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Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.
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Given and representation
* Speech recognition: Given a sound clip of a person or people speaking, determine the textual representation of the speech.
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.
Given a context pertaining to a specific OSI layer, PDU is sometimes used as a synonym for its representation at that layer.
Given the hexadecimal representation 3FD5 5555 5555 5555 < sub > 16 </ sub >,
Given that, the Culture had intended all along to use him to discredit the brutality of the Azadian system by publicly giving the lie to the game's representation of social reality.
Given the mixed member proportional representation system used to elect its members, it is difficult for a single party to gain an overall majority of seats in the Scottish Parliament.
Given a state ρ of A, there is a *- representation π of A with distinguished cyclic vector ξ such that its associated state is ρ, i. e.
For a finite group G, the left regular representation λ ( over a field K ) is a linear representation on the K-vector space V whose basis is the elements of G. Given g ∈ G, λ ( g ) is the linear map determined by its action on the basis by left translation by g, i. e.
A more general class are flat G-bundles with for a manifold F. Given a representation, the flat-bundle with monodromy is given by, where acts on the universal cover by deck transformations and on F by means of the representation.
Given an arbitrary category C, a representation of G in the category C is a functor from G to C. Such a functor selects an object of C and a subgroup of automorphisms of that object.
Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:
However, the modelling study also states that " Given the complex, three-dimensional nature of the real Pine Island glacier ... it should be clear that the [...] model is a very crude representation of reality.
Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.
Given a faithful irreducible representation ρ of G, the lattice Yang-Mills action is the sum over all lattice sites of the ( real component of the ) trace over the n links e < sub > 1 </ sub >, ..., e < sub > n </ sub > in the Wilson loop,
The Government believes that it must remain so .” Given the prospect of " incineration without representation ", Canadians seemed to feel that the doctrine which most encouraged restraint was the strategically soundest one to support.
Given this constraint, we can write in the following representation:
Given such a choice of scale, a central representation of the Heisenberg group is a map of-algebras which is the formal way of saying that it sends the center to a chosen scale.
Given a presentation of a group G by generators and relations and a subgroup H of G, the algorithm enumerates the cosets of H on G and describes the permutation representation of G on the space of the cosets.
Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra ( which form a base for a topology ).
Given an interior algebra, by replacing the topology of its Stone representation with the corresponding canonical preorder ( specialization preorder ) we obtain a representation of the interior algebra as a canonical preorder field.
Given the common view of knitting as work associated with middle-and lower-class women-typically older women-the groups hope that this representation will help to counter stereotypes.
Given and T
Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:
Given a probability space, a filtration is a weakly increasing collection of sigma-algebras on,, indexed by some totally ordered set T, and bounded above by.
* Given the popularity of the radio show at the time many catch phrases appear frequently in Warner Bros. cartoon shorts from the 1930s and 1940s, such as Molly's " T ' ain't funny, McGee!
Given a core geometry, the B field needed for a given force can be calculated from ( 2 ); if it comes out to much more than 1. 6 T, a larger core must be used.
So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.
The isomorphism is not canonical ; it depends on a choice of basis in V. Given a basis ( e < sub > 1 </ sub >, ..., e < sub > n </ sub >) of V and an automorphism T in GL ( V ), we have
: Given two sets, A and T, of equal size, together with a weight function C: A × T → R. Find a bijection f: A → T such that the cost function:
Given the time reversal operator T, it does nothing to the x-operator, TxT < sup >− 1 </ sup > =
Given Western Electric / AT & T Unix licensing, DEC ( and others ) were restricted to selling binary-only licenses.
Given a dynamical system ( T, M, Φ ) with T a group, M a set and Φ the evolution function
Given a linear differential operator T
Given a circle k, with a center O, and a point P outside of the circle, we want to construct the ( red ) tangent ( s ) to k that pass through P. Suppose the ( as yet unknown ) tangent t touches the circle in the point T. From symmetry, it is clear that the radius OT is orthogonal to the tangent.
* Pukui, Mary Kawena ; Elbert, Samuel Hoyt ; Mookini, Esther T. & Nishizawa, Yu Mapuana ( 1992 ): New Pocket Hawaiian Dictionary with a Concise Grammars and Given Names in Hawaiian.
* Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T.
Given a torus T ( not necessarily maximal ), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, Fix a maximal torus in G ; then the corresponding Weyl group is called the Weyl group of G ( it depends up to isomorphism on the choice of T ).
Given and multiplication
* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.
Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.
** Closure axiom for multiplication: Given two integers a and b, their product, a · b is also an integer.
** Associativity of multiplication: Given any integers, a, b and c, ( a · b ) · c
Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).
Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition
* Given two complex vectors x and y, multiplication by U preserves their inner product ; that is,
Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as
Given a Boolean algebra B, we turn B into a Boolean ring by using the symmetric difference as addition and the meet operation as multiplication.
Case: Given positive ( or more generally irreducible non-negative matrix ) A, the Perron-Frobenius eigenvector is the only ( up to multiplication by constant ) non-negative eigenvector for A.
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