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In this paper we generalise and unify the results and methods used by Benson, Liardet, Evetts, and Evetts & Levine, to show that rational sets in a virtually abelian group G have rational (relative) growth series with respect to any generating set for G. We prove equivalences between the structures used in the literature, and establish the rationality of important classes of sets in G: definable sets, algebraic sets, conjugacy representatives and coset representatives (of any fixed subgroup), among others. Furthermore, we show that any rational set, when written as words over the generating set of G, has several EDT0L representations., 24 pages, 3 figures. To appear in L'Enseignement Mathematique

Let T be the Cayley graph of a finitely generated free group F. Given two vertices in T consider all the walks of a given length between these vertices that at a certain time must follow a number of predetermined steps. We give formulas for the number of such walks by expressing the problem in terms of equations in F and solving the corresponding equations.

Let F be a free group of rank at least three. We show that some retracts of F previously studied by Martino-Ventura are not equal to the fixed subgroup of any group of automorphisms of F. This shows that, in F, there exist subgroups that are equal to the fixed subgroup of some set of endomorphisms but are not equal to the fixed subgroup of any set of automorphisms. Moreover, we determine the Galois monoids of these retracts, where, by the Galois monoid of a subgroup H of F, we mean the monoid consisting of all endomorphisms of F that fix H.

One of the key elements for achieving sustainable development in the Romanian rural areas is investing in education, which, in a long term approach, will lead to the formation of farmers able to face the challenges of the competitive European and international markets. This paper aims to present, the result of a theoretical analysis of two aspects concerning the problem of education, namely: the current level of education of the rural population and the main difficulties faced by the educational rural system.

The European Union enlargement to 27 Member States in the year 2007 changed the European agricultural reality, which at present includes a wide range of agriculture types, with large development gaps between the rural areas. Taking into consideration this reality, reforming the Common Agricultural Policy became again a necessity.The new Romanian Agricultural Policy should take this into consideration and try to adjust to the new international context, which is mainly characterized by the instability of markets and price volatility, and provide an integrating vision of the European agricultural policy.

Let F be a free group of finite rank. We say that the monomorphism problem in F is decidable if there is an algorithm such that, for any two elements u and v in F, it determines whether there exists a monomorphism of F that sends u to v. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the problem.

"Vegeu el resum a l'inici del document del fitxer adjunt".

"Vegeu el resum a l’inici del document del fitxer adjunt."

Let F be a free group of rank at least three. We show that some retracts of F previously studied by Martino-Ventura are not equal to the fixed subgroup of any group of automorphisms of F. This shows that, in F, there exist subgroups that are equal to the fixed subgroup of some set of endomorphisms but are not equal to the fixed subgroup of any set of automorphisms. Moreover, we determine the Galois monoids of these retracts, where, by the Galois monoid of a subgroup H of F, we mean the monoid consisting of all endomorphisms of F that fix H.

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism Φ of F sending W to U. This work analyzes an approach due to C. Edmunds and improved by C. Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two- generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.

In this paper we introduce the geodesic conjugacy language and ge odesic conjugacy growth series for a finitely generated group. We study the ef fects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular we show that regularity of the geodesic conjugacy language is preserved by the graph product construction and rationality of the geodesic conjugacy growth series is preserved by both direct and free products.