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.
摘要：
Abstract: This paper used factor analysis and analytic hierarchy process to discuss the influencing factors of regional economic development in Changsha, Zhuzhou and Xiangtan urban agglomerations, and obtained five first-level indicators—economic development indicators, research education indicators, human resources indicators, openness indicators, and infrastructure indicators. The empirical analysis of the index provides an important reference for regional economic devel-opment.#@#@#摘要: 本文采用了因子分析和层次分析法讨论了长株潭城市群区域经济发展的影响因素，得到五个一级指标--经济发展指标、科研教育指标、人力资源指标、开放程度指标、基础设施指标，并对该指标进行实证分析，为区域经济发展提供了重要参考。
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.
摘要：
Conjugate gradient methods can be used with advantages such as fast convergence and low memory requirement in real applications. A conjugate gradient-based neuro-fuzzy learning algorithm for zero-order Takagi-Sugeno inference systems is proposed in this paper. Compared with the existing gradient-based algorithm, this method enhances the learning performance.
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.
