Geometric convergence of rational approximations toe −z in infinite sectors

Summary In this paper, we study the geometric convergence of rational approximations toe ^–z in infinite sectors symmetric about the positive real axis.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research CouncilResearch supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT (11-1)-2075     

On the sharpness of theorems concerning zero-free regions for certain sequences of polynomials

Summary In this paper, we establish the sharpness of a theorem concerning zero-free parabolic regions for certain sequences of polynomials satisfying a three-term recurrence relation. Similarly, we establish the sharpness of a zero-free sectorial region for certain sequences of Padé approximants toe ^z .Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(11-1)-2075     

The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method

Summary The paper addresses the problem of the implementation of nonhomogeneous essential Dirichlet type boundary conditions in thep-version of the finite element method.Partially supported by the Office of Naval Research under Grant N-00014-85-K-0169Research partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0322     

On the zeros and poles of Padé approximants toe z

Summary In this paper, we study the location of the zeros and poles of general Padé approximats toe ^z. The location of these zeros and poles is useful in the analysis of stability for related numerical methods for solving systems of ordinary differential equations.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Fla. Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT(11-1)-2075.     

On the zeros and poles of Padé approximants toe z . III

Summary In this paper, we continue our study of the location of the zeros and poles of general Padé approximants toe ^z . We state and prove here new results for the asymptotic location of the normalized zeros and poles for sequences of Padé approximants toe ^z , and for the asymptotic location of the normalized zeros for the associated Padé remainders toe ^z . In so doing, we obtain new results for nontrivial zeros of Whittaker functions, and also generalize earlier results of Szegö and Olver.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant EY-76-S-02-2075     

Equivalent factor matroids of graphs

The factor matroid of a graphG is the matric matroid of the vertex-edge incidence matrix ofG interpreted over the real numbers. This paper presents a constructive characterization of the graphs hat have the same factor matroid as a given 4-connected bipartite graph.Research partially supported by NSF Grant ESS-8307796 and Office of Naval Research Grant N00014-86-K-0689.     

Maximal Local Time of a <Emphasis Type="Italic">d</Emphasis>-dimensional Simple Random Walk on Subsets

Strong theorems are given for the maximal local time on balls and subspaces for the d-dimensional simple symmetric random walk.Endre Csáki - Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 037886 and T 043037.Pál Révész - Research supported by a PSC CUNY Grant, No. 65685-0034.     

A new proof of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the parabolic polyharmonic equations

In this paper we obtain local L ^p estimates for the parabolic polyharmonic equations by a straightforward approach. Yao was supported by the Innovation Foundation of Shanghai University (Grant No. A10-0101-08-905), Shanghai Leading Academic Discipline Project (Grant No. J50101) and Key Disciplines of Shanghai Municipality (Grant No. S30104). Zhou was supported by the National Basic Research Program of China (Grant No. 2006CB705700), National Natural Science Foundation of China (Grant No. 60532080), and the Key Project of Chinese Ministry of Education (Grant No. 306017)     

Geometric convergence to e−z by rational functions with real poles

Summary In this paper, we show that there exists a sequence of rational functions of the formR _n(z)=p_n–1(z)/(1+z/n)ⁿ,n=1, 2, ..., with degp _n–1n–1, which converges geometrically toe ^–z in the uniform norm on [0, +), as well as on some infinite sector symmetric about the positive real axis. We also discuss the usefulness of such rational functions in approximating the solutions of heat-conduction type problems.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(11-1)-2075.     

Nonparametric tests for bounds on the derivative of a regression function

We consider two tests of the null hypothesis that the k-th derivative of a regression function is uniformly bounded by a specified constant. These tests can be used to study the shape of the regression function. For instance, we can test for convexity of the regression function by setting k=2 and the constant equal to zero. Our tests are based on k-th order divided difference of the observations. The asymptotic distribution and efficacies of these tests are computed and simulation results presented.Research supported by Natural Sciences and Engineering Research Council of Canada Grant OGP0007969.Research supported by National Science Foundation Grant DMS-9306738.     

A class of martingales with non-symmetric limit distributions

Summary A class of non-pathological martingales is exhibited to which current martingale central limit theory does not apply. The problem consists in the convergence of the normalized sums of squares being only in distribution. The class is that of degenerate U-statistics.This work was done while the first two authors were on Sabbatical leave at the Technion.Work supported in part by National Research Council of Canada, Grant No. A-9068Work supported in part by National Research Council of Canada, Grant No. A-9076     

Finite groups in which elements of the same order outside the center are conjugate

We prove that if G is a finite group in which the elements of the same order outside the center are conjugate,then either G is abelian or G(?)S_3.     

Quasi-planar graphs have a linear number of edges

A graph is calledquasi-planar if it can be drawn in the plane so that no three of its edges are pairwise crossing. It is shown that the maximum number of edges of a quasi-planar graph withn vertices isO(n).Work on this paper by Pankaj K. Agarwal, Boris Aronov and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work on this paper by Pankaj K. Agarwal has also been supported by NSF Grant CCR-93-01259, by an Army Research Office MURI grant DAAH04-96-1-0013, by an NYI award, and by matching funds from Xerox Corporation. Work on this paper by Boris Aronov has also been supported by NSF Grant CCR-92-11541 and by a Sloan Research Fellowship. Work on this paper by János Pach, Richard Pollack, and Micha Sharir has been supported by NSF Grants CCR-91-22103 and CCR-94-24398. Work by János Pach was also supported by Grant OTKA-4269 and by a CUNY Research Award. Work by Richard Pollack was also supported by NSF Grants CCR-94-02640 and DMS-94-00293. Work by Micha Sharir was also supported by NSF Grant CCR-93-11127, by a Max-Planck Research Award, and by grants from the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Part of the work on this paper was done during the participation of the first four authors in the Special Semester on Computational and Combinatorial Geometry organized by the Mathematical Research Institute of Tel Aviv University, Spring 1995.     

Some universal graphs

It is shown that various classes of graphs have universal elements. In particular, for eachn the class of graphs omitting all paths of lengthn and the class of graphs omitting all circuits of length at leastn possess universal elements in all infinite powers. Research partially supported by Hungarian Science Research Fund No. 1805. Research partially supported by NSERC of Canada Grant #A8948.     

Hammersley's interacting particle process and longest increasing subsequences

Summary In a famous paper [8] Hammersley investigated the lengthL _n of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly “soft” arguments that limn ^′1/2 EL _n =2. This is a known result, but previous proofs [14, 11] relied on hard analysis of combinatorial asymptotics. Research supported by NSF Grant MCS 92-24857 and the Miller Institute for Basic Research in Science Research supported by NSF Grant DMS92-04864     

k-sets and random hulls

We re-examine the probabilistic analysis of Clarkson and Shor [5] involvingk-sets of point sets and related structures. By studying more carefully the equations that they derive, we are able to obtain refined analysis of these quantities, which lead to a collection of interesting relationships involvingk-sets, convex hulls of random samples, and generalizations of these constructs.Work on this paper has been supported by Office of Naval Research Grant N00014-89-J-3042 and N00014-90-J-1284, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Use of dynamic trees in a network simplex algorithm for the maximum flow problem

Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on ann-vertex,m-arc network in at mostnm pivots and O(n ² m) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, 1985) to reduce the running time of this algorithm to O(nm logn). This bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem. Our extension of dynamic trees is interesting in its own right and may well have additional applications.Research partially supported by a Presidential Young Investigator Award from the National Science Foundation, Grant No. CCR-8858097, an IBM Faculty Development Award, and AT&T Bell Laboratories.Research partially supported by the Office of Naval Research, Contract No. N00014-87-K-0467.Research partially supported by the National Science Foundation, Grant No. DCR-8605961, and the Office of Naval Research, Contract No. N00014-87-K-0467.     

On strongly-non-reflexive groups

A strong negative answer is given to the old question of whether every dual group is reflexive. Using ◊_ω1 a groupA is constructed so thatA, A*, A**, andA*** are weakly ω₁-separable groups of cardinalityω ₁ andA* is not isomorphic toA***. Research partially supported by NSF Grant No. DMS-8400451. Research partially supported by NSERC Grant No. A8948.     

Laws of the Iterated Logarithm for the Local U-Statistic Process

Laws of the iterated logarithm are established for the local U-statistic process. This entails the development of probability inequalities and moment bounds for U-processes that should be of separate interest. The local U-statistic process is based upon an estimator of the density of a function of several i.i.d. variables proposed by Frees (J. Am. Stat. Assoc. 89, 517–525, 1994). As a consequence, our results are directly applicable to the derivation of exact rates of uniform in bandwidth consistency in the sup and in the L _p norms for these estimators. Research of E. Giné partially supported by NSA Grant H98230-04-1-0075. Research of D.M. Mason partially supported by NSA Grant MDA904-02-1-0034 and NSF Grant DMS-0503908.     

On the possible number of elements of given order in a finite group

The main motivation of this paper is to introduce a problem of some combinatorial flavor about finite groups which seems to be new in the literature. Letk>1 be a fixed positive integer and denote byf(k, G) the number of elements of orderk in the groupG. We examine the setF(k)={f(k, G)| G a finite group}/{0}. We give a complete characterization ofF(k) if 4|k ork=6 and show some modest partial results for certain other values ofk. It seems to us that the question is surprisingly difficult even in such simple cases ask=3, which we investigate in detail. Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1901. Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1903.