中国科学院数学与系统科学研究院期刊网

15 May 2008, Volume 51 Issue 5
    

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  • He SHI
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 833-840. https://doi.org/10.12386/A2008sxxb0098
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    The reguralization of $SU(3)$ Yang-Mills gauge fields are presented using the method of Mathematics Mechanization in this paper. Two fundamental concepts are introduced: exact Yang-Mills equation and characteristic transformation of Yang-Mills gauge fields. Based on the two concepts, the specializations of $SU(3)$ gauge fields are realized, that is, the linearizations of exact Yang-Mills equations are obtained via characteristic transformations. Thus, the existence of $SU(3)$ Yang-Mills gauge fields is proved.
  • Yu Lei WANGLIU He-GuoJi Ping ZHANG
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 841-846. https://doi.org/10.12386/A2008sxxb0099
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    In this paper, conjugate separability problem in a finitely generated nilpotent group is researched. Let $G$ be a finitely generated nilpotent group, $\pi$ be a nonempty proper subset of set $\omega$ of all primes, then the following three results are equivalent: (i) If $x$ and $y$ aren't conjugate in $G$, then $x$ and $y$ aren't conjugate in some finite $p$-quotient group of $G$, where $p\in \pi$; (ii) If $x$ and $y$ aren't conjugate in $G$, then $x$ and $y$ aren't conjugate in some finite $\pi$-quotient group of $G$; (iii) The torsion subgroup $T(G)$ of $G$ is a $\pi$-group and $G/T(G)$ is abelian.\ Furthermore, an example is given, i.e, let $G$ be a finitely generated torsion-free nilpotent group, $x$ and $y$ are conjugate in the quotient group $G/G^{p}$ for arbitrary prime $p$, but $x$ and $y$ aren't conjugate in $G$.
  • Rui Dong WANG
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 847-852. https://doi.org/10.12386/A2008sxxb0100
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    We study the extension of isometries between the unit spheres $S_1(E)$ and $S_1(F)$, where $E$ and $F$ are both two-dimensional strictly convex real normed spaces. Using the property of the unit sphere of two-dimensional strictly convex real normed space, we obtain that if the isometry $V_0$ from $S_1(E)$ to $S_1(F)$ satisfies some properties, then $V_0$ can be extended to be a real linearly isometric map $V$ from $E$ to $F$.
  • Gong Xiang LIU
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 853-863. https://doi.org/10.12386/A2008sxxb0101
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    We firstly introduce the concept of generalized path coalgebra through assigning a $k$-coalgebra to each vertex of a given quiver. Then some elementary properties of generalized path coalgebras are given. Moreover, we discuss the isomorphism problem. It is shown that two normal generalized path coalgebras are isomorphic if and only if their quivers and the simple coalgebras over the corresponding vertices are isomorphic. For a coalgebra with Codim $C_{0}\leq 1$, the Wedderburn--Malcev Theorem is proven. As an application of the generalized path coalgebras, the Dual Gabriel Theorem for pointed coalgebras is generalized.
  • Yong Ming LI
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 863-876. https://doi.org/10.12386/A2008sxxb0102
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    The complete constructions of scale generalized effect algebras and scale effect algebras are studied in this paper. We introduce the concept of total scale generalized algebra, then we show that scale generalized effect algebras on the interval $[0,1)$ and scale effect algebras on the unit interval $[0,1]$ are completely determined by the Archimedean co-norm on the unit interval $[0,1]$. Scale generalized effect algebras are exactly the lower set of total scale generalized algebras. Furthermore, if a scale generalized effect algebra is locally finite, then it is isomorphic to a sub-algebra of real additive group. Scale effect algebra satisfying (S) condition is isomorphic to the lexicographic product of a sub-algebra of real additive group and a total scale generalized algebra.
  • Jun Fang CHENGDeng Feng LI
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 877-888. https://doi.org/10.12386/A2008sxxb0103
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    Let $E=\binom{1\ \ 1}{1 \ -1}$ or $\binom{ \,0 \ \,2\,}{ \,1 \ \,0\,}$, $\psi(x)\in L^2(R^2)\ \mbox{and}\ \psi_{jk}(x)=2^{\frac{j}{2}}\psi(E^jx -k),$ where $ j\in Z,\, k\in Z^2.$ $\psi(x)$ is called an $E$-tight frame wavelet if $\{\psi_{jk}\,|\,j\in Z,\, k\in Z^2\}$ is a tight frame for $L^2(R^2)$. In this paper, a sufficient and necessary condition for an $E$-tight frame wavelet to be an MRA $E$-tight frame wavelet is presented. Precisely, an $E$-tight frame wavelet $\psi$ is an MRA $E$-tight frame wavelet if and only if the dimension of a particular linear space $F_\psi(\xi)$ is either $0$ or $1$, where $F_\psi(\xi)=\overline{\rm span}\{\Psi_j(\xi)\,|\,j\geq 1\},\, \Psi_j(\xi)=\{\hat{\psi}({(E^T)}^j(\xi+2k\pi))\}_{k\in Z^2},\, j\geq 1.$
  • Yu Feng ZHANGFu Kui Guo
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 889-900. https://doi.org/10.12386/A2008sxxb0104
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    By introducing a higher-dimensional loop algebra, a new integrable coupling of the AKNS-KN soliton hierarchy (called AKNS-KN-SH, for short) is obtained under the framework of zero curvature equations, whose Hamiltonian structure is worked out by using the quadratic-form identity. Finally we give a new Lie algebra $A_4$ so that its various loop algebras and its equivalent colummn-vector Lie algebra are introduced respectively for which multi-component integrable couplings and their Hamiltonian structure of the the AKNS-KN-SH could be generated.
  • Yan QimingZhu Lei
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 900-910. https://doi.org/10.12386/A2008sxxb0105
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    We consider the degeneracy of holomorphic curve $f$ from $\mathbb{C}$ to a complex nonsingular projective variety $X$ of dimension 3. Let $D_1,\ldots,D_r$ be distinct irreducible effective and nef divisors on $X$ located in general position. Assume that there exist positive integers $n_1,\ldots,n_r,c$, such that $n_in_jn_k(D_i.D_j.D_k)=c$ for any $i,j,k$. If $r\ge 11$ and the image of $f$ omits $D_1,\ldots,D_r$, then $f$ is algebraically degenerate, i.e., its image is contained in a proper algebraic subset of $X$.
  • Xi Yong ZHANGHua GUO
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 911-922. https://doi.org/10.12386/A2008sxxb0106
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    By using the techniques of Galois rings, Bent functioins and partial character sums, we construct a class of splitting relative difference sets and non-splitting building sets in Abelian groups with exponent not exceeding 4.
  • Wei Li HE
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 923-926. https://doi.org/10.12386/A2008sxxb0107
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    We correct an error in the paper ``The Lower Bounds of the Maximum Genus on Graphs in Terms of Diameter and Girth'' (Acta Mathematica Sinica 2004, 47(6): 1201--1204), and obtain the following result: Let $G$ be a simple graph with diameter $d$. If its girth is not less than $d$, then the Betti deficiency number of $G$, $\xi(G) \leq 2$, i.e. the maximum genus of $G$, $\gamma_M(G)\geq \frac {1}{2}\beta (G)-1$.
  • Rui Xiang DAI
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 927-932. https://doi.org/10.12386/A2008sxxb0108
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    In this paper, the entwining structures of and monads, comonads, entwined modules and the relations with those of algebras and coalgebras in categories are investigated. We also define group-like elements of comonads and get some interesting results. Furthermore we build two functors between two categories of entwined modules, and prove that the two functors are adjoint.
  • LI JingZhi Hong JIANG
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 933-946. https://doi.org/10.12386/A2008sxxb0109
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    In this note, all projective indecomposable modules over generalized Witt algebra $W(2,\mathbf{1})$ with character height $\leq 0$ in characteristic 2 are realized, and all related Cartan invariants are computed. Furthermore, we discuss the representation type of the reduced enveloping algebra $u(W(2,\mathbf{1}),\chi)$.
  • LI xing min
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 947-954. https://doi.org/10.12386/A2008sxxb0110
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    There are exactly four normed division algebras: the real numbers $R,$ complex numbers $C,$ quaternions $H$ and octonions $O.$ However, due to their noncommtativity and nonassociativity, for a octonionic matrix, how to define the determinant that satisfies nice calculating properties becomes very difficult. Recently, according to the mathematical and physical requirements that ``the determinant of any hermitian octonion matrix should be a real number", by choosing the multiplication orders and the associative methods for $n$ octonioin numbers, the definition of determinant for the octonionic matrix is first given by Li xingmin and Li li. But, compared with the corresponding cases of real, complex and quaternion, such a definition has less calculating properties. In this paper, we give a new definition, which satisfies almost all the properties, and automatically, the determinant for any hermitian octonionic matrix is a real number.
  • wei yi ZHU
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 955-958. https://doi.org/10.12386/A2008sxxb0111
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    For any positive integer $n$, the famous F.Smarandache LCM function $SL(n)$ is defined as the smallest positive integer $k$ such that $n\, |\,[1, 2,\ldots,k]$, where $[1,2,\ldots,k]$ denotes the least common multiple of $1, 2, \ldots, k$. The main purpose of this paper is to use using the elementary methods to study a conjecture involving the F.Smarandache LCM function, which is proposed by Zhang Wenpeng in his book ``Elementary Number Theory'', thus making some substantial progress for this conjecture.
  • Yu Cheng LI
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 959-964. https://doi.org/10.12386/A2008sxxb0112
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    Let $H^{\infty}(D)$ denote the algebra of bounded analytic functions on unit disk $D$. For $\varphi\in H^{\infty}(D)$, under some condition, we prove that $T=M^{*}_{\varphi}$ is a Cowen--Douglas operator with index $n$, then give a sufficient condition such that $\mathscr{ A}'(T)/{\rm rad}\,\mathscr{ A}'(T)$ is commutative. For $n=1$, we characterize the commutant of $T$.
  • Han LIJin Xuan FANG
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 965-970. https://doi.org/10.12386/A2008sxxb0113
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    The relationship between exchangeable translation spaces and sub-normed {\bf Z}-linear spaces is further studied. The Hahn-Banach theorem on Abel group is established. As its consequence, Hahn-Banach theorem in sub-normed {\bf Z}-linear spaces is obtained.
  • Yan Hong DINGYan ZHAOShan Shan LI
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 971-978. https://doi.org/10.12386/A2008sxxb0114
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    Let $(M,\,T)$ be a smooth closed manifold with a smooth involution $T$ whose fixed point set is $F=\{x\,|\,T(x)=x,\,x\in M\}$, then $F$ is the disjoint union of smooth closed submanifold of $M$. In this paper, we discuss: for $F=P(2m,\,2l+1)\sqcup P(2m,\,2n{+}1)$, $n>l\geq m,\,m\neq1,\,3$, then $(M,\,T)$ is bounded.
  • Tong Yi MA
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 979-992. https://doi.org/10.12386/A2008sxxb0115
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    A new deviation metric which is called the deviation regular metric of a simplice is introduced. Utilizing the deviation metric, the author establishes stability theorems of Veljan-Korchmaros's inequality for simplexes. As application, stability theorems of the well-known Weitzenb\"{o}ck's inequality and Euler's inequality are considered. Moreover, some open questions are put forward.
  • Lin Na SHANG
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 993-100. https://doi.org/10.12386/A2008sxxb0116
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    The value distribution of analytic functions defined by Laplace-Stieltjes transforms in the right half-plane is considered in this paper. We prove the existence of Borel points of such analytic functions of finite non-zero order and infinite order.
  • Shen Zhou ZHENG
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 1001-101. https://doi.org/10.12386/A2008sxxb0117
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    We shall establish that the derivatives of weak solutions for $P$-Harmonic systems under the subcritical growth belong to everywhere interior H\"older continuity spaces with some H\"older exponent. This conclusion is the best situation as for the lower order items with the subcritical growth index.
  • Chang Jian ZHAOZai Xiu JI
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 1015-102. https://doi.org/10.12386/A2008sxxb0118
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    The main purpose of the present article is to establish inverse Hilbert-Pachpatte's type inequalities. As applications, we generalize and improve the disperse and continuous Pachpatte's inequalities.
  • Jing Shi XU
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 1021-103. https://doi.org/10.12386/A2008sxxb0119
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    In this paper, multilinear commutators generated multilinear singular integrals and BMO functions are introduced. Then the boundedness of this class of multilinear commutators from products of Lebesgue spaces to Lebesgue spaces is obtained. Moreover, weighted and vector-valued inequalities are also given.
  • SHI Xian-lianghaiying zhang
    Acta Mathematica Sinica, Chinese Series. 2008, 51(5): 1035-104. https://doi.org/10.12386/A2008sxxb0120
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    We give a complete characterization of generators for multiresolution analysis. Precisely, we prove the following results: $\phi\in L^2(\mathbb{R})$ is a genarator of a dyadic multireso-lution if and only if (1) there exists $\{a_k\}\in l^2$ such that $\phi(x)=\sum_{k\in \mathbb{Z}}{a_k \phi(2x-k)};$ (2) there exists positive numbers $A$ and $B$ such that $A \leq \Phi(\omega)\leq B$, a.e., where $\Phi(\omega)=\sum_{l\in \mathbb{Z}}{|\hat{\phi}(\omega+2l\pi)|}^2;$ (3) the function $F(x,y)=\frac{1}{y-x}\int_x^y{|\hat{\phi}(\omega)|}^2d\omega$ is dyadicly away from zero at the origin.