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 infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

noted, " Given that the bandwidth for conducting crawls is neither infinite nor free, it is becoming essential to crawl the Web in not only a scalable, but efficient way, if some reasonable measure of quality or freshness is to be maintained.

* Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.

Given any input, GEN generates an infinite number of candidates, or possible realizations of that input.

Given two points A and B, with A not lower than B, there is just one upside down cycloid that passes through A with infinite slope, passes also through B and does not have maximum points between A and B.

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 we have and are everything, and there's nothing we have to do, there are an infinite number of ways to experience this, not just the one way we may have chosen so far.

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

Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the precision of approximate equality might be the most obvious way to assess mathematical coincidences ; but there is no standard, and the strong law of small numbers is the sort of thing one has to appeal to with no formal opposing mathematical guidance.

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 a Hilbert space ( either finite or infinite dimensional ), its complex conjugate is the same vector space as its continuous dual space.

Given two real numbers, say x and y, with we define an uncountably infinite family of open sets denoted by S < sub > x, y </ sub > as follows:

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.

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 our formula φ, we group strings of quantifiers of one kind together in blocks:

Given that this scale is much smaller than any cosmological scale these strings are often studied in the zero width, or Nambu-Goto approximation.

Given two strings, equally taut and heavy, one twice as long as the other, the longer would vibrate with a pitch one octave lower than the shorter.

Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that

Given of a monoid M of every strings over some alphabet, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating.

Given the context of this conversation ( Blackadder is attempting to get transferred into the Royal Flying Corps ), and Darling's character in general, he has presumably pulled some strings to allow him to remain in the army without having to serve in battle.

Given two strings, of length and of length, find the longest strings which are substrings of both and.

Given the set of strings, where and Σ.

Given a sequence " qrj ," " aty ," " qur ," " dim ," " ofu ," " gcl ," " rhv ," " clq ," " ecd ," " qsu " of randomly generated three character long strings, the following table would be generated ( using Bob Jenkins ' One-at-a-Time hash algorithm ) with a table of size 10:

Given a binary relation R between fixed strings in the alphabet, called rewrite rules, denoted by, an SRS extends the rewriting relation to all strings in which the left-and right-hand side of the rules appear as substrings, that is, where s, t, u, and v are strings.