On quantum vertex algebras and their modules

We give a survey on the developments in a certain theory of quantum vertex algebras, including a conceptual construction of quantum vertex algebras and their modules and a connection of double Yangians and Zamolodchikov-Faddeev algebras with quantum vertex algebras.Comment: 18 pages; contribution to the proceedings of the conference in honor of Professor Geoffrey Maso

A guided Monte Carlo method for optimization problems

We introduce a new Monte Carlo method by incorporating a guided distribution function to the conventional Monte Carlo method. In this way, the efficiency of Monte Carlo methods is drastically improved. To further speed up the algorithm, we include two more ingredients into the algorithm. First, we freeze the sub-patterns that have high probability of appearance during the search for optimal solution, resulting in a reduction of the phase space of the problem. Second, we perform the simulation at a temperature which is within the optimal temperature range of the optimization search in our algorithm. We use this algorithm to search for the optimal path of the traveling salesman problem and the ground state energy of the spin glass model and demonstrate that its performance is comparable with more elaborate and heuristic methods.Comment: 4 pages, ReVTe

Modules-at-infinity for quantum vertex algebras

This is a sequel to \cite{li-qva1} and \cite{li-qva2} in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian $DY_{\hbar}(sl_{2})$, denoted by $DY_{q}(sl_{2})$ and $DY_{q}^{\infty}(sl_{2})$ with $q$ a nonzero complex number. For each nonzero complex number $q$, we construct a quantum vertex algebra $V_{q}$ and prove that every $DY_{q}(sl_{2})$-module is naturally a $V_{q}$-module. We also show that $DY_{q}^{\infty}(sl_{2})$-modules are what we call $V_{q}$-modules-at-infinity. To achieve this goal, we study what we call $\S$-local subsets and quasi-local subsets of \Hom (W,W((x^{-1}))) for any vector space $W$, and we prove that any $\S$-local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with $W$ as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.Comment: Latex, 48 page

A sequence based genetic algorithm with local search for the travelling salesman problem

The standard Genetic Algorithm often suffers from slow convergence for solving combinatorial optimization problems. In this study, we present a sequence based genetic algorithm (SBGA) for the symmetric travelling salesman problem (TSP). In our proposed method, a set of sequences are extracted from the best individuals, which are used to guide the search of SBGA. Additionally, some procedures are applied to maintain the diversity by breaking the selected sequences into sub tours if the best individual of the population does not improve. SBGA is compared with the inver-over operator, a state-of-the-art algorithm for the TSP, on a set of benchmark TSPs. Experimental results show that the convergence speed of SBGA is very promising and much faster than that of the inver-over algorithm and that SBGA achieves a similar solution quality on all test TSPs

A general framework of multi-population methods with clustering in undetectable dynamic environments

Copyright @ 2011 IEEETo solve dynamic optimization problems, multiple population methods are used to enhance the population diversity for an algorithm with the aim of maintaining multiple populations in different sub-areas in the fitness landscape. Many experimental studies have shown that locating and tracking multiple relatively good optima rather than a single global optimum is an effective idea in dynamic environments. However, several challenges need to be addressed when multi-population methods are applied, e.g., how to create multiple populations, how to maintain them in different sub-areas, and how to deal with the situation where changes can not be detected or predicted. To address these issues, this paper investigates a hierarchical clustering method to locate and track multiple optima for dynamic optimization problems. To deal with undetectable dynamic environments, this paper applies the random immigrants method without change detection based on a mechanism that can automatically reduce redundant individuals in the search space throughout the run. These methods are implemented into several research areas, including particle swarm optimization, genetic algorithm, and differential evolution. An experimental study is conducted based on the moving peaks benchmark to test the performance with several other algorithms from the literature. The experimental results show the efficiency of the clustering method for locating and tracking multiple optima in comparison with other algorithms based on multi-population methods on the moving peaks benchmark

Convolutions of slanted half-plane harmonic mappings

Let ${\mathcal S^0}(H_{\gamma})$ denote the class of all univalent, harmonic, sense-preserving and normalized mappings $f$ of the unit disk \ID onto the slanted half-plane $H_\gamma :=\{w:\,{\rm Re\,}(e^{i\gamma}w) >-1/2\}$ with an additional condition $f_{\bar{z}}(0)=0$. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in \ID is not necessarily even univalent in \ID. In this paper, we fix $f_0\in{\mathcal S^0}(H_{0})$ and show that the convolutions of $f_0$ and some slanted half-plane harmonic mapping are still convex in a particular direction. The results of the paper enhance the interest among harmonic mappings and, in particular, solves an open problem of Dorff, et. al. \cite{DN} in a more general setting. Finally, we present some basic examples of functions and their corresponding convolution functions with specified dilatations, and illustrate them graphically with the help of MATHEMATICA software. These examples explain the behaviour of the image domains.Comment: 15 pages, preprint of December 2011 (submitted to a journal for publication

Schemes and estimates for the long-time numerical solution of Maxwell’s equations for Lorentz metamaterials

We consider time domain formulations of Maxwell's equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall's inequality. The resulting estimates are, in this sense, sharp. We also give fully discrete formulations for each case and extend the sharp data-stability to these. Since the physical problem is linear it follows (and we show this with examples) that this stability property is also reflected in the constants appearing in the a priori error bounds. By removing the exponential growth in time from these estimates we conclude that these schemes can be used with confidence for the long-time numerical simulation of Lorentz metamaterials.This work was supported in part by NSFC Project 11271310, NSF grant DMS-1416742, and a grant from the Simons Foundation (#281296 to Li), in part by scheme 4 London Mathematical Society funding and in part by the Engineering and Physical Sciences Research Council (EP/H011072/1 to Shaw)