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Adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoint or adjunction may mean: * Adjoint of a linear map, also called its transpose * Hermitian adjoint (adjoint of a linear operator) in functional analysis * Adjoint endomorphism of a Lie algebra * Adjoint representation of a Lie group * Adjoint functors in category theory * Adjunction (field theory) * Adjunction formula (algebraic geometry) * Adjunction space in topology * Conjugate transpose of a matrix in linear algebra * Adjugate matrix, related to its inverse * Adjoint equation * The upper and lower adjoints of a Galois connection in order theory * The adjoint of a differential operator with general polynomial coefficients * Kleisli adjunction * Monoidal adjunction * Quillen adjunction * Axiom of adjunction In mathematical set theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects X in \mathcal and Y in \mathcal a bijection between the respective morphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Endomorphism
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation Of A Lie Group
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g(h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Adjoint
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \langle \cdot,\cdot \rangle is the inner product on the vector space. The adjoint may also be called the Hermitian conjugate or simply the Hermitian after Charles Hermite. It is often denoted by in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators are represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose). The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H. The definition has been further extended to include unbounded '' densely defined'' operators whose domain is topologically dense in—but not necessarily equal to—H. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transpose Of A Linear Map
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors. Definition Let X^ denote the algebraic dual space of a vector space X. Let X and Y be vector spaces over the same field \mathcal. If u : X \to Y is a linear map, then its algebraic adjoint or dual, is the map ^ u : Y^ \to X^ defined by f \mapsto f \circ u. The resulting functional ^ u(f) := f \circ u is called the pullback of f by u. The continuous dual space of a topological vector space (TVS) X is denoted by X^. If X and Y are TVSs then a linear map u : X \to Y is weakly continuous if and only if ^ u\left(Y^\right) \subseteq X^, in which case we let ^t u : Y^ \to X^ denote the restriction of ^ u to Y^. The map ^t u is called the transpose or algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Adjunction
Suppose that (\mathcal C,\otimes,I) and (\mathcal D,\bullet,J) are two monoidal categories. A monoidal adjunction between two lax monoidal functors :(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J) and (G,n):(\mathcal D,\bullet,J)\to(\mathcal C,\otimes,I) is an adjunction (F,G,\eta,\varepsilon) between the underlying functors, such that the natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...s :\eta:1_\Rightarrow G\circ F and \varepsilon:F\circ G\Rightarrow 1_{\mathcal D} are monoidal natural transformations. Lifting adjunctions to monoidal adjunctions Suppose that :(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J) is a lax monoidal functor such that the underlying functor F:\mathcal C\to\mathcal D has a right adjoint G:\mathcal D\to\mathcal C. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). It is often denoted as \boldsymbol^\mathrm or \boldsymbol^* or \boldsymbol'. H. W. Turnbull, A. C. Aitken, "An Introduction to the Theory of Canonical Matrices," 1932. For real matrices, the conjugate transpose is just the transpose, \boldsymbol^\mathrm = \boldsymbol^\mathsf. Definition The conjugate transpose of an m \times n matrix \boldsymbol is formally defined by where the subscript ij denotes the (i,j)-th entry, for 1 \le i \le n and 1 \le j \le m, and the overbar denotes a scalar complex conjugate. This definition can also be written as :\boldsymbol^\mathrm = \left(\overline\right)^\mathsf = \overline where \boldsymbol^\mathsf denotes the transpose and \overline denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Equation
An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization, fluid flow control and uncertainty quantification. For example dX_t = a(X_t)dt + b(X_t)dW this is an Itō stochastic differential equation. Now by using Euler scheme, we integrate the parts of this equation and get another equation, X_ = X_n + a \Delta t + \zeta b \sqrt, here \zeta is a random variable, later one is an adjoint equation. Example: Advection-Diffusion PDE Consider the following linear, scalar advection-diffusion equation for the primal solution u(\vec), in the domain \Omega with Dirichlet boundary conditions In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleisli Adjunction
In category theory, a Kleisli category is a category naturally associated to any monad ''T''. It is equivalent to the category of free ''T''-algebras. The Kleisli category is one of two extremal solutions to the question ''Does every monad arise from an adjunction?'' The other extremal solution is the Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli. Formal definition Let ⟨''T'', ''η'', ''μ''⟩ be a monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', a ... over a category ''C''. The Kleisli category of ''C'' is the category ''C''''T'' whose objects and morphisms are given by :\begin\mathrm() &= \mathrm(), \\ \mathrm_(X,Y) &= \mathrm_(X,TY).\end That is, every morphism ''f: X → T Y'' in ''C'' (with codomain ''TY'') can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quillen Adjunction
In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen. Formal definition Given two closed model categories C and D, a Quillen adjunction is a pair :(''F'', ''G''): C \leftrightarrows D of adjoint functors with ''F'' left adjoint to ''G'' such that ''F'' preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that ''G'' preserves fibrations and trivial fibrations. In such an adjunction ''F'' is called the left Quillen functor and ''G'' is called the right Quillen functor. Properties It is a consequence of the axioms that a left (right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects. The total derived func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois. A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections. A Galois connection is rather weak compared to an order isomorphism between the involved posets, but every Galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below. The term Galois correspondence is sometimes used to mean a bijective ''Galois connection' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
