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 cross-references 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

In this paper, it is shown that a certain class of Petri nets called event graphs can be represented as linear "time-invariant" finite-dimensional systems using some particular algebras. This sets the ground on which a theory of these systems can be developped in a manner which is very analogous to that of conventional linear system theory. Part 2 of the paper is devoted to showing some preliminary basic developments in that direction. Indeed, there are several ways in which one can consider event graphs as linear systems: these ways correspond to approaches in the time domain, in the event domain and in a two-dimensional domain. In each of these approaches, a di#erent algebra has to be used for models to remain linear. However, the common feature of these algebras is that they all fall into the axiomatic definition of "dioids". Therefore, Part 1 of the paper is devoted to a unified presentation of basic algebraic results on dioids. 1 Introduction Definitions and examples of Discrete ...

We prove a new combinatorial characterization of the determinant. The characterization yields a simple combinatorial algorithm for computing the determinant. Hitherto, all (known) algorithms for the determinant have been based on linear algebra. Our combinatorial algorithm requires no division, and works over arbitrary commutative rings. It also lends itself to e#cient parallel implementations. It has been known for some time now that the complexity class GapL characterizes the complexity of computing the determinant of matrices over the integers. We present a direct proof of this characterization.

We show the first efficient combinatorial algorithm for the computation of the determinant. Hitherto, all (known) algorithms for determinant have been based on linear algebra. In contrast, our algorithm and its proof of correctness are totally combinatorial in nature. The algorithm requires no division and works on arbitrary commutative rings. It also lends itself to efficient sequential and parallel implementations. 1 Introduction The Determinant has been the subject of study for over 200 years. Its history can be traced back to Leibnitz, Crammer, Vandermode, Binet, Cauchy, Jacobi, Gauss and others. Given its importance in linear algebra in particular and in geometry in general, it is not surprising that a galaxy of great mathematicians investigated the determinant from varied viewpoints. The algorithmic history of the determinant is as old as the mathematical concept itself. After all, the determinant was invented to solve systems of linear equations. Much of the initial effort was...

Abstract. In this paper we approach the problem of computing the characteristic polynomial of a matrix from the combinatorial viewpoint. We present several combinatorial characterizations of the coefficients of the characteristic polynomial in terms of walks and closed walks of different kinds in the underlying graph. We develop algorithms based on these characterizations and show that they tally with well-known algorithms arrived at independently from considerations in linear algebra.

The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so called formal inverse approach, that of perturbative quantum field theory.

5 1. Introduction: the (max, +) and tropical semirings 5 2. Seven good reasons to use the (max, +) semiring 6 3. Solving Linear Equations in the (max, +) Semiring 11 Chapter 2. Exotic Semirings: Examples and General Results 21 1. Definitions and Zoology 21 2. Combinatorial Formul in Semirings 25 3. Naturally Ordered Semirings 32 4. Semimodules and Linear Maps 34 5. Images and Kernels 36 6. Factorization of Linear Maps and Linear Extension Theorem 37 7. Finiteness Theorems for Semimodules 39 8. Minimal Generating Families, Convex Geometries, and Max-Plus Projective Geometry 41 9. Equivalence of matrices and Green classes 44 10. Hints and Answers to Exercises 47 The max-plus policy improvement algorithm 49 Bibliography 53 Index 57 3 4 CONTENTS CHAPTER 1 Motivations and Survey of Results This introductive chapter is taken verbatim from the survey "Methods and Applications of (max,+) Linear Algebra", S. Gaubert and M. Plus 1 , which appeared in the Proceedings of STACS'1997, L...

Abstract. The Cayley-Hamilton theorem (CHT) is a classic result in linear algebra over fields which states that a matrix satisfies its own characteristic polynomial. CHT has been extended from fields to commutative semirings by Rutherford in 1964. However, to the best of our knowledge, no result is known for noncommutative semirings. This is a serious limitation, as the class of regular languages, with finite automata as their recognizers, is a noncommutative idempotent semiring. In this paper we extend the CHT to noncommutative semirings. We also provide a simpler version of CHT for noncommutative idempotent semirings. 1

Max-algebra, where the classical arithmetic operations of addition and multiplication are replaced by a b := max(a; b) and b := a + b oers an attractive way for modelling discrete event systems and optimization problems in production and transportation. Moreover, it shows a strong similarity to classical linear algebra: for instance, it allows a consideration of linear equation systems and the eigenvalue problem. The max-algebraic permanent of a matrix A corresponds to the maximum value of the classical linear assignment problem with cost matrix A. The analogue of van der Waerden's conjecture in max-algebra is proved. Moreover the role of the linear assignment problem in max-algebra is elaborated, in particular with respect to the uniqueness of solutions of linear equation systems, regularity of matrices and the minimal-dimensional realisation of discrete event systems. Further, the eigenvalue problem in max-algebra is discussed. It is intimately related to the best principal submatrix problem which is nally investigated: Given an integer k, 1 k n, nd a (k k) principal submatrix of the given (n n) matrix which yields among all principal submatrices of the same size the maximum (minimum) value for an assignment. For k = 1; 2; :::; n, the maximum assignment problem values of the principal (k k) submatrices are the coecients of the max-algebraic characteristic polynomial of the matrix for A. This problem can be used to model job rotations.

independence over tropical semirings and beyond