A Shy Invariant of Graphs

 Moving from a well known result of Hammer, Hansen, and Simeone, we introduce a new graph invariant, say λ(G) referring to any graph G. It is a non-negative integer which is non-zero whenever G contains particular induced odd cycles or, equivalently, admits a particular minimum clique-partition. We show that λ(G) can be efficiently evaluated and that its determination allows one to reduce the hard problem of computing a minimum clique-cover of a graph to an identical problem of smaller size and special structure. Furthermore, one has α(G)≤θ(G)−λ(G), where α(G) and θ(G) respectively denote the cardinality of a maximum stable set of G and of a minimum clique-partition of G. Received: April 12, 1999 Final version received: September 15, 2000     

Coalescence of synchronous couplings

 We consider a pair of reflected Brownian motions in a Lipschitz planar domain starting from different points but driven by the same Brownian motion. First we construct such a pair of processes in a certain weak sense, since it is not known whether a strong solution to the Skorohod equation in Lipschitz domains exists. Then we prove that the distance between the two processes converges to zero with probability one if the domain has a polygonal boundary or it is a ``lip domain', i.e., a domain between the graphs of two Lipschitz functions with Lipschitz constants strictly less than 1. Received: 2 March 2001 / Revised version: 6 March 2001 / Published online: 1 July 2002     

Optimal markovian couplings and applications

This paper is devoted to studying a new topic: optimal Markovian couplings, mainly for time-continuous Markov processes. The study emphasizes the analysis of the coupling operators rather than the processes. Some constructions of optimal Markovian couplings for Markov chains and diffusions are presented, which are often unexpected. Then, the results are applied to study theL ²-convergence for Markov chains and for a diffusion on compact manifold. The estimate of the convergent rate provided by this method can be sharp. Supported in part by NSFC, the State Education Commission of China, the NSERC operating grant of D.A. Dawson and Centro Vito Volterra.     

Nonlinear continuous integrable Hamiltonian couplings

Based on a kind of special non-semisimple Lie algebras, a scheme is presented for constructing nonlinear continuous integrable couplings. Variational identities over the corresponding loop algebras are used to furnish Hamiltonian structures for the resulting continuous integrable couplings. The application of the scheme is illustrated by an example of nonlinear continuous integrable Hamiltonian couplings of the AKNS hierarchy of soliton equations.     

Synchronization of networks with time-varying couplings

In this paper, we present a review of our recent works on complete synchro-nization analyses of networks of the coupled dynamical systems with time-varying cou-plings. The main approach is composed of algebraic graph theory and dynamic system method. More precisely, the Hajnal diameter of matrix sequence plays a key role in in-vestigating synchronization dynamics and the joint graph across time periods possessing spanning tree is a doorsill for time-varying topologies to reach synchronization. These techniques with proper modification count for diverse models of networks of the cou-pled systems, including discrete-time and continuous-time models, linear and nonlinear models, deterministic and stochastic time-variations. Alternatively, transverse stability analysis of general time-varying dynamic systems can be employed for synchronization study as a special case and proved to be equivalent to Hajnal diameter.     

Concentration of measures via size-biased couplings

Let Y be a nonnegative random variable with mean??? and finite positive variance ?? ², and let Y ^s , defined on the same space as Y, have the Y size-biased distribution, characterized by $$ E[Yf(Y)]=\mu E f(Y^s) \quad {\rm for\,all\,functions}\,f\,{\rm for\,which\,these\,expectations\,exist}. $$ Under a variety of conditions on Y and the coupling of Y and Y ^s , including combinations of boundedness and monotonicity, one sided concentration of measure inequalities such as $$ P\left(\frac{Y-\mu}{\sigma} \ge t\right)\le {\rm exp}\left(-\frac{t^2}{2(A+Bt)} \right) \quad {\rm for\,all}\,t\, > 0 $$ hold for some explicit A and B. The theorem is applied to the number of bulbs switched on at the terminal time in the so called lightbulb process of Rao et?al. (Sankhy?? 69:137?C161, 2007).     

Mirror and synchronous couplings of geometric Brownian motions

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.     

Integrable couplings,bi‐integrable couplings and their Hamiltonian structures of the Giachetti–Johnson soliton hierarchy

On the basis of zero curvature equations from semi‐direct sums of Lie algebras, we construct integrable couplings of the Giachetti–Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi‐integrable couplings of the Giachetti–Johnson hierarchy and their Hamiltonian structures by applying a class of non‐semisimple matrix loop algebras consisting of triangular block matrices. Copyright © 2014 John Wiley & Sons, Ltd.     

Successful couplings for diffusion processes with state-dependent switching

This work is concerned with successful couplings for a class of multidimensional diffusion processes with state-dependent switching. We construct a type of couplings for this class of processes, and give some sufficient conditions to guarantee this type of couplings to be successful. Besides, two illustrative examples are provided.     

Nonlinear supersymmetry and goldstino couplings to the MSSM

We briefly review the nonlinear supersymmetry formalisms in the standard realization and superfield methods. We then evaluate the goldstino couplings to the minimal supersymmetric standard model (MSSM) superfields and discuss their phenomenological consequences. These relate to the tree-level Higgs mass and to invisible Higgs- and Z-boson decays. The Higgs mass is increased from its MSSM tree-level value and brought above the LEP2 mass bound for a low scale of supersymmetry breaking √f ∼ 2 TeV to 7 TeV. The invisible decay rates of the Higgs and Z bosons into goldstino and neutralino are computed and shown to bring stronger constraints on f than their decays into goldstino pairs, which are subleading in 1/f.     

Successful couplings for difusion processes with state-dependent switching

This work is concerned with successful couplings for a class of multidimensional difusion processes with state-dependent switching.We construct a type of couplings for this class of processes,and give some sufcient conditions to guarantee this type of couplings to be successful.Besides,two illustrative examples are provided.     

Recent efforts in the computation of string couplings

We review recent advances towards the computation of string couplings. Duality symmetry, mirror symmetry, Picard-Fuchs equations, etc. are some of the tools.Partially supported by the Deutsche Forschungsgemeinschaft.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 293–306, May, 1993.     

Self-synchronization of the integrate-and-fire pacemaker model with continuous couplings

The integrate-and-fire cardiac pacemaker model of the pulse coupled oscillators was introduced by C. Peskin. Due to the function of the pacemaker, two famous synchronization conjectures for identical and not identical oscillators were formulated. There are still many issues related to the nature and types of couplings. The couplings may be impulsive, continuous, delayed or advanced, and oscillators may be locally or globally connected. Consequently, it is reasonable to consider various ways of synchronization, if one wants the biological and mathematical analyses to interact productively. We investigate the integrate-and-fire model in both cases-one with identical, and another with not quite identical oscillators. A combination of continuous and pulse couplings that sustain the firing in unison is carefully constructed. Moreover, we obtain conditions on the parameters of continuous couplings that make possible a rigorous mathematical investigation of the problem. The technique developed for differential equations with discontinuities at non-fixed moments and a special continuous map lies on the basis of the analysis. This is the first analytically derived synchronization result for a model with continuous couplings. Illustrative examples are provided.     

Broer-Kaup-Kupershmidt族的非线性双可积耦合及其自相容源

基于新的非半单矩阵李代数,介绍了构造孤子族非线性双可积耦合的方法,由相应的变分恒等式给出了孤子族非线性双可积耦合的Hamilton结构.作为应用,给出了Broer-Kaup-Kupershmidt族的非线性双可积耦合及其Hamilton结构.最后指出了文献中的一些错误,利用源生成理论建立了新的公式,并导出了带自相容源Broer-Kaup-Kupershmidt族的非线性双可积耦合方程.