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...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.

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

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

...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

''Cauchy–Fantappié formula'' which generalizes the Cauchy integral formula (see [[Cauchy integral|Cauchy integral]]).

...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

...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

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

...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,

$#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

...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

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

...{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

.../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]]

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,

...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

...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.

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]]].

one has a corresponding Cauchy problem yields a strong solution of the Cauchy problem (*) for $ x \in D ( A) $,

...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 $

...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

...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

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.

be a solution of the [[Cauchy problem|Cauchy problem]] $ \dot{x} = f ( x , t ) $, the Cauchy problem $ \dot{y} = g ( y , t ) $,

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 $

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

...o all of $\textbf{R}^+=[0,\infty)$ as the solution of the [[Cauchy problem|Cauchy problem]] ==Exponential and product formulas.==

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

...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

...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) $.

...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

is called the indefinite inner product of the Krein space $ {\mathcal K} $. a Hilbert inner product $ ( \cdot , \cdot ) $

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

...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:

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

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 $

...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

...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.==

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.

...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

$\def\f#1{\mathfrak{#1}}\f A$ of subsets of the product $X\times X$. A uniform space $X$ is called complete if every Cauchy filter in $X$

...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

...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

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

from the direct product $ G \times G $ is the direct product of $ n $

...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

...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.

...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

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

...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]].

...$v _ { \infty } ( f ) = - \operatorname { log } | f |$, 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 $\mathbf{Q}$.

obtained by taking the product of $ n $ is Cauchy in $ {\mathsf P} $-

...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

...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

...(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

If one solves the Cauchy problem for it on $ 0 \leq \lambda \leq 1 $ with the new scalar product

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.'

...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).

...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 $

...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.

...ram that came to be known as the [[Arithmetization of analysis]], Bolzano, Cauchy, Weierstrass, Dedekind, Cantor, and others succeeded in “reducing” anal * ''Cartesian product'' of $A$ and $B$, denoted $A \times B$, is the set whose members are all po