Submathematics and tropical mathematics. (English) Zbl 1468.00005
Summary: The notions of “tropical mathematics” and “subtropical mathematics” are studied. The main principles of tropical analysis and examples of their application to various problems are considered.
MSC:
| 00A30 | Philosophy of mathematics |
| 01A80 | Sociology (and profession) of mathematics |
| 14Txx | Tropical geometry |
| 15A80 | Max-plus and related algebras |
Keywords:
tropical mathematics; submathematics; tropical relations; invariants; idempotent correspondence principle; dequantization; nonlinear differential equations; tropical topologyReferences:
| [1] | Maslov, V. P., On Tropical Analysis, Math. Notes, 98, 5, 798-804 (2015) · Zbl 1384.14001 · doi:10.1134/S0001434615110085 |
| [2] | Simon, I., Recognizable sets with multiplicities in the tropical semiring, Lecture Notes in Computer Science, Vol. 324: Mathematical Foundations of Computer Science, 0, 107-120 (1988) · Zbl 0656.68086 · doi:10.1007/BFb0017135 |
| [3] | Pin, J. E., Tropical semirings, Publ. Newton Institute, Vol. 11: Idempotency, 0, 50-60 (1998) · Zbl 0909.16028 |
| [4] | Maslov, V. P., Motivation and essence of the term “Tropical mathematics”, Russian J. Math. Phys., 27, 4, 482-487 (2020) · doi:10.1134/S106192082004007X |
| [5] | Maslov, V. P., Méthodes Opératorielles (1987), Moscow: Mir, Moscow · Zbl 0651.47039 |
| [6] | V. P. Maslov and V. N. Kolokoltsov, Idempotent Analysis and Its Applications to Optimal Control (Fizmatlit, Moscow, 1994) [in Russian]. · Zbl 0857.49022 |
| [7] | Kolokoltsov, V. N., Maslov’s arithmetic in general topology, Geometry, Topology, and Applications, 0, 64-68 (1990) |
| [8] | Kolokoltsov, V. N., Introduction of a new Maslov-type currency (coupons) as a means of solving a market game under non-equilibrium prices, Sov. Math. Dokl., 44, 2, 624-629 (1992) · Zbl 0792.90012 |
| [9] | Kolokoltsov, V. N., Stochastic Bellman equation as a nonlinear equation in Maslov spaces. Perturbation theory, Sov. Math. Dokl., 45, 2, 294-300 (1992) · Zbl 0802.49022 |
| [10] | Maslov, V. P., Thermodynamics and tropical mathematics. Definition of quasistatistical processes, Russian J. Math. Phys., 23, 1, 101-114 (2016) · Zbl 1342.82057 · doi:10.1134/S1061920816010088 |
| [11] | Viro, O., Dequantization of real algebraic geometry on a logarithmic paper, 3rd European Congress of Mathematics (Barcelona, 2000 ), 0, 0 (0000) |
| [12] | Akian, M.; Gaubert, S.; Guterman, A., Tropical polyhedra are equivalent to mean payoff games, Internat. J. Algebra Comput., 22, 1-1250001, 0 (2012) · Zbl 1239.14054 |
| [13] | Akian, M.; Gaubert, S.; Hochart, A., Minimax representation of nonexpansive functions and application to zero-sum recursive games, J. Convex Anal., 25, 1, 225-240 (2018) · Zbl 1386.49009 |
| [14] | Gaubert, S.; McEneaney, W., Min-max spaces and complexity reduction in min-max expansions, Appl. Math. Optim., 65, 3, 315 (2012) · Zbl 1244.93051 · doi:10.1007/s00245-011-9158-5 |
| [15] | Allamigeon, X.; Benchimol, P.; Gaubert, S.; Joswig, M., Tropicalizing the simplex algorithm, SIAM J. Discrete Math., 29, 2, 751-795 (2015) · Zbl 1334.14033 · doi:10.1137/130936464 |
| [16] | Maslov, V. P., The Complex WKB Method for Nonlinear Equations. I (1994), Basel-Boston-Berlin: Birkhäuser, Basel-Boston-Berlin · Zbl 0811.35088 · doi:10.1007/978-3-0348-8536-2 |
| [17] | Noel, V.; Grigoriev, D.; Vakulenko, S.; Radulescu, O., Tropicalization and tropical equilibration of chemical reactions, Contemporary Math., 616, 261-275 (2014) · Zbl 1320.92091 · doi:10.1090/conm/616/12316 |
| [18] | Krivulin, N., Tropical optimization problems, Advances in Economics and Optimization. Economic Issues : Problems and Perspectives, 0, 195-214 (2014) |
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