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 fact that the Lakeport area has an economy that is based somewhat on agricultural activities occurring in the Big Valley and Scotts Valley area which involve the growing and harvesting of food and nut crops as well as the increasing wine industry, there are a significant number of transient and seasonal farm workers.

Given the commune's proximity and good transport links to the airport, to NATO and to the European Quarter, an increasing number of non-Belgians have settled in the commune and this trend looks set to continue as the few remaining greenfield sites are developed.

Given the ( particularly modern ) prevalence of crowd events, and the potential safety issues associated with such large gatherings of people, the topic is receiving increasing attention from agencies responsible for crowd management and also from governments.

Given any random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >..., X < sub > n </ sub >, the order statistics X < sub >( 1 )</ sub >, X < sub >( 2 )</ sub >, ..., X < sub >( n )</ sub > are also random variables, defined by sorting the values ( realizations ) of X < sub > 1 </ sub >, ..., X < sub > n </ sub > in increasing order.

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 the epidemic proportions of obesity in the Western world, and the fact that it is increasing rapidly in some poorer countries, observers expect developments in this area to snowball in the near future.

Given the increasing unpopularity of the executive duo, and with a view to the next presidential election, the RPR became more and more critical.

Given that they are paying to be there, some clubs allow as many dancers as possible to appear on any given night, increasing competition among the dancers.

Given that Hg ( CN )< sub > 2 </ sub > molecules remain undissociated in pure water and in basic solutions, it makes sense that dissociation would increase with increasing acidity.

Given the seriousness that drug resistant pathogens have now become, there is increasing research into PDT as a new antimicrobial therapy.

Given his increasing profile and popularity within the party due to his role as the manifesto chair, he was seen as a probable candidate for the social liberal grouping's vote in the future leadership election.

Given the small overall amount of power produced per cubic meter, there is a substantial case for increasing water flow to the natural channel, now that Arnacrusha is producing so small a proportion of ESB's power.

Given the increasing role of the European Union in the political arena, the National Academies acknowledge that the scope of their advisory tasks must exceed the national borders and that it's important not to lose sight of the European level.

Given its rural location, much of the focus of the school has been on undergraduate studies in education and agriculture, although many other courses of study are offered, particularly in the liberal arts, and in recent years there has been an increasing emphasis on business, engineering, and music.

Given the increasing ranges of shore launched anti-ship missiles, the EFV's range for amphibious landing may no longer provide the anticipated protection predicted for an over the horizon launch.

Given the prevalence of crowd events, and the potential safety issues associated with such large gatherings of people, the topic is receiving increasing attention from agencies responsible for crowd management and also from governments.

Given that inflation is increasing at the same pace as wages, an individual cannot actually afford to increase their consumption.

Given that average yearly temperatures are currently increasing, and the outlook is for this trend to continue, it is actually possible that the glacier might experience a period of stabilization or slight advance during its retreating march.

Given the increasing seismic and volcanic activity, Johnston and the other volcanologists working for the USGS in its Vancouver branch prepared to observe any impending eruption.

For example, more people will likely answer " yes " to the question " Given the increasing burden of taxes on middle-class families, do you support cuts in income tax?

Given the costs associated with such policies ( e. g., global climate disruption, widespread habitat loss and species extinctions, consumption of natural resources, pollution, urban congestion, intensifying competition for remaining resources, and increasing disparity between the wealthy and the poor ), some economists, scientists, and philosophers have questioned the biophysical limits to growth, and the desirability of continuous growth.

Given the rise to dominance of Hebrew as a means of communication among Jewish communities in the Middle East, and the increasing prestige of English, as well as the economic importance of local vernaculars ( especially Turkish ), the situation appears grim.

Given the high spectacularity of the freeride and downhill action, an increasing number of films are released every year, covering the annual major competitions and promoting particular trails and champions.

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

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 a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given is each element's name, atomic number, year of first report, name of the discoverer, and some notes related to the discovery.

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

: Given any positive number ε, there is a sequence

Given the number of homes, it seems likely that no more than fifty people lived in Skara Brae at any given time.

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given that detecting a price cut becomes harder as the number of firms increases, the bigger are the gains from price cutting.

Given the ( period, cash flow ) pairs (, ) where is the total number of periods, the net present value is given by:

* Aggregate signature-a signature scheme that supports aggregation: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users.

Given the ( period, cash flow ) pairs (, ) where is a positive integer, the total number of periods, and the net present value, the internal rate of return is given by in:

( Please note that Japan has a guideline of a limit of 1 % of GDP on defense spending, however, Japan defines a number of activities as non-defense spending ) Given these circumstances, some have viewed Article 9 as increasingly irrelevant.

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Given that the Masoretes would not alter the sacred consonantal text, the Kethiv-Qere notes were a way of " correcting " or commenting on the text for any number of reasons ( grammatical, theological, aesthetic, etc.

Given the number of unpleasant, harmful or even fatal consequences of coming out in world societies, it is questionable to call being closeted a choice.

Given any input, GEN generates an infinite number of candidates, or possible realizations of that input.

Given the number of times tea is poured in a meal, the tapping is a timesaver in loud restaurants or lively company, as an individual being served might be speaking to someone else or have food in their mouth.

Given an input consisting of any number of bytes, it produces as output a single byte that is strongly dependent on every byte of the input.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.