Related papers: Tropical limit and the micro-macro correspondence
A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal…
This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…
We reconcile the discrepancy between the complex and tropical counts of some enumerative problems reducing to positive characteristic. Each problem that we consider suggests a prime with special behaviour. Modulo this prime, the solutions…
A multiscale theory of interacting continuum mechanics and thermodynamics of mixtures of fluids, electrodynamics, polarization and magnetization is proposed. The mechanical (reversible) part of the theory is constructed in a purely…
Using the interpretation of the ultradiscretization procedure as a non-Archimedean valuation, we use results of tropical geometry to show how roots and poles manifest themselves in piece-wise linear systems as points of…
This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's…
Understanding the assembly of ecosystems to estimate the number of species at different spatial scales is a challenging problem. Until now, maximum entropy approaches have lacked the important feature of considering space in an explicit…
We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly…
We show that various types of equilibrium play an important part in the behaviour of the troposphere. In analogy to the electro-chemical potential (well-known in solid-state physics and electro-chemistry) a gravito-chemical potential and a…
The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications…
We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a…
The theory of small-system thermodynamics was originally developed to extend the laws of thermodynamics to length scales of nanometers. Here we review this "nanothermodynamics," and stress how it also applies to large systems that subdivide…
We propose a definition of tropical linear series that isolates some of the essential combinatorial properties of tropicalizations of not-necessarily-complete linear series on algebraic curves. The definition combines the Baker-Norine…
Bayesian maxent lets one integrate thermal physics and information theory points of view in the quantitative study of complex systems. Since net surprisal (a free energy analog for measuring "departures from expected") allows one to place…
The thermodynamic definition of entropy can be extended to nonequilibrium systems based on its relation to information. To apply this definition in practice requires access to the physical system's microstates, which may be prohibitively…
The relationship between microsystems and macrosystems is considered in the context of quantum field formulation of statistical mechanics: it is argued that problems on foundations of quantum mechanics can be solved relying on this…
The set theory relations \in, \backslash, \Delta, \cap, and \cup have corollaries in subspace relations. Geometric Algebra is introduced as the ideal framework to explore these subspace operations. The relations \in, \backslash, and \Delta…
The metriplectic formalism couples Poisson brackets of the Hamiltonian description with metric brackets for describing systems with both Hamiltonian and dissipative components. The construction builds in asymptotic convergence to a…
This paper proposes a tropical geometry-based edge detection framework that reformulates convolution and gradient computations using min-plus and max-plus algebra. The tropical formulation emphasizes dominant intensity variations,…
We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical…