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Page "Stone–Čech compactification" ¶ 38
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Given and bounded
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 representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.
Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold:
Having a measure on G allows us to define integration of bounded functions on G. Given a bounded function, the integral
Given a differentiable function defined on a bounded open set, the total variation of has the following expression
Given a polynomially bounded functional F over the field configurations, then, for any state vector ( which is a solution of the QFT ), | ψ >, we have
Given a set of points,, and the set of hyperplanes,, which are each spanned by, the number of incidences between and is bounded above by
Given a topological vector space ( X, τ ) over a field F, S is called bounded if for every neighborhood N of the zero vector there exists a scalar α so that
* Theorem ( by Mackey ): Given a dual pair, the bounded sets under any dual topology are identical.
Given the extent of the parish, it is bounded by several other parishes and villages.

Given and sequence
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 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 and there
: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.
* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.
Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).
Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.
Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).
Given its different forms, there are various ways of representing uncertainty and modelling economic agents ' responses to it.
* Given any set X, there is an equivalence relation over the set of all possible functions X → X.
* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.
Given a left neutral element and for any given then A4 ’ says there exists an such that.
Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force.
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 that non-violence has priority, all other principles yield to it whenever there is a conflict.
Given this assumption, there are four categories in which a firm's profit may be considered to be.
Given the overall size of trade between Mexico and the United States, there are remarkably few trade disputes, involving relatively small dollar amounts.
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 infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.
Given the diversity of functions performed by neurons in different parts of the nervous system, there is, as expected, a wide variety in the shape, size, and electrochemical properties of neurons.
Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.
Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!
Given the immense expanse of the entire Universe, it has been argued that there is a higher probability that there exists ( or has existed ) another Earth-like planet that has yielded life ( geogenesis ) than not.
Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).
# Given any two distinct points, there is exactly one line incident with both of them.

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