[permalink] [id link]
Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.
from
Wikipedia
Some Related Sentences
Given and subset
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 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 >).
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 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 ).
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 and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.
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 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 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 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 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 >
0.304 seconds.