14,061 research outputs found
Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space
Consider a linear impulsive equation in a Banach space
with . Suppose each solution of
the corresponding semi-homogeneous equation
(2) is bounded for any bounded sequence .
The conditions are determined ensuring
(a) the solution of the corresponding homogeneous equation has an exponential
estimate;
(b) each solution of (1),(2) is bounded on the half-line for any bounded
and bounded sequence ;
(c) for any tending to
zero;
(d) exponential estimate of implies a similar estimate for .Comment: 19 pages, LaTex-fil
Homogenization of linear transport equations. A new approach
The paper is devoted to a new approach of the homogenization of linear
transport equations induced by a uniformly bounded sequence of vector fields
, the solutions of which agree at with a
bounded sequence of for some .
Assuming that the sequence is compact in
( conjugate of ) for some gradient field
bounded in , and that
there exists a uniformly bounded sequence such that
is divergence free if or is a cross
product of bounded gradients in if
, we prove that the sequence
converges weakly to a solution to a linear transport equation. It turns out
that the compactness of is a substitute to
the ergodic assumption of the classical two-dimensional periodic case, and
allows us to deal with non-periodic vector fields in any dimension. The
homogenization result is illustrated by various and general examples
The eigenvalues of limits of radial Toeplitz operators
Let be the Bergman space on the unit disk. A bounded operator on
is called radial if for all , where
is a bounded sequence of complex numbers. We characterize the
eigenvalues of radial operators that can be approximated by Toeplitz operators
with bounded symbols.Comment: 14 page
The cohesive principle and the Bolzano-Weierstra{\ss} principle
The aim of this paper is to determine the logical and computational strength
of instances of the Bolzano-Weierstra{\ss} principle (BW) and a weak variant of
it.
We show that BW is instance-wise equivalent to the weak K\"onig's lemma for
-trees (-WKL). This means that from every bounded
sequence of reals one can compute an infinite -0/1-tree, such that
each infinite branch of it yields an accumulation point and vice versa.
Especially, this shows that the degrees d >> 0' are exactly those containing an
accumulation point for all bounded computable sequences.
Let BW_weak be the principle stating that every bounded sequence of real
numbers contains a Cauchy subsequence (a sequence converging but not
necessarily fast). We show that BW_weak is instance-wise equivalent to the
(strong) cohesive principle (StCOH) and - using this - obtain a classification
of the computational and logical strength of BW_weak. Especially we show that
BW_weak does not solve the halting problem and does not lead to more than
primitive recursive growth. Therefore it is strictly weaker than BW. We also
discuss possible uses of BW_weak.Comment: corrected typos, slightly improved presentatio
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