Routing complexity of faulty networks

One of the fundamental problems in distributed computing is how to efficiently perform routing in a faulty network in which each link fails with some probability. This article investigates how big the failure probability can be, before the capability to efficiently find a path in the network is lost. Our main results show tight upper and lower bounds for the failure probability, which permits routing both for the hypercube and for the d‐dimensional mesh. We use tools from percolation theory to show that in the d‐dimensional mesh, once a giant component appears—efficient routing is possible. A different behavior is observed when the hypercube is considered. In the hypercube there is a range of failure probabilities in which short paths exist with high probability, yet finding them must involve querying essentially the entire network. Thus the routing complexity of the hypercube shows an asymptotic phase transition. The critical probability with respect to routing complexity lies in a different location than that of the critical probability with respect to connectivity. Finally we show that an oracle access to links (as opposed to local routing) may reduce significantly the complexity of the routing problem. We demonstrate this fact by providing tight upper and lower bounds for the complexity of routing in the random graph G_n,p. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The diameter of long‐range percolation clusters on finite cycles

Bounds for the diameter and expansion of the graphs created by long‐range percolation on the cycle ?/N? are given. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 102–111, 2001     

Starting point to molecular design: efficient automated 3D model builder Key3D

Obtaining three-dimensional (3D) structures from structural formulae is a crucial process in molecular design. We have developed a new 3D model builder, Key3D, in which the simplified distance geometry technique and structure optimization based on the MMFF force field are combined. In an evaluation study using 598 crystal structures, the high performance and accuracy of Key3D were demonstrated. In the "flexible-fitting" test, which is focused on practical usefulness in the molecular design process, 88% of the Key3D structures acceptably reproduced the reference crystal structures (root-mean-square deviation <0.6 A) upon rotation of acyclic bonds. These results indicate that Key3D will be very effective in providing starting points for practical molecular design.     

Slip, yield, and bands in colloidal crystals under oscillatory shear

We study dense colloidal crystals under oscillatory shear using a confocal microscope. At large strains the crystals yield and the suspensions form shear bands. The pure harmonic response exhibited by the suspension rules out the applicability of nonlinear rheology models typically used to describe shear banding in other types of complex fluids. Instead, we show that a model based on the coexistence of linearly responding phases of the colloidal suspension accounts for the observed flows. These results highlight a new use of oscillatory measurements in distinguishing the contribution of linear and nonlinear local rheology to a globally nonlinear material response.     

Boundary trace of reflecting Brownian motions

We establish a uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains. Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are determined. Extensions to stable-like jump processes and to symmetric reflecting diffusions are also given.Mathematics Subject Classification (2000):Primary 60G17, 60J60, Secondary 28A80, 30C35, 60G52, 60J50     

Uniqueness of positive solutions for singular problems involving the -Laplacian

We study existence and uniqueness of positive eigenfunctions for the singular eigenvalue problem: on a bounded smooth domain with zero boundary condition. We also characterize all positive solutions of in .

     

Local Nash Equilibrium in Multiparty Politics

We present a model of multi-party, spatial competition under proportional rule with both electoral and coalitional risk. Each party consists of a set of delegates with heterogeneous policy preferences. These delegates choose one delegate as leader or agent. This agent announces the policy declaration (or manifesto) to the electorate prior to the election. The choice of the agent by each party elite is assumed to be a local Nash equilibrium to a game form . This game form encapsulates beliefs of the party elite about the nature of both electoral risk and the post-election coalition bargaining game. It is demonstrated, under the assumption that is smooth, that, for almost all parameter values, a locally isolated, local Nash equilibrium exists.In the final section of the paper some empirical work is reviewed in order to obtain some insights into why parties do not simply converge to an electoral center in order to maximize expected vote shares.