$\def\f#1{\mathfrak{#1}}\f A$ of subsets of the product $X\times X$.
A uniform space $X$ is called complete if every Cauchy filter in $X$
16 KB (2,875 words) - 21:57, 12 October 2014
...ust given provides sufficient conditions for the unique solvability of the Cauchy problem (2), (3).
The Cauchy problem for equation (5) is to find the solution satisfying the initial con
33 KB (4,933 words) - 01:50, 23 January 2022
...c functions originated in the 19th century, mainly due to the work of A.L. Cauchy, B. Riemann and K. Weierstrass. The "transition to the complex domain" had
...e concept of analyticity. One definition, which was originally proposed by Cauchy, and was considerably advanced by Riemann, is based on a structural propert
61 KB (9,850 words) - 19:04, 20 January 2022
and a scalar product is defined on the bundles $ E $
...operators are the [[Mixed problem|mixed problem]] and the [[Cauchy problem|Cauchy problem]] with conditions at infinity. The class of hypo-elliptic linear di
25 KB (3,768 words) - 09:07, 14 June 2022
from the direct product $ G \times G $
is the direct product of $ n $
14 KB (2,197 words) - 16:40, 31 March 2020
...R$, and it can be endowed with a multiplication by induction of the tensor product. The characteristic or Frobenius mapping [[#References|[a1]]] $\operatornam
The Cauchy identity and its dual are
14 KB (2,001 words) - 10:09, 11 November 2023
...which the concept of a limit of a sequence historically arose first (see [[Cauchy criteria]]). For such sequences the following formulas hold:
...ces of points in linear topological spaces, the property of the limit of a product — to sequences of points in a topological group, etc.
32 KB (5,224 words) - 19:36, 25 March 2023
...ategory $\mathcal{V}$ and was shown to be exactly what arose when a tensor product (independent of specific axioms) was present on the one-object bicategory $
...ategories enriched in the monoidal category $2$-'''Cat''' where the tensor product is a pseudo-version of that defined in [[#References|[a20]]]. The coherence
18 KB (2,710 words) - 00:41, 15 February 2024
The combination of these assertions gives a convolution, dual to the product:
(Sato's fundamental theorem). Hence Cauchy data for solutions can be specified on a non-characteristic manifold. Holmg
13 KB (1,805 words) - 19:06, 9 January 2024
...sequence of elements, convergence of a series, convergence of an infinite product, convergence of a continued fraction, convergence of an integral, etc. The
...e in a complete metric space it is necessary and sufficient that it be a [[Cauchy sequence]].
22 KB (3,726 words) - 10:31, 2 September 2017
...$v _ { \infty } ( f ) = - \operatorname { log } | f |$, is similar to the Cauchy residue formula
...s good functoriality properties and is equipped with a graded intersection product, at least after tensoring it by $\mathbf{Q}$.
8 KB (1,219 words) - 21:00, 13 July 2020
obtained by taking the product of $ n $
is Cauchy in $ {\mathsf P} $-
11 KB (1,627 words) - 06:28, 26 March 2023
...onsidered to be the cord of the sector; its area is therefore equal to the product of the length of the cord and one-half of the radius; if these areas are su
...nt must be credited to J.L. Lagrange (1736–1813), and was finally fixed by Cauchy; the latter also gave a rigorous definition of an integral as a limit of su
22 KB (3,357 words) - 17:34, 1 January 2021
...ial of degree greater than zero and with real coefficients factorizes as a product of polynomials of degrees one and two with real coefficients" (Euler, J. d
...), and K. Weierstrass (1872). Here Cantor and Meray used [[Cauchy sequence|Cauchy sequences]] of rational numbers, Dedekind used cuts in the field of rationa
23 KB (3,482 words) - 08:03, 6 June 2020
...(areas, volumes, angles) were represented by the lengths of lines and the product of two such quantities was represented by a rectangle with sides representi
...the absolute value $|x|$ (K. Weierstrass, 1841), the vector $\vec{v}$ (A. Cauchy, 1853), the determinant
18 KB (2,697 words) - 13:11, 13 December 2013
If one solves the Cauchy problem for it on $ 0 \leq \lambda \leq 1 $
with the new scalar product
20 KB (2,830 words) - 19:25, 9 January 2024
As a substitute for the resolvent one can take a continuous operator whose product with $ A - \lambda I $,
<TR><TD valign="top">[8]</TD> <TD valign="top"> J. Chazarain, "Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes" ''J. Funct. Anal.'
34 KB (5,024 words) - 09:12, 21 January 2024
...ion of problems for ordinary differential equations. These studies applied Cauchy's method of contour integration to the resolvent.
...$ n $-fold completeness is, naturally, connected with the solution of the Cauchy problem for the non-stationary equation corresponding to (1).
35 KB (5,059 words) - 04:12, 9 May 2022
...work of J. Fourier, N.I. Lobachevskii, P. Dirichlet, B. Bolzano, and A.L. Cauchy, where the notion of a function as a correspondence between two sets of num
is called the product of the sets $ X $
34 KB (5,509 words) - 22:06, 28 January 2020
...he hydrodynamics of a viscous fluid. The uniqueness of the solution of the Cauchy problem for them has not been proved (proofs have only been found for two-d
...w is played here by the so-called potential vorticity, which is the scalar product of the vorticity of the absolute velocity and the gradient of the entropy.
21 KB (3,004 words) - 08:26, 6 June 2020