...tions were laid by S.L. Sobolev [[#References|[2]]] in 1936 by solving the
Cauchy problem for hyperbolic equations, while in the 1950-s L. Schwartz (see [[#R
...encyclopediaofmath.org/legacyimages/g/g043/g043810/g043810176.png" />. The
product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
74 KB (9,823 words) - 19:33, 9 November 2014
...ry]]). In the case of one complex variable, the familiar [[Cauchy integral|Cauchy integral]] formula plays a dominant and unique role in the theory of functi
..._ { j } | < r _ { j } , j = 1 , \dots , n \}$ in $\mathbf{C} ^ { n }$ (the product of $n$ discs), one obtains
15 KB (2,167 words) - 16:10, 11 February 2024
A generalization of the concept of a [[Cauchy integral]] to a certain class of discontinuous functions; introduced by B.
...ntegrability over $[a,b]$ of both functions $f$ and $g$ implies that their product $fg$ is integrable over this interval.
6 KB (964 words) - 08:25, 25 April 2016
be a solution of the [[Cauchy problem|Cauchy problem]] $ \dot{x} = f ( x , t ) $,
the Cauchy problem $ \dot{y} = g ( y , t ) $,
15 KB (2,177 words) - 16:07, 5 February 2022
This is the Cauchy formula, a generalization of which to the case of an arbitrary (non-integer
if it can be represented as the product of an operator $ q $
12 KB (1,635 words) - 14:54, 7 June 2020
Consider the [[Cauchy problem|Cauchy problem]] for the [[Wave equation|wave equation]]
which is considered in the product space $V \times L ^ { 2 } ( \Omega )$. Since the equation is of first order
10 KB (1,449 words) - 17:45, 1 July 2020
...o all of $\textbf{R}^+=[0,\infty)$ as the solution of the [[Cauchy problem|Cauchy problem]]
==Exponential and product formulas.==
24 KB (3,989 words) - 20:19, 11 January 2021
where $\langle \, .\, ,\, . \, \rangle$ is the inner product,
...[#References|[a1]]]. The functional representation of the solution for the Cauchy problem depends on symmetry properties of the Hamiltonian and on initial di
10 KB (1,427 words) - 07:38, 7 February 2024
...b120/b120230/b12023034.png" />, whose horizontal composition is the tensor
product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
...24]]]. The generalization of
Cauchy completion (cf. also [[
Cauchy sequence|
Cauchy sequence]]) from the case of metric spaces is fundamental [[#References|[a2
24 KB (3,338 words) - 17:29, 7 February 2011
...ld of smooth functions which satisfy (2) in the norm induced by the scalar product
where the round brackets denote the scalar product in $ H ( V) $.
21 KB (3,008 words) - 17:33, 5 June 2020
...the power series, whereas the power series of the product is the [[Cauchy product]] of the power series. More complicated formulas hold for the quotient and
6 KB (1,048 words) - 21:19, 14 January 2021
is called the indefinite inner product of the Krein space $ {\mathcal K} $.
a Hilbert inner product $ ( \cdot , \cdot ) $
32 KB (4,628 words) - 10:55, 20 January 2024
partial sums, then in this sense the product of the two given series will converge to the sum $ C = AB $.
...a result of the transformation defining the summation method. For example, Cauchy's theorem establishes that $ ( s _ {0} + \dots + s _ {n} )/( n+ 1) \right
10 KB (1,530 words) - 08:24, 6 June 2020
...ows from this. The determinant of a triangular matrix is also equal to the product of its diagonal entries. For a matrix
...{ij})$ be an $(n\times m)$-matrix over $R$, and let $C=AB$. Then the Binet–Cauchy formula holds:
11 KB (1,876 words) - 20:27, 30 November 2016
semi-module is the direct sum (product) $ A ^ {n} = \{ {( a _ {1} \dots a _ {n} ) } : {a _ {j} \in A } \} $.
...ional analysis]] to idempotent analysis. For example, an idempotent scalar product can be defined as
18 KB (2,598 words) - 22:11, 5 June 2020
are thought of as having the product topologies). Entirely analogously, one can define topological left and righ
is complete if every [[Cauchy filter|Cauchy filter]] in $ E $
41 KB (6,085 words) - 08:26, 6 June 2020
...y $r \geq 0$, let $\otimes ^ { r } \mathcal{E}$ denote the $r$-fold tensor product $\cal E \otimes \ldots \otimes E$ over $C ^ { \infty } ( M )$. In particula
...M ) [ s , t ]$ into a product of two linear homogeneous factors leads to a product $\theta \otimes \varphi \in \otimes ^ { 2 } \mathcal{E}$ of linearly indepe
40 KB (5,895 words) - 17:45, 1 July 2020
...2$ and $\partial K$ is a closed simple curve was already considered by A. Cauchy in 1837 (the [[Winding number|winding number]]). After several interesting
==Product theorem.==
12 KB (1,815 words) - 17:42, 1 July 2020
called the product of the series (2) and the number $ \lambda $,
...a series which does not use the notion of its sum is the [[Cauchy criteria|Cauchy criterion]] for the convergence of a series.
29 KB (4,393 words) - 19:21, 27 January 2020
...or concerns operators acting on a Banach space with a so-called semi-inner product. Another generalization concerns operators acting on a [[Hilbert space with
...e><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.O. Fattorini, "The Cauchy problem" , Addison-Wesley (1983) pp. 120–125; 154–159</TD></TR><TR><T
7 KB (1,051 words) - 20:01, 27 February 2021