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Conjecture Four color theorem Conjecture > Important examples > Four color theorem 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 cou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conjecture Four color theorem Conjecture > Important examples > Four color theorem 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 counterex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Four-color conjecture Summary Map-coloring_problem 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 s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Four-color conjecture Summary Map-coloring_problem 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 Ken... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Four-spiral semigroup Summary Four-spiral_semigroup 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 prope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fractional Laplacian Summary Fractional_Laplacian In mathematics, the fractional Laplacian is an operator, which generalizes the notion of Laplacian spatial derivatives to fractional powers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free category Summary Free_category 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free category Summary Free_category 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_{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free factor complex Summary Free_factor_complex 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 Kare... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free group Summary Free_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free matroid Summary Free_matroid 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orientation homology class Summary Fundamental_class 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 \mat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fundamental group scheme Summary Fundamental_group_scheme 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 it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fundamental theorem of Galois theory Summary Fundamental_theorem_of_Galois_theory 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 mos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fundamental Theorem of Arithmetic Summary Unique_factorization_theorem 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fundamental Theorem of Arithmetic Summary Unique_factorization_theorem 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 theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fuzzy sphere Summary Fuzzy_sphere 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 commutat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fuzzy sphere Summary Fuzzy_sphere 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. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gamma function Summary Euler_Gamma_Function 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 eve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gamma function Summary Euler_Gamma_Function 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite general linear group Summary Full_linear_monoid 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite general linear group Summary Full_linear_monoid 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 number... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite general linear group Summary Full_linear_monoid 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite general linear group Summary Full_linear_monoid 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 li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized Pochhammer symbol Summary Generalized_Pochhammer_symbol 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized dihedral group Summary Generalized_dihedral_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized minimal residual method Summary Generalized_minimal_residual_method 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 subsp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized minimal residual method Summary Generalized_minimal_residual_method 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 specia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized symmetric group Summary Generalized_symmetric_group 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized taxicab number Summary Generalized_taxicab_number 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 ,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized taxicab number Summary Generalized_taxicab_number 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^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Genus of a quadratic form Summary Genus_of_a_quadratic_form 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 equi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geodesics as Hamiltonian flows Summary Geodesics_as_Hamiltonian_flows 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometric Langlands correspondence Summary Geometric_Langlands_correspondence 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 fr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometric mean Summary Geometric_mean 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometric mean Summary Geometric_mean 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 benchmarkin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometric mean Summary Geometric_mean 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometric topology (object) Summary Geometric_topology_(object) In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometric-harmonic mean Summary Geometric-harmonic_mean 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gluing axiom Summary Gluing_axiom 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 is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lattice reduction Summary Lattice_reduction 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 l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gonality of an algebraic curve Summary Gonality_of_an_algebraic_curve 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gonality of an algebraic curve Summary Gonality_of_an_algebraic_curve 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 equat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gonality of an algebraic curve Summary Gonality_of_an_algebraic_curve 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), wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gradient conjecture Summary Gradient_conjecture 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 con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Grand Riemann hypothesis Summary Grand_Riemann_hypothesis 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}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph Fourier Transform Summary Graph_Fourier_Transform 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph (function) Summary Surface_plot_(mathematics) 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph (function) Summary Surface_plot_(mathematics) 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph (function) Summary Surface_plot_(mathematics) 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph (function) Summary Surface_plot_(mathematics) 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 sho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph structure theorem Summary Graph_structure_theorem 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Greatest Common Divisor Summary Greatest_Common_Divisor 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 ) {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Greedy algorithm for Egyptian fractions Summary Greedy_algorithm_for_Egyptian_fractions 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 f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Greedy algorithm for Egyptian fractions Summary Greedy_algorithm_for_Egyptian_fractions 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Group Hopf algebra Summary Group_Hopf_algebra 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rotations in 4-dimensional Euclidean space Summary Rotations_in_4-dimensional_Euclidean_space 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 r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hafnian Summary Hafnian 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hafnian Summary Hafnian . . , 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 ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Half-period ratio Summary Half-period_ratio 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 e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Half-period ratio Summary Half-period_ratio 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 te... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weighted harmonic mean Summary Subcontrary_mean 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating harmonic series Summary Harmonic_series_(mathematics) 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ulm's theorem Summary Ulm's_theorem 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 divisi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Height zeta function Summary Height_zeta_function 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cohomology of algebras Summary Cohomology_of_algebras 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 hom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homotopy category of topological spaces Summary Homotopy_category 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. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homotopy principle Summary Homotopy_principle 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, i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homotopy principle Summary Homotopy_principle 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 Sma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Horizontal line test Summary Horizontal_line_test In mathematics, the horizontal line test is a test used to determine whether a function is injective (i.e., one-to-one). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hypergeometric function of a matrix argument Summary Hypergeometric_function_of_a_matrix_argument 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 multivari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hypergraph regularity method Summary Hypergraph_regularity_method 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 meth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hypergraph regularity method Summary Hypergraph_regularity_method 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 le... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperkähler quotient Summary Hyperkähler_quotient 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperoperation Summary Hyperoperation 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), multi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hypograph (mathematics) Summary Hypograph_(mathematics) 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free functor Summary Cofree_functor 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Descent theory Summary Descent_(mathematics) 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scale ratio Mathematics Scale_ratio > Mathematics 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 bet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Identity theorem for Riemann surfaces Summary Identity_theorem_for_Riemann_surfaces 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inverse image Summary Image_(mathematics) 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} u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Imaginary time Origins Imaginary_time > Origins 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. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Imaginary time Origins Imaginary_time > Origins 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Imaginary time Origins Imaginary_time > Origins 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 al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Imaginary time Origins Imaginary_time > Origins 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Immanant Summary The_immanant_of_a_matrix 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incomplete Fermi–Dirac integral Summary Incomplete_Fermi–Dirac_integral 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.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incompressibility method Summary Incompressibility_method 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pro-category Summary Pro-category 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 con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indefinite orthogonal group Summary Generalized_orthogonal_group 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 ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indefinite orthogonal group Summary Generalized_orthogonal_group 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 gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indefinite orthogonal group Summary Generalized_orthogonal_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indefinite product Summary Indefinite_product 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 auth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indefinite product Summary Indefinite_product {\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 give... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indicator vector Summary Indicator_vector 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 {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
AM–GM inequality Summary Arithmetic-geometric_mean_inequality 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
AM–GM inequality Summary Arithmetic-geometric_mean_inequality 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 perimet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infimum and supremum Summary Least_upper_bound 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 elemen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infimum and supremum Summary Least_upper_bound 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 conc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infimum and supremum Summary Least_upper_bound 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 char... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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