173,264 research outputs found
Singular value decay of operator-valued differential Lyapunov and Riccati equations
We consider operator-valued differential Lyapunov and Riccati equations,
where the operators and may be relatively unbounded with respect to
(in the standard notation). In this setting, we prove that the singular values
of the solutions decay fast under certain conditions. In fact, the decay is
exponential in the negative square root if generates an analytic semigroup
and the range of has finite dimension. This extends previous similar
results for algebraic equations to the differential case. When the initial
condition is zero, we also show that the singular values converge to zero as
time goes to zero, with a certain rate that depends on the degree of
unboundedness of . A fast decay of the singular values corresponds to a low
numerical rank, which is a critical feature in large-scale applications. The
results reported here provide a theoretical foundation for the observation
that, in practice, a low-rank factorization usually exists.Comment: Corrected some misconceptions, which lead to more general results
(e.g. exponential stability is no longer required). Also fixed some
off-by-one errors, improved the presentation, and added/extended several
remarks on possible generalizations. Now 22 pages, 8 figure
Remarks on Entropy Formulae for Linear Heat Equation
In this note, we prove some new entropy formula for linear heat equation on
static Riemannian manifold with nonnegative Ricci curvature. The results are
analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type
equations.Comment: Refined and revised versio
Examples illustrating some aspects of the weak Deligne-Simpson pro blem
We consider the variety of -tuples of matrices (resp. )
from given conjugacy classes (resp. ) such that (resp. ). This
variety is connected with the weak {\em Deligne-Simpson problem: give necessary
and sufficient conditions on the choice of the conjugacy classes (resp. ) so that there exist
-tuples with trivial centralizers of matrices (resp.
) whose sum equals 0 (resp. whose product equals ).} The
matrices (resp. ) are interpreted as matrices-residua of Fuchsian
linear systems (resp. as monodromy operators of regular linear systems) on
Riemann's sphere. We consider examples of such varieties of dimension higher
than the expected one due to the presence of -tuples with non-trivial
centralizers; in one of the examples the difference between the two dimensions
is O(n).Comment: Research partially supported by INTAS grant 97-164
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