Related papers: Supertropical Matrix Algebra II: Solving tropical …
We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic, we…
The superalgebra of observables of the rational Calogero model based on the root system R is the associative superalgebra generated by polynomials in N indeterminates, the differential-difference Dunkl's operators and the group algebra of…
This is the fourth article in a series where we succeed in enlarging the class of exactly solvable quantum systems. We do that by working in a complete set of square integrable basis that carries a tridiagonal matrix representation for the…
We show that the submonoid of all nxn triangular tropical matrices satisfies a nontrivial semigroup identity and provide a generic construction for classes of such identities. The utilization of the Fibonacci number formula gives us an…
A generalization of the second main theorem of tropical Nevanlinna theory is presented for noncontinuous piecewise linear functions and for tropical hypersurfaces without requiring a growth condition. The method of proof is novel and…
We provide an explicit algorithm to solve the idempotent analogue of the discrete Monge-Kantorovich optimal mass transportation problem with the usual real number field replaced by the tropical (max-plus) semiring, in which addition is…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure…
An exactly solvable position-dependent mass Schr\"odinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems…
We define tropical rational function semifields $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and prove that a tropical curve $\varGamma$ is realized (except for points at infinity) as the congruence variety $V \subset \boldsymbol{R}^n$…
We show that super Gromov-Witten invariants can be defined and computed by methods of tropical geometry. When the target is a point, the super invariants are descendant invariants on the moduli space of curves, which can be computed…
We discuss a novel analysis method for reaction network systems with polynomial or rational rate functions. This method is based on computing tropical equilibrations defined by the equality of at least two dominant monomials of opposite…
We present simple compact proofs of the strong and weak duality theorems of tropical linear programming. It follows that there is no duality gap for a pair of tropical primal-dual problems. This result together with known properties of…
This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concept. A special notion of determinant is introduced and a Cramer formula is derived for a class of limit systems derived from the…
We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we…
We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to…
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…
In this note I will explain how relative/log Gromov-Witten invariants of pairs $(X,D)$ with very ample smooth anticanonical divisor $D$ can be computed using algebro-combinatorial objects called scattering diagrams. The underlying principle…
The purpose of this article is to investigate triangularization and simultaneous triangularization of matrices over max algebras using graph theoretic methods. We establish a connection between commutators and commutants with simultaneous…
Continuing [5], this paper investigates finer points of supertropical vector spaces, including dual bases and bilinear forms, with supertropical versions of standard classical results such as the Gram-Schmidt theorem and Cauchy-Schwarz…