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Unitary Representation Theory
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Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigroup Politics * Unitary authority * Unitary state See also * Unital (other) * Unitarianism Unitarianism () is a Nontrinitarianism, nontrinitarian sect of Christianity. Unitarian Christians affirm the wikt:unitary, unitary God in Christianity, nature of God as the singular and unique Creator deity, creator of the universe, believe that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Divisor
In mathematics, a natural number ''a'' is a unitary divisor (or Hall divisor) of a number ''b'' if ''a'' is a divisor of ''b'' and if ''a'' and \frac are coprime, having no common factor other than 1. Equivalently, a divisor ''a'' of ''b'' is a unitary divisor if and only if every prime factor of ''a'' has the same multiplicity in ''a'' as it has in ''b''. The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931), who used the term block divisor. Example The integer 5 is a unitary divisor of 60, because 5 and \frac=12 have only 1 as a common factor. On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and \frac=10 have a common factor other than 1, namely 2. Sum of unitary divisors The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(''n''). The sum of the ''k''-th powers of the unitary divisors is denoted by σ*''k''(''n''): :\sigma_k^*(n) = \sum_ \!\! d^k. It is a multiplicative function. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Element
In mathematics, an element of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element. Definition Let \mathcal be a *-algebra with unit An element a \in \mathcal is called unitary if In other words, if a is invertible and a^ = a^* holds, then a is unitary. The set of unitary elements is denoted by \mathcal_U or A special case from particular importance is the case where \mathcal is a complete normed *-algebra. This algebra satisfies the C*-identity (\left\, a^*a \right\, = \left\, a \right\, ^2 \ \forall a \in \mathcal) and is called a C*-algebra. Criteria * Let \mathcal be a unital C*-algebra and a \in \mathcal_N a normal element. Then, a is unitary if the spectrum \sigma(a) consists only of elements of the circle group \mathbb, i.e. Examples * The unit e is unitary. Let \mathcal be a unital C*-algebra, then: * Every projection, i.e. every element a \in \mathcal with a = a^* = a^2, is unitary. For the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Group
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{{disambiguation ...
Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigroup Politics * Unitary authority * Unitary state See also * Unital (other) * Unitarianism Unitarianism () is a Nontrinitarianism, nontrinitarian sect of Christianity. Unitarian Christians affirm the wikt:unitary, unitary God in Christianity, nature of God as the singular and unique Creator deity, creator of the universe, believe that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Matrix
In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if U^* U = UU^* = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (), so the equation above is written U^\dagger U = UU^\dagger = I. A complex matrix is special unitary if it is unitary and its matrix determinant equals . For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes. Properties For any unitary matrix of finite size, the following hold: * Given two complex vectors and , multiplication by preserves their inner product; that is, . * is normal (U^* U = UU^*). * is diagonalizable; that is, is unitarily similar to a diagonal matrix, as a consequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Morphism
In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution) is a category equipped with a certain structure called ''dagger'' or ''involution''. The name dagger category was coined by Peter Selinger. Formal definition A dagger category is a category \mathcal equipped with an involutive contravariant endofunctor \dagger which is the identity on objects. In detail, this means that: * for all morphisms f: A \to B, there exists its adjoint f^\dagger: B \to A * for all morphisms f, (f^\dagger)^\dagger = f * for all objects A, \mathrm_A^\dagger = \mathrm_A * for all f: A \to B and g: B \to C, (g \circ f)^\dagger = f^\dagger \circ g^\dagger: C \to A Note that in the previous definition, the term "adjoint" is used in a way analogous to (and inspired by) the linear-algebraic sense, not in the category-theoretic sense. Some sources define a category with involution to be a dagger category with the additional proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitary matrices. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the concept of isomorphism ''between'' Hilbert spaces. Definition Definition 1. A ''unitary operator'' is a bounded linear operator on a Hilbert space that satisfies , where is the adjoint of , and is the identity operator. The weaker condition defines an ''isometry''. The other weaker condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry, or, equivalently, a surjective isometry. An equivalent definition is the following: Definition 2. A ''unitary operator'' is a bounded linear operator on a Hilbert space for which the followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Transformation
In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, a unitary transformation is an isometric isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a ''unitary transformation'' is a bijective function :U : H_1 \to H_2 between two inner product spaces, H_1 and H_2, such that :\langle Ux, Uy \rangle_ = \langle x, y \rangle_ \quad \text x, y \in H_1. It is a linear isometry, as one can see by setting x=y. Unitary operator In the case when H_1 and H_2 are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator. Antiunitary transformation A closely related notion is that of antiunitary transformation, which is a bijective function :U:H_1\to H_2\, between two complex Hilbert spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Representation
In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in the case that ''G'' is a locally compact ( Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book '' Gruppentheorie und Quantenmechanik''. One of the pioneers in constructing a general theory of unitary representations, for any group ''G'' rather than just for particular groups useful in applications, was George Mackey. Context in harmonic analysis The theory of unitary representations of topological groups is closely connected with harmonic analysis. In the case of an abelian group ''G'', a fairly complete picture of the representation theory of ''G'' is given by Pontryagin duality. In genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitarity (physics)
In quantum physics, unitarity is (or a unitary process has) the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. A unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that time evolution preserves inner products in Hilbert space. Hamiltonian evolution Time evolution described by a time-independent Hamiltonian is represented by a one-parameter family of unitary operators, for which the Hamiltonian is a generator: U(t) = e^. In the Schrödinger picture, the unitary operators are taken to act upon the system's quantum state, whereas in the Heisenberg picture, the time dependence is incorporated into the observable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E-unitary Inverse Semigroup
In group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that and , i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries. (The convention followed in this article will be that of writing a function on the right of its argument, e.g. ''x'' ''f'' rather than ''f''(''x''), and composing functions from left to right—a convention often observed in semigroup theory.) Origins Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in 1952, and by Gordon Preston in the United Kingdom in 1954. Both authors arrived at inverse semigroups via the study of partial bijections of a set: a partial transformation ''α'' of a set ''X'' is a function from ''A'' to ''B'', whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Authority
A unitary authority is a type of local government, local authority in New Zealand and the United Kingdom. Unitary authorities are responsible for all local government functions within its area or performing additional functions that elsewhere are usually performed by a multiple tiers of local government. Typically unitary authorities cover towns or city, cities which are large enough to function independently of a council or other authority. An authority can be a unit of a county or combined authority. New Zealand In local government in New Zealand, New Zealand, a unitary authority is a territorial authorities of New Zealand, territorial authority (district, city or metropolitan area) that also performs the functions of a regions of New Zealand, regional council (first-level division). There are five unitary authorities, they are (with the year they were constituted): Gisborne District Council (1989), Tasman District Council (1992), Nelson City Council (1992), Marlborough Distric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary State
A unitary state is a (Sovereign state, sovereign) State (polity), state governed as a single entity in which the central government is the supreme authority. The central government may create or abolish administrative divisions (sub-national or sub-state units). Such units exercise only the powers that the central government chooses to delegate. Although Power (social and political), political power may be delegated through devolution to regional or local governments by statute, the central government may alter the statute, to override the decisions of Devolution, devolved governments or expand their powers. The modern unitary state concept originated in France; in the aftermath of the Hundred Years' War, national feelings that emerged from the war unified France. The war accelerated the process of transforming France from a feudal monarchy to a unitary state. The French people, French then later spread unitary states by conquests, throughout Europe during and after the Napoleoni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
