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 also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

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 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 a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

* 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 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.

* Given a category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

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 the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums: the product IJ is again a fractional ideal.

Given the finite supply of natural resources at any specific cost and location, agriculture that is inefficient or damaging to needed resources may eventually exhaust the available resources or the ability to afford and acquire them.

Given a function w on U × Y, with finite integral of its modulus for any input function u and initial state x ( 0 ) over any finite time t, called the " supply rate ", a system is said to be dissipative if there exist a continuous nonnegative function V ( x ), with x ( 0 ) = 0, called the storage function, such that for any input u and initial state x ( 0 ) the difference V ( x ( t )) − V ( x ( 0 )) does not exceed the integral of the supply over ( 0, t ) for any t ( dissipation inequality ).

Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil ( based on working out some examples ) was striking and novel.

Given a finite set

Given a finite observation set S, one can simply select the measure for all.

For a finite group G, the left regular representation λ ( over a field K ) is a linear representation on the K-vector space V whose basis is the elements of G. Given g ∈ G, λ ( g ) is the linear map determined by its action on the basis by left translation by g, i. e.

Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance.

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F ( S ) whose underlying set is N < sup > n </ sup > where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of N < sup > n </ sup >; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms.

Given a base scheme S, an algebraic torus over S is defined to be a group scheme over S that is fpqc locally isomorphic to a finite product of multiplicative groups.

* Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble

:: Given a recursive presentation P

Given that many of the talented young artists had been raised in housing projects and were short on social and dress skills, this Motown department was not only necessary, it created an elegant style of presentation long associated with the label.

Given their role in antigen presentation, HSPs are useful as immunologic adjuvants in boosting the response to a vaccine.

Given the initial presentation of Guellen, Claire's power, the history between Claire and Ill, and the nature of Claire's decree, Ill's death appears almost inevitable.

" Given this presentation, it is hardly surprising that it has been taken to be a retelling of an actual legend ; Gallico writes that " the person and character of the painter are wholly fictional as is the story itself, although I am told that in some quarters the snow goose appearing over Dunkirk has been accepted as legend and I have been compelled to reply to many correspondents that it was sheer invention.

Given a presentation of a group G by generators and relations and a subgroup H of G, the algorithm enumerates the cosets of H on G and describes the permutation representation of G on the space of the cosets.

Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.

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 points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

Given the fact that the period P of an object in circular orbit around a spherical object obeys

Given the abundance of such optimization problems in everyday life, a positive answer to the " P vs. NP " question would likely have profound practical and philosophical consequences.

Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line.

Given two points P and Q on C, let s ( P, Q ) be the arc length of the portion of the curve between P and Q and let d ( P, Q ) denote the length of the line segment from P to Q.

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.

: Given any x and y, x = y if, given any predicate P, P ( x ) if and only if P ( y ).

Given to films dealing with science and technology by the Alfred P. Sloan Foundation each year at the Sundance Film Festival.

Given two permutations π and σ of m elements and the corresponding permutation matrices P < sub > π </ sub > and P < sub > σ </ sub >