Related papers: Tropical planar networks
Networks with a high degree of symmetry are useful models for parallel processor networks. In earlier papers, we defined several global communication tasks (universal exchange, universal broadcast, universal summation) that can be critical…
This thesis delves into the geometry of abstract tropical curves, exploring their complete linear system and associated tropical submodules. We establish a lower bound on the dimension of tropical submodules in terms of the Baker-Norine…
The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative…
In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…
We present a simple proof of the factorization of (complex) symmetric matrices into a product of a square matrix and its transpose, and discuss its application in establishing a uniqueness property of certain antilinear operators.
In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian…
We study a notion of tropical linear series on metric graphs that combines two essential properties of tropicalizations of linear series on algebraic curves: the Baker-Norine rank and the independence rank. Our main results relate the local…
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 show that if a strictly positive joint probability distribution for a set of binary random variables factors according to a tree, then vertex separation represents all and only the independence relations enclosed in the distribution. The…
In this note it is proved that every rational matrix which lies in the interior of the cone of completely positive matrices also has a rational cp-factorization.
Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of…
We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed…
We introduce a faithful tropical linear representation of the Chinese monoid, and thus prove that this monoid admits all the semigroup identities satisfied by tropical triangular matrices.
The set of infinite upper-triangular totally positive Toeplitz matrices has a classical parametrisation proved by Edrei et al and originally conjectured by Schoenberg, that involves pairs of sequences of positive real parameters. These…
Deep neural networks show great success when input vectors are in an Euclidean space. However, those classical neural networks show a poor performance when inputs are phylogenetic trees, which can be written as vectors in the tropical…
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…
Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces…
We propose a gradient-based Jacobi algorithm for a class of maximization problems on the unitary group, with a focus on approximate diagonalization of complex matrices and tensors by unitary transformations. We provide weak convergence…
In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call `tropical polynomials' and `sign polynomials', respectively. We classify all irreducible polynomials in either case. We…
Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that $2\times 2$ minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial…