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Synchronization and linearity: an algebra for discrete event systems
, 2001
"... The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific ..."
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Cited by 248 (10 self)
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The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX crossreferences are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The
Theoremes Asymptotiques En Programmation Dynamique
, 1990
"... . On montre l'analogie existant entre le calcul des probabilites et la programmation dynamique. Dans la premiere situation les convolutions iterees de lois de probabilite jouent un role central, dans la seconde les infconvolutions de fonctions couts ont un role similaire. L'outil d'analyse privileg ..."
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Cited by 15 (7 self)
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. On montre l'analogie existant entre le calcul des probabilites et la programmation dynamique. Dans la premiere situation les convolutions iterees de lois de probabilite jouent un role central, dans la seconde les infconvolutions de fonctions couts ont un role similaire. L'outil d'analyse privilegiedelapremiere situation est la transformee de Fourier, celui de la seconde devrait etre la transform ee de Fenchel. Aux lois gaussiennes  stables par convolution  correspondent les formes quadratiques  stables par infconvolution. A la loi des grands des nombres et au theoreme de la limite centrale correspondent des theoremes asymptotiques pour la programmation dynamique  convergence de la fonction valeur de l'etat moyenne vers la fonction caracteristique du minimum du cout instantane, convergencede la fonction valeur de l'ecart au minimun renormalise vers une forme quadratique. ABSTRACT. Asymptotic Theorems in Dynamic Programming We show the analogy between probability calculu...
Bellman Processes
 In 11th International Conference on Analysis and Optimization of Systems : Discrete Event Systems. Lecture notes in Control and Information Sciences
, 1994
"... this paper is to make a presentation of known and new results of optimal control with this morphism in mind. The emphasis of this talk is i) on the trajectory point of view by opposition to the cost point of view and ii) on the optimization counterpart of processes with independent increments. 2. IN ..."
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Cited by 11 (4 self)
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this paper is to make a presentation of known and new results of optimal control with this morphism in mind. The emphasis of this talk is i) on the trajectory point of view by opposition to the cost point of view and ii) on the optimization counterpart of processes with independent increments. 2. INFCONVOLUTION AND CRAMER TRANSFORM
Theory of Cost Measures: Convergence of Decision Variables
 INRIA REPORT N
, 1995
"... Considering probability theory in which the semifield of positive real numbers is replaced by the idempotent semifield of real numbers (union infinity) endowed with the min and plus laws leads to a new formalism for optimization. Probability measures correspond to minimums of functions that we call ..."
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Cited by 11 (6 self)
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Considering probability theory in which the semifield of positive real numbers is replaced by the idempotent semifield of real numbers (union infinity) endowed with the min and plus laws leads to a new formalism for optimization. Probability measures correspond to minimums of functions that we call cost measures, whereas random variables correspond to constraints on these optimization problems that we call decision variables. We review in this context basic notions of probability theory  random variables, convergence of random variables, characteristic functions, L p norms. Whenever it is possible, results and definitions are stated in a general idempotent semiring.
Maxplus algebra and applications to system theory and optimal control
 in Proceedings of the International Congress of Mathematicians
, 1994
"... In the modeling of human activities, in contrast to natural phenomena, quite frequently only the operations max (or min) and � are needed. A typical example is the performance evaluation of synchronized processes such as those encountered in manufacturing (dynamic systems made up of storage and queu ..."
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Cited by 7 (1 self)
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In the modeling of human activities, in contrast to natural phenomena, quite frequently only the operations max (or min) and � are needed. A typical example is the performance evaluation of synchronized processes such as those encountered in manufacturing (dynamic systems made up of storage and queuing networks). Another typical example is the computation of a path of maximum weight in a graph and more generally of the optimal control of dynamical systems. We give examples of such situations. The maxplus algebra which is a mathematical framework well suited to handle such situations. We present results on i � linear algebra, ii � system theory, iii � duality between probability and optimization based on this algebra. 1 MaxPlus Linear Algebra Definition 1. 1. A abelian monoid K is a set endowed with one operation � which is associative, commutative and has a zero element �. 2. A semiring is an abelian monoid endowed with a second operation � which is associative and distributive with respect to � which has an identity element denoted e, with � absorbing (that is � � a � a � � � �).
WHAT SHAPE IS YOUR CONJUGATE? A SURVEY OF COMPUTATIONAL CONVEX ANALYSIS AND ITS APPLICATIONS
"... Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental t ..."
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Cited by 5 (1 self)
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Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolicnumeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms, and to further communications between several fields benefiting from the same techniques. We survey applications of the algorithms which have been applied to problems arising from image processing (distance transform, generalized distance transform, mathematical morphology), partial differential equations (solving HamiltonJacobi equations, and using differential equations numerical schemes to compute the convex envelope), maxplus algebra, multifractal analysis, and several others. They span a wide range of applications in computer vision, robot navigation, phase transition in thermodynamics, electrical networks,
Computing rank convolutions with a mask
, 2006
"... Rankconvolutions have important applications in a variety of areas such as signal processing and computer vision. We define a “mask ” as a function taking only values zero and infinity. Rankconvolutions with masks are of special interest to image processing. We show how to compute the rankk convo ..."
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Cited by 3 (0 self)
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Rankconvolutions have important applications in a variety of areas such as signal processing and computer vision. We define a “mask ” as a function taking only values zero and infinity. Rankconvolutions with masks are of special interest to image processing. We show how to compute the rankk convolution of a function over an interval of length n with an arbitrary mask of length m in O(n √ m log m) time. The result generalizes to the ddimensional case. Previously no algorithm performing significantly better than the brute force O(nm) bound was known. Our algorithm seems to perform well in practice. We describe an implementation, illustrating its application to a problem in image processing. Already on relatively small images, our experiments show a signficant speedup compared to brute force.
Necklaces, Convolutions, and X + Y
"... Abstract. We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓp norm of the vector of distances between pairs of beads from opposite nec ..."
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Cited by 1 (0 self)
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Abstract. We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓp norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p = 2, and p = ∞. For p = 2, we reduce the problem to standard convolution, while for p = ∞ and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n 2) time, whereas the obvious algorithms for these problems run in Θ(n 2) time. 1
MaxPlus Algebra and Applications to System Theory and Optimal Control
"... 7.25> 3.Adioidisasemiringwhichisidempotent(thatisa#a=a,#a#K). 4.AsemifieldisasemiringhavingitssecondoperationinvertibleonK # =K\{#}. 5.Asemifieldwhichisalsoadioidiscalledanidempotentsemifield. 6.Wewillsaythatthesestructuresarecommutativewhentheproductisalsocommutative. 7.WecallR max [resp.R min ] ..."
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7.25> 3.Adioidisasemiringwhichisidempotent(thatisa#a=a,#a#K). 4.AsemifieldisasemiringhavingitssecondoperationinvertibleonK # =K\{#}. 5.Asemifieldwhichisalsoadioidiscalledanidempotentsemifield. 6.Wewillsaythatthesestructuresarecommutativewhentheproductisalsocommutative. 7.WecallR max [resp.R min ]thesetR#{#}[resp.R#{+#}]endowedwiththe twooperations#=max[resp#=min]and#=+. 8.WecallR nn max andanalogouslyR nn min thesetofnnmatriceswithentriesbelonging toR max endowedwith#denotingthemaxentrybyentryand#definedby<F9.9
Target Tracking using Irregularly Spaced Detectors and a ContinuousState Viterbi Algorithm
, 2000
"... of the CSVA, a recently developed algorithm for MAP state sequence estimation, to the problem of tracking a target constrained to two dimensional movement using detections from a field of stationary, arbitrarily located, identical sensors. Models for target motion and detector performance are develo ..."
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of the CSVA, a recently developed algorithm for MAP state sequence estimation, to the problem of tracking a target constrained to two dimensional movement using detections from a field of stationary, arbitrarily located, identical sensors. Models for target motion and detector performance are developed for this problem, providing a system model to which the MAP estimator is applied to create a target trajectory estimator. The performance of the MAP target trajectory estimator is illustrated through simulation.