Given n = p × q a Blum integer, Q < sub > n </ sub > the set of all quadratic residues modulo n, and a ∈ Q < sub > n </ sub >.

Given the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

Given the complexity of NLP problems, it is often difficult to predict performance only on the basis of glass-box evaluation, but this type of evaluation is more informative with respect to error analysis or future developments of a system.

Given that one motive for the operational approach is stability, systems that relax the operational factor can be problematic, for several reasons, as the operational is a means to manage complexity.

Given the complexity of perioperative management, and the potential for catastrophic intra and postoperative complications, such surgery should be performed only at centers experienced in the management of this disorder.

Given the complexity of putting ideals into practice, and resolving conflicts between them, it is not uncommon to see them reduced to dogma.

Given the complexity of entering punctuation on the Spectrum's tiny keyboard, this was cumbersome.

Given the nature of post 9 / 11 counter terrorism strategies employed by the United States of America the operations of the Special Collections Service have experienced significant growth in scale and complexity.

Given the complexity of this process, software is increasingly used to simplify the task, while better supporting reproducible and more justifiable results.

Given this complexity, early games using the system tended to be quite simple in layout, using the TIA to create simple symmetric playfields with players on top.

Given the complexity of kelp forests – their variable structure, geography and interactions – they pose a considerable challenge to environmental managers.

Given a pattern T, the number of other patterns may have Kolmogorov complexity no larger than that of T is denoted by φ ( T ).

Given the special effects complexity of landing a giant starship each week, " rarely " was quickly changed to " never ".

Given the complexity of the mechanisms regulating heart rate, it is reasonable to assume that applying HRV analysis based on methods of non-linear dynamics will yield valuable information.

Given the complexity of its refining facilities, SPC is able to refine heavy, medium and light crudes.

Given the complexity of RWA, there are two general methodologies for solving the problem:

Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

noted, " Given that the bandwidth for conducting crawls is neither infinite nor free, it is becoming essential to crawl the Web in not only a scalable, but efficient way, if some reasonable measure of quality or freshness is to be maintained.

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 any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure.

Given the potential for sampling biases of individual survey reports, researchers and investors try sometimes to average the values of different index reports into a single aggregated measure of consumer confidence.

When the measure was sent in May to the parliament they said " Given the damage it produces on those rules which allow the life in community, ensure the dignity of the person and equality between sexes, this practice, even if it is voluntary, cannot be tolerated in any public place ".

Given that deviance is a measure of the difference between a given model and the saturated model, smaller values indicate better fit.

Given a finite observation set S, one can simply select the measure for all.

Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?

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

Given a left-invariant measure, the function is a right-invariant measure.

Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of and its analytic capacity, it might be

Given a σ-finite measure space ( S, Σ, μ ), consider the Banach space L < sup > p </ sup >( μ ).

Given a countably additive measure μ on X, a measurable set is one that differs from a Borel set by a null set.

Given a sequence ( f < sub > n </ sub >) of M-valued measurable functions on some measure space ( X, Σ, μ ), and a measurable subset A of finite μ-measure such that ( f < sub > n </ sub >) converges μ-almost everywhere on A to a limit function f, the following result holds: for every ε > 0, there exists a measurable subset B of A such that μ ( B ) < ε, and ( f < sub > n </ sub >) converges to f uniformly on the relative complement A

Given two objects, A and B, each with n binary attributes, the Jaccard coefficient is a useful measure of the overlap that A and B share with their attributes.

The most recent measure of social dominance orientation ( see SDO-6 above ) focuses on the “ general desire for unequal relations among social groups, regardless of whether this means ingroup domination or ingroup subordination ” ( p. 312 ) Given these changes, Rubin and Hewstone believe that evidence for social dominance theory should be considered “ as supporting three separate SDO hypotheses, rather than one single theory ” ( p. 22 ).

Given a fixed total concentration of one or more species over the measurement time, the scattering signal is a direct measure of the weight-averaged molar mass of the solution, which will vary as complexes form or dissociate.

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 this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

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 high fixed costs, the new price was below average total cost, resulting in a loss.

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.

# Minimisation operator: Given a ( k + 1 )- ary total function:

Given a total circulation of 56 billion coins, counterfeit coins are relatively rare.

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

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 a precomplete numbering then for any partial computable function with two parameters there exists a total computable function with one parameter such that

Given an inelastic reaction with total incoming energy larger than the total detected energy ( i. e. not all outgoing particles are detected in the experiment ), the invariant mass ( also known as the " missing mass ") of the reaction is defined as follows:

Given a total interaction cross section σ ( typically measured in barns ), the mean free path of a prompt neutron is where n is the nuclear number density.

Given Bluestein's algorithm, such a transform can be used, for example, to obtain a more finely spaced interpolation of some portion of the spectrum ( although the frequency resolution is still limited by the total sampling time ), enhance arbitrary poles in transfer-function analyses, etcetera.

Given its production function and cost structure, it knows its average total costs at that output level will be represented as point A.

Given this, sabermetricians generally regard Cy Young's total of 749 complete games as the career baseball record most unlikely to ever be broken.

Given an oriented manifold M of dimension n with fundamental class, and a G-bundle with characteristic classes, one can pair a product of characteristic classes of total degree n with the fundamental class.

Given at the discretion of the Council of the Academy on the recommendation of a nominating committee, the prize is awarded to a person for his or her total literary achievement rather than for a specific work.

Given the total energy E and the length ( magnitude ) p of the total momentum vector, the invariant mass is given by:

The original problem was stated in the form that has become known as the Euclidean Steiner tree problem or geometric Steiner tree problem: Given N points in the plane, the goal is to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments.

Given the near total destruction of its original habitat, and the lack of any confirmed sightings in over 50 years, most ornithologists believe the Imperial Woodpecker must be extinct.

Given the operational name Storm-333 and involving a combination of Spetsnaz, GRU, KGB, and Soviet paratroopers were a total of 24 men from the « Гром » (" Thunder ") detachment of Alpha Group.