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.

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 polynomially bounded functional F over the field configurations, then, for any state vector ( which is a solution of the QFT ), | ψ >, we have

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 a graph Λ ( for example, a d-dimensional lattice ), per each lattice site j ∈ Λ there is a discrete variable σ < sub > j </ sub > such that σ < sub > j </ sub > ∈

* Given a positive integer n, the set of all positive divisors of n forms a distributive lattice, again with the greatest common divisor as meet and the least common multiple as join.

Given a simple polygon constructed on a grid of equal-distanced points ( i. e., points with integer coordinates ) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:

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,

Given a-dimensional lattice, per each lattice site there is a two-dimensional, unit-length vector

Given a basis with n-dimensional integer coordinates, for a lattice L in R < sup > n </ sup > with, the LLL algorithm outputs an LLL-reduced ( short, nearly orthogonal ) lattice basis in time

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given the opportunity, adults eat hatchlings, which may be protected by separating the two groups with a net, breeding box or separate tank.

Love and Erlandson continued to pursue with the band, and released the single " Be A Man "— an outtake from the Celebrity Skin sessions — for the soundtrack of the Oliver Stone film Any Given Sunday ( 1999 ).

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

Given the bias of the Victorian age, female students were denied direct study in anatomy and could not attend drawing classes with live models ( who were often prostitutes ) until a decade later.

Given the small population of the region ( compared with the Central Canadian provinces or the New England states ), the regional economy is a net exporter of natural resources, manufactured goods, and services.

Given the rigorous admission process it is not surprising that CUNY BA boasts a 70 % graduation rate within an average of 2. 2 years and that 60 % graduate with academic honors.

Given Chrétien's known tendency to create new stories and characters, being the first to mention the hero Lancelot and his love affair with Queen Guinevere for example, the name might also be entirely invented.

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 Mountbatten's determination, Nehru and Patel's inability to deal with the Muslim League and lastly Jinnah's obstinacy, all Indian party leaders ( except Gandhi ) acquiesced to Jinnah's plan to divide India, which in turn eased Mountbatten's task.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Given limited defense budgets, air forces tended to be conservative in their aircraft purchases, and biplanes remained popular with pilots because of their agility, and remained in service long after they had ceased to be competitive.

Given Guatemala's large expatriate community in the United States, it is the top remittance recipient in Central America, with inflows serving as a primary source of foreign income equivalent to nearly two-thirds of exports.

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.

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.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given the limited amount of good quality agricultural land to begin with, that policy has resulted in continual deforestation and subsequent erosion.

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 inefficiencies in the heat transfer, the actual NERVA designs were planned to produce about 5 GW, which would make them the largest nuclear reactors in the world.

Given that the U. S. is still home to the world's largest economy, there's no reason it shouldn't have the most vibrant equity markets — unless regulation is holding back the creation of new public companies.

Given the size of Merchant's capital outlay — including construction in Boswell of the world's largest tipple, a central business district and a high school — the company planned big things for Boswell.

Dr. Tennison states: " Given the account of Elliot Paul, and given that Lead Belly witnessed boogie-woogie in 1899 in the Arklatex ; and given the North to South migration of the Thomas family ; and given the Texas & Pacific headquarters in Marshall in the early 1870s ; and given that Harrison County had the largest slave population in the state of Texas ; and given the fact that the best-documented and largest-scale turpentine camps in Texas did not occur until after 1900 in Southeast Texas, it is most probable that boogie-woogie spread from Northeast to Southeast Texas, rather than from Southeast to Northeast Texas, or by having developed diffusely with an even density over all of the Piney Woods of East Texas.

Given these orbital elements and the physical characteristics known so far, Ananke is thought to be the largest remnant of an original break-up forming the Ananke group.

Given that UTV had the third largest audience in the Republic, the carriers declined to drop them in favour of TV3.

Given its length and a basin that covers an area of 102, 000 km², it is one of the largest rivers of the Caucasus.

Given the fact that, at one time, Llanrwst was one of the ten largest towns in Wales, it can be seen that the Conwy Valley had great historical significance.

Given that California's automotive market is one of the largest in the world, CARB wields enormous influence over the emissions requirements that major automakers must meet if they wish to sell into that market.

Given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm.

Given that National Education is France's largest employer, and employs more than half of the French state civil servants, the position is traditionally a fairly strategic one.

Given its platform mate, the Navigator is the largest vehicle Lincoln currently offers, having the greatest cargo capacity and seating for seven to eight people, depending on the model.

Given its close proximity to Asia and constant flow of Asian immigrants, Asian food has the largest foothold in Los Angeles after Mexican cuisine.

Given its size it is the largest park in Estonia and one of Europe's biggest national parks.

Given the excellent road and rail networks that connect to Bohicon, Bohicon is an important trade centre that is the third largest in the country after Cotonou and Parakou.

Given the fertility of the soils and favourbale climate, the locals were able to export wheat, barley, wine and local spirits, maintaining an active commerce with Terceira, then the largest port, center of commerce and the administration of the archipelago.

3: Given the bounds presented in ( 1 ) and ( 2 ), choose N to be the largest number possible.

Given that omniverse is the largest set conceptualized by humanity, such a view is sometimes called " the end of infinity ".

: Given positive integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > such that gcd ( a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub >) = 1, find the largest integer that cannot be expressed as an integer conical combination of these numbers, i. e., as a sum

Given a graph, one may write for the largest book ( of the kind being considered ) contained within.