Free subgroups of free products and combinatorial hypermaps

We derive a generating series for the number of free subgroups of finite index in $\Delta^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $\Delta^+$ and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy. We also study the generating series for conjugacy classes of free subgroups of finite index in $\Delta^+$, which correspond to isomorphism classes of hypermaps. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, or, equivalently, various types of hypermaps and their isomorphism classes.Comment: 27 pages, 3 figures; supplementary SAGE worksheets available at http://sashakolpakov.wordpress.com/list-of-papers

Aspects of Technical Progress in Romanian Agriculture Compared to Some EU Countries

The importance of using technical progress factors in agriculture varies from one country to another and the influences of these factors on yields and agricultural production are essential. Thus, in this paper we considered necessary and appropriate to achieve the dynamic analysis of the main factors of technical progress in agriculture referring to agricultural technical endowment, fertilizers and irrigation. On the other hand, we accomplished the dynamic analysis of the outcome indicators in agricultural production in Romania compared to some EU countries, as well as the gaps registered between countries under study. We also point out the positive and negative effects of technical progress in agriculture which had developed under the revealed indicators evolution.technical progress, irrigation, mechanization, soil fertilization, economic effects, ecological effect.