Minimax Algebra

R.C. Backhouse and B.A. Carré, "Regular algebra applied to path-finding problems", J. Inst. Math. Appl. 15 (1975) 161 – 186. R.A. Cuninghame-Green, "Minimax algebra I: Bands and belts", Memorandum no. 70, Dept. of Applied Mathematics, ...

Author: R. A. Cuninghame-Green

Publisher: Springer Science & Business Media

ISBN: 9783642487088

Category: Business & Economics

Page: 258

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A number of different problems of interest to the operational researcher and the mathematical economist - for example, certain problems of optimization on graphs and networks, of machine-scheduling, of convex analysis and of approx imation theory - can be formulated in a convenient way using the algebraic structure (R,$,@) where we may think of R as the (extended) real-number system with the binary combining operations x$y, x®y defined to be max(x,y),(x+y) respectively. The use of this algebraic structure gives these problems the character of problems of linear algebra, or linear operator theory. This fact hB.s been independently discovered by a number of people working in various fields and in different notations, and the starting-point for the present Lecture Notes was the writer's persuasion that the time had arrived to present a unified account of the algebra of linear transformations of spaces of n-tuples over (R,$,®),to demonstrate its relevance to operational research and to give solutions to the standard linear-algebraic problems which arise - e.g. the solution of linear equations exactly or approximately, the eigenvector eigenvalue problem andso on.Some of this material contains results of hitherto unpublished research carried out by the writer during the years 1970-1977.

Advances in Imaging and Electron Physics

(III.1) This is essentially the problem of the solution of linear equations in max algebra. It is clear that by starting early enough, ... From the meaning of the operators max and min, it MINIMAX ALGEBRA AND APPLICATIONS 27.

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Publisher: Academic Press

ISBN: 0080577563

Category: Science

Page: 441

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Advances in Imaging and Electron Physics

Minimax Algebra

A number of different problems of interest to the operational researcher and the mathematical economist - for example, certain problems of optimization on graphs and networks, of machine-scheduling, of convex analysis and of approx imation ...

Author: R. A. Cuninghame-Green

Publisher: Springer

ISBN: 3540091130

Category: Business & Economics

Page: 258

View: 710

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A number of different problems of interest to the operational researcher and the mathematical economist - for example, certain problems of optimization on graphs and networks, of machine-scheduling, of convex analysis and of approx imation theory - can be formulated in a convenient way using the algebraic structure (R,$,@) where we may think of R as the (extended) real-number system with the binary combining operations x$y, x®y defined to be max(x,y),(x+y) respectively. The use of this algebraic structure gives these problems the character of problems of linear algebra, or linear operator theory. This fact hB.s been independently discovered by a number of people working in various fields and in different notations, and the starting-point for the present Lecture Notes was the writer's persuasion that the time had arrived to present a unified account of the algebra of linear transformations of spaces of n-tuples over (R,$,®),to demonstrate its relevance to operational research and to give solutions to the standard linear-algebraic problems which arise - e.g. the solution of linear equations exactly or approximately, the eigenvector eigenvalue problem andso on.Some of this material contains results of hitherto unpublished research carried out by the writer during the years 1970-1977.

Advances in Electronics and Electron Physics

It is exactly these linear algebraic concepts that provide tools to help solve linear image processing problems, and the minimax algebra will undoubtedly prove to be as useful, Within the minimax algebra, there are notions of matrix ...

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Publisher: Academic Press

ISBN: 0080577504

Category: Computers

Page: 358

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Advances in Electronics and Electron Physics

Introduction to Lattice Algebra

4.2 MINIMAX ALGEBRA For the remainder of this chapter we discuss lattice - based algebras from a viewpoint that differs from the one presented in most current textbooks and research papers on the ...

Author: Gerhard X. Ritter

Publisher: CRC Press

ISBN: 9781000412567

Category: Mathematics

Page: 432

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Lattice theory extends into virtually every branch of mathematics, ranging from measure theory and convex geometry to probability theory and topology. A more recent development has been the rapid escalation of employing lattice theory for various applications outside the domain of pure mathematics. These applications range from electronic communication theory and gate array devices that implement Boolean logic to artificial intelligence and computer science in general. Introduction to Lattice Algebra: With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks lays emphasis on two subjects, the first being lattice algebra and the second the practical applications of that algebra. This textbook is intended to be used for a special topics course in artificial intelligence with a focus on pattern recognition, multispectral image analysis, and biomimetic artificial neural networks. The book is self-contained and – depending on the student’s major – can be used for a senior undergraduate level or first-year graduate level course. The book is also an ideal self-study guide for researchers and professionals in the above-mentioned disciplines. Features Filled with instructive examples and exercises to help build understanding Suitable for researchers, professionals and students, both in mathematics and computer science Contains numerous exercises.

A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences

Applied Math., T.H. Twente (Netherlands) 1970. [1976 Projections in minimax algebra, Math. Programming 10 (1976), 111-123. [1978] An algebra for the absolute centre of a graph, Report, Dept. Math. Statist. Univ. of Birmingham, 1978.

Author: K. Glazek

Publisher: Springer Science & Business Media

ISBN: 9789401599641

Category: Mathematics

Page: 392

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This volume presents a short guide to the extensive literature concerning semir ings along with a complete bibliography. The literature has been created over many years, in variety of languages, by authors representing different schools of mathematics and working in various related fields. In many instances the terminology used is not universal, which further compounds the difficulty of locating pertinent sources even in this age of the Internet and electronic dis semination of research results. So far there has been no single reference that could guide the interested scholar or student to the relevant publications. This book is an attempt to fill this gap. My interest in the theory of semirings began in the early sixties, when to gether with Bogdan W ~glorz I tried to investigate some algebraic aspects of compactifications of topological spaces, semirings of semicontinuous functions, and the general ideal theory for special semirings. (Unfortunately, local alge braists in Poland told me at that time that there was nothing interesting in investigating semiring theory because ring theory was still being developed). However, some time later we became aware of some similar investigations hav ing already been done. The theory of semirings has remained "my first love" ever since, and I have been interested in the results in this field that have been appearing in literature (even though I have not been active in this area myself).

Minimax and Applications

J. M. Borwein and D. Zhuang, On Fan's minimax theorem, Math. Programming 34(1986), 232–234. [7]D. G. Bourgin, Fixed point and min-max theorems, Pac. J. Math. 45(1973), 403-412. H. Brézis, L. Nirenberg and G. Stampacchia, A remark on Ky ...

Author: Ding-Zhu Du

Publisher: Springer Science & Business Media

ISBN: 9781461335573

Category: Computers

Page: 296

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Techniques and principles of minimax theory play a key role in many areas of research, including game theory, optimization, and computational complexity. In general, a minimax problem can be formulated as min max f(x, y) (1) ",EX !lEY where f(x, y) is a function defined on the product of X and Y spaces. There are two basic issues regarding minimax problems: The first issue concerns the establishment of sufficient and necessary conditions for equality minmaxf(x,y) = maxminf(x,y). (2) "'EX !lEY !lEY "'EX The classical minimax theorem of von Neumann is a result of this type. Duality theory in linear and convex quadratic programming interprets minimax theory in a different way. The second issue concerns the establishment of sufficient and necessary conditions for values of the variables x and y that achieve the global minimax function value f(x*, y*) = minmaxf(x, y). (3) "'EX !lEY There are two developments in minimax theory that we would like to mention.

Algebraic and Combinatorial Methods in Operations Research

[12] [13] [14] [15] [16] [17] [18] Wongseelashote, A., Path algebras: A multiset-theoretic approach, Ph.D. thesis (University of Southampton, 1976). Cuninghame-Green, R.A., Minimax Algebra (Lecture Notes in Economics and Mathematical ...

Author: R.E. Burkard

Publisher: Elsevier

ISBN: 0080872069

Category: Mathematics

Page: 380

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For the first time, this book unites different algebraic approaches for discrete optimization and operations research. The presentation of some fundamental directions of this new fast developing area shows the wide range of its applicability. Specifically, the book contains contributions in the following fields: semigroup and semiring theory applied to combinatorial and integer programming, network flow theory in ordered algebraic structures, extremal optimization problems, decomposition principles for discrete structures, Boolean methods in graph theory and applications.

Semirings and Affine Equations over Them

[130 —, Projections in minimax algebra, Math. Programming 10 (1976), 111123. [131] –, Minimaa Algebra, Lecture Notes in Economics and Mathematical Systems #166, Springer-Verlag, Berlin, 1979. [132 –, Minimax algebra and its applications ...

Author: Jonathan S. Golan

Publisher: Springer Science & Business Media

ISBN: 9789401703833

Category: Mathematics

Page: 241

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Semiring theory stands with a foot in each of two mathematical domains. The first being abstract algebra and the other the fields of applied mathematics such as optimization theory, the theory of discrete-event dynamical systems, automata theory, and formal language theory, as well as from the allied areas of theoretical computer science and theoretical physics. Most important applications of semiring theory in these areas turn out to revolve around the problem of finding the equalizer of a pair of affine maps between two semimodules. In this volume, we chart the state of the art on solving this problem, and present many specific cases of applications. This book is essentially the third part of a trilogy, along with Semirings and their Applications, and Power Algebras over Semirings, both written by the same author and published by Kluwer Academic Publishers in 1999. While each book can be read independently of the others, to get the full force of the theory and applications one should have access to all three. This work will be of interest to academic and industrial researchers and graduate students. The intent of the book is to bring the applications to the attention of the abstract mathematicians and to make the abstract mathematics available to those who are using these tools in an ad-hoc manner without realizing the full force of the theory.

Hamilton Jacobi Equations Approximations Numerical Analysis and Applications

arXiv.org/abs/math.FA/0212294 G.E. Coxson, Computing exact bounds on the elements of an inverse interval matrix is NP-hard. Reliable Comput. 5, 137–142 (1999) R.A. Cuninghame-Green, in Minimax Algebra. Lecture Notes in Economics and ...

Author: Yves Achdou

Publisher: Springer

ISBN: 9783642364334

Category: Mathematics

Page: 301

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These Lecture Notes contain the material relative to the courses given at the CIME summer school held in Cetraro, Italy from August 29 to September 3, 2011. The topic was "Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications". The courses dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman equations, properties of viscosity solutions, asymptotic behaviors, mean field games, approximation and numerical methods, idempotent analysis. The content of the courses ranged from an introduction to viscosity solutions to quite advanced topics, at the cutting edge of research in the field. We believe that they opened perspectives on new and delicate issues. These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for Mean Field Games), Guy Barles (An Introduction to the Theory of Viscosity Solutions for First-order Hamilton-Jacobi Equations and Applications), Hitoshi Ishii (A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and Grigory Litvinov (Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations).