摘要：
For $$b\in L_{\mathrm{loc}}({\mathbb {R}}^n)$$ and $$0<\alpha <1$$, we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely $$\begin{aligned} g_{\Omega ,\alpha ;b}(f )(x) =\bigg (\int _0^\infty \bigg |\frac{1}{t} \int _{|x-y|\le t}\frac{\Omega (x-y)}{|x-y|^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg |^2\frac{dt}{t}\bigg )^{1/2}. \end{aligned}$$Here, we obtain the necessary and sufficient conditions for the function b to guarantee that $$g_{\Omega ,\alpha ;b}$$ is a bounded operator on $$L^2({\mathbb {R}}^n)$$. More precisely, if $$\Omega \in L(\log ^+ L)^{1/2}{(S^{n-1})}$$ and $$b\in I_{\alpha }(BMO)$$, then $$g_{\Omega ,\alpha ;b}$$ is bounded on $$L^2({\mathbb {R}}^n)$$. Conversely, if $$g_{\Omega ,\alpha ;b}$$ is bounded on $$L^2({\mathbb {R}}^n)$$, then $$b \in Lip_\alpha ({\mathbb {R}}^n)$$ for $$0<\alpha < 1$$.
期刊：
Complex Analysis and Operator Theory,2019年13(4):2049-2068 ISSN：1661-8254
通讯作者：
Chen, Shaolin
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Kalaj, David] Univ Montenegro, Fac Nat Sci & Math, Cetinjski Put B B, Podgorica 81000, Montenegro.
通讯机构：
[Chen, Shaolin] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
关键词：
Landau type theorem;Poisson’s equation;Schwarz’s Lemma
摘要：
For a given continuous function
$$g:~\overline{\mathbb {D}}\rightarrow {\mathbb {C}}$$
and a given continuous function
$$\psi :~{\mathbb {T}}\rightarrow {\mathbb {C}}$$
, we establish some Schwarz type Lemmas for mappings f satisfying the PDE:
$$\Delta f=g$$
in
$${\mathbb {D}}$$
, and
$$f=\psi $$
in
$${\mathbb {T}}$$
, where
$${\mathbb {D}}$$
is the unit disk of the complex plane
$${\mathbb {C}}$$
and
$${\mathbb {T}}=\partial {\mathbb {D}}$$
is the unit circle. Then we apply these results to obtain a Landau type theorem, which is a partial answer to the open problem in Chen and Ponnusamy (Bull Aust Math Soc 97: 80–87, 2018).
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
关键词：
K-quasiconformal harmonic mapping;Radial John disk;Radial length;Pommerenke interior domain
摘要：
In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain a sharp growth theorem of the radial length of K-quasiconformal harmonic mappings. Then we present an alternate characterization of radial John disks. In addition, we investigate the linear measure distortion and the Lipschitz continuity on K-quasiconformal harmonic mappings of the unit disk onto a radial John disk. Finally, using Pommerenke interior domains, we characterize certain differential properties of K-quasiconformal harmonic mappings.
期刊：
Computers & Mathematics with Applications,2019年78(8):2560-2574 ISSN：0898-1221
通讯作者：
Wang, Qisheng
作者机构：
[Wang, Keyan] Hengyang Normal Univ, Sch Math & Stat, Hengyang 421008, Hunan, Peoples R China;[Wang, Qisheng] Wuyi Univ, Sch Math & Computat Sci, Jiangmen 529020, Guangdong, Peoples R China
通讯机构：
[Wang, Qisheng] W;Wuyi Univ, Sch Math & Computat Sci, Jiangmen 529020, Guangdong, Peoples R China.
关键词：
Hyperbolic equations;Expanded mixed finite element method;Error estimates
摘要：
In this paper, an expanded mixed finite element method is developed to approximate the solution of second order hyperbolic equations. We discuss and theoretically prove the existence and uniqueness of solution for the semidiscrete formulation, and obtain the optimal order error estimates in the L-infinity(L-2) norm. In addition, we also construct the fully discrete approximations and provide the corresponding error analysis. Some numerical examples are presented to demonstrate the theoretical analysis. (C) 2019 Elsevier Ltd. All rights reserved.
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
通讯机构：
[Chen, Shaolin] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
期刊：
Results in Mathematics,2019年74(3):1-16 ISSN：1422-6383
通讯作者：
Chen, Shaolin
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
通讯机构：
[Chen, Shaolin] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
关键词：
Length;area;modulus of continuity;Poisson's equation
摘要：
In this paper, we discuss the modulus of continuity of solutions to Poisson’s equation, and give bounds of length and area distortion for some classes of K-quasiconformal mappings satisfying Poisson’s equations. The obtained results are the extension of the corresponding classical results.
摘要：
In order to broaden the study of the most popular and general Takagi-Sugeno (TS) system, we propose a complex-valued neuro-fuzzy inference system which realises the zero-order TS system in the complex-valued network architecture and develop it. In the complex domain, boundedness and analyticity cannot be achieved together. The splitting strategy is given by computing the gradients of the real-valued error function with respect to the real and the imaginary parts of the weight parameters independently. Specifically, this system has four layers: in the Gaussian layer, the L-dimensional complex-valued input features are mapped to a Q-dimensional real-valued space, and in the output layer, complex-valued weights are employed to project it back to the complex domain. Hence, split-complex valued gradients of the real-valued error function are obtained, forming the split-complex valued neuro-fuzzy (split-CVNF) learning algorithm based on gradient descent. Another contribution of this paper is that the deterministic convergence of the split-CVNF algorithm is analysed. It is proved that the error function is monotone during the training iteration process, and the sum of gradient norms tends to zero. By adding a moderate condition, the weight sequence itself is also proved to be convergent.
作者机构：
[Deng, Yi-Hua] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Peoples R China.
通讯机构：
[Deng, Yi-Hua] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Peoples R China.
关键词：
p-Laplace equation;variational methods;infinitely many solutions;nonnegative solution
摘要：
In this paper, we study a class of p-Laplace equations. Using variational methods, we prove that there are two solutions and one of these solutions is nonnegative. Using recurrence method, we prove that there are infinitely many solutions to this class of equations.
作者机构：
[Liu, Gang] Hengyang Normal Univ, Coll Math & Stat, Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang 421002, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol, Dept Math, Madras 600036, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Inst Technol, Dept Math, Madras 600036, Tamil Nadu, India.
关键词：
Uniformly locally univalent;Analytic functions;Harmonic and stable harmonic mappings;Pre-Schwarzian and Schwarzian;Uniformly hyperbolic radius;Distortion;Hardy space;Bloch space;BMOA;BMOH;Subordination
摘要：
In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates. (C) 2017 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
期刊：
Journal of Computational and Applied Mathematics,2018年336:468-476 ISSN：0377-0427
通讯作者：
Ponnusamy, Saminathan
作者机构：
[Liu, Gang] Hengyang Normal Univ, Coll Math & Stat, Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang 421002, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Madras 600036, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Inst Technol Madras, Dept Math, Madras 600036, Tamil Nadu, India.
关键词：
König’s method;Cycle;Fatou set;Attracting basin;Parabolic basin;Siegel disk
摘要：
We consider König’s method for finding roots of polynomials. Let Kf,n be the function defined in König’s method for polynomial f of order n (n≥2). For any given complex number λ∈C and any given set Ω≔{z1,z2,…,zk} of k (k≥2) distinct complex numbers, we present a procedure of constructing a polynomial fn,λ for which Ω is a k-cycle of Kfn,λ,n with multiplier λ.
作者机构：
[Huang, Haiwu] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Peoples R China.;[Huang, Haiwu] Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang 421002, Peoples R China.
通讯机构：
[Huang, Haiwu] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Peoples R China.;Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang 421002, Peoples R China.
关键词：
ANA random variables;complete convergence;weighted sums
摘要：
In this paper, the author investigates the complete convergence for weighted sums of asymptotically negatively associated (ANA) random variables with different distributions, and obtains some equivalent conditions of complete convergence theorem for weighted sums as well as summation of ANA cases. These results generalize and improve the corresponding ones obtained by Baum and Katz (1965), Peligrad and Gut (1999) and Cai (2006), respectively.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2018年2018(1):1-13 ISSN：1029-242X
通讯作者：
Huang, Haiwu
作者机构：
[Wu, Xiongtao; Huang, Haiwu] Hengyang Normal Univ, Coll Math & Stat, Hengyang, Peoples R China.;[Huang, Haiwu] Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang, Peoples R China.;[Zhang, Qingxia] Southwest Petr Univ, Sch Sci, Chengdu, Sichuan, Peoples R China.
通讯机构：
[Huang, Haiwu] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang, Peoples R China.;Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang, Peoples R China.
关键词：
ANA random variables;Complete convergence;Weighed sums;Equivalent conditions
摘要：
In this investigation, some sufficient and necessary conditions of the complete convergence for weighted sums of asymptotically negatively associated (ANA, in short) random variables are presented without the assumption of identical distribution. As an application of the main results, the Marcinkiewicz-Zygmund type strong law of large numbers based on weighted sums of ANA cases is obtained. The results of this paper extend and generalize some well-known corresponding ones.
期刊：
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY,2018年97(1):80-87 ISSN：0004-9727
通讯作者：
Ponnusamy, Saminathan
作者机构：
[Chen, Shaolin] Soochow Univ, Dept Math, Suzhou 215006, Jiangsu, Peoples R China.;[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Stat Inst, Chennai Ctr, Stat Math Unit, 110 Nelson Manickam Rd, Madras 600029, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Stat Inst, Chennai Ctr, Stat Math Unit, 110 Nelson Manickam Rd, Madras 600029, Tamil Nadu, India.
关键词：
Landau's theorem;(partial derivative)over-bar-equation;Dirichlet-type space
