## Session 1: Algebra, Number Theory, Crytography The FP-projective Dimensions of Modules and Rings

Lixin Mao and Nanqing Ding*
Department of Mathematics
Nanjing University
Nanjing 210093, P.R. China
nqding@nju.edu.cn

Abstract : We define a dimension, called FP-projective dimension, for modules and rings. It measures how far away a finitely generated module is from being finitely presented, and how far away a ring is from being Noetherian. This dimension has nice properties when the ring in question is coherent. The relations between the FP-projective dimension and other homological dimensions are discussed.

## Some Aspects of Exploring Algebraic Structuref

Lianggui Feng
Department of Mathematics and System Science
National University of Defense Technology
Changsha 410073, P. R. China
fenglg2002@yahoo.com

Abstract : Some results obtained in our exploring the related algebraic structure these years are presented in this exposition. It consists of four aspects: (1) The study of radical theory of rings (modules) and generalization; (2) Descriptions and computations on homological dimensions; (3) The structure of K group with lower rank; (4) Large subdirect products and the related structure of prufer rings. Finally, two questions are posed.

Keywords: ring, module, homological dimension, K group

AMS subject classification: 16E50; 16S90

f The author is supported in part by NNSFC(A0324660)

## Some new optimal quaternary constant weight codes

Gennian Ge
Department of Mathematics
Zhejiang University
Hangzhou 310027, Zhejiang
P. R. China
gnge@zju.edu.cn

Abstract : Constant weight codes (CWCs) are an important class of codes in coding theory. Generalized Steiner systems GS(2,k,v,g) were first introduced by Etzion and used to construct optimal nonlinear CWCs over an alphabet of size g+1 with minimum Hamming distance 2k-3, in which each codeword has length v and weight k. In this talk, Weil's theorem on character sum estimates is used to show that there exists a GS(2,4,v,3) for any prime v º 1  (mod  4) and v > 13. From the coding theory point of view, we can get that an optimal nonlinear quaternary (v,5,4) CWC exists for such a prime v.

## Hn-Normal Subgroups

Department of Mathematics
Yazd University
Yazd, Iran

Abstract : The largest class of algebraic hyperstructures satisfying the group-like axiom is called the Hn-group. Here, the notion of an Hn-normal subgroup and anHn- quotient group of an Hn-group are defined. Then the fundamental equivalence relation for Hn-quotient group and the first isomorphism theorem for Hn-groups is obtained.

## Equivalence Classes of Vogan Diagrams

Meng-Kiat Chuah
Department of Applied Mathematics
National Chiao Tung University
Hsinchu, Taiwan
chuah@math.nctu.edu.tw

Chu-Chin Hu*
Department of Applied Mathematics
National Chiao Tung University
Hsinchu, Taiwan
fox.am89g@nctu.edu.tw

Abstract : A Vogan diagram is a Dynkin diagram with an involution, and the vertices fixed by the involution may be painted. They represent real simple Lie algebras, and two diagrams are said to be equivalent if they represent the same Lie algebra. In this talk we classify the equivalence classes of all Vogan diagrams. In doing so, we find that the underlying Dynkin diagrams have certain properties in graph painting. We show that this combinatorial property provides an easy classification for most of the simply-laced Dynkin diagrams.

## New Hadamard Matrices of Order 4p2 Obtained from Jacobi Sums of Order 16

Ka Hin Leung*
Department of Mathematics
National University of Singapore
Kent Ridge, Singapore 119260
Republic of Singapore
matlkh@@nus.edu.sg

Siu Lun Ma
Department of Mathematics
National University of Singapore
Kent Ridge, Singapore 119260
Republic of Singapore
matmasl@@nus.edu.sg

Bernhard Schmidt
Institut für Mathematik
Universität Augsburg
86135 Augsburg
Germany
schmidt@@math.uni-augsburg.de

Abstract : A Hadamard matrix of order v is a v×v matrix H with entries ±1 such HHt = vI where I is the identity matrix. A Hadamard matrix is called regular if each row contains the same number of entries 1. It is conjectured that a Hadamard matrix of order v > 2 exists if and only if v is divisible by 4.

While the construction of Hadamard matrices of order 4t for arbitrary t seems out of reach at the present time, there may be some hope to construct Hadamard matrices of order 4q2 for all prime powers q. For q º 1\mod 4 and q º 3\mod 8 this already has been accomplished by the marvelous work of Mingyuan Xia and Gang Liu. The constructions of Xia and Liu are based on cyclotomy, namely, the use of 4th, 8th and (q+1)th cyclotomic classes in \Bbb Fq2. However, it seems that the difficulty of implementing the approach using cyclotomy increases with the exact power of 2 dividing q+1, In fact, up to our knowledge no general constructions for Hadamard matrices of order 4q2 with q º 7\mod 8 have been known.

In the present paper, we obtain two putative infinite families of Hadamard matrices of order 4q2 with q º 7\mod 8 prime. We believe that, for any large enough n, our constructions yield at least 5/8n2/5 primes q < n, q º 7\mod16 such that a regular Hadamard matrices of order 4q2 exists. Our approach is based on 16th and (q+1)th cyclotomic classes.

## The l-Groebner Bases under Composition

Jinwang Liu
College of Mathematics and Computation
Hunan Science and Technology University,
Xiangtan,Hunan,China, 411201
College of Mathematics and Computation
Hunan Normal University
Changsha, Hunan,China, 410081
e-mail: jwliu@xtnu.edu.cn

Abstract : In several recent papers Hoon Hong and Gutieerz addressed the problem of the behaviour of groebner bases under composition of polynomials..Adressing the problem of some Groebner bases under composition is extraordinary significant,we mainly probe into the behaviour of l-Groebner bases under composition.

Let k be a field,k[x1,x2,¼,xn] be a polynomial ring in the variables x1,x2,¼,xn with coefficient from k, > be a term ordering.Composition Q is the operation of replacing variable in a polynomial with other polynomials.

Let F be a finite set of polynomials in k[x1,x2,¼,xn,] and let G be a Groebner basis of the ideal generated by F under the term ordering > .Let Q = (q1,q2,¼,qn) be a list of n polynomials in k[x1,x2,¼,xn].F°Q be the set obtained from F by replacing xi by qi.One ponders whether G°Q is also a Groebner basis of F°Q.If for any F,G;G°Q is also a Groebner basis of F°Q,we say Groebner basis computation commute with composition by Q.We ponder whether Groebner basis l-Groebner basis computation commute with composition b Q for some F's and G's.

Lemma 1[6]    Groebner basis computation commute with composition by Q if and only if

(a)    "p "q,p > qÞ p°lt(q) > q°lt(q) > q°lt(q);and

(b)    the list lt(q) is a 'permuted powering',the ltqi and ltqj are pairwise relatively prime.

Let f,g Î k[x1,x2,¼,xn],we denote the leading power product of f by lp(f),the leading coefficient of f by lc(f),the leading term of f by lt(f),that is,lt(f) = lc(f)lp(f).

Lemma 2 [1]    Let d be a greatest common divisor of f and g,then {f,g} is a Groebner basis if and only if lp(g/d) and lp(g/d) is relatively prime.

Theorem 1.    Let lp(f) = x1u1x2u2¼xnun, lp(g) = x1v1x2v2¼xnvn,and there is k such that degxklp(f) = uk ³ degxklp(g);then {f,g} is not a Groebner basis,or {f+h,g} is not a Groebner basis,here lp(h) < lp(f),degxklp(h) < degxklp(g).

Theorem 2.    If degxklp(f) ¹ 0,  degxklp(g) ¹ 0,for any integer l > 0,then there are integers l,m,s such that {fm,gs} is not a Groebner basis,and or {fm+fl,gs} is not a Groebner basis.

Let t1,t2,¼,ts are all items of f,if degti ³ l,"i;we call that f l-degree.Let G = {g1,g2.¼,gs} be a Groebner basis,if "gi,gi¢s are l-degree,we call that G l-Groebner basis.

Theorem 3.    l-Groebner basis computation commute with composition by Q if and only if

(a)   "p "q,  degp.degq ³ l,  p > qÞ p°lt(q) > q°lt(q);and

(b)   the list lt(q) is a 'powering'.

Reference

[1] Adams,W., Loustaunau, P., An Introduction to Gröbner Bases, Graduate Studies in Mathematics 3, Amer.Math.Soc., Providence, 1994.

[2] Becker,T. and Weispfenning,V.: Gröbner Bases-A Computational Approach to Commutative Algebra, GTM 141, New York: Springer, 1993.

[3] Gutierrez,J.,etc. Reduced Gröbner bases under composition, J. Symb. Comput. 26, 433-444, 1998.

[4] Greuel,G., Pfister,G., Advances and improvements in the theory of standars bases and syzybies, Arch.Math., Vol.66, 163-176, 1906.

[5] Gräbe,H., The tangent cone algorithm and homogeization, J.pure Applied Algebra, Vol.97, 303-312, 1994.

[6] Hong,H: Gröner Bases Under Composition I, J. Symb. Comput., 25, 643-663, 1978.

[7] Jinwang,L.: The term orderings which are compatible with comosition II, J. Smb. Comput,(2003).35,153-168.

## Representations of IT-biordered Sets and their Regular Semigroups

Xilin Tang
Department of Mathematics
South China University of Technology
Guangzhou, Guangdong, P.R.China
Xilintang@21cn.com

Abstract : Let S be a regular semigroup. An inverse subsemigroup S° of S is called an inverse transversal of S if S° contains a unique inverse x° of each element x of S. An inverse transversal S° of S is called a Q-inverse transversal of S if S° is a quasi-ideal of S. A regular biordered set E is an IT-biordered set if there is a regular semigroup S with sets of idempotent E and S has inverse transversals. The class of regular semigroups with inverse transversals contains inverse semigroups, completely simple semigroups, etc..

The representations for regular semigroups by means their biordered sets are very important in theoretical of semigroups. In this paper, we gives representations for the class of regular biordered sets firstly. On the basis of it, and give the representations for IT-biordered sets and their regular semigroups. The representation for a fully fundamental regular semigroups of an IT-biordered set is an analogue of the Munn representations on semilattices or the Hall representations on bands.

## On Non-abelian Clemens-Schmid Exact Sequences

Yen-Lung Tsai
National Center for Theoretical Sciences
Mathematics Division
3rd General Building, National Tsing Hua University
Hsinchu 30043, Taiwan, R.O.C.
yenlung@math.cts.nthu.edu.tw

Abstract : Let f be a degeneration of Kähler manifolds. The Clemens-Schmid exact sequence studies the Picard-Lefchetz transformation and gain some important properties of the degeneration. In hope of gaining more information of the degeneration, we survey the possibility of generalizing classical (abelian) Clemens-Schmid exact sequence to non-abelian one. We will provide ``counterexamples'' to show that we cannot use the same formulation as in abelian case. Some notions of non-abelian Clemens-Schmid exact sequences will be discussed.

## Unique Range sets, Uniqueness Polynomials, and some Related Problems

Julie Tzu-Yueh Wang
Institute of Mathematics
Nankang, Taipei 115
Taiwan
jwang@math.sinica.edu.tw

Abstract : We discuss unique range sets, uniqueness polynomials, and other uniqueness problems for entire functions and meromophic functions. We also study their analogous problems for number fields.

## On Hyperlattice

Xiao-long Xin
Department of Mathematics
Northwest University
Xi'an, Shaanxi 710069, P.R.China
xlxin@nwu.edu.cn

Abstract : In this paper, we give the concepts of hyperlattice, subhyperlattice, and complemented hyperlattice, and study a few results in these aspects. We also give the definitions of homomorphism and isomorphism of hyperlattices, then investigate a series of properties of hyperlattice.

Keywords: hyperlattice; distributive hyperlattice; complemented hyperlattice

MSC: 06B99

## On Primitive Roots for Rank One Drinfeld Modules

Wei-Chen Yao
Department of Mathematics and Computer Science Education
Taipei Municipal Teachers College
Taipei, Taiwan, ROC
yao@mail1.tmtc.edu.tw

This is a joint work with Jing Yu

Abstract : Let k be a function field of one variable over a finite field, ¥ a fixed prime divisor of k, and let A be the ring of the elements of k which are integral at all primes of k other than ¥. Let K be an A-field of finite A-characteristic and let f be a rank one Drinfeld A-module over K. Given a Î K, we show that the set

 M a f = {\frakP place of K \mid a is a primitive root modulo \frakP for f}
possesses a Dirichlet density. Furthermore, we also give a criterion for this density to be positive. This is an analogue of Bilharz' version of Artin's primitive roots conjecture.

## Relatively Projective Modules and Generalized Charactersf

Jiwen Zeng*
The Department of Mathematics
Xiamen University
Xiamen, 361005, P. R. China
jwzeng@jingxian.xmu.edu.cn

Zhang Japing
Mathematics Institute
Beijing University
Beijing, 100871, P. R. China

Abstract : In this paper, we introduce several matrices to describe the relations among the Külshammer-Robinson basis, projective modules, the ordinary irreducible characters and Brauer irreducible characters. We show that there are various equalities of Cartan matrices, decomposition matrices of the blocks according to the local blocks via Brauer correspondence. By using these equalities, we prove that there are several inequalities about the ordinary irreducible characters and Brauer irreducible characters, which imply their new relations and pure group results. Some results are given to describe the relations between generalized characters and Külshammer-Robinson basis.

f Research partially supported by CNSF and CSC.

## On higher-power Moments of D(x)

Wenguang Zhai
School of Mathematics Sciences
Shandong Normal University
Jian, Shandong 250014
P.R.China
zhaiwg@hotmail.com

Abstract : Let D(x) denote the error term of the Dirichlet divisor problem . Dirichlet first proved D(x) = O(x1/2). The exponent 1/2 was improved by many authors. The latest result is D(x) << x131/416(logx)26957/8320 proved by Huxley. The conjectured bound is D(x) = O(x1/4+e), which is supported by the classical mean-square formula of D(x). Tsang established the asymptotic formulas of the third- and fourth-power moments of D(x). Now we established the asymptotic formula of the k-th power moment of D(x) for each k Î {5,6,7,8,9}.

File translated from TEX by TTH, version 2.00.
On 09 Dec 2004, 09:46.