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Algebra homomorphism Summary 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 β 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebra (ring theory) Summary Algebra_over_a_commutative_ring 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 mul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebra (ring theory) Summary Algebra_over_a_commutative_ring 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 multiplic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebra (ring theory) Summary Algebra_over_a_commutative_ring 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 gener... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homotopy associative algebra Summary Homotopy_associative_algebra In mathematics, an algebra such as ( R , + , β
) {\displaystyle (\mathbb {R} ,+,\cdot )} has multiplication β
{\displaystyle \cdot } whose associativity is well-defined on the nose. This means for any real numbers a , b , c β R {\displaystyle a,b,c\in \m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homotopy associative algebra Summary Homotopy_associative_algebra The study of A β {\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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homotopy associative algebra Summary Homotopy_associative_algebra When looking at the underlying cohomology algebra H ( A β , 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 , β¦ {\displaystyle m_{3},m_{4},\ldots } should be ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homotopy associative algebra Summary Homotopy_associative_algebra Their structure was originally discovered by Jim Stasheff while studying Aβ-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 Aβ structure keep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic cycle Summary Algebraic_cycle In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic cycle Summary Algebraic_cycle The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic cycle Summary Algebraic_cycle While divisors on higher-dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider. The behavior of these cycles is strikingly different from tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic cycle Summary 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 β‘ ( S ) {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic cycle Summary 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 Γ©tale cohomology. Alexander Grot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomial vector field Summary 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 forme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomial vector field Summary 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 wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomial equation Summary Algebraic_equations 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomial equation Summary Algebraic_equations For example, x 5 β 3 x + 1 = 0 {\displaystyle x^{5}-3x+1=0} is an algebraic equation with integer coefficients and y 4 + x y 2 β 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 polynomi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic expression Summary Algebraic_formula 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 β 2xy ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic expression Summary Algebraic_formula Usually, Ο is constructed as a geometric relationship, and the definition of e requires an infinite number of algebraic operations. A rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations (com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic expression Summary Algebraic_formula Thus, 3 x β 2 x y + c y β 1 {\displaystyle {\frac {3x-2xy+c}{y-1}}} is a rational expression, whereas 1 β 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic expression Summary Algebraic_formula These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic extension Summary Algebraic_field_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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic function field Summary 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 m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic function Summary Algebraic_functions 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, multiplica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic function Summary Algebraic_functions 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic function Summary Algebraic_functions 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 β‘... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic function Summary Algebraic_functions A composition of transcendental functions can give an algebraic function: f ( x ) = cos β‘ arcsin β‘ x = 1 β 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 f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic function Summary Algebraic_functions {\displaystyle y^{2}+x^{2}=1.\,} This determines y, except only up to an overall sign; accordingly, it has two branches: y = Β± 1 β 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 , β¦ , x m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic function Summary Algebraic_functions 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 fiel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Group variety Summary 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 ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Group variety Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex projective manifold Summary Projective_algebraic_manifold In mathematics, an algebraic manifold is an algebraic variety which is also a manifold. As such, algebraic manifolds are a generalisation of the concept of smooth curves and surfaces defined by polynomials. An example is the sphere, which can be defined ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex projective manifold Summary Projective_algebraic_manifold For an algebraic manifold, the ground field will be the real numbers or complex numbers; in the case of the real numbers, the manifold of real points is sometimes called a Nash manifold. Every sufficiently small local patch of an algebraic manifold is is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic matroid Summary Algebraic_matroid In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power basis Summary Algebraic_number_field 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 al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic representation Summary Algebraic_representation In mathematics, an algebraic representation of a group G on a k-algebra A is a linear representation Ο: G β G L ( A ) {\displaystyle \pi :G\to GL(A)} such that, for each g in G, Ο ( g ) {\displaystyle \pi (g)} is an algebra automorphism. Equipped with such a rep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic stack Summary 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 use... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ternary ring Summary Planar_ternary_ring In mathematics, an algebraic structure ( R , T ) {\displaystyle (R,T)} consisting of a non-empty set R {\displaystyle R} and a ternary mapping T: R 3 β R {\displaystyle T\colon R^{3}\to R\,} may be called a ternary system. A planar ternary ring (PTR) or ternary field is special ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ternary ring Summary Planar_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 te... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Structure (algebraic) Summary Pointed_unary_system 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, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Structure (algebraic) Summary Pointed_unary_system 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 al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic surface Summary 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 manif... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic torus Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic torus Summary 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 β = Spec ( C ) {\displaystyle \mathbb {C} ^{*}={\text{Spec}}(\mathbb {C} )} , which is the scheme theoreti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete intersection Summary Complete_intersection In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pn, there should exist n β m homogeneous polynomials: F i ( X 0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebroid function Summary 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 + β¦ + a 0 ( z ) = 0 , {\displaystyle a_{d}(z)y^{d}+\ldots +a_{0}(z)=0,} where a k ( z ) {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Aliquot sequence Summary Aliquot_sequence In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
All one polynomial Summary All_one_polynomial In mathematics, an all one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are known, which allow this polynomial to be used to define efficient algorithms and circuits for mult... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Almost complex manifold Summary 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. Almo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Almost perfect number Summary 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 Ο(n)) is equal to 2n β 1, the sum of all proper divisors of n, s(n)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Almost periodic functions Summary 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 lat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Almost periodic functions Summary 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 ve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating algebra Summary Alternating_product In mathematics, an alternating algebra is a Z-graded algebra for which xy = (β1)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 |
Alternating factorial Summary Alternating_factorial In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials of positive integers. This is the same as their sum, with the odd-indexed factorials multiplied by β1 if n is even, and the even-indexed factorials multipli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating factorial Summary Alternating_factorial or with the recurrence relation af β‘ ( n ) = n ! β af β‘ ( n β 1 ) {\displaystyle \operatorname {af} (n)=n!-\operatorname {af} (n-1)} in which af(1) = 1. The first few alternating factorials are 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating factorial Summary Alternating_factorial β 2! + 3!. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating factorial Summary Alternating_factorial The fourth alternating factorial is β1! + 2! β 3! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating factorial Summary Alternating_factorial + 4! = 19. Regardless of the parity of n, the last (nth) summand, n!, is given a positive sign, the (n β 1)th summand is given a negative sign, and the signs of the lower-indexed summands are alternated accordingly. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating factorial Summary Alternating_factorial This pattern of alternation ensures the resulting sums are all positive integers. Changing the rule so that either the odd- or even-indexed summands are given negative signs (regardless of the parity of n) changes the signs of the resulting sums but not their absolute... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating groups Summary Alternating_group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt(n). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating series Summary Alternating_sum 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 con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating sign matrix Summary Alternating_sign_matrix In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and β1s 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Amenable group Summary 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Amenable group Summary Amenable_group An intuitive way to understand this version is that the support of the regular representation is the whole space of irreducible representations. In discrete group theory, where G has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Amicable triple Summary Amicable_triple In mathematics, an amicable triple is a set of three different numbers so related that the restricted sum of the divisors of each is equal to the sum of other two numbers.In another equivalent characterization, an amicable triple is a set of three different numbers so related tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Analytical function Summary Analytic_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 prope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Analytical function Summary Analytic_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 p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real-analytic manifold Summary Real-analytic_manifold In mathematics, an analytic manifold, also known as a C Ο {\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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real-analytic manifold Summary Real-analytic_manifold ( x β x 0 ) Ξ± {\displaystyle T_{f}(\mathbf {x} )=\sum _{|\alpha |\geq 0}{\frac {D^{\alpha }f(\mathbf {x_{0}} )}{\alpha ! }}(\mathbf {x} -\mathbf {x_{0}} )^{\alpha }} converges to f ( x ) {\displaystyle f(\mathbf {x} )} in a neighborhood of x 0 {\displaystyle \mathbf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Analytic proof Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Analytic semigroup Summary Analytic_semigroup In mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ancient solution Summary 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 (ββ, T). "The term was introduced by Richard Hamil... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Punctured disk Summary Punctured_disk In mathematics, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word anulus or annulus meaning 'little ring'. The adjectival form is annula... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Anti-diagonal matrix Summary 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 (β), known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Antiautomorphism Summary 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 antiauto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Antimatroid Summary 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 equivalen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Antiunitary operator Summary Antiunitary_operator In mathematics, an antiunitary transformation, is a bijective antilinear map U: H 1 β H 2 {\displaystyle U:H_{1}\to H_{2}\,} between two complex Hilbert spaces such that β¨ U x , U y β© = β¨ x , y β© Β― {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Anyonic Lie algebra Summary 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 Γ L β L {\displaystyle \colon L\times L\rightarrow L} and linear maps Ξ΅: L β C {\displaystyle \varepsilon \co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Aperiodic semigroup Summary Aperiodic_semigroup In mathematics, an aperiodic semigroup is a semigroup S such that every element is aperiodic, that is, for each x in S there exists a positive integer n such that xn = xn+1. An aperiodic monoid is an aperiodic semigroup which is a monoid. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Approximate group Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
AF C*-algebra Summary Approximately_finite_dimensional_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 Brat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
AF C*-algebra Summary Approximately_finite_dimensional_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 |
AF C*-algebra Summary Approximately_finite_dimensional_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 Murra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Argument of a function Summary Function_argument 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Argument of a function Summary Function_argument 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Argument of a function Summary Function_argument 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 |
Argument of a function Summary Function_argument 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 β‘ ( x ) , {\displaystyle f(x)=\log _{b}(x),} the base b {\displaystyle b} is considered a pa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arithmetic group Summary Arithmetic_subgroup 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arithmetic surface Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arithmetic variety Summary Arithmetic_variety In mathematics, an arithmetic variety is the quotient space of a Hermitian symmetric space by an arithmetic subgroup of the associated algebraic Lie group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stasheff polytope Summary Stasheff_polytope In mathematics, an associahedron Kn is an (n β 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of n letters, and the edges correspond to single application of the associativity rule. E... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linear associative algebra Summary 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 togethe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linear associative algebra Summary 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-bi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Asymmetric norm Summary 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 |
Asymmetric relation Summary Asymmetric_relation In mathematics, an asymmetric relation is a binary relation R {\displaystyle R} on a set X {\displaystyle X} where for all a , b β X , {\displaystyle a,b\in X,} if a {\displaystyle a} is related to b {\displaystyle b} then b {\displaystyle b} is not related to a . {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Asymptotic series Summary Asymptotic_expansions In mathematics, an asymptotic expansion, asymptotic series or PoincarΓ© expansion (after Henri PoincarΓ©) 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 ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Asymptotic series Summary Asymptotic_expansions 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Asymptotic series Summary Asymptotic_expansions 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Asymptotic series Summary Asymptotic_expansions The error is then typically of the form ~ exp(βc/Ξ΅) where Ξ΅ 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Atoroidal Summary Atoroidal In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non-boundary parallel, incompressible torus, or it may be defined algebraically, as a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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