作者机构：
[Guo, Qing; Wang, Xuewen] Minzu Univ China, Coll Sci, Beijing, Peoples R China;[Wang, Hua] Cent China Normal Univ, Sch Math & Stat, Wuhan, Hubei, Peoples R China;[Wang, Hua] Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan, Hubei, Peoples R China
摘要：
We show that assuming some initial datum leads to a blow-up solution of the inter-critical defocusing nonlinear Schrodinger equation, there must exist an initial datum with the minimal -norm which will produce blow-up. Moreover, up to the invariant transformations, the set of such data is compact in . The main tool is the profile decomposition combined with the perturbation argument.
期刊：
Applied Mathematics and Computation,2019年361:232-245 ISSN：0096-3003
通讯作者：
Wang, Shujing
作者机构：
[Wang, Shujing; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hua] Georgia Southern Univ, Dept Math Sci, Statesboro, GA 30460 USA.
通讯机构：
[Wang, Shujing] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Tree;Subtree;Wiener index;Ratio
摘要：
Among many topological indices of trees the sum of distances sigma (T) and the number of subtrees F(T) have been a long standing pair of graph invariants that are well known for their negative correlation. That is, among various given classes of trees, the extremal structures maximizing one usually minimize the other, and vice versa. By introducing the "local" versions of these invariants, sigma(T) (nu) for the sum of distance from nu to all other vertices and sigma(T) (nu) for the number of subtrees containing nu, extremal problems can be raised and studied for vertices within a tree. This leads to the concept of "middle parts" of a tree with respect to different indices. A challenging problem is to find extremal values of the ratios between graph indices and corresponding local functions at middle parts or leaves. This problem also provides new opportunities to further verify the the correlation between different indices such as sigma(T) and F(T). Such extremal ratios, along with the extremal structures, were studied and compared for the distance and subtree problems for general trees (Barefoot, 1997; Szekely and Wang, 2013, 2014). In this paper, this study is extended to binary trees, a class of trees with numerous practical applications in which the extremal ratio problems appear to be even more complicated. After justifying some basic properties on the distance and subtree problems in trees and binary trees, characterizations are provided for the extremal structures achieving two extremal ratios in binary trees of given order. The generalization of this work to k-ary trees is also briefly discussed. The findings are compared with the previous established extremal structures in general trees. Lastly some potential future work is mentioned. (C) 2019 Elsevier Inc. All rights reserved.
作者机构：
[Wang, Shujing; Li, Shuchao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hua] Nankai Univ, Coll Software, Tianjin 300071, Peoples R China.;[Wang, Hua] Georgia Southern Univ, Dept Math Sci, Statesboro, GA 30460 USA.
通讯机构：
[Wang, Shujing] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Distance;Tree;Subtree;Full binary tree;Span
摘要：
The sum of distances between all pairs of vertices (denoted by sigma(.) and called the Wiener index) and the number of subtrees (denoted by F(.) and called the subtree index) of a graph G are two representative graph invariants that have been extensively studied. The "local" version of these graph invariants (i.e. sum of distances from a given vertex, called the distance of the vertex, and the number of subtrees containing such a vertex, called the local subtree index of the vertex) have been studied. The distance of a vertex v in a tree T, denoted by sigma(T)(v), attains its minimum at one or two adjacent vertices called the centroid while the maximum sigma(T)(v) occurs at one or more leaves. On the other hand, the local subtree index, denoted by F-T(v), attains its maximum at one or two adjacent vertices called the subtree core and the minimum F-T(v) occurs at one ore more leaves. In this paper we study the difference between the values of sigma(T)(v) at a centroid vertex and a leaf, called the a-span, and similarly the F-span for the difference in values of the local subtree index at the subtree core and at a leaf. Among trees and full binary trees (trees in which each vertex has degree 1 or 3) on a given number of vertices we study the maximum and minimum possible values of the sigma-span and F-span. The extremal structures corresponding to some of these extremal values are also presented. Some unsolved problems are also discussed and proposed as open questions. (C) 2019 Elsevier B.V. All rights reserved.
期刊：
Mathematical Methods in the Applied Sciences,2019年42(1):237-249 ISSN：0170-4214
通讯作者：
Wang, Hua
作者机构：
[Yang, Qixiang] Wuhan Univ, Sch Math & Stat, Wuhan 430072, Hubei, Peoples R China.;[Wang, Hua] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hua] Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Wang, Hua] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Hubei, Peoples R China.
摘要：
In this paper, we apply wavelets to consider local norm function spaces with the Lorentz index. Triebel–Lizorkin–Lorentz spaces are based on the real interpolation of the Triebel–Lizorkin spaces. Triebel–Lizorkin–Morrey spaces are based on local norm of the Triebel–Lizorkin spaces. We give a unified depict of spaces that include these two kinds of spaces. Each index of the five index spaces represents a property of functions. We prove the wavelet characterization of the Triebel–Lizorkin–Lorentz–Morrey spaces and use such characterization to study some basic properties of these spaces.
期刊：
Journal of Mathematical Physics,2019年60(12):121508 ISSN：0022-2488
通讯作者：
Wang, Hua
作者机构：
[Wang, Hua] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Wang, Hua] Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Peoples R China.;[Yang, Qixiang] Wuhan Univ, Sch Math & Stat, Wuhan 430072, Peoples R China.
通讯机构：
[Wang, Hua] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Peoples R China.
摘要：
In this paper, we consider a 5D quadratic nonlinear Schrodinger system: iu(t) + Delta u = -2v (u) over bar, iv(t) + k Delta v = -u(2), and (t, x) is an element of R x R-5, k > 0. If k=1/2, we say the system satisfies the mass-resonance condition. Using the argument of Dodson and Murphy [Math. Res. Lett. 25, 1805 (2018)], we prove that the solution (u.v) scatters if initial data (u(0), v(0)) is an element of H-1(R-5) x H-1(R-5) satisfy E(u(0), v(0))M(u(0), v(0)) < E(Q(1), Q(2))M(Q(1), Q(2)) and K(u(0), v(0))M(u(0), v(0)) < K(Q(1), Q(2))M(Q(1), Q(2)) and k lies in some small neighborhood of 1/2. Under the condition of radial initial data, the range of k can be extended to k > 0 based on the idea of Dodson and Murphy [Proc. Am. Math. Soc. 145, 4859 (2017)].
期刊：
Journal of Combinatorial Optimization,2018年36(1):65-80 ISSN：1382-6905
通讯作者：
Li, Shuchao
作者机构：
[Huang, Jing; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hua] Georgia Southern Univ, Dept Math Sci, Statesboro, GA 30460 USA.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Skew-rank;Oriented graph;Evenly-oriented;Independence number
摘要：
An oriented graph
$$G^\sigma $$
is a digraph without loops or multiple arcs whose underlying graph is G. Let
$$S\left( G^\sigma \right) $$
be the skew-adjacency matrix of
$$G^\sigma $$
and
$$\alpha (G)$$
be the independence number of G. The rank of
$$S(G^\sigma )$$
is called the skew-rank of
$$G^\sigma $$
, denoted by
$$sr(G^\sigma )$$
. Wong et al. (Eur J Comb 54:76–86, 2016) studied the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. In this paper, the correlation involving the skew-rank, the independence number, and some other parameters are considered. First we show that
$$sr(G^\sigma )+2\alpha (G)\geqslant 2|V_G|-2d(G)$$
, where
$$|V_G|$$
is the order of G and d(G) is the dimension of cycle space of G. We also obtain sharp lower bounds for
$$sr(G^\sigma )+\alpha (G),\, sr(G^\sigma )-\alpha (G)$$
,
$$sr(G^\sigma )/\alpha (G)$$
and characterize all corresponding extremal graphs.
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2018年38(4):2207-2228 ISSN：1078-0947
通讯作者：
Wang, Hua
作者机构：
[Sun, Chenmin; Zheng, Jiqiang] Univ Cote dAzur, LJAD, F-06100 Nice, France.;[Wang, Hua; Yao, Xiaohua] Cent China Normal Univ, Dept Math, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hua; Yao, Xiaohua] Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Wang, Hua] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Hubei, Peoples R China.
摘要：
The aim of this paper is to adapt the strategy in [8] [ See, B. Dodson, J. Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for certain focusing fractional NLS. The main ingredient is to apply the fractional virial identity proved in [3] [ See, T. Boulenger, D. Himmelsbach, E. Lenzmann, Blow up for fractional NLS,J. Func. Anal, 271(2016), 2569-2603 ] to exclude the concentration of mass near the origin.
作者机构：
[Feng, Hong Liang; Yao, Xiao Hua; Wang, Hua] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Yao, Xiao Hua; Wang, Hua] Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Yao, Xiao Hua] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Hubei, Peoples R China.
关键词：
Fourth order NLS;potential;morawetz estimate;scattering
作者机构：
[Wang, Hongzhuan; Hua, Hongbo] Huaiyin Inst Technol, Fac Math & Phys, Huaian 223003, Jiangsu, Peoples R China.;[Hu, Xiaolan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Hua, Hongbo] H;Huaiyin Inst Technol, Fac Math & Phys, Huaian 223003, Jiangsu, Peoples R China.
摘要：
The eccentric distance sum (EDS) and degree distance (DD) are two distance-based graph invariants which have been well-studied in recent years. The study on relationships between various graph invariants has received much attention over the past few decades, and some of these research are associated with Graffiti conjectures (Fajtlowicz and Waller, 1987) or AutoGraphiX conjectures (Aouchiche et al., 2006). More recently, several groups of authors have investigated the relationships between several distance-based graph invariants along this line, see e.g., Klavžar and Nadjafi-Arani (2014), Hua et al. (2015), and Zhang and Li (0000), and so on. In this paper, we investigate the relationship between the eccentric distance sum and degree distance. First, we establish several sufficient conditions for a connected graph to have a larger/smaller EDS than DD, respectively. Second, we investigate extremal problems on the difference between EDS and DD for general connected graphs, trees, and self-centered graphs, respectively. More specifically, we present sharp upper and lower bounds on the difference between EDS and DD among all connected graphs, trees and self-centered graphs, respectively. In addition, we characterize all extremal graphs attaining those upper or lower bounds.
作者机构：
[Qin, Hong; Yan, Ting; Yan, T; Qin, H] Cent China Normal Univ, Dept Stat, Wuhan 430079, Peoples R China.;[Wang, Hansheng] Peking Univ, Dept Business Stat & Econometr, Guanghua Sch Management, Beijing 100871, Peoples R China.
通讯机构：
[Yan, T; Qin, H] C;[Wang, Hansheng] P;Cent China Normal Univ, Dept Stat, Wuhan 430079, Peoples R China.;Peking Univ, Dept Business Stat & Econometr, Guanghua Sch Management, Beijing 100871, Peoples R China.
关键词：
Asymptotical normality;consistency;increasing number of parameters;moment estimators;undirected network models
摘要：
To capture the heterozygosity of vertex degrees of networks and understand their distributions, a class of random graph models parameterized by the strengths of vertices is proposed. These models have a framework of mutually independent edges, where the number of parameters matches the size of the network. The asymptotic properties of the maximum likelihood estimator have been derived in such models as the beta-model, but general results are lacking. In these models, the likelihood equations are identical to the moment equations. Here, we establish a unified asymptotic result that includes the consistency and asymptotic normality of the moment estimator instead of the maximum likelihood estimator, when the number of parameters goes to infinity. We apply it to the generalized beta-model, maximum entropy models, and Poisson models.
期刊：
Journal of Differential Equations,2011年250(9):3559-3583 ISSN：0022-0396
通讯作者：
Wang, Hua
作者机构：
[Wang, Hua] Huazhong Normal Univ, Dept Math, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hua] Ecole Normale Super, Dept Math & Applicat, F-75230 Paris 05, France.;[Cui, Shangbin] Sun Yat Sen Univ, Dept Math, Guangzhou 510275, Guangdong, Peoples R China.
通讯机构：
[Wang, Hua] H;Huazhong Normal Univ, Dept Math, Wuhan 430079, Hubei, Peoples R China.
关键词：
35B65;35Q55;35Q60;Bilinear estimates;Local well-posedness;Schrödinger-Korteweg-de Vries system
摘要：
In this paper we prove that in the general case (i.e. beta not necessarily vanishing) the Cauchy problem for the Schrodinger-Korteweg-de Vries system is locally well-posed in L(2) x H(-3/4), and if beta = 0 then it is locally well-posed in H(s) x H(-3/4) with -3/16 < s <= 1/4. These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space G(s) x F(s), where G(s) and F(s) are dyadic Bourgain-type spaces related to the Schrodinger operator and the Airy operator at, respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator. (C) 2011 Elsevier Inc. All rights reserved.
