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Page "Diffeomorphism" ¶ 8
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Given and subset
Given a set of integers, does some nonempty subset of them sum to 0?
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 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.
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 X
: 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 any Banach space X, the continuous linear operators A: X X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.
* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.
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 any set X, there is an equivalence relation over the set of all possible functions X X.
Given a topological space X, let G < sub > 0 </ sub > be the set X.
Given a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as
# 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 X such that
Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).
Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.

Given and manifold
Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.
Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.
Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping
Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:
Given a local coordinate system x < sup > i </ sup > on the manifold, the reference axes for the coordinate system are the vector fields
Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.
Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define
Given a Riemannian manifold with metric tensor, we can compute the Ricci tensor, which collects averages of sectional curvatures into a kind of " trace " of the Riemann curvature tensor.
Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M,
Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:
Given a manifold M representing ( continuous / smooth / with certain boundary conditions / etc.
* Given the action of a Lie algebra g on a manifold M, the set of g-invariant vector fields on M is a Lie algebroid over the space of orbits of the action.
* Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other.
* Given a Lie group acting on a manifold, there is a Lie groupoid called the translation groupoid with one morphism for each triple with.
Given an oriented manifold M of dimension n with fundamental class, and a G-bundle with characteristic classes, one can pair a product of characteristic classes of total degree n with the fundamental class.
Given a manifold with a submanifold, one sometimes says can be knotted in if there exists an embedding of in which is not isotopic to.
A more general class are flat G-bundles with for a manifold F. Given a representation, the flat-bundle with monodromy is given by, where acts on the universal cover by deck transformations and on F by means of the representation.
Given a smooth 4n-dimensional manifold M and a collection of natural numbers
Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.
Given a function on, one may " geometrize " it by taking it to define a new manifold.
Given a statistical manifold, with coordinates given by, one writes for the probability distribution.

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