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  2. Universal approximation theorem - Wikipedia

    en.wikipedia.org/wiki/Universal_approximation...

    Also, certain non-continuous activation functions can be used to approximate a sigmoid function, which then allows the above theorem to apply to those functions. For example, the step function works. In particular, this shows that a perceptron network with a single infinitely wide hidden layer can approximate arbitrary functions.

  3. Complex analysis - Wikipedia

    en.wikipedia.org/wiki/Complex_analysis

    Complex analysis, traditionally known as the theory of functions of a complex variable, ... Similarly, any complex-valued function f on an arbitrary set X ...

  4. Stone–Weierstrass theorem - Wikipedia

    en.wikipedia.org/wiki/Stone–Weierstrass_theorem

    The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on are shown to suffice, as is detailed below.

  5. Function (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Function_(mathematics)

    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).

  6. Littlewood's three principles of real analysis - Wikipedia

    en.wikipedia.org/wiki/Littlewood's_three...

    Littlewood stated the principles in his 1944 Lectures on the Theory of Functions [1] as: . There are three principles, roughly expressible in the following terms: Every set is nearly a finite sum of intervals; every function (of class L p) is nearly continuous; every convergent sequence of functions is nearly uniformly convergent.

  7. Gauge fixing - Wikipedia

    en.wikipedia.org/wiki/Gauge_fixing

    A particular choice of the scalar and vector potentials is a gauge (more precisely, gauge potential) and a scalar function ψ used to change the gauge is called a gauge function. [citation needed] The existence of arbitrary numbers of gauge functions ψ(r, t) corresponds to the U(1) gauge freedom of this theory. Gauge fixing can be done in many ...

  8. Automata theory - Wikipedia

    en.wikipedia.org/wiki/Automata_theory

    Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science with close connections to mathematical logic. The word automata comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving".

  9. Blum's speedup theorem - Wikipedia

    en.wikipedia.org/wiki/Blum's_speedup_theorem

    Each computable function has an infinite number of different program representations in a given programming language. In the theory of algorithms one often strives to find a program with the smallest complexity for a given computable function and a given complexity measure (such a program could be called optimal). Blum's speedup theorem shows ...