For general domains \ Omega in \ mathbf { R } ^ n the inequality can be stated as follows : If \ omega is a bounded domain with \ bar { \ omega } \ subset \ Omega, then there is a constant C such that
22.
The Lie algebra of a connected Lie group acting transitively on a homogeneous bounded domain is a " j "-algebra, which is not surprising as " j "-algebras are defined to have the obvious properties of such a Lie algebra.
23.
Whereas the genome is a bounded domain of only ~ 3, 000, 000, 000 nucleotides and ~ 25, 000 genes ( in man ), there is essentially no limit to the number of G2P relationships that can be examined, each by multiple different procedures.
24.
The second proof is based on the method of, where a sharpening of the maximum modulus inequality was established for bounded holomorphic functions " h " defined on a bounded domain " V " : if " a " lies in " V ", then
25.
Is it ok ? ( PS : in our link, Rouch?s theorem is quoted with a bounded domain K with a smooth boundary, but in fact no smoothness is needed, and a rectangle works perfectly ) .-- a 20 : 15, 3 April 2010 ( UTC)
26.
One of the earliest results of this kind was due to Hermann Weyl who used David Hilbert's theory of integral equation in 1911 to show that the volume of a bounded domain in Euclidean space can be determined from the asymptotic behavior of the eigenvalues for the isometry class of a manifold ( see isospectral ).
27.
Gaetano quickly answered : " Non solo ti dar?la condizione sufficiente, ma ti dar?anche la condizione necessaria e pure per insiemi non limitati ( I can give you not only a sufficient condition, but also a necessary condition, and not only for bounded domains, but also for unbounded domains ) " .
28.
Furthermore, for a smooth, bounded domain \ Omega, since the Rayleigh quotient for the Laplace operator in the space W ^ { 1, 2 } _ 0 ( \ Omega ) is minimized by the eigenfunction corresponding to the minimal eigenvalue ? 1 of the ( negative ) Laplacian, it is a simple consequence that, for any u \ in W ^ { 1, 2 } _ 0 ( \ Omega ),
29.
Let \ Omega be an open bounded domain of \ mathbb { R } ^ d, with d \ geq 2, which is subject to a nonsmooth perturbation confined in a small region \ omega _ \ varepsilon ( \ tilde { x } ) = \ tilde { x } + \ varepsilon \ omega of size \ varepsilon with \ tilde { x } an arbitrary point of \ Omega and \ omega a fixed domain of \ mathbb { R } ^ d.
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