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183
A positive systems model of TCPlike congestion control: Asymptotic results
 IEEE/ACM Transactions on Networking
, 2004
"... In this paper we study communication networks that employ droptail queueing and AdditiveIncrease MultiplicativeDecrease (AIMD) congestion control algorithms. We show that the theory of nonnegative matrices may be employed to model such networks. In particular, we show that important network p ..."
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Cited by 64 (10 self)
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In this paper we study communication networks that employ droptail queueing and AdditiveIncrease MultiplicativeDecrease (AIMD) congestion control algorithms. We show that the theory of nonnegative matrices may be employed to model such networks. In particular, we show that important network properties such as: (i) fairness; (ii) rate of convergence; and (iii) throughput; can be characterised by certain nonnegative matrices that arise in the study of AIMD networks. We demonstrate that these results can be used to develop tools for analysing the behaviour of AIMD communication networks. The accuracy of the models is demonstrated by means of several NSstudies.
On mixing properties of compact group extensions of hyperbolic systems
 Israel J. Math
"... Abstract. We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is alw ..."
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Cited by 46 (7 self)
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Abstract. We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is always the case for ergodic semisimple extensions as well as for stably ergodic extensions of Anosov diffeomorphisms of infranilmanifolds. 1.
Affine linear sieve, expanders, and sumproduct
"... This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and ..."
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Cited by 42 (8 self)
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This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and
Performance Evaluation of (max,+) Automata
 IEEE Trans. on Automatic Control
, 1993
"... Automata with multiplicities over the (max,+) semiring can be used to represent the behavior of timed discrete event systems. This formalism which extends both conventional automata and (max,+) linear representations covers a class of systems with synchronization phenomena and variable schedules. Pe ..."
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Cited by 40 (8 self)
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Automata with multiplicities over the (max,+) semiring can be used to represent the behavior of timed discrete event systems. This formalism which extends both conventional automata and (max,+) linear representations covers a class of systems with synchronization phenomena and variable schedules. Performance evaluation is considered in the worst, mean, and optimal cases. A simple algebraic reduction is provided for the worst case. The last two cases are solved for the subclass of deterministic series (recognized by deterministic automata). Deterministic series frequently arise due to the finiteness properties of (max,+) linear projective semigroups. The mean performance is given by the Kolmogorov equation of a Markov chain. The optimal performance is given by a HamiltonJacobiBellman equation. KeywordsDiscrete Event Systems, (max,+) algebra, Automata, Rational Series, Performance Evaluation I. INTRODUCTION A UTOMATA with multiplicities [10] over the (max,+) or the dual (min,+) s...
Evolutionary Formalism for Products of Positive Random Matrices
 Ann. Appl. Probab
, 1994
"... We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter we establish a random version of the PerronFrobenius theory which extends al ..."
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Cited by 36 (6 self)
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We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter we establish a random version of the PerronFrobenius theory which extends all known results and enables us to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. We develop a thermodynamic formalism for random dynamical systems, and in this framework prove that the top Lyapunov exponent is an analytic function of the generator of the cocycle. On this basis a fluctuation theory for products of positive random matrices can be developed which leads to an inequality in dynamical entropy that can be interpreted as a directionality principle for the mutation and selection process in evolutionary dynamics. Key words: evolutionary theory, random dynamical system, products of random matrices, PerronFrobenius theo...
On dynamics of mostly contracting diffeomorphisms
 Comm. Math. Phys
"... Abstract. Mostly contracting dieomorphisms are the simplest examples of robustly nonuniformly hyperbolic systems. This paper studies the mixing properties of mostly contracting dieomorphisms. 1. Introduction. This paper treats a class of partially hyperbolic systems with nonzero Lyapunov exponent ..."
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Cited by 35 (6 self)
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Abstract. Mostly contracting dieomorphisms are the simplest examples of robustly nonuniformly hyperbolic systems. This paper studies the mixing properties of mostly contracting dieomorphisms. 1. Introduction. This paper treats a class of partially hyperbolic systems with nonzero Lyapunov exponents. Before stating our result let us recall some recent work motivating our research. In recent years there were several advances in understanding
Localization for onedimensional, continuum, BernoulliAnderson models
 Duke Math. J
"... We use scattering theoretic methods to prove strong dynamical and exponential localization for onedimensional, continuum, Andersontype models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two ..."
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Cited by 33 (15 self)
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We use scattering theoretic methods to prove strong dynamical and exponential localization for onedimensional, continuum, Andersontype models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported singlesite perturbations of a periodic background which we use to verify the necessary hypotheses of multiscale analysis. We show that nonreflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs. 1.
Recent Results About Stable Ergodicity
 In Smooth ergodic theory and its applications
, 2000
"... this paper, has been directed toward extending their results beyond Axiom A. ..."
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Cited by 31 (8 self)
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this paper, has been directed toward extending their results beyond Axiom A.
Delocalization in Random Polymer Models
 Commun. Math. Phys
"... A random polymer model is a onedimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer ..."
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Cited by 31 (7 self)
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A random polymer model is a onedimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have purepoint spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.
The pressure function for products of nonnegative matrices
 Math. Research Letter
"... Abstract. Let (ΣA,σ) be a subshift of finite type and let M(x) be a continuous function on ΣA takingvalues in the set of nonnegative matrices. We extend the classical scalar pressure function to this new settingand prove the existence of the Gibbs measure and the differentiability of the pressure f ..."
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Cited by 27 (15 self)
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Abstract. Let (ΣA,σ) be a subshift of finite type and let M(x) be a continuous function on ΣA takingvalues in the set of nonnegative matrices. We extend the classical scalar pressure function to this new settingand prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where M(x) takes finite values M1, ·· ·,Mm. The pressure function reduces to P (q): = limn→ ∞ 1 log n J∈Σ ‖MJ ‖ A,n q. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps. 1.