through modern economics, many with explicit economic or technical constraints. In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem for a given level of utility, are economic optimization problems. | Mathematical economics |
c_00rdwv5mpa4x | Theory posits that consumers maximize their utility, subject to their budget constraints and that firms maximize their profits, subject to their production functions, input costs, and market demand.Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that in principle could be tested against empirical data. Newer developments have occurred in dynamic programming and modeling optimization with risk and uncertainty, including applications to portfolio theory, the economics of information, and search theory.Optimality properties for an entire market system may be stated in mathematical terms, as in formulation of the two fundamental theorems of welfare economics and in the Arrow–Debreu model of general equilibrium (also discussed below). More concretely, many problems are amenable to analytical (formulaic) solution. | Mathematical economics |
c_9t48php9z2ac | Many others may be sufficiently complex to require numerical methods of solution, aided by software. Still others are complex but tractable enough to allow computable methods of solution, in particular computable general equilibrium models for the entire economy.Linear and nonlinear programming have profoundly affected microeconomics, which had earlier considered only equality constraints. Many of the mathematical economists who received Nobel Prizes in Economics had conducted notable research using linear programming: Leonid Kantorovich, Leonid Hurwicz, Tjalling Koopmans, Kenneth J. Arrow, Robert Dorfman, Paul Samuelson and Robert Solow. Both Kantorovich and Koopmans acknowledged that George B. Dantzig deserved to share their Nobel Prize for linear programming. Economists who conducted research in nonlinear programming also have won the Nobel prize, notably Ragnar Frisch in addition to Kantorovich, Hurwicz, Koopmans, Arrow, and Samuelson. | Mathematical economics |
c_hcwlrf1omvva | In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R , {\displaystyle f:U\to \mathbb {R} ,} where U is an open subset of R n , {\displaystyle \mathbb {R} ^{n},} that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 + ∂ 2 f ∂ x 2 2 + ⋯ + ∂ 2 f ∂ x n 2 = 0 {\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0} everywhere on U. This is usually written as ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} or Δ f = 0 {\displaystyle \Delta f=0} | Harmonic function |
c_qpt52melv7a2 | In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general relativity and special relativity, as well as quantum mechanics, relativistic quantum mechanics, and quantum field theory. The special Euclidean group SE(d) of direct isometries is generated by translations and rotations. Its Lie algebra is written s e ( d ) {\displaystyle {\mathfrak {se}}(d)} . This article uses Cartesian coordinates and tensor index notation. | Spin current |
c_lfzk2gwvc9qx | In mathematics, mathematical structures can have more than one definition. Therefore, there are several definitions of named sets, each representing a specific construction of named set theory. The informal definition is the most general. | Named set theory |
c_zb4c1jl3mfky | In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, v → {\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec {v}}\!} | Matrix addition |
c_jr01n7nvk9yp | , then adding the transformed vectors. A v → + B v → = ( A + B ) v → {\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {v}}\!} However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum. | Matrix addition |
c_rwg52ymvkxlx | In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. | Matrix differentiation |
c_xcrvj73oxhgk | Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is made that vectors should be treated as column vectors when combined with matrices (rather than row vectors). | Matrix differentiation |
c_rzfcx1okzrdd | A single convention can be somewhat standard throughout a single field that commonly uses matrix calculus (e.g. econometrics, statistics, estimation theory and machine learning). However, even within a given field different authors can be found using competing conventions. | Matrix differentiation |
c_gcl3hm1toiew | Authors of both groups often write as though their specific conventions were standard. Serious mistakes can result when combining results from different authors without carefully verifying that compatible notations have been used. Definitions of these two conventions and comparisons between them are collected in the layout conventions section. | Matrix differentiation |
c_kuxfcby0lihg | In mathematics, mimesis is the quality of a numerical method which imitates some properties of the continuum problem. The goal of numerical analysis is to approximate the continuum, so instead of solving a partial differential equation one aims to solve a discrete version of the continuum problem. Properties of the continuum problem commonly imitated by numerical methods are conservation laws, solution symmetries, and fundamental identities and theorems of vector and tensor calculus like the divergence theorem. Both finite difference or finite element method can be mimetic; it depends on the properties that the method has. | Mimesis (mathematics) |
c_l08bdnkoda7q | For example, a mixed finite element method applied to Darcy flows strictly conserves the mass of the flowing fluid. The term geometric integration denotes the same philosophy. == References == | Mimesis (mathematics) |
c_jz9asmz6qaly | In mathematics, mimetic interpolation is a method for interpolating differential forms. In contrast to other interpolation methods, which estimate a field at a location given its values on neighboring points, mimetic interpolation estimates the field's k {\displaystyle k} -form given the field's projection on neighboring grid elements. The grid elements can be grid points as well as cell edges or faces, depending on k = 0 , 1 , 2 , ⋯ {\displaystyle k=0,1,2,\cdots } . Mimetic interpolation is particularly relevant in the context of vector and pseudo-vector fields as the method conserves line integrals and fluxes, respectively. | Mimetic interpolation |
c_kyvj8dpzzmo6 | In mathematics, minimum polynomial extrapolation is a sequence transformation used for convergence acceleration of vector sequences, due to Cabay and Jackson.While Aitken's method is the most famous, it often fails for vector sequences. An effective method for vector sequences is the minimum polynomial extrapolation. It is usually phrased in terms of the fixed point iteration: x k + 1 = f ( x k ) . {\displaystyle x_{k+1}=f(x_{k}).} | Minimum polynomial extrapolation |
c_1o9zsgwyhvt2 | Given iterates x 1 , x 2 , . . | Minimum polynomial extrapolation |
c_qwnl112xv8nk | . , x k {\displaystyle x_{1},x_{2},...,x_{k}} in R n {\displaystyle \mathbb {R} ^{n}} , one constructs the n × ( k − 1 ) {\displaystyle n\times (k-1)} matrix U = ( x 2 − x 1 , x 3 − x 2 , . . | Minimum polynomial extrapolation |
c_buz9skidjuua | . , x k − x k − 1 ) {\displaystyle U=(x_{2}-x_{1},x_{3}-x_{2},...,x_{k}-x_{k-1})} whose columns are the k − 1 {\displaystyle k-1} differences. Then, one computes the vector c = − U + ( x k + 1 − x k ) {\displaystyle c=-U^{+}(x_{k+1}-x_{k})} where U + {\displaystyle U^{+}} denotes the Moore–Penrose pseudoinverse of U {\displaystyle U} . | Minimum polynomial extrapolation |
c_pzew8enn8564 | The number 1 is then appended to the end of c {\displaystyle c} , and the extrapolated limit is s = X c ∑ i = 1 k c i , {\displaystyle s={Xc \over \sum _{i=1}^{k}c_{i}},} where X = ( x 2 , x 3 , . . . , x k + 1 ) {\displaystyle X=(x_{2},x_{3},...,x_{k+1})} is the matrix whose columns are the k {\displaystyle k} iterates starting at 2. The following 4 line MATLAB code segment implements the MPE algorithm: == References == | Minimum polynomial extrapolation |
c_dzlcktdr1hk1 | In mathematics, mirror descent is an iterative optimization algorithm for finding a local minimum of a differentiable function. It generalizes algorithms such as gradient descent and multiplicative weights. | Mirror descent |
c_3f20pqp5oymf | In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold (encoded as Gromov–Witten invariants) to integrals from a family of varieties (encoded as period integrals on a variation of Hodge structures). In short, this means there is a relation between the number of genus g {\displaystyle g} algebraic curves of degree d {\displaystyle d} on a Calabi-Yau variety X {\displaystyle X} and integrals on a dual variety X ˇ {\displaystyle {\check {X}}} . These relations were original discovered by Candelas, de la Ossa, Green, and Parkes in a paper studying a generic quintic threefold in P 4 {\displaystyle \mathbb {P} ^{4}} as the variety X {\displaystyle X} and a construction from the quintic Dwork family X ψ {\displaystyle X_{\psi }} giving X ˇ = X ~ ψ {\displaystyle {\check {X}}={\tilde {X}}_{\psi }} . Shortly after, Sheldon Katz wrote a summary paper outlining part of their construction and conjectures what the rigorous mathematical interpretation could be. | Mirror symmetry conjecture |
c_lv3zwtaf9fyk | In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures through the six functor formalism. Essentially, these objects are a pair of a filtered D-module ( M , F ∙ ) {\displaystyle (M,F^{\bullet })} together with a perverse sheaf F {\displaystyle {\mathcal {F}}} such that the functor from the Riemann–Hilbert correspondence sends ( M , F ∙ ) {\displaystyle (M,F^{\bullet })} to F {\displaystyle {\mathcal {F}}} . This makes it possible to construct a Hodge structure on intersection cohomology, one of the key problems when the subject was discovered. This was solved by Morihiko Saito who found a way to use the filtration on a coherent D-module as an analogue of the Hodge filtration for a Hodge structure. This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves. | Mixed Hodge module |
c_sn5d0h88oto7 | In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: e.g. mixing paint, mixing drinks, industrial mixing. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including strong mixing, weak mixing and topological mixing, with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" condition than ergodicity). | Strong mixing |
c_06v8so3gbog6 | In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and their properties were the subject of investigation since at least the time of Euclid. In fact, Euclid's Elements contains description of the four special points – centroid, incenter, circumcenter and orthocenter - associated with a triangle. Even though Pascal and Ceva in the seventeenth century, Euler in the eighteenth century and Feuerbach in the nineteenth century and many other mathematicians had made important discoveries regarding the properties of the triangle, it was the publication in 1873 of a paper by Emile Lemoine (1840–1912) with the title "On a remarkable point of the triangle" that was considered to have, according to Nathan Altschiller-Court, "laid the foundations...of the modern geometry of the triangle as a whole." | Modern triangle geometry |
c_nnsu4nsf38w6 | The American Mathematical Monthly, in which much of Lemoine's work is published, declared that "To none of these more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement of modern triangle geometry". The publication of this paper caused a remarkable upsurge of interest in investigating the properties of the triangle during the last quarter of the nineteenth century and the early years of the twentieth century. A hundred-page article on triangle geometry in Klein's Encyclopedia of Mathematical Sciences published in 1914 bears witness to this upsurge of interest in triangle geometry.In the early days, the expression "new triangle geometry" referred to only the set of interesting objects associated with a triangle like the Lemoine point, Lemoine circle, Brocard circle and the Lemoine line. | Modern triangle geometry |
c_nyyvsj1xdtvu | Later the theory of correspondences which was an offshoot of the theory of geometric transformations was developed to give coherence to the various isolated results. With its development, the expression "new triangle geometry" indicated not only the many remarkable objects associated with a triangle but also the methods used to study and classify these objects. Here is a definition of triangle geometry from 1887: "Being given a point M in the plane of the triangle, we can always find, in an infinity of manners, a second point M' that corresponds to the first one according to an imagined geometrical law; these two points have between them geometrical relations whose simplicity depends on the more or less the lucky choice of the law which unites them and each geometrical law gives rise to a method of transformation a mode of conjugation which it remains to study." | Modern triangle geometry |
c_sg66u3a8c8jq | (See the conference paper titled "Teaching new geometrical methods with an ancient figure in the nineteenth and twentieth centuries: the new triangle geometry in textbooks in Europe and USA (1888–1952)" by Pauline Romera-Lebret presented in 2009.) However, this escalation of interest soon collapsed and triangle geometry was completely neglected until the closing years of the twentieth century. In his "Development of Mathematics", Eric Temple Bell offers his judgement on the status of modern triangle geometry in 1940 thus: "The geometers of the 20th Century have long since piously removed all these treasures to the museum of geometry where the dust of history quickly dimmed their luster." | Modern triangle geometry |
c_1b7ofgyjrq3z | (The Development of Mathematics, p. 323) Philip Davis has suggested several reasons for the decline of interest in triangle geometry. These include: The feeling that the subject is elementary and of low professional status. | Modern triangle geometry |
c_u8pk5mttn3f3 | The exhaustion of its methodologic possibilities. The visual complexity of the so-called deeper results of the subject. The downgrading of the visual in favor of the algebraic. | Modern triangle geometry |
c_h1xf066m1d5a | A dearth of connections to other fields. Competition with other topics with a strong visual content like tessellations, fractals, graph theory, etc.A further revival of interest was witnessed with the advent of the modern electronic computer. | Modern triangle geometry |
c_qis3099ghamt | The triangle geometry has again become an active area of research pursued by a group of dedicated geometers. As epitomizing this revival, one can point out the formulation of the concept of a "triangle centre" and the compilation by Clark Kimberling of an encyclopedia of triangle centers containing a listing of nearly 50,000 triangle centers and their properties and also the compilation of a catalogue of triangle cubics with detailed descriptions of several properties of more than 1200 triangle cubics. The open access journal Forum Geometricorum founded by Paul Yiu of Florida Atlantic University in 2001 also provided a tremendous impetus in furthering this new found enthusiasm for triangle geometry. Unfortunately, since 2019, the journal is not accepting submissions although back issues are still available online. | Modern triangle geometry |
c_3kkiw7hg3w8i | In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. | Residue class |
c_kzwxs0sk6lsp | If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. | Residue class |
c_mgffrvidrx3p | In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime p, there is an analogous theory to the classical theory of complex modular forms and the p-adic theory of modular forms. | Modular forms modulo p |
c_vqf71okvcn44 | In mathematics, modular symbols, introduced independently by Bryan John Birch and by Manin (1972), span a vector space closely related to a space of modular forms, on which the action of the Hecke algebra can be described explicitly. This makes them useful for computing with spaces of modular forms. | Modular symbol |
c_3b7na91axprk | In mathematics, modular units are certain units of rings of integers of fields of modular functions, introduced by Kubert and Lang (1975). They are functions whose zeroes and poles are confined to the cusps (images of infinity). | Modular unit |
c_fsvf4zu4782m | In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation by polynomials and splines. | Modulus of smoothness |
c_wk4lyvv3nkl6 | In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a function which is rather irregular, by convolving it with a mollifier the function gets "mollified", that is, its sharp features are smoothed, while still remaining close to the original nonsmooth (generalized) function.They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them. | Cutoff function |
c_19ie6vc748ah | In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of monodromy comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a monodromy group: a group of transformations acting on the data that encodes what happens as we "run round" in one dimension. Lack of monodromy is sometimes called polydromy. | Monodromy theory |
c_8q0oe3oiv71i | In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979.The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. | Monstrous Moonshine |
c_fuxhgzne6s5v | In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. The monus operator may be denoted with the − symbol because the natural numbers are a CMM under subtraction; it is also denoted with the − ˙ {\displaystyle \mathop {\dot {-}} } symbol to distinguish it from the standard subtraction operator. | Truncated subtraction |
c_0dxfs6u273p9 | In mathematics, more particularly in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (n-p,n-p)-form over an n-dimensional complex manifold, taking values in distributions. For a formal definition, consider a manifold M. Currents on M are (by definition) differential forms with coefficients in distributions; integrating over M, we may consider currents as "currents of integration", that is, functionals η ↦ ∫ M η ∧ ρ {\displaystyle \eta \mapsto \int _{M}\eta \wedge \rho } on smooth forms with compact support. This way, currents are considered as elements in the dual space to the space Λ c ∗ ( M ) {\displaystyle \Lambda _{c}^{*}(M)} of forms with compact support. | Positive current |
c_v1d0lmzeh433 | Now, let M be a complex manifold. The Hodge decomposition Λ i ( M ) = ⨁ p + q = i Λ p , q ( M ) {\displaystyle \Lambda ^{i}(M)=\bigoplus _{p+q=i}\Lambda ^{p,q}(M)} is defined on currents, in a natural way, the (p,q)-currents being functionals on Λ c p , q ( M ) {\displaystyle \Lambda _{c}^{p,q}(M)} . A positive current is defined as a real current of Hodge type (p,p), taking non-negative values on all positive (p,p)-forms. | Positive current |
c_blnuo4a1kvpq | In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of M. | Integral current |
c_64j2qvseek3v | In mathematics, more particularly in the field of algebraic geometry, a scheme X {\displaystyle X} has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map f: Y → X {\displaystyle f\colon Y\rightarrow X} from a regular scheme Y {\displaystyle Y} such that the higher direct images of f ∗ {\displaystyle f_{*}} applied to O Y {\displaystyle {\mathcal {O}}_{Y}} are trivial. That is, R i f ∗ O Y = 0 {\displaystyle R^{i}f_{*}{\mathcal {O}}_{Y}=0} for i > 0 {\displaystyle i>0} .If there is one such resolution, then it follows that all resolutions share this property, since any two resolutions of singularities can be dominated by a third. For surfaces, rational singularities were defined by (Artin 1966). | Rational singularities |
c_qgz4jokuizjn | In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems. Anosov diffeomorphisms were introduced by Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense generic (when they exist at all). | Anosov system |
c_7886d52170vy | In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a topological group, every open neighborhood of the identity contains a normal subgroup whose corresponding quotient group is a solvable group. | Prosolvable group |
c_xhn8hyi091wg | In mathematics, more precisely in algebra, an étale group scheme is a certain kind of group scheme. | Étale group scheme |
c_n378i4k4ndi2 | In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of contact with a certain model Klein geometry at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by Charles Ehresmann in 1950. | Solder form |
c_v0zq84e3zsof | In mathematics, more precisely in formal language theory, the profinite words are a generalization of the notion of finite words into a complete topological space. This notion allows the use of topology to study languages and finite semigroups. For example, profinite words are used to give an alternative characterization of the algebraic notion of a variety of finite semigroups. | Profinite word |
c_wm6ognc9rkea | In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for the name comes from physics, as in many physical problems the energy of a system can be expressed in terms of the energetic inner product. An example of this will be given later in the article. | Energetic space |
c_sgfpluuislro | In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} by an arithmetic Kleinian group. | Arithmetic hyperbolic 3-manifold |
c_i9cv3osdcr4y | In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures μ {\displaystyle \mu } and ν {\displaystyle \nu } on a measurable space ( Ω , Σ ) , {\displaystyle (\Omega ,\Sigma ),} there exist two σ-finite signed measures ν 0 {\displaystyle \nu _{0}} and ν 1 {\displaystyle \nu _{1}} such that: ν = ν 0 + ν 1 {\displaystyle \nu =\nu _{0}+\nu _{1}\,} ν 0 ≪ μ {\displaystyle \nu _{0}\ll \mu } (that is, ν 0 {\displaystyle \nu _{0}} is absolutely continuous with respect to μ {\displaystyle \mu } ) ν 1 ⊥ μ {\displaystyle \nu _{1}\perp \mu } (that is, ν 1 {\displaystyle \nu _{1}} and μ {\displaystyle \mu } are singular).These two measures are uniquely determined by μ {\displaystyle \mu } and ν . {\displaystyle \nu .} | Lebesgue's decomposition theorem |
c_o9uhh9v9c65w | In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometrically, a discrete measure (on the real line, with respect to Lebesgue measure) is a collection of point masses. | Discrete measure |
c_x9o2o4onk71d | In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless. | Atom (measure theory) |
c_n5eyiyic4cyz | In mathematics, more precisely in operator theory, a sectorial operator is a linear operator on a Banach space, whose spectrum in an open sector in the complex plane and whose resolvent is uniformly bounded from above outside any larger sector. Such operators might be unbounded. Sectorial operators have applications in the theory of elliptic and parabolic partial differential equations. | Sectorial operator |
c_nvjls03pr4d7 | In mathematics, more precisely in symplectic geometry, a hypersurface Σ {\displaystyle \Sigma } of a symplectic manifold ( M , ω ) {\displaystyle (M,\omega )} is said to be of contact type if there is 1-form α {\displaystyle \alpha } such that j ∗ ( ω ) = d α {\displaystyle j^{*}(\omega )=d\alpha } and ( Σ , α ) {\displaystyle (\Sigma ,\alpha )} is a contact manifold, where j: Σ → M {\displaystyle j:\Sigma \to M} is the natural inclusion. The terminology was first coined by Alan Weinstein. | Contact type |
c_xzkqlcbhwug0 | In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cn. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let G ⊂ C n {\displaystyle G\subset {\mathbb {C} }^{n}} be a domain, that is, an open connected subset. One says that G {\displaystyle G} is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function φ {\displaystyle \varphi } on G {\displaystyle G} such that the set { z ∈ G ∣ φ ( z ) < x } {\displaystyle \{z\in G\mid \varphi (z) | Pseudoconvex domain |
c_ngd1kdo3n15y | In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group). Hyperbolic 3-manifolds of finite volume have a particular importance in 3-dimensional topology as follows from Thurston's geometrisation conjecture proved by Perelman. The study of Kleinian groups is also an important topic in geometric group theory. | Hyperbolic 3-manifold |
c_kd21i4taaxcu | In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The Dold–Kan correspondence says that a simplicial abelian group may be identified with a chain complex. | Simplicial group |
c_w7l901ys5fs5 | In fact it can be shown that any simplicial abelian group A {\displaystyle A} is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces, ∏ i ≥ 0 K ( π i A , i ) . {\displaystyle \prod _{i\geq 0}K(\pi _{i}A,i).} A commutative monoid in the category of simplicial abelian groups is a simplicial commutative ring. Eckmann (1945) discusses a simplicial analogue of the fact that a cohomology class on a Kähler manifold has a unique harmonic representative and deduces Kirchhoff's circuit laws from these observations. | Simplicial group |
c_46y322zp0zm4 | In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the n {\displaystyle n} th homology group of a chain complex is the n {\displaystyle n} th homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.) | Dold–Kan correspondence |
c_wzezoqjy09ap | Example: Let C be a chain complex that has an abelian group A in degree n and zero in all other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space K ( A , n ) {\displaystyle K(A,n)} . There is also an ∞-category-version of the Dold–Kan correspondence.The book "Nonabelian Algebraic Topology" cited below has a Section 14.8 on cubical versions of the Dold–Kan theorem, and relates them to a previous equivalence of categories between cubical omega-groupoids and crossed complexes, which is fundamental to the work of that book. | Dold–Kan correspondence |
c_k56x1wz129mx | In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring. The lemma is named after the Japanese mathematician Tadashi Nakayama and introduced in its present form in Nakayama (1951), although it was first discovered in the special case of ideals in a commutative ring by Wolfgang Krull and then in general by Goro Azumaya (1951). | Nakayama's lemma |
c_037fpmcgcw50 | In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley–Hamilton theorem, an observation made by Michael Atiyah (1969). The special case of the noncommutative version of the lemma for right ideals appears in Nathan Jacobson (1945), and so the noncommutative Nakayama lemma is sometimes known as the Jacobson–Azumaya theorem. The latter has various applications in the theory of Jacobson radicals. | Nakayama's lemma |
c_amktnhowub3l | In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history. Although rings have more structure than groups do, the theory of finite rings is simpler than that of finite groups. | Finite ring |
c_kcu0cgkdpkl7 | For instance, the classification of finite simple groups was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, it has been known since 1907 that any finite simple ring is isomorphic to the ring M n ( F q ) {\displaystyle \mathrm {M} _{n}(\mathbb {F} _{q})} – the n-by-n matrices over a finite field of order q (as a consequence of Wedderburn's theorems, described below). The number of rings with m elements, for m a natural number, is listed under OEIS: A027623 in the On-Line Encyclopedia of Integer Sequences. | Finite ring |
c_8ph5i4mlgjkq | In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. | Abstract Algebra |
c_mjokp6f9s8h8 | Presently, the term "abstract algebra" is typically used for naming courses in mathematical education, and is rarely used in advanced mathematics. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the variety of groups. | Abstract Algebra |
c_szvx7u1ky2ro | In mathematics, more specifically algebraic topology, a pair ( X , A ) {\displaystyle (X,A)} is shorthand for an inclusion of topological spaces i: A ↪ X {\displaystyle i\colon A\hookrightarrow X} . Sometimes i {\displaystyle i} is assumed to be a cofibration. A morphism from ( X , A ) {\displaystyle (X,A)} to ( X ′ , A ′ ) {\displaystyle (X',A')} is given by two maps f: X → X ′ {\displaystyle f\colon X\rightarrow X'} and g: A → A ′ {\displaystyle g\colon A\rightarrow A'} such that i ′ ∘ g = f ∘ i {\displaystyle i'\circ g=f\circ i} . | Pair of spaces |
c_cgaofwn7f62p | A pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace (with the subspace topology). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of X by A. Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains in A {\displaystyle A} are made equivalent to 0, when considered as chains in X {\displaystyle X} . Heuristically, one often thinks of a pair ( X , A ) {\displaystyle (X,A)} as being akin to the quotient space X / A {\displaystyle X/A} . | Pair of spaces |
c_nuojnr6iqome | There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space X {\displaystyle X} to the pair ( X , ∅ ) {\displaystyle (X,\varnothing )} . A related concept is that of a triple (X, A, B), with B ⊂ A ⊂ X. Triples are used in homotopy theory. Often, for a pointed space with basepoint at x0, one writes the triple as (X, A, B, x0), where x0 ∈ B ⊂ A ⊂ X. | Pair of spaces |
c_564421d2iqsi | In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. The study of such generalizations is known as higher category theory. Quasi-categories were introduced by Boardman & Vogt (1973). | Weak Kan complex |
c_uq237w7z2eoe | André Joyal has much advanced the study of quasi-categories showing that most of the usual basic category theory and some of the advanced notions and theorems have their analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by Jacob Lurie (2009). Quasi-categories are certain simplicial sets. | Weak Kan complex |
c_9a3edbb3dloh | Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined. All the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms (2-simplices thought of as "homotopies"). | Weak Kan complex |
c_djze62t81f0m | These higher order morphisms can also be composed, but again the composition is well-defined only up to still higher order invertible morphisms, etc. The idea of higher category theory (at least, higher category theory when higher morphisms are invertible) is that, as opposed to the standard notion of a category, there should be a mapping space (rather than a mapping set) between two objects. This suggests that a higher category should simply be a topologically enriched category. The model of quasi-categories is, however, better suited to applications than that of topologically enriched categories, though it has been proved by Lurie that the two have natural model structures that are Quillen equivalent. | Weak Kan complex |
c_wpyytfng4sy2 | In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f: C ∖ { a k } k → C {\displaystyle f\colon \mathbb {C} \setminus \{a_{k}\}_{k}\rightarrow \mathbb {C} } that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem. | Residue (complex analysis) |
c_n5gj6tvepoep | In mathematics, more specifically differential algebra, a p-derivation (for p a prime number) on a ring R, is a mapping from R to R that satisfies certain conditions outlined directly below. The notion of a p-derivation is related to that of a derivation in differential algebra. | P-derivation |
c_nkfw33wf2ig9 | In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. | Local diffeomorphism |
c_9awprpjuqtxk | In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently. | Degree of an extension |
c_bmdrwd65vlhm | In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should sometimes be understood as "not necessarily bounded"; "operator" should be understood as "linear operator" (as in the case of "bounded operator"); the domain of the operator is a linear subspace, not necessarily the whole space; this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general topological vector spaces are possible. | Unbounded operator |
c_q2u2fuzxhu14 | In mathematics, more specifically general topology, the divisor topology is a specific topology on the set X = { 2 , 3 , 4 , . . . } {\displaystyle X=\{2,3,4,...\}} of positive integers greater than or equal to two. The divisor topology is the poset topology for the partial order relation of divisibility of integers on X {\displaystyle X} . | Divisor topology |
c_qb9z3fz9la5s | In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R2 with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set X = R2 ∐ {0*}, where ∐ denotes the disjoint union. | Double origin topology |
c_uui4u9c7r1yg | In mathematics, more specifically general topology, the nested interval topology is an example of a topology given to the open interval (0,1), i.e. the set of all real numbers x such that 0 < x < 1. The open interval (0,1) is the set of all real numbers between 0 and 1; but not including either 0 or 1. To give the set (0,1) a topology means to say which subsets of (0,1) are "open", and to do so in a way that the following axioms are met: The union of open sets is an open set. The finite intersection of open sets is an open set. The set (0,1) and the empty set ∅ are open sets. | Nested interval topology |
c_r60qzwvw13c5 | In mathematics, more specifically general topology, the rational sequence topology is an example of a topology given to the set R of real numbers. | Rational sequence topology |
c_5wlkjotuq6sp | In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity. | Three subgroups lemma |
c_q0rhll722cv9 | In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers) C (the complex numbers) H (the quaternions).These algebras have real dimension 1, 2, and 4, respectively. Of these three algebras, R and C are commutative, but H is not. | Frobenius theorem (real division algebras) |
c_lsctl5bth4zm | In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G / N {\displaystyle G/N} is abelian if and only if N {\displaystyle N} contains the commutator subgroup of G {\displaystyle G} . So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. | Transfinite derived series |
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