Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

Given the difficulty of this position, it is not uncommon for the best player on the team to play Number Two so long as another strong player is available to play Three.

Given the difficulty in treating scleroderma, treatments with a smaller evidence base are often tried to control the disease.

Given this view, the difficulty or impossibility of finding a plausible explanation for evil in a world created by God is to be expected, and so the argument from evil is assumed to fail unless it can be proven that God's reasons would be comprehensible to us.

Given a desired backbone conformation, side chain packing can be designed using variations of the dead-end elimination algorithm ; however, attempts to design proteins of novel folds have difficulty using this method due to an absence of plausible backbone models.

Given the difficulty of distinguishing these differences, an accurate and reliable schizoaffective disorder diagnosis usually requires an extended period of observation and treatment.

Given this fundamental logical difficulty, there has been very little experimental investigation by parapsychologists of retrocognition.

Given the difficulty of scoring directly from the point due to the distance to the goal, goals scored from the point are typically either on screens, or are tipped goals.

Given his adept use of angle, depth, and pace, his opponents often have difficulty breaking down one particular side.

Given the difficulty of identifying the Messianic figure, Jews also pray for the coming of the prophet Elijah to serve as the Messiah's herald and to properly identify him.

Given the Narodniks ' generally middle-and upper-middle-class social background, they found difficulty relating to the impoverished peasants and their culture.

Given the difficulty of this task as proven in Book I, Socrates in Book II leads his interlocutors into a discussion of justice in the city, which Socrates suggests may help them see justice not only in the person, but on a larger scale, " first in cities searching for what it is ; then thusly we could examine also in some individual, examining the likeness of the bigger in the idea of the littler "

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a " general " solution to the Einstein field equation, or even a " general " solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions.

Given how difficult all concerned found discussing the details of the crimes at issue, their language characterizes the questionable activities repeatedly without ever specifying the actions themselves.

The wavelets forming a continuous wavelet transform ( CWT ) are subject to the uncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event.

Given that Rothko had known in advance about the luxury decor of the restaurant and the social class of its future patrons, the exact motives for his abrupt repudiation remain mysterious.

Given a short exact sequence with maps q and r:

Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences.

Given that there are many historical definitions of the keystone species concept, and without a consensus on its exact definition, a list of examples best illustrates the concept of keystone species.

The Mastermind satisfiability problem is a decision problem that asks, " Given a set of guesses and the number of colored and white pegs scored for each guess, is there at least one secret pattern that generates those exact scores?

Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,

Given a module, M, a projective resolution of M is an infinite exact sequence of modules

Given approximations of A from three distinct step sizes h, h / t, and h / s, the exact relationship

Given R-modules A and B, an extension of A by B is a short exact sequence of R-modules

Given the tendency of the legends of different seafarers – Greek, Norse, Irish, Arab and Iberian – to cross-fertilize and influence each other, the exact source of some legendary Atlantic islands – such as the mythical islands of Brasil and the Isle of Mam – are impossible to disentangle.

Given a subspace, one may form the short exact sequence

Given the possibility that symptoms of a known disease may have mutated over time, or that the plague was caused by a disease which no longer exists, the exact nature of the Athenian plague may never be known.

The exponential map is surjective on the stalks: Given a germ g of an holomorphic function at a point P such that g ( P ) ≠ 0, one can take the logarithm of g in a neighborhood of P. The long exact sequence of sheaf cohomology shows that we have an exact sequence

Given an exact differential equation defined on some simply connected and open subset D of R < sup > 2 </ sup > with potential function F then a differentiable function f with ( x, f ( x )) in D is a solution if and only if there exists real number c so that

Given a parameterized family of probability density functions ( or probability mass functions in the case of discrete distributions )

Given that the derivatives as Schwartz distributions are symmetric, this kind of example belongs in the ' fine ' theory of real analysis.

Given its importance, < tt > dvips </ tt > is a standard part of most TeX distributions, such as teTeX, and TeX Live.

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 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

Given a total sample size greater than 1.