Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

: Given two sets, P (" facilities ") and L (" locations "), of equal size, together with a weight function w: P × P → R and a distance function d: L × L → R. Find the bijection f: P → L (" assignment ") such that the cost function:

Given the assignment by then-Seton Hall University president Monsignor James Kelly to create a radio station, Monsignor Gillhooly got WSOU up and running in just three months and provided a steady hand during the station ’ s early days.

Given a group of n volunteers, one possible design would be to give n / 2 of them shoes with the new soles and n / 2 of them shoes with the ordinary soles, randomizing the assignment of the two kinds of soles.

Given its ineffectiveness on that occasion, the strategy was then largely forgotten, aside from the fact that Maverick player Bubba Wells, whose assignment it had been to foul Rodman, set the all-time NBA record for fewest minutes played ( 3 ) in fouling out of a game.

Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.

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 an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

# Given a large enough beta-tester and co-developer base, almost every problem will be characterized quickly and the fix obvious to someone.

: Given any set A and any set B, if for every set C, C is a member of A if and only if C is a member of B, then A is equal to B.

Given expectations about returns on fixed investment, every level of the real interest rate ( i ) will generate a certain level of planned fixed investment and other interest-sensitive spending: lower interest rates encourage higher fixed investment and the like.

Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.

Given an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.

The law states that " given enough eyeballs, all bugs are shallow "; or more formally: " Given a large enough beta-tester and co-developer base, almost every problem will be characterized quickly and the fix will be obvious to someone.

Given an input consisting of any number of bytes, it produces as output a single byte that is strongly dependent on every byte of the input.

Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin.

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

* Given 27 same-size cubes whose nominal values progress from 1 to 27, a 3 × 3 × 3 magic cube can be constructed such that every row, column, and corridor, and every diagonal passing through the center, is composed of 3 cubes whose sum of values is 42.

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 that it would be complicated to perform this test on every patient with sleep-related breathing problems, some suggest that measuring bicarbonate levels in normal ( venous ) blood would be a reasonable screening test.

Given that there are around 700, 000 live births in Britain each year, Meadow argued that a double cot death would be expected to occur once every hundred years.

Given an arbitrary ( n ; m ; p ) machine S, such that every two of its states are distinguishable from one another, then there exists an experiment of length

* Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T.

Given a labelled state transition system ( S, Λ, →), a simulation relation is a binary relation R over S ( i. e. R ⊆ S × S ) such that for every pair of elements p, q ∈ S, if ( p, q )∈ R then for all α ∈ Λ, and for all p ' ∈ S,

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

Given a function f of a real variable x and an interval of the real line, the definite integral

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 a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given a set of variable symbols X =

Given its location at the base of the Black Hills and its proximity to the High Plains, the climate in Spearfish is highly variable at any time of the year.

Given the limited market for a very expensive product of variable quality, Morganite also decided that carbon-fiber production was peripheral to its core business, leaving Courtaulds as the only big UK manufacturer.

Given a probability space with, the indicator random variable is defined by if otherwise

Given a graph Λ ( for example, a d-dimensional lattice ), per each lattice site j ∈ Λ there is a discrete variable σ < sub > j </ sub > such that σ < sub > j </ sub > ∈

Given an unobservable function that relates the independent variable to the dependent variable – say, a line – the deviations of the dependent variable observations from this function are the errors.

Given that the toxins alter acetylcholine transmission which causes the paralysis, some patients have been successfully treated with cholinesterase inhibitors such as physostigmine or neostigmine, but success is variable and may be species dependent as well.

Given the variable half-life of flecainide and the characteristic QRS prolongation on ECG elicited in flecainide toxicity, especially at rapid heart rates, starting flecainide or changing the level of the drug is done under telemetry monitoring ( preferably in a hospital telemetry unit ) until a steady state plasma level has been achieved, typically three to five days after the dose has been increased.

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms.

Given discrete random variable with support and with support, the conditional entropy of given is defined as:

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 a set of independent identically distributed data points where according to some probability distribution parameterized by θ, where θ itself is a random variable described by a distribution, i. e. the marginal likelihood in general asks what the probability is, where θ has been marginalized out ( integrated out ):

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

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

Given two random variables and, with correlation, and a known reliability for each variable, and, the correlation between and corrected for attenuation is

Given a Banach space, a subset of, and a functional from to the dual space of the space, the variational inequality problem is the problem of solving respect to the variable ' belonging to the following inequality:

Given that diabetic women show a significant variability in their response to this medical disorder, it is not surprising that the disease ’ s influence on arousal is also highly variable.

Given any x and any y, if all the molecules in y are perfectly elastic and spherical, possess equal masses and volumes, have negligible size, and exert no forces on one another except during collisions, then if x is a gas and y is a given mass of x which is trapped in a vessel of variable size and the temperature of y is kept constant, then any decrease of the volume of y increases the pressure of y proportionally, and vice versa.