Central extensions of generalized orthosymplectic Lie superalgebras

We completely determine the universal central extension of the generalized orthosymplectic Lie superalgebra osp _m|2n (R,-) that is coordinatized by an arbitrary unital associative superalgebra (R,-) with superinvolution. As a result, an identification between the second homology group of the Lie superalgebra osp _m|2n (R,-) and the first skew-dihedral homology group of the associative superalgebra (R,-) with superinvolution is created for positive integers m and n with (m,n) ≠ (1,1) and (m, n) ≠ (2,1). The second homology groups of the Lie superalgebras osp_1|2(R,-) and osp_2|2(R,-) are also characterized explicitly.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.     

A boundary Schwarz lemma on the classical domain of type <Emphasis Type="Italic">I</Emphasis>

Let R _I (m, n) be the classical domain of type I in ? ^m×n with 1 ≤ m ≤ n. We obtain the optimal estimates of the eigenvalues of the Fréchet derivative Df(\(\mathop Z\limits^ \circ \)) at a smooth boundary fixed point \(\mathop Z\limits^ \circ \)of R _I (m, n) for a holomorphic self-mapping f of R _I (m, n). We provide a necessary and sufficient condition such that the boundary points of R _I (m, n) are smooth, and give some properties of the smooth boundary points of R _I (m, n). Our results extend the classical Schwarz lemma at the boundary of the unit disk Δ to R _I (m, n), which may be applied to get some optimal estimates in several complex variables.     

Eulerian graphs and automorphisms of a maximal graph

Let R be a commutative ring with identity. Let Γ(R) denote the maximal graph corresponding to the non-unit elements of R, i.e., Γ(R) is a graph with vertices the non-unit elements of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. In this paper, we have shown that, for any finite ring R which is not a field, Γ(R) is a Euler graph if and only if R has odd cardinality. Moreover, for any finite ring R ? R ₁×R ₂× · · · ×R _n, where the R _i is a local ring of cardinality p _i ^αi for all i, and the p _i’s are distinct primes, it is shown that Aut(Γ(R)) is isomorphic to a finite direct product of symmetric groups. We have also proved that clique(G(R)’) = χ(G(R)’) for any semi-local ring R, where G(R)’ denote the comaximal graph associated to R.     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

Structural properties and enumeration of 1-generator generalized quasi-cyclic codes

Let F _q be a finite field of cardinality q, m ₁, m ₂, . . . , m _l be any positive integers, and \({A_i=F_q[x]/(x^{m_i}-1)}\) for i = 1, . . . , l. A generalized quasi-cyclic (GQC) code of block length type (m ₁, m ₂, . . . , m _l ) over F _q is defined as an F _q [x]-submodule of the F _q [x]-module \({A_1\times A_2\times\cdots\times A_l}\). By the Chinese Remainder Theorem for F _q [x] and enumeration results of submodules of modules over finite commutative chain rings, we investigate structural properties of GQC codes and enumeration of all 1-generator GQC codes and 1-generator GQC codes with a fixed parity-check polynomial respectively. Furthermore, we give an algorithm to count numbers of 1-generator GQC codes.     

A New Method of Approximating the Probability of Matching Common Words in Multiple Random Sequences

In this paper we consider R independent sequences of length T formed by independent, not necessarily uniformly distributed letters drawn from a finite alphabet. We first develop a new and efficient method of calculating the expectation \(\mathbb{E}(N_{R}) = \mathbb{E}(N_{R}(m,T))\) of the number of distinct words of length m, N _R (m, T), which are common to R such sequences. We then consider the case of four uniformly distributed letters. We determine a b _R ?=?b _R (m, T)?≥?0 such that the interval \([\mathbb{E}(N_{R}) - b_{R}; \mathbb{E}(N_{R})]\) contains the probability p _R ?=??(N _R ?≥?1) that there exists a word of length m common to the R sequences. We show that \(b_{R} \approx 0.07\mathbb{E}(N_{R})\) if R?=?3 and \(b_{R} \leq 0.05 \mathbb{E}(N_{R})\) if R?≥?4. Thus, for unusual common words, i.e. such that p _R is small, E(N _R ) provides a very accurate approximation of this probability. We then compare numerically the intervals \([\mathbb{E}(N_{R})-b_{R}, \mathbb{E}(N_{R})]\) with former approximations of p _R provided by Karlin and Ost (Ann Probab 16:535–563, 1988) and Naus and Sheng (Bull Math Biol 59(3):483–495, 1997).     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Some <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M ⁿ , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m? the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m? goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L ^p -norm of R m? is finite. Moreover, If R is positive, then (M ⁿ , g) is compact. As applications, we prove that (M ⁿ , g) is isometric to a spherical space form if for \(p\geq \frac n2\), R is positive and the L ^p -norm of R m? is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).     

Some results on the annihilator graph of a commutative ring

Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann _R (xy) ≠ ann _R (x) ∪ ann _R (y), where for z ∈ R, ann _R (z) = {r ∈ R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n ? 1.     

Hopf Modules and Representations of Finite Wreath Products

For a finite group G and nonnegative integer n ≥ 0, one may consider the associated tower \(G \wr S_{n} := S_{n} \ltimes G^{n}\) of wreath product groups. Zelevinsky associated to such a tower the structure of a positive self-adjoint Hopf algebra (PSH-algebra) R(G) on the direct sum over integers n ≥ 0 of the Grothendieck groups K ₀(R e p?G?S _n ). In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) associated to finite groups H ? G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k ^{t h} -power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms R(H) → R(G) and R(G) → R(H) arising from induction and restriction. In an appendix, we consider a family of subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n).     

Homological invariants of modules over contracting endomorphisms

It is proved that when R is a local ring of positive characteristic, \({\phi\colon R{\to} R}\) is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through \({\phi}\) , then R is regular. This broad generalization of Kunz’s characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If \({\phi}\) is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over \({\phi}\) of any non-zero finitely generated R-module grow at the same rate, on an exponential scale, as the Betti numbers of k over R.     

On decompositions of trigonometric polynomials

Let R _t [θ] be the ring generated over R by cosθ and sinθ, and R _t (θ) be its quotient field. In this paper we study the ways in which an element p of R _t [θ] can be decomposed into a composition of functions of the form p = R ? q, where R ∈ R(x) and q ∈ R _t (θ). In particular, we describe all possible solutions of the functional equation R ₁ ? q ₁ = R ₂ ? q ₂, where R ₁,R ₂ ∈ R[x] and q ₁, q ₂ ∈ R _t [θ].     

Isometric embeddings of finite metric spaces

It is proved that there exists a metric on a Cantor set such that any finite metric space whose diameter does not exceed 1 and the number of points does not exceed n can be isometrically embedded into it. It is also proved that for any m, n ∈ N there exists a Cantor set in R^m that isometrically contains all finite metric spaces which can be embedded into R^m, contain at most n points, and have the diameter at most 1. The latter result is proved for a wide class of metrics on R^m and, in particular, for the Euclidean metric.     

The regular digraph of ideals of a commutative ring

Let R be a commutative ring and Max?(R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by \(\overrightarrow{\Gamma_{\mathrm{reg}}}(R)\), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γ_reg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R, we prove that |Max?(R)|?1≦ω(Γ_reg(R))≦|Max?(R)| and \(\chi(\Gamma_{\mathrm{ reg}}(R)) = 2|\mathrm{Max}\, (R)| -k-1\), where k is the number of fields, appeared in the decomposition of R to local rings. Among other results, we prove that \(\overrightarrow{\Gamma_{\mathrm{ reg}}}(R)\) is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined.     

On soluble groups of module automorphisms of finite rank

Let R be a commutative ring, M an R-module and G a group of R-automorphisms of M, usually with some sort of rank restriction on G. We study the transfer of hypotheses between M/C _M (G) and [M,G] such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose [M,G] is R-Noetherian. If G has finite rank, then M/C _M (G) also is R-Noetherian. Further, if [M,G] is R-Noetherian and if only certain abelian sections of G have finite rank, then G has finite rank and is soluble-by-finite. If M/C _M (G) is R-Noetherian and G has finite rank, then [M,G] need not be R-Noetherian.     

Classification of finite commutative rings with planar,toroidal, and projective line graphs associated with Jacobson graphs

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by J _R , is a graph with vertex-set R J(R), such that two distinct vertices a and b in R J(R) are adjacent if and only if 1 ? ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L(J _R ). In this paper, we characterize all finite commutative rings R such that the graphs L(J _R ) are planar, toroidal or projective.     

Realizability of Two-dimensional Linear Groups over Rings of Integers of Algebraic Number Fields

Given the ring of integers O _K of an algebraic number field K, for which natural numbers n there exists a finite group G???GL(n, O _K ) such that O _K G, the O _K -span of G, coincides with M(n, O _K ), the ring of (n?×?n)-matrices over O _K ? The answer is known if n is an odd prime. In this paper we study the case n?=?2; in the cases when the answer is positive for n?=?2, for n?=?2m there is also a finite group G???GL(2m, O _K ) such that O _K G?=?M(2m, O _K ).     

On control of the prime spectrum of a finite simple group

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H ₁, …, H _m of some special form such that π(H ₁)∪…∪π(H _m ) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections H _i so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H ₁,…,H _m } controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.     

Several Generalizations of the Wedderburn-Artin Theorem with Applications

We say that an R-module M is virtually semisimple if each submodule of M is isomorphic to a direct summand of M. A nonzero indecomposable virtually semisimple module is then called a virtually simple module. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules . Our study provides several natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring R: (i) Every finitely generated left (right) R-module is virtually semisimple; (ii) Every finitely generated left (right) R-module is a direct sum of virtually simple R-modules; (iii) \(R\cong {\prod }_{i = 1}^{k} M_{n_{i}}(D_{i})\) where k,n ₁,…,n _k ∈ ? and each D _i is a principal ideal V-domain; and (iv) Every nonzero finitely generated left R-module can be written uniquely (up to isomorphism and order of the factors) in the form R m ₁ ⊕… ⊕ R m _k where each R m _i is either a simple R-module or a virtually simple direct summand of R.