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95
Solution of LargeScale Lyapunov Equations via the Block Modified Smith Method
, 2006
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The kinetic limit of a system of coagulating Brownian particles. To appear in Arch. Rational Mech. Anal. Available at www.arxiv.org/math.PR/0408395. 29 Alan Hammond and Fraydoun Rezakhanlou. The kinetic limit of a system of coagulating planar Brownian par
"... Understanding the evolution in time of macroscopic quantities such as pressure or temperature is a central task in nonequilibrium statistical mechanics. We study this problem rigorously for a model of massbearing Brownian particles that are prone to coagulate when they are close, where the macrosc ..."
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Cited by 17 (7 self)
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Understanding the evolution in time of macroscopic quantities such as pressure or temperature is a central task in nonequilibrium statistical mechanics. We study this problem rigorously for a model of massbearing Brownian particles that are prone to coagulate when they are close, where the macroscopic quantity in this case is the density of particles of a given mass. Brownian motion
Convergence analysis of Krylov subspace iterations with methods from potential theory
 SIAM Review
"... Abstract. Krylov subspace iterations are among the bestknown and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on t ..."
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Cited by 16 (2 self)
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Abstract. Krylov subspace iterations are among the bestknown and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix. This leads to an extremal problem in polynomial approximation theory: how small can a monic polynomial of a given degree be on the spectrum? This survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices. It is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure. Then depending on the number of iterations, the Lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region, which is characterized by means of a constrained equilibrium problem from potential theory. The same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems. Key words. Krylov subspace iterations, Ritz values, eigenvalue distribution, equilibrium measure, contrained equilibrium, potential theory AMS subject classifications. 15A18, 31A05, 31A15, 65F15 1. Introduction. Krylov
MONGEAMPÈRE MEASURES ON PLURIPOLAR SETS
, 2008
"... In this article we solve the complex MongeAmpère equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Ko̷lodziej’s s ..."
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Cited by 13 (4 self)
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In this article we solve the complex MongeAmpère equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Ko̷lodziej’s subsolution theorem. More precisely, we prove that if a nonnegative Borel measure is dominated by a complex MongeAmpère measure, then it is a complex MongeAmpère measure.
EBRP: Energybalanced routing protocol for data gathering in wireless sensor networks
 IEEE Trans. Parallel Distrib. Syst
, 2011
"... Abstract—Energy is an extremely critical resource for batterypowered wireless sensor networks (WSN), thus making energyefficient protocol design a key challenging problem. Most of the existing energyefficient routing protocols always forward packets along the minimum energy path to the sink to me ..."
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Abstract—Energy is an extremely critical resource for batterypowered wireless sensor networks (WSN), thus making energyefficient protocol design a key challenging problem. Most of the existing energyefficient routing protocols always forward packets along the minimum energy path to the sink to merely minimize energy consumption, which causes an unbalanced distribution of residual energy among sensor nodes, and eventually results in a network partition. In this paper, with the help of the concept of potential in physics, we design an EnergyBalanced Routing Protocol (EBRP) by constructing a mixed virtual potential field in terms of depth, energy density, and residual energy. The goal of this basic approach is to force packets to move toward the sink through the dense energy area so as to protect the nodes with relatively low residual energy. To address the routing loop problem emerging in this basic algorithm, enhanced mechanisms are proposed to detect and eliminate loops. The basic algorithm and loop elimination mechanism are first validated through extensive simulation experiments. Finally, the integrated performance of the full potentialbased energybalanced routing algorithm is evaluated through numerous simulations in a random deployed network running eventdriven applications, the impact of the parameters on the performance is examined and guidelines for parameter settings are summarized. Our experimental results show that there are significant improvements in energy balance, network lifetime, coverage ratio, and throughput as compared to the commonly used energyefficient routing algorithm. Index Terms—Wireless sensor networks, balancing energy consumption, energyefficient routing, potential field. Ç 1
Estimates and structure of αharmonic functions
 Kumagai: On Heat Kernel Estimates and Parabolic Harnack Inequality for Jump Processes on Metric Measure Spaces. Acta
"... We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D c ∪{∞} satisfying an integrability condition. The corresponding Martin bound ..."
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We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D c ∪{∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test. 1 Main results and introduction Let d = 1, 2,..., and 0 < α < 2. The boundary Harnack principle (BHP) for nonnegative harmonic functions of the fractional Laplacian on R d ∆ α/2 ϕ(x) = lim ε→0 + y>ε was proved for Lipschitz domains in 1997 ([9]). Here [ϕ(y) − ϕ(x)] ν(x, y)dy, (1) ν(x, y) = Ad,−αy − x  −d−α, Ad,γ = Γ((d − γ)/2)/(2γπd/2 Γ(γ/2)) for −2 < γ < 2, and, say, ϕ ∈ C ∞ c (Rd). BHP was extended to all open sets in 1999 ([41]), with the constant in the estimate depending on local geometry of their boundary (compare Corollary 1 below). The question whether the constant may be chosen independently of the domain, or uniformly, was since open. In what follows D is a domain i.e. an open nonempty subset of Rd. Let GD be the Green function of D for ∆α/2 ([34], [7], [38]). We define the Poisson kernel of D: PD(x, y) = GD(x, v)ν(v, y) dv, x ∈ R d, y ∈ D c. (2)
Boundary Harnack principle for ∆ + ∆ α/2
, 2009
"... For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo differential operators { ∆ +b ∆ α/2; b ∈ [0, 1]} on R d that evolves continuously from ∆ to ∆ + ∆ α/2. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are ..."
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Cited by 6 (5 self)
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For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo differential operators { ∆ +b ∆ α/2; b ∈ [0, 1]} on R d that evolves continuously from ∆ to ∆ + ∆ α/2. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to ∆+b ∆ α/2 (or equivalently, the sum of a Brownian motion and an independent symmetric αstable process with constant multiple b 1/α) in C 1,1 open sets. Here a “uniform ” BHP means that the comparing constant in the BHP is independent of b ∈ [0, 1]. Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to ∆ + b ∆ α/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.
HARMONIC FUNCTIONS, ENTROPY, AND A CHARACTERIZATION OF THE HYPERBOLIC SPACE
"... Abstract. Let (M n; g) be a compact Riemannian manifold with Ric (n 1). It is well known that the bottom of spectrum 0 of its unverversal covering satis…es 0 (n 1) 2 =4. We prove that equality holds i ¤ M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. 1. ..."
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Abstract. Let (M n; g) be a compact Riemannian manifold with Ric (n 1). It is well known that the bottom of spectrum 0 of its unverversal covering satis…es 0 (n 1) 2 =4. We prove that equality holds i ¤ M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. 1.