Injective and Projective Properties of <Emphasis Type="Italic">R</Emphasis>[<Emphasis Type="Italic">x</Emphasis>]-Modules

We study whether the projective and injective properties of left R-modules can be implied to the special kind of left R[x]-modules, especially to the case of inverse polynomial modules and Laurent polynomial modules.     

Multiple <Emphasis Type="Italic">q</Emphasis>-zeta functions and multiple <Emphasis Type="Italic">q</Emphasis>-polylogarithms

For every positive integer d we define the q-analog of multiple zeta function of depth d and study its properties, generalizing the work of Kaneko et al. who dealt with the case d=1. We first analytically continue it to a meromorphic function on ℂ ^d with explicit poles. In our Main Theorem we show that its limit when q ↑1 is the ordinary multiple zeta function. Then we consider some special values of these functions when d=2. At the end of the paper we also propose the q-analogs of multiple polylogarithms by using Jackson’s q-iterated integrals and then study some of their properties. Our definition is motivated by those of Koornwinder and Schlesinger although theirs are slightly different from ours. Partially supported by NSF grant DMS0139813 and DMS0348258.     

On the prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis>) where <Emphasis Type="Italic">p</Emphasis> > 3 is a prime number

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph. The third author was supported in part by a grant from IPM (No. 84200024).     

2-(<Emphasis Type="Italic">v,k</Emphasis>,1) Designs and PSL (3,<Emphasis Type="Italic">q</Emphasis>) where <Emphasis Type="Italic">q</Emphasis> is ODD

Let G be a block-primitive automorphism group of a 2-(v,k,1) design. If G is isomorphic to PSL (3,q) where q is odd, then G is also point-primitive. Supported by the National Natural Science Foundation of China (10171089).     

<Emphasis Type="Italic">q</Emphasis>-Schur Subalgebras

In Fayers and Martin [J. Algebra 240 (2001), 859–873], the author and Martin constructed embeddings of Schur algebras S(2,r)↪S(2,R). Here, we generalise to the q-Schur algebras S_q(2,r). Presented by A. Verschoren Mathematics Subject Classifications (2000) 20G05, 20C33.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

The <Emphasis Type="Italic">p</Emphasis>-rank of Ext<Subscript>ℤ</Subscript>(<Emphasis Type="Italic">G</Emphasis>, ℤ) in certain models of <Emphasis Type="Italic">ZFC</Emphasis>

We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext _ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal μ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π₀ and Π₁, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2^μ = μ⁺ such that the p-rank of Ext _ℤ(G, ℤ) equals 2^μ = μ⁺ for every p ∈ Π₀ and 0 otherwise, that is, for p ∈ Π₁. Number 874 in Shelah’s list of publications. Supported by the German-Israeli Foundation for Scientific Research & Development project No. I-706-54.6/2001. Supported by a grant from the German Research Foundation DFG. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 369–397, May–June, 2007.     

SOME NEW PROPERTIES ON L^p（Ω, H^n）

Let L^p（Ω, H^n） indicate the L^P-space of the maps for Heisenberg group target. In this paper some new properties are obtained for the space L^p（Ω, H^n）     

A group with <Emphasis Type="Italic">H</Emphasis>-Frobenius element of even order

We settle Question 10.61 in the Kourovka Notebook for the case where the order of an element a is even.Supported by RFBR grant No. 03-01-00356 and by the Krasnoyarsk Science Foundation, project 11F0202C.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 70–80, January–February, 2005.     

On generalized k-syzygy modules

In this paper, we first introduce the notion of generalized k-syzygy modules, and then give an equivalent characterization that the class of generalized k-syzygy modules coincides with that ofω-k-torsionfree modules. We further study the extension closure of the category consisting of generalized k-syzygy modules. Some known results are obtained as corollaries.     

Composite Integers n for Which φ（n）｜n- 1

In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.     

Constructions for generalized Steiner systems <Emphasis Type="Italic">GS</Emphasis>(3, 4, <Emphasis Type="Italic">v</Emphasis>, 2)

Generalized Steiner systems GS (3, 4, v, 2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance three, in which each codeword has length v and weight four. Not much is known for GS (3, 4, v, 2)s except for a recursive construction and two small designs for v = 8,10 given by Etzion. In this paper, more small designs are found by computer search and also given are direct constructions based on finite fields and rotational Steiner quadruple systems and recursive constructions using three-wise balanced designs. Some infinite families are also obtained.      

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

Multiplicities of simple modules in the Sp(4, <Emphasis Type="Italic">q</Emphasis>)-permutation module on <Emphasis Type="Italic">P</Emphasis>(3, <Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">q</Emphasis> even

In this paper we shall determine the multiplicities of simple modules in characteristic 2 in the Sp(4, q)-permutation module on projective 3-space P(3, q), q = 2 ⁿ .     

Sturmian morphisms,the braid group <Emphasis Type="Italic">B</Emphasis><Subscript>4</Subscript>, Christoffel words and bases of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F ₂) of automorphisms of the rank two free group F ₂ and show that it can be realized as a monoid in the group B ₄ of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F ₂ lifting any given basis of the free abelian group Z ². We further give an algorithm allowing to decide whether two elements of F ₂ form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes. Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15     

Lacunary Strong (<Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">σ</Emphasis></Subscript><Emphasis Type="Italic">, p</Emphasis>)-Convergence

The definition of lacunary strongly convergence is extended to the definition of lacunary strong (A _σ , p)-convergence with respect to invariant mean when A is an infinite matrix and p = (p _i ) is a strictly positive sequence. We study some properties and inclusion relations.     

<Emphasis Type="Italic">q</Emphasis>-Enumeration of Salié Permutations

Salié permutations are defined to be first alternating, and then monotone. We q-enumerate them by considering words (instead of permutations), equipped with geometric probabilities. Received May 26, 2005     

An efficient algorithm for the generalized centro-symmetric solution of matrix equation <Emphasis Type="Italic">A X B</Emphasis> = <Emphasis Type="Italic">C</Emphasis>

In this paper, an iterative algorithm is constructed for solving linear matrix equation AXB = C over generalized centro-symmetric matrix X. We show that, by this algorithm, a solution or the least-norm solution of the matrix equation AXB = C can be obtained within finite iteration steps in the absence of roundoff errors; we also obtain the optimal approximation solution to a given matrix X ₀ in the solution set of which. In addition, given numerical examples show that the iterative method is efficient.     

Extensions of <Emphasis Type="Italic">GM</Emphasis>-rings

It is shown that a ring R is a GM-ring if and only if there exists a complete orthogonal set { e ₁,...,e _n } of idempotents such that all e _i Re _i are GM-rings. We also investigate GM-rings for Morita contexts, module extensions and power series rings.This work was supported by the Natural Science Foundation of Zhejiang Province.