Variable window width kernel estimates of probability densities

Kernel density estimators which allow different amounts of smoothing at different locations are studied. Modifications of estimators proposed by Breiman, Meisel and Purcell (1977) and Abramson (1982a), which have variable window widths, are seen to have very fast rates of convergence. These rates have traditionally been obtained using a less natural higher order kernel, which has the disadvantage of allowing an estimator which takes on negative values.Research partially supported by NSF Grants DMS-8400602 and DMS-8701201     

Sharp trace estimates of solutions to Kirchhoff and Euler-Bernoulli equations

This paper gives sharp estimates of second-order traces of solutions to mixed problems for Kirchhoff equations and Euler-Bernoulli equations with second- and third-order nonhomogeneous boundary conditions. The approach uses pseudodifferential analysis. These sharp trace regularity results have, among others, important implications in corresponding exact controllability/uniform stabilization theories; they allow the removal of geometrical conditions made in prior literature.This research was partially supported by the National Science Foundation under Grant DMS-89-02811.     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

Adaptive estimates of linear functionals

Summary Given a collection of nested closed, convex symmetric sets and a linear functional, we find estimates which are within a logarithm term of being simultaneously asymptotically minimax. Moreover, these estimates can be constructed so that the loss of this logarithm term only occurs on a small subset of functions. These estimates are quasi-optimal since there do not exist estimators which do not lose a logarithm term on some part of the parameter spaces.This author was partially supported by an NSF Grant DMS-9123956This author was supported by an NSF Postdoctoral Research Fellowship     

Analysis of blockwise shrinkage wavelet estimates via lower bounds for no-signal setting

A blockwise shrinkage is a popular procedure of adaptation that has allowed the statisticians to establish an impressive bouquet of asymptotic mathematical results and develop softwares for solving practical problems. Traditionally risks of the estimates are studied via upper bounds that imply sufficient conditions for a blockwise shrinkage procedure to be minimax. This article suggests to analyze the estimates via exact (non-asymptotic) lower bounds established for a no-signal setting. The approach complements the familiar minimax, Bayesian and numerical analysis, it allows to find necessary conditions for a procedure to attain desired rates, and it sheds a new light on popular choices of blocks and thresholds recommended in the literature. Mathematical results are complemented by a numerical study. Supported in part by NSF Grants DMS-9971051 and DMS-0243606.     

Small ball estimates for Brownian motion and the Brownian sheet

Small ball estimates are obtained for Brownian motion and the Brownian sheet when balls are given by certain Hölder norms. As an application of these results we include a functional form of Chung's LIL in this setting.Both authors were supported in part by NSF Grant Number DMS-9024961.     

Weighted weak type estimates involving spaces generated by blocks

The problem studied in this paper is how to establish weighted estimates of weak type near L¹ for those operators that are not of weak type (1,1). The first author is supported by the NSFC (Tian Yuan) and the second is supported by NSF grant DMS-9007491.     

Compactness of isospectral conformal metrics onS 3

Research supported by NSF grants DMS-88-16321 and DMS-87-02871.     

Rank estimates in a class of semiparametric two-sample models

We consider a two-sample semiparametric model involving a real parameter and a nuisance parameter F which is a distribution function. This model includes the proportional hazard, proportional odds, linear transformation and Harrington-Fleming models (1982, Biometrika, 69, 533–546). We propose two types of estimates based on ranks. The first is a rank approximation to Huber's M-estimates (1981, Robust Statistics, Wiley) and the second is a Hodges-Lehmann type rank inversion estimate (1963, Ann. Math. Statist., 34, 598–611). We obtain asymptotic normality and efficiency results. The estimates are consistent and asymptotically normal generally but fully efficient only for special cases.Research partially supported by National Science Foundation Grant DMS-86-02083 and National Institute of General Medical Sciences Grant SSS-Y1RO1GM35416-01     

Morera and mean-value type theorems in the hyperbolic disk

Some Morera and mean-value type theorems are proved in the hyperbolic disk. Partially supported by NSF grants DMS-9000619 and CDR-8803012. Partially supported by NSF grants DMS-8703072 and DMS-9000619, and DGI-CYT grant PB 89-0311.     

On the scaling limit of a singular integral operator

The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems. Research supported in part by National Science Foundation grants DMS-02-45371 and DMS-04-05519.     

Rank two vector bundles over regular elliptic surfaces

Research partially supported by NSF grants DMS-85-03743 and DMS-87-03569 and the Alfred P. Sloan Foundation     

Proof of a conjecture on multisets of hook numbers

A conjecture of Regev and Vishik on the equality of two multisets of hook numbers is proved. Supported in part by N.S.F. Grant No. DMS-94-01197. Supported in part by N.S.F. Grant No. DMS-95-00646.     

The structure of singly-periodic minimal surfaces

The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grants DMS-8611574 and DMS-8802858     

On the eigenvalues of matrices with given upper triangular part

We give necessary and sufficient conditions for a set of numbers to be the eigenvalues of a completion of a matrix prescribed in its upper triangular part.Partially supported by the NSF Grant DMS-8701615-02Partially supported by the NSF Grant DMS-8802836 and United States-Israel Binational Science Foundation Grant 88-00304/I.     

Lattice size estimates for Gabor decompositions

It is shown that the Gabor frame operator for frames with lattice such thatab=1/N for some integerN1 is continuous on the modulation spaces ([9], [20]) whenever the basic functiong is a Schwartz function and (g, a, b) generates a frame forL ² (R). From this a sufficient condition on the lattice is given which guarantees that one has a Banach frame for the modulation spaces of Feichtinger.This work was supported by NSF grant # DMS-8916968 while the author was visiting Yale University.     

Embedded minimal surfaces with an infinite number of ends

The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation grants DMS-8503350 and DMS-8611574     

A Markov-modulated M/G/1 queue II: Busy period and time for buffer overflow

We consider an M/G/1 queue where the arrival and service processes are modulated by a two state Markov chain. We assume that the arrival rate, service time density and the rates at which the Markov chain switches its state, are functions of the total unfinished work (buffer content) in the queue. We compute asymptotic approximations to performance measures such as the mean residual busy period, mean length of a busy period, and the mean time to reach capacity.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

Robust convex quadratically constrained programs

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. In contrast to [4], where it is shown that such robust problems can be formulated as semidefinite programs, our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second-order cone programs (SOCP). We propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit SOCP and present examples where these classes of uncertainty sets are natural. Research partially supported by DOE grant GE-FG01-92ER-25126, NSF grants DMS-94-14438, CDA-97-26385, DMS-01-04282 and ONR grant N000140310514.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

Affine systems inL 2 (ℝ d ) II: Dual systems

The fiberization of affine systems via dual Gramian techniques, which was developed in previous papers of the authors, is applied here for the study of affine frames that have an affine dual system. Gramian techniques are also used to verify whether a dual pair of affine frames is also a pair of bi-orthogonal Riesz bases. A general method for a painless derivation of a dual pair of affine frames from an arbitrary MRA is obtained via the mixed extension principle. This work was partially sponsored by the National Science Foundation under Grants DMS-9102857, DMS-9224748, and DMS-9626319, by the United States Army Research Office under Contracts DAAL03-G-90-0090, DAAH04-95-1-0089, and by the Strategic Wavelet Program Grant from the National University of Singapore.