Classification of lattice-regular lattice convex polytopes

We completely describe lattice convex polytopes in ℝ ⁿ (for any dimension n) that are regular with respect to the group of affine transformations preserving the lattice. Supported in part by the RFBR (Grant Nos. SS-1972.2003.1 and 05-01-01012a) and the NWO-RFBR (Grant No. 047.011.2004.026/RFBR No. 05-02-89000-NWO_a).     

The Word Problem for Polynilpotent Groups with a Single Primitive Defining Relation

It is shown that polynilpotent groups with a single primitive defining relation have a decidable word problem. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 28–43, January–February, 2006.     

The cyclic structure of maximal tori of the finite classical groups

Maximal tori of all finite simple classical groups, as well as of special and general projective linear and unitary groups, are treated. For every such torus, its expression as a direct sum of cyclic groups is obtained in an explicit form. Supported by RFBR grant Nos. 05-01-00797 and 06-01-39001, and by SB RAS Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 129–156, March–April, 2007.     

On conformal invariants for elliptic systems with multiple critical exponents

In this article we study multiple solutions of the prescribed curvatures problem for a compact Riemannian manifold with smooth boundary and with indefinite signs of the energy function and nonlinearities. We generalize the conception of the critical Palais–Smale level and find new type of conformally invariant necessary and sufficient conditions for existence of multiple solutions of the problem. The second author was supported in part by grants INTAS 03-51-5007, RFBR 05-01-00370, 05-01-00515.     

Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy

We investigate differences in isomorphism types for Rogers semilattices of computable numberings of families of sets lying in different levels of the arithmetical hierarchy. Supported by RFBR grant No. 05-01-00819 and by INTAS grant No. 00-499. Supported by NSFC grant No. 60310213. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 637–654, November–December, 2006.     

A weak form of interpolation in equational logic

The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety. Supported by RFBR (grant Nos. 06-01-00358 and 05-01-04003-NNIOa) and by INTAS (grant No. 04-77-7080). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 94–107, January–February, 2008.     

Slender partial quadrangles and their automorphisms

The partial quadrangle PQ(s, t, μ) is an incidence system consisting of points and lines in which every line contains s + 1 points, every point sits on t + 1 lines (two lines meet in at most one point), and the meet of the neighborhoods of any two non-adjacent points in the collinearity graph is a μ-coclique. We provide a classification for partial quadrangles PQ(s, t, μ) with t ⩽ 6, and study into their automorphisms. Supported by RFBR grant No. 05-01-00046 and by RFBR-NSFC grant No. 05-01-39000. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 603–619, September–October, 2006.     

Gröbner-Shirshov bases for the lie algebra <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We estimate Gröbner-Shirshov bases for the Lie algebra A_n given arbitrary orders of generators (nodes of a Dynkin graph). Previously, the Gröbner-Shirshov basis was computed in [1] for the particular case where nodes of the Dynkin graph are ordered successively.Supported by RFBR grant Nos. 05-01-00230 and 02-01-00258 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 131–147, March–April, 2005.     

Simple Filippov superalgebras of type B(m, n)

This account is a first step toward a classification of finite-dimensional simple Filippov superalgebras over an algebraically closed field of characteristic 0. Here, n-ary Filippov superalgebras with nonzero even and odd parts are treated for the case n ⩾ 3. Supported by RFBR (grant No. 05-01-00230) and by SB RAS (Integration project No. 1.9 and Young Researchers Support grant No. 29). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 240–261, March–April, 2008.     

Hochschild cohomologies for associative conformal algebras

We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel. Supported by RFBR grant No. 05-01-00230 and via SB RAS Integration project No. 1.9. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 688–706, November–December, 2007.     

Reducibility on families

A reducibility on families of subsets of natural numbers is introduced which allows the family per se to be treated without its representation by natural numbers being fixed. This reducibility is used to study a series of problems both in classical computability and on admissible sets: for example, describing index sets of families belonging to , generalizing Friedberg’s completeness theorem for a suitable reducibility on admissible sets, etc. *Supported by RFBR (project No. 05-01-00605) and by the Council for Grants (under RF President) and State Aid of Young Candidates of Science (grant MK-4314.2008.1). **Supported by RFBR (projects No. 08-01-00442 and 06-04002-DFGa), by the Council for Grants (under RF President) of Leading Scientific Schools (grant NSh-335.2008.1), and by the Russian Foundation for Support of Domestic Science. Translated from Algebra i Logika, Vol. 48, No. 1, pp. 31-53, January-February, 2009.     

Stably Definable Classes of Theories

A question is studied as to which properties (classes) of elementary theories can be defined via generalized stability. We present a topological account of such classes. It is stated that some well-known classes of theories, such as strongly minimal, o-minimal, simple, etc., are stably definable, whereas, for instance, countably categorical, almost strongly minimal, ω-stable ones, are not. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 583–600, September–October, 2005. Supported by RFBR grant Nos. 02-01-00540 and 05-01-00411, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.     

Computing test rank for a free solvable group

It is proved that test rank of a free solvable non-Abelian group of finite rank is 1 less than the rank of that group. This gives the answer to Question 14.88 posed in the Kourovka Notebook by Fine and Shpilrain, asking whether or not a free solvable group of rank 2 and solvability index n ≥ 3 has test elements. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 447–457, July–August, 2006.     

Recognition of simple groups U 3(Q) by element orders

An exhaustive solution is given to the recognition-by-spectrum problem for finite, simple, three-dimensional unitary groups. For every such group, the number of non-isomorphic, finite, isospectral groups is determined. In particular, a new counterexample to Problem 13.63 in the Kourovka Notebook is furnished. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 185–202, March–April, 2006.     

Complexity of finite sequences of zeros and ones and geometry of finite spaces of functions

The sequence 001 001 001 001 is simpler than 01 001 0111 001. We present a formal mathematical theory that gives an exact meaning to the preceding (intuitively clear) statement. Supported in part by the RFBR (Grant No. 05-01-00104). Read at the Moscow Mathematical Society meeting 22 November 2005, homepage: .     

On the separation of spectra in the analysis of Berezin kernels

The problem of restricting a highest weight representation of the group U(p, q) to the subgroup O(p, q) is considered. This restriction has an intricate spectrum that contains representations of different types. We construct a decomposition of this representation into reducible representations each of which has a single-type spectrum. Some integrals over classical groups are also calculated; these integrals generalize those of Hua. Partially supported by grant RFBR 98-01-00303 and the Russian program of support of leading scientific schools (grant RFBR 96-01-96249). Moscow Institute of Electronics and Mathematics. Translated from Funktsional'nyi analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 49–62, July–September, 2000. Translated by Yu. A. Neretin     

Algebraic sets in metabelian groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = F_n is a free metabelian group of rank n; (2) G = W_n,k is a wreath product of free Abelian groups of ranks n and k. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007.     

Recognition of finite groups by the prime graph

We obtain the first example of an infinite series of finite simple groups that are uniquely determined by their prime graph in the class of all finite groups. We also show that there exist almost simple groups for which the number of finite groups with the same prime graph is equal to 2. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 390–408, July–August, 2006.