Implicit function theorem over free groups

We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the Implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lifting solutions of equations into generic points. From the model theoretic view-point we claim the existence of simple Skolem functions for particular $\forall\exists$-formulas over free groups. Proving these theorems we describe in details a new version of the Makanin-Razborov process for solving equations in free groups. We also prove a weak version of the Implicit function theorem for NTQ systems which is one of the key results in the solution of the Tarski's problems about the elementary theory of a free group.Comment: 144 pages, 16 figure

Heuristics for The Whitehead Minimization Problem

In this paper we discuss several heuristic strategies which allow one to solve the Whitehead's minimization problem much faster (on most inputs) than the classical Whitehead algorithm. The mere fact that these strategies work in practice leads to several interesting mathematical conjectures. In particular, we conjecture that the length of most non-minimal elements in a free group can be reduced by a Nielsen automorphism which can be identified by inspecting the structure of the corresponding Whitehead Graph

A new microscopic nucleon-nucleon interaction derived from relativistic mean field theory

A new microscopic nucleon-nucleon (NN) interaction has been derived for the first time from the popular relativistic mean field theory (RMFT) Lagrangian. The NN interaction so obtained remarkably relate to the inbuilt fundamental parameters of RMFT. Furthermore, by folding it with the RMFT-densities of cluster and daughter nuclei to obtain the optical potential, it's application is also examined to study the exotic cluster radioactive decays, and results obtained found comparable with the successfully used M3Y phenomenological effective NN interactions. The presently derived NN-interaction can also be used to calculate a number of other nuclear observables.Comment: 4 Pages 2 Figure

Structural changes in Russian economy and objectives of investment policy

© 2015, Asian Social Science. All rights reserved. Issues of Russian economy development in the period of economic reforms, tendency of sectorial structure change of national production on different stages of market reformation, reductions of production volumes of advanced technology products in the industries with a high value added are examined in the article. The influence of government policy on the state finances of Russia is considered, the criteria applied for the estimation of reforms realization success are exposed. The analysis of lack of internal credit resources reasons in an economy, forming of national and corporate external debt, and also strengthening of technological dependence of the Russian producers on foreign partners and insufficient stability of the Russian economy in a global competition is conducted

Development of russian economy in conditions of globalization and investment policy

© 2015, Mediterranean Center of Social and Educational Research. All rights reserved. Theoretical issues of Russian economy development in the context of globalization, internal and external factors, forming tendencies of structural changes of national production are considered in the article. The situation on the Russian market of exported goods, and its impact on the financial system of Russia are considered. Conditions for investment business are analyzed, the role of state finance, taxation and monetary policies on the performance of business in Russia are explored. A comparative analysis of investment conditions in Russia and in Western developed countries is made. Possible changes in macroeconomic policies, aimed at improving the conditions for investments in Russia, are determined

Primitive Words, Free Factors and Measure Preservation

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I: A New Algorithm", and "On Primitive Words II: Measure Preservation". 42 pages, 14 figures. Some parts of the paper reorganized towards publication in the Israel J. of Mat

Activity of Somatosensory-Responsive Neurons in High Subdivisions of SI Cortex during Locomotion

Responses of neurons in the primary somatosensory cortex during movements are poorly understood, even during such simple tasks as walking on a flat surface. In this study, we analyzed spike discharges of neurons in the rostral bank of the ansate sulcus (areas 1–2) in 2 cats while the cats walked on a flat surface or on a horizontal ladder, a complex task requiring accurate stepping. All neurons (n = 82) that had receptive fields (RFs) on the contralateral forelimb exhibited frequency modulation of their activity that was phase locked to the stride cycle during simple locomotion. Neurons with proximal RFs (upper arm/shoulder) and pyramidal tract-projecting neurons (PTNs) with fast-conducting axons tended to fire at peak rates in the middle of the swing phase, whereas neurons with RFs on the distal limb (wrist/paw) and slow-conducting PTNs typically showed peak firing at the transition between swing and stance phases. Eleven of 12 neurons with tactile RFs on the volar forepaw began firing toward the end of swing, with peak activity occurring at the moment of foot contact with floor, thereby preceding the evoked sensory volley from touch receptors. Requirement to step accurately on the ladder affected 91% of the neurons, suggesting their involvement in control of accuracy of stepping. During both tasks, neurons exhibited a wide variety of spike distributions within the stride cycle, suggesting that, during either simple or ladder locomotion, they represent the cycling somatosensory events in their activity both predictively before and reflectively after these events take place