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Что (кто) такое Cauchy sequences - определение

SEQUENCE WHOSE ELEMENTS BECOME ARBITRARILY CLOSE TO EACH OTHER

Cauchy sequences; Regular Cauchy sequence; Cauchy Sequences; Cauchy Sequence

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Cauchy sequence

<mathematics> A sequence of elements from some vector space that converge and stay arbitrarily close to each other (using the norm definied for the space). (2000-03-10)

Cauchy sequence

In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

https://en.wikipedia.org/wiki/Cauchy_sequence

Uniformly Cauchy sequence

SEQUENCE FUNCTION

Uniformly cauchy; Uniformly Cauchy

In mathematics, a sequence of functions \{f_{n}\} from a set S to a metric space M is said to be uniformly Cauchy if:

https://en.wikipedia.org/wiki/Uniformly_Cauchy_sequence

Augustin-Louis Cauchy

FRENCH MATHEMATICIAN (1789–1857)

Augustin Cauchy; Cauchy; Augustin Louis, Baron Cauchy; A. L. Cauchy; Augustin-Louis, Baron Cauchy; Cauchy, Augustin Louis; A. L. de Cauchy; Augustin Louis Baron Cauchy; Augustin louis cauchy; Baron Augustin-Louis Cauchy; Augustin Louis Cauchy; Augustine Louis Cauchy

Baron Augustin-Louis Cauchy (, koh-SHEE; French: [oɡystɛ̃ lwi koʃi]; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra.

A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics.

https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy

Cauchy–Kowalevski theorem

LOCAL EXISTENCE AND UNIQUENESS THEOREM FOR ANALYTIC PARTIAL DIFFERENTIAL EQUATIONS ASSOCIATED WITH CAUCHY INITIAL VALUE PROBLEMS

Cauchy-Kovalevsky theorem; Cauchy-Kowalewski Theorem; Cauchy-Kowalewski theorem; Cauchy-Kovalevskaya theorem; Cauchy-Kowalevsky theorem; Cauchy-Kovalevski theorem; Cauchy-Kowalevski theorem; Cauchy-Kowalevski Theorem; Cauchy–Kovalevskaya theorem

In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by , and the full result by .

https://en.wikipedia.org/wiki/Cauchy–Kowalevski_theorem

Cauchy–Schwarz inequality

A USEFUL INEQUALITY ENCOUNTERED IN MANY DIFFERENT SETTINGS, SUCH AS LINEAR ALGEBRA, ANALYSIS, PROBABILITY THEORY, VECTOR ALGEBRA AND OTHER AREAS. IT IS CONSIDERED TO BE ONE OF THE MOST IMPORTANT INEQUALITIES IN ALL OF MATHEMATICS

Cauchy-Schwartz inequality; Cauchy-Schwarz Inequality; Cauchy-Schwarz; Schwarz inequality; Cauchy-schwarz; Schwarz's inequality; Schwartz Inequality; Cauchy-Schwartz; Buniakowsky inequality; Cauchy-Bunyakowski-Schwarz inequality; Bunyakovskii inequality; Cauchy Schwarz Inequality; Cauchy-Schwarz inequality; Bunyakovsky inequality; Cauchy–Bunyakovski–Schwarz inequality; Cauchy-schwartz inequality; Cauchy-Bunyakovski-Schwarz inequality; Cauchy–Schwartz inequality; Cauchy Schwarz; Cauchy–Schwarz; Cauchy-Schwartz-Buniakowsky Inequality; Cauchy–Schwarz–Bunyakovsky inequality; CBS inequality; Cauchy-Schwarz-Bunyakovsky inequality; Cauchyschwartz Inequality; Cauchy schwarz; Cauchy schwartz; Cauchy–Bunyakovsky–Schwarz inequality; Cauchy-Bunyakovsky inequality; Cauchy-Bunyakovsky-Schwarz inequality; Covariance inequality

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.

https://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality

Cauchy distribution

PROBABILITY DISTRIBUTION

Lorentz distribution; Lorentzian function; Cauchy-Lorentz distribution; Lorentzian lineshape; Lorentzian Lineshape; Cauchy Distribution; Lorentzian distribution; Cauchy Random Variable; Cauchy noise; Lorentzian profile; Lorentz profile; Lorentzian Function; Cauchy–Lorentz distribution; Multivariate Cauchy distribution; Lorentz function; Lorenz distribution; Cauchy random variable

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution.

https://en.wikipedia.org/wiki/Cauchy_distribution

Cauchy–Hadamard theorem

THEOREM

Cauchy-Hadamard Theorem; Cauchy-hadamard theorem; Cauchy-Hadamard theorem

In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy,.

https://en.wikipedia.org/wiki/Cauchy–Hadamard_theorem

Cauchy–Riemann equations

SYSTEM OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS CHARACTERIZING HOLOMORPHIC (COMPLEX DIFFERENTIABLE) FUNCTIONS

Cauchy Riemann equations; Cauchy-Riemann equation; Cauchy-Riemann; Cauchy-riemann equations; Cauchy-Riemann Equations; Cauchy riemann equations; Cauchy-Riemann conditions; Cauchy-Riemann Equation; Cauchy reimann equations; Cauchy-Riemann system; Cauchy riemann; Cauchy-Riemann equations; Cauchy–Riemann differential equations; Cauchy–Riemann conditions; Cauchy–Riemann equation; Cauchy-Riemann differential equations; Cauchy-Riemann Relation; Cauchy–Riemann; Cauchy-Riemann operator; Cauchy–Riemann Equations; Cauchy–Riemann operator; Inhomogeneous Cauchy–Riemann equations; Inhomogeneous Cauchy-Riemann equations; Riemann-cauchy equations

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be holomorphic (complex differentiable). This system of equations first appeared in the work of Jean le Rond d'Alembert.

https://en.wikipedia.org/wiki/Cauchy–Riemann_equations

Séquences

CANADIAN MAGAZINE

Séquences is a French-language film magazine originally published in Montreal, Quebec by the Commission des ciné-clubs du Centre catholique du cinéma de Montréal, a Roman Catholic film society. It is the third oldest French film magazine in publication after Les Cahiers du cinéma and Positif.

https://en.wikipedia.org/wiki/Séquences

Википедия

Cauchy sequence

In mathematics, a Cauchy sequence (French pronunciation: [koʃi]; English: KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

It is not sufficient for each term to become arbitrarily close to the preceding term. For instance, in the sequence of square roots of natural numbers:

the consecutive terms become arbitrarily close to each other: However, with growing values of the index n, the terms

a_{n}

become arbitrarily large. So, for any index n and distance d, there exists an index m big enough such that

a_{m}-a_{n}>d.

As a result, no matter how far one goes, the remaining terms of the sequence never get close to each other; hence the sequence is not Cauchy.

The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination.

Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets.

https://en.wikipedia.org/wiki/Cauchy sequence