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Maslov dequantization, idempotent and tropical mathematics: A brief introduction

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Abstract

This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as the result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant ℏ tends to zero taking imaginary values. Bibliography: 187 titles.

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References

  1. M. Akian, “Densities of idempotent measures and large deviations,” Trans. Amer. Math. Soc., 351, 4515–4543 (1999).

    MATH  MathSciNet  Google Scholar 

  2. M. Akian and S. Gaubert, “Spectral theorem for convex monotone homogeneous maps and ergodic control,” Nonlinear Anal., 52, 637–679 (2003); see also arXiv:math.SP/0110108.

    MATH  MathSciNet  Google Scholar 

  3. M. Akian, S. Gaubert, and V. Kolokoltsov, “Set coverings and invertibility of functional Galois connections,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 19–52.

    Google Scholar 

  4. M. Akian, S. Gaubert, and C. Walsh, “The max-plus Martin boundary,” arXiv:math.MG/0412408 (2004).

  5. M. Akian, S. Gaubert, and C. Walsh, “Discrete max-plus spectral theory,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 53–78.

    Google Scholar 

  6. M. Akian, J. P. Quadrat, and M. Viot, “Duality between probability and optimization,” in: Idempotency, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge (1998)[69], pp. 331–353.

    Google Scholar 

  7. D. Alessandrini, “Amoebas, tropical varieties and compactification of Teichmüller spaces,” arXiv:math.AG/0505269 (2005).

  8. S. M. Avdoshin, V. V. Belov, V. P. Maslov, and A. M. Chebotarev, “Design of computational media: mathematical aspects,” in: Mathematical Aspects of Computer Engineering [in Russian], Mir, Moscow (1988)[124], pp. 9–145.

    Google Scholar 

  9. F. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, John Wiley & Sons Publishers, New York (1992).

    MATH  Google Scholar 

  10. M. Bardi, I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston-Basel-Berlin (1997).

    MATH  Google Scholar 

  11. A. Berenstein, S. Fomin, and A. Zelevinsky, “Parametrizations of canonical bases and totally positive matrices,” Adv. Math., 122, 49–149 (1996).

    MATH  MathSciNet  Google Scholar 

  12. A. Berenstein and A. Zelevinsky, “Tenzor product multiplicities, canonical bases and totally positive varieties,” Invent. Math., 143, 77–128 (2001).

    MATH  MathSciNet  Google Scholar 

  13. P. Bernhard, “Minimax versus stochastic partial information control,” in: Proceedings of the 33rd Conference on Decision and Control, Lake Buena Vista, Florida, December 1994, IEEE (1994), pp. 2572–2577.

  14. F. Block and J. Yu, “Tropical convexity via cellular resolutions,” arXiv:math.MG/0503279 (2005).

  15. Y. Brenier, U. Frisch, M. Hénon, G. Loeper, S. Matarrese, R. Mohayaee, and A. Sobolevskii, “Reconstruction of the early Universe as a convex optimization problem,” Mon. Not. R. Astron. Soc., 346, 501–524 (2003).

    Google Scholar 

  16. V. M. Bukhshtaber, “The Yang-Baxter mappings,” Uspekhi Mat. Nauk, 53, 241–242 (1998).

    MathSciNet  Google Scholar 

  17. P. Butkovič, “Strong regularity of matrices — a survey of results,” Discrete Appl. Math., 48, 45–68 (1994).

    MathSciNet  MATH  Google Scholar 

  18. P. Butkovič, “On the combinatorial aspects of max-algebra,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 93–104.

    Google Scholar 

  19. I. Capuzzo Dolcetta and P.-L. Lions (eds.), Viscosity Solutions and Applications, Lectures given at the 2nd session of the C.I.M.E. held in Montecatini Terme, Italy, June 12–20, 1995. Lect. Notes Math., 1660 (1997).

  20. B. A. Carré, “An algebra for network routing problems,” J. Inst. Math. Appl., 7, 273–294 (1971).

    MATH  MathSciNet  Google Scholar 

  21. B. A. Carré, Graphs and Networks, The Clarendon Press/Oxford Univ. Press, Oxford (1979).

    MATH  Google Scholar 

  22. K. Cechlárová and R. A. Cuninghame-Green, “Interval systems of max-separable linear equations,” Linear Algebra Appl., 340, 215–224 (2002).

    MATH  MathSciNet  Google Scholar 

  23. W. Chou and R. J. Duffin, “An additive eigenvalue problem of physics related HJB equation to linear programming,” Adv. Appl. Math., 8, 486–498 (1987).

    MATH  MathSciNet  Google Scholar 

  24. G. Choquet, “Theory of capacities,” Ann. Inst. Fourier, 5, 131–295 (1955).

    MathSciNet  Google Scholar 

  25. G. Cohen, S. Gaubert, and J. P. Quadrat, “Max-plus algebra and system theory: where we are and where to go now,” Annual Rev. Control, 23, 207–219 (1999).

    Google Scholar 

  26. G. Cohen, S. Gaubert, and J.-P. Quadrat, “Duality and separation theorems in idempotent semimodules,” Linear Algebra Appl., 379, 395–422 (2004); arXiv:math.FA/0212294.

    MATH  MathSciNet  Google Scholar 

  27. G. Cohen, S. Gaubert, J.-P. Quadrat, and I. Singer, “Max-plus convex sets and functions,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 105–130.

    Google Scholar 

  28. G. Cohen and J.-P. Quadrat (eds.), 11th International Conference on Analysis and Optimization Systems, Lect. Notes Control Information Sci, 199, Springer (1994).

  29. R. A. Cuninghame-Green, Minimax Algebra, Springer Lect. Notes Econom. Math. Systems, 166, Springer-Verlag, Berlin-Heidelberg-New York (1979).

    MATH  Google Scholar 

  30. R. A. Cuninghame-Green, “Minimax algebra and applications,” Adv. Imag. Elect. Phys., 90, 1–121 (1995).

    Article  Google Scholar 

  31. R. Cuninghame-Green and P. Meijer, “An algebra for piesewise linear minimax problems,” Discr. Appl. Math., 2, 267–294 (1980).

    MATH  MathSciNet  Google Scholar 

  32. V. I. Danilov and G. A. Koshevoi, “Discrete convexity,” Zap. Nauchn. Semin. POMI, 312, 86–93 (2004).

    MATH  Google Scholar 

  33. V. Danilov, G. Koshevoi, and K. Murota, “Discrete convexity and equilibria in economics with indivisible goods and money,” Math. Soc. Sci., 11, 251–273 (2001).

    MathSciNet  Google Scholar 

  34. P. Dehornoy, I. Dynnikov, D. Rolfsen, and B. Wiest, Why are Braids Orderable?, Panoramas et Synthèses, 14, Société Math’ematique de France, Paris (2002).

    Google Scholar 

  35. P. Del Moral, “A survey of Maslov optimization theory,” in: Idempotent Analysis and Applications, V. N. Kolokoltsov and V. P. Maslov (eds.), Kluwer Acad. Publ., Dordrecht (1997), pp. 243–302 (Appendix).

    Google Scholar 

  36. P. Del Moral, Feynman-Kac Formulas. Genealogical and Interacting Particle Systems with Applications, Springer, New York (2004).

    Google Scholar 

  37. P. Del Moral and M. Doisy, “Maslov idempotent probability calculus. I, II,” Theor. Probab. Appl., 43, No. 4, 735–751 (1998); 44, No. 2, 384–400 (1999).

    Google Scholar 

  38. P. Del Moral and M. Doisy, “On the applications of Maslov optimization theory,” Math. Notes, 69, 232–244 (2001).

    MATH  MathSciNet  Google Scholar 

  39. M. Develin, “The moduli space of n tropically collinear points in R d,” arXiv:math.CO/0401224 (2004).

  40. M. Develin and B. Sturmfels, “Tropical convexity,” arXiv:math.MG/0308254 (2003).

  41. M. Develin, F. Santos, and B. Sturmfels, “On the rank of a tropical matrix,” arXiv:math.CO/0312114v2 (2004).

  42. A. Di Nola, B. Gerla, “Algebras of Lukasiewicz’s logic and their semiring reducts,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 131–144.

    Google Scholar 

  43. D. Dubois, H. Prade, and R. Sabbadin, “Decision-theory foundations of qualitative possibility theory,” European J. Oper. Res., 128, 459–478 (2001).

    MATH  MathSciNet  Google Scholar 

  44. P. S. Dudnikov and S. N. Samborskii, “Endomorphisms of semimodules over semirings with an idempotent operation,” Preprint Math. Inst. Ukr. Acad. Sci., Kiev (1987); Izv. Akad. Nauk SSSR, Ser. Mat., 55, No. 1, 93–109 (1991).

    Google Scholar 

  45. I. A. Dynnikov, “On a Yang-Baxter mapping and the Dehornoy ordering,” Uspekhi Mat. Nauk, 57, 151–152 (2002).

    MathSciNet  Google Scholar 

  46. I. A. Dynnikov and B. Wiest, “On the complexity of braids,” arXiv:math.GT/0403177 (2004).

  47. M. Einsiedler, M. Kapranov, and D. Lind, Non-archimedean amoebas and tropical varieties, arXiv:math.AG/0408311 (2004).

  48. R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965).

    MATH  Google Scholar 

  49. A. V. Finkelstein and M. A. Roytberg, “Computation of biopolymers: a general approach to different problems,” BioSystems, 30, 1–20 (1993).

    Google Scholar 

  50. W. H. Fleming, “Max-plus stochastic control,” in: Stochastic Theory and Control, B. Pasik-Duncan (ed.), Lect. Notes Control Inform. Sci., 280, Springer (2002), pp. 111–119.

  51. W. H. Fleming, “Max-plus stochastic processes,” Appl. Math. Optim., 48, 159–181 (2004).

    MathSciNet  Google Scholar 

  52. W. H. Fleming and W. M. McEneaney, “A max-plus-based algorithm for a Hamilton-Jacobi-Bellman equation of nonlinear filtering,” SIAM J. Control Optim., 38, 683–710 (2000).

    MATH  MathSciNet  Google Scholar 

  53. W. H. Fleming and W. M. McEneaney, “Max-plus approaches to continuous space control and dynamic programming,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 145–160.

    Google Scholar 

  54. W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York (1993).

    MATH  Google Scholar 

  55. V. V. Fock and A. B. Goncharov, “Moduli spaces of local systems and higher Teichmuller theory,” arXiv:math.AG/0311149 (2003).

  56. V. V. Fock and A. B. Goncharov, “Cluster ensembles, quantization and the dilogarithm,” arXiv:math.AG/0311245 (2003).

  57. S. Fomin and A. Zelevinsky, “Cluster algebras: Notes for the CDM-03 Conference,” in: Conference “Current Developments in Mathematics 2003” held at Harvard University on November 21-22, 2003, arXiv:math.RT/0311493v2 (2003).

  58. U. Frisch, S. Matarrese, R. Mohayaee, and A. Sobolevskii, “A reconstruction of the initial conditions of the Universe by optimal mass transportation,” Nature, 417, 260–262 (2002).

    Google Scholar 

  59. A. Gathmann and H. Markwig, “The Caporaso-Harris formula and plane relative Gromov-Witten invariants in tropical geometry,” arXiv:math.AG/0504392 (2005).

  60. A. Gathmann and H. Markwig, “The numbers of tropical plane curves through points in general position,” arXiv:math.AG/0406099 (2005).

  61. I. M. Gelfand, M. M. Kapranov, and A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston (1994).

    MATH  Google Scholar 

  62. K. Glazek, A Guide to the Literature on Semirings and Their Applications in Mathematics and Information Sciences. With complete bibliography. Kluwer Acad. Publ., Dordrecht (2002).

    MATH  Google Scholar 

  63. J. A. Goguen, “L-fuzzy sets,” J. Math. Anal. Appl., 18, 145–174 (1967).

    MATH  MathSciNet  Google Scholar 

  64. J. S. Golan, Semirings and Their Applications, Kluwer Acad. Publ., Dordrecht (1999).

    MATH  Google Scholar 

  65. J. S. Golan, Power Algebras Over Semirings, Kluwer Acad. Publ., Dordrecht (1999).

    MATH  Google Scholar 

  66. J. S. Golan, Semirings and Affine Equations Over Them: Theory and Applications, Kluwer Acad. Publ., Dordrecht (2003).

    MATH  Google Scholar 

  67. M. Gondran and M. Minoux, Graphes et Algorithmes, Éditions Eyrolles, Paris (1979, 1988).

    MATH  Google Scholar 

  68. M. Gondran and M. Minoux, Graphes, Dioïdes et Semi-Anneaux, Éditions TEC&DOC, Paris (2001).

    Google Scholar 

  69. J. Gunawardena (ed.), Idempotency, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge (1998).

    MATH  Google Scholar 

  70. J. Gunawardena, “An introduction to idempotency,” in: Idempotency, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge (1998)[69], pp. 1–49.

    Google Scholar 

  71. E. Hopf, “The partial differential equation u t +uu x = μu xx ,” Comm. Pure Appl. Math., 3, 201–230 (1950).

    MATH  MathSciNet  Google Scholar 

  72. I. Itenberg, V. Kharlamov, and E. Shustin, “Welschinger invariant and enumeration of real rational curves,” Int. Math. Res. Not., 49, 26–39 (2003).

    MathSciNet  Google Scholar 

  73. I. Itenberg, V. Kharlamov, and E. Shustin, Appendix to “Welschinger invariants and enumeration of real rational curves,” arXiv:math.AG/0312142v2 (2003).

  74. I. Itenberg, V. Kharlamov, and E. Shustin, “Logarithmic equivalence of Welschinger and Gromov-Witten invariants,” Uspekhi Mat. Nauk, 59, 85–110 (2004); arXiv:math.AG/0407188.

    MathSciNet  Google Scholar 

  75. I. Itenberg, V. Kharlamov, and E. Shustin, “Logarithmic asymptotics of the genus zero Gromov-Witten invariants of the blown up plane,” Geom. Topol., 9, 483–491 (2005); see also arXiv:math.AG/0412533 (2004).

    MATH  MathSciNet  Google Scholar 

  76. Z. Izhakian, “Duality of tropical curves,” arXiv:math.AG/0503691 (2005).

  77. Z. Izhakian, “Tropical arithmetic and algebra of tropical matrices,” arXiv:math.AG/0505458 (2005).

  78. M. Joswig, “Tropical halfspaces,” arXiv:math.CO/0312068 (2004).

  79. L. V. Kantorovich, “On the translocation of masses,” Dokl. Akad. Nauk SSSR, 37, 227–229 (1942).

    Google Scholar 

  80. M. M. Kapranov, “Amoebas over non-Archimedean fields,” Preprint (2000).

  81. Y. Katsov, “On flat semimodules over semirings,” Algebra Univers., 51, 287–299 (2004).

    MATH  MathSciNet  Google Scholar 

  82. K. Khanin, D. Khmelëv, and A. Sobolevskii, “A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 161–180.

    Google Scholar 

  83. K. Khanin, D. Khmelëv, and A. Sobolevskii, “On velocities of Lagrangian minimizers,” Moscow Math. J., 5, No. 1, 157–169 (2005).

    MATH  Google Scholar 

  84. K. H. Kim and F. W. Roush, “Inclines and incline matrices: a survey,” Linear Algebra Appl., 379, 457–473 (2004).

    MATH  MathSciNet  Google Scholar 

  85. K. H. Kim and F. W. Roush, “Kapranov rank vs. tropical rank,” arXiv:math.CO/0503044 (2005).

  86. K. H. Kim and F. W. Roush, “Factorization of polynomials in one variable over the tropical semiring,” arXiv:math.CO/0501167 (2005).

  87. A. N. Kirillov, “Introduction to tropical combinatorics,” in: Physics and Combinatorics. Proceedings of the Nagoya 2000 2nd International Workshop, A. N. Kirillov and N. Liskova (eds.), World Scientific, Singapore (2001), pp. 82–150.

    Google Scholar 

  88. S. C. Kleene, “Representation of events in nerve sets and finite automata,” in: Automata Studies, J. McCarthy and C. Shannon (eds), Princeton Univ. Press, Princeton (1956), pp. 3–40.

    Google Scholar 

  89. E. P. Klement, R. Mesiar, and E. Pap, Triangular Norms, Kluwer Acad. Publ., Dordrecht (2000).

    MATH  Google Scholar 

  90. E. P. Klement and E. Pap (eds.), Mathematics of Fuzzy Systems, 25th Linz Seminar on Fuzzy Set Theory, Linz, Austria, February 3–7 (2004), Abstracts, J. Kepler Univ., Linz (2004).

  91. V. N. Kolokoltsov, “Stochastic Hamilton-Jacobi-Bellman equations and stochastic Hamiltonian systems,” J. Dynam. Control Systems, 2, 299–319 (1996).

    MATH  MathSciNet  Google Scholar 

  92. V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, Lect. Notes Math., 1724, Springer, Berlin (2000).

    MATH  Google Scholar 

  93. V. N. Kolokoltsov, “Idempotency structures in optimization,” J. Math. Sci., 104, 847–880 (2001).

    Google Scholar 

  94. V. N. Kolokoltsov, “Small diffusion and fast dying out asymptotics for superprocesses as non-Hamiltonian quasiclassics for evalution equations,” Electr. J. Probab., 6, Paper 21 (2001); http://www.math.washington.edu/∼ejpecp).

  95. V. N. Kolokoltsov and O. A. Malafeyev, “A turnpike theorem in conflict-control processes with many participants,” in: Conflict Models in Economics and Finance, O. Malafeyev (ed.), St.Petersburg Univ. Press, St.Petersburg (1997).

    Google Scholar 

  96. V. Kolokoltsov and V. Maslov, Idempotent Analysis and Applications, Kluwer Acad. Publ., Dordrecht (1997).

    MATH  Google Scholar 

  97. V. Kolokoltsov and A. Tyukov, “Small time amd semiclassical asymptotics for stochastic heat equation driven by Lévy noise,” Stoch. Stoch. Rep., 75, 1–38 (2003).

    MathSciNet  Google Scholar 

  98. M. Kontsevich and Y. Soibelman, “Homological mirror symmetry and torus fibration,” in: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, New Jersey (2001), pp. 203–263.

    Google Scholar 

  99. M. Kontsevich and Y. Soibelman, “Affine structures and non-Archimedean analytic spaces,” arXiv:math.AG/0406564 (2004).

  100. G. L. Litvinov, “Dequantization of mathematics, idempotent semirings and fuzzy sets,” in: Mathematics of Fuzzy Systems, 25th Linz Seminar on Fuzzy Set Theory, Linz, Austria, February 3–7 (2004)[90], pp. 113–117.

  101. G. L. Litvinov, “Maslov dequantization, idempotent, and tropical mathematics: a very brief introduction,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 1–17.

    Google Scholar 

  102. G. L. Litvinov and V. P. Maslov, “Correspondence principle for idempotent calculus and some computer applications,” Preprint IHES/M/95/33, Institut des Hautes Études Scientifiques, Bures-sur-Yvette (1995); arXiv:math.GM/0101021.

    Google Scholar 

  103. G. L. Litvinov and V. P. Maslov, “Idempotent mathematics: correspondence principle and applications,” Russian Math. Surveys, 51, 1210–1211 (1996).

    MATH  MathSciNet  Google Scholar 

  104. G. L. Litvinov and V. P. Maslov, “The correspondence principle for idempotent calculus and some computer applications,” in: Idempotency, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge (1998)[69], pp. 420–443.

    Google Scholar 

  105. G. L. Litvinov and V. P. Maslov (eds.), Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005).

    MATH  Google Scholar 

  106. G. L. Litvinov and E. V. Maslova, “Universal numerical algorithms and their software implementation,” Program. Comput. Software, 26, 275–280 (2000); arXiv:math.SC/0102114.

    MATH  MathSciNet  Google Scholar 

  107. G. L. Litvinov, V. P. Maslov, and A. Ya. Rodionov, “A unifying approach to software and hardware design for scientific calculations and idempotent mathematics,” International Sophus Lie Centre, Moscow (2000); arXiv:math.SC/0101069.

    Google Scholar 

  108. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, “Linear functionals on idempotent spaces: an algebraic approach,” Dokl. Math., 58, 389–391 (1998); arXiv:math.FA/0012268.

    Google Scholar 

  109. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, “Tensor products of idempotent semimodules. An algebraic approach,” Math. Notes, 65, 497–489 (1999); arXiv:math.FA/0101153.

    MathSciNet  Google Scholar 

  110. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, “Idempotent functional analysis. An algebraic approach,” Math. Notes, 69, 696–729 (2001); arXiv:math.FA/0009128.

    MATH  MathSciNet  Google Scholar 

  111. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, “Idempotent (asymptotic) analysis and the representation theory,” in: Asymptotic Combinatorics with Applications to Mathematical Physics, V. A. Malyshev and A. M. Vershik (eds.), Kluwer Academic Publ., Dordrecht (2002), pp. 267–278; arXiv:math.RT/0206025.

    Google Scholar 

  112. G. L. Litvinov and G. B. Shpiz, “Nuclear semimodules and kernel theorems in idempotent analysis: an algebraic approach,” Dokl. Math., 66, 197–199 (2002); arXiv:math.FA/0202026.

    MATH  Google Scholar 

  113. G. L. Litvinov and G. B. Shpiz, “The dequantization transform and generalized Newton polytopes,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 181–186.

    Google Scholar 

  114. G. L. Litvinov and A. N. Sobolevskii, “Exact interval solutions of the discrete Bellman equation and polynomial complexity of problems in interval idempotent linear algebra,” Dokl. Math., 62, 199–201 (2000); arXiv:math.LA/0101041.

    Google Scholar 

  115. G. L. Litvinov and A. N. Sobolevskii, “Idempotent interval analysis and optimization problems,” Reliab. Comput., 7, 353–377 (2001); arXiv:math.SC/0101080.

    MATH  MathSciNet  Google Scholar 

  116. P. Loreti and M. Pedicini, “An object-oriented approach to idempotent analysis: integral equations as optimal control problems,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 187–208.

    Google Scholar 

  117. P. Lotito, J.-P. Quadrat, and E. Mancinelli, “Traffic assignment and Gibbs-Maslov semirings,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 209–220.

    Google Scholar 

  118. G. G. Magaril-Il’yaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, Transl. Math. Monographs, 222, Amer. Math. Soc., Providence, Rhode Insland (2003).

    MATH  Google Scholar 

  119. V. P. Maslov, “New superposition principle for optimization problems,” in: Seminaire sur les Equations aux Dérivées Partielles 1985/86, Centre Math. de l’École Polytechnique, Palaiseau (1986), exposé 24.

    Google Scholar 

  120. V. P. Maslov, “On a new superposition principle for optimization problems,” Uspekhi Mat. Nauk, 42, No. 3, 39–48 (1987).

    MathSciNet  Google Scholar 

  121. V. P. Maslov, Méthodes Opératorielles, Mir, Moscow (1987).

    MATH  Google Scholar 

  122. V. P. Maslov and V. N. Kolokoltsov, Idempotent Analysis and Its Application in Optimal Control, Nauka, Moscow (1994).

    Google Scholar 

  123. V. P. Maslov and S. N. Samborskii (eds), Idempotent Analysis, Adv. Sov. Math., 13, Amer. Math. Soc., Providence, Rhode Insland (1992).

    MATH  Google Scholar 

  124. V. P. Maslov and K. A. Volosov (eds.), Mathematical Aspects of Computer Engineering [in Russian], Mir, Moscow (1988).

    Google Scholar 

  125. D. McCaffrey, “Viscosity solutions on Lagrangian manifolds and connections with tunneling operators,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 221–238.

    Google Scholar 

  126. G. Mikhalkin, “Amoebas of algebraic varieties,” Notes for the Real Algebraic and Analytic Geometry Congress, June 11–15 (2001), Rennes, France; arXiv:math.AG/0108225.

  127. G. Mikhalkin, “Counting curves via lattice path in polygons,” C. R. Acad. Sci. Paris, 336, 629–634 (2003).

    MATH  MathSciNet  Google Scholar 

  128. G. Mikhalkin, “Amoebas of algebraic varieties and tropical geometry,” in: Different Faces in Geometry, to appear, arXiv:math.AG/0403015 (2004).

  129. G. Mikhalkin, “Enumerative tropical algebraic geometry in R 2,” J. Amer. Math. Soc., 18, No. 2, 313–377 (2005); arXiv:math.AG/0312530.

    MATH  MathSciNet  Google Scholar 

  130. E. Nelson, “Probability theory and Euclidean field theory,” Lect. Notes Phys., 25, 94–124 (1973).

    Google Scholar 

  131. T. Nishinou and B. Siebert, “Toric degenerations of toric varieties and tropical curves,” arXiv:math.AG/0409060 (2004).

  132. M. Noumi and Y. Yamada, “Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions,” arXiv:math-ph/0203030 (2002).

  133. R. D. Nussbaum, “Convergence of iterates of a nonlinear operator arising in statistical mechanics,” Nonlinearity, 4, 1223–1240 (1991).

    MATH  MathSciNet  Google Scholar 

  134. L. Pachter and B. Sturmfels, “Tropical geometry of statistical models,” arXiv:q-bio.QM/0311009v2 (2004).

  135. L. Pachter and B. Sturmfels, “The mathematics of phylogenomics,” arXiv:math.ST/0409132 (2004).

  136. S. N. N. Pandit, “A new matrix calculus,” SIAM J. Appl. Math., 9, 632–639 (1961).

    MATH  MathSciNet  Google Scholar 

  137. E. Pap, “Pseudo-additive measures and their applications,” in: Handbook of Measure Theory, E. Pap (ed.), Elsevier, Amsterdam (2002), pp. 1403–1465.

    Google Scholar 

  138. E. Pap, “Applications of the generated pseudo-analysis to nonlinear partial differential equations,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 239–260.

    Google Scholar 

  139. M. Passare and A. Tsikh, “Amoebas: their spines and their contours,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 275–288.

    Google Scholar 

  140. J. E. Pin, “Tropical semirings,” in: Idempotency, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge (1998)[69], pp. 50–60.

    Google Scholar 

  141. A. A. Puhalskii, Large Deviations and Idempotent Probability, Chapman and Hall/CRC Press, London-Boka Raton, Florida (2001).

    MATH  Google Scholar 

  142. A. A. Puhalskii, “On large deviations convergence of invariant measures,” J. Theor. Probab., 16, 689–724 (2003).

    MATH  MathSciNet  Google Scholar 

  143. J.-P. Quadrat, “Théorèmes asymptotiques en programmation dynamique,” C. R. Acad. Sci. Paris, 311, 745–748 (1990).

    MATH  MathSciNet  Google Scholar 

  144. J.-P. Quadrat and Max-Plus Working Group, “Max-plus algebra and applications to system theory and optimal control,” in: Proceedings of the Internatational Congress of Mathematicians, Zürich, 1994, Vol. II, Birkhäuser, Basel (1995), pp. 1511–1522.

    Google Scholar 

  145. J.-P. Quadrat and Max-Plus Working Group, “Min-plus linearity and statistical mechanics,” Markov Process. Related Fields, 3, 565–587 (1997).

    MATH  MathSciNet  Google Scholar 

  146. J. Richter-Gebert, B. Sturmfels, and T. Theobald, “First steps in tropical geometry,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 289–318; arXiv:math.AG/0306366.

    Google Scholar 

  147. K. I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics Series, 234, Longman, Harlow (1990).

    MATH  Google Scholar 

  148. K. I. Rosenthal, The Theory of Quantaloids, Pitman Research Notes in Mathematics Series, 348, Longman, Harlow (1996).

    MATH  Google Scholar 

  149. I. V. Roublev, “On minimax and idempotent generalized weak solutions to the Hamilton-Jacobi equation,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 319–338.

    Google Scholar 

  150. M. A. Roytberg, “Fast algorithm for optimal aligning of symbol sequences,” DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 8, 113–126 (1992).

    MathSciNet  Google Scholar 

  151. H. Rullgåard, “Polynomial amoebas and convexity,” Preprint, Stockholm University (2001).

  152. S. N. Samborskii and A. A. Tarashchan, “The Fourier transform and semirings of Pareto sets,” in: Idempotent Analysis, Adv. Sov. Math., 13, Amer. Math. Soc., Providence, Rhode Insland (1992)[123], pp. 139–150.

    Google Scholar 

  153. E. Schrödinger, “Quantization as an eigenvalue problem,” Ann. Phys., 364, 361–376 (1926).

    Google Scholar 

  154. G. B. Shpiz, “Solution of algebraic equations over idempotent semifields,” Uspekhi Mat. Nauk, 55, 185–186 (2000).

    MathSciNet  Google Scholar 

  155. E. Shustin, “Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry,” arXiv:math.AG/0211278.

  156. E. Shustin, “A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces,” arXiv:math.AG/0504390 (2004).

  157. I. Singer, Abstract Convex Analysis, John Wiley & Sons Inc., New York (1997).

    MATH  Google Scholar 

  158. I. Singer, “Some relations between linear mappings and conjugations in idempotent analysis,” J. Math. Sci., 115, 2610–2630 (2003).

    MATH  MathSciNet  Google Scholar 

  159. A. N. Sobolevskii, “Aubry-Mather theory and idempotent eigenfunctions of Bellman operator,” Commun. Contemp. Math., 1, 517–533 (1999).

    MATH  MathSciNet  Google Scholar 

  160. A. N. Sobolevskii, “Periodic solutions of the Hamilton-Jacobi equation with a periodic nonhomogeneity, and the Aubry-Mather theory,” Mat. Sb., 190, 87–104 (1999).

    MathSciNet  Google Scholar 

  161. F. Sottile, “Tropical interpolation,” arXiv:math.AG/0501146 (2005).

  162. D. E. Speyer, “Tropical linear spaces,” arXiv:math.CO/0410455 (2004).

  163. D. Speyer and B. Sturmfels, “The tropical Grassmannian,” arXiv:math.AG/0304218 (2003).

  164. D. Speyer and B. Sturmfels, “Tropical mathematics,” arXiv:math.CO/0408099 (2004).

  165. D. Speyer and L. K. Williams, “The tropical totally positive Grassmannian,” arXiv:math.CO/0312297 (2003).

  166. B. Sturmfels, Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics, 97, Amer. Math. Soc., Providence, Phode Island (2002).

    MATH  Google Scholar 

  167. A. I. Subbotin, Generalized Solutions of First Order PDEs: The Dynamic Optimization Perspective, Birkhäuser, Boston (1996).

    Google Scholar 

  168. A. Szenes and M. Vergne, “Mixed toric residues and tropical degenerations,” arXiv:math.AG/0410064 (2004).

  169. J. Tevelev, “Tropical compactifications,” arXiv:math.AG/0412329 (2004).

  170. T. Theobald, “On the frontiers of polynomial computations in tropical geometry,” arXiv:math.CO/0411012 (2004).

  171. M. D. Vigeland, “The group law on a tropical elliptic curve,” arXiv:math.AG/0411485 (2004).

  172. O. Viro, “Dequantization of real algebraic geometry on a logarithmic paper,” in: 3rd European Congress of Mathematics, Barcelona 2000, Vol. I, Birkhäuser, Basel (2001), pp. 135–146; arXiv:math/0005163.

    Google Scholar 

  173. O. Viro, “What is an amoeba?,” Notices Amer. Math. Soc., 49, 916–917 (2002).

    Google Scholar 

  174. N. N. Vorobjev, “The extremal matrix algebra,” Dokl. Akad. Nauk SSSR, 4, 1220–1223 (1963).

    Google Scholar 

  175. N. N. Vorobjev, “Extremal algebra of positive matrices,” Elektr. Inform. Kybern., 3, 39–57 (1967).

    Google Scholar 

  176. N. N. Vorobjev, “Extremal algebra of nonnegative matrices,” Elektr. Inform. Kybern., 6, 302–312 (1970).

    Google Scholar 

  177. E. Wagneur, “Dequantization: semi-direct sums of idempotent semimodules,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 339–357.

    Google Scholar 

  178. C. Walsh, “Minimum representing measures in idempotent analysis,” arXiv:math.MG/0503716 (2005).

  179. J. van der Woude and G. J. Olsder, “On (min, max, +)-inequalities,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 358–363.

    Google Scholar 

  180. L. A. Zadeh, “Fuzzy sets,” Inform. Control, 8, 338–353 (1965).

    MATH  MathSciNet  Google Scholar 

  181. L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, 1, 3–28 (1978).

    MATH  MathSciNet  Google Scholar 

  182. K. Zimmermann, Extremální Algebra [in Czech], Ekonomický ùstav ČSAV, Praha, 46 (1976).

    Google Scholar 

  183. K. Zimmermann, “A general separation theorem in extremal algebras,” Ekonom.-Mat. Obzor, 13, 179–210 (1977).

    MathSciNet  Google Scholar 

  184. K. Zimmermann, “Extremally convex functions,” Wiss. Z. Päd. Hochschule “N. K. Krupskaya,” 17, 147–158 (1979).

    Google Scholar 

  185. K. Zimmermann, “A generalization of convex functions,” Ekonom.-Mat. Obzor, 15, 147–158 (1979).

    MathSciNet  Google Scholar 

  186. K. Zimmermann, “Solution of some max-separable optimization problems with inequality constraints,” in: Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005)[105], pp. 363–370.

    Google Scholar 

  187. U. Zimmermann, “Linear and combinatorial optimization in ordered algebraic structures,” Ann. Discrete Math., 10, 1–380 (1981).

    Article  Google Scholar 

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  1. Independent University of Moscow, Moscow, Russia

    G. L. Litvinov

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To Anatoly Vershik with admiration and gratitude

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 145–182.

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Litvinov, G.L. Maslov dequantization, idempotent and tropical mathematics: A brief introduction. J Math Sci 140, 426–444 (2007). https://doi.org/10.1007/s10958-007-0450-5

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