Mathematical analysisAnalysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
Analyticity of holomorphic functionsIn complex analysis, a complex-valued function of a complex variable : is said to be holomorphic at a point if it is differentiable at every point within some open disk centered at , and is said to be analytic at if in some open disk centered at it can be expanded as a convergent power series (this implies that the radius of convergence is positive). One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa.
Local class field theoryIn mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Qp (where p is any prime number), or the field of formal Laurent series Fq((T)) over a finite field Fq
Continuum hypothesisIn mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that there is no set whose cardinality is strictly between that of the integers and the real numbers, or equivalently, that any subset of the real numbers is finite, is countably infinite, or has the same cardinality as the real numbers. In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: , or even shorter with beth numbers: .
Continuum (set theory)In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by . Georg Cantor proved that the cardinality is larger than the smallest infinity, namely, . He also proved that is equal to , the cardinality of the power set of the natural numbers. The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers, , or alternatively, that .
Continuum percolation theoryIn mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space (often Euclidean space Rn). More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components.
Cardinality of the continuumIn set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by (lowercase Fraktur "c") or . The real numbers are more numerous than the natural numbers . Moreover, has the same number of elements as the power set of Symbolically, if the cardinality of is denoted as , the cardinality of the continuum is This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities.
