The random key graph is a random graph naturally associated with the random key predistribution scheme introduced by Eschenauer and Gligor in the context of wireless sensor networks (WSNs). For this class of random graphs, we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to WSNs.
Blackburn and Gerke @cite_17 also succeeded in generalizing the one-law result by Di in a number of directions: Under the additional conditions they showed [Thm. 5] BlackburnGerke that This result is weaker than the one-law in the conjecture )-). However, in the process of establishing ), they also show [Thm. 3] BlackburnGerke that the conjecture does hold in the special case @math for all @math without any constraint on the size of the key pools, say @math or @math . Specifically, the one-law in ) is shown to hold whenever the scaling is done according to with @math . As pointed out by these authors, it is now easy to conclude that the one-law in ) holds whenever @math and @math ; this corresponds to a constraint @math .
@cite_17
2591592591 2211648829 2059725015 2102318662
This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . Elementary approaches to classic strong laws of large numbers use a monotonicity argument or a Tauberian argument of summability theory. Together with results on variance of sums of dependent random variables they allow to establish various strong laws of large numbers in case of dependence, especially under mixing conditions. Strong consistency of nonparametric regression estimates of local averaging type (kernel and nearest neighbor estimates), pointwise as well as in L 2, can be considered as a generalization of strong laws of large numbers. Both approaches can be used to establish strong universal consistency in the case of independence and, mostly by sharpened integrability assumptions, consistency under ρ-mixing or α-mixing. In a similar way Rosenblatt-Parzen kernel density estimates are treated. also with probability 1. This result is proved in ?5. The difficulty in proving lower bounds like (1.2) is that one has to consider all possible coverings of the path by small convex sets in the definition of 4-measure. In the past the only successful method has been to use the connection between Hausdorff measures and generalized capacities. The first result of this kind appears in [16] where it is proved that for k >2, the Hausdorff measure with respect to ta is infinite for all a 3 we obtain a "law of the iterated logarithm" for the total time Tk(a, co) spent by the path X in a sphere of radius a as a-* 0+. We also determine the local behavior of the first passage time Pk(a, co) out of a sphere of radius a for k > 1. In order to obtain the required asymptotic results we required good estimates of the distribution functions for the random variables Tk(a, co), Pk(a, c). We use the method developed by Mark Kac to compute these dis- We study the maximum of a Gaussian field on @math ( @math ) whose correlations decay logarithmically with the distance. Kahane Kah85 introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas DRSV12a DRSV12b extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in DRSV12a several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in DRSV12a : we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale.