We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule Sn over a semiring S is rational if it has a generating family that is a rational subset of Sn, Sn being thought of as a monoid under the entrywise prod...

Energy problems are important in the formal analysis of embedded or autonomous systems. With the purpose of unifying a number of approaches to energy problems found in the literature, we introduce energy automata. These are finite automata whose edges are labeled with energy functions that define how energy levels evolve during transitions. Motivated by this application and in order to compute ...

Abstract. An optimization problem is formulated in the tropical mathematics setting to maximize a nonlinear objective function defined by conjugate transposition on vectors in a semimodule over a general idempotent semifield. The study is motivated by problems from project scheduling, where the deviation between completion times of activities is to be maximized subject to precedence constraints...

We consider a multidimensional optimization problem in the framework of tropical mathematics. The problem is formulated to minimize a nonlinear objective function that is defined on vectors in a finitedimensional semimodule over an idempotent semifield and calculated by means of multiplicative conjugate transposition. We offer two complete direct solutions to the problem. The first solution con...

We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring. We say that a subsemimodule of the free semimodule Sn over a semiring S is rational if it has a generating family that is a rational subset of Sn, Sn being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, r...

We examine two multidimensional optimization problems that are formulated in terms of tropical mathematics. The problems are to minimize nonlinear objective functions, which are defined through the multiplicative conjugate vector transposition on vectors of a finitedimensional semimodule over an idempotent semifield, and subject to boundary constraints. The solution approach is implemented, whi...

We consider a multidimensional extremal problem formulated in terms of tropical mathematics. The problem is to minimize a nonlinear objective function, which is defined on a finite-dimensional semimodule over an idempotent semifield, subject to linear inequality constraints. An efficient solution approach is developed which reduces the problem to that of solving a linear inequality with an exte...

We consider multidimensional optimization problems, which are formulated and solved in terms of tropical mathematics. The problems are to minimize (maximize) a linear or nonlinear function defined on vectors of a finite-dimensional semimodule over an idempotent semifield, and may have constraints in the form of linear equations and inequalities. The aim of the paper is twofold: first to give a ...

The analytic moduli of equisingular plane branches has the semimodule differential values as most relevant system discrete invariants. Focusing in case cusps, minimal generators this is reached by attached to $1$-forms so-called standard bases. We can complete a basis an enlarged one adding last $1$-form that considered cusp invariant branch and ``correct'' divisorial order. elements such bases...

Automatic differentiation (AD) has been a topic of interest for researchers in many disciplines, with increased popularity since its application to machine learning and neural networks. Although appreciate know how apply AD, it remains challenge truly understand the underlying processes. From an algebraic point view, however, AD appears surprisingly natural: originates from laws. In this work w...