Given any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

Given a color name to match the name of amber mutants, ochre mutant viruses had a similar property in that they recovered infectious ability within certain suppressor strains of bacteria.

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that

Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis.

Given a set of n + 1 data points ( x < sub > i </ sub >, y < sub > i </ sub >) where no two x < sub > i </ sub > are the same, one is looking for a polynomial p of degree at most n with the property

Given a graph G with n vertices, the closure cl ( G ) is uniquely constructed from G by repeatedly adding a new edge uv connecting a nonadjacent pair of vertices u and v with until no more pairs with this property can be found.

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A < sub > L </ sub > there exists a unique unital algebra homomorphism g: U → A such that: f (-) = g < sub > L </ sub > ( h (-)).

Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).

On the issue of the second home he wrote: " I have concluded that Mr MacKay was wrong to designate the Bromsgrove property as his main home Given that, it would be unreasonable to hold that Ms Kirkbride could not make legitimate claims from parliamentary resources for the home in her constituency which she had identified as her second home ".

Given a map, and a space, one says that has the homotopy lifting property, or that has the homotopy lifting property with respect to, if:

Given additionally a map, one says that has the homotopy lifting extension property if:

Given a graph property P, an invariant u, and a set of graphs H, we wish to find the minimum value of m such that every graph in H which has u larger than m possess property P. In the example above, H was the set of n-vertex graphs, P was the property of being cyclic, and u was the number of edges in the graph.

Given the preceding jurisprudence regarding due process, Chief Justice Taney was not entirely breaking ground in his Dred Scott opinion when, without elaboration, he pronounced the Missouri Compromise unconstitutional because an " act of Congress that deprived a citizen of his liberty or property merely because he came himself or brought his property into a particular territory of the United States, and who had committed no offence against the laws, could hardly be dignified with the name of due process of law.

Given an exterior connection D satisfying this compatibility property, there exists a unique extension of D:

Given these definitions, the physical property is:

Given the previous conclusion and noting that depends only on y, the non-decreasing property of R < sup > 2 </ sup > follows directly from the definition above.

Given the new importance that IBD alleles are assuming in genetic association studies, a new definition of IBD that would better suit the underlying inheritance property would be that two or more alleles are identical by descent ( IBD ) if they have been inherited from the same ancestral allele without recombination events.

Given the value of riverside property now that flooding is better managed, factories are now being sold to make way for upscale waterfront apartments.

Given scalar functions f and h, and a real number a, the Laplacian has the property:

Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime,, ( also called simply the lifetime ) is the expected value of the amount of time before an object is removed from the assembly.

Given a mean AGB lifetime of one Myr and an outer velocity of, its maximum radius can be estimated to be roughly ( 30 light years ).

Given that the Tenn-Tom Waterway has an estimated lifetime of well over 50 years, the Waterway is expected to play a significant role in southern interstate trade for years to come.

Given that Classical scholarship was more widespread among the potential audience for his paintings during his lifetime than in the present day, meticulous research of detail was important in order to attain a standing as an artist in this genre.

Without doubt, the single most significant paper concerning the distribution of prime numbers was Riemann's 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Given their wide historical distribution, red wolves probably utilized a large suite of habitat types at one time.

Given the date of his publication and the widespread, permanent distribution of his work, it appears that he should be regarded as the originator of the concept of space sailing by light pressure, although he did not develop the concept further.

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval ( 0, 1 ), the variate

Solomonoff's universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:

Given testable information, the maximum entropy procedure consists of seeking the probability distribution which maximizes information entropy, subject to the constraints of the information.

Given the global nature of gonococcal and meningococcal diseases, the worldwide distribution of antibiotics, differing social practices in controlling and monitoring antibiotic availability, and geographical differences in treatment regimens, it is likely that the global problem of antibiotic resistance will continue in the foreseeable future.

Given that there are only two possible outcomes in a binomial distribution, one can determine the proportion of " noncases " from the proportion of " cases " and vice versa.

Given its native distribution the species is also a potential natural vagrant to Western Europe and there have been records in areas such as Cornwall and the Isles of Scilly which some observers consider may relate to wild birds ; however, given the Wood Duck's popularity in captivity it would be extremely difficult to prove their provenance one way or another.

Given a random variate U drawn from the uniform distribution in the interval < nowiki >( 0, 1 )</ nowiki >, then the variate

Given a random variate U drawn from the uniform distribution in the interval < nowiki ></ nowiki >, the variate

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 the unequal distribution of talent, energy, and luck, Classical legal thought posited that a self-executing, decentralized, competitive market economy would inevitably result in organized inequality.

The universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

Given a specified distribution of matter and energy in the form of a stress – energy tensor, the EFE are understood to be equations for the metric tensor, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner.

Given the mean arrival and service rates, then actual rates vary around these average values randomly and hence have to be determined using a cumulative probability distribution function.

Given our current estimate of the parameters θ < sup >( t )</ sup >, the conditional distribution of the Z < sub > i </ sub > is determined by Bayes theorem to be the proportional height of the normal density weighted by τ:

Given a statistical manifold, with coordinates given by, one writes for the probability distribution.

Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value.

Given two random variables X and Y whose joint distribution is known, the marginal distribution of X is simply the probability distribution of X averaging over information about Y.