Chinese Quarterly Journal of Mathematics

Kac Determinant Formula for the q-deformed Virasoro Algebra of Hom-type

QI Huan-ge, CHENG Yong-sheng, WANG Meng-ping, WU Lin-li

2016, 31(4): 331-339. doi:10.13371/j.cnki.chin.q.j.m.2016.04.001

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We investigate the highest weight representations of the q-deformed Virasoro algebra of Hom-type. In order to determine its unitarity and irreducible highest weight representations, we present its Kac determinant formula when q is nonzero and non-root of unity.

Composition and Multiplication Operators on L^p,∞(M)

HAN Ya-zhou

2016, 31(4): 340-348. doi:10.13371/j.cnki.chin.q.j.m.2016.04.002

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We give some properties of the composition and multiplication operators on Lp,∞（M）, where M is a semifinite von Neumann algebra with a normal semifinite faithful trace τ.

Some Inequalities for the L_p-polar Curvature Images of Star Bodies

LI Xin-hong, MA Tong-yi

2016, 31(4): 349-358. doi:10.13371/j.cnki.chin.q.j.m.2016.04.003

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Zhu,Lü and Leng extended the concept of L_p-polar curvature image. We continuously study the L_p-polar curvature image and mainly expound the relations between the volumes of star bodies and their L_p-polar curvature images in this article. We first establish the L_p-affine isoperimetric inequality associated with L_p-polar curvature image. Secondly,we give a monotonic property for L_p-polar curvature image. Finally, we obtain an interesting equation related to L_p-projection body of L_p-polar curvature image and L_p-centroid body.

Complete Convergence for Weighted Sums of Negatively Superadditive Dependent Random Variables

WANG Qiang, ZHOU De-xia, DU Ling, XIAO Ru, WANG Xue-jun

2016, 31(4): 359-368. doi:10.13371/j.cnki.chin.q.j.m.2016.04.004

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In the paper, the complete convergence for the maximum of weighted sums of negatively superadditive dependent(NSD, in short) random variables is investigated by using the Rosenthal type inequality. Some sufficient conditions are presented to prove the complete convergence. The result obtained in the paper generalizes some corresponding ones for independent random variables and negatively associated random variables.

On the Growth of Solutions of a Class of Higher Order Linear Differential Equations

XIONG Hui, LIU Hui-fang

2016, 31(4): 369-378. doi:10.13371/j.cnki.chin.q.j.m.2016.04.005

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In this paper, we investigate the growth of solutions of the differential equations f^（k）+ A_k-1（z）f^（k-1）+ ··· + A₀（z）f = 0, where A_j（z）（j = 0, ···, k-1） are entire functions.When there exists some coefficient A_s（z）（s ∈ {1, ···, k-1}） being a nonzero solution of f’’+P（z）f = 0, where P（z） is a polynomial with degree n（≥ 1） and A₀（z） satisfies σ（A₀） ≤1/2 or its Taylor expansion is Fabry gap, we obtain that every nonzero solution of such equations is of infinite order.

Certain Subclasses of Analytic Functions Involving the Generalized Dziok-Srivastava Operator

LI Shu-hai, TANG Huo, MA Li-na

2016, 31(4): 379-389. doi:10.13371/j.cnki.chin.q.j.m.2016.04.006

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The generalized Dziok-Srivastava operator is used here to introduce a class of analytic functions in the open unit disc. We provide convolution properties, some subordination relations and the problem of radius. The results presented here extend some of the earlier results.

Common Fixed Points for Mappings with Quasi-Lipschitz Conditions on TVS-valued Cone Metric Spaces

PIAO Yong-jie, JIN Zhe-zhi

2016, 31(4): 390-398. doi:10.13371/j.cnki.chin.q.j.m.2016.04.007

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A new unique common fixed point result for a pair of mappings satisfying certain quasi-Lipschitz type conditions on a topological vector space-valued cone metric space is obtained, and its particular forms and a more general form are given. Our main results generalize and improve some well-known recent results in the literature.

Rainbow Vertex-connection Number of Ladder and MÄobius Ladder

LIU Hui-min, MAO Ya-ping

2016, 31(4): 399-405. doi:10.13371/j.cnki.chin.q.j.m.2016.04.008

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A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc（G）, is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc（G）. Observe that rvc（G） ≤ srvc（G） for any nontrivial connected graph G. In this paper, for a Ladder Ln,we determine the exact value of srvc（L_n） for n even. For n odd, upper and lower bounds of srvc（L_n） are obtained. We also give upper and lower bounds of the（strong） rainbow vertex-connection number of Mbius Ladder.

On µ¤-completions of F-abnormal Subgroups

GAO Hui, GAO Sheng-zhe, YIN Li

2016, 31(4): 406-411. doi:10.13371/j.cnki.chin.q.j.m.2016.04.009

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Let F be a saturated formation of finite groups. Given a proper subgroup H of a group G, a subgroup H of G is called F-normal in G if G/HGbelongs to F; otherwise H is said to be F-abnormal in G. In this paper, we investigate the structure of a finite group by θ*completions of F-abnormal subgroups.

Expression for the Group Inverse of the 2 ×2 Block Matrix

DU Fa-peng

2016, 31(4): 412-421. doi:10.13371/j.cnki.chin.q.j.m.2016.04.010

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In this paper, we present a method how to get the expression for the group inverse of 2×2 block matrix and get the explicit expressions of the block matrix (A C B D) under some conditions.

Stability of Neutral Fractional Neural Networks with Delay

LI Yan, JIANG Wei, HU Bei-bei

2016, 31(4): 422-429. doi:10.13371/j.cnki.chin.q.j.m.2016.04.011

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This paper studies stability of neutral fractional neural networks with delay. By introducing the definition of norm and using the uniform stability, the sufficient condition for uniform stability of neutral fractional neural networks with delay is obtained.

When is an Almost Locally Compact Space Supercomplete?

LI Xiao-ting, LIN Fu-cai

2016, 31(4): 430-434. doi:10.13371/j.cnki.chin.q.j.m.2016.04.012

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It is proved in this paper that(1) the topological sum of a family of supercomplete spaces is supercomplete;(2) if X is a metacompact and almost locally compact space then X is supercomplete. Moreover, some questions on supercomplete spaces are posed in the paper.

Adaptive Image Zooming Algorithm Combining Total Variation Minimization and a Second-order Functional

WANG Yan-yan, GAO Ran, GU Cong

2016, 31(4): 435-440. doi:10.13371/j.cnki.chin.q.j.m.2016.04.013

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An image zooming algorithm by using partial differential equations（PDEs） is proposed here. It combines the second-order PDE with a fourth-order PDE. The combined algorithm is able to preserve edges and at the same time avoid the blurry effect in smooth regions. An adaptive function is used to combine the two PDEs. Numerical experiments illustrate advantages of the proposed model.

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