...ed. The product of complete uniform spaces is complete; conversely, if the product of non-empty uniform spaces is complete, then all the spaces are complete.
1,014 bytes (147 words) - 17:31, 9 December 2013
A formula aimed at expressing the determinant of the product of two matrices $A\in\mathrm{M}_{m,n}(\mathbb{R})$
and $p\leq\min\{m,q\}$, then any minor of order $p$ of the product matrix $AB$ can be expressed
4 KB (615 words) - 16:20, 24 November 2012
The condensed formulation of a [[Cauchy problem|Cauchy problem]] (as phrased by J. Hadamard) in an infinite-dimensional [[Topologi
Narrowly, but loosely speaking, the abstract Cauchy problem consists in solving a linear abstract differential equation (cf. al
5 KB (689 words) - 07:45, 27 January 2024
...y and sufficient condition for the absolute convergence of a series is the Cauchy's criterion (cp. with Theorem 3.22 of {{Cite|Ru}}): for each $\varepsilon >
series. [[Cauchy products]] of absolutely convergent series are
5 KB (821 words) - 09:35, 16 August 2013
''Cauchy–Fantappié formula''
which generalizes the Cauchy integral formula (see [[Cauchy integral|Cauchy integral]]).
8 KB (1,209 words) - 10:51, 20 January 2024
...of different permutations of a set $X$ with $|X|=n$ is equal to $n!$. The product of the permutations $\def\a{\alpha}\a$ and $\def\b{\beta}\b$ of a set $X$ i
...permutations, and also of two odd ones, is an even permutation, while the product of an even and an odd permutation (in either order) is odd. The even permut
7 KB (1,262 words) - 20:15, 27 September 2016
...w.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565057.png" /> and
product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
Cauchy's intermediate value theorem: A function that is continuous on a closed int
26 KB (3,622 words) - 17:12, 7 February 2011
A metric space is called complete if each [[Cauchy sequence]] in it converges. In the same sense one understands the completen
...has a countable base and is metrizable. Paracompactness is retained in the product operation when the spaces are Čech complete. Čech completeness is also pr
5 KB (764 words) - 17:23, 9 December 2013
...is property can be regarded as a generalization of the [[Cauchy inequality|Cauchy inequality]]:
...inequality is that the volume of the parallelotope is not larger than the product of the volumes of complementary faces. In particular,
5 KB (733 words) - 19:42, 5 June 2020
$#C+1 = 247 : ~/encyclopedia/old_files/data/C020/C.0200890 Cauchy integral
A Cauchy integral is a definite integral of a continuous function of one real variab
26 KB (3,804 words) - 08:11, 13 February 2022
...e [[Uniform distribution|uniform distribution]], the [[Cauchy distribution|Cauchy distribution]], the [[Student distribution|Student distribution]], and the
...al with mode at zero if and only if it is the distribution function of the product of two independent random variables one of which has a [[Uniform distributi
5 KB (663 words) - 07:36, 10 April 2023
there exists a unique number, known as their product and denoted by $ ab $,
...that the field of rational numbers only is no longer complete: It contains Cauchy sequences which do not converge to any rational number. The continuity (or
26 KB (4,086 words) - 09:51, 4 April 2020
...{T} )$ is the orthogonal projection given by the [[Cauchy integral theorem|Cauchy integral theorem]]. The [[C*-algebra|$C ^ { * }$-algebra]] ${\cal T} ({\bf
...)$. In this way the well-developed representation theory of (co-) crossed product $C ^ { * }$-algebras [[#References|[a4]]] can be applied to obtain Toeplitz
8 KB (1,186 words) - 16:46, 1 July 2020
.../legacyimages/h/h046/h046320/h04632029.png" /> by an integral of
Cauchy or
Cauchy–Stieltjes type belong, generally speaking, only to the classes <img align
.../h046/h046320/h04632068.png" /> of a canonical [[Blaschke
product|Blaschke
product]]
37 KB (5,073 words) - 18:20, 1 December 2014
the metric product of $ k $
...r, the main results there relate to convex polyhedra (see [[Cauchy theorem|Cauchy theorem]] on polyhedra), and to surfaces in Riemannian spaces, for example,
5 KB (791 words) - 08:11, 6 June 2020
...s allow of a sort of generalized metrization by means of écarts satisfying Cauchy's condition. There is also a convenient characterization of spaces with dev
...a Moore space), is solved now. In 1978 P. Nyikos showed that, assuming the product measure extension axiom (PMEA), every normal Moore space is metrizable. To
6 KB (1,004 words) - 19:52, 3 February 2021
...t A } x = S ( t ) x$ is the unique strong solution to the [[Cauchy problem|Cauchy problem]] $y ^ { \prime } ( t ) = - A y ( t )$, $y ( 0 ) = x$. If $A$ is un
...t A } x = S ( t ) x$ is said to be a mild (or generalized) solution to the Cauchy problem above.
8 KB (1,236 words) - 17:03, 1 July 2020
It follows from a property of the [[product topology]] that every [[continuous function]] $f:X\times Y\to Z$ between [[
...hen the set $C(f)$ is the complement of an $F_\sigma$-set contained in the product of two sets of the first Baire category [[#References|[a8]]].
6 KB (979 words) - 11:10, 21 December 2020
one has a corresponding Cauchy problem
yields a strong solution of the Cauchy problem (*) for $ x \in D ( A) $,
15 KB (2,235 words) - 08:13, 6 June 2020
...s which was systematically studied was completeness: attempts to introduce Cauchy filters or fundamental sequences in terms of compactifications were unsucce
Completeness defined using Cauchy filters — filters $ \Phi $
20 KB (2,990 words) - 16:25, 15 October 2023