860 research outputs found
The asymptotic spectrum of graphs and the Shannon capacity
We introduce the asymptotic spectrum of graphs and apply the theory of
asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new
dual characterisation of the Shannon capacity of graphs. Elements in the
asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional
clique cover number, the complement of the fractional orthogonal rank and the
fractional Haemers bounds
Network Coding in a Multicast Switch
We consider the problem of serving multicast flows in a crossbar switch. We
show that linear network coding across packets of a flow can sustain traffic
patterns that cannot be served if network coding were not allowed. Thus,
network coding leads to a larger rate region in a multicast crossbar switch. We
demonstrate a traffic pattern which requires a switch speedup if coding is not
allowed, whereas, with coding the speedup requirement is eliminated completely.
In addition to throughput benefits, coding simplifies the characterization of
the rate region. We give a graph-theoretic characterization of the rate region
with fanout splitting and intra-flow coding, in terms of the stable set
polytope of the 'enhanced conflict graph' of the traffic pattern. Such a
formulation is not known in the case of fanout splitting without coding. We
show that computing the offline schedule (i.e. using prior knowledge of the
flow arrival rates) can be reduced to certain graph coloring problems. Finally,
we propose online algorithms (i.e. using only the current queue occupancy
information) for multicast scheduling based on our graph-theoretic formulation.
In particular, we show that a maximum weighted stable set algorithm stabilizes
the queues for all rates within the rate region.Comment: 9 pages, submitted to IEEE INFOCOM 200
Realizations of uncertain systems and formal power series
Rational functions of several noncommuting indeterminates arise naturally in robust control when studying systems with structured uncertainty. Linear fractional transformations (LFTs) provide a convenient way of obtaining realizations of such systems and a complete realization theory of LFTs is emerging. This paper establishes connections between a minimal LFT realization and minimal realizations of a formal power series, which have been studied extensively in a variety of disciplines. The result is a fairly complete generalization of standard minimal realization theory for linear systems to the formal power series and LFT setting
A new property of the Lov\'asz number and duality relations between graph parameters
We show that for any graph , by considering "activation" through the
strong product with another graph , the relation between the independence number and the Lov\'{a}sz number of
can be made arbitrarily tight: Precisely, the inequality
becomes asymptotically an equality for a suitable sequence of ancillary graphs
.
This motivates us to look for other products of graph parameters of and
on the right hand side of the above relation. For instance, a result of
Rosenfeld and Hales states that with the fractional
packing number , and for every there exists that makes the
above an equality; conversely, for every graph there is a that attains
equality.
These findings constitute some sort of duality of graph parameters, mediated
through the independence number, under which and are dual
to each other, and the Lov\'{a}sz number is self-dual. We also show
duality of Schrijver's and Szegedy's variants and
of the Lov\'{a}sz number, and explore analogous notions for the chromatic
number under strong and disjunctive graph products.Comment: 16 pages, submitted to Discrete Applied Mathematics for a special
issue in memory of Levon Khachatrian; v2 has a full proof of the duality
between theta+ and theta- and a new author, some new references, and we
corrected several small errors and typo
Computability Theory
Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work
On the Reduction of Singularly-Perturbed Linear Differential Systems
In this article, we recover singularly-perturbed linear differential systems
from their turning points and reduce the rank of the singularity in the
parameter to its minimal integer value. Our treatment is Moser-based; that is
to say it is based on the reduction criterion introduced for linear singular
differential systems by Moser. Such algorithms have proved their utility in the
symbolic resolution of the systems of linear functional equations, giving rise
to the package ISOLDE, as well as in the perturbed algebraic eigenvalue
problem. Our algorithm, implemented in the computer algebra system Maple, paves
the way for efficient symbolic resolution of singularly-perturbed linear
differential systems as well as further applications of Moser-based reduction
over bivariate (differential) fields.Comment: Keywords: Moser-based Reduction, Perturbed linear Differential
systems, turning points, Computer algebr
Model reduction of behavioural systems
We consider model reduction of uncertain behavioural models. Machinery for gap-metric model reduction and multidimensional model reduction using linear matrix inequalities is extended to these behavioural models. The goal is a systematic method for reducing the complexity of uncertain components in hierarchically developed models which approximates the behavior of the full-order system. This paper focuses on component model reduction that preserves stability under interconnection
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