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Page "Algebraic variety" ¶ 14
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Given and subset
Given a set of integers, does some nonempty subset of them sum to 0?
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 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 any set A, there is a set such that, given any set B, B is a member of if and only if B is a subset of A.
Given a bounded sequence, there exists a closed ball that contains the image of ( is a subset of the scalar field ).
Given a set of integers, FIND-SUBSET-SUM is the problem of finding some nonempty subset of the integers that adds up to zero ( or returning the empty set if there is no such subset ).
Given a set of integers, SUBSET-SUM is the problem of finding whether there exists a subset summing to zero.
Given any subset F =
Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).
* Rural postman problem: Given is also a subset of the edges.
for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:
Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.
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 a subset V of A < sup > n </ sup >, we define I ( V ) to be the ideal of all functions vanishing on V:
Given a ring R and a subset S, one wants to construct some ring R * and ring homomorphism from R to R *, such that the image of S consists of units ( invertible elements ) in R *.
; Generating set: Given a field extension E / F and a subset S of E, we write F ( S ) for the smallest subfield of E that contains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations +,-,*,/ on the elements of F and S. If E = F ( S ) we say that E is generated by S over F.
Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.
Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?
Given a compact subset K of X and an open subset U of Y, let V ( K, U ) denote the set of all functions such that Then the collection of all such V ( K, U ) is a subbase for the compact-open topology on C ( X, Y ).
Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of and its analytic capacity, it might be

Given and V
Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).
If V is finite-dimensional, then V * has the same dimension as V. Given a basis
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 vector space V over the field R of real numbers, a function is called sublinear if
:( a ) Given player 2's strategy, the best payoff possible for player 1 is V, and
:( b ) Given player 1's strategy, the best payoff possible for player 2 isV.
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.
Given a class function G: VV, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that
Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:
Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.
A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that
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 subspaces U and W of a vector space V, then their intersection U ∩ W :=

Given and P
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 a recursive presentation P
Given a finite presentation P =
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 >

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