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 a bounded sequence, there exists a closed ball that contains the image of ( is a subset of the scalar field ).

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.

Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold:

Having a measure on G allows us to define integration of bounded functions on G. Given a bounded function, the integral

Given a differentiable function defined on a bounded open set, the total variation of has the following expression

Given a set of points,, and the set of hyperplanes,, which are each spanned by, the number of incidences between and is bounded above by

Given a topological vector space ( X, τ ) over a field F, S is called bounded if for every neighborhood N of the zero vector there exists a scalar α so that

* Theorem ( by Mackey ): Given a dual pair, the bounded sets under any dual topology are identical.

Given the extent of the parish, it is bounded by several other parishes and villages.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

The functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place.

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

Given a strictly increasing integer sequence / function ( n ≥ 1 ) we can produce a faster growing sequence ( where the superscript n denotes the n < sup > th </ sup > functional power ).

Given that X, Y, and Z are sets of attributes in a relation R, one can derive several properties of functional dependencies.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Given a functional of the form

The general scenario is the following: Given a class S of computable functions, is there a learner ( that is, recursive functional ) which for any input of the form ( f ( 0 ), f ( 1 ),..., f ( n )) outputs a hypothesis ( an index e with respect to a previously agreed on acceptable numbering of all computable functions ; the indexed function should be consistent with the given values of f ).

Given a multilinear functional M < sub > n </ sub > of degree n ( that is, M < sub > n </ sub > is n-linear ), we can define a polynomial p as

Given an operator T, the range of the continuous functional calculus h → h ( T ) is the ( abelian ) C *- algebra C ( T ) generated by T. The Borel functional calculus has a larger range, that is the closure of C ( T ) in the weak operator topology, a ( still abelian ) von Neumann algebra.

Given the partition function Z in terms of the source field J, the energy functional is its logarithm.

Given a Banach space, a subset of, and a functional from to the dual space of the space, the variational inequality problem is the problem of solving respect to the variable ' belonging to the following inequality:

Given the unique mapping between densities and wave function, Runge and Gross then treated the Dirac action as a density functional,

Given a characteristic functional on a nuclear space A, the Bochner – Minlos theorem ( after Salomon Bochner and Robert Adol ' fovich Minlos ) guarantees the existence and uniqueness of the corresponding probability measure on the dual space, given by

Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that

Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.

Given a natural transformation Φ from h < sup > A </ sup > to F, the corresponding element of F ( A ) is.

Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

* Shay Given – former goalkeeper for both the Republic of Ireland and Newcastle United F. C ..

Given a topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F.

Given any subset F =

" Given its general release in January 1931, City Lights proved to be a popular and financial success — eventually grossing over $ 5 million.

Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.

Given that the ColecoVision could produce near arcade-quality ports, industry magazines like Electronic Games were unanimous in their enthusiasm over the console.

Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").

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.

Given the differences of opinion within the Lord Aberdeen cabinet over the direction of foreign policy with regard to relations between Britain and the French under Napoleon III, it is not surprising that debate raged within the government as Louis Bonaparte, now assuming the title of Emperor Napoleon III of France.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given the extraordinary competition and technological craftsmanship required for top search engine placement, the implication of the term " search engine optimization " has deteriorated over the last decade.

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

Given a function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is

The Fascist government had direct rule only over the Benadir territory Given the defeat of the Dervish movement in the early 1920s and the rise of fascism in Europe, on 10 July 1925, Mussolini gave the green light to De Vecchi to start the takeover of the north-eastern sultanates.

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 particular wing and its velocity through the air, debates over which mathematical approach is the most convenient to use can be misperceived by novices as differences of opinion about the basic principles of flight.

" Given their 3: 00 a. m. start time and non-inclusion ( at frontman John Fogerty's insistence ) in the Woodstock film, members have expressed bitterness over their experiences at the famed festival.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed ( maximum ) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK-at the same BER.

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that