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Adjoint Pair
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 in case of matrices * 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 ... [...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 (mathematics), 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 tra ... [...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. 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: : \begin \nabla \cdot \left(\vec u - \mu \nabla u \right) &= f, \qquad \vec \in \Omega, \\ u &= b, \qquad \vec \in \partial \Omega. \end Let the output of interest be the following linear functional: : J(u) = \int_\Omega g u \ dV. Derive the weak form by multiplying the primal equation with a weighting function w(\vec) and performing integration by parts: : \begin B(u, w) &= L(w), \end where, : \beg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Adjunction
In mathematical set theory, the axiom of adjunction states that for any two sets ''x'', ''y'' there is a set ''w'' = ''x'' ∪ given by "adjoining" the set ''y'' to the set ''x''. It is stated as :\forall x. \forall y. \exists w. \forall z. \big( z \in w \leftrightarrow (z \in x \lor z=y) \big). introduced the axiom of adjunction as one of the axioms for a system of set theory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as general set theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursive set functions. Interpretability of arithmetic Tarski and Szmielew showed that Robinson arithmetic () can be interpreted in a weak set theory whose axioms are extensionality, the existence of the empty set, and the axiom of adjunction . In fact, empty set and adjunction alone (without extensionality) suffice to interpret . (They are mutually interpreta ... [...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 funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Adjunction
A monoidal adjunction is an adjunction in mathematics between monoidal categories which respects the monoidal structure. 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_ 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 funct ... [...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 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 also be regarded as a morphism in ''C''''T'' (but with codomain ''Y''). Composition of morphisms in ''C''''T'' is given by :g\circ_T f = \mu_Z \circ Tg \circ f : X \to T Y \to T^2 Z \to T Z where ''f: X → T Y'' and ''g: Y → T Z''. The identi ... [...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 Given a nonnegative integer ''m'', an order-m linear differential operator is a map P from a function space \mathcal_1 on \mathbb^n to another function space \mathcal_2 that can be written as: P = \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 ... [...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] |
Adjugate Matrix
In linear algebra, the adjugate or classical adjoint of a square matrix , , is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose. The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix: :\mathbf \operatorname(\mathbf) = \det(\mathbf) \mathbf, where is the identity matrix of the same size as . Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant. Definition The adjugate of is the transpose of the cofactor matrix of , :\operatorname(\mathbf) = \mathbf^\mathsf. In more detail, suppose is a ( unital) commutative ring and is an matrix with entries from . The -'' minor'' of , denoted , is the determinant of the matrix that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Adjoint
In mathematics, specifically in operator theory, each linear operator A on an inner product 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 dimension (vector space), finite dimensions where operators can be represented by Matrix (mathematics), 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 operator, bounded linear operators on Hilbert spaces H. The definition has been further extended to include unbounded ''Densely defined operator, densely def ... [...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 \mathbf is an n \times m matrix obtained by transposing \mathbf and applying complex conjugation to each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). There are several notations, such as \mathbf^\mathrm or \mathbf^*, \mathbf', or (often in physics) \mathbf^. For real matrices, the conjugate transpose is just the transpose, \mathbf^\mathrm = \mathbf^\operatorname. Definition The conjugate transpose of an m \times n matrix \mathbf is formally defined by where the subscript ij denotes the (i,j)-th entry (matrix element), 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 :\mathbf^\mathrm = \left(\overline\right)^\operatorname = \overline where \mathbf^\operatorname denotes the transpose and \overline denotes the matrix with complex conjugated entries. Other na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjunction Space
In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f : A \rightarrow X be a continuous map (called the attaching map). One forms the adjunction space X \cup_f Y (sometimes also written as X +_f Y) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. Formally, :X\cup_f Y = (X\sqcup Y) / \sim where the equivalence relation \sim is generated by a\sim f(a) for all a in A, and the quotient is given the quotient topology. As a set, X \cup_f Y consists of the disjoint union of X and ( Y-A). The topology, however, is specified by the quotient construction. Intuitively, one may think of Y as being glued onto X via the map f. Examples *A common example of an adjunction space is given when ''Y'' is a closed ''n''-ball (or ''cell'') and ''A'' is the boundary of the ball ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
