In algebraic geometry, the theorem on formal functions states the following: Let f : X → S {\displaystyle f:X\to S} be a proper morphism of noetherian...

formal functions, which is used to deduce theorems of interest for usual schemes. A locally Noetherian scheme is a locally Noetherian formal scheme in...

mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic". The theorem applies more generally...

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These...

inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. Lagrange...

f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}} . One then uses the theorem on formal functions to show that the last equality implies f ′ {\displaystyle f'}...

intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval [a, b], then it takes on any given value...

is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may...

analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves;...

complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile...

In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses...

Liouville theorem holds for them in analogy to the corresponding theorems in complex functions theory. Some important properties of harmonic functions can be...

In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about...

preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to...

This theorem is about the existence of solutions to a system of m differential equations in n dimensions when the coefficients are analytic functions. The...

sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions. Formal languages are...

fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every point on its...

connectedness theorem Fulton–Hansen connectedness theorem Grothendieck's connectedness theorem Stein factorization Theorem on formal functions Danilov, V...

theorem. The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using...

automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As a Logic for Computable Functions (LCF) style...

(see above), abstract interpretation, automated theorem proving, type systems, and lightweight formal methods. A promising type-based verification approach...

holomorphic functions on C {\displaystyle \mathbb {C} } have unbounded images. The theorem is considerably improved by Picard's little theorem, which says...

A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In...

fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f {\displaystyle...

the addition theorem for elliptic functions. To "classify" addition theorems it is necessary to put some restriction on the type of function G admitted...

squeeze theorem is formally stated as follows. Theorem— Let I be an interval containing the point a. Let g, f, and h be functions defined on I, except...

objective functions that do not change while optimization is in progress, and the second hypothesizes objective functions that may change. Theorem—For any...

transcendental functions such as the exponential function and trigonometric functions. It is the starting point of the study of analytic functions, and is fundamental...

algebraic functions on k. More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions...

Standard conjectures on algebraic cycles Sketch of a program Tannakian formalism Theorem of absolute purity Theorem on formal functions Ultrabornological...