Let us assign independent, exponentially distributed random edge lengths to the edges of an undirected graph. Lyons, Pemantle and Peres [11] proved that the expected length of the shortest path between two given nodes is bounded from below by the resistance between these nodes, where the resistance of an edge is the expectation of its length. They remarked that instead of exponentially distribu...

The article presents conditions under which the probability of a linear combination random vectors falling into polyhedral cone is Schur-concave function coefficients combination. It required that contains point 0, its edges are parallel to coordinate axes, and distribution density logarithmically concave sign-invariant function.

A novel N,N'-diphosphanyl-imidazol-2-ylidene acts as a stable, hybrid PC(NHC)P ligand for M2 or linear M3 (M = Cu, Ag, Au) arrays with metallophilic interactions.

In the last lecture we discussed applications of log concavity of real rooted polynomials in designing a deterministic approximation algorithm for permanent. We also introduced the class H-stable polynomial. Recall that a polynomial p(z1, . . . , zn) is H -stable if p(z1, . . . , zn) 6= 0 whenever =zi > 0 for all i. In the last lecture we saw that any univariate H-stable polynomial (with real c...

In 1990, E. Baum gave an elegant polynomial-time algorithm for learning the intersection of two origin-centered halfspaces with respect to any symmetric distribution (i.e., any D such that D(E) = D(−E)) [3]. Here we prove that his algorithm also succeeds with respect to any mean zero distribution D with a log-concave density (a broad class of distributions that need not be symmetric). As far as...

The EM algorithm is a popular tool for clustering observations via a parametric mixture model. Two disadvantages of this approach are that its success depends on the appropriateness of the assumed parametric model, and that each model requires a different implementation of the EM algorithm based on model-specific theoretical derivations. We show how this algorithm can be extended to work with t...

This article deals with the Bayesian inference of the unknown parameters of the Weibull distribution based on the left truncated and right censored data. It is assumed that the scale parameter of the Weibull distribution has a gamma prior. The shape parameter may be known or unknown. If the shape parameter is unknown, it is assumed that it has a very general log-concave prior distribution. When...