THE CLASSES OF SOME POSITIVE DEFINITE UNIMODULAR LATTICES OVER Z[3~(1/2)]AND Z[6~(1/2)]

In this paper,the author generalizes Kneser's method which was used by Kneser,R.Salamon and Y.Minura,and applies this method to determine the classes of some positivedefinite unimodular lattices over Z[3~(1/2)]and Z[6~(1/2].     

A coupled continuous-discontinuous FEM approach for convection diffusion equations

In this article, we introduce a coupled approach of local discontinuous Galerkin and standard finite element method for solving convection diffusion problems. The whole domain is divided into two disjoint subdomains. The discontinuous Galerkin method is adopted in the subdomain where the solution varies rapidly, while the standard finite element method is used in the other subdomain due to its lower computational cost. The stability and a priori error estimate are established. We prove that the coupled method has O((ε1 / 2 + h 1 / 2 )h k ) convergence rate in an associated norm, where ε is the diffusion coefficient, h is the mesh size and k is the degree of polynomial. The numerical results verify our theoretical results. Moreover, 2k-order superconvergence of the numerical traces at the nodes, and the optimal convergence of the errors under L 2 norm are observed numerically on the uniform mesh. The numerical results also indicate that the coupled method has the same convergence order and almost the same errors as the purely LDG method.     

A Spectral Element/Laguerre Coupled Method to the Elliptic Helmholtz Problem on the Half Line

A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results. The advantage of this method is demonstrated by a numerical comparison with the pure Laguerre method.     

C^1 C^2 INTERPOLATION OF SCATTERED DATA POINTS

In this paper an error in [4] is pointed out and a method for constructing surface interpolating scattered data points is presented, The main feature of the method in this paper is that the surface so constructed is polynomial, which makes the construction simple and the calculation easy.     

A Switching Algorithm Based on Modified Quasi-Newton Equation

In this paper, a switching method for unconstrained minimization is proposed. The method is based on the modified BFGS method and the modified SR1 method. The eigenvalues and condition numbers of both the modified updates are evaluated and used in the switching rule. When the condition number of the modified SR1 update is superior to the modified BFGS update, the step in the proposed quasi-Newton method is the modified SR1 step. Otherwise the step is the modified BFGS step. The efficiency of the proposed method is tested by numerical experiments on small, medium and large scale optimization. The numerical results are reported and analyzed to show the superiority of the proposed method.     

块图的2-步长控制问题的一个线性时间算法

The 2-step domination problem is to find a minimum vertex set D of a graph such that every vertex of the graph is either in D or at distance two from some vertex of D.In the present paper,by using a labeling method,we provide an O(m) time algorithm to solve the2-step domination problem on block graphs,a superclass of trees.     

Burgers方程的带有限制算子的Chebyshev谱方法

In this paper, a restraint operator is used to improve the stability of the Chebyshev spectral method. The generalized stability of this new method is proved and the rate of convergence is analyzed. The numerical results show the advantage of the method.     

An Improved Hybrid Method for Inverse Obstacle Scattering Problems

An improved hybrid method is introduced in this paper as a numerical method to reconstruct the scatterer by far-field pattern for just one incident direction with unknown physical properties of the scatterer. The improved hybrid method inherits the idea of the hybrid method by Kress and Serranho which is a combination of Newton and decomposition method, and it improves the hybrid method by introducing a general boundary condition. The numerical experiments show the feasibility of this method.     

ON DEGENERATE WEAKLY （K1, K2）-QUASIREGULAR MAPPINGS

The authors first give the definition of degenerate weakly （K1,K2）-quasiregular mappings using the technique of exterior power and exterior differential forms, and then, using the method of McShane extension, a useful inequality is obtained, which can be used to derive the self-improving regularity.     

WEIGHTED NONLINEAR REGRESSION

Minimization of the weighted nonlinear sum of squares of differences may be converted to the minimization of sum of squares. The Gauss-Newton method is recalled and the length of the step of the steepest descent method is determined by substituting the steepest descent direction in the Gauss-Newton formula. The existence of minimum is shown.     

一类$L^2(\mathbb{R}^n)$的紧支撑不可分正交小波

本篇文章给出一类$L^{2}(\mathbb{R}^{n})$, $n\geq2$的紧支撑不可分正交小波基的具体构造算法,其中正交小波的伸缩矩阵为$\alpha I_{n}~(\alpha\geq2,\ \alpha \in \mathbb{Z})$, $I_{n}$是$n$阶单位矩阵.最后给出两个不可分正交小波基的构造算例.     

Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring

The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.     

A characterization of $$\mathbb {Z}_{2}\mathbb {Z}_{2}[u]$$-linear codes

We prove that the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is exactly the class of \(\mathbb {Z}_2\)-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which have a nontrivial \(\mathbb {Z}_2\mathbb {Z}_2[u]\) structure. Moreover, we exhibit some examples of \(\mathbb {Z}_2\)-linear codes which are not \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear. Also, we state that the duality of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is the same as the duality of \(\mathbb {Z}_2\)-linear codes. Finally, we prove that the class of \(\mathbb {Z}_2\mathbb {Z}_4\)-linear codes which are also \(\mathbb {Z}_2\)-linear is strictly contained in the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes.     

psl(2|2) 非线性α -模型超对称性质研究

由$\widehat{psl(2|2)^{(2)}}_{k}$非线性$\sigma$ -模型加上WZ -项得到的WZW模型是共形场论,它具有李超代数$psl(2|2)$对称性.该文用向量相干态方法给出了李超代数$psl(2|2)$的微分算子表示.并在此基础上给出了扭曲Kac-Moody李超代数 $\widehat{psl(2|2)^{(2)}}_{k}$自由场实现,相应共形场论的中心荷为$-2$.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

关于一类中心循环的有限P-群的自同构群

确定了一类中心循环的有限p-群G的自同构群.设G=X_3(p~m)~(*n)*Z_(p~(m+r)),其中m≥1,n≥1和r≥0,并且X_3(p~m)=x,y|x~(p~m)=y~(p~m)=1,[x,y]~(p~m)=1,[x,[x,y]]=[y,[x,y]]=1.Aut_nG表示Aut G中平凡地作用在N上的元素形成的正规子群,其中G'≤N≤ζG,|N|=p~(m+s),0≤s≤r,则(i)如果p是一个奇素数,那么AutG/Aut_nG≌Z_(p~((m+s-1)(p-1))),Aut_nG/InnG≌Sp(2n,Z_(p~m))×Z_(p~(r-s)).(ii)如果p=2,那么AutG/Aut_nG≌H,其中H=1(当m+s=1时)或者Z_(2~(m+s-2))×Z_2(当m+s≥2时).进一步地,Aut_nG/InnG≌K×L,其中K=Sp(2n,Z_(2~m))(当r0时)或者O(2n,Z_(2~m))(当r=0时),L=Z_(2~(r-1))×Z_2(当m=1,s=0,r≥1时)或者Z_(2~(r-s)).     

ISOGENOUS OF THE ELLIPTIC CURVES OVER THE RATIONALS

AbstractAn elliptic curve is a pair (E,O), where ?is a smooth projective curve of genus 1 and O is a point of E, called the point at infinity. Every elliptic curve can be given by a Weierstrass equationE:y2 a1xy a3y = x3 a2x2 a4x a6.Let Q be the set of rationals. E is said to be dinned over Q if the coefficients ai, i = 1,2,3,4,6 are rationals and O is defined over Q.Let E/Q be an elliptic curve and let E(Q)tors be the torsion group of points of E denned over Q. The theorem of Mazur asserts that E(Q)tors is one of the following 15 groupsE(Q)tors　Z/mZ, m = 1,2,..., 10,12,Z/2Z × Z/2mZ, m = 1,2,3,4.We say that an elliptic curve E'/Q is isogenous to the elliptic curve E if there is an isogeny, i.e. a morphism : E E' such that (O) = O, where O is the point at infinity.We give an explicit model of all elliptic curves for which E(Q)tors is in the form Z/mZ where m= 9,10,12 or Z/2Z × Z/2mZ where m = 4, according to Mazur's theorem. Morever, for every family of such elliptic curves, we give an explicit m     

一个特殊6点图Q与nK_1,P_n及C_n的联图交叉数

图的交叉数是图的一个重要参数,研究图的交叉数问题是拓扑图论中的前沿难题.确定图的交叉数是NP-难问题,因为其难度,能够确定交叉数的图类很少.通过圆盘画法途径,确定了一个特殊6点图与n个孤立点nK_1,路P_n及圈C_n的联图的交叉数分别是cr(Q+nK_1)=Z(6,n)+2[n/2],cr(Q+P_n)=Z(6,n)+2[n/2]+1及cr(Q+C_n)=Z(6,n)+2[n/2]+3.     

整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} be a function such that f-1(0) is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets A  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.     

DIRECT EXPANSIONS FOR THE DISTRIBUTION FUNCTIONS AND DENSITY FUNCTIONS OF χ2-TYPE AND t-TYPE DISTRIBUTED RANDOM VARIABLES

Suppose that Z1,Z2…,Zn are independent normal random variables with common mean μ and variance σ^2. Then S^2=∑n n=1 (zi-z)^2/σ^2 and T =（n-1的平方根）-Z/（S^2/n的平方根） have x2n-1 distribution and tn-1 distribution respectively. If the normal assumption fails, there will be the remainders of the distribution functions and density functions. This paper gives the direct expansions of distribution functions and density functions of S^2 and T up to o(n^-1). They are more intuitive and convenient than usual Edgeworth expansions.