Bootstrapped confidence bands for percentile lifetime

Summary We propose bootstrapped confidence bands for the percentile lifetime function. Our method is based on a joint approximation of the empirical and quantile processes and their bootstrapped counterparts. Modest simulations support the results. Confidence bands are applied to quantile and median residual lifetimes of tractor rear brakes. Research done while at Carleton University Norman Paterson Centre: also partially supported by NSERC Canada grants of M. Cs?rgő. Research supported in part by a NSERC Canada operating grant. Research done while at Carleton University; also partially supported by NSERC Canada grants of m. Cs?rgő and D. A. Dawson and by an EMR Canada grant of M. Cs?gő. Research supported in part by a United States Department of Agriculture CSRS grant and a United States NSF grant of D. M. Bates, G. Wahba and B. S. Yandell.     

Characterization of the barrier parameter of homogeneous convex cones

We characterize the smallest (best) barrier parameter of self-concordant barriers for homogeneous convex cones. In particular, we prove that this parameter is the same as the rank of the cone which is the number of steps in a recursive construction of the cone (Siegel domain construction). We also provide lower bounds on the barrier parameter in terms of the Carathéodory number of the cone. The bounds are tight for homogeneous self-dual cones. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research supported in part by an operating grant from NSERC of Canada.Research supported in part by the National Science Foundation under grant DMS-9306318.     

Automorphisms of split Cayley algebras over real closed fields

Research supported in part by NSERC Canada grant A7251.     

Large deviations for Markov processes with mean field interaction and unbounded jumps

Summary An N-particle system with mean field interaction is considered. The large deviation estimates for the empirical distributions as N goes to infinity are obtained under conditions which are satisfied, by many interesting models including the first and the second Schlögl models.Supported partially by a scholarship from the Faculty of Graduate Studies and Research of Carleton University and the NSERC operating grant of D.A. Dawson     

A note on the existence of subgradients

We describe an apparently novel way of constructing the subgradient of a convex function defined on a finite dimensional vector space.Research partially funded on NSERC grant A5116.     

Super efficiency in convex vector optimization

We establish a Lagrange Multiplier Theorem for super efficiency in convex vector optimization and express super efficient solutions as saddle points of appropriate Lagrangian functions. An example is given to show that the boundedness of the base of the ordering cone is essential for the existence of super efficient points.Research is supported partially by NSERC.Research is supported partially by NSERC and Mount St. Vincent University grant.     

Primitive subspaces of Hopf algebras of finite depth

This research was partially supported by an NSERC operating grant, and by a NATO travel grant     

Integral global minimization: Algorithms,implementations and numerical tests

The theoretical foundation of integral global optimization has become widely known and well accepted [4],[24],[25]. However, more effort is needed to demonstrate the effectiveness of the integral global optimization algorithms. In this work we detail the implementation of the integral global minimization algorithms. We describe how the integral global optimization method handles nonconvex unconstrained or box constrained, constrained or discrete minimization problems. We illustrate the flexibility and the efficiency of integral global optimization method by presenting the performance of algorithms on a collection of well known test problems in global optimization literature. We provide the software which solves these test problems and other minimization problems. The performance of the computations demonstrates that the integral global algorithms are not only extremely flexible and reliable but also very efficient.Research supported partially by NSERC grant and Mount St Vincent University research grant.     

Variational Methods in Convex Analysis

We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth subdifferential theory. Research was supported by NSERC and by the Canada Research Chair Program and National Science Foundation under grant DMS 0102496.     

Amenability: A survey for statistical applications of hunt-stein and related conditions on groups

Summary A number of conditions on groups have appeared in the literature of invariant statistical models in connection with minimaxity, approximation of invariant Bayes priors by proper priors, the relationship between Bayesian and classical inference, ergodic theorems, and other matters. In the last decade, rapid development has occurred in the field and many of these conditions are now known to be equivalent. We survey the subject, make the equivalences explicit, and list some groups of statistical interest which do, and also some which do not, have these properties. In particular, it is shown that the existence of the asymptotically invariant sequence of probabilities in the hypothesis of the Hunt-Stein theorem is equivalent to amenability, a condition that has been much studied by functional analysts.Research supported in part by NSERC grant A7857     

Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces

We consider nested sequences of linear or convex closed sets of the form arising in estimation and other inverse problems. We show that such sequences may fail to converge in any of the recently studied set convergences other than Mosco convergence. We also provide a positive result concerning the epislice convergence of related sequences of functions.Research partially supported by NSERC operating grants.     

Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.Research of the first author was supported in part by NSERC grant A5149.Research of the second author was supported in part by NSF grant DMS93-02991.     

A two parameter mixed interior-exterior penalty algorithm

In this paper, we analyze the mixed penalty methods introduced in the classic book of Fiacco and McCormick usingtwo distinct penalty parametersr, t. The two penalty coefficients induce a two-parameter differentiable trajectory. We analyze the numerical behaviour of an extrapolation strategy that follows the path of the two-parameter trajectory. We show also how to remove the ill-conditioning by suitable transformations of the equations. In the resulting theory, we show that function values as well as distances to the optimum are both governed by the same behaviour as interior methods (two-step superlinearly convergent, with limiting exponent 4/3).Research partially supported by NSERC grant OGP0036512.Research partially supported by NSERC grant OGP0005491.     

Overlarge sets and partial geometries

We extend our earlier work on overlarge sets of Fano planes, obtaining three results of particular interest. We find seven new partial geometries pg(8,7,4) and nine new strongly regular graphs, by means of switching cliques of points with spreads of lines. One of these new strongly regular graphs supports four different partial geometries. Then we give a new construction of the recently discovered eightfold cover of the complete graph K₁₆.Supported by NSERC grant OGP0008651Supported by ARC grant A49130102 and an Australian Senior Research Fellowship     

Simultaneous dominance representation of multiple posets

We characterize the codominance pairs-pairs of posets that admit simultaneous dominance representations in the (x, y)-and (–x, y)-coordinate systems-and present a linear algorithm to recognize them and construct codominance representations. We define dominance polysemy as a generalization of codominance and describe several related problems and preliminary results.This author's research, supported in part by NSF grant CCR-9300079, also appears in his doctoral thesis [15], written, at the Johns Hopkins University under the supervision of Professors Edward R. Scheinerman and Michael T. Goodrich.This author's research, supported in part by an NSERC operating grant and an FCAR team grant, was performed in part during a visit at INRIA Sophia Antipolis.     

Further arguments for slice convergence in nonreflexive spaces

It is shown that no notion of set convergence at least as strong as Wijsman convergence but not as strong as slice convergence can be preserved in superspaces. We also show that such intermediate notions of convergence do not always admit representations analogous to those given by Attouch and Beer for slice convergence, and provide a valid reformulation. Some connections between bornologies and the relationships between certain gap convergences for nonconvex sets are also observed.Research supported in part by an NSERC research grant and by the Shrum endowment.NSERC postdoctoral fellow.     

Stable theories without dense forking chains

Summary We define a generalized notion of rank for stable theories without dense forking chains, and use it to derive that every type is domination-equivalent to a finite product of regular types. We apply this to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisation of a forking extension.Research supported by the Deutsche ForschungsgemeinschaftThanks as always to Alistair. Research supported by NSERC and FCARPartially supported by NSF grant DMS90 06628     

On the failure of maximum entropy reconstruction for Fredholm equations and other infinite systems

We analyse several examples where the maximum entropy solution to a system of equations exists but fails to satisfy the natural (dual) formula. These examples highlight the role that finiteness of the number of constraints has in the efficacy of maximum entropy type estimation and reconstruction. We also provide two regularization processes which repair the problem.Research partially supported by NSERC grant OGP005116.     

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.Research supported by NSERC under grant no. 90749-00.Research supported by NSERC under grant no. 227043-00.