23,253 research outputs found
Interpolation sets in spaces of continuous metric-valued functions
Let and be a topological space and metric space, respectively. If
denotes the set of all continuous functions from X to M, we say that a
subset of is an \emph{-interpolation set} if given any function
with relatively compact range in , there exists a map such that . In this paper, motivated by a result of Bourgain
in \cite{Bourgain1977}, we introduce a property, stronger than the mere
\emph{non equicontinuity} of a family of continuous functions, that isolates a
crucial fact for the existence of interpolation sets in fairly general
settings. As a consequence, we establish the existence of sets in every
nonprecompact subset of a abelian locally -groups. This implies
that abelian locally -groups strongly respects compactness
A posteriori error estimates for the Electric Field Integral Equation on polyhedra
We present a residual-based a posteriori error estimate for the Electric
Field Integral Equation (EFIE) on a bounded polyhedron. The EFIE is a
variational equation formulated in a negative order Sobolev space on the
surface of the polyhedron. We express the estimate in terms of
square-integrable and thus computable quantities and derive global lower and
upper bounds (up to oscillation terms).Comment: Submitted to Mathematics of Computatio
Quantitative Approximation of the Probability Distribution of a Markov Process by Formal Abstractions
The goal of this work is to formally abstract a Markov process evolving in
discrete time over a general state space as a finite-state Markov chain, with
the objective of precisely approximating its state probability distribution in
time, which allows for its approximate, faster computation by that of the
Markov chain. The approach is based on formal abstractions and employs an
arbitrary finite partition of the state space of the Markov process, and the
computation of average transition probabilities between partition sets. The
abstraction technique is formal, in that it comes with guarantees on the
introduced approximation that depend on the diameters of the partitions: as
such, they can be tuned at will. Further in the case of Markov processes with
unbounded state spaces, a procedure for precisely truncating the state space
within a compact set is provided, together with an error bound that depends on
the asymptotic properties of the transition kernel of the original process. The
overall abstraction algorithm, which practically hinges on piecewise constant
approximations of the density functions of the Markov process, is extended to
higher-order function approximations: these can lead to improved error bounds
and associated lower computational requirements. The approach is practically
tested to compute probabilistic invariance of the Markov process under study,
and is compared to a known alternative approach from the literature.Comment: 29 pages, Journal of Logical Methods in Computer Scienc
A dichotomy property for locally compact groups
We extend to metrizable locally compact groups Rosenthal's theorem describing
those Banach spaces containing no copy of . For that purpose, we transfer
to general locally compact groups the notion of interpolation () set,
which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact
abelian groups. Thus we prove that for every sequence in a locally compact group , then either has a weak Cauchy subsequence or contains a subsequence
that is an set. This result is subsequently applied to obtain sufficient
conditions for the existence of Sidon sets in a locally compact group , an
old question that remains open since 1974 (see [32] and [20]). Finally, we show
that every locally compact group strongly respects compactness extending
thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this
property for abelian locally compact groups.Comment: To appear in J. of Functional Analysi
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