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Frontiers of Mathematics in China

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, Volume 13 Issue 4 Previous Issue   
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RESEARCH ARTICLE
Finite p-groups whose non-normal subgroups have few orders
Lijian AN
Front. Math. China. 2018, 13 (4): 763-777.  https://doi.org/10.1007/s11464-018-0693-0
Abstract   PDF (294KB)

Suppose that G is a nite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use pM(G) and pm(G) to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that M(G)2m(G)1: As a by-product, we also classify p-groups whose orders of non-normal subgroups are pk and pk+1:

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Pseudo almost automorphic solution to stochastic differential equation driven by Lévy process
Xinwei FENG, Gaofeng ZONG
Front. Math. China. 2018, 13 (4): 779-796.  https://doi.org/10.1007/s11464-018-0715-y
Abstract   PDF (279KB)

Almost automorphic is a particular case of the recurrent motion, which has been studied in differential equations for a long time. We introduce square-mean pseudo almost automorphic and some of its properties, and then study the pseudo almost automorphic solution in the distribution sense to stochastic differential equation driven by Lévy process.

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Results of Diophantine approximation by unlike powers of primes
Gaiyun GAO, Zhixin LIU
Front. Math. China. 2018, 13 (4): 797-808.  https://doi.org/10.1007/s11464-018-0713-0
Abstract   PDF (274KB)

Let k be an integer with k6: Suppose that λ1,λ2,...,λ5 be nonzero real numbers not all of the same sign, satisfying that λ1/λ2 is irrational, and suppose that η is a real number. In this paper, for any ε0; we consider the inequality |λ1p1+λ2p22+λ3p33+λ4p44+λ5p55+η|(max?pj)σ(k)+ε has innitely many solutions in prime variables p1,p2,...,p5, where σ(k) depends on k: Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can rene some results on Diophantine approximation by unlike powers of primes, and get the related problem.

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Moderate deviations for Euler-Maruyama approximation of Hull-White stochastic volatility model
Yunshi GAO, Hui JIANG, Shaochen WANG
Front. Math. China. 2018, 13 (4): 809-832.  https://doi.org/10.1007/s11464-018-0705-0
Abstract   PDF (438KB)

We consider the Euler-Maruyama discretization of stochastic volatility model dSt=σtStdWt,dσt=ωσtdZt,t[0,T] which has been widely used in nancial practice, where Wt,Zt,t[0,T] are two uncorrelated standard Brownian motions. Using asymptotic analysis techniques, the moderate deviation principles for log Sn (or log |Sn| in case Sn is negative) are obtained as n under different discretization schemes for the asset price process St and the volatility process σt: Numerical simulations are presented to compare the convergence speeds in different schemes.

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Some remarks on one-sided regularity
Tai Keun KWAK, Yang LEE, Young Joo SEO
Front. Math. China. 2018, 13 (4): 833-847.  https://doi.org/10.1007/s11464-018-0711-2
Abstract   PDF (270KB)

A ring is said to be right (resp., left) regular-duo if every right (resp., left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.

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Uniqueness and perturbation bounds for sparse non-negative tensor equations
Dongdong LIU, Wen LI, Michael K. NG, Seak-Weng VONG
Front. Math. China. 2018, 13 (4): 849-874.  https://doi.org/10.1007/s11464-018-0707-y
Abstract   PDF (652KB)

We discuss the uniqueness and the perturbation analysis for sparse non-negative tensor equations arriving from data sciences. By two different techniques, we may get better ranges of parameters to guarantee the uniqueness of the solution of the tensor equation. On the other hand, we present some perturbation bounds for the tensor equation. Numerical examples are given to show the effciency of the theoretical results.

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Torsion pairs in recollements of abelian categories
Xin MA, Zhaoyong HUANG
Front. Math. China. 2018, 13 (4): 875-892.  https://doi.org/10.1007/s11464-018-0712-1
Abstract   PDF (285KB)

For a recollement (A ;ℬ; C ) of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in ℬ; and the converse holds true under certain conditions.

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Generalized inverses of tensors via a general product of tensors
Lizhu SUN, Baodong ZHENG, Yimin WEI, Changjiang BU
Front. Math. China. 2018, 13 (4): 893-911.  https://doi.org/10.1007/s11464-018-0695-y
Abstract   PDF (308KB)

We define the {i}-inverse (i = 1; 2; 5) and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.

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Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth
Yongqiang SUO, Jin TAO, Wei ZHANG
Front. Math. China. 2018, 13 (4): 913-933.  https://doi.org/10.1007/s11464-018-0710-3
Abstract   PDF (311KB)

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coeffcients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coeffcients are polynomial growth with respect to the delay variables.

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Solution structures of tensor complementarity problem
Xueyong WANG, Haibin CHEN, Yiju WANG
Front. Math. China. 2018, 13 (4): 935-945.  https://doi.org/10.1007/s11464-018-0675-2
Abstract   PDF (260KB)

We introduce two new types of tensors called the strictly semi-monotone tensor and the range column Sufficient tensor and explore their structure properties. Based on the obtained results, we make a characterization to the solution of tensor complementarity problem.

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Irreducible +-modules of near-group fusion ring K(3, 3)
Chengtao YUAN, Ruju ZHAO, Libin LI
Front. Math. China. 2018, 13 (4): 947-966.  https://doi.org/10.1007/s11464-018-0709-9
Abstract   PDF (313KB)

The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible ?+-modules over the near-group fusion ring K(?3, 3) are explicitly classified. It turns out that there are only four inequivalent irreducible ?+-modules of rank 2 and two inequivalent irreducible ?+-modules of rank 4 over K(?3, 3).

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COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inated count data
Huiming ZHANG, Kai TAN, Bo LI
Front. Math. China. 2018, 13 (4): 967-998.  https://doi.org/10.1007/s11464-018-0714-z
Abstract   PDF (624KB)

We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein's identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inated data sets. With the aid of ratio regression, we employ maximum likeli-hood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.

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Realization of Poisson enveloping algebra
Can ZHU, Yaxiu WANG
Front. Math. China. 2018, 13 (4): 999-1011.  https://doi.org/10.1007/s11464-018-0708-x
Abstract   PDF (179KB)

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.

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