Poisson-Lie group of pseudodifferential symbols

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators (with scalar or matrix coefficients) on the line and on the circle. This defines a Poisson-Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, GL _n -KdV (or GL _n -Adler-Gelfand-Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this universal Poisson-Lie group. Moreover, the reduced (=SL _n ) versions of these manifolds (orW _n -algebras in physical terminology) can be viewed as certain subspaces of the quotient of this Poisson-Lie group by the dressing action of the group of functions on the circle (or as a result of a Poisson reduction). Finally we define an infinite set of commuting functions on the Poisson-Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning ofW as a limit of Poisson algebrasW as 0.     

离散时间序列的网络模体分析

时间序列可以被转换成网络的形式,复杂网络理论也因此可以用于刻画时间序列的时域和相空间特性.本文针对可视图算法和相空间重构算法这两种时间序列的转换算法,研究了它们的伴生网络在倍周期分岔和混沌等各种类型时间序列的模体分布特征,分析了这两种算法各自的优点.     

MAP序列多帧图像超分辨率复原

序列多帧图像的超分辨率复原是由一序列低分辨率图像来估计一幅高分辨率图像的技术。介绍了序列图像的成像观测模型,给出了一种随机微扫描亚像素运动估计的方法,采用最大后验概率法进行了图像的超分辨率复原,对仿真结果进行了分析和比较。     

Synchronization between different motifs

In this paper,we study the synchronization between different motifs.First,the synchronization between two networks with different topology structures and different dynamical behaviours is studied.With the open-plus-closed-loop(OPCL) method,conditions for two different networks to realize synchronization are given.Then based on the theoretical results achieved,the synchronization between different motifs is studied,which verifies the effectiveness and feasibility of the synchronization scheme.     

The method of covariant symbols in curved space-time

Diagonal matrix elements of pseudodifferential operators are needed in order to compute effective Lagrangians and currents. For this purpose the method of symbols is often used, which however lacks manifest covariance. In this work the method of covariant symbols, introduced by Pletnev and Banin, is extended to curved space-time with arbitrary gauge and coordinate connections. For the Riemannian connection we compute the covariant symbols corresponding to external fields, the covariant derivative and the Laplacian, to fourth order in a covariant derivative expansion. This allows one to obtain the covariant symbol of general operators to the same order. The procedure is illustrated by computing the diagonal matrix element of a nontrivial operator to second order. Applications of the method are discussed. PACS 04.62.+v; 11.15.-q; 11.15.Tk     

Operator symbols in the description of observable-state systems

For an observable-state system with finite degrees of freedom N topological properties of the kernels and symbols belonging to the considered operators are investigated. For the operators of $\text{L}$⁺($\text{S}$) kernels and symbols are distributions and for density matrices o? they are smooth functions.     

Kernels and symbols of operators for fermion fields

The functional method is studied in a mathematical framework, using same formalism for boson fields and fermion fields. Nuclear triplets are built up and regularity properties of operators are characterized by a kernels theorem.Talk given at the ldSymposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.     

Multi-scaling Networks from Vertex Intrinsic Fitness

Defining and detecting the community structure is an important topic on exploring the complex networks. Previous works were mostly based on the so-called modularity method, in which vertices that interconnect to each other form modules in the network, In many real-world networks, however, vertices are grouped not by their connectivity but by their functions. To demonstrate this idea, we propose a new kind of network, in which the vertices are catalogued into several types and are assigned intrinsic fitness, Each type of vertices may satisfy a different fitness distribution. It is found that the whole network exhibits a multi-scaling degree distribution. The clustering coefficients of the subnetworks are modified by the interlinks between vertices in the subnetworks.     

Estimation of network structures only from spike sequences

A neuron, the fundamental element of neural systems, interacts with other neurons, often producing very complicated behavior. To analyze, model, or predict such complicated behavior, it is important to understand how neurons are connected as well as how they behave. In this paper, we propose two measures, the spike time metric coefficient and the partial spike time metric coefficient, to estimate the network structure, that is, the topological connectivity between neurons. The proposed measures are based on the spike time metric and partialization analysis. To check the validity, we applied the proposed measures to asynchronous spike sequences that are produced by a mathematical neural network model. It was found that the proposed measure has high performance for estimating the network structures even though the structures have a complex topology such as a small-world structure or a scale-free structure.     

Complex networks constructed from irrational number sequences

In this paper, we construct (d,r) networks from sequences of different irrational numbers. In detail, segment an irrational number sequence of length M into groups of d digits which represent the nodes while two consecutive groups overlap by r digits (r=0,1,…,d−1), and the undirected edges indicate the adjacency between two consecutive groups. (3,r) and (4,r) networks are respectively constructed from 14 different irrational numbers and their topological properties are examined. By observation, we find that network topologies change with different values of d, r and even sequence length M instead of the types of irrational numbers, although they share some similar features with traditional random graphs. We make a further investigation to explain these interesting phenomena and propose the identical-degree random graph model. The results presented in this paper provide some insight into distributions of irrational number digits that may help better understanding of the nature of irrational numbers.     

量子论中狄拉克符号积分的意义

量子的引入最先是普朗克在1900年为理论“凑合”黑体辐射实验曲线的无奈之举(曲线拟合),然此举如招幡令旗,呼风唤雨,聚溪成流,乘奔御风,浩浩汤汤,终成今日量子流行的漫山遍野之势,是几个能人的集灵思积广益而相辅相成,还是时势造英雄,还是两者兼而有之！普朗克以能量分离的观点看待微观世界,是他在理论推导拟合实验结果逐渐形成的信仰。物理学家狄拉克指出,伟大的物理学家如牛顿和爱因斯坦是靠基本信仰“从上到下”推导出一些大自然的定律的。狄拉克自己的信仰是相信方程的美有时比实验结果更重要,因为实验会有误差。量子的时髦,自然引来众说纷纭,惟在量子园地里“种过树”的人才可能有较深刻的体会。
作者历经50多年的理论探索,首创了有序算符内的积分理论,对发展量子力学数理基础——狄拉克的符号法略有建树,既能抑制爱因斯坦认为量子力学数学不够完美的抱怨,为爱因斯坦的量子纠缠思想提供纠缠态表象,也从数学上将量子力学几率假说落实到有序算符的正态分布,从而推陈出新、别开生面地丰富量子力学、量子统计力学和量子光学的内容。     

Christoffel symbols and inertia in flat space-time theory

A necessary and sufficient criterion of inertia is presented, for the flat space-time theory of general frames of reference, in terms of the vanishing of some typical components of the affine connection pertaining to curvilinear coordinate systems. The physical identification of inertial forces thus arises in the context of the special theory of relativity.     

Invertibility of parabolic pseudodifferential operators with rapidly increasing symbols

We consider pseudodifferential operators with rapidly increasing double symbols analytic with respect to the variable dual to the time on the lower complex half-plane. We construct invertibility theory for these operators in weighted Sobolev spaces with weights related to growths of symbols and give applications to heat equations with potentials of power, exponential, and superexponential growths. Dedicated to the memory of Professor Leonid Romanovich Volevich     

The bases of authority for our symbols and units

This article, which is in two parts, surveys the wide application of scattering experiments to studies of the structures of solids and liquids. Part 1 showed how the angular distribution of a scattered beam of photons or neutrons is related by Fourier transform to the space and time-dependent distribution of electrons and nuclei in the scattering target. The use of X-rays and of neutrons in determining time-averaged density distributions was examined. Part 2 discusses the time-dependent aspects of the distributions for solids and liquids, including superfluid helium, and also of dynamical distributions of magnetism (or angular momentum) density, and examines the present limitations and future possibilities of scattering experiments.     

Networks with time structure from time series

We propose a method of constructing a network, in which its time structure is directly incorporated, based on a deterministic model from a time series. To construct such a network, we transform a linear model containing terms with different time delays into network topology. The terms in the model are translated into temporal nodes of the network. On each link connecting these nodes, we assign a positive real number representing the strength of relationship, or the “distance,” between nodes specified by the parameters of the model. The method is demonstrated by a known system and applied to two actual time series.     

The motifs of crystals and their nano-aggregates

Metal-organic-hybrid bulk chain-like, block, tubular, and plate crystals have been synthesized by a wet chemical method. Single crystal X-ray diffraction and TEM reveal that all the nano-structures are associated with their crystal motifs, which provide a successful and effective access to assemble controlled specific nano-structures from bulk crystal of which the motifs are much desired.     

Complex Networks: from Graph Theory to Biology

The aim of this text is to show the central role played by networks in complex system science. A remarkable feature of network studies is to lie at the crossroads of different disciplines, from mathematics (graph theory, combinatorics, probability theory) to physics (statistical physics of networks) to computer science (network generating algorithms, combinatorial optimization) to biological issues (regulatory networks). New paradigms recently appeared, like that of ‘scale-free networks’ providing an alternative to the random graph model introduced long ago by Erdös and Renyi. With the notion of statistical ensemble and methods originally introduced for percolation networks, statistical physics is of high relevance to get a deep account of topological and statistical properties of a network. Then their consequences on the dynamics taking place in the network should be investigated. Impact of network theory is huge in all natural sciences, especially in biology with gene networks, metabolic networks, neural networks or food webs. I illustrate this brief overview with a recent work on the influence of network topology on the dynamics of coupled excitable units, and the insights it provides about network emerging features, robustness of network behaviors, and the notion of static or dynamic motif.     

Cost-Effective Migration from Sub-Lambda to Wavelength-Based Metro Networks

This article will discuss the importance of hybrid coexistence of sub-lambda grooming with wavelength-based services for driving down costs and maximizing the use of metro bandwidth, as well as enabling next-generation lambda-based services. The article will also address migration to all optical metro networks.     

Cost-Effective Migration from Sub-Lambda to Wavelength-Based Metro Networks

This article will discuss the importance of hybrid coexistence of sub-lambda grooming with wavelength-based services for driving down costs and maximizing the use of metro bandwidth, as well as enabling next-generation lambda-based services. The article will also address migration to all optical metro networks.