**Contemporary Mathematics**

Volume: 616;
2014;
300 pp;
Softcover

MSC: Primary 15; 16; 05; 14; 52; 35; 90; 65; 92; 82;

Print ISBN: 978-0-8218-9496-5

Product Code: CONM/616

List Price: $102.00

AMS Member Price: $81.60

MAA member Price: $91.80

**Electronic ISBN: 978-1-4704-1684-3
Product Code: CONM/616.E**

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AMS Member Price: $81.60

MAA member Price: $91.80

# Tropical and Idempotent Mathematics and Applications

Share this page *Edited by *
*G. L. Litvinov; S. N. Sergeev*

This volume contains the proceedings of the International Workshop on
Tropical and Idempotent Mathematics, held at the Independent
University of Moscow, Russia, from August 26–31, 2012. The main
purpose of the conference was to bring together and unite researchers
and specialists in various areas of tropical and idempotent
mathematics and applications.

This volume contains articles on algebraic foundations of tropical
mathematics as well as articles on applications of tropical
mathematics in various fields as diverse as economics,
electroenergetic networks, chemical reactions, representation theory,
and foundations of classical thermodynamics.

This volume is intended for graduate students and researchers
interested in tropical and idempotent mathematics or in their
applications in other areas of mathematics and in technical
sciences.

#### Readership

Graduate students and research mathematicians interested in tropical and idempotent mathematics and its applications.

# Table of Contents

## Tropical and Idempotent Mathematics and Applications

- Preface vii8 free
- In Memory of Grigory Litvinov ix10 free
- Tropical Cramer determinants revisited 114 free
- 1. Introduction 114
- 2. Semirings with a symmetry and a modulus 518
- 3. Combinatorial properties of semirings 1326
- 4. Elimination in semirings and Cramer theorem 1730
- 5. Existence of solutions of tropical linear systems 2235
- 6. Homogeneous systems: the generalized Gondran-Minoux theorem 3245
- 7. Systems of balances and intersections of signed hyperplanes 3649
- 8. Computing all Cramer Permanents: tropical Jacobi versus transportation approach 3851
- 9. Computing determinants 4255
- Acknowledgment 4356
- References 4356

- An approximation of Hopf-Lax type formula via idempotent analysis 4760
- Ideals of MV-semirings and MV-algebras 5972
- 1. Introduction 5972
- 2. Preliminaries 6174
- 3. Ideals, congruences and quotients 6275
- 4. The frame of radical ideals of an MV-semiring 6780
- 5. The frame of radical ideals of an MV-algebra 6982
- 6. The frames of open sets of \Spec(𝐴) and of \Spec(𝑆) 6982
- 7. The frame of open sets of \Max(𝐴) 7285
- References 7588

- Tropical Plücker functions and Kashiwara crystals 7790
- A metric on max-min algebra 101114
- Optimization on the range of a max-separable operator 115128
- Algebraic structures of tropical mathematics 125138
- Parametric dequantization, tropical reduction of hyperfields and steady states of AC electrical networks 151164
- 1. Introduction 151164
- 2. Real tropical polynomials over a semifield 152165
- 3. Parametric limits of polynomials with various conditions on their coefficients 153166
- 4. Complex tropical polynomials over hyperfield and tropical reduction 155168
- 5. The power balance equations in AC network 156169
- 6. Tropical reduction of the power balance equations, Foster coefficients and minimal spanning trees 158171
- 7. Concluding remarks 160173
- References 161174

- A constrained tropical optimization problem: Complete solution and application example 163176
- On the mathematical foundations of classical thermodynamics 179192
- 1. Introduction 179192
- 2. The new ideal gas: Mathematical statement 184197
- 3. The new ideal gas: Physical statement 188201
- 4. Phase transition to the ideal liquid 189202
- 5. Holes in the Bose condensate 192205
- 6. Taking attraction into account and the Zeno line. The role of vanishingly small viscosity 193206
- 7. Determination of the new critical point 199212
- 8. Phase transition in the ideal Boltzmann–Maxwell gas 201214
- 9. Consideration of an infinitely small dissipation and the Maxwell rule 202215
- 10. The law of corresponding states taking the Zeno line into account. The role of crude measuring devices 204217
- Appendix A. On a mathematical mistake 206219
- References 207220

- Tropical support sets in analysis of weak links and complementarity 211224
- Idempotent algebra models of single-agent and multi-agent dynamics 221234
- Tropical convexity over max-min semiring 241254
- Tropicalization and tropical equilibration of chemical reactions 261274
- 1. Introduction 261274
- 2. Tropically truncated chemical kinetics 261274
- 3. Tropical equilibration and permanence 263276
- 4. Geometry of tropical equilibrations 268281
- 5. Tropical approach to the permanency problem 269282
- 6. Application to chemical reactions kinetics 271284
- 7. Conclusion 274287
- Acknowledgements 274287
- References 274287

- An overview of transience bounds in max-plus algebra 277290
- 1. Introduction 277290
- 2. Definitions and Preliminaries 278291
- 3. The Boolean Case: Index of Convergence 279292
- 4. Nachtigall Decomposition 280293
- 5. Bound by Hartmann and Arguelles 282295
- 6. A Bound for Primitive Matrices 284297
- 7. When All Entries Are Finite 285298
- 8. Inclusion of Cyclicity and Girth 285298
- 9. From Systems to Matrices 287300
- 10. Conclusion 287300
- Acknowledgments 287300
- References 288301

- Strong independence and injectivity in tropical modules 291304