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Table of Content

      
    20 February 2013
    Volume 48 Issue 2
    Articles
    THE TWO GOLDBACH CONJECTURES
    Ming-Chit Liu
    J4. 2013, 48(2):  1-14. 
    Abstract ( 415 )   PDF (888KB) ( 863 )   Save
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    The article is for general readers who may have heard the name, Goldbach or Jingrun Chen (陈景润) and would like to know some mathematical problems related to them. There are four sections in the article. The first two sections are for those who are only interested in the stories, developments, latest results of the two Goldbach conjectures, their differences in difficulty and Chen’s achievement. If after having read the first two sections, one would like to know more about mathematical information and explanation on the contents in the first two sections, one may go further to the last two sections.

    On the fibre product of Zn and its property
    CHENG Zhi1,2, SUN Cui-fang2, WANG Ning1, DU Xian-neng1
    J4. 2013, 48(2):  15-19. 
    Abstract ( 347 )   PDF (552KB) ( 1304 )   Save
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    The fibre product of Zn in the category of Zmodules is described with the methods of algebra and number theory. The results also apply a way to construct the equalities of Euler function φ(n).

    The fundamental theorem for weak Hopf module in  Yetter-Drinfeld module categories
    DONG Li-hong1,2, GUO Shuang-jian1
    J4. 2013, 48(2):  20-22. 
    Abstract ( 379 )   PDF (527KB) ( 958 )   Save
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    The definitions of weak Hopf algebra and weak Hopf module in Yetter-Drinfeld module categories are introduced. And the fundamental theorem for weak Hopf modules in Yetter-Drinfeld module categories is obtained.

    Strongly Ω-Gorenstein injective modules
    WANG Xin-xin
    J4. 2013, 48(2):  23-26. 
    Abstract ( 406 )   PDF (539KB) ( 707 )   Save
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    The strongly Ω-Gorenstein injective modules are defined and the properties of strongly Ω-Gorenstein injective modules are discussed by homological method. Some examples are given to show that the class of Ω-Gorenstein injective modules contains the class of strongly Ω-Gorenstein injective modules. At last, it is proved that M is Ω-Gorenstein injective modules if and only if M is a direct summand of a strongly Ω-Gorenstein injective modules.

    The structions of finite groups of order p3q
    CHEN Song-liang, LI Jing-lei, OUYANG Jian-xin
    J4. 2013, 48(2):  27-31. 
    Abstract ( 480 )   PDF (603KB) ( 1339 )   Save
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    Let p, q be odd primes such that p>q, let G be a finite group of order p3q. It is discussed that the isomorphic classification of G, and their structions are completely described.

    Sufficient and necessary conditions for field extensions  to be simple radical towers
    ZHANG Wen-hua1, JIANG Xiao-long2, WANG Zhen1
    J4. 2013, 48(2):  32-35. 
    Abstract ( 486 )   PDF (540KB) ( 1246 )   Save
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    Let F be a field and K be an extension field of F. Some sufficient and necessary conditions such that K is a simple radical tower of F are given by using primitive root of unity and Galois theory. Moreover, it is proved that under certain conditions K is a simple radical tower of F if and only if K is Galois over F.

    FS-basis under one-sided monomial orderings
    ZHAO Zhi-qin
    J4. 2013, 48(2):  36-41. 
    Abstract ( 438 )   PDF (610KB) ( 1049 )   Save
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     Let K be a field, R a K-algebra with an SM-basis B, and let - be a onesided (i.e. left or right) monomial ordering on B. Then the classical Factor-SAGBI basis theory (w.r.t a twosided monomial ordering) for subalgebra of commutative polynomial algebras and noncommutative free algebras can be completely generalized to subalgebras of any quotient algebra R/I of R. In particular, for a class of N-graded quotient algebras, finite n-truncated FS-bases for subalgebras can be computed by means of effective algorithm, thereby the feasibility of constructing FS-bases under one-sided monomial ordering is clarified.

    On the rank and idempotent rank of each star ideal of the semigroup PHn
    LUO Yong-gui, XU Bo, YOU Tai-jie
    J4. 2013, 48(2):  42-48. 
    Abstract ( 473 )   PDF (578KB) ( 1024 )   Save
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     Let PHn be the semigroups of order-decreasing and orderpreserving finite partial singular transformations on the natural order set Xn={1,2,3,…,n} if n≥3, and let P(n,r)={α∈PHn:|imα|≤r} be the two-sided star ideal of the semigroup PHn if 0≤r≤n-1. By analyzing the idempotent elements, the minimal idempotent generating set, rank and idempotent rank of the semigroup P(n,r) are characterized, respectively. Furthermore, it is shown that for 0≤l≤r, the relative rank of the semigroup P(n,r) with respect to itself each star ideal P(n,l).

    The total number of matchings of L*n,p
    ZOU Jin-yu, REN Hai-zhen
    J4. 2013, 48(2):  49-52. 
    Abstract ( 440 )   PDF (1237KB) ( 755 )   Save
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    Let Kp be a complete graph of order p. Let L*n,p be the graph with n vertices obtained by identifying the vertex ui of Kp with the vertex vi of the tree Ti, where i=1,2,…,r,1≤r≤p. The graphs of L*n,p with maximal and minimal, with secondlargest and thirdlargest total number of matchings are obtained respectively.

    On the Wiener index of triangular chains
    WEN Chang-kun, REN Hai-zhen
    J4. 2013, 48(2):  53-56. 
    Abstract ( 429 )   PDF (828KB) ( 879 )   Save
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    The Wiener index of the graph G is defined as the sum over all unordered pairs of distinct vertices in G.  The Wiener index of geometrically planar triangular chains is characterized. It is showed that the linear triangular chain and helicene triangular chain attain the maximum Wiener index and minimum Wiener index, respectively.

    he timelike axis surface of revolution with pointwise 1-type gauss map in Minkowski 3space
    JIN Ming-hao1,2, PEI Dong-he1*
    J4. 2013, 48(2):  57-61. 
    Abstract ( 796 )   PDF (601KB) ( 1044 )   Save
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    The pointwise 1-type Gauss map of surface of revolution in Minkowski 3-space is introduced. The results state that timelike axis surfaces of revolution with pointwise 1-type Gauss map of first kind coincide with surfaces of revolution with constant mean curvature; the nonlitelike Lorentzian right cones are the only rational timelike axis surfaces of revolution with pointwise 1-type Gauss map of the second kind.

    The existence of solutions for a elliptic equations with Hardy potential
    CHENG Yong-kuang, YAO Yang-xin, HAN Ya-die
    J4. 2013, 48(2):  62-66. 
    Abstract ( 427 )   PDF (547KB) ( 987 )   Save
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    We establish a new Hilbert space and the existence of nontrivial solutions of nonlinear elliptic equation with Hardy potential and critical potential in the new Hilbert space by using the Mountain pass theorem with (PS) condition. Also,  the existence of multiple solutions are proved by using Fountain theorem.

    Boundedness of Marcinkiewicz integral higher commutators with variable kernels on Hardy spaces
    YAN Yan-zong, SHAO Xu-kui, WANG Su-ping
    J4. 2013, 48(2):  67-71. 
    Abstract ( 451 )   PDF (605KB) ( 1590 )   Save
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    The boundedness results on Hardy spaces were established for a class of generalized higher commutators  μ

    Uniform topological spaces on R0 algebras
    LUO Qing-jun1,2
    J4. 2013, 48(2):  72-78. 
    Abstract ( 433 )   PDF (581KB) ( 796 )   Save
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    It is proved that the uniform topological space induced by a filter in an R0 algebra is a disconnected, zero-dimensional, locally compact, completely regular and firstcountable space, and a T0 space if and only if the filter is equal to {1}. Also, we proved that the negative operation, the join operation and the implication operation are continuous in the uniform topological space. Finally, some properties of the uniform topology are discussed on the quotient R0 algebra under the equivalence relation induced by a filter.

    he submesocompactness and the H\hereditarily submesocompactness
    CAI Qi-rong1, SU Shu-hua2
    J4. 2013, 48(2):  79-80. 
    Abstract ( 459 )   PDF (550KB) ( 699 )   Save
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     It is proved that the  space X is a submesocompact space if and only if every scattered patition of X has a θ-cf-open expansions. If every scattered patition of X has a θcfopen expansions, than X is a hereditarily submesocompact space, but not established and given example. One result of submesocompact space is given.

    Localic nuclei and conuclei on Quantales
    LIU Min, ZHAO Bin*
    J4. 2013, 48(2):  81-87. 
    Abstract ( 484 )   PDF (640KB) ( 1234 )   Save
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    This paper deals with localic nuclei and conuclei on quantales and the unital quantale Q[e]. Firstly, it is proved that a localic nucleus on a quantale Q can be uniquely extended to the unital quantale Q[e]. Also, it is proved that a twosided commutative quantale has a largest localic subquantale, which is applied to study the extension of the largest localic quantic conucleus on Q to the unitale quantale Q[e]. All the extensions of a largest localic quantic conucleus to the unitale quantale Q[e] are given. And it is proved that Q[e] has a largest localic subquantale if and only if Q is the trivial quantale.

    Toward a model for solid materials with memory by use of  the fractionalorder derivatives
    LI Ming, XU Ming-yu
    J4. 2013, 48(2):  88-92. 
    Abstract ( 473 )   PDF (885KB) ( 969 )   Save
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    A constitutive equation on viscoelastic materials with different fractional order derivatives is discussed, and its solution is obtained by using Laplace transform techniques and can be expressed in terms of H-Fox functions.The solution is consistent with the experimental data.The model behavior in the frequency domain is also discussed, the limit of the loss tangent is governed by the difference between the order of time derivatives of strain and stress.

    Fully implicit finite difference scheme for the variable-order  nonlinear fractional diffusion equation
    MA Wei-yuan, ZHANG Hai-dong, SHAO Ya-bin
    J4. 2013, 48(2):  93-97. 
    Abstract ( 467 )   PDF (742KB) ( 1474 )   Save
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    A fully implicit finite difference scheme for the variable-order nonlinear fractional diffusion equation is considered. It is shown that the method is unconditional stable by discrete energy method. The convergence order of the method is O(τ+h). Numerical results demonstrate that the method is efficient and reliable.

    The steptype contrast structure for a singularly perturbed system with slow and fast variables
    WANG Ai-feng1,2
    J4. 2013, 48(2):  98-104. 
    Abstract ( 430 )   PDF (585KB) ( 852 )   Save
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    The step-type contrast structure for a singularly perturbed system with slow and fast variables is considered. We not only construct the heteroclinic orbit by first integral, but also determine the internal transition time t. The asymptotic expansion of this problem with a step-type contrast structure is constructed by the boundary function method. By sewing orbit smooth, the existence of the step-type contrast structure is shown and the asymptotic solution is proved to be uniformly in the whole interval.

    The solution of the operators equation in Banach space
    FAN Da-fu
    J4. 2013, 48(2):  105-110. 
    Abstract ( 484 )   PDF (554KB) ( 1535 )   Save
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    We consider the equivalence for the operator equation XA+AXT=B, and give some simple form equivalent equations of the operator equation XA+AXT=B.