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.     

Real analyticity of Hausdorff dimension for higher dimensional hyperbolic graph directed Markov systems

In this paper we prove that the Hausdorff dimension function of the limit sets of strongly regular, hyperbolic, conformal graph directed Markov systems living in higher dimensional Euclidean spaces , , and with an underlying finitely irreducible incidence matrix is real-analytic. Research of M. Roy was supported by NSERC (Natural Sciences and Engineering Research Council of Canada). Research of M. Urbański was supported in part by the NSF Grant DMS 0400481.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

Some comments on Wolfe's ‘away step’

We give a detailed proof, under slightly weaker conditions on the objective function, that a modified Frank-Wolfe algorithm based on Wolfe's away step strategy can achieve geometric convergence, provided a strict complementarity assumption holds.Research supported by FCAC (Québec) and NSERC (Canada).     

A numerical method using the Prfer transformation for the calculation of eigenpairs of two-parameter Sturm-Liouville problems

A shooting method for coupled Prfer equations is discussedfor numerical solution of two-parameter Sturm-Liouville problems. Research supported in part by grants from the NSERC of Canada.     

Specialized Block-Colourings of Steiner Triple Systems and the Upper Chromatic Index

Specialized block-colourings of Steiner triple systems, in which the blocks incident at each point have a restricted colour pattern, are examined. Valid block-colourings, in which every element is incident to at least two blocks of the same colour, and to at least two blocks of different colours, are shown to exist for all admissible orders and numbers of colours. Acknowledgements.Research of the first author is supported by the U.S. Army Research Office. Research of the second author is supported by NSERC Canada.     

Topological spaces and quasi-varieties

We describe Top ^op and Sob ^op as quasi-varieties by means of suitable schizophrenic objects.Research of the first author supported by grants from the NSERC of Canada and the FCAR du Québec. Research of the second author supported by the Topology grant 40% and by the NATO grant CRG 941330.     

Thin-tall Boolean algebras and their automorphism groups

Our main result is a general construction to show that for each countable groupG there are 2¹ pairwise non-isomorphic thin-tall superatomic Boolean algebras, all of them representingG as a group of non-trivial automorphisms.Presented by J. Mycielski.Research supported by the NSERC of Canada.     

Congruence lattices ofp-algebras

T. Kartriák proved the following theorem:Every finite distributive lattice is the congruence lattice of a finite p-algebra. We provide a short proof, and a generalization, of this result.Presented by J. Berman.The research of the first author was supported by the NSERC of Canada.The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

On additive functionals of Markov chains

Let{S _n } _n=0 ^∞ be a Harris-recurrent Markov chain on a measurable state space. We prove strong approximation results for the additive functionals . Research supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O.B. 127, H-1364, Hungary. Research supported by an NSERC Canada Grant, Carleton University. Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6.     

A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences

Summary With the help of an extension of Hoeffding's equality, we develop a way for estimating the covariance structures for empirical functions of associated sequences in terms of covariances of the original random variables. Based on these estimations, a Glivenko-Cantelli lemma for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained, all under the conditions on covariances of the original random variables.This research has been supported partially by a scholarship from the Faculty of Graduate Studies and Research of Carleton University, Ottawa, and by an NSERC Canada grant of M. Csörg     

EP Theorem for Dual Linear Complementarity Problems

The linear complementarity problem (LCP) belongs to the class of -hard problems. Therefore, we cannot expect a polynomial time solution method for LCPs without requiring some special property of the matrix of the problem. We show that the dual LCP can be solved in polynomial time if the matrix is row sufficient; moreover, in this case, all feasible solutions are complementary. Furthermore, we present an existentially polytime (EP) theorem for the dual LCP with arbitrary matrix. The research of Tibor Illés and Marianna Nagy has been supported by the Hungarian National Research Fund OTKA T 049789 and by the Hungarian Science and Technology Foundation TéT SLO-4/2005. Tamàs Terlaky has been supported by an NSERC Discovery grant, MITACS and the Canada Research Chair program.     

Strong and weak approximations of k-spacings processes

Summary We obtain strong approximation results for the empirical process and the quantile process based on non-overlapping k-spacings using a characterization of uniform spacings in terms of independent exponential random variables. Moreover the weak convergence of these processes in /q-metrics is discussed. Applications to spacing statistics are included.Part of this research was done while the author was a visiting scientist at the Department of Mathematics and Statistics, Carleton University, supported by Canadian NSERC operating grant of M. Csörg. This research was also supported by a research grant of the Research Center of the College of Science, King Saud University     

Classes of Carleson-type measures generated by capacities

We introduce classes of measures in the half-space generated by Riesz, or Bessel, or Besov capacities in , and give a geometric characterization of these as Carleson-type measures. G. Dafni was supported in part by NSERC, the CRM, Barcelona, and the CRM, Montreal. G. E. Karadzhov was supported in part by Memorial and Concordia Universities, NSERC, and the CRM, Montreal. J. Xiao was supported in part by NSERC of Canada and Dean of Science Start-up Funds of Memorial University.     

Brownian motion on the Sierpinski gasket

Summary We construct a Brownian motion taking values in the Sierpinski gasket, a fractal subset of ², and study its properties. This is a diffusion process characterized by local isotropy and homogeneity properties. We show, for example, that the process has a continuous symmetric transition density, p _t(x,y), with respect to an appropriate Hausdorff measure and obtain estimates on p _t(x,y).Research partially supported by an NSERC of Canada operating grantResearch partially supported by an SERC (UK) Visiting Fellowship     

On large deviation expansion of distribution of maximum likelihood estimator and its application in large sample estimation

For estimating an unknown parameter , the likelihood principle yields the maximum likelihood estimator. It is often favoured especially by the applied statistician, for its good properties in the large sample case. In this paper, a large deviation expansion for the distribution of the maximum likelihood estimator is obtained. The asymptotic expansion provides a useful tool to approximate the tail probability of the maximum likelihood estimator and to make statistical inference. Theoretical and numerical examples are given. Numerical results show that the large deviation approximation performs much better than the classical normal approximation.This work is supported in part by the Natural Science and Engineering Research Council of Canada under grant NSERC A-9216.This author is also partially supported by the National Science Foundation of China.     

Products of quasireflections and transvections over local rings

Let R be a commutative local ring and M a free R-module of rank n. The module M is endowed with a metric structure given by a sesquilinear form or by a quadratic form. We show, every isometry of M is a product of quasireflections or transvections. We determine the minimal number of factors needed in any factorization of if the path of is a subspace. For all other isometries we obtain only an upper bound.Research supported in part by NSERC Canada grant A7251To Helmut Karzel on his 60th birthday     

L_{\infty _\omega }free algebras

In this paper the study of which varieties, in a countable similarity type, have non-free (or equivalently ₁-free) algebras is completed. It was previously known that if a variety satisfies a property known as the construction principle then there are such algebras. If a variety does not satisfy the construction principle then either every -free algebra is free or for every infinite cardinalk, there is a k⁺-free algebra of cardinality k⁺ which is not free. Under the set theoretic assumption V=L, for any varietyV in a countable similarity type, either the class of free algebras is definable in or it is not definable in any .In Memory of Evelyn NelsonPresented by Ralph McKenzie.Research partially supported by NSERC of Canada Grant #A8948.Research partially supported by NSERC of Canada. The research for this paper was begun while the second author was visiting Simon Fraser University.     

Reductions of an elliptic curve and their Tate-Shafarevich groups

In this paper we study the Tate-Shafarevich groups III_p of the reductions modulo primes p of an elliptic curve E/ considered as being defined over their function fields. Assuming GRH when E has no CM, we show that III_p is trivial for a positive proportion of primes p, provided E has an irrational point of order two.Research supported in part by an NSERC postdoctoral fellowship.Research supported in part by NSF grant DMS-98-01642.     

On the Liu–Floudas Convexification of Smooth Programs

It is well known that a twice continuously differentiable function can be convexified by a simple quadratic term. Here we show that the convexification is possible also for every Lipschitz continuously differentiable function. This implies that the Liu–Floudas convexification works for, loosely speaking, almost every smooth program occurring in practice. Research supported by an NSERC of Canada grant