Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.

: Given any positive number ε, there is a sequence

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that

Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure.

* Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.

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 a bounded sequence, there exists a closed ball that contains the image of ( is a subset of the scalar field ).

Given a testing procedure E applied to each prepared system, we obtain a sequence of values

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 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 a short exact sequence with maps q and r:

Given an ordered sequence of real numbers: the first difference is defined as

Given a linearly recursive sequence, let C be the transpose of the companion matrix of its characteristic polynomial, that is

Given this hypothesis that a novel FOXP2 sequence can aid echolocation, echolocating and non echolocating cetaceans might be predicted to display differences in their FOXP2 sequences.

Given a strictly increasing integer sequence / function ( n ≥ 1 ) we can produce a faster growing sequence ( where the superscript n denotes the n < sup > th </ sup > functional power ).

Given a sequence of positive integers, the Gödel encoding of the sequence is the product of the first n primes raised to their corresponding values in the sequence:

Given the observation space, the state space, a sequence of observations, transition matrix of size such that stores the transition probability of transiting from state to state, emission matrix of size such that stores the probability of observing from state, an array of initial probabilities of size such that stores the probability that. We say a path is a sequence of states that generate the observations.

Given two sequences X and Y, a sequence G is said to be a common subsequence of X and Y, if G is a subsequence of both X and Y.

Given an isogeny f of degree n, one can prove using linear algebra on weights and faithfully flat descent that there exists a dual isogeny g such that gf is the nth power map on the source torus.

Given good circumstances one might be able to discern the result of some human activity such as the changing of the Netherlands ' coast or the partial drying out of the Aral Sea, but even that would not be easy.

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.

Given a partial derivative, it allows for the partial recovery of the original function.

Given the particular differential operators involved, this is a linear partial differential equation.

Given a natural number x, h outputs the index of the partial computable function that performs the following computation:

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 a domain D we define a tuple over D as a partial function

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:

Given a subset of the index set, the partial hypergraph generated by is the hypergraph

Given a subset, the section hypergraph is the partial hypergraph

Given two polynomials and, where the α < sub > i </ sub > are distinct constants and deg P < n, partial fractions are generally obtained by supposing that

* Given a partial function f from the natural numbers into the natural numbers, f is a partial recursive function if and only if the graph of f, that is, the set of all pairs such that f ( x ) is defined, is recursively enumerable.

Given a partial solution to the puzzle, they use dynamic programming within each row or column to determine whether the constraints of that row or column force any of its squares to be white or black, and whether any two squares in the same row or column can be connected by an implication relation.

Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of and its analytic capacity, it might be

Given a Gödel numbering of recursive functions, there is a primitive recursive function s of two arguments with the following property: for every Gödel number p of a partial computable function f with two arguments, the expressions and are defined for the same combinations of natural numbers x and y, and their values are equal for any such combination.

Given a set of integers, does some nonempty subset of them sum to 0?

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

** Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.

Given two ultrafilters and on, we define their sum by

Given a circle A, find a circle B such that the area of the intersection of A and B is equal to the area of the symmetric difference of A and B ( the sum of the area of A − B and the area of B − A ).

Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line.

Given the sheer power of leverage the Rothschild family had at its disposal, this profit was an enormous sum.

* 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.

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 the rapid growth in absolute value of Γ ( z + k ) when k → ∞, and the fact that the reciprocal of Γ ( z ) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex s and x.

Given a faithful irreducible representation ρ of G, the lattice Yang-Mills action is the sum over all lattice sites of the ( real component of the ) trace over the n links e < sub > 1 </ sub >, ..., e < sub > n </ sub > in the Wilson loop,

Given a finite set of probability density functions p < sub > 1 </ sub >( x ), …, p < sub > n </ sub >( x ), or corresponding cumulative distribution functions P < sub > 1 </ sub >( x ), …, P < sub > n </ sub >( x ) and weights w < sub > 1 </ sub >, …, w < sub > n </ sub > such that and the mixture distribution can be represented by writing either the density, f, or the distribution function, F, as a sum ( which in both cases is a convex combination ):

Given a large sum of money by the team to set up a design office in England, Barnard founded the Ferrari Guildford Technical Office and began work on returning Ferrari to regular winning.

* Given two representations ρ < sub > 1 </ sub >, ρ < sub > 2 </ sub > we may construct their direct sum ρ < sub > 1 </ sub > ⊕ ρ < sub > 2 </ sub > by ( ρ < sub > 1 </ sub > ⊕ ρ < sub > 2 </ sub >) ( g )( v, w )

Given two I-graded vector spaces V and W, their direct sum has underlying vector space V ⊕ W with gradation

Given two positive integers N and i, there is a unique way to expand N as a sum of binomial coefficients as follows:

Given two objects A and B, their direct sum is written as.

Given a indexed family of objects A < sub > i </ sub >, indexed with i ∈ I from an index set I, one may write their direct sum as.

Given two abelian groups ( A, ∗) and ( B, ·), their direct sum A ⊕ B is the same as their direct product, i. e. its underlying set is the Cartesian product A × B with the group operation ○ given componentwise:

Given a finite family of rings R < sub > 1 </ sub >, ..., R < sub > n </ sub >, the direct product of the R < sub > i </ sub > is sometimes called the direct sum.