On random walks arising in queueing systems: ergodicity and transience via quadratic forms as lyapounov functions — Part I

A simple and quite general approach is proposed to derive criteria for transience and ergodicity of a certain class of irreducibleN-dimensional Markov chains in ₊ ^N assuming a boundedness condition on the second moment of the jumps. The method consists in constructing convenient smooth supermartingales outside some compact set. The Lyapounov functions introduced belong to the set of quadratic forms in ₊ ^N and do not always have a definite sign. Existence and construction of these forms is shown to be basically equivalent to finding vectors satisfying a system of linear inequalities.Part I is restricted toN=2, in which case a complete characterization is obtained for the type of random walks analyzed by Malyshev and Mensikov, thus relaxing their condition of boundedness of the jumps. The motivation for this work is partly from a large class of queueing systems that give rise to random walks in ₊ ^N      

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Solvable isotherms for a two-component system of charged rods on a line

The Coulomb system consisting of an equal number of positive and negative charged rods confined to a one-dimensional lattice is studied. The grand partition function can be calculated exactly at two values of the coupling constant=q ²/k _B T (q denoting the magnitude of the charges). The exact results lead to the conjecture that in the complex scaled fugacity plane, all the zeros of the grand partition function lie on the negative real axis for<2, on the point=–1 for=2, and on the unit circle for>2. In addition, for>4, we conjecture in general and prove at=4 that the zeros pinch the real axis in the thermodynamic limit, with an essential singularity in the pressure at the reduced density 1/2.     

Vicious random walkers in the limit of a large number of walkers

The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that thep walkers will survive a timet (denotedP _t ^(p) ) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas forP _t ^(p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet, given that allp walkers survive, is calculated in the limittp.     

Copolymer of Di (methacryloyloxymethyl) naphthalene and divinylbenzene as a column packing for high-performance liquid chromatography

Summary A new porous polyaromatic ester packing was synthetized for high performance liquid chromatography. The relationship between retention and chain length of the members of homologous series of alkylbenzenes, N-alkylanilines, alkylarylethers, alkylbenzoates and alkylarylketones on this new stationary phase using different eluents was investigated. Using the alkylarylketone scale the retention indices of the homologues and test compounds were calculated. The results were compared with those obtained for poly (styrene-divinylbenzene) polymers. For both types of packing the first members of each homologous series gave non-linear behaviour. The methylene group index increments are different for the studied homologous series; thus there is no simple additivity of the retention indices. The efficiency of the porous polymeric columns is a function of the capacity factor of the solute and the organic component of the eluent.     

Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

On the existence of thermodynamics for the generalized random energy model

Derrida's generalized random energy model is considered. Almost sure andL ^p convergence of the free energy at any inverse temperature are proven for an arbitrary numbern of hierarchical levels. The explicit form of the free energy is given in the most general case and the limitn is discussed.     

Lifschitz singularities for periodic operators plus random potentials

UsingX-bounding (lower bounds by Laplacians with mixed boundary conditions and discrete analogs), we obtain the Lifschitz exponent at the bottom of the spectrum for random operators of typeH =T+V , withT a (periodic) generator of a positivity-preserving semigroup, extending results by Kirsch and Simon.     

二维高分子链形态的计算机模拟

介绍在“高分子物理实验”教学中新开设的又一个计算机模拟实验，即应用自编的改进型四位置模型，模拟二维空间中的自回避行走链和无规行走链，并验算均方末端距和均方回转半径与聚合度的标度关系，所得结果与de Gennes的理论符合良好。     

Cu离子A位掺杂对LaSrCoO4复合氧化物氧化性能的影响

采用聚丙烯酰胺法制备了La1-xCuxSrCoO4(x=0.2-0.8)复合氧化物,考察了Cu离子掺杂量(x)对CO及C3H8氧化反应活性的影响,并运用XRD、IR、TPR和TPD等多种手段对复合氧化物催化剂进行了表征。结果表明,催化剂活性随x的变化而变化,Cu部分取代A位La能提高LaSrCoO4催化剂的CO及C3H8氧化反应活性,当x=0.4时La1-xCuxSrCoO4催化剂的晶格氧和Co3 含量较多,晶格氧的活动性较高,催化活性最佳。