Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given the definition of above, we might fix ( or ' bind ') the first argument, producing a function of type.

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Given that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given the above commonalities there appear to be only two string theories: the heterotic string theory ( which is also the type I string theory ) and the type II theory.

Given the above fact, one can ask:

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

Given any of the above definitions of " myth ", the myths of many religions, both ancient and modern, share common elements.

Given the opportunity, for example during an illness that suppresses the immune system, the virus is reactivated and travels to the end of the nerve cell, where it causes the symptoms described above.

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

Given the above, the 39 melakhot are not so much activities as " categories of activity.

Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.

Given the earlier meanings above, this story is probably apocryphal.

Given the rapid orbital decay of objects below approximately, the commonly accepted definition for LEO is between and above the Earth's surface.

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 dimensions of the ellipsoid, the conversion from lat / lon / height-above-ellipsoid coordinates to X-Y-Z is straightforward — calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid.

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

Given all of the above premises are true, the angle B may be acute or obtuse ; meaning, one of the following is true:

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

Given an SVD of M, as described above, the following two relations hold:

Given this s, we can compute the coordinates of 2 ( 2P ), just as we did above: 4P =( 259851, 116255 ).

Given a transfer function in the same form as above:

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given the way black music has been named by ( usually ) outsiders ever since the blues, the reaction to the name by artists who ostensibly fit into the ' neo-soul ' category represents a wonderful example of black self-determination in an industry that is still defiantly wedded to narrow definitions and images of black folks ".

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.

Given these definitions, we can state the second part of the Peter – Weyl theorem:

Given a set, a set is defined to be if it is definable by a formula in which the symbol is interpreted as ; similar definitions for and apply.

Given these definitions, the physical property is:

Given the size and nature of the mortgage market, many observers classify mortgage lending as a separate category of personal borrowing, and consequently residential mortgages are excluded from some definitions of consumer credit-such as the one adopted by the Federal Reserve in the US.

Given the above definitions the interaction Hamiltonian is

Given the controversy on the etiology of PANDAS ( i. e., whether caused by strep throat ) and the apparent ambiguity in the definitions of the clinical presentation leading to different research results, a new definition was proposed that would clarify the presentation of symptoms to ensure that research studies were studying the same group of children.

Given an unramified finite extension L / K of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.

Case: Given a positive ( or more generally irreducible non-negative matrix ) A, for all non-negative non-zero vectors x and f ( x ) as the minimum value of < sub > i </ sub > / x < sub > i </ sub > taken over all those i such that x < sub > i </ sub > ≠ 0, then f is a real valued function whose maximum is the Perron – Frobenius eigenvalue r.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

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.

Given an algebra, a homomorphism h thus defines two algebras homomorphic to, the image h () and The two are isomorphic, a result known as the homomorphic image theorem.

Given a source with a fixed voltage and fixed source impedance, the maximum power theorem says this is the only way to extract the maximum power from the source.

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 the Reynolds operator, Hilbert's theorem is proved as follows.

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 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 that Euler's rotation theorem states that there is always an instantaneous axis of rotation, the angular velocity,, can be given by a vector over this axis

Given two functions, in order to apply the Banach fixed point theorem we want

Given a twice continuously differentiable function f of one real variable, Taylor's theorem for the case n = 1 states that

Given a very ample sheaf L on X and a coherent sheaf F, a theorem of Serre shows that ( the coherent sheaf ) F ⊗ L < sup >⊗ n </ sup > is generated by finitely many global sections for sufficiently large n. This in turn implies that global sections and higher ( Zariski ) cohomology groups

Given a pair, a new index can be constructed using the s < sub > mn </ sub > theorem such that the program coded by ignores its input and merely simulates the computation of the machine with index e on input n.

The invariance theorem follows: Given any description language, our optimal description language is at least as efficient as, with some constant overhead.

Given the RG theorem, the next step in developing a computationally useful method is to determine the fictitious non-interacting system which has the same density as the physical ( interacting ) system of interest.

Given any desired positive integer c, this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by c. This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as schemes.

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