摘要：
This paper investigates the split complex gradient descent based neuro-fuzzy algorithm with self-adaptive momentum and L-2 regularizer for training TSK (Takagi-Sugeno-Kang) fuzzy inference models. The major threat for disposing complex data with fuzzy system is contradiction of boundedness and analyticity in the complex domain, as expressed by Liouville's theorem. The proposed algorithm operates a couple of real-valued functions and splits the complex variables into real part and imaginary part. Dynamical momentum is included in the learning mechanism to promote learning speed. L-2 regularizer is also added to control the magnitude of the weight parameters. Furthermore, a detailed convergence analysis of the proposed algorithm is fully studied. The monotonic decreasing property of the error function and convergence of the weight sequence are guaranteed. Plus a mild condition, strong convergence of the weight sequence is deduced. Finally, the simulation results are also demonstrated to verify the theoretical analysis results. (C) 2018 Elsevier B.V. All rights reserved.
摘要：
We prove that, the so called total energy functional defined on the class of radial stretchings between annuli attains its minimum on a total energy diffeomorphism between annuli on R-n. This involves a subtle analysis of some special ODE. The result is an extension of the corresponding 2-dimensional case obtained by Iwaniec and Onninen (2009). (C) 2017 Elsevier Ltd. All rights reserved.
摘要：
The aim of this paper is twofold. One of them is to introduce the class of harmonic
$$\nu $$
-Bloch-type mappings as a generalization of harmonic
$$\nu $$
-Bloch mappings and thereby we generalize some recent results of harmonic 1-Bloch-type mappings investigated recently by Efraimidis et al. (Complex Var Elliptic Equ 62:1081–1092, 2017). The other is to investigate some subordination principles for harmonic Bloch mappings and then establish Bohr’s theorem for these mappings and in a general setting, in some cases.
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] ISI, Chennai Ctr, SETS, MGR Knowledge City, CIT Campus, Madras 600113, Tamil Nadu, India.
通讯机构：
[Chen, Shaolin] S;[Chen, Shaolin] H;Soochow Univ, Dept Math, Suzhou 215006, Jiangsu, Peoples R China.;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
作者机构：
[Yang, Liu] Hengyang Normal Univ, Dept Math & Comp Sci, Hengyang 421008, Hunan, Peoples R China.;[Ouyang, Zigen; Liu, Zhisu] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
通讯机构：
[Liu, Zhisu] U;Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
关键词：
Multiplicity;Kirchhoff type equation;Critical growth
摘要：
In this paper, we study the following Kirchhoff type equation with critical growth {-(a + b integral(Omega) vertical bar del u vertical bar(2)dx) del u = lambda u + mu vertical bar u vertical bar(2)u + vertical bar u vertical bar(4)u in Omega, u = 0 on partial derivative Omega, where a > 0, b >= 0 and Omega is a smooth bounded domain in R-3. When the real parameter mu is larger than some positive constant, we investigate the multiplicity of nontrivial solutions for the above problem with parameter lambda belonging to a left neighborhood of the Dirichlet eigenvalue of the Laplacian operator -Delta. (C) 2016 Elsevier Ltd. All rights reserved.
期刊：
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY,2017年145(2):833-846 ISSN：0002-9939
通讯作者：
Li, Liulan
作者机构：
[Li, Liulan] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Ponnusamy, Saminathan] ISI, Chennai Ctr, SETS, CIT Campus, Madras 600113, Tamil Nadu, India.
通讯机构：
[Li, Liulan] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.
关键词：
Univalent;convex;starlike and close-to-convex functions;Fekete-Szegö inequality;Zalcman and generalized Zalcman functionals
摘要：
Let S denote the class of all functions f(z) = z + Sigma(infinity)(n=2) a(n)az(n) 2 anzn analytic and univalent in the unit disk D. For f. S, Zalcman conjectured that vertical bar a(n)(2) - a(2n-1)vertical bar <= (n -1)(2) for n >= 3. This conjecture has been verified for only certain values of n for f is an element of S and for all n >= 4 for the class C of close-to-convex functions (and also for a couple of other classes). In this paper we provide bounds of the generalized Zalcman coefficient functional vertical bar lambda a(n)(2) n-a2(n-1)vertical bar for functions in C and for all n >= 3, where. is a positive constant.
作者机构：
[Chen, Yuan; Li, Long] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Peoples R China.;[Qiao, Zhijun] Univ Texas Rio Grande Valley, Sch Math & Stat Sci, Edinburg, TX 78539 USA.;[Liu, Yan] Dalian Polytech Univ, Dept Appl Math, Dalian 116034, Peoples R China.
通讯机构：
[Li, Long] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Peoples R China.
摘要：
In this paper, a smooth function is constructed to approximate the nonsmooth output of max-min fuzzy neural networks (FNNs) and its approximation is also presented. In place of the output of max-min FNNs by its smoothing approximation function, the error function, defining the discrepancy between the actual outputs and desired outputs of max-min FNNs, becomes a continuously differentiable function. Then, a smoothing gradient decent-based algorithm with Armijo-Goldstein step size rule is formulated to train max-min FNNs. Based on the existing convergent result, the convergence of our proposed algorithm can easily be obtained. Furthermore, the proposed algorithm also provides a feasible procedure to solve fuzzy relational equations with max-min composition. Finally, some numerical examples are implemented to support our results and demonstrate that the proposed smoothing algorithm has better learning performance than other two gradient decent-based algorithms. (C) 2017 Elsevier B.V. All rights reserved.
期刊：
JOURNAL OF GEOMETRIC ANALYSIS,2017年27(2):1468-1488 ISSN：1050-6926
通讯作者：
Chen, Shaolin
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Stat Inst, Chennai Ctr, SETS, MGR Knowledge City, CIT Campus, Chennai 600113, Tamil Nadu, India.
通讯机构：
[Chen, Shaolin] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
摘要：
The crossing number problem is in the forefront of topological graph theory. At present, there are only a few results concerning crossing numbers of join of some graphs. In this paper, for the special graph Q on six vertices we give the crossing numbers of its join with n isolated vertices as well as with the path Pn on n vertices and with the cycle Cn.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2016年2016(1):1-16 ISSN：1029-242X
通讯作者：
Huang, Haiwu
作者机构：
[Ouyang, Mengqian; 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.;[Jiang, Yuanying] Guilin Univ Technol, Coll Sci, Guilin 541004, 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：
arrays of rowwise ANA random variables;complete convergence;complete moment convergence;mean convergence
摘要：
In this paper, some complete convergence, complete moment convergence, and mean convergence results for arrays of rowwise asymptotically negatively associated (ANA) random variables are obtained. These theorems not only generalize some well-known ones to ANA cases, but they also improve them.
作者机构：
[Li, L.] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Ponnusamy, S.] Indian Stat Inst, Chennai Ctr, Soc Elect Transact & Secur, MGR Knowledge City, CIT Campus, Madras 600113, Tamil Nadu, India.;[Qiao, J.] Hebei Univ, Dept Math, Baoding 071002, Hebei, Peoples R China.
关键词：
univalent;starlike;convex and close-to-convex function;extreme point;closed convex hull and subordination;Zalcman conjecture
摘要：
Let
$${\mathcal S}$$
denote the class of all functions of the form
$${f(z)=z+a_2z^2+a_3z^3+\cdots}$$
which are analytic and univalent in the open unit disk
$${{\mathbb{D}} }$$
and, for
$${\lambda > 0}$$
, let
$${\Phi_\lambda (n,f)=\lambda a_n^2-a_{2n-1}}$$
denote the generalized Zalcman coefficient functional. Zalcman conjectured that if
$${f\in \mathcal S}$$
, then
$${|\Phi_1 (n,f)|\leq (n-1)^2}$$
for
$${{n\ge 3}}$$
. The functional of the form
$${\Phi_\lambda (n,f)}$$
is indeed related to Fekete–Szegő functional of the
$${n}$$
-th root transform of the corresponding function in
$${\mathcal S}$$
. This conjecture has been verified for a certain special geometric subclasses of
$${\mathcal S}$$
but it remains open for
$${f\in {\mathcal S}}$$
and for
$${n > 6}$$
. In the present paper, we prove sharp bounds on
$${|\Phi_\lambda (n,f)|}$$
for
$${f\in \mathcal{F}(\alpha )}$$
and for all
$${n\geq 3}$$
, in the case that
$${\lambda}$$
is a positive real parameter, where
$${ \mathcal{F}(\alpha )}$$
denotes the family of all functions
$${f\in {\mathcal S}}$$
satisfying the condition
$${\rm{Re}} \Big( 1+\frac{zf''(z)}{f'(z)}
\Big) > \alpha \quad \mbox{for } z\in {\mathbb{D}} ,$$
where
$${-1/2\leq \alpha < 1}$$
. Thus, the present article proves the generalized Zalcman conjecture for convex functions of order
$${\alpha}$$
,
$${\alpha \in [-1/2,1)}$$
.
作者机构：
[Zhang, Hanjun; Huang, Haiwu; Zhang, HJ] Xiangtan Univ, Sch Math & Computat Sci, Xiangtan 411105, Peoples R China.;[Huang, Haiwu] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Peoples R China.;[Zhang, Qingxia] Southwest Petr Univ, Sch Sci, Chengdu 610500, Peoples R China.
通讯机构：
[Huang, HW; Zhang, HJ] X;[Huang, Haiwu] H;[Zhang, Qingxia] S;Xiangtan Univ, Sch Math & Computat Sci, Xiangtan 411105, Peoples R China.;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Peoples R China.
关键词：
AANA random variables;Complete convergence;Complete moment convergence;Weighted sums
摘要：
In this work, a complete moment convergence theorem is obtained for weighted sums of asymptotically almost negatively associated (AANA) random variables without assumption of identical distribution under some mild moment conditions. As an application, the complete convergence theorems for weighted sums of negatively associated (NA) and AANA random variables are obtained. The result not only generalizes the corresponding ones of Sung [13] and Huang et al. [8], but also improves them.
摘要：
In this paper, we first establish a Schwarz-Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.
作者机构：
[Li, Liulan] Hengyang Normal Univ, Coll Math & Stat, E Ring Rd, Hengyang 421002, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Stat Inst, Chennai Ctr, Soc Elect Transact & Secur, CIT Campus, Madras 600113, Tamil Nadu, India.
通讯机构：
[Li, Liulan] H;Hengyang Normal Univ, Coll Math & Stat, E Ring Rd, Hengyang 421002, Hunan, Peoples R China.
摘要：
We consider the class H
0 of sense-preserving harmonic functions
$$f = h + \bar g$$
defined in the unit disk |z| < 1 and normalized so that h(0) = 0 = h′(0) − 1 and g(0) = 0 = g′(0), where h and g are analytic in the unit disk. In the first part of the article we present two classes P
H
0(α) and G
H
0(β) of functions from H
0 and show that if f ∈ P
H
0(α) and F ∈ G
H
0(β), then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial sums)
$${s_{n,n}}\left( f \right)\left( z \right) = {s_n}\left( h \right)\left( z \right) + \overline {{s_n}\left( g \right)\left( z \right)} ,$$
where
$$f = h + \bar g$$
∈ H
0, s
n
(h) and s
n
(g) denote the n-th partial sums of h and g, respectively. We prove, among others, that if
$$f = h + \bar g$$
∈ H
0 is a univalent harmonic convex mapping, then s
n
,n(f) is univalent and close-to-convex in the disk |z| < 1/4 for n ≥ 2, and s
n
,n(f) is also convex in the disk |z| < 1/4 for n ≥ 2 and n ≠ 3. Moreover, we show that the section s
3,3(f) of f ∈ C
H
0 is not convex in the disk |z| < 1/4 but it is convex in a smaller disk.
