Consider the subset sum problem, an example of a problem that is easy to verify, but whose answer may be difficult to compute.

That means that some member of P ( S ), i. e., some subset of S, is not in the image of f. Consider the set:

Consider a function with its corresponding graph as a subset of the Cartesian product.

Consider an open subset U of the complex plane C. Let a be an element of U, and f: U

Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1 / n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε.

Consider the subset C of B, consisting of those B elements, which satisfy monic polynomial equations over A:

Consider the translates t + N of this subset.

Consider the set, subset of on which the multidimensional definite integral

Consider a wave packet as a function of position x and time t: α ( x, t ).

Consider a pseudo random number generator ( PRNG ) function P ( key ) that is uniform on the interval 2 < sup > b </ sup > − 1.

Consider the logarithm function: For any fixed base b, the logarithm function log < sub > b </ sub > maps from the positive real numbers R < sup >+</ sup > onto the real numbers R ; formally:

Consider a function from a metric space M to a topological space V, and a point c of M. We direct the set M

Consider a function that takes no parameters and returns input from the keyboard.

Consider a wave function that is a sum of many waves, however, we may write this as

Consider the function f, piecewise defined by f ( x ) = – 1 for x < 0 and f ( x ) = 1 for x ≥ 0.

Consider the recursion equations for the factorial function f:

Consider a function that reads the next line of text from a given file:

Consider the complex Hilbert space L < sup > 2 </ sup >( R ), and the operator which multiplies a given function by x:

Consider a function of jump process.

Consider a graph G with vertices V, each numbered 1 through N. Further consider a function shortestPath ( i, j, k ) that returns the shortest possible path from i to j using vertices only from the set

Consider a differentiable function ƒ ( x ) whose derivative is ƒ '( x ).

Consider the vector-valued function F from R < sup > 2 </ sup > to R < sup > 2 </ sup > defined by

Consider an open set on the real line and a function f defined on that set with real values.

Consider, for purposes of illustration, a mountainous landscape M. If f is the function sending each point to its elevation, then the inverse image of a point in ( a level set ) is simply a contour line.

Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their ( f and the reversed and shifted g ) product for each possible amount of sliding.

Consider the complex logarithm function log z.

Consider the function

The Mind in Consider Phlebas is also described as having internal power sources which function as back-up shield generators and space propulsion, and seeing the rational, safety-conscious thinking of Minds, it would be reasonable to assume that all Minds have such features, as well as a complement of drones and other remote sensors as also described.

In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of, then there exists a biholomorphic ( bijective and holomorphic ) mapping from onto the open unit disk

Kernel may mean a subset associated with a mapping, which measures how far it is from being injective or being a monomorphism:

The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of admits a bijective conformal map to the open unit disk in.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

In terms closer to those that Hilbert would have used, near the identity element e of the group G in question, we have some open set U in Euclidean space containing e, and on some open subset V of U we have a continuous mapping

Let U be an open subset of R < sup > n </ sup > and T: U → R < sup > n </ sup > be a bi-Lipschitz mapping.

This mapping is surjective only when is a non-empty proper subset of.

Furthermore, suppose the subset is a disjoint union of open sets each of which is diffeomorphic with by the mapping.

The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient.

A birational mapping between irreducible varieties V and W is a morphism such that its restriction to an open subset U of V is an isomorphism.

It asserts that if is a convex subset of a topological vector space and is a continuous mapping of into itself so that is contained in a compact subset of, then has a fixed point.

If E is a Borel subset of R, and 1 < sub > E </ sub > is the indicator function of E, then 1 < sub > E </ sub >( T ) is a self-adjoint projection on H. Then mapping

More generally, especially as applied to symplectic geometry and singularity theory, a caustic is the critical value set of a Lagrangian mapping where is a Lagrangian immersion of a Lagrangian submanifold L into a symplectic manifold M, and is a Lagrangian fibration of the symplectic manifold M. The caustic is a subset of the Lagrangian fibration's base space B.

A subset S of the domain U of a mapping T is an invariant set under the mapping when Note that the elements of S are not fixed, but rather the set S is fixed in the power set of U.

Concept mapping and mind mapping are a subset of diagramming software aimed to represent collections of ideas.

The Information Exchanges Mapping Tool allows the user to specify metadata and upload XMI domain models associated with NIEM IEPDs, map components within domain models to NIEM components, and generate artifacts based on mappings, including mapping reports, wantlist, exchange schemas, extension schemas, and subset schemas.

In mathematics, and in particular in group theory, a cycle is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing ( i. e., mapping to themselves ) all other elements of X.