Unilateral <Emphasis Type="Italic">o</Emphasis>-Groups

For a big number of varieties of groups close to Engelian, it is proved that a variety of lattice-ordered groups generated by all linearly ordered groups in the class does not coincide with the variety of all o-approximable lattice-ordered groups. Supported by FP “Universities of Russia” grant No. UR.04.01.001. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 20–27, January–February, 2006.     

Periodic groups saturated with <Emphasis Type="Italic">L</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let be a set of finite groups. A group G is saturated with groups from if every finite subgroup of G is contained in a subgroup isomorphic to some member of . It is proved that a periodic group G saturated with groups from the set {L₃(2^m)|m = 1, 2, …} is isomorphic to L₃(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 606–626, September–October, 2007.     

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras of Bergman Type Operators with Piecewise Continuous Coefficients

The C ^*-algebra generated by the n poly-Bergman and m antipoly-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L ²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional convolution operators with symbols admitting homogeneous discontinuities we reduce the study to simpler C ^*-algebras associated with points and pairs . Applying a symbol calculus for the abstract unital C ^*-algebras generated by N orthogonal projections sum of which equals the unit and by M = n + m one-dimensional orthogonal projections and using relations for the Gauss hypergeometric function, we study local algebras at points being the discontinuity points of coefficients. A symbol calculus for the C ^*-algebra is constructed and a Fredholm criterion for the operators is obtained.     

Optimal Domains and Integral Representations of Convolution Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">G</Emphasis>)

Given , a compact abelian group G and a function , we identify the maximal (i.e. optimal) domain of the convolution operator (as an operator from L^p(G) to itself). This is the largest Banach function space (with order continuous norm) into which L^p(G) is embedded and to which has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and g. Whereas is compact, this is not always so for the extension of to its optimal domain. Several characterizations of precisely when this is the case are presented.     

Distribution of <Emphasis Type="Italic">q</Emphasis>-additive functions on the set of primes

We characterize the q-multiplicative functions which belong to or = the set of uniformly summable functions on the set of primes. Dedicated to Professor Jonas Kubilius on his 85th anniversary Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 29–38, January–March, 2007.     

Removing vertices from a <Emphasis Type="Italic">k</Emphasis>-connected graph without losing its <Emphasis Type="Italic">k</Emphasis>-connectivity

The paper studies the problem of removing vertices from a k-connected graph without losing its k-connectivity. It is proved that certain inner vertices can be removed from k-blocks, provided that the interior of each block is sufficiently large with respect to its boundary, and the degree of every vertex of the graph is no less than or . Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 340, 2006, pp. 103–116.     

Approximation by K-finite functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Let Γ ⊂ ℝⁿ, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in L ^p (ℝⁿ) for . A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators. The first author was supported in part by a grant from UGC via DSA-SAP Phase IV.     

Dual <Emphasis Type="Italic">R</Emphasis>-matrix integrability

Using the R-operator on a Lie algebra satisfying the modified classical Yang-Baxter equation, we define two sets of functions that mutually commute with respect to the initial Lie-Poisson bracket on . We consider examples of the Lie algebras with the Kostant-Adler-Symes and triangular decompositions, their R-operators, and the corresponding two sets of mutually commuting functions in detail. We answer the question for which R-operators the constructed sets of functions also commute with respect to the R-bracket. We briefly discuss the Euler-Arnold-type integrable equations for which the constructed commutative functions constitute the algebra of first integrals. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 147–160, April, 2008.     

Higher-Rank Conformal Fields in the <Emphasis Type="Italic">Sp</Emphasis>(2<Emphasis Type="Italic">M</Emphasis>)-Symmetric Generalized Space-Time

We study various Sp(2M)-invariant field equations corresponding to rank-r tensor products of the Fock (singleton) representation of Sp(2M). These equations describe localization on “branes” of different dimensions embedded in the generalized space-time with matrix (i.e., “central charge”) coordinates. We consider the case of a bilinear tensor product in detail and show that the conserved currents built from bilinears of rank-1 fields in satisfy the field equations for rank-2 fields in . We also show that rank-2 fields in are equivalent to rank-1 fields in . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 35–65, October, 2005.     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

A cohomological property of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be a prime, a finite p-group, any finite group with order divisible by p, and any action of on . We show that the cardinality of the set of all derivations with respect to this action is a multiple of p. This generalises theorems of Frobenius and Hall. Received: 16 June 2003     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

The solutions of the<Emphasis Type="Italic">n</Emphasis>-dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution

In this article, the operator is introduced and named as the Bessel diamond operator iteratedk times and is defined by

Slightly larger than a graph<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-algebra

To certain higher rank Cuntz algebras including the classical cases we trace a certain partial isometryU in its strong closure. AdjoiningU to we obtain a kind of uniqueness property for this largerC ^*-algebra. Its explanation is not entirely “Cuntz’s uniqueness argumentation”. The author is supported by the Austrian Research Foundation (FWF) project S8308.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Boundedness of Marcinkiewicz Integral

A weighted norm inequality for the Marcinkiewicz integral operator is proved when belongs to . We also give the weighted L^p-boundedness for a class of Marcinkiewicz integral operators with rough kernels and related to the Littlewood-Paley -function and the area integral S, respectively.     

Soluble Groups and Varieties of <Emphasis Type="Italic">l</Emphasis>-Groups

A sufficient condition is given under which factors of a system of normal convex subgroups of a linearly ordered (l.o.) group are Abelian. Also, a sufficient condition is specified subject to which factors of a system of normal convex subgroups of an l.o. group are contained in a group variety . In particular, for every soluble l.o. group G of solubility index n, n ⩾ 2, factors of a system of normal convex subgroups are soluble l.o. groups of solubility index at most n − 1. It is proved that the variety of all lattice-ordered groups, approximable by linearly ordered groups, does not coincide with a variety generated by all soluble l.o. groups. It is shown that if is any o-approximable variety of l-groups, and if every identity in the group signature is not identically true in , then contains free l.o. groups.Supported by FP “Universities of Russia” grant UR. 04. 01. 001.__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 355–367, May–June, 2005.     

<Emphasis Type="Italic">c</Emphasis>*-Supplemented subgroups and <Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

A subgroup H of a finite group G is said to be c*-supplemented in G if there exists a subgroup K such that G = HK and H ⋂ K is permutable in G. It is proved that a finite group G that is S ₄-free is p-nilpotent if N _G (P) is p-nilpotent and, for all x ∈ G\N _G (P), every minimal subgroup of is c*-supplemented in P and (if p = 2) one of the following conditions is satisfied: (a) every cyclic subgroup of of order 4 is c*-supplemented in P, (b) , (c) P is quaternion-free, where P a Sylow p-subgroup of G and is the p-nilpotent residual of G. This extends and improves some known results. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1011–1019, August, 2007.