text stringlengths 9 3.55k | source stringlengths 31 280 |
|---|---|
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://en.wikipedia.org/wiki/Algebra_homomorphism |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Algebra_(ring_theory) |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Homotopy_associative_algebra |
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 insights into the structure of the varie... | https://en.wikipedia.org/wiki/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 regular differentials on a compact Riemann... | https://en.wikipedia.org/wiki/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 that of divisors. For example, every curve ... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Algebraic_cycle |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Polynomial_vector_field |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Polynomial_equation |
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://en.wikipedia.org/wiki/Algebraic_expression |
Usually, π 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 (commutative properties and associative properties ... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Algebraic_expression |
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://en.wikipedia.org/wiki/Algebraic_expression |
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://en.wikipedia.org/wiki/Algebraic_extension |
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://en.wikipedia.org/wiki/Algebraic_function_field |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Algebraic_function |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Group_variety |
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 as the zero set of the polynomial x2 + y2 + z2 – 1, and hence is a... | https://en.wikipedia.org/wiki/Complex_projective_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 isomorphic to km where k is the ground field. Equivalently the varie... | https://en.wikipedia.org/wiki/Complex_projective_manifold |
In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence. | https://en.wikipedia.org/wiki/Algebraic_matroid |
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://en.wikipedia.org/wiki/Power_basis |
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://en.wikipedia.org/wiki/Algebraic_representation |
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://en.wikipedia.org/wiki/Algebraic_stack |
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 → R {\displaystyle T\colon R^{3}\to R\,} may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall H... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Ternary_ring |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Structure_(algebraic) |
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://en.wikipedia.org/wiki/Algebraic_surface |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Algebraic_torus |
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 − m homogeneous polynomials: F i ( X 0 , ⋯ , X n ) , 1 ≤ i ≤ n − m , {\displaystyle F_{i}(... | https://en.wikipedia.org/wiki/Complete_intersection |
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://en.wikipedia.org/wiki/Algebroid_function |
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://en.wikipedia.org/wiki/Aliquot_sequence |
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 multiplication in finite fields of characteristic ... | https://en.wikipedia.org/wiki/All_one_polynomial |
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://en.wikipedia.org/wiki/Almost_complex_manifold |
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://en.wikipedia.org/wiki/Almost_perfect_number |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Almost_periodic_functions |
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://en.wikipedia.org/wiki/Alternating_algebra |
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 −1 if n is even, and the even-indexed factorials multiplied by −1 if n is odd, resulting in an alternation of... | https://en.wikipedia.org/wiki/Alternating_factorial |
or with the recurrence relation af ( n ) = n ! − af ( n − 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, 5784634181, 81393657019 (sequence A005165 in the OEI... | https://en.wikipedia.org/wiki/Alternating_factorial |
– 2! + 3!. | https://en.wikipedia.org/wiki/Alternating_factorial |
The fourth alternating factorial is −1! + 2! − 3! | https://en.wikipedia.org/wiki/Alternating_factorial |
+ 4! = 19. Regardless of the parity of n, the last (nth) summand, n!, is given a positive sign, the (n – 1)th summand is given a negative sign, and the signs of the lower-indexed summands are alternated accordingly. | https://en.wikipedia.org/wiki/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 values. Miodrag Zivković proved in 1999 that there ... | https://en.wikipedia.org/wiki/Alternating_factorial |
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://en.wikipedia.org/wiki/Alternating_groups |
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://en.wikipedia.org/wiki/Alternating_series |
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://en.wikipedia.org/wiki/Alternating_sign_matrix |
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://en.wikipedia.org/wiki/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 can say what proportion of G any given s... | https://en.wikipedia.org/wiki/Amenable_group |
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 that the sum of the divisors of each is equ... | https://en.wikipedia.org/wiki/Amicable_triple |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Analytical_function |
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://en.wikipedia.org/wiki/Real-analytic_manifold |
( x − x 0 ) α {\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 {x_{0}} } , for all x 0 ∈ U {\displaystyle \mathbf {x... | https://en.wikipedia.org/wiki/Real-analytic_manifold |
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://en.wikipedia.org/wiki/Analytic_proof |
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 to initial value problems, better results conc... | https://en.wikipedia.org/wiki/Analytic_semigroup |
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://en.wikipedia.org/wiki/Ancient_solution |
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 annular (as in annular eclipse). The open an... | https://en.wikipedia.org/wiki/Punctured_disk |
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://en.wikipedia.org/wiki/Anti-diagonal_matrix |
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://en.wikipedia.org/wiki/Antiautomorphism |
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://en.wikipedia.org/wiki/Antimatroid |
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://en.wikipedia.org/wiki/Antiunitary_operator |
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://en.wikipedia.org/wiki/Anyonic_Lie_algebra |
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://en.wikipedia.org/wiki/Aperiodic_semigroup |
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://en.wikipedia.org/wiki/Approximate_group |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/AF_C*-algebra |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Argument_of_a_function |
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://en.wikipedia.org/wiki/Arithmetic_group |
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://en.wikipedia.org/wiki/Arithmetic_surface |
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://en.wikipedia.org/wiki/Arithmetic_variety |
In mathematics, an associahedron Kn is an (n – 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. Equivalently, the vertices of an associahedro... | https://en.wikipedia.org/wiki/Stasheff_polytope |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Linear_associative_algebra |
In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm. | https://en.wikipedia.org/wiki/Asymmetric_norm |
In mathematics, an asymmetric relation is a binary relation R {\displaystyle R} on a set X {\displaystyle X} where for all a , b ∈ 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 . {\displaystyle a.} | https://en.wikipedia.org/wiki/Asymmetric_relation |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Asymptotic_series |
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 subgroup Z × Z {\displaystyl... | https://en.wikipedia.org/wiki/Atoroidal |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.