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Arie Leizarowitz

Technion – Israel Institute of Technology

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Alain Haurie

University of Geneva

Dean A. Carlson

American Mathematical Society

Yacov Tsur

Hebrew University of Jerusalem

Alexander J. Zaslavski

Technion – Israel Institute of Technology

Assaf Schuster

Technion – Israel Institute of Technology

Iddo Kan

Hebrew University of Jerusalem

Ran Wolff

Technion – Israel Institute of Technology

Adam Shwartz

Jacobs Institute

Noam Palatin

Technion – Israel Institute of Technology

Victor J. Mizel

Carnegie Mellon University

Cooperative Institutions

Technion – Israel Institute of Technology

Hebrew University of Jerusalem

Cornell University

University of Haifa

National University of Ireland, Maynooth

University of Maryland, College Park

Israel Institute

Rutgers, The State University of New Jersey

Weizmann Institute of Science

University of Wisconsin–Madison

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On the Lyapunov Exponent of a Harmonic Oscillator Driven by a Finite-State Markov Process

SIAM Journal on Applied Mathematics (1989)

Arie Leizarowitz

This paper considers the asymptotic behavior of the random harmonic oscillator driven by a Markov process with a finite-state space. For a certain choice of variables, the motion of the system follows circles in the plane, with jumps from one circle to another at random times. The Lyapunov exponent is expressed in terms of the equilibrium measure of a related Markov chain. The latter satisfies an integralequation whose form permits an easy derivation of estimates in various limiting cases. This is demonstrated by computing the first-order approximation in a limit of weak noise. It is shown how some of the results can be extended to a certain class of two-dimensional systems.

Harmonic

Exponent

10.1137/0149025

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Exact finite approximations of average-cost countable Markov Decision Processes

arXiv (Cornell University) (2007)

Arie Leizarowitz Adam Shwartz

For a countable-state Markov decision process we introduce an embedding which produces a finite-state Markov decision process. The finite-state embedded process has the same optimal cost, and moreover, it has the same dynamics as the original process when restricting to the approximating set. The embedded process can be used as an approximation which, being finite, is more convenient for computation and implementation.

Finite set

Markov kernel

10.48550/arxiv.0711.2185

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Exact finite approximations of average-cost countable Markov decision processes

Automatica (2008)

Arie Leizarowitz Adam Shwartz

Approximations of π

10.1016/j.automatica.2007.09.013

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Infinite horizon stochastic regulation and tracking with the overtaking criterion^∗

Stochastics (1987)

Arie Leizarowitz

This paper considers the standard LQG problem on an infinite horizon with overtaking criterion. A control u*is overtaking optimal if for every other control u there is a time T'u' such that Cr'u∗'T'u', where Cr'u' is the cost over the time interval [0, T] when employing u. We show that the separation principle holds and that the usual stationary optimal control is also optimal in the overtaking sense. We describe certain limiting properties of the optimal controlled process. The relation between overtaking optimal solutions and optimal solutions on finite large intervals is studied

Overtaking

Limiting

10.1080/17442508708833470

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Iterated logarithm approximations to the distribution of the largest prime divisor

arXiv (Cornell University) (2009)

Arie Leizarowitz

The paper is concerned with estimating the number of integers smaller than $x$ whose largest prime divisor is smaller than $y$, denoted $ψ(x,y)$. Much of the related literature is concerned with approximating $ψ(x,y)$ by Dickman's function $ρ(u)$, where $u=\ln x/\ln y$. A typical such result is that $$ ψ(x,y)=xρ(u)(1+o(1)) \eqno (1) $$ in a certain domain of the parameters $x$ and $y$. In this paper a different type of approximation of $ψ(x,y)$, using iterated logarithms of $x$ and $y$, is presented. We establish that $$ \ln (\frac ψ{x})=-u [\ln ^{(2)}x-\ln ^{(2)}y+\ln ^{(3)}x-\ln ^{(3)}y+\ln ^{(4)}x-a] \eqno (2) $$ where $\underbar{a}

Iterated logarithm

Iterated function

10.48550/arxiv.0903.2712

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Optimal Solutions of Linear Control Systems with Nonperiodic Convex Integrands

Mathematics of Operations Research (1997)

Alexander J. Zaslavski Arie Leizarowitz

In this work we study the existence and asymptotic behaviour of overtaking optimal trajectories for a linear system with a nonperiodic convex cost function. We extend the results obtained by Leizarowitz (Leizarowitz, A. 1986. Tracking nonperiodic trajectories with the overtaking criterion. Appl. Math. Opt. 14 155–171.) for a problem of tracking a prescribed nonperiodic trajectory with a quadratic cost function and establish the existence and uniqueness of optimal trajectories on an infinite horizon. The asymptotic dynamics of finite time optimizers is examined.

Overtaking

10.1287/moor.22.3.726

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A square shape of the graph of iterates of multifunctions: a complete controllability result

Proceedings of the American Mathematical Society (1990)

Arie Leizarowitz

We consider a set valued function T : K → 2 K T:K \to {2^K} from a domain K K into itself. We look for conditions under which the graph of T ( n ) {T^{(n)}} (the n n th iterate of T T ) will be equal to K × K K \times K for some integer n n . When the graph of T T is convex a sufficient (though not necessary) condition is that neither T ( x ) T(x) nor T − 1 ( x ) {T^{ - 1}}(x) are contained in the boundary of K K whenever x x is there. We show that a necessary and sufficient condition is that there are neither forward nor backward trajectories which remain in the boundary for all times. In the introduction, we remark on the significance of this problem for the study of infinite-horizon control systems.

10.1090/s0002-9939-1990-1023351-2

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An Algorithm to Identify and Compute Average Optimal Policies in Multichain Markov Decision Processes

Mathematics of Operations Research (2003)

Arie Leizarowitz

This paper concerns discrete-time, finite state multichain MDPs with compact action sets. The optimality criterion is long-run average cost. Simple examples illustrate that optimal stationary Markov policies do not always exist. We establish the existence of ε-optimal policies that are stationary Markovian, and develop an algorithm that computes these approximate optimal policies. We establish a necessary and sufficient condition for the existence of an optimal policy that is stationary Markovian, and in case that such an optimal policy exists the algorithm computes it.

Forward algorithm

10.1287/moor.28.3.553.16388

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Mining for Misconfigured Machines in Grid Systems

Noam Palatin Arie Leizarowitz Assaf Schuster Ran Wolff

This chapter contains sections titled: Introduction Preliminaries and related work Acquiring, pre-processing and storing data Data analysis The GMS Evaluation Conclusions and future work References

Data Processing

10.1002/9780470699904.ch5

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Radially symmetric minimizers of variational problems in the plane

Advances in Differential Equations (2005)

Arie Leizarowitz

The functional $J_{\Omega }=\frac {1}{|\Omega |}\int _{\Omega }[(\Delta u)^2-b|\nabla |^2+\psi (u)]dxdy$ is considered on bounded domains $\Omega \subset R^2$ and configurations $u$, which are functions belonging to $H^2_{loc}(R^2)$. For such $u$ the value of $J_{\Omega }(u)$ is studied in the limit where $\Omega $ extends to the whole plane $R^2$, which we denote $J(u)$. The integrand defining $J_{\Omega }$ is radially symmetric, and we raise the question whether $J(u)$ has a minimizer $u^{\star }$ which is radially symmetric, namely has the form $u^{\star }(x,y)=v(\sqrt {x^2+y^2})$. It is shown that the minimal value of $J(u)$ over the radially symmetric configurations is equal to its minimal value over 1-dimensional configurations, which are functions of the form $u(x,y)=v(x)$. Restricting to 1-dimensional configurations yields the 1-dimensional model, hence $\lambda _2\leq \lambda _1$, where $\lambda _1$ and $\lambda _2$ are the values of the 1 and 2-dimensional models respectively. Then the question which we address is whether strict inequality holds in $\lambda _2\leq \lambda _1$. We establish a sufficient condition for a configuration $u$ to satisfy $J(u)\geq \lambda _1$, and use it to conclude that the minimal value of $J(u)$ on the class of quasi-convex configurations is $\lambda _1$.

Star (game theory)

Nabla symbol

10.57262/ade/1355867878

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