Structure of random 312‐avoiding permutations

We evaluate the probabilities of various events under the uniform distribution on the set of 312‐avoiding permutations of . We derive exact formulas for the probability that the i^th element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312‐avoiding permutation has k specified decreasing points, and we show that for large N the points below the diagonal look like trajectories of a random walk. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 599–631, 2016     

The threshold for combs in random graphs

For let denote the tree consisting of an ‐vertex path with disjoint ‐vertex paths beginning at each of its vertices. An old conjecture says that for any the threshold for the random graph to contain is at . Here we verify this for with any fixed . In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for (with ). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 794–802, 2016     

Discovery Through Gossip

We study randomized gossip‐based processes in dynamic networks that are motivated by information discovery in large‐scale distributed networks such as peer‐to‐peer and social networks. A well‐studied problem in peer‐to‐peer networks is resource discovery, where the goal for nodes (hosts with IP addresses) is to discover the IP addresses of all other hosts. Also, some of the recent work on self‐stabilization algorithms for P2P/overlay networks proceed via discovery of the complete network. In social networks, nodes (people) discover new nodes through exchanging contacts with their neighbors (friends). In both cases the discovery of new nodes changes the underlying network — new edges are added to the network — and the process continues in the changed network. Rigorously analyzing such dynamic (stochastic) processes in a continuously changing topology remains a challenging problem with obvious applications. This paper studies and analyzes two natural gossip‐based discovery processes. In the push discovery or triangulation process, each node repeatedly chooses two random neighbors and connects them (i.e., “pushes” their mutual information to each other). In the pull discovery process or the two‐hop walk, each node repeatedly requests or “pulls” a random contact from a random neighbor and connects itself to this two‐hop neighbor. Both processes are lightweight in the sense that the amortized work done per node is constant per round, local, and naturally robust due to the inherent randomized nature of gossip. Our main result is an almost‐tight analysis of the time taken for these two randomized processes to converge. We show that in any undirected n‐node graph both processes take rounds to connect every node to all other nodes with high probability, whereas is a lower bound. We also study the two‐hop walk in directed graphs, and show that it takes time with high probability, and that the worst‐case bound is tight for arbitrary directed graphs, whereas Ω(n²) is a lower bound for strongly connected directed graphs. A key technical challenge that we overcome in our work is the analysis of a randomized process that itself results in a constantly changing network leading to complicated dependencies in every round. We discuss implications of our results and their analysis to discovery problems in P2P networks as well as to evolution in social networks. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 48, 565–587, 2016     

Efficient Spatial Modeling Using the SPDE Approach With Bivariate Splines

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to nonstationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency. Supplementary materials are available online.     

On the convergence of the product of independent random operators

In this paper, we study infinite products of independent random operators. Conditions for the existence a.s. and the convergence in mean of order p of such infinite products are established.     

Complete moment convergence for arrays of rowwise NSD random variables

In this paper, the complete convergence and complete moment convergence for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables are investigated. Some sufficient conditions to prove the complete convergence and the complete moment convergence are presented. The results obtained in the paper generalize and improve some corresponding ones for independent random variables and negatively associated random variables.     

Synergistic toughening effect of β-nucleating agent and long chain branching on polypropylene random copolymer

β-nucleated long chain branched (LCB) polypropylene random copolymers (PPRs) were prepared via reactive extrusion by introducing β nucleating agent (NA), dicumyl peroxide (DCP) and various contents of 1, 6-Hexanediol diacrylate (HDDA) into PPR. Results of Fourier infrared spectroscopy (FTIR) and the rheological properties demonstrate the existence of LCB polypropylene. Mechanical properties including impact strength, tensile strength and elongation at break were studied. The crystal structure, morphology and crystallization behavior were investigated via wide-angle X-ray diffraction (WAXD), polarized light microscopy (PLM) and differential scanning calorimetry (DSC). Non-isothermal crystallization kinetics using the Jeziorny method was also studied. The results show an increased LCB degree with increasing HDDA amount. For the sample with a moderate LCB level, synergistic toughening effect of NA and LCB is achieved without deterioration of yield strength or elongation at break, partially because of its higher β-phase content and much smaller spherulite size. It exhibits the minimum values of T_c and Z_c1, and the maximum values of t_1/2 and n₁ in the primary stage of crystallization, regardless of the cooling rate, indicating a slower crystallization rate and more complicated nucleation and crystal growth model.     

Multi walled carbon nanotubes induced viscoelastic response of polypropylene copolymer nanocomposites: Effect of filler loading on rheological percolation

Polypropylene random copolymer nanocomposites having 0.2–7.0 vol% multi-walled carbon nanotubes (MWCNTs) were prepared via melt processing. Transmission electron microscopy (TEM) was employed to determine the nano scale dispersion of carbon nanotubes. Linear viscoelastic behavior of these nanocomposites was investigated using parallel plate rheometry. Incorporation of carbon nanotubes in the polymer matrix resulted in higher complex viscosity (η*), storage (G′) and loss modulus (G″) as compared to neat polymer, especially in the low-frequency region, suggesting a change from liquid to solid-like behavior in the nanocomposites. By plotting storage modulus vs. carbon nanotube loading and fitting with a power law function, the rheological percolation threshold in these nanocomposites was observed at a loading of ∼0.27 vol% of MWCNTs. However, electrical percolation threshold was reported at ∼0.19 vol% of MWCNTs loading. The difference in the percolation thresholds is understood in terms of nanotube connectivity with nanotubes and polymer chain required for electrical conductivity and rheological percolation.     

对称随机变量部分和的概率不等式

在对称随机变量分布函数关于原点的值大于或等于二分之一的基础上,阐明对称随机变量的部分和仍是对称随机变量,进一步,给出关于对称随机变量序列部分和的概率不等式.