Related papers: Minimal solutions of tropical linear differential …
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…
A tropical (or min-plus) semiring is a set $\mathbb{Z}$ (or $\mathbb{Z \cup \{\infty\}}$) endowed with two operations: $\oplus$, which is just usual minimum, and $\odot$, which is usual addition. In tropical algebra the vector $x$ is a…
In this work we prove the lower bound for the number of $T$-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, $T$-periodic in time, with $T$-Maslov indices $i_0,i_\infty$ at the…
A system of quasilinear elliptic equations on an unbounded domain is considered. The existence of a sequence of radially symmetric weak solutions is proved via variational methods.
A linear vector equation is considered defined in terms of idempotent mathematics. To solve the equation, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces and reduces the solution…
Tropical circuits are circuits with Min and Plus, or Max and Plus operations as gates. Their importance stems from their intimate relation to dynamic programming algorithms. The power of tropical circuits lies somewhere between that of…
Nonlinear dispersionless equations arise as the dispersionless limit of well know integrable hierarchies of equations or by construction, such as the system of hydrodynamic type. Some of these equations are integrable in the Hamiltonian…
We establish the existence of weak solutions of coupled systems of elliptic partial differential equations with quasimonotone nonlinearities in the domain interior and on the boundary. When the nonlinearities satisfy some monotonicity…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
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 examine a multidimensional optimisation problem in the tropical mathematics setting. The problem involves the minimisation of a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear…
We consider small nonlinear perturbations of linear systems on a time scale with the phase space being finite or infinite-dimensional. For $\Delta$-differential operators, corresponding to linear dynamic systems we consider their…
We show that if any four distinct solutions of a rational difference equation are algebraically independent, then any number of distinct solutions to the equation are independent. A nontrivial variant of this result is given for autonomous…
We consider multidimensional optimization problems that are formulated in the framework of tropical mathematics to minimize functions defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible…
The present work includes some of the author's original researches on integer solutions of Diophantine liner equations and systems. The notion of "general integer solution" of a Diophantine linear equation with two unknowns is extended to…
Let ${\mathcal D}^{k,l}(m,n)$ be the set of all the integer points in the transportation polytope of $kn\times ln$ matrices with row sums $lm$ and column sums $km$. In this paper we find the sharp lower bound on the tropical determinant…
An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…