Maslov dequantization, idempotent and tropical mathematics: A brief introduction
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M. Akian, “Densities of idempotent measures and large deviations,” Trans. Amer. Math. Soc., 351, 4515–4543 (1999).
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.
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.
M. Akian, S. Gaubert, and C. Walsh, “The max-plus Martin boundary,” arXiv:math.MG/0412408 (2004).
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.
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.
D. Alessandrini, “Amoebas, tropical varieties and compactification of Teichmüller spaces,” arXiv:math.AG/0505269 (2005).
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.
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).
M. Bardi, I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston-Basel-Berlin (1997).
A. Berenstein, S. Fomin, and A. Zelevinsky, “Parametrizations of canonical bases and totally positive matrices,” Adv. Math., 122, 49–149 (1996).
A. Berenstein and A. Zelevinsky, “Tenzor product multiplicities, canonical bases and totally positive varieties,” Invent. Math., 143, 77–128 (2001).
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.
F. Block and J. Yu, “Tropical convexity via cellular resolutions,” arXiv:math.MG/0503279 (2005).
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).
P. Butkovič, “Strong regularity of matrices — a survey of results,” Discrete Appl. Math., 48, 45–68 (1994).
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.
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).
B. A. Carré, Graphs and Networks, The Clarendon Press/Oxford Univ. Press, Oxford (1979).
K. Cechlárová and R. A. Cuninghame-Green, “Interval systems of max-separable linear equations,” Linear Algebra Appl., 340, 215–224 (2002).
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).
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).
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.
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.
G. Cohen and J.-P. Quadrat (eds.), 11th International Conference on Analysis and Optimization Systems, Lect. Notes Control Information Sci, 199, Springer (1994).
R. A. Cuninghame-Green, Minimax Algebra, Springer Lect. Notes Econom. Math. Systems, 166, Springer-Verlag, Berlin-Heidelberg-New York (1979).
R. A. Cuninghame-Green, “Minimax algebra and applications,” Adv. Imag. Elect. Phys., 90, 1–121 (1995).
R. Cuninghame-Green and P. Meijer, “An algebra for piesewise linear minimax problems,” Discr. Appl. Math., 2, 267–294 (1980).
V. I. Danilov and G. A. Koshevoi, “Discrete convexity,” Zap. Nauchn. Semin. POMI, 312, 86–93 (2004).
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).
P. Dehornoy, I. Dynnikov, D. Rolfsen, and B. Wiest, Why are Braids Orderable?, Panoramas et Synthèses, 14, Société Math’ematique de France, Paris (2002).
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).
P. Del Moral, Feynman-Kac Formulas. Genealogical and Interacting Particle Systems with Applications, Springer, New York (2004).
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).
P. Del Moral and M. Doisy, “On the applications of Maslov optimization theory,” Math. Notes, 69, 232–244 (2001).
M. Develin, “The moduli space of n tropically collinear points in R d ,” arXiv:math.CO/0401224 (2004).
M. Develin, F. Santos, and B. Sturmfels, “On the rank of a tropical matrix,” arXiv:math.CO/0312114v2 (2004).
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.
D. Dubois, H. Prade, and R. Sabbadin, “Decision-theory foundations of qualitative possibility theory,” European J. Oper. Res., 128, 459–478 (2001).
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).
I. A. Dynnikov, “On a Yang-Baxter mapping and the Dehornoy ordering,” Uspekhi Mat. Nauk, 57, 151–152 (2002).
I. A. Dynnikov and B. Wiest, “On the complexity of braids,” arXiv:math.GT/0403177 (2004).
M. Einsiedler, M. Kapranov, and D. Lind, Non-archimedean amoebas and tropical varieties, arXiv:math.AG/0408311 (2004).
R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965).
A. V. Finkelstein and M. A. Roytberg, “Computation of biopolymers: a general approach to different problems,” BioSystems, 30, 1–20 (1993).
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.
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).
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.
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York (1993).
V. V. Fock and A. B. Goncharov, “Moduli spaces of local systems and higher Teichmuller theory,” arXiv:math.AG/0311149 (2003).
V. V. Fock and A. B. Goncharov, “Cluster ensembles, quantization and the dilogarithm,” arXiv:math.AG/0311245 (2003).
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).
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).
A. Gathmann and H. Markwig, “The Caporaso-Harris formula and plane relative Gromov-Witten invariants in tropical geometry,” arXiv:math.AG/0504392 (2005).
A. Gathmann and H. Markwig, “The numbers of tropical plane curves through points in general position,” arXiv:math.AG/0406099 (2005).
I. M. Gelfand, M. M. Kapranov, and A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston (1994).
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).
J. S. Golan, Semirings and Affine Equations Over Them: Theory and Applications, Kluwer Acad. Publ., Dordrecht (2003).
M. Gondran and M. Minoux, Graphes et Algorithmes, Éditions Eyrolles, Paris (1979, 1988).
M. Gondran and M. Minoux, Graphes, Dioïdes et Semi-Anneaux, Éditions TEC&DOC, Paris (2001).
J. Gunawardena (ed.), Idempotency, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge (1998).
J. Gunawardena, “An introduction to idempotency,” in: Idempotency, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge (1998)[69], pp. 1–49.
E. Hopf, “The partial differential equation u t +uu x = μu xx ,” Comm. Pure Appl. Math., 3, 201–230 (1950).
I. Itenberg, V. Kharlamov, and E. Shustin, “Welschinger invariant and enumeration of real rational curves,” Int. Math. Res. Not., 49, 26–39 (2003).
I. Itenberg, V. Kharlamov, and E. Shustin, Appendix to “Welschinger invariants and enumeration of real rational curves,” arXiv:math.AG/0312142v2 (2003).
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.
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).
Z. Izhakian, “Duality of tropical curves,” arXiv:math.AG/0503691 (2005).
Z. Izhakian, “Tropical arithmetic and algebra of tropical matrices,” arXiv:math.AG/0505458 (2005).
M. Joswig, “Tropical halfspaces,” arXiv:math.CO/0312068 (2004).
L. V. Kantorovich, “On the translocation of masses,” Dokl. Akad. Nauk SSSR, 37, 227–229 (1942).
M. M. Kapranov, “Amoebas over non-Archimedean fields,” Preprint (2000).
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.
K. Khanin, D. Khmelëv, and A. Sobolevskii, “On velocities of Lagrangian minimizers,” Moscow Math. J., 5, No. 1, 157–169 (2005).
K. H. Kim and F. W. Roush, “Inclines and incline matrices: a survey,” Linear Algebra Appl., 379, 457–473 (2004).
K. H. Kim and F. W. Roush, “Kapranov rank vs. tropical rank,” arXiv:math.CO/0503044 (2005).
K. H. Kim and F. W. Roush, “Factorization of polynomials in one variable over the tropical semiring,” arXiv:math.CO/0501167 (2005).
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.
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.
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).
V. N. Kolokoltsov, “Stochastic Hamilton-Jacobi-Bellman equations and stochastic Hamiltonian systems,” J. Dynam. Control Systems, 2, 299–319 (1996).
V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, Lect. Notes Math., 1724, Springer, Berlin (2000).
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 ).
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).
V. Kolokoltsov and V. Maslov, Idempotent Analysis and Applications, Kluwer Acad. Publ., Dordrecht (1997).
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).
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.
M. Kontsevich and Y. Soibelman, “Affine structures and non-Archimedean analytic spaces,” arXiv:math.AG/0406564 (2004).
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.
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.
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.
G. L. Litvinov and V. P. Maslov, “Idempotent mathematics: correspondence principle and applications,” Russian Math. Surveys, 51, 1210–1211 (1996).
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.
G. L. Litvinov and V. P. Maslov (eds.), Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, Rhode Island (2005).
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.
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.
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.
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.
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.
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.
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.
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.
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.
G. L. Litvinov and A. N. Sobolevskii, “Idempotent interval analysis and optimization problems,” Reliab. Comput., 7, 353–377 (2001); arXiv:math.SC/0101080.
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.
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.
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).
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.
V. P. Maslov, “On a new superposition principle for optimization problems,” Uspekhi Mat. Nauk, 42, No. 3, 39–48 (1987).
V. P. Maslov, Méthodes Opératorielles, Mir, Moscow (1987).
V. P. Maslov and V. N. Kolokoltsov, Idempotent Analysis and Its Application in Optimal Control, Nauka, Moscow (1994).
V. P. Maslov and S. N. Samborskii (eds), Idempotent Analysis, Adv. Sov. Math., 13, Amer. Math. Soc., Providence, Rhode Insland (1992).
V. P. Maslov and K. A. Volosov (eds.), Mathematical Aspects of Computer Engineering [in Russian], Mir, Moscow (1988).
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.
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.
G. Mikhalkin, “Counting curves via lattice path in polygons,” C. R. Acad. Sci. Paris, 336, 629–634 (2003).
G. Mikhalkin, “Amoebas of algebraic varieties and tropical geometry,” in: Different Faces in Geometry, to appear, arXiv:math.AG/0403015 (2004).
G. Mikhalkin, “Enumerative tropical algebraic geometry in R 2,” J. Amer. Math. Soc., 18, No. 2, 313–377 (2005); arXiv:math.AG/0312530.
T. Nishinou and B. Siebert, “Toric degenerations of toric varieties and tropical curves,” arXiv:math.AG/0409060 (2004).
M. Noumi and Y. Yamada, “Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions,” arXiv:math-ph/0203030 (2002).
R. D. Nussbaum, “Convergence of iterates of a nonlinear operator arising in statistical mechanics,” Nonlinearity, 4, 1223–1240 (1991).
L. Pachter and B. Sturmfels, “Tropical geometry of statistical models,” arXiv:q-bio.QM/0311009v2 (2004).
L. Pachter and B. Sturmfels, “The mathematics of phylogenomics,” arXiv:math.ST/0409132 (2004).
E. Pap, “Pseudo-additive measures and their applications,” in: Handbook of Measure Theory, E. Pap (ed.), Elsevier, Amsterdam (2002), pp. 1403–1465.
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.
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.
J. E. Pin, “Tropical semirings,” in: Idempotency, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge (1998)[69], pp. 50–60.
A. A. Puhalskii, Large Deviations and Idempotent Probability, Chapman and Hall/CRC Press, London-Boka Raton, Florida (2001).
A. A. Puhalskii, “On large deviations convergence of invariant measures,” J. Theor. Probab., 16, 689–724 (2003).
J.-P. Quadrat, “Théorèmes asymptotiques en programmation dynamique,” C. R. Acad. Sci. Paris, 311, 745–748 (1990).
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.
J.-P. Quadrat and Max-Plus Working Group, “Min-plus linearity and statistical mechanics,” Markov Process. Related Fields, 3, 565–587 (1997).
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.
K. I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics Series, 234, Longman, Harlow (1990).
K. I. Rosenthal, The Theory of Quantaloids, Pitman Research Notes in Mathematics Series, 348, Longman, Harlow (1996).
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.
M. A. Roytberg, “Fast algorithm for optimal aligning of symbol sequences,” DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 8, 113–126 (1992).
H. Rullgåard, “Polynomial amoebas and convexity,” Preprint, Stockholm University (2001).
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.
G. B. Shpiz, “Solution of algebraic equations over idempotent semifields,” Uspekhi Mat. Nauk, 55, 185–186 (2000).
E. Shustin, “Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry,” arXiv:math.AG/0211278.
E. Shustin, “A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces,” arXiv:math.AG/0504390 (2004).
I. Singer, Abstract Convex Analysis, John Wiley & Sons Inc., New York (1997).
I. Singer, “Some relations between linear mappings and conjugations in idempotent analysis,” J. Math. Sci., 115, 2610–2630 (2003).
A. N. Sobolevskii, “Aubry-Mather theory and idempotent eigenfunctions of Bellman operator,” Commun. Contemp. Math., 1, 517–533 (1999).
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).
F. Sottile, “Tropical interpolation,” arXiv:math.AG/0501146 (2005).
D. E. Speyer, “Tropical linear spaces,” arXiv:math.CO/0410455 (2004).
D. Speyer and B. Sturmfels, “Tropical mathematics,” arXiv:math.CO/0408099 (2004).
D. Speyer and L. K. Williams, “The tropical totally positive Grassmannian,” arXiv:math.CO/0312297 (2003).
B. Sturmfels, Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics, 97, Amer. Math. Soc., Providence, Phode Island (2002).
A. I. Subbotin, Generalized Solutions of First Order PDEs: The Dynamic Optimization Perspective, Birkhäuser, Boston (1996).
A. Szenes and M. Vergne, “Mixed toric residues and tropical degenerations,” arXiv:math.AG/0410064 (2004).
J. Tevelev, “Tropical compactifications,” arXiv:math.AG/0412329 (2004).
T. Theobald, “On the frontiers of polynomial computations in tropical geometry,” arXiv:math.CO/0411012 (2004).
M. D. Vigeland, “The group law on a tropical elliptic curve,” arXiv:math.AG/0411485 (2004).
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.
O. Viro, “What is an amoeba?,” Notices Amer. Math. Soc., 49, 916–917 (2002).
N. N. Vorobjev, “The extremal matrix algebra,” Dokl. Akad. Nauk SSSR, 4, 1220–1223 (1963).
N. N. Vorobjev, “Extremal algebra of positive matrices,” Elektr. Inform. Kybern., 3, 39–57 (1967).
N. N. Vorobjev, “Extremal algebra of nonnegative matrices,” Elektr. Inform. Kybern., 6, 302–312 (1970).
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.
C. Walsh, “Minimum representing measures in idempotent analysis,” arXiv:math.MG/0503716 (2005).
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.
L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, 1, 3–28 (1978).
K. Zimmermann, Extremální Algebra [in Czech], Ekonomický ùstav ČSAV, Praha, 46 (1976).
K. Zimmermann, “A general separation theorem in extremal algebras,” Ekonom.-Mat. Obzor, 13, 179–210 (1977).
K. Zimmermann, “Extremally convex functions,” Wiss. Z. Päd. Hochschule “N. K. Krupskaya,” 17, 147–158 (1979).
K. Zimmermann, “A generalization of convex functions,” Ekonom.-Mat. Obzor, 15, 147–158 (1979).
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.