Let us consider a 3-numerical semigroup S = 〈a, b,N 〉. Given m ∈ S, the triple (x, y, z) ∈ N3 is a factorization of m in S if xa+ yb+ zN = m. This work is focused on finding the full set of factorizations of any m ∈ S and as an application we compute the catenary degree of S. To this end, we relate a 2D tessellation to S and we use it as a main tool.

Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S)) and the tame degree of S (denoted t(S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe met...

The purpose of this paper is to describe the possibilities of studying the influence of forces external variables on the stability of motion of dynamical systems modeled by systems of differential equations with periodic coefficients of stability. The method is applied to analyze the influence of external harmonic forces on the stability of motion of the pantograph couple contact wire electric ...

In this paper, a new contact force control technique, based on the modified Takagi-Sugeno model and parallel distributed compensation concept is developed to suppress vibrations between pantograph catenary by regulating reference value, thereby achieving stable current collection. The proposed method uses simple standard PID fuzzy controllers. two controllers guarantee designed system's robust ...

The modal analysis, which decouples the contact wire and rail track conductors, is used to develop an integrated model of single rail track and contact wire. The characteristic impedance and propagation constant matrices of integrated model have been computed. The authors have studied effects of various parameters of interest on characteristic impedance and propagation constant using sensitivit...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...