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Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.
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).
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V.In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
The function field of the n-dimensional space over a field F is F(x 1, ..., x n), i.e., the field consisting of ratios of polynomials in n indeterminates. The function field of X is the same as the one of any open dense subvariety. In other words, the function field is insensitive to replacing X by a (slightly) smaller subvariety.
Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex topological vector space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other ...
Similarly, any complex-valued function f on an arbitrary set X (is isomorphic to, and therefore, in that sense, it) can be considered as an ordered pair of two real-valued functions: (Re f, Im f) or, alternatively, as a vector-valued function from X into .
The most general notion is the intersection of an arbitrary nonempty collection of sets. If M {\displaystyle M} is a nonempty set whose elements are themselves sets, then x {\displaystyle x} is an element of the intersection of M {\displaystyle M} if and only if for every element A {\displaystyle A} of M , {\displaystyle M,} x {\displaystyle x ...