特征不为 2 的有限域上酉群的极小生成元集

设K=Fq2为含有q2个元素的有限域,q为奇素数的幂,*：a→a*=aq是Fq2的一个二阶自同构．本文用几何方法证明了除K为F32而n=4的情形外,Fq2上的酉群Un（V）可由2个元素生成．     

关于有限域上的置换二项式

设l为一奇正整数,q是某素数的方幂,二者满足l|q-1,记s=(q-1)/l;又设Fq是一个q元有限域,r,e为正整数,(e,l)=1.本文应用序列{an=∑(l-1)/2t=1(2(-1)t-1cos(tπ/l))n}∞n=-∞的性质给出了当l=9时Fq上的二项式f(x)=xr(1+xes)成为Fq上的置换多项式的充要条件.     

理想类群指数≤2的虚二次函数域

本文给出虚二次函数域K=k~（1/2）D（k=Fq（x）,2+q）的理想类群指数≤2的一个必要条件.在某些条件下,这个条件也是充分的.特别地,给出了Louboutin[1]一个结果在函数域情形的模拟.     

有限域上本原多项式系数的研究

设Fq表示有q个元素的有限域,q为素数的方幂,f(x)=xn+a1xn-1+…+an-1x+an∈Fq[x].当n≥7时,文[8]指出存在Fq上可预先指定a1,a2的n次本原多项式.本文讨论了剩余的n=5,6两种情形,利用有限域上的两类特征和估计及Cohen筛法(见[4,6]),改进了文[8]中关于本原解个数的下界,并得到当n=5,6时,在特征为奇的有限域上存在可预先指定前两项系数的n次本原多项式.     

有限域上本原多项式系数的研究

设Fq表示有q个元素的有限域,q为素数的方幂,f(x)=xn+a1xn-1+…+an-1x+an∈Fq[x].当n(≥)7时,文[8]指出存在Fq上可预先指定a1,a2的n次本原多项式.本文讨论了剩余的n=5,6两种情形,利用有限域上的两类特征和估计及Cohen筛法(见[4,6]),改进了文[8]中关于本原解个数的下界,并得到当n=5,6时,在特征为奇的有限域上存在可预先指定前两项系数的n次本原多项式.     

Gauss Sum of Index Four

Let p be a prime number, q = pf and m| q - 1. For a multiplicative character χ of Fq with order m, the Gauss sum G(χ) over Fq is defined by G(χ)=∑(X()Fq)χ(x)ζTP(x),ζp=e2πi/p,(1)where T is the mapping for Fq/Fp.     

Gauss Sum of Index 4： （1） Cyclic Case

Let p be a prime, m ≥ 2, and （m,p（p - 1）） = 1. In this paper, we will calculate explicitly the Gauss sum G（X） = ∑x∈F＊qX（x）ζ^Tp^（x） in the case of [（Z/mZ）＊ ： （p）] = 4, and -1 （不属于） （p）, where q P^f, f =φ（m）/4, X is a multiplicative character of Fq with order m, and T is the trace map for Fq/Fp. Under the assumptions [（Z/mZ）＊ ： （p）] = 4 and 1（不属于） （p）, the decomposition field of p in the cyclotomic field Q（ζm） is an imaginary quartic （abelian） field. And G（X） is an integer in K. We deal with the case where K is cyclic in this oaDer and leave the non-cvclic case to the next paper.     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

有限域上一类方程解数的直接公式

本文给出有限域F=Fq上一类方程(?)当指数满足一定条件时,在Fn2上解数的一个直接公式,这里dij＞0,ai∈F*,b∈F,q=pf,f≥1,p是一个奇素数,0＜n1 ≤ n2.     

有限域上存在弱自对偶正规基的一个充要条件

对于将有限域上的自对偶基概念推广到了更一般的弱自对偶的情形,给出了有限域上存在这类正规基的一个充要条件设q为素数幂,E=Fqn为q元域F=Fq的n次扩张,N={αi=αq2| i=0,1,…,n-1}为E在F上的一组正规基.则存在c∈F*及r,0≤r≤n-1,使得β=cαr生成N的对偶基的充要条件是以下三者之一成立(1)q为偶数且n≠0(mod 4);(2)n与q均为奇数;(3)q为奇数,n为偶数,(-1)为F中的非平方元且r为奇数.     

The Collineation Groups of Generalized Twisted Field Planes

We investigate autotopisms and isotopisms of generalized twisted field planes. A revision of Alberts results is developed using a mixture of group-theoretical and algebraic methods. In particular, transitivity conditions of the autotopism group on the nonvertex points of one or more sides of the autotopism triangle are determined. Also, correlations and polarities in generalized twisted field planes are examined in detail. These results allow us to obtain several improvements of recent results in semifield planes.     

关于拟Einstein场方程

In this paper some properties of a symmetric tensor field T(X, Y) = g(A(X), Y) on a Riemannian manifold (M,g) without boundary which satisfies the quasi-Einstein equation Rij-S/2gij=Tij+bξiξj are given. The necessary and sufficient conditions for this tensor to satisfy the quasi-Einstein equation are also obtained.     

模糊一致体

定义了模糊一致体概念,研究了它与模糊拓扑体的关系及它与模糊一致空间的关系;给出了借助于体的模糊子集族对模糊一致体的刻画,还引入了模糊一致子体,模糊一致商体与模糊一致体的直积;并讨论了它们的分离性.     

Symmetries of Field Theories on the Noncommutative Plane

We report new developments concerning the symmetry properties and their actions on special solutions allowed by certain field theory models on the noncommutative plane. In particular, we seek Galilean-invariant models. The analysis indicates that this requirement strongly restricts the admissible interactions. Moreover, if a scalar field is coupled to a gauge field, then a geometric phase emerges for vortexlike solutions transformed by Galilean boosts.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 64–73, July, 2005.     

Higgs场的虫洞解

已有不少人讨论过引力场与各种物质场耦合下的虫洞解,有鉴于Ｈｉｇｇｓ场在现代场论中的极端重要性,探求引力场与Ｈｉｇｇｓ场耦合下是否存在虫洞解就是一个有意义的问题,该文指出在一定条件下,虫洞解是存在的。     

Instanton in the Field of a Pointlike Source of a Euclidean Non-Abelian Field

We consider the behavior of an (anti)instanton in the field of a pointlike source of a Euclidean non-Abelian field and investigate (anti)instanton deformations described by variations of its characteristic parameters. We formulate the variational problem of seeking the corresponding “crumpled” topological configurations and solve it algebraically using the Ritz method (of multipole decomposition of deformation fields). We investigate the domain of parameters specific for the instanton liquid model. We propose a simple method of taking contributions of such configurations in the functional integral into account approximately. In the framework of superposition analysis, we obtain an estimate for the mean energy of the source in the instanton liquid, and the estimate increases linearly with distance. In the case of a dipole in the color-singlet state, the energy is a linear function of the distance between the sources with the “strength” coefficient agreeing well with the known model and lattice estimates. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 267–298, February, 2006.     

典型电磁现象的可视化实现

介绍了典型电磁现象的可视化实现过程.用计算机模拟计算出典型电磁现象在空间的分布情况,因此我们可以用图形方式来观察典型电磁现象.同时给出了几个具体实例.     

Species Over a Finite Field

We generalize Joyals theory of species to the case of functors from the groupoid of finite sets to the category of varieties over F_q. These have cycle index series defined by counting fixed points of twisted Frobenius maps. We give an application to configuration spaces.     

有限域的子空间表示及其应用

本文研究了有限域在其子域上向量子空间的表示问题,推广了文献[3]中的一个结果,并给出了关于Singer差集的一个构造性证明；最后,利用有限域的所有超平面作为集合,构作了一类结合方案,并计算了它们的参数.     

关于向量场旋度的一个附注

若在某单连通区域V∈R3上,向量函数A是某个数量函数u的梯度,则其旋度rotA在所定义的区域上为零.本文证明了其旋度的任意两个分量在所定义的区域V上为零,另一个分量满足在区域V的边界S上为零与其三个分量在所定义的区域V上为零是等价的,解惑教学中学生的问题.