Orthosymplectic Quantum Function Superalgebras OSPq（2l＋ 1｜2n）

By the R-matrix of orthosymplectic quantum superalgebra U _q (osp(2l+1|2n)) in the vector representation, we establish the corresponding quantum Hopf superalgebra OSP _q (2l + 1|2n). Furthermore, it is shown that OSP _q (2l + 1|2n) is coquasitriangular.     

On (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-semigroups

We generalize Green’s lemma and Green’s theorem for usual binary semigroups to (n,m)-semigroups, define and describe the regularity for an element of an (n,m)-semigroup, give some criteria for an element of an (n,m)-semigroup to be invertible, and further apply the invertibility for (n,m)-semigroups to (n,m)-groups and give some equivalent characterizations for (n,m)-groups. We establish Hosszú-Gluskin theorems for (n,m)-semigroups in two cases, as generalizations of the corresponding theorems for n-groups.     

The <Emphasis Type="Italic">s</Emphasis>-energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S ². A spherical n-design is a point set on S ² that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E _s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X _m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).      

Compact embedded rotation hypersurfaces of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

In this paper, we prove that and round geodesic spheres are the only n-dimensional compact embedded rotation hypersurfaces with H_m = 0 (1 ≤ m ≤ n − 1) in a unit sphere Sⁿ⁺¹(1). When m = 1, our result reduces to the result of T. Otsuki [O1], [O2], Brito and Leite [BL]. The project is supported by the grant No. 10531090 of NSFC.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

<Emphasis Type="Italic">N</Emphasis>-commutators

The N-commutator is conjecturally a well-defined nontrivial operation on for x = (x ₁, ... , x _n ) if and only if N = n ² + 2n - 2. This is proved for n = 2 and confirmed by computer experiments for n < 5. Under 2- and 5-commutators the algebra of divergence-free vector fields in two dimensions is an sh-Lie (strong homotopic Lie) algebra in the sense of Stasheff. Similarly, W(2) is an sh-Lie algebra with respect to 2- and 6-commutators.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> boundedness for multilinear fractional integral on product spaces

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).     

On Resolvable Multipartite <Emphasis Type="Italic">G</Emphasis>-Designs

A necessary and sufficient condition for the existence of a _km–factorization of the complete symmetric k–partite multi-digraph K*(n₁,n₂,...,n_k) is obtained for odd k. As a consequence, a resolvable (k,n,km,) multipartite _km–design exists for odd k if and only if m|n. This deduces a result of Ushio when m=1 and k=3. Further, a necessary and sufficient condition for the existence of a _km–factorization of is established for even k, where denotes the wreath product of graphs. Finally, a simple and short proof for the non-existence of a _k–factorization of is obtained for odd k.Acknowledgments.The author thanks Dr. P. Paulraja for his useful ideas in writing this paper and the Department of Science and Technology, New Delhi, for its support (Project Grant No. DST/MS/103/99).Final version received: November 17, 2003     

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

Generalizing Pancyclic and <Emphasis Type="Italic">k</Emphasis>-Ordered Graphs

Given positive integers kmn, a graph G of order n is (k,m)-pancyclic if for any set of k vertices of G and any integer r with mrn, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k,m)-pancylic are proved. If the additional property that the k vertices must appear on the cycle in a specified order is required, then the graph is said to be (k,m)-pancyclic ordered. Minimum degree conditions and minimum sum of degree conditions for nonadjacent vertices that imply a graph is (k,m)-pancylic ordered are also proved. Examples showing that these constraints are best possible are provided.Acknowledgments. The authors would like to thank the referees for their careful reading of the paper and their useful suggestions.Final version received: January 26, 2004     

A Family of Binary (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">m</Emphasis>,<Emphasis Type="Italic">s</Emphasis>)-Nets of Strength 5

(t,m,s)-Nets were defined by Niederreiter [Monatshefte fur Mathematik, Vol. 104 (1987) pp. 273–337], based on earlier work by Sobol’ [Zh. Vychisl Mat. i mat. Fiz, Vol. 7 (1967) pp. 784–802], in the context of quasi-Monte Carlo methods of numerical integration. Formulated in combinatorial/coding theoretic terms a binary linear (m−k,m,s)₂-net is a family of ks vectors in F₂^m satisfying certain linear independence conditions (s is the length, m the dimension and k the strength: certain subsets of k vectors must be linearly independent). Helleseth et al. [5] recently constructed (2r−3,2r+2,2^r−1)₂-nets for every r. In this paper, we give a direct and elementary construction for (2r−3,2r+2,2^r+1)₂-nets based on a family of binary linear codes of minimum distance 6.Communicated by: T. Helleseth     

Compromise 4<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> plans with clear two-factor interactions

This paper gets some necessary conditions for the existence of some kinds of clear 4^m2^n compromise plans which allow estimation of all main effects and some specified two-factor interactions without assuming the remaining two-factor interactions being negligible. Some methods for constructing clear 4^m2^n compromise plans are introduced.     

<Emphasis Type="Italic">X</Emphasis>-decomposable finite groups for <Emphasis Type="Italic">X</Emphasis>={1, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis> + 1, <Emphasis Type="Italic">m</Emphasis> + 2}

A normal subgroup N of a finite group G is called n-decomposable in G if N is the union of n distinct G-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is m-decomposable, m+1-decomposable, or m+2-decomposable for some positive integer m. Furthermore, we give classification for the soluble case.     

The Dirichlet series for the exterior square <Emphasis Type="Italic">L</Emphasis>-function on GL(<Emphasis Type="Italic">n</Emphasis>)

We present an elementary derivation of the Jacquet–Shalika construction for the exterior square L-function on GL(n), as a classical Dirichlet series in the Fourier coefficients A(m ₁,…,m _n−1).     

Some cyclic codes of length 2<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Explicit expressions for 4n + 2 primitive idempotents in the semi-simple group ring $R_{2p^{n}}\equiv \frac{GF(q)[x]}{p and q are distinct odd primes; n ≥ 1 is an integer and q has order \fracf(2pⁿ)2{\frac{\phi(2p^{n})}{2}} modulo 2p ⁿ . The generator polynomials, the dimension, the minimum distance of the minimal cyclic codes of length 2p ⁿ generated by these 4n + 2 primitive idempotents are discussed. For n = 1, the properties of some (2p, p) cyclic codes, containing the above minimal cyclic codes are analyzed in particular. The minimum weight of some subset of each of these (2p, p) codes are observed to satisfy a square root bound.     

<Emphasis Type="Italic">L</Emphasis>-fuzzy roughness of <Emphasis Type="Italic">n</Emphasis>-ary polygroups

In this paper, we consider the relations among L-fuzzy sets, rough sets and n-ary polygroup theory. Some properties of (normal) TL-fuzzy n-ary subpolygroups of an n-ary polygroup are first obtained. Using the concept of L-fuzzy sets, the notion of ϑ-lower and T-upper L-fuzzy rough approximation operators with respect to an L-fuzzy set is introduced and some related properties are presented. Then a new algebraic structure called (normal) TL-fuzzy rough n-ary polygroup is defined and investigated. Also, the (strong) homomorphism of ϑ-lower and T-upper L-fuzzy rough approximation operators is studied.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where