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wiki_3700_chunk_0 | Sigma additive | Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity. The term modul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3701_chunk_0 | Adherent point | This definition differs from that of a limit point of a set, in that for a limit point it is required that every neighborhood of x {\displaystyle x} contains at least one point of A {\displaystyle A} different from x . {\displaystyle x.} Thus every limit point is an adherent point, but the converse is not true. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3702_chunk_0 | Adherent point | An adherent point of A {\displaystyle A} is either a limit point of A {\displaystyle A} or an element of A {\displaystyle A} (or both). An adherent point which is not a limit point is an isolated point. Intuitively, having an open set A {\displaystyle A} defined as the area within (but not including) some boundary, the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3703_chunk_0 | Adjoint bundle | In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles have important applications in the theory of connections as well as in gauge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3704_chunk_0 | Admissible algebra | In mathematics, an admissible algebra is a (possibly non-associative) commutative algebra whose enveloping Lie algebra of derivations splits into the sum of an even and an odd part. Admissible algebras were introduced by Koecher (1967). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3705_chunk_0 | Affine Hecke algebra | In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3706_chunk_0 | Affine Kac–Moody algebra | In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, af... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3707_chunk_0 | Affine Kac–Moody algebra | As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3708_chunk_0 | Affine Kac–Moody algebra | More generally, if σ is an automorphism of the simple Lie algebra g {\displaystyle {\mathfrak {g}}} associated to an automorphism of its Dynkin diagram, the twisted loop algebra L σ g {\displaystyle L_{\sigma }{\mathfrak {g}}} consists of g {\displaystyle {\mathfrak {g}}} -valued functions f on the real line which sati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3709_chunk_0 | Algebraic curves | In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3710_chunk_0 | Algebraic curves | This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula). A non-plane curve is often called a space curve or a skew curve. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3711_chunk_0 | Affine combination | In mathematics, an affine combination of x1, ..., xn is a linear combination ∑ i = 1 n α i ⋅ x i = α 1 x 1 + α 2 x 2 + ⋯ + α n x n , {\displaystyle \sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n},} such that ∑ i = 1 n α i = 1. {\displaystyle \sum _{i=1}^{n}{\alpha _{i}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3712_chunk_0 | Affine combination | This concept is fundamental in Euclidean geometry and affine geometry, because the set of all affine combinations of a set of points forms the smallest subspace containing the points, exactly as the linear combinations of a set of vectors form their linear span. The affine combinations commute with any affine transform... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3713_chunk_0 | Affine representation | In mathematics, an affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3714_chunk_0 | Affine line | In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3715_chunk_0 | Affine line | Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3716_chunk_0 | Affine line | In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3717_chunk_0 | Affine line | One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3718_chunk_0 | Affine line | Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3719_chunk_0 | Algebra bundle | In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that the transition functions are algebra isomorphisms. Since algebras are also vector spaces, every algebra bundle is a vector bundle. Examples include the tensor-algebra bu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3720_chunk_0 | Algebra homomorphism | In mathematics, an algebra homomorphism is a homomorphism between two algebras. More precisely, if A and B are algebras over a field (or a ring) K, it is a function F: A → B {\displaystyle F\colon A\to B} such that, for all k in K and x, y in A, one has F ( k x ) = k F ( x ) {\displaystyle F(kx)=kF(x)} F ( x + y ) = F ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3721_chunk_0 | Algebra (ring theory) | In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3722_chunk_0 | Algebra (ring theory) | An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order n forms a unital algebra since the identity matrix of order n is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a (un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3723_chunk_0 | Algebra (ring theory) | Many authors use the term algebra to mean associative algebra, or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra. Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Algebras are not to be co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3724_chunk_0 | Homotopy associative algebra | In mathematics, an algebra such as ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} has multiplication ⋅ {\displaystyle \cdot } whose associativity is well-defined on the nose. This means for any real numbers a , b , c ∈ R {\displaystyle a,b,c\in \mathbb {R} } we have a ⋅ ( b ⋅ c ) − ( a ⋅ b ) ⋅ c = 0 {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3725_chunk_0 | Homotopy associative algebra | The study of A ∞ {\displaystyle A_{\infty }} -algebras is a subset of homotopical algebra, where there is a homotopical notion of associative algebras through a differential graded algebra with a multiplication operation and a series of higher homotopies giving the failure for the multiplication to be associative. Loos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3726_chunk_0 | Homotopy associative algebra | When looking at the underlying cohomology algebra H ( A ∙ , m 1 ) {\displaystyle H(A^{\bullet },m_{1})} , the map m 2 {\displaystyle m_{2}} should be an associative map. Then, these higher maps m 3 , m 4 , … {\displaystyle m_{3},m_{4},\ldots } should be interpreted as higher homotopies, where m 3 {\displaystyle m_{3}} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3727_chunk_0 | Homotopy associative algebra | Their structure was originally discovered by Jim Stasheff while studying A∞-spaces, but this was interpreted as a purely algebraic structure later on. These are spaces equipped with maps that are associative only up to homotopy, and the A∞ structure keeps track of these homotopies, homotopies of homotopies, and so fort... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3728_chunk_0 | Algebraic cycle | The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group H 2 ( S ) {\displaystyle H^{2}(S)} contains transcendental information, and in effect Mumford's theorem implies that, despite CH 2 ( S ) {\displaystyle \operatorname {CH} ^{2}(S)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3729_chunk_0 | Algebraic cycle | The Hodge conjecture, one of the Clay Mathematics Institute's Millennium Prize Problems, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algeb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3730_chunk_0 | Polynomial vector field | In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used. The intention is to include equations formed by means of differential operators, in which the coeff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3731_chunk_0 | Polynomial vector field | Algebraic differential equations are widely used in computer algebra and number theory. A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3732_chunk_0 | Polynomial equation | In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0 {\displaystyle P=0} where P is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term algebraic equation refers only to univariate equations, that is polynomial equati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3733_chunk_0 | Polynomial equation | For example, x 5 − 3 x + 1 = 0 {\displaystyle x^{5}-3x+1=0} is an algebraic equation with integer coefficients and y 4 + x y 2 − x 3 3 + x y 2 + y 2 + 1 7 = 0 {\displaystyle y^{4}+{\frac {xy}{2}}-{\frac {x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0} is a multivariate polynomial equation over the rationals. Some but not all... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3734_chunk_0 | Algebraic expression | In mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression. Since taking th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3735_chunk_0 | Algebraic expression | Thus, 3 x − 2 x y + c y − 1 {\displaystyle {\frac {3x-2xy+c}{y-1}}} is a rational expression, whereas 1 − x 2 1 + x 2 {\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}} is not. A rational equation is an equation in which two rational fractions (or rational expressions) of the form P ( x ) Q ( x ) {\displaystyle {\frac ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3736_chunk_0 | Algebraic extension | In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is, every element of L is a root of a non-zero polynomial with coefficients in K. A field extension that is not algebraic, is said to be transcendental, and must cont... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3737_chunk_0 | Algebraic function field | In mathematics, an algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K =... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3738_chunk_0 | Algebraic function | In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional pow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3739_chunk_0 | Algebraic function | It can be shown that the same class of functions is obtained if algebraic numbers are accepted for the coefficients of the ai(x)'s. If transcendental numbers occur in the coefficients the function is, in general, not algebraic, but it is algebraic over the field generated by these coefficients. The value of an algebrai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3740_chunk_0 | Algebraic function | Sometimes, coefficients a i ( x ) {\displaystyle a_{i}(x)} that are polynomial over a ring R are considered, and one then talks about "functions algebraic over R". A function which is not algebraic is called a transcendental function, as it is for example the case of exp x , tan x , ln x , Γ ( x ) {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3741_chunk_0 | Algebraic function | A composition of transcendental functions can give an algebraic function: f ( x ) = cos arcsin x = 1 − x 2 {\displaystyle f(x)=\cos \arcsin x={\sqrt {1-x^{2}}}} . As a polynomial equation of degree n has up to n roots (and exactly n roots over an algebraically closed field, such as the complex numbers), a polynomia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3742_chunk_0 | Algebraic function | {\displaystyle y^{2}+x^{2}=1.\,} This determines y, except only up to an overall sign; accordingly, it has two branches: y = ± 1 − x 2 . {\displaystyle y=\pm {\sqrt {1-x^{2}}}.\,} An algebraic function in m variables is similarly defined as a function y = f ( x 1 , … , x m ) {\displaystyle y=f(x_{1},\dots ,x_{m})} whic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3743_chunk_0 | Algebraic function | It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field of rational functions K(x1, ..., xm). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3744_chunk_0 | Group variety | In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, ort... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3745_chunk_0 | Group variety | An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called linear algebraic groups. Another class is formed by the abelian varieties, wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3746_chunk_0 | Power basis | In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle \mathbb {Q} } such that the field extension K / Q {\displaystyle K/\mathbb {Q} } has finite degree (and hence is an algebraic field extension). Thus K {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3747_chunk_0 | Algebraic representation | In mathematics, an algebraic representation of a group G on a k-algebra A is a linear representation π: G → G L ( A ) {\displaystyle \pi :G\to GL(A)} such that, for each g in G, π ( g ) {\displaystyle \pi (g)} is an algebra automorphism. Equipped with such a representation, the algebra A is then called a G-algebra. For... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3748_chunk_0 | Algebraic stack | In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of point... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3749_chunk_0 | Ternary ring | There is wide variation in the terminology. Planar ternary rings or ternary fields as defined here have been called by other names in the literature, and the term "planar ternary ring" can mean a variant of the system defined here. The term "ternary ring" often means a planar ternary ring, but it can also simply mean a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3750_chunk_0 | Structure (algebraic) | In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic str... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3751_chunk_0 | Structure (algebraic) | The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomorphisms). In universal algebra, an algebraic structure is called an algebra; this term m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3752_chunk_0 | Algebraic surface | In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is muc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3753_chunk_0 | Algebraic torus | In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by G m {\displaystyle \mathbf {G} _{\mathbf {m} }} , G m {\displaystyle \mathbb {G} _{m}} , or T {\displaystyle \mathbb {T} } , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3754_chunk_0 | Algebraic torus | For example, over the complex numbers C {\displaystyle \mathbb {C} } the algebraic torus G m {\displaystyle \mathbf {G} _{\mathbf {m} }} is isomorphic to the group scheme C ∗ = Spec ( C ) {\displaystyle \mathbb {C} ^{*}={\text{Spec}}(\mathbb {C} )} , which is the scheme theoretic analogue of the Lie group U ( 1 ) ⊂ C ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3755_chunk_0 | Algebroid function | In mathematics, an algebroid function is a solution of an algebraic equation whose coefficients are analytic functions. So y(z) is an algebroid function if it satisfies a d ( z ) y d + … + a 0 ( z ) = 0 , {\displaystyle a_{d}(z)y^{d}+\ldots +a_{0}(z)=0,} where a k ( z ) {\displaystyle a_{k}(z)} are analytic. If this eq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3756_chunk_0 | Almost complex manifold | In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in sym... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3757_chunk_0 | Almost perfect number | In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. The only kno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3758_chunk_0 | Almost periodic functions | In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3759_chunk_0 | Almost periodic functions | Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a planetary system, with planets in orbits moving with periods that are not commensurable (i.e., with a period vector that is not proportional to a vector of integers). A th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3760_chunk_0 | Alternating algebra | In mathematics, an alternating algebra is a Z-graded algebra for which xy = (−1)deg(x)deg(y)yx for all nonzero homogeneous elements x and y (i.e. it is an anticommutative algebra) and has the further property that x2 = 0 for every homogeneous element x of odd degree. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3761_chunk_0 | Alternating series | In mathematics, an alternating series is an infinite series of the form or with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3762_chunk_0 | Alternating sign matrix | In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3763_chunk_0 | Amenable group | In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of G, was introduced by John von Neumann in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3764_chunk_0 | Analytical function | In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real anal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3765_chunk_0 | Analytical function | It is important to note that it's a neighborhood and not just at some point x 0 {\displaystyle x_{0}} , since every differentiable function has at least a tangent line at every point, which is its Taylor series of order 1. So just having a polynomial expansion at singular points is not enough, and the Taylor series mus... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3766_chunk_0 | Real-analytic manifold | In mathematics, an analytic manifold, also known as a C ω {\displaystyle C^{\omega }} manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic geometry, analytic spaces are a generalization of ana... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3767_chunk_0 | Analytic proof | In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods. The term was first used by Bernard Bolzano, who first provided a non-analytic proof of his intermediate value theorem and the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3768_chunk_0 | Ancient solution | In mathematics, an ancient solution to a differential equation is a solution that can be extrapolated backwards to all past times, without singularities. That is, it is a solution "that is defined on a time interval of the form (−∞, T). "The term was introduced by Richard Hamilton in his work on the Ricci flow. It has ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3769_chunk_0 | Anti-diagonal matrix | In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗), known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diago... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3770_chunk_0 | Antiautomorphism | In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From bijectivity it follows that antiautomorphisms have inverses, and that the inve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3771_chunk_0 | Antimatroid | In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids are commonly axiomatized in two equivalent ways, either as a set system m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3772_chunk_0 | Antiunitary operator | In mathematics, an antiunitary transformation, is a bijective antilinear map U: H 1 → H 2 {\displaystyle U:H_{1}\to H_{2}\,} between two complex Hilbert spaces such that ⟨ U x , U y ⟩ = ⟨ x , y ⟩ ¯ {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}} for all x {\displaystyle x} and y {\displaystyle y... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3773_chunk_0 | Anyonic Lie algebra | In mathematics, an anyonic Lie algebra is a U(1) graded vector space L {\displaystyle L} over C {\displaystyle \mathbb {C} } equipped with a bilinear operator : L × L → L {\displaystyle \colon L\times L\rightarrow L} and linear maps ε: L → C {\displaystyle \varepsilon \colon L\to \mathbb {C} } (some authors use | ⋅ |: ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3774_chunk_0 | Approximate group | In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correct). For example, it is required that the set of products of elements in the subset be not much bigger than the subset it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3775_chunk_0 | AF C*-algebra | In mathematics, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3776_chunk_0 | AF C*-algebra | Its proof divides into two parts. The invariant here is K0 with its natural order structure; this is a functor. First, one proves existence: a homomorphism between invariants must lift to a *-homomorphism of algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3777_chunk_0 | AF C*-algebra | Second, one shows uniqueness: the lift must be unique up to approximate unitary equivalence. Classification then follows from what is known as the intertwining argument. For unital AF algebras, both existence and uniqueness follow from the fact the Murray-von Neumann semigroup of projections in an AF algebra is cancell... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3778_chunk_0 | Argument of a function | In mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable.For example, the binary function f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} has two arguments, x {\displaystyle x} and y {\displaystyle y} , in an ordered pair ( x , y... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3779_chunk_0 | Argument of a function | A function that takes a single argument as input, such as f ( x ) = x 2 {\displaystyle f(x)=x^{2}} , is called a unary function. A function of two or more variables is considered to have a domain consisting of ordered pairs or tuples of argument values. The argument of a circular function is an angle. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3780_chunk_0 | Argument of a function | The argument of a hyperbolic function is a hyperbolic angle. A mathematical function has one or more arguments in the form of independent variables designated in the definition, which can also contain parameters. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3781_chunk_0 | Argument of a function | The independent variables are mentioned in the list of arguments that the function takes, whereas the parameters are not. For example, in the logarithmic function f ( x ) = log b ( x ) , {\displaystyle f(x)=\log _{b}(x),} the base b {\displaystyle b} is considered a parameter. Sometimes, subscripts can be used to den... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3782_chunk_0 | Arithmetic group | In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example S L 2 ( Z ) . {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} ).} They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3783_chunk_0 | Arithmetic surface | In mathematics, an arithmetic surface over a Dedekind domain R with fraction field K {\displaystyle K} is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes. When R is the ring of integers Z, this intuition depends on the prime ideal spectrum Spec(Z) b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3784_chunk_0 | Linear associative algebra | In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field K. The addition and multiplication operations together give A the structure of a ring; the addition and scalar mult... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3785_chunk_0 | Linear associative algebra | We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3786_chunk_0 | Asymmetric norm | In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3787_chunk_0 | Asymptotic series | In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particula... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3788_chunk_0 | Asymptotic series | Repeated integration by parts will often lead to an asymptotic expansion. Since a convergent Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3789_chunk_0 | Asymptotic series | The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asympt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3790_chunk_0 | Asymptotic series | The error is then typically of the form ~ exp(−c/ε) where ε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3791_chunk_0 | Automorphic L-function | In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive group G over a global field and a finite-dimensional complex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet cha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3792_chunk_0 | Automorphic factor | In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article 'factor of automorphy'. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3793_chunk_0 | Automorphic function | In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3794_chunk_0 | Automorphic number | In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base b {\displaystyle b} whose square "ends" in the same digits as the number itself. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3795_chunk_0 | Autonomous convergence theorem | In mathematics, an autonomous convergence theorem is one of a family of related theorems which specify conditions guaranteeing global asymptotic stability of a continuous autonomous dynamical system. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3796_chunk_0 | Non-autonomous system (mathematics) | In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } over R {\displaystyle \mathbb {R} } . For instance, this is the case of non-autonomous mechanics. An r-order differential equati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3797_chunk_0 | Non-autonomous system (mathematics) | A dynamic equation on Q → R {\displaystyle Q\to \mathbb {R} } is a differential equation which is algebraically solved for a higher-order derivatives. In particular, a first-order dynamic equation on a fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } is a kernel of the covariant differential of some connection Γ {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3798_chunk_0 | Non-autonomous system (mathematics) | {\displaystyle q_{t}^{i}=\Gamma (t,q^{i}).} For instance, this is the case of Hamiltonian non-autonomous mechanics. A second-order dynamic equation q t t i = ξ i ( t , q j , q t j ) {\displaystyle q_{tt}^{i}=\xi ^{i}(t,q^{j},q_{t}^{j})} on Q → R {\displaystyle Q\to \mathbb {R} } is defined as a holonomic connection ξ {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3799_chunk_0 | Non-autonomous system (mathematics) | This equation also is represented by a connection on an affine jet bundle J 1 Q → Q {\displaystyle J^{1}Q\to Q} . Due to the canonical embedding J 1 Q → T Q {\displaystyle J^{1}Q\to TQ} , it is equivalent to a geodesic equation on the tangent bundle T Q {\displaystyle TQ} of Q {\displaystyle Q} . A free motion equation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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