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

Humanitarian Bases of Teaching Specialists in Forestry at Tomsk University

AbstractIn this work the authors examine the reasons which served as an incentive to modernize the educational tasks for future specialists in forestry at Tomsk State University. Methodological bases of teaching the course of introduction to the profession on the basis of humanistic values are determined. The authors emphasize the main focus of this work on methodological bases of course design in Propedeutics (introduction to the profession). The content side of the course is considered and a sequence of thematic modules and references are substantiated

Genetic organization and heterogeneity of the Siberian cedar pine (Pinus sibirica Du Tour) population in the Western Siberia (Tomsk region)

Cedar forest is the most complex and dynamic formation of the Siberian Taiga. It clearly demonstrates the processes of natural regeneration dynamics, sustainability, space-time structure and biodiversity of Siberian forests (Drozdov and Grishenkov, 2003). Within its range, ​​Siberian stone pine grows in various environmental conditions, including high mountains and marshland, which led to the formation of various ecotypes, forms, biotypes that has been created by nature for centuries (Nikolaeva and Savchuk, 2013). Much attention is currently being paid to biological and seed productivity of cedar cultures depending on their geographical origin (Bratilova and Kalinin, 2012; Matveeva et al, 2012; Dragavcev, 2008). This study involved research on the growth of the Siberian cedar pine (Pinus sibirica Du Tour) in the process of artificial plantation formation. In order to develop methods for the genetic evaluation of plant structures and search for indicators, patterns of growth for 100 model trees were analyzed in the experimental population

Conjugacy in Baumslag's group, generic case complexity, and division in power circuits

The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the complexity of the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS(1,2) and the Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is TC^0-complete. To the best of our knowledge BS(1,2) is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that the conjugacy problem is decidable (which has been known before); but our results go far beyond decidability. In particular, we are able to show that conjugacy in G(1,2) can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs conjugacy in G(1,2) can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in G(1,2) by reducing the division problem in power circuits to the conjugacy problem in G(1,2). The complexity of the division problem in power circuits is an open and interesting problem in integer arithmetic.Comment: Section 5 added: We show that an HNN extension G = < H, b | bab^-1 = {\phi}(a), a \in A > has a non-amenable Schreier graph with respect to the base group H if and only if A \neq H \neq

Groups Whose Universal Theory Is Axiomatizable by Quasi-Identities

Discriminating groups were introduced in [3] with an eye toward applications to the universal theory of various groups. In [6] it was shown that if G is any discriminating group, then the universal theory of G coincides with that of its direct square G x G. In this paper we explore groups G whose universal theory coincides with that of their direct square. These are called square-like groups. We show that the class of square-like groups is first-order axiomatizable and contains the class of discriminating groups as a proper subclass. Further we show that the class of discriminating groups is not first-order axiomatizable

Algebraic Geometry over Free Metabelian Lie Algebra I: U-Algebras and Universal Classes

This paper is the first in a series of three, the aim of which is to lay the foundations of algebraic geometry over the free metabelian Lie algebra $F$. In the current paper we introduce the notion of a metabelian Lie $U$-algebra and establish connections between metabelian Lie $U$-algebras and special matrix Lie algebras. We define the $\Delta$-localisation of a metabelian Lie $U$-algebra $A$ and the direct module extension of the Fitting's radical of $A$ and show that these algebras lie in the universal closure of $A$.Comment: 34 page

Discriminating Groups

A group G is termed discriminating if every group separated by G is discriminated by G. In this paper we answer several questions concerning discrimination which arose from [2]. We prove that a finitely generated equationally Noetherian group G is discriminating if and only if the quasivariety generated by G is the minimal universal class containing G. Among other results, we show that the non-abelian free nilpotent groups are non-discriminating. Finally we list some open problems concerning discriminating groups

Quadratic equations over free groups are NP-complete

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete

EXTENDED COREY-CHAYKOVSKY REACTION AS A PATHWAY FOR THE SYNTHESIS OF SUBSTITUTED FURANS

This work was supported by RSF № 21-73-10063