# Given any elements a and a ' of G, if a ~ a ', then a < sup >− 1 </ sup > ~ a ' < sup >− 1 </ sup > ( this can actually be proven from the other four, so is strictly redundant ).

Given that the conduct of conclaves is strictly confidential and that any cardinal revealing such details would face automatic excommunication, no documentary evidence has ever substantiated or disproved the widely claimed rumour.

Given a system, we say that a physical event is a stimulus for the system if and only if the probability that the system suffers a change or be perturbed ( in its elements or in its processes ) when the event occurs is strictly greater than the prior probability that suffers a change independently of:

Given that borrowed money was no longer strictly for consumption but for production as well, it could not be viewed in the same manner.

Given a non-negative eigenvector v, and that at least one of its components say j-th is strictly positive, the corresponding eigenvalue is strictly positive, indeed, given n such that ( A < sup > n </ sup >)< sub > ii </ sub > > 0, hence: r < sup > n </ sup > v < sub > i </ sub >

Given a strictly positive eigenvector v corresponding to r and another eigenvector w with the same eigenvalue.

Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional bloch sphere.

Given the increasing number of scientists who study the nervous system, several prominent neuroscience organizations have been formed to provide a forum to all neuroscientists and educators.

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 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 a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Informally it solves the following problem: Given an integer N, find its prime factors.

** Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.

** Closure axiom for multiplication: Given two integers a and b, their product, a · b is also an integer.

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:

Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm.

Given any closed spin network, a non-negative integer can be calculated which is called the norm of the spin network.

Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k:

: Given an integer n, choose some integer a coprime to n and calculate a < sup > n − 1 </ sup > modulo n. If the result is different from 1, then n is composite.

Given an integer n, choose some integer a < n. Let 2 < sup > s </ sup > d = n − 1 where d is odd.

The Solovay – Strassen primality test uses another equality: Given an odd number n, choose some integer a < n, if

Given a bin size and a list of items with sizes to pack, find an integer and a-partition of such that for all.

* 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 positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups of order n there are.

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 an integer k, unrank ( k ) is the kth permutation.

Given an integer n > 1, let H be any subgroup of the multiplicative group

Given a particular ℓ, m is entitled to be any integer from-ℓ up to ℓ.

Given an integer n ( throughout this article, n refers to " the integer to be factored "), trial division consists of testing whether n is divisible by any number.