中国科学院数学与系统科学研究院期刊网

15 November 2017, Volume 60 Issue 6
    

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  • Dong Lei TANG, Rui HU, Wei Yi SU
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 889-896. https://doi.org/10.12386/A2017sxxb0076
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    We define the Dirichlet form and its related harmonic functions on a homogeneous hierarchical gasket by the method of averages, and show that they enjoy the same properties as those defined in the pointwise sense.

  • Tao LI, Bian WU
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 897-910. https://doi.org/10.12386/A2017sxxb0077
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    We propose an alternative ranked set sampling scheme-no overlap k-tuple ranked set sample, in which k units are quantified in each set and the units in different sets have sample mean and show that the correlations between the k-units and show that the correlations between the k-units in each set decreases the efficiency of the sample mean. The stronger the correlation structure is, the more the efficiency of the sample mean decreases. The purpose of this work is to find the optimal k-tuple ranked set sample and prove that the no overlap k-tuple ranked set sample with the optimal allocation is more efficient than generalized k-tuple ranked set sample and simple sample and simple random sample. Although the statistical efficiency of no overlap k-tuple ranked set sample is not as much as the classic ranked set sample, under the consider of the cost for sampling and ranking, the no overlap k-tuple ranked set sample is more efficient than the classic ranked set sample.

  • Li Qiong LIN, Yun Nan ZHANG, Yu Can ZHU
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 911-918. https://doi.org/10.12386/A2017sxxb0078
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    In the research of frame theory, the questions that what kinds of invertible operators can make some frame properties remain unchanged are essential and important. This paper study the following two questions on infinite dimensional Hilbert spaces. Question one:what kind of invertible operators can make the frame operators remain unchanged; Question two:what kind of invertible operators can makes the norms of frames are a multiple by a fixed sequence of the norms of original frames. This paper gives the answer of Question one by the abstract operator theory and by the concrete constructions. When limited the class of invertible operators to a smaller class of operators, this paper obtain the answer of Question two by using the concept of admissible operators of frames.

  • Yue Feng LIN
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 919-930. https://doi.org/10.12386/A2017sxxb0079
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    We study any 4-valent connected planar graph without cut vertices whose facial cycle lengths are of 2, 3 or 4 only, called them I-hedrites. We prove that in the equal sense every I-hedrite is uniquely embedded in the plane. This uniqueness conclusion means that any two i-hedrites (i.e. 4-valent connected plane graph whose faces are 2-, 3-and 4-gons only) are equal if and only if they are isomorphic. The result solves the problem of certainty in isomorphism structure of i-hedrite in the equal sense.

  • Shui Li ZHANG, Shao Yi ZHANG
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 931-946. https://doi.org/10.12386/A2017sxxb0080
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    We study the strong ergodicity for jump processes in general state spaces, and obtain some equivalence conditions of strongly ergodicity by minimal nonnegative solutions and strongly Markov property. which extended the results of countable state to the jump processes on genersl state spaces.

  • He SONG, Kai Nan XIANG
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 947-954. https://doi.org/10.12386/A2017sxxb0081
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    We prove that a simple random walk on quasi-transitive graphs with the volume growth being at least as fast as a polynomial of degree 5 has a.s. infinitely many cut times, and hence infinitely many cutpoints. This confirms a conjecture raised by Benjamini, Gurel-Gurevich and Schramm[2011, Cutpoints and resistance of random walk paths, Ann. Probab., 39(3):1122-1136] that PATH of a simple random walk on any transient vertex-transitive graph has a.s. infinitely many cutpoints in the corresponding case.

  • Li Hua MENG, Jian Hua ZHANG
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 955-960. https://doi.org/10.12386/A2017sxxb0082
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    Let T=Tri(A, M, B) be a triangular algebra, and Q={TT:T2=0}. We prove that if a map δ:TT satisfies δ(ABC)=δ(A)BC+Aδ(B)C+ABδ(C) for any A, B, CT with ABCQ, then δ is an additive derivation.

  • Hong Chang HU, Zhen ZENG
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 961-976. https://doi.org/10.12386/A2017sxxb0083
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    The paper studies a generalized linear model (GLM) yi=h(xiTβ)+ei,i=1,2,...,n, where ei=G(...,εi-1, εi), h is a continuous differentiable function, εi's are independent and identically distributed (i.i.d.) random errors with zero mean and finite variance σ2. We consider the M-estimator of parameter β in the model, and investigate the Bahadur representations, which extend the correspondingly results of linear regression models to the generalized linear model. Moreover, by our results, it is easy to obtain the asymptotic properties of βn in some models, including linear regression models with dependent errors, poisson models with dependent errors, logistic models with dependent errors, and GLMs with i.i.d. errors.

  • Feng Juan CHEN
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 977-982. https://doi.org/10.12386/A2017sxxb0084
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    In 1913, Frobenius posed a famous conjecture on the Markoff equation a2+b2+c2=3abc:if a < b < c in positive integers is a solution triple of the Markoff equation, then a, b are determined uniquely by the largest number c. The conjecture is still open. In this paper, the author proved the following result:let si,ti,w,u,v (i=1,2) be positive integers, suppose that (ai,bi,c) in positive integers are two distinct solution triples of the Markoff equation, and ai < bi < c (i=1, 2), then gcd(s1a1+s2a2+t1b1+t2b2+w, uc+v) ≤ K(uc+v)13/14, where K is a positive constant depending only on si,ti,w,u,v (i=1, 2).

  • Jin Ke HAI, Xin Xin LV
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 983-992. https://doi.org/10.12386/A2017sxxb0085
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    In this paper, using the Luthar-Passi method, we investigate the normalized torsion units of the integral group ring of the direct product of the alternative group A5 and the dihedral group D6. As a consequence, we confirm the Zassenhaus's conjecture about this group.

  • Qian Qian ZHAO, Sheng Li ZHAO
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 993-1002. https://doi.org/10.12386/A2017sxxb0086
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    Mixed-level designs are widely used in factorial experiments. Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs. This paper considers mixed-level designs with one eight-level factor, one four-level factor, and some two-level factors, denoted by 8×4×2n designs, and derives the upper and lower bounds on the maximum number of clear two-factor interaction components in 8×4×2n designs with resolution Ⅲ and IV. The lower bounds are achieved by constructing specific designs.

  • Yan LIU, Jian Lin LI, Qi WANG
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 1003-1012. https://doi.org/10.12386/A2017sxxb0087
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    Let μM,D be the self-affine measure uniquely determined by an expanding matrix MMn(Z) and a finite digit set D ⊂ Zn through the affine iterated function system (IFS){φd(x)=M-1(x+d)}dD. The non-spectrality of μM,D is directly connected with the finiteness or infiniteness of orthogonal exponentials in the Hilbert space L2(μM,D). We provide a better estimate on the cardinality of μM,D-orthogonal exponentials by characterizing the zero set Z(mD) of the symbol function mD(x) and its middle points. The results here extend the corresponding results of Dutkay, Jorgensen and others.

  • Jiong DONG, Xiao Hong CAO, Jun Hui LIU
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 1013-1024. https://doi.org/10.12386/A2017sxxb0088
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    A variant of the essential approximate point spectrum is discussed. Using the new spectrum, we investigate the stability of a-Weyl's theorem under compact perturbations, and we characterize the sufficient condition that TnB(H) does not satisfy the stability of a-Weyl's theorem for each positive integer nN, where H denotes an infinite dimensional separable complex Hilbert space and B(H) denotes the algebra of all bounded linear operators on H.

  • Shao Lin CHEN, Miodrag MATELJEVIC, Saminathan PONNUSAMY, Xian Tao WANG
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 1025-1036. https://doi.org/10.12386/A2017sxxb0089
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    We investigate some properties on harmonic functions and solutions to Poisson equations. We will discuss the Lipschitz type spaces on harmonic functions. Secondly, we establish the Schwarz-Pick type lemma for harmonic functions in the unit ball Bn of Rn, and then we apply it to obtain a Landau-Bloch type theorem for harmonic functions in Hardy spaces. At last, we use a normal family argument to extend the Landau-Bloch type theorem to functions which are solutions to Poisson equations.

  • Rong Kun ZHUANG, Bao Guo JIA, Qi Ru WANG
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 1037-1046. https://doi.org/10.12386/A2017sxxb0090
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    We first establish some differential inequalities and then some new Sturm type comparison theorems for the second order nonlinear damped elliptic differential equations are obtained by using them. Our results generalize some classicial Sturm Comparison Theorems. Some examples are presented to illustrate their applications.

  • Dan ZHAO, Hua WANG, Jun Jie HUANG
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 1047-1056. https://doi.org/10.12386/A2017sxxb0091
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    We discuss the existence and the representation of the Drazin inverse of the sum for two bounded linear operators on a Banach space. Under the assumptions PQ3=0,P2Q=0,QPQ2=0, the existence of the Drazin inverse of P+Q is proved and the representation of (P+Q)D is obtained by using the Laurent expansion of resolvent. At the same time, we obtain ind (P+Q) ≤ 2t+r+s-1. Finally, a numerical example is given to illustrate its effectiveness.

  • Wei Xing CHEN
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 1057-1064. https://doi.org/10.12386/A2017sxxb0092
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    In this paper, an example of a central reversible ring which is not central semicommutative is constructed, answering a question of Jung et al. [Bull. Korean Math. Soc., 2015, 52(1):247-261] negatively. Moreover, it is proved that if R is a central reduced ring, then n, R[x]/(xn) is a central symmetric ring for any positive integer n.

  • He Guo LIU, Xiao Liang LUO, Xin QIN, Jun MIAO
    Acta Mathematica Sinica, Chinese Series. 2017, 60(6): 1065-1074. https://doi.org/10.12386/A2017sxxb0093
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    The purposes of this study is to extend as many decomposition theorems of abelian groups (finite or infinite) as possible to modules over a principal ideal domain (PID) and then using these theorems of modules to study vector spaces and linear transformations on vector spaces and hence obtain decomposition theorems of vector spaces (finite or infinite dimension). As the first of a series of articles, this paper aims to present basic concepts for the whole study, such as pure submodule, bounded module, locally cyclic module etc. The main content of this paper is as follows:
    (1) determining the structures of divisible modules, bounded modules and locally cyclic modules over a PID;
    (2) giving generating properties of the quasicyclic module over a PID, which plays a very important role in the future study;
    (3) characterizing modules over a PID with minimal condition or with minimax condition; (4) offering two nonisomorphic Z[i]-modules, but isomorphic as Abelian groups.