Convergence may refer to:
Convergence may refer to:
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers is not compact either, because it excludes the two limiting values and . However, the extended real number linewould be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences.
In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize the concept of a sequence in a metric space. Nets are primarily used in the fields of Analysis and Topology, where they are used to characterize many important topological properties that, sequences are unable to characterize. Nets are in one-to-one correspondence with filters.
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.
In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space . The theorem states that each infinite bounded sequence in has a convergent subsequence. An equivalent formulation is that a subset of is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem.
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence, in the sense that the convergence is uniform over the domain. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every pointin. Described in an informal way, if converges to uniformly, then how quickly the functions approach is "uniform" throughout in the following sense: in order to guarantee that differs from by less than a chosen distance , we only need to make sure that is larger than or equal to a certain , which we can find without knowing the value of in advance. In other words, there exists a number that could depend on but is independent of , such that choosing will ensure that for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find such that, for that particular, falls within of whenever .
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.
Technological convergence is the tendency for technologies that were originally unrelated to become more closely integrated and even unified as they develop and advance. For example, watches, telephones, television, computers, and social media platforms began as separate and mostly unrelated technologies, but have converged in many ways into an interrelated telecommunication, media, and technology industry.
In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
The DC Universe (DCU) is the shared universe in which most stories in American comic book titles published by DC Comics take place. In context, the term "DC Universe" usually refers to the main DC continuity. It contains such well-known superheroes as Batman, Superman, Wonder Woman, the Flash, Green Lantern, Aquaman, Green Arrow, Captain Marvel, Martian Manhunter, and Cyborg; as well as teams such as the Justice League, the Justice Society of America, the Suicide Squad, Doom Patrol, and the Teen Titans. It also contains well-known supervillains, including the Joker, Lex Luthor, the Cheetah, the Reverse-Flash, Sinestro, Black Manta, Deathstroke, Black Adam, Brainiac, and Darkseid.
A uniform is a standard set of clothing identifying the wearer as a member of an organisation.
An invasion is a military action of soldiers entering a foreign land.
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of countability, and all first-countable spaces are sequential.
Inheritor or Inheritors may refer to:
Wijsman convergence is a variation of Hausdorff convergence suitable for work with unbounded sets. Intuitively, Wijsman convergence is to convergence in the Hausdorff metric as pointwise convergence is to uniform convergence.
In mathematics, a limit is the value that a function approaches as the input approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
Convergence proof techniques are canonical components of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.