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c_9v9w2d8aw146 | Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if they did appear, one could make a smaller counter-example). Appel and Haken used a special-purpose computer program to confirm t... | Conjecture |
c_36j9auhtytsn | Showing this with hundreds of pages of hand analysis, Appel and Haken concluded that no smallest counterexample exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was no... | Conjecture |
c_jztjq75425bm | In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more reg... | Four-color conjecture |
c_3vsa5a3xle5k | Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubters remain.The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken after many false proo... | Four-color conjecture |
c_l9vw4byq7t1i | In mathematics, the four-spiral semigroup is a special semigroup generated by four idempotent elements. This special semigroup was first studied by Karl Byleen in a doctoral dissertation submitted to the University of Nebraska in 1977. It has several interesting properties: it is one of the most important examples of b... | Four-spiral semigroup |
c_67n0bug6x55t | In mathematics, the fractional Laplacian is an operator, which generalizes the notion of Laplacian spatial derivatives to fractional powers. | Fractional Laplacian |
c_jl2vaxmgkgff | In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next. More precisely, the objects of the category are the vertices of the quiver, and the morphisms are... | Free category |
c_7u5kyvqtjkv9 | For every vertex V {\displaystyle V} of the quiver, there is an "empty path" which constitutes the identity morphisms of the category. The composition operation is concatenation of paths. Given paths V 0 → E 0 ⋯ → E n − 1 V n , V n → F 0 W 0 → F 1 ⋯ → F n − 1 W m , {\displaystyle V_{0}{\xrightarrow {E_{0}}}\cdots {\xri... | Free category |
c_01fuqo5sczlh | In mathematics, the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type surface. The free factor complex was originally introduced in a 1998 paper of Allen Hatcher and Karen Vogtmann. Like the curve complex, the free fac... | Free factor complex |
c_weca6hliojc2 | In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st = suu−1t, but s ≠ t−1 for s,t,u ∈ S). The members of S are called generators of FS, and the number of generat... | Free group |
c_r40chmwjzr1x | In mathematics, the free matroid over a given ground-set E is the matroid in which the independent sets are all subsets of E. It is a special case of a uniform matroid. The unique basis of this matroid is the ground-set itself, E. Among matroids on E, the free matroid on E has the most independent sets, the highest ran... | Free matroid |
c_ga98cafvnqel | In mathematics, the fundamental class is a homology class associated to a connected orientable compact manifold of dimension n, which corresponds to the generator of the homology group H n ( M , ∂ M ; Z ) ≅ Z {\displaystyle H_{n}(M,\partial M;\mathbf {Z} )\cong \mathbf {Z} } . The fundamental class can be thought of a... | Orientation homology class |
c_rg82tjosa6k7 | In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental group. Although its existence was conjectured by Alexander Grothendieck, the... | Fundamental group scheme |
c_iv8mv4ktyvvj | In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basic form, the theorem asserts that given a field extension E/F that is finite ... | Fundamental theorem of Galois theory |
c_ojer51rlb94l | In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example, 1200 = 2 4 ⋅ 3 1 ⋅ 5 2 = ( 2 ⋅ 2 ⋅ 2 ⋅ 2 )... | Fundamental Theorem of Arithmetic |
c_92c21miwo6p8 | This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example, 2 = 2 ⋅ 1 = 2 ⋅ 1 ⋅ 1 = … {\displaystyle 2=2\cdot 1=2\cdot 1\cdot 1=\ldots } The theorem generalizes to other algebraic structures that are called unique fact... | Fundamental Theorem of Arithmetic |
c_x547xslrsa3v | In mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is generated by spherical ... | Fuzzy sphere |
c_ust3xm8mln1y | This truncation replaces an infinite-dimensional commutative algebra by a j 2 {\displaystyle j^{2}} -dimensional non-commutative algebra. The simplest way to see this sphere is to realize this truncated algebra of functions as a matrix algebra on some finite-dimensional vector space. Take the three j-dimensional matric... | Fuzzy sphere |
c_b16i1qlq66oz | In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Ber... | Gamma function |
c_lc660jph5xx0 | The gamma function has no zeros, so the reciprocal gamma function 1/Γ(z) is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a compon... | Gamma function |
c_xmsud78yx17e | In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix a... | Infinite general linear group |
c_ggwq4dl6xx4l | For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GLn(R) or GL(n, R). More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the se... | Infinite general linear group |
c_j4mjxc1ldn2k | More generally still, the general linear group of a vector space GL(V) is the automorphism group, not necessarily written as matrices. The special linear group, written SL(n, F) or SLn(F), is the subgroup of GL(n, F) consisting of matrices with a determinant of 1. The group GL(n, F) and its subgroups are often called l... | Infinite general linear group |
c_7ndaz7wdb2wm | These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear group SL(2, Z). If n ≥ 2, then the group GL(n, F) is... | Infinite general linear group |
c_z608xvc1mf5h | In mathematics, the generalized Pochhammer symbol of parameter α > 0 {\displaystyle \alpha >0} and partition κ = ( κ 1 , κ 2 , … , κ m ) {\displaystyle \kappa =(\kappa _{1},\kappa _{2},\ldots ,\kappa _{m})} generalizes the classical Pochhammer symbol, named after Leo August Pochhammer, and is defined as ( a ) κ ( α ) =... | Generalized Pochhammer symbol |
c_9a64irsykn9s | In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group O(2). Dihedral groups play an important role in group theory, geometry, and chemistry. | Generalized dihedral group |
c_7gxfmjsyt2u8 | In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector. Th... | Generalized minimal residual method |
c_xzshb662lwji | It is a generalization and improvement of the MINRES method due to Paige and Saunders in 1975. The MINRES method requires that the matrix is symmetric, but has the advantage that it only requires handling of three vectors. GMRES is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is applicable t... | Generalized minimal residual method |
c_lyzd2dw9wniy | In mathematics, the generalized symmetric group is the wreath product S ( m , n ) := Z m ≀ S n {\displaystyle S(m,n):=Z_{m}\wr S_{n}} of the cyclic group of order m and the symmetric group of order n. | Generalized symmetric group |
c_xria0rwdu66r | In mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number — if it exists — that can be expressed as the sum of j kth positive powers in n different ways. For k = 3 and j = 2, they coincide with taxicab number. T a x i c a b ( 1 , 2 , 2 ) = 4 = 1 + 3 = 2 + 2. {\displaystyle \mathrm {Taxicab}... | Generalized taxicab number |
c_27nhn2g3hwc2 | T a x i c a b ( 2 , 2 , 2 ) = 50 = 1 2 + 7 2 = 5 2 + 5 2 . {\displaystyle \mathrm {Taxicab} (2,2,2)=50=1^{2}+7^{2}=5^{2}+5^{2}.} T a x i c a b ( 3 , 2 , 2 ) = 1729 = 1 3 + 12 3 = 9 3 + 10 3 {\displaystyle \mathrm {Taxicab} (3,2,2)=1729=1^{3}+12^{3}=9^{3}+10^{3}} — 1729 (number) by Ramanujan.Euler showed that T a x i c ... | Generalized taxicab number |
c_1ppm1xdbegl2 | In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Zn, or equivalently a free Z-module of finite rank. Two such forms are in the same genus if they are equivalent over the local rings Zp for each prime p and also equ... | Genus of a quadratic form |
c_da7v10tqx2gh | In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equations, in the form of Hamilton's equations. This latter formulation is develope... | Geodesics as Hamiltonian flows |
c_9mnjwpjogmj5 | In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic geometry. The geometric Langlands correspondence relates algebrai... | Geometric Langlands correspondence |
c_ss9a5cc885to | In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers ... | Geometric mean |
c_prdz3c1vyh8z | The geometric mean is often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as a set of growth figures: values of the human population or interest rates of a financial investment over time. It also applies to benchmarking, where it is particularly useful for... | Geometric mean |
c_rlrpq0watli1 | The geometric mean of two numbers, a {\displaystyle a} and b {\displaystyle b} , is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a {\displaystyle a} and b {\displaystyle b} . Similarly, the geometric mean of three numbers, a {\displaystyle a} , b {\displaystyle... | Geometric mean |
c_0c5b8cgr7hwp | In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume. | Geometric topology (object) |
c_oba2aq2p9vtn | In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we form the geometric mean of g0 = x and h0 = y and call it g1, i.e. g1 is the square root of xy. We also form the harmonic mean of x and y and call it h1, i.e. h1 is the reciprocal of the arithmetic mean of ... | Geometric-harmonic mean |
c_j2b8ah0w7hwv | For example, a vector field is a section of a tangent bundle on a smooth manifold; this says that a vector field on the union of two open sets is (no more and no less than) vector fields on the two sets that agree where they overlap. Given this basic understanding, there are further issues in the theory, and some will ... | Gluing axiom |
c_7898drhu74l2 | In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice. | Lattice reduction |
c_gmrd3upc4btr | In mathematics, the gonality of an algebraic curve C is defined as the lowest degree of a nonconstant rational map from C to the projective line. In more algebraic terms, if C is defined over the field K and K(C) denotes the function field of C, then the gonality is the minimum value taken by the degrees of field exten... | Gonality of an algebraic curve |
c_50rutav1w46v | The gonality of the generic curve of genus g is the floor function of (g + 3)/2.Trigonal curves are those with gonality 3, and this case gave rise to the name in general. Trigonal curves include the Picard curves, of genus three and given by an equation y3 = Q(x)where Q is of degree 4. The gonality conjecture, of M. Gr... | Gonality of an algebraic curve |
c_8udkqor64b2h | In many cases the gonality is two more than the Clifford index. The Green–Lazarsfeld conjecture is an exact formula in terms of the graded Betti numbers for a degree d embedding in r dimensions, for d large with respect to the genus. Writing b(C), with respect to a given such embedding of C and the minimal free resolut... | Gonality of an algebraic curve |
c_jgi5on83b7ak | In mathematics, the gradient conjecture, due to René Thom (1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka (University of Savoie, France), Tadeusz Mostowski (Warsaw University, Poland) and Adam Parusiński (University of Angers, France). The conjecture states that given a real-valued analytic... | Gradient conjecture |
c_fbse5kgnnyh8 | In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic L-functions lie on the critical line 1 2 + i t {\displaystyle {\frac {1}{2}}+it} with t {\displaystyle t} a real number variable and i {\d... | Grand Riemann hypothesis |
c_wqh6u4xi980r | In mathematics, the graph Fourier transform is a mathematical transform which eigendecomposes the Laplacian matrix of a graph into eigenvalues and eigenvectors. Analogously to the classical Fourier transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. The Graph F... | Graph Fourier Transform |
c_emfn9md2r7tr | In mathematics, the graph of a function f {\displaystyle f} is the set of ordered pairs ( x , y ) {\displaystyle (x,y)} , where f ( x ) = y . {\displaystyle f(x)=y.} In the common case where x {\displaystyle x} and f ( x ) {\displaystyle f(x)} are real numbers, these pairs are Cartesian coordinates of points in two-dim... | Graph (function) |
c_hog2kmyml4gx | This set is a subset of three-dimensional space; for a continuous real-valued function of two real variables, it is a surface. In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. | Graph (function) |
c_p7smdlnt4r90 | In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see Plot (graphics) for details. A graph of a function is a special case of a relation. In the modern foundations of mathematics, and, typically, in set theory, a function is actually equal to its graph. | Graph (function) |
c_bbjzoa37wvdm | However, it is often useful to see functions as mappings, which consist not only of the relation between input and output, but also which set is the domain, and which set is the codomain. For example, to say that a function is onto (surjective) or not the codomain should be taken into account. The graph of a function o... | Graph (function) |
c_6n95iilqgzna | In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its pro... | Graph structure theorem |
c_seckvp0280xm | In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted gcd ( x , y ) {\displaystyle \gcd(x,y)} . For example, the GCD of 8 and 12 is 4, ... | Greatest Common Divisor |
c_9rko59eqfwgd | In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions, such as 5/6 = 1/2 + 1/3. As the name indica... | Greedy algorithm for Egyptian fractions |
c_bzwt06opl263 | Fibonacci actually lists several different methods for constructing Egyptian fraction representations. He includes the greedy method as a last resort for situations when several simpler methods fail; see Egyptian fraction for a more detailed listing of these methods. As Salzer (1948) details, the greedy method, and ext... | Greedy algorithm for Egyptian fractions |
c_77ehnohk34vq | In mathematics, the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational in the theory of quantum groups. | Group Hopf algebra |
c_9c4q4x5nnpha | In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the s... | Rotations in 4-dimensional Euclidean space |
c_55xmx8z3460q | In mathematics, the hafnian of an adjacency matrix of a graph is the number of perfect matchings in the graph. It was so named by Eduardo R. Caianiello "to mark the fruitful period of stay in Copenhagen (Hafnia in Latin). "The hafnian of a 2 n × 2 n {\displaystyle 2n\times 2n} symmetric matrix is computed as haf ( A ... | Hafnian |
c_qel729740wyy | . . , 2 n } {\displaystyle =\{1,2,...,2n\}} .Equivalently, haf ( A ) = ∑ M ∈ M ∏ ( u , v ) ∈ M A u , v {\displaystyle \operatorname {haf} (A)=\sum _{M\in {\mathcal {M}}}\prod _{\scriptscriptstyle (u,v)\in M}A_{u,v}} where M {\displaystyle {\mathcal {M}}} is the set of all 1-factors (perfect matchings) on the complete... | Hafnian |
c_o3504z2spp8p | In mathematics, the half-period ratio τ of an elliptic function is the ratio τ = ω 2 ω 1 {\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}} of the two half-periods ω 1 2 {\displaystyle {\frac {\omega _{1}}{2}}} and ω 2 2 {\displaystyle {\frac {\omega _{2}}{2}}} of the elliptic function, where the elliptic functio... | Half-period ratio |
c_acrsai65rql4 | Hence, the period ratio is the same as the "half-period ratio". Note that the half-period ratio can be thought of as a simple number, namely, one of the parameters to elliptic functions, or it can be thought of as a function itself, because the half periods can be given in terms of the elliptic modulus or in terms of t... | Half-period ratio |
c_hyupe0wpp43k | In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observatio... | Weighted harmonic mean |
c_1t0igtu3e2wr | In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: The first n {\displaystyle n} terms of the series sum to approximately ln n + γ {\displaystyle \ln n+\gamma } , where ln {\displaystyle \ln } is the natural logarithm and γ ≈ 0.577 {\displaystyle \gamma \approx 0... | Alternating harmonic series |
c_2ioa9w6bdv82 | In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solution x ∈ A, or the symbol ∞ if there is no such N. The p-height considers only divisibility properties by the powers of a... | Ulm's theorem |
c_zv3n84t3a1ya | In mathematics, the height zeta function of an algebraic variety or more generally a subset of a variety encodes the distribution of points of given height. | Height zeta function |
c_qtwkb8yir1e6 | In mathematics, the homology or cohomology of an algebra may refer to Banach algebra cohomology of a bimodule over a Banach algebra Cyclic homology of an associative algebra Group cohomology of a module over a group ring or a representation of a group Hochschild homology of a bimodule over an associative algebra Lie al... | Cohomology of algebras |
c_0tzruz2v3rdf | In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topologic... | Homotopy category of topological spaces |
c_w5yyznjzwk8i | In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, an... | Homotopy principle |
c_efbypmc0mk8f | It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash–Kuiper isometric C1 embedding theorem and the Smale–Hirsch immersion theorem. | Homotopy principle |
c_wh1kr1nmvcob | In mathematics, the horizontal line test is a test used to determine whether a function is injective (i.e., one-to-one). | Horizontal line test |
c_4ui4m6owr1s5 | In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals. Hypergeometric functions of a matrix argument have applications in random matrix th... | Hypergeometric function of a matrix argument |
c_zqzx1fz6qot7 | In mathematics, the hypergraph regularity method is a powerful tool in extremal graph theory that refers to the combined application of the hypergraph regularity lemma and the associated counting lemma. It is a generalization of the graph regularity method, which refers to the use of Szemerédi's regularity and counting... | Hypergraph regularity method |
c_7xlc9xoqak0h | This is an extension of Szemerédi's regularity lemma that partitions any given graph into bounded number parts such that edges between the parts behave almost randomly. Similarly, the hypergraph counting lemma is a generalization of the graph counting lemma that estimates number of copies of a fixed graph as a subgraph... | Hypergraph regularity method |
c_vz9ogddeopve | In mathematics, the hyperkähler quotient of a hyperkähler manifold acted on by a Lie group G is the quotient of a fiber of a hyperkähler moment map M → g ⊗ R 3 {\displaystyle M\to {\mathfrak {g}}\otimes \mathbb {R} ^{3}} over a G-fixed point by the action of G. It was introduced by Nigel Hitchin, Anders Karlhede, Ulf L... | Hyperkähler quotient |
c_ionutjgoxq2m | In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation ... | Hyperoperation |
c_arftbnopqkwh | In mathematics, the hypograph or subgraph of a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph. The domain (rather than... | Hypograph (mathematics) |
c_gx9yrx8i1rh0 | In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the alg... | Free functor |
c_qipc3pil9sca | In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. | Descent theory |
c_0ng7yx54i3l8 | In mathematics, the idea of geometric scaling can be generalized. The scale between two mathematical objects need not be a fixed ratio but may vary in some systematic way; this is part of mathematical projection, which generally defines a point by point relationship between two mathematical objects. (Generally, these m... | Scale ratio |
c_ganxp5szmxlf | In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on any subset of its domain that has a limit point. | Identity theorem for Riemann surfaces |
c_uo9apq0uxrnm | In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f {\displaystyle f} at each element of a given subset A {\displaystyle A} of its domain produces a set, called the "image of A {\displaystyle A} under (or through) f {\displaystyle f} ". S... | Inverse image |
c_dli7z8u6zpag | In mathematics, the imaginary unit i {\displaystyle i} is the square root of − 1 {\displaystyle -1} , such that i 2 {\displaystyle i^{2}} is defined to be − 1 {\displaystyle -1} . A number which is a direct multiple of i {\displaystyle i} is known as an imaginary number. : Chp 4 In certain physical theories, periods of... | Imaginary time |
c_6ivd4bk5vtxf | Mathematically, an imaginary time period τ {\textstyle \tau } may be obtained from real time t {\textstyle t} via a Wick rotation by π / 2 {\textstyle \pi /2} in the complex plane: τ = i t {\textstyle \tau =it} . : 769 Stephen Hawking popularized the concept of imaginary time in his book The Universe in a Nutshell. "On... | Imaginary time |
c_iohptqa4hf9c | From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been... | Imaginary time |
c_0av87a3myb99 | So what is real and what is imaginary? Is the distinction just in our minds?" In fact, the terms "real" and "imaginary" for numbers are just a historical accident, much like the terms "rational" and "irrational": "...the words real and imaginary are picturesque relics of an age when the nature of complex numbers was no... | Imaginary time |
c_jrsoo0q0zo55 | In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent. Let λ = ( λ 1 , λ 2 , … ) {\displaystyle \lambda =(\lambda _{1},\lambda _{2},\ldots )} be a partition of an integer n {\displaystyle n} and let χ λ... | Immanant |
c_mvk02jvmptw3 | In mathematics, the incomplete Fermi–Dirac integral for an index j is given by F j ( x , b ) = 1 Γ ( j + 1 ) ∫ b ∞ t j exp ( t − x ) + 1 d t . {\displaystyle F_{j}(x,b)={\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }{\frac {t^{j}}{\exp(t-x)+1}}\,dt.} This is an alternate definition of the incomplete polylogarithm. | Incomplete Fermi–Dirac integral |
c_aarlqvxtx6lw | In mathematics, the incompressibility method is a proof method like the probabilistic method, the counting method or the pigeonhole principle. To prove that an object in a certain class (on average) satisfies a certain property, select an object of that class that is incompressible. If it does not satisfy the property,... | Incompressibility method |
c_k8k75pkpuoxb | In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category C. The objects in this ind-completed category, denoted Ind(C), are known as direct systems, they are functors from a small filtered category I to C. The dual concept is the pro-completion, Pro(C)... | Pro-category |
c_qxa403fo9qi5 | In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. It is also called the pseudo-orthogonal group or generalized orthogonal group.... | Indefinite orthogonal group |
c_m1ba0u13zs25 | The signature of the form determines the group up to isomorphism; interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. If either p or q equals zero, then the group is isomorphic to the ordinary orthogonal group O(n). We assume in what follows that both p and q are positiv... | Indefinite orthogonal group |
c_9dqzi3ckqkd8 | The group O(p, q) is defined for vector spaces over the reals. For complex spaces, all groups O(p, q; C) are isomorphic to the usual orthogonal group O(p + q; C), since the transform z j ↦ i z j {\displaystyle z_{j}\mapsto iz_{j}} changes the signature of a form. This should not be confused with the indefinite unitary ... | Indefinite orthogonal group |
c_tmlwwzqhrf78 | In mathematics, the indefinite product operator is the inverse operator of Q ( f ( x ) ) = f ( x + 1 ) f ( x ) {\textstyle Q(f(x))={\frac {f(x+1)}{f(x)}}} . It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Some authors use term discrete multiplicative integrati... | Indefinite product |
c_qgkp52l6g6o6 | {\displaystyle Q\left(\prod _{x}f(x)\right)=f(x)\,.} More explicitly, if ∏ x f ( x ) = F ( x ) {\textstyle \prod _{x}f(x)=F(x)} , then F ( x + 1 ) F ( x ) = f ( x ) . {\displaystyle {\frac {F(x+1)}{F(x)}}=f(x)\,.} If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. T... | Indefinite product |
c_kioply7olxld | In mathematics, the indicator vector or characteristic vector or incidence vector of a subset T of a set S is the vector x T := ( x s ) s ∈ S {\displaystyle x_{T}:=(x_{s})_{s\in S}} such that x s = 1 {\displaystyle x_{s}=1} if s ∈ T {\displaystyle s\in T} and x s = 0 {\displaystyle x_{s}=0} if s ∉ T . {\displaystyle s\... | Indicator vector |
c_abyl708kd3z2 | In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the lis... | AM–GM inequality |
c_ytxg2usbyyq8 | Similarly, a square with all sides of length √xy has the perimeter 4√xy and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that 2x + 2y ≥ 4√xy and that only the square has the smallest perimeter amongst all rectangles of equal area. Extensions of the AM–... | AM–GM inequality |
c_63dnxbcslfny | In mathematics, the infimum (abbreviated inf; plural infima) of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the greatest element in P {\displaystyle P} that is less than or equal to each element of S , {\displaystyle S,} if such an element exists. In other words, it is the greatest el... | Infimum and supremum |
c_cw95laexzdfl | In other words, it is the least element of P {\displaystyle P} that is greater than or equal to the greatest element of S {\displaystyle S} . Consequently, the supremum is also referred to as the least upper bound (or LUB).The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real n... | Infimum and supremum |
c_wwt737wrq7ey | However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum... | Infimum and supremum |
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