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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 375 (11 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
Maxplus algebra and system theory: Where we are and where to go now
 Annu. Rev. Control
, 1999
"... Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison ..."
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Cited by 70 (19 self)
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Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory, to provide another important insight to systemtheoretic and controlsynthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the maxplus algebra and their use for system theory is proposed in the last part of the paper. Résumé: Plus de seize ans après le début d’une théorie linéaire de certains systèmes à événements discrets dans laquelle l’algèbre maxplus et autres outils algébriques assimilés jouent un rôle central, ce papier cherche àdécrire quelques uns des principaux résultats obtenus de façon informelle, en s’appuyant sur des exemples. Par comparaison avec la théorie classique des systèmes linéaires, il existe des domaines pratiquement vierges, surtout en raison du fait que les outils mathématiques correspondants restent à forger. C’est en particulier le cas de l’approche géométrique des systèmes qui, dans la théorie classique, est connue pour apporter un autre regard important sur les questions de théorie des systèmes et de synthèse de lois de commandes àcôté de la machinerie purement algébrique. Une discussion préliminaire sur les aspects géométriques de l’algèbre maxplus et leur utilité pour la théorie des systèmes est proposée dans la dernière partie du papier.
Minplus methods in eigenvalue perturbation theory and generalised LidskiiVishikLjusternik theorem
, 2005
"... Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of minplus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturba ..."
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Cited by 21 (2 self)
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Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of minplus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulæ, extending the classical ones and solving cases which where singular in previous approaches. Our results include general weak majorisation inequalities, relating leading exponents of eigenvalues of perturbed matrices and minplus analogues of eigenvalues. 1.
GENERIC ASYMPTOTICS OF EIGENVALUES AND MINPLUS ALGEBRA
, 2004
"... Abstract. We consider a square matrix Aǫ whose entries have asymptotics of the form (Aǫ)ij = aijǫ A ij + o(ǫ A ij) when ǫ goes to 0, for some complex coefficients aij and real exponents Aij. We look for asymptotics of the same type for the eigenvalues of Aǫ. We show that the sequence of exponents of ..."
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Cited by 3 (1 self)
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Abstract. We consider a square matrix Aǫ whose entries have asymptotics of the form (Aǫ)ij = aijǫ A ij + o(ǫ A ij) when ǫ goes to 0, for some complex coefficients aij and real exponents Aij. We look for asymptotics of the same type for the eigenvalues of Aǫ. We show that the sequence of exponents of the eigenvalues of Aǫ is weakly (super) majorized by the sequence of corners of the minplus characteristic polynomial of the matrix A = (Aij), and that the equality holds for generic values of the coefficients aij. We derive this result from a variant of the NewtonPuiseux theorem which applies to asymptotics of the preceding type. We also introduce a sequence of generalized minimal circuit means of A, and show that this sequence weakly majorizes the sequence of corners of the minplus characteristic polynomial of A. We characterize the equality case in terms of perfect matching. When the equality holds, we show that the coefficients of all the eigenvalues of Aǫ can be computed generically by Schur complement formulæ, which extend the perturbation formulæ of Viˇsik, Ljusternik and Lidskiĭ, and have fewer singular cases. 1.
Second Order Theory of MinLinear Systems and its Application to Discrete Event Systems
, 1991
"... A Second Order Theory is developed for linear systems over the (min, +)algebra; in particular the classical notion of correlation is extended to this algebraic structure. It turns out that if we model timed event graphs as linear systems in this algebra, this new notion of correlation can be used t ..."
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A Second Order Theory is developed for linear systems over the (min, +)algebra; in particular the classical notion of correlation is extended to this algebraic structure. It turns out that if we model timed event graphs as linear systems in this algebra, this new notion of correlation can be used to study stocks and sojourn times, and thus to characterize internal stability (boundedness of stocks and sojourn times). This theory relies heavily on the algebraic notion of residuation which is briefly presented. 1 Introduction In [4, 6, 5], a linear system theory analogous to the conventional theory has been developed for a particular class of Discrete Event Dynamic Systems (DEDS) called Timed Event Graphs (TEG). This theory extends the notions of state space, impulse response and transfer function to TEG's. The periodic behavior of these systems has been characterized and a spectral theory has been developed. The key feature which allows extending all these classical concepts is a gene...
An essay on the Riemann Hypothesis
"... Abstract The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explic ..."
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Abstract The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is “analytic ” and is based on Riemannian spaces and Selberg’s work on the trace formula and its comparison with the explicit formulas. The second is based on algebraic geometry and the RiemannRoch theorem. We establish a framework in which one can transpose many of the ingredients of the Weil proof as reformulated by Mattuck, Tate and Grothendieck. This framework is elaborate and involves noncommutative geometry, Grothendieck toposes and tropical geometry. We point out the remaining difficulties and show that RH gives a strong motivation to develop algebraic geometry in the emerging world of characteristic one. Finally we briefly discuss a third strategy based on the development of a suitable “Weil cohomology”, the role of Segal’s Γrings and of topological cyclic homology as a model for “absolute algebra ” and as a cohomological tool. 1
Geometry of the Arithmetic Site
"... We introduce the Arithmetic Site: an algebraic geometric space deeply related to the noncommutative geometric approach to the Riemann Hypothesis. We prove that the noncommutative space quotient of the adele class space of the eld of rational numbers by the maximal compact subgroup of the idele cla ..."
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We introduce the Arithmetic Site: an algebraic geometric space deeply related to the noncommutative geometric approach to the Riemann Hypothesis. We prove that the noncommutative space quotient of the adele class space of the eld of rational numbers by the maximal compact subgroup of the idele class group, which we had previously shown to yield the correct counting function to obtain the complete Riemann zeta function as HasseWeil zeta function, is the set of geometric points of the arithmetic site over the semield of tropical real numbers. The action of the multiplicative group of positive real numbers on the adele class space corresponds to the action of the Frobenius automorphisms on the above geometric points. The underlying topological space of the arithmetic site is the topos of functors from the multiplicative semigroup of nonzero natural numbers to the category of sets. The structure sheaf is made by semirings of characteristic one and is given globally by the semield of tropical integers. In spite of the countable combinatorial nature of the arithmetic site, this space admits a one parameter semigroup of Frobenius correspondences obtained as subvarieties of the square of the site. This square is a semiringed topos whose structure sheaf involves Newton polygons. Finally, we show that the arithmetic site is intimately related to the structure of the (absolute) point in noncommutative geometry.