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 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 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 a group G, a factor group G / N is abelian if and only if ≤ N.

* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given a topological space X, let G < sub > 0 </ sub > be the set X.

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 ).

Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

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

noted, " Given that the bandwidth for conducting crawls is neither infinite nor free, it is becoming essential to crawl the Web in not only a scalable, but efficient way, if some reasonable measure of quality or freshness is to be maintained.

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 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 the potential for sampling biases of individual survey reports, researchers and investors try sometimes to average the values of different index reports into a single aggregated measure of consumer confidence.

When the measure was sent in May to the parliament they said " Given the damage it produces on those rules which allow the life in community, ensure the dignity of the person and equality between sexes, this practice, even if it is voluntary, cannot be tolerated in any public place ".

Given that deviance is a measure of the difference between a given model and the saturated model, smaller values indicate better fit.

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

Having a measure on G allows us to define integration of bounded functions on G. Given a bounded function, the integral

Given a left-invariant measure, the function is a right-invariant measure.

Given a σ-finite measure space ( S, Σ, μ ), consider the Banach space L < sup > p </ sup >( μ ).

Given a countably additive measure μ on X, a measurable set is one that differs from a Borel set by a null set.

Given a sequence ( f < sub > n </ sub >) of M-valued measurable functions on some measure space ( X, Σ, μ ), and a measurable subset A of finite μ-measure such that ( f < sub > n </ sub >) converges μ-almost everywhere on A to a limit function f, the following result holds: for every ε > 0, there exists a measurable subset B of A such that μ ( B ) < ε, and ( f < sub > n </ sub >) converges to f uniformly on the relative complement A

Given two objects, A and B, each with n binary attributes, the Jaccard coefficient is a useful measure of the overlap that A and B share with their attributes.

The most recent measure of social dominance orientation ( see SDO-6 above ) focuses on the “ general desire for unequal relations among social groups, regardless of whether this means ingroup domination or ingroup subordination ” ( p. 312 ) Given these changes, Rubin and Hewstone believe that evidence for social dominance theory should be considered “ as supporting three separate SDO hypotheses, rather than one single theory ” ( p. 22 ).

Given a fixed total concentration of one or more species over the measurement time, the scattering signal is a direct measure of the weight-averaged molar mass of the solution, which will vary as complexes form or dissociate.

Given a characteristic functional on a nuclear space A, the Bochner – Minlos theorem ( after Salomon Bochner and Robert Adol ' fovich Minlos ) guarantees the existence and uniqueness of the corresponding probability measure on the dual space, given by

Given a Blum complexity measure and a total computable function with two parameters, then there exists a total computable predicate ( a boolean valued computable function ) so that for every program for, there exists a program for so that for almost all