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The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem).
A partition of unity can be used to define the integral (with respect to a volume form) of a function defined over a manifold: one first defines the integral of a function whose support is contained in a single coordinate patch of the manifold; then one uses a partition of unity to define the integral of an arbitrary function; finally one shows ...
The first modern and more precise definition of a vector space was introduced by Peano in 1888; [5] by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century when many ideas and methods of previous centuries were generalized as ...
Analytic continuation of natural logarithm (imaginary part) Analytic continuation is a technique to extend the domain of a given analytic function.Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
In the mathematical theory of artificial neural networks, universal approximation theorems are theorems [1] [2] of the following form: Given a family of neural networks, for each function from a certain function space, there exists a sequence of neural networks ,, … from the family, such that according to some criterion.
On the other hand, if X is the domain of a function f(x) and if the limit as n approaches infinity of f(x n) is L for every arbitrary sequence of points {x n} in X − x 0 which converges to x 0, then the limit of the function f(x) as x approaches x 0 is equal to L. [10] One such sequence would be {x 0 + 1/n}.
In type theory and in outgrowths thereof such as the axiomatic set theory NF, the Quine–Rosser pair has the same type as its projections and hence is termed a "type-level" ordered pair. Hence this definition has the advantage of enabling a function , defined as a set of ordered pairs, to have a type only 1 higher than the type of its arguments.
Another example. A structure of a (simple) graph on a set V = { 1, 2, ..., n } of vertices may be described by means of its adjacency matrix, a (0,1)-matrix of size n×n (with zeros on the diagonal). More generally, for arbitrary V an adjacency function on V × V may be used. The canonical equivalence is given by the rule: "1" means "connected ...