Related papers: Tropical limit and the micro-macro correspondence
Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted…
Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the…
In a series of articles, we have been developing a theory of tropical diagrams of probability spaces, expecting it to be useful for information optimization problems in information theory and artificial intelligence. In this article, we…
We define arroids as an abstract axiom set encoding the intersection properties of arrangements of curves. The tropicalization of the complement of arrangement of curves meeting pairwise transversely is shown to be determined by the…
We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise…
After endowing the space of diagrams of probability spaces with an entropy distance, we study its large-scale geometry by identifying the asymptotic cone as a closed convex cone in a Banach space. We call this cone the tropical cone, and…
Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic…
We study representations of tropical linear spaces as intersections of tropical hyperplanes of circuits. For several classes of matroids, we describe minimal tropical bases. We also show that every realizable tropical linear space has a…
In this survey, we discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in the field, with an emphasis on…
Trophic coherence and non-normality are both ways of describing the overall directionality of directed graphs, or networks. Trophic coherence can be regarded as a measure of how neatly a graph can be divided into distinct layers, whereas…
Asymptotic properties of matrices are, in general, difficult to analyze with classical mathematical techniques. In very specific cases, there is a well-known connection between the asymptotic behavior of a matrix's leading eigenvector and…
Understanding future changes in temperature variability and extremes is an important scientific challenge with societal impacts. Here the responses of daily near-surface temperature distributions to climate warming is explored using an…
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in…
In our derivation of the second law of thermodynamics from the relation of adiabatic accessibility of equilibrium states we stressed the importance of being able to scale a system's size without changing its intrinsic properties. This…
Consistent statistical physical description is given for systems where the elementary excitations are composite objects. Explicit calculational scheme is constructed for the energy density and the total number of thermodynamical degrees of…
Tropical Geometry, an established field in pure mathematics, is a place where String Theory, Mirror Symmetry, Computational Algebra, Auction Theory, etc, meet and influence each other. In this paper, we report on our discovery of a tropical…
We consider multidimensional optimization problems in the framework of tropical mathematics. The problems are formulated to minimize a nonlinear objective function that is defined on vectors over an idempotent semifield and calculated by…
We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the…
A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field. Here, we ask: Where can classical intersection points of planar curves tropicalize to? An answer should have two parts: first, identifying…
Symmetries and isomorphisms play similar conceptual roles when we consider how models represent physical situations, but they are formally distinct, as two models related by symmetries are not typically isomorphic. I offer a rigorous…