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Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules
, 2002
"... In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the a ..."
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Cited by 597 (42 self)
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In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its \neighbors." In their paper, Vicsek et. al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models.
The Generalized Spectral Radius and Extremal Norms
, 2000
"... The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm alway ..."
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Cited by 26 (4 self)
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The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm always exists. This approach lends itself easily to the analysis of further properties of the generalized spectral radius. We prove that the generalized spectral radius is locally Lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius. Sufficient conditions for the existence of extremal norms are obtained.
Approximating the spectral radius of sets of matrices in the maxalgebra is NPhard
 THE IEEE TRANS. ON AUTOMATIC CONTROL
, 1999
"... The lower and average spectral radii measure the minimal and average growth rates, respectively, of long products of matrices taken from a finite set. The logarithm of the average spectral radius is traditionally called the Lyapunov exponent. When one forms these products in the maxalgebra, we obtai ..."
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Cited by 11 (2 self)
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The lower and average spectral radii measure the minimal and average growth rates, respectively, of long products of matrices taken from a finite set. The logarithm of the average spectral radius is traditionally called the Lyapunov exponent. When one forms these products in the maxalgebra, we obtain quantities that measure the performance of Discrete Event Systems. We show that approximating the lower and average maxalgebraic spectral radii is NPhard.
Subdivision Schemes and Refinement Equations with Nonnegative Masks
, 2000
"... We consider the twoscale renement equation f(x) = P N n=0 c n f(2x n) with P n c 2n = P n c 2n+1 = 1 where c 0 ; c N 6= 0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all c n 0. It has long been conjectured that ..."
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Cited by 6 (0 self)
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We consider the twoscale renement equation f(x) = P N n=0 c n f(2x n) with P n c 2n = P n c 2n+1 = 1 where c 0 ; c N 6= 0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all c n 0. It has long been conjectured that under such an assumption the subdivision algorithm converge, as well as the cascade algorithm converge uniformly to a continuous function, if and only if only if 0 < c 0 ; c N < 1 and the greatest common divisor of S = fn : c n > 0g is 1. We prove the conjecture for a large class of renement equations. Keywords: Nonnegative mask, cascade algorithm, subdivision scheme, renement equation, renable function. 1 Introduction The twoscale renement equation f(x) = X n2Z c n f(2x n); X n c 2n = X n c 2n+1 = 1 (1.1) plays a central role in the construction of orthonormal wavelet bases and in the subdivision scheme for curve and surface generations. An important question is the cont...
A geometric approach to ergodic nonhomogeneous Markov chains
 in: T.X. He (Ed.), Proc. Wavelet Analysis and Multiresolution Methods
, 2000
"... Inspired by the work of Daubechies and Lagarias on a set of matrices with convergent infinite products, we study the geometric approach to the classical problem of (weakly) ergodic nonhomogeneous Markov chains. The existing key inequalities (related to the Hajnal inequality) in the literature are u ..."
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Cited by 5 (3 self)
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Inspired by the work of Daubechies and Lagarias on a set of matrices with convergent infinite products, we study the geometric approach to the classical problem of (weakly) ergodic nonhomogeneous Markov chains. The existing key inequalities (related to the Hajnal inequality) in the literature are unified in this geometric picture. A more general inequality is established. Important quantities introduced by various authors are easily interpreted. A quantitative connection is established between the classical work of Hajnal and the more recent one of Daubechies and Lagarias. The results here may also be useful for studying nonnegative masks in wavelet theory.