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 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     

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.     

Estimation of elastic parameters in beams and certain plates:H 1 regularization

The estimation of elastic parameters in beams and certain types of plates is discussed using anH ¹-regularization technique that easily accommodates pointwise constraints. The optimal coefficient is shown to enjoy more regularity than that assumed in the formulation of the problem. This additional smoothness is useful for analyzing the limit behavior of finite-dimensional problems. Numerical results are presented.This work was supported in part by the Air Force Office of Scientific Research, Grant AFOSR-84-0271.     

Polynomial dual network simplex algorithms

We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy leads to an O(m ² logn) bound on the number of pivots, wheren andm denotes the number of nodes and arcs in the input network. If the demands are integral and at mostB, we also give an O(m(m+n logn) min(lognB, m logn))-time implementation of a strategy that requires somewhat more pivots.Research supported by AFOSR-88-0088 through the Air Force Office of Scientific Research, by NSF grant DOM-8921835 and by grants from Prime Computer Corporation and UPS.Research supported by NSF Research Initiation Award CCR-900-8226, by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by ONR Contract N00014-88-K-0166.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

Subharmonics near an equilibrium point for hamitonian systems

This work is devoted to the study of subharmonic solutions near an equilibrium for certain Hamiltonian systems. We impose a weak condition on the Floquet exponents of the linearized system, and a superquadratic condition on the higher order term. This last condition is reduced to the center manifold. This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-87-0202.     

The bilinear field of values

A complete characterization is given of a bilinear field of values involving orthonormal sets of vectors.To Olga Taussky-ToddThis research was supported by the Air Force Office of Scientific Research under Grant AFOSR-72-2164.     

Truncation error analysis by means of approximant systems and inclusion regions

Summary A general approach to truncation error analysis is described, in which bounds for the truncation error are determined by means of inclusion regions, and the notion of bestness is meaningfully formulated. A new mathematical structure (approximant system) is introduced and developed. It consists of a family of infinite processes having a natural structure for truncation error analysis. Applications of the methods are included for infinite series, Cesaro sums, approximate integration, an iterative method for solving equations, Padé approximants and continued fractions.Research supported in part by the National Science Foundation under Grant No. MPS 74-22111 and by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-70-1888. The United States Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon     

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     

Martingale representation and the Malliavin calculus

Using stochastic flows and the Itô differentiation rule, the integrand in the representation of a martingale as a stochastic integral is identified. By iterating this representation result a homogeneous chaos type expansion is obtained. Using the stochastic integral representation, an integration by parts formula is obtained without using any calculus of variations in function space. If the inverse of the Malliavin matrix belongs to all spacesL ^p() it follows that a random variable has a smooth density.The research of R. J. Elliott was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-7964 and the Air Force Office of Scientific Research, United States Air Force, under Grant AFOSR-86-0332. The research of M. Kohlmann was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-7964.     

Continuity of the spectrum on closed similarity orbits

We present a useful case when the spectral radius of a norm limit of operator similar to a fixed operatorT still equals that ofT.This work was supported in part by grants from the National Science Foundation DMS-8811084, ECS-9001371, ECS-9122106, by the Air Force Office of Scientific Research AFOSR-90-0024 and AFOSR-90-0053, and by the Army Research Office DAAL03-91-G-0019.     

On the computation of the optimal cost function for discrete time Markov models with partial observations

We consider several applications of two state, finite action, infinite horizon, discrete-time Markov decision processes with partial observations, for two special cases of observation quality, and show that in each of these cases the optimal cost function is piecewise linear. This in turn allows us to obtain either explicit formulas or simplified algorithms to compute the optimal cost function and the associated optimal control policy. Several examples are presented.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, in part by the Advanced Technology Program of the State of Texas, and in part by the DoD Joint Services Electronics Program through the Air Force Office of Scientific Research (AFSC) Contract F49620-86-C-0045.     

Adaptive control of Markov processes with incomplete state information and unknown parameters

Recent results for parameter-adaptive Markov decision processes (MDP's) are extended to partially observed MDP's depending on unknown parameters. These results include approximations converging uniformly to the optimal reward function and asymptotically optimal adaptive policies.This research was supported in part by the Consejo del Sistema Nacional de Educación Tecnologica (COSNET) under Grant 178/84, in part by the Air Force Office of Scientific Research under Grant AFOSR-84-0089, in part by the National Science Foundation under Grant ECS-84-12100, and in part by the Joint Services Electronics Program under Contract F49602-82-C-0033.     

An augmented Lagrangian technique for variational inequalities

A general framework for the treatment of a class of elliptic variational inequalities by an augmented Lagrangian method, when inequalities with infinite-dimensional image space are augmented, is developed. Applications to the obstacle problem, the elastoplastic torsion problem, and the Signorini problem are given.The research of the first author was supported in part by the Air Force Office of Scientific Research under Grants AFOSR-84-0398 and AFOSR-85-0303, by the National Aeronautics and Space Administration under Grant NAG-1-1517, and by NSF under Grant No. UINT-8521208. The second author's research was supported in part by the Fonds zur Förderung der wissenschaftlichen Forschung under S3206 and P6005.     

On superlinear convergence in univariate nonsmooth minimization

This note demonstrates a new result on superlinear convergence in nonsmooth univariate minimization. It also gives other concepts of rapid convergence for minimization of functions that may have discontinuous derivatives.Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Numbers AFOSR-83-0210 and AFOSR-88-0180.     

On the average cost optimality equation and the structure of optimal policies for partially observable Markov decision processes

We consider partially observable Markov decision processes with finite or countably infinite (core) state and observation spaces and finite action set. Following a standard approach, an equivalent completely observed problem is formulated, with the same finite action set but with anuncountable state space, namely the space of probability distributions on the original core state space. By developing a suitable theoretical framework, it is shown that some characteristics induced in the original problem due to the countability of the spaces involved are reflected onto the equivalent problem. Sufficient conditions are then derived for solutions to the average cost optimality equation to exist. We illustrate these results in the context of machine replacement problems. Structural properties for average cost optimal policies are obtained for a two state replacement problem; these are similar to results available for discount optimal policies. The set of assumptions used compares favorably to others currently available.This research was supported in part by the Advanced Technology Program of the State of Texas, in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, and in part by the Air Force Office of Scientific Research (AFSC) under Contract F49620-89-C-0044.     

Expansions for statistics involving the mean absolute deviations

Expansion for the difference of mean absolute deviations from the sample mean and the population mean is derived. This result is used to obtain strong representations for mean absolute deviations from the sample mean and the sample median. Edgeworth expansions for some scale invariant statistics involving the mean absolute deviations are studied. These expansions are shown to be valid in spite of the presence of a lattice variable.Research supported in part by NSA Grant MDA904-90-H-1001.Research supported by the Air Force Office of Scientific Research under Grant AFOSR-89-0279.     

Hyperbolic equations with Dirichlet boundary feedback via position vector: Regularity and almost periodic stabilization—Part I

A hyperbolic equation defined on a bounded domain is considered, with input acting in theDirichlet boundary condition and expressed as a specifiedfeedback of theposition vector only. Two main results are established. First, we prove a well-posedness and regularity result of the feedback solutions. Second, we specialize our equation to the case when the original differential operator withzero boundary conditions is self-adjoint and unstable. Here, under certain natural algebraic conditions based on the finitely many unstable eigenvalues, we establish the existence ofboundary vectors, for which the corresponding feedback solutions have the same desirablestructural property of astable free system: They can be expressed as an infinite linear combination of sines and cosines (special case of almost periodicity). A cosine operator approach is employed.This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-78-3350 (I.L.) and Grant AFOSR-77-3338 (R.T.) through ISU.This research was performed while the author was visiting the Department of Systems Science, University of California, Los Angeles.     

Inequalities for E k(X,Y) when the marginals are fixed

When k(x, y) is a quasi-monotone function and the random variables X and Y have fixed distributions, it is shown under some further mild conditions that k(X, Y) is a monotone functional of the joint distribution function of X and Y. Its infimum and supremum are both attained and correspond to explicitly described joint distribution functions.Research supported by the Air Force Office of Scientific Research under Grant AFOSR-75-2796Research supported by the National Science Foundation