1n Xi Research Articles

Distributed Averaging Using Compensation

In a sensor network of n nodes, each node i is assigned with a time-dependent scalar state xi(t) and is able to exchange information with a certain number of other nodes called its neighbor nodes. The distributed averaging algorithm is to drive each xi(t), 1≤ i≤ n, to converge to 1/nΣi=1n xi(0), the average of the initial state values of all nodes, only by communication between each node and its neighbor nodes. This paper proposes a new request-based distributed averaging protocol using the idea of compensation, in which at each time t each node i first updates its state to a weighted average of its own current state xi(t) and the current states of some neighbor nodes and then compensates its updated value by the change in the sums, before and after the update, of the states of all the nodes involved. It is proved that the proposed protocol drives all the xi(t) to converge to 1/nΣi=1n xi(0). Simulations on random networks are also provided to show that this new protocol converges much faster and consumes less energy than the deterministic gossiping protocol in [1].

D.C. Versus Copositive Bounds for Standard QP

The standard quadratic program (QPS) is minx??xTQx, where $$\Delta\subset\Re^n$$ is the simplex ? = {x ? 0 ? ?i=1n xi = 1}. QPS can be used to formulate combinatorial problems such as the maximum stable set problem, and also arises in global optimization algorithms for general quadratic programming when the search space is partitioned using simplices. One class of `d.c.' (for `difference between convex') bounds for QPS is based on writing Q=S?T, where S and T are both positive semidefinite, and bounding xT Sx (convex on ?) and ?xTx (concave on ?) separately. We show that the maximum possible such bound can be obtained by solving a semidefinite programming (SDP) problem. The dual of this SDP problem corresponds to adding a simple constraint to the well-known Shor relaxation of QPS. We show that the max d.c. bound is dominated by another known bound based on a copositive relaxation of QPS, also obtainable via SDP at comparable computational expense. We also discuss extensions of the d.c. bound to more general quadratic programming problems. For the application of QPS to bounding the stability number of a graph, we use a novel formulation of the Lovasz ? number to compare ?, Schrijver's ??, and the max d.c. bound.

Some results on starlike trees and sunlike graphs

A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike trees G and H are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, let G be a simple graph of order n with vertex set V(G) = {1,2 ....,n} and let H = {H1,H2,...,Hn} be a family of rooted graphs. According to [2], the rooted product G(H) is the graph obtained by identifying the root of Hi with the i-th vertex of G. In particular, if H is the family of the paths Pk1, Pk2, ..., Pkn with the rooted vertices of degree one, in this paper the corresponding graph G(H) is called the sunlike graph and is denoted by G(k1,k2,...,kn). For any (x1,x2,...,xn) ∈ I*n, where I* = {0,1}, let G(x1,x2,.....xn) be the subgraph of G which is obtained by deleting the vertices i1,i2,...,ij ∈ V(G) (0 ≤ j ≤ n), provided that xi1 = xi2 =..... = xij = 0. Let G[x1, x2 .... , xn] be the characteristic polynomial of G(x1, x2,.., xn), understanding that G[0,0,..., 0] ≡ 1. We prove that G[k1,k2 ....., kn] = Σx∈I*n [Πi=1n Pki+xi-2(λ)](-1)n-(Σi=1n xi) G[x1,x2 ....., xn] where X = (x1,x2,...,xn); G[k1,k2,...,kn] and Pn(λ denote the characteristic polynomial of G(k1,k2 ....,kn) and Pn, respectively. Besides, if G is a graph with λ1(G) ≥ 1 we show that λ1(G) ≤ λ1(G(k1,k2,...,kn)) > λ1(G) + λ1-1(G) for all positive integers k1, k2....,kn, where λ1 denotes the largest eigenvalue.

Large-deviation probability and the local dimension of sets

Let X1, X2, ..., Xn be n independent identically distributed real random variables and Sn = Σn=1n Xi. We obtain precise asymptotics forP (Sn ∈ nA) for rather arbitrary Borel sets A1 in terms of the density of the dominating points in A. Our result extends classical theorems in the field of large deviations for independent samples. We also obtain asymptotics forP (Sn ∈ γnA), with γn/n → ∞.

On a Problem of Simultaneous Quasi-Uniform Extension

Let T be a topology on a set X and X = ∪1n Xi, Xi ∩ Xj = 0 (i ≠ j), Ui a quasi-uniformity on Xi (i = 1,...,n). The paper establishes necessary and sufficient conditions for the existence of a quasi-uniformity U on X with the property that the topology induced by U coincides with T and the restriction of U to Xi is equal to Ui. The rather complicated condition in the general case can be simplified in the special case n = 2.

On the rate of Poisson convergence

Let X1, …, Xn be independent Bernoulli random variables, and let pi = P[Xi = 1], λ = Σi=1n pi and Σi=1n Xi. Successively improved estimates of the total variation distance between the distribution ℒ(W) of W and a Poisson distribution Pλ with mean λ have been obtained by Prohorov[5], Le Cam [4], Kerstan[3], Vervaat[8], Chen [2], Serfling[7] and Romanowska[6]. Prohorov, Vervaat and Romanowska discussed only the case of identically distributed Xi's, whereas Chen and Serfling were primarily interested in more general, dependent sequences. Under the present hypotheses, the following inequalities, here expressed in terms of the total variation distancewere established respectively by Le Cam, Kerstan and Chen:(Kerstan's published estimate of ([3], p.174, equation (1)) is a misprint for , the constant 2·1 appearing twice on p. 175 of his paper.) Here, we use Chen's [2] elegant adaptation of Stein's method to improve hte estimates given in (1·1), and we complement these estimates with a reverse inequality expressed in similar terms. Second order estimates, and the case of more general non-negative integer valued X's, are also discussed.

Serum lipoprotein distribution, flotation rates and protein analysis.

AbstractFlotation rates of the major Sf 0舑12 lowdensity lipoprotein component in human serum may be calculated from ultracentrifuge data utilizing two computer programs. One program calculates a classical moving boundary uncorrected flotation rate by a best fit straight line for the points (1n xi, ३2ti). The other program permits erratum for concentration dependence and erratum to standard reference conditions. Preliminary application of these methods indicates significantly greater flotation rates in normal human females than in males for the 35舑49 year age group.The significance of interrelationships between the serum lipoprotein spectra, the serum lipids and the serum proteins is considered, resulting in the development of a revised method of measuring serum proteins by precision refractometry. The refractometric measurement is corrected in accordance with (any of various) lipid measurements in order to account for the contribution of lipoproteins to the total refractive increment. Such a technique, giving potentially a very accurate protein measurements, has application in screening studies involving abnormalities of both serum lipoprotein and serum protein metabolism.

A General Algorithm for the Optimal Distribution of Effort

The problem of distribution of effort is that of the optimal allocation of a given resource among various activities, that is, maximize ∑i=1n fi(xi) subject to ∑i=1n xi = w, xi ≥ 0. This paper presents an algorithm for obtaining the optimal allocation as a function of w, when the fi are arbitrary piecewise-linear, continuous functions. The method is recursive and appropriate for hand or machine computation. It is an extension of the well-known technique of ordering the segments of all the fi according to decreasing slope which applies in the special case when each fi has decreasing slopes (that is, is marginally decreasing).