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## Tropical formulae for summation over a part of SL(2,Z)

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Abstract Let f(a,b,c,d) = a2+b2 + c2+d2 − (a+c)2+(b+d)2, let

(a,b,c,d) stand for a,b,c,d ∈ Z0 such that ad − bc = 1. Defines

In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one.

We prove that F(s) converges when s > 1 and diverges at s = 1/2. We also prove that

1=1,(a,b,c,d) (a+c)2(b+d)2(a+b+c+d)2 3

Toric geometry exhibited a profound relation between algebra and topology on one side and combinatorics and convex geometry on the other side. In the last decades, the interplay between algebraic and convex geometry has been explored and used successfully in a much more general setting: first, for varieties with an algebraic group action (such as spherical varieties) and recently for all algebraic varieties (construction of Newton-Okounkov bodies). The main goal of the conference is to survey recent developments in these directions. Main topics of the conference are: Theory of Newton polytopes and Newton-Okounkov bodies; Toric geometry, geometry of spherical varieties, Schubert calculus, geometry of moduli spaces; Tropical geometry and convex geometry; Real algebraic geometry and fewnomial theory; Polynomial vector fields and the Hilbert 16th problem.

Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation.

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

Suppose that there exists a hypersurface with the Newton polytope Δ, which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ from below. As a particular application of our method we consider a planar algebraic curve *C* which passes through generic points p1,…,pn with prescribed multiplicities m1,…,mn. Suppose that the minimal lattice width ω(Δ) of the Newton polygon Δ of the curve *C* is at least max(mi). Using tropical floor diagrams (a certain degeneration of p1,…,pn on a horizontal line) we prove that

area(Δ)≥12n∑i=1m2i−S, where S=12max(n∑i=1s2i|si≤mi,n∑i=1si≤ω(Δ)).

In the case m1=m2=⋯=m≤ω(Δ) this estimate becomes area(Δ)≥12(n−ω(Δ)m)m2. That rewrites as d≥(√n−12−12√n)m for the curves of degree *d*. We consider an arbitrary toric surface (i.e. arbitrary Δ) and our ground field is an infinite field of any characteristic, or a finite field large enough. The latter constraint arises because it is not *a priori* clear what is *a collection of generic points* in the case of a small finite field. We construct such collections for fields big enough, and that may be also interesting for the coding theory.

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove tropical Nullstellensatz and moreover we show effective formulation of this theorem. Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and effective version is relevant for computational aspects of this field.

Let K be a field with non-Archimedean valuation v, and assume A is a matrix of size m×n and rank k over K. Richter- Gebert, Sturmfels, and Theobald proved that the rows of A are a tropical basis of the corresponding linear space whenever m = k and any submatrix formed by k columns of v(A) has tropical rank k. We show that this result is no longer true without assumption m = k, refuting the conjecture proposed by Yu and Yuster in 2007.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.