Entropy, independent sets and antichains: A new approach to Dedekind's problem

For -regular, -vertex bipartite graphs with bipartition , a precise bound is given for the sum over independent sets of the quantity . (In other language, this is bounding the partition function for certain instances of the hard-core model.) This result is then extended to graded partially ordered sets, which in particular provides a simple proof of a well-known bound for Dedekind's Problem given by Kleitman and Markowsky in 1975.

     

Uniform dimension results of multi-parameter stable processes

The problem of uniform dimensions for multi-parameter processes, which may not possess the uniform stochastic Hölder condition, is investigated. The problem of uniform dimension for multi-parameter stable processes is solved. That is, ifZ is a stable (N,d, α)-process and αN ?d, then $\forall E \subseteq \mathbb{R}_ + ^N , \dim Z\left( E \right) = \alpha \cdot \dim E$ holds with probability 1, whereZ(E) = {x : ?t ∈E,Z _t =x} is the image set ofZ onE. The uniform upper bounds for multi-parameter processes with independent increments under general conditions are also given. Most conclusions about uniform dimension can be considered as special cases of our results.     

Cumulants are universal homomorphisms into Hausdorff groups

This is a contribution to the theory of sums of independent random variables at an algebraico-analytical level: Let Prob denote the convolution semigroup of all probability measures on with all moments finite, topologized by polynomially weighted total variation. We prove that the cumulant sequence regarded as a function from Prob into the additive topological group ofall real sequences, is universal among continuous homomorphisms from Prob into Hausdorff topological groups, in the usual sense that every other such homomorphism factorizes uniquely through . An analogous result, referring to just the first cumulants,holds for the semigroup of all probability measures with existing rth moments. In particular, there is no nontrivial continuous homomorphism from the convolution semigroup of all probability measures, topologized by the total variation metric, into any Hausdorff topological group.Mathematics Subject Classification (2000): 60B15, 60E10, 60G50     

Extending the Balas-Yu bounds on the number of maximal independent sets in graphs to hypergraphs and lattices

A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most ^p+1, where is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching in G. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors in the product of n lattices =₁××_n, where the notion of an induced matching in G is replaced by a certain binary tree each internal node of which is mapped into . We show that our bounds may be nearly sharp for arbitrarily large hypergraphs and lattices. As an application, we prove that the number of maximal infeasible vectors x=₁××_n for a system of polymatroid inequalities does not exceed max{Q,^logt/c(2Q,)}, where is the number of minimal feasible vectors for the system, , , and c(,) is the unique positive root of the equation 2^c(^c/log–1)=1. This bound is nearly sharp for the Boolean case ={0,1}ⁿ, and it allows for the efficient generation of all minimal feasible sets to a given system of polymatroid inequalities with quasi-polynomially bounded right-hand sides . This research was supported by the National Science Foundation (Grant IIS-0118635), and by the Office of Naval Research (Grant N00014-92-J-1375). The second and third authors are also grateful for the partial support by DIMACS, the National Science Foundation's Center for Discrete Mathematics and Theoretical Computer Science.Mathematics Subject Classification (2000):20E28, 20G40, 20C20     

Worst Case Traffic from Regulated Sources

We address the problem of finding the worst possible traffic a user of a telecommunications network can send. We take worst to mean having the highest effective bandwidth, a concept that arises in the Large Deviation theory of queueing networks. The traffic is assumed to be stationary and to satisfy leaky bucket constraints, which represent the a priori knowledge the network operator has concerning the traffic. Firstly, we show that this optimization problem may be reduced to an optimization over periodic traffic sources. Then, using convexity methods, we show that the realizations of a worst case source must have the following properties: at each instant the transmission rate must be either zero, the peak rate, or the leaky bucket rate; it may only be the latter when the leaky bucket is empty or full; each burst of activity must either start with the leaky bucket empty or end with it full.     

On Lower Bounds for Lp Norms in the Central Limit Theorem for Independent and m-Dependent Random Variables

We derive lower bounds for L_p norms , in the central limit theorem for independent and m–dependent random variables with finite fifth order absolute moments and for independent and m–dependent identically distributed random variables with fourth order moments.     

理性、有序地建设探究实验室

建设探究实验室是提高学生对学习、学科兴趣的重要举措.本文提出了一套探究实验室评价方案以期对实验室的建设和发展有所帮助.总体评价和导向性评价基于对一个个实验就一些指标进行评估以及其后的统计.我们还对一些深层次的要求和指标进行了探讨.     

Influences of the Current Transverse Profile on the Transverse Mode Competition of VCSELs Diodes

A spatially independent model of vertical-cavity surface-emitting lasers (VCSELs) is derived by integrating the spatially dependent rate equations over the cross section of the cavity of a VCSEL. The angular and radial non-uniformities of the injection current are taken into account. The well-known LP modes of a weakly-guiding cylindrical waveguide are employed to describe the transverse modal structure in the VCSEL cavity. This model is solved in a self-consistent way by using the 4th order Runge-Kutta method. The dependence of transverse mode competition on the current intensity, the angular and radial non-uniformities of the injection current, and the geometrical parameters of the electrical contact are thoroughly investigated and analyzed. The results are useful to the optimum design of the optical transverse modal structure of VCSELs.     

Constructing the time independent Hamiltonian from a time dependent one

In this paper we introduce a method for finding a time independent Hamiltonian of a given Hamiltonian dynamical system by canonoid transformation of canonical momenta. We find a condition that the system should satisfy to have an equivalent time independent formulation. We study the example of a damped harmonic oscillator and give the new time independent Hamiltonian for it, which has the property of tending to the standard Hamiltonian of the harmonic oscillator as damping goes to zero.      

Path coupling using stopping times and counting independent sets and colorings in hypergraphs

We analyse the mixing time of Markov chains using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Δ of a vertex and the minimum size m of an edge satisfy m ≥ 2Δ + 1. We also show that the Glauber dynamics for proper q‐colorings of a hypergraph mixes rapidly if m ≥ 4 and q > Δ, and if m = 3 and q ≥ 1.65Δ. We give related results on the hardness of exact and approximate counting for both problems. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008