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.

Given a fixed budget, this can allow an increased sample size.

Given a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as

Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.

Given samples from a population, the equation for the sample skewness above is a biased estimator of the population skewness.

Given a sample of wood, the variation of the tree ring growths provides not only a match by year, it can also match location because the climate across a continent is not consistent.

Given a sample of DNA, the instrument determines the order of the four bases — adenine, guanine, cytosine, and thymine — composing pieces of the sample.

Given the observed percentage difference p − q ( 2 % or 0. 02 ) and the standard error of the difference calculated above (. 03 ), any statistical calculator may be used to calculate the probability that a sample from a normal distribution with mean 0. 02 and standard deviation 0. 03 is greater than 0.

Given any set of points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.

Given a sufficiently large sample size, a statistical comparison will always show a significant difference unless the population effect size is exactly zero.

Given a sample, the expectation function is approximated by the sample average

Given a random sample ( X < sub > i </ sub >, Y < sub > i </ sub >), i = 1, ..., n, each pair ( X < sub > i </ sub >, Y < sub > i </ sub >) satisfies

Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

Given a sample consisting of n independent observations x < sub > 1 </ sub >,..., x < sub > n </ sub > of a p-dimensional random vector X ∈ R < sup > p × 1 </ sup > ( a p × 1 column-vector ), an unbiased estimator of the ( p × p ) covariance matrix

Given a sample of n independent observations x < sub > 1 </ sub >,..., x < sub > n </ sub > of a p-dimensional zero-mean Gaussian random variable X with covariance R, the maximum likelihood estimator of R is given by

Given the estimation of from, the error bars of can be estimated by the sample variance using the unbiased estimate of the variance:

Given a random sample of T observations from this process, the ordinary least squares estimator is

Given the sample mean () and sample coefficient of variation () the parameters and can be estimated:

: Given that there are almost eight times as many data points in this sample as in the previous analysis by Burbidge & Napier ( 2001 ), we must conclude that the previous detection of a periodic signal arose from the combination of noise and the effects of the window function.

Given that v < sup > 2 </ sup > ignores direction, it is logical to assume that the formula can be extended to the entire sample, replacing m with the entire sample's mass, equal to the molar mass times the number of moles n yielding

Given the sample ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, …, x < sub > n </ sub >), it is natural to estimate the characteristic function as