Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature

Let (X, d, μ) be a metric measure space with non-negative Ricci curvature. This paper is concerned with the boundary behavior of harmonic function on the (open) upper half-space X\times {\text{ℝ}}_{+}. We derive that a function f of bounded mean oscillation (BMO) is the trace of harmonic function u(x,t) on X\times {\text{ℝ}}_{+},u(x,0)=f\left(x\right), whenever u satisfies the following Carleson measure condition where \nabla =({\nabla }_x,{\partial }_t) denotes the total gradient and B(x_B,r_B) denotes the (open) ball centered at x_B with radius r_B. Conversely, the above condition characterizes all the harmonic functions whose traces are in BMO space.     

Perpetual cutoff method and CDE′(K,N) condition on graphs

By using the perpetual cutoff method, we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CD{E}^{\prime }(K,N). This generalizes a main result of F. Münch who considers the case of CD(K, \infty) curvature. Hence, we answer a question raised by Münch. For that purpose, we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded Laplacian \text{Δ} and perpetual cutoff semigroup P_t^W in our setting.     

A survey on book-embedding of planar graphs

The book-embedding problem arises in several area, such as very large scale integration (VLSI) design and routing multilayer printed circuit boards (PCBs). It can be used into various practical application fields. A book embedding of a graph G is an embedding of its vertices along the spine of a book, and an embedding of its edges to the pages such that edges embedded on the same page do not intersect. The minimum number of pages in which a graph G can be embedded is called the pagenumber or book-thickness of the graph G. It is an important measure of the quality for book-embedding. It is NP-hard to research the pagenumber of book-embedding for a graph G. This paper summarizes the studies on the book-embedding of planar graphs in recent years.     

Convergent sequences of sparse graphs: A large deviations approach

Models based on sparse graphs are of interest to many communities: they appear as basic models in combinatorics, probability theory, optimization, statistical physics, information theory, and more applied fields of social sciences and economics. Different notions of similarity (and hence convergence) of sparse graphs are of interest in different communities. In probability theory and combinatorics, the notion of Benjamini‐Schramm convergence, also known as left‐convergence, is used quite frequently. Statistical physicists are interested in the the existence of the thermodynamic limit of free energies, which leads naturally to the notion of right‐convergence. Combinatorial optimization problems naturally lead to so‐called partition convergence, which relates to the convergence of optimal values of a variety of constraint satisfaction problems. The relationship between these different notions of similarity and convergence is, however, poorly understood. In this paper we introduce a new notion of convergence of sparse graphs, which we call Large Deviations or LD‐convergence, and which is based on the theory of large deviations. The notion is introduced by “decorating” the nodes of the graph with random uniform i.i.d. weights and constructing corresponding random measures on and . A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on . The corresponding large deviations rate function can be interpreted as the limit object of the sparse graph sequence. In particular, we can express the limiting free energies in terms of this limit object. We then establish that LD‐convergence implies the other three notions of convergence discussed above, and at the same time establish several previously unknown relationships between the other notions of convergence. In particular, we show that partition‐convergence does not imply left‐ or right‐convergence, and that right‐convergence does not imply partition‐convergence. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 52–89, 2017     

Comment on “Kirchhoff index in line,subdivision and total graphs of a regular graph”

Let G$G$ be a connected regular graph. Denoted by t(G)$t\left(G\right)$ and Kf(G)$Kf\left(G\right)$ the total graph and Kirchhoff index of G$G$, respectively. This paper is to point out that Theorem 3.7 and Corollary 3.8 from “Kirchhoff index in line, subdivision and total graphs of a regular graph” [X. Gao, Y.F. Luo, W.W. Liu, Kirchhoff index in line, subdivision and total graphs of a regular graph, Discrete Appl. Math. 160(2012) 560–565] are incorrect, since the conclusion of a lemma is essentially wrong. Moreover, we first show the Laplacian characteristic polynomial of t(G)$t\left(G\right)$, where G$G$ is a regular graph. Consequently, by using Kf(G)$Kf\left(G\right)$, we give an expression on Kf(t(G))$Kf\left(t\left(G\right)\right)$ and a lower bound on Kf(t(G))$Kf\left(t\left(G\right)\right)$ of a regular graph G$G$, which correct Theorem 3.7 and Corollary 3.8 in Gao et al. (2012)  [2].     

截断调和Bergman空间上Toeplitz算子的代数性质

本文首先讨论调截断和Bergman空间 $b_{n}^{2}$ 上拟齐次函数为符号的Toeplitz算子的有限秩乘积问题,其次考察两个以拟齐次函数为符号的Toeplitz算子的换位子与半换位子的有限秩问题.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

一类特殊分块矩阵为循环矩阵的循环分块矩阵的几个性质

本文给出一类特殊分块矩阵为循环矩阵的循环分块矩阵的几个性质。     

一种复杂车辆图像中的多车牌定位方法

针对复杂背景中多个车牌的定位问题,提出一种新的定位方法.该方法综合利用边缘检测、连通域分析、倾斜矫正等多种方法,解决了复杂背景中定位难的问题.能够准确定位杂乱背景中的车牌,对天气、光照变化、车牌在图像中的移动和旋转等,具有良好的适应能力.该方法为后续的字符分割和字符识别提供旋转角度、字符区域定位信息.     

基于ABAQUS的微细电火花超声振动主轴仿真研究

微细电火花技术凭借自身的特点和优势,在航空航天、医疗器械等领域发挥着重要的作用.超声振动辅助电火花加工在传统电火花加工中引入超声振动,加工精度和加工效率明显提高.作为核心部件,超声振动主轴的设计尤为重要.但传统设计过程,大大延长了研发周期和增加了成本,因此,采用仿真方式进行研究具有重要的意义.该文对设计的微细电火花超声...