860 research outputs found

    The asymptotic spectrum of graphs and the Shannon capacity

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    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

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    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

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    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

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    We show that for any graph GG, by considering "activation" through the strong product with another graph HH, the relation α(G)≤ϑ(G)\alpha(G) \leq \vartheta(G) between the independence number and the Lov\'{a}sz number of GG can be made arbitrarily tight: Precisely, the inequality α(G×H)≤ϑ(G×H)=ϑ(G) ϑ(H) \alpha(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H) becomes asymptotically an equality for a suitable sequence of ancillary graphs HH. This motivates us to look for other products of graph parameters of GG and HH on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that α(G×H)≤α∗(G) α(H), \alpha(G \times H) \leq \alpha^*(G)\,\alpha(H), with the fractional packing number α∗(G)\alpha^*(G), and for every GG there exists HH that makes the above an equality; conversely, for every graph HH there is a GG that attains equality. These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which α\alpha and α∗\alpha^* are dual to each other, and the Lov\'{a}sz number ϑ\vartheta is self-dual. We also show duality of Schrijver's and Szegedy's variants ϑ−\vartheta^- and ϑ+\vartheta^+ 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

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    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

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    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

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    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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