浅析图论教学

图论是《离散数学》课程的重要组成部分,也是数学专业高年级的选修课程.本文介绍了从事图论教学和研究的一些心得,探讨了如何在该课程的教学过程中激发学生的学习兴趣和培养学生的发现问题及解决问题能力,从而为学生今后作毕业论文或者进一步从事科学研究打下基础.     

耦合相振子系统同步的序参量理论

节律行为,即系统行为呈现随时间的周期变化,在我们的周围随处可见.不同节律之间可以通过相互影响、相互作用产生自组织,其中同步是最典型、最直接的有序行为,它也是非线性波、斑图、集群行为等的物理内在机制.不同的节律可以用具有不同频率的振子(极限环)来刻画,它们之间的同步可以用耦合极限环系统的动力学来加以研究.微观动力学表明,随着耦合强度增强,振子同步伴随着动力学状态空间降维到一个低维子空间,该空间由序参量来描述.序参量的涌现及其所描述的宏观动力学行为可借助于协同学与流形理论等降维思想来进行.本文从统计物理学的角度讨论了耦合振子系统序参量涌现的几种降维方案,并对它们进行了对比分析.序参量理论可有效应用于耦合振子系统的同步自组织与相变现象的分析,通过进一步研究序参量的动力学及其分岔行为,可以对复杂系统的涌现动力学有更为深刻的理解.     

An ergodic theorem of a parabolic Anderson model driven by L��vy noise

In this paper, we study an ergodic theorem of a parabolic Andersen model driven by Lévy noise. Under the assumption that A = (a(i, j)) _i,j∈S is symmetric with respect to a σ-finite measure gp, we obtain the long-time convergence to an invariant probability measure ν _h starting from a bounded nonnegative A-harmonic function h based on self-duality property. Furthermore, under some mild conditions, we obtain the one to one correspondence between the bounded nonnegative A-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Lévy noise, which is an extension of the result of Y. Liu and F. X. Yang.     

弦图的L(3,2,1)}-标号

图 G 的一个 L(3,2,1)- 标号是指从 V(G) 到非负整数集的一个映射 f, 满足: 当 d_G(u,v)=1 时, |f(u)-f(v)|\geq 3; 当 d_G(u,v)=2 时, |f(u)-f(v)|\geq 2; 当 d_G(u,v)=1 时, |f(u)-f(v)|\geq 1. L(3,2,1)-标号问题就是确定出最小的整数 \lambda_3(G) 使得 G存在最大标号不超过该数的 L(3,2,1)- 标号. 本文研究了弦图的 L(3,2,1)- 标号问题,获得了弦图及其一些子类, 如扇, r- 路,r- 树等的 \lambda_3 数的界.     

Inversion of Trace Formulas for a Sturm-Liouville Operator

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.     

Real-time positioning technology of train based on optical fiber coherent Rayleigh backscattering北大核心CSCD

With the rapid growth of railway mileage in China, the degree of heavy load of railway wagons continues to increase. An interference technology based on Rayleigh backscattering signal in optical fiber was proposed by using existing communication optical cables along railway lines. When the fiber vibrated slightly, the phase and refractive index of the fiber at disturbed position changed, which resulting in the Rayleigh backscattering light. By performing differential calculation on the Rayleigh signal curves before and after the operation, the location of the interference light intensity signal corresponded to the location of the disturbances was obtained. Based on this method, the recognition and positioning of railway vehicles were realized. By collecting and analyzing the time-domain and frequency-domain signal, the signal strength, train length, the number of carriages and other characteristics were extracted, and the model of railway vehicles were accurately recognized. Compared with traditional positioning technologies, this technology could realize long-distance monitoring, and the sensing fiber was buried underground on both sides of the railway, which was conducive to the concealment and protection of optical fiber. Experimental results show that the positioning error of the system is within ±10 m, and the detection of railway speed and position within 25 km can be realized. © 2022 Editorial office of Journal of Applied Optics. All rights reserved.     

Anisotropic and valley-resolved beam-splitter based on a tilted Dirac system

We investigate theoretically valley-resolved lateral shift of electrons traversing an n–p–n junction bulit on a typical tilted Dirac system (8-Pmmn borophene). A gauge-invariant formula on Goos–Hänchen (GH) shift of transmitted beams is derived, which holds for any anisotropic isoenergy surface. The tilt term brings valley dependence of relative position between the isoenergy surface in n region and that in the p region. Consequently, valley double refraction can occur at the n–p interface. The exiting positions of two valley-polarized beams depend on the incident angle and energy of incident beam and barrier parameters. Their spatial distance D can be enhanced to be ten to a hundred times larger than the barrier width. Due to tilting-induced high anisotropy of the isoenergy surface, D depends strongly on the barrier orientation. It is always zero when the junction is along the tilt direction of Dirac cones. Thus GH effect of transmitted beams in tilted Dirac systems can be utilized to design anisotropic and valley-resolved beam-splitter.     

A centrality measure for communication ability in weighted network

This paper proposes a new node centrality measurement in a weighted network, the communication centrality, which is inspired by Hirsch’s h$h$-index. We investigated the properties of the communication centrality, and proved that the distribution of the communication centrality has the power-law upper tail in weighted scale-free networks. Relevant measures for node and network are discussed as extensions. A case study of a scientific collaboration network indicates that the communication centrality is different from other common centrality measures and other h$h$-type indexes. Communication centrality displays moderate correlation with other indexes, and contains a well-balanced mix of other centrality measures and cannot be replaced by any of them.     

Structural rearrangement of β-lactoglobulin at different oil-water interfaces and its effect on emulsion stability

Understanding the factors that control protein structure and stability at the oil-water interface continues to be a major focus to optimize the formulation of protein-stabilized emulsions. In this study, a combination of synchrotron radiation circular dichroism spectroscopy, front-face fluorescence spectroscopy, and dual polarization interferometry (DPI) was used to characterize the conformation and geometric structure of β-lactoglobulin (β-Lg) upon adsorption to two oil-water interfaces: a hexadecane-water interface and a tricaprylin-water interface. The results show that, upon adsorption to both oil-water interfaces, β-Lg went through a β-sheet to α-helix transition with a corresponding loss of its globular tertiary structure. The degree of conformational change was also a function of the oil phase polarity. The hexadecane oil induced a much higher degree of non-native α-helix compared to the tricaprylin oil. In contrast to the β-Lg conformation in solution, the non-native α-helical-rich conformation of β-Lg at the interface was resistant to further conformational change upon heating. DPI measurements suggest that β-Lg formed a thin dense layer at emulsion droplet surfaces. The effects of high temperature and the presence of salt on these β-Lg emulsions were then investigated by monitoring changes in the ζ-potential and particle size. In the absence of salt, high electrostatic repulsion meant β-Lg-stabilized emulsions were resistant to heating to 90 °C. Adding salt (120 mM NaCl) before or after heating led to emulsion flocculation due to the screening of the electrostatic repulsion between colloidal particles. This study has provided insight into the structural properties of proteins adsorbed at the oil-water interface and has implications in the formulation and production of emulsions stabilized by globular proteins.