Applied Mathematics Commons^™

Berry-Esseen-Type Bounds For Signed Linear Rank Statistics With A Broad Range Of Scores, Munsup Seoh

Mathematics and Statistics Faculty Publications

The Berry-Esseen-type bounds of order N^−1/2 for the rate of convergence to normality are derived for the signed linear rank statistics under the hypothesis of symmetry. The results are obtained with a broad range of regression constants and scores (allowed to be generated by discontinuous score generating functions, but not necessarily) restricted by only mild conditions, while almost all previous results are obtained with continuously differentiable score generating functions. Furthermore, the proof is very short and elementary, based on the conditioning argument.

Bimodules Over Cartan Subalgebras, Richard Mercer

Mathematics and Statistics Faculty Publications

Given a Cartan subalgebra A of a non Neumann algebra M, the techniques of Feldman and Moore are used to analyze the partial isometries v in M such that v* Av is contained in A. Orthonormal bases for M consisting of such partial isometries are discussed, and convergence of the resulting generalized fourier series is shown to take place in the Bures A-topology. The Bures A-topology is shown to be equivalent to the strong topology on the unit ball of M. These ideas are applied to A-bimodules and to give a simplified and intuitive proof of the Spectral Theorem …

On The Equivalence Of The Operator Equations Xa + Bx = C And X - P(-B)Xp(A)(-1) = W In A Hilbert-Space, P A Polynomial, Tapas Mazumdar, David Miller

Mathematics and Statistics Faculty Publications

We consider the solution of (*) XA+BX = C for bounded operators A,B,C and X on a Hilbert space, A normal. We establish the existence of a polynomial p and a bounded operator W with the property that the unique solution X of (*) also solves X − p(−B)Xp(A)⁻¹ = W uniquely. A known iterative algorithm can be applied to the latter equation to solve (*).

The Truncated Cauchy Distribution: Estimation Of Parameters And Application To Stock Returns, Paul G. Staneski

Mathematics & Statistics Theses & Dissertations

The problem addressed in this dissertation is the existence and estimation of the parameters of a truncated Cauchy distribution. It is known that when a number of distributions with infinite support are truncated to a finite interval that the maximum likelihood estimator of the scale parameter fails to exist with positive probability. In particular, necessary and sufficient conditions which give rise to instances of non-existence have been found for the exponential (Deemer and Votaw (1955)), gamma (Broeder (1955), Hegde and Dahiya (1989)), Weibull (Mittal and Dahiya (1989)) and normal distribution (Barndorff-Nielsen (1978), Mittal and Dahiya (1987), Hegde and Dahiya (1989)). …

On The Optimal Reward Function Of The Continuous Time Multiarmed Bandit Problem, José Luis Menaldi, Maurice Robin

Mathematics Faculty Research Publications

The optimal reward function associated with the so-called "multiarmed bandit problem" for general Markov-Feller processes is considered. It is shown that this optimal reward function has a simple expression (product form) in terms of individual stopping problems, without any smoothness properties of the optimal reward function neither for the global problem nor for the individual stopping problems. Some results relative to a related problem with switching cost are obtained.

Spatial Critical Points Of Solutions Of A One-Dimensional Nonlinear Parabolic Problem, Larry Turyn

Mathematics and Statistics Faculty Publications

The number of spatial critical points is nonincreasing in time, for positive, analytic solutions of a scalar, nonlinear, parabolic partial differential equation in one space dimension. While proving this, we answer the question: What happens to a critical point which loses simplicity?

Mathematical Models Of Prevascular Tumor Growth By Diffusion, Sophia A. Maggelakis

Mathematics & Statistics Theses & Dissertations

A study of several complementary mathematical models that describe the early, prevascular stages of solid tumor growth by diffusion under various simplifying assumptions is presented. The advantage of these models is that their degree of complexity is relatively low, which ensures fairly straightforward comparisons with experimental or clinical data (as it becomes available), yet they are mathematically sophisticated enough to capture the main biological phenomena of interest.

The tumor growth and cell proliferation rate are assumed to depend on the local concentrations of nutrients and inhibitory factors. The effects of geometry and spatially non-uniform inhibitor production and non-uniform nutrient consumption …

Some Estimates For Finite Difference Approximations, José-Luis Menaldi

Mathematics Faculty Research Publications

Some estimates for the approximation of optimal stochastic control problems by discrete time problems are obtained. In particular an estimate for the solutions of the continuous time versus the discrete time Hamilton-Jacobi-Bellman equations is given. The technique used is more analytic than probabilistic.

On Asymptotic Behavior Of Stopping Time Problems, Jose Luis Menaldi, Maurice Robin

Mathematics Faculty Research Publications

No abstract provided.

Pixley-Roy Hyperspaces Of Ω-Graphs, Joe Mashburn

Mathematics Faculty Publications

The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two ω-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of Rⁿ are also shown to be homeomorphic.

On The Existence And Symmetry Properties Of Finite Total Mass Solutions Of The Matukuma Equation, The Eddington Equation And Their Generalizations, Yi Li, Wei-Ming Ni

Mathematics and Statistics Faculty Publications

No abstract provided.

Remarks On A Semilinear Elliptic Equation On Rn, Yi Li

Mathematics and Statistics Faculty Publications

No abstract provided.

Spectral Measures, Orthogonal Polynomials, And Absolute Continuity, Joanne Dombrowski

Mathematics and Statistics Faculty Publications

This paper studies the spectral measure of an unbounded tridiagonal matrix operator for which the matrix entries satisfy a certain growth condition, and presents a sufficient condition for the existence of an absolutely continuous part. The results are related to a class of orthogonal polynomials with exponential weights.

On Conformal Scalar Curvature Equations In Rn, Yi Li, Wei-Ming Ni

Mathematics and Statistics Faculty Publications

No abstract provided.

Periodic Solutions Of Volterra Integral Equations, Muhammad Islam

Mathematics Faculty Publications

Consider the system of equations

x(t)=f(t)+∫−∞tk(t,s)x(s)ds,           (1)

and

x(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds.        (2)

Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are obtained using the …

Cyclic Operators, Commutators, And Absolutely Continuous Measures, Joanne Dombrowski

Mathematics and Statistics Faculty Publications

Commutator equations are used to study the relationship between the tridiagonal matrix structure of an unbounded cyclic selfadjoint operator and its spectrum. Sufficient conditions are given for absolute continuity. Results are related to the study of systems of orthogonal polyomials for which the measure of orthogonality is supported on an unbounded subset of the real line.

Orthogonal Polynomials, Measures And Recurrence Relations, Joanne Dombrowski, Paul Nevai

Mathematics and Statistics Faculty Publications

Properties of measures associated with orthogonal polynomials are investigated in terms of the coefficients of the three term recurrence formula satisfied by the orthogonal polynomials.

L(P) Estimates For Maximal Functions And Hilbert-Transforms Along Flat Convex Curves In R(2), Hasse Carlsson, Michael Christ, Antonio Cordoba, Javier Duoandikoetxea, Jose L. Rudio De Francia, Jim Vance, Stephen Wainger, David Weinberg

Mathematics and Statistics Faculty Publications

No abstract provided.

Reliable Computation In The Presence Of Noise, Nicholas Pippenger

All HMC Faculty Publications and Research

This talk concerns computation by systems whose components exhibit noise (that is, errors committed at random according to certain probabilistic laws). If we aspire to construct a theory of computation in the presence of noise, we must possess at the outset a satisfactory theory of computation in the absence of noise.

A theory that has received considerable attention in this context is that of the computation of Boolean functions by networks (with perhaps the strongest competition coming from the theory of cellular automata; see [G] and [GR]). The theory of computation by networks associates with any two sets Q and …

Cramer Type Large Deviations For Generalized Rank Statistics, Munsup Seoh, Stegan S. Ralescu, Madan L. Puri

Mathematics and Statistics Faculty Publications

A Cramer type large deviation theorem is proved under alternatives as well as under hypothesis for the generalized linear rank statistic which includes as special cases (unsigned) linear rank statistics, signed linear rank statistics, linear combination of functions of order statistics, and a rank combinatorial statistic.

Perturbation Of Periodic Boundary-Conditions, Larry Turyn

Mathematics and Statistics Faculty Publications

We consider perturbations of the problem (*) - x'' + bx = lambda ax, x(0) - x(1) = 0 = x'(0) - x'(1) both by changes of the boundary conditions and by addition of nonlinear terms. We assume that at lambda = lambda 0 there are two linearly independent solutions of the unperturbed problem (*) and that a(dot) is bounded away from zero. When only the boundary conditions are perturbed either the Hill’s discriminant or the method of Lyapunov–Schmidt reduces the problem to 0 = det ((lambda - lambda 0)A - epsilon …

A Note On Irreducibility And Weak Covering Properties, Joe Mashburn

Mathematics Faculty Publications

A space X is irreducible if every open cover of X has a minimal open refinement. Interest in irreducibility began when Arens and Dugendji used this property to show that metacompact countably compact spaces are compact. It was natural, then, to find out what other types of spaces would be irreducible and therefore compact in the presence of countable compactness or Lindelof in the presence of N₁-compactness. …

It is shown in this paper that T₁ δθ -refinable spaces and T1 weakly δθ-refinable spaces are irreducible. Since examples of Lindelof spaces that are neither T₁ nor …

The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn

Mathematics Faculty Publications

A partially ordered set P is ω-chain complete if every countable chain (including the empty set) in P has a supremum. … Notice that an ω-chain continuous function must preserve order. P has the (least) fixed point property for ω-chain continuous functions if every ω-chain continuous function from P to itself has (least) fixed point.

It has been shown that a partially ordered set does not have to be ω-chain complete to have the least fixed point property for ω-chain continuous functions. This answers a question posed by G. Plotkin in 1978. I.I. Kolodner has shown that an ω-chain complete …

Evidence Of Intrinsic Double Acceptor In Gaas, Phil Won Yu, W. C. Mithel, M. G. Mier, S. S. Li, Weizhen Wang

Mathematics and Statistics Faculty Publications

Acceptors present in undoped p‐type conducting GaAs have been studied with photoluminescence, temperature‐dependent Hall measurements, deep level transient spectroscopy, and spark source mass spectrometry. It is shown that p‐type conduction is due to presence of the shallow acceptor C_As and the cation antisite double acceptor Ga_As. The first and second ionization energies determined for Ga_As are 77 and 230 meV from the valence‐band edge.

Dissertation: The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn

Mathematics Faculty Publications

The basic definitions are given in the first section, including those for ω-chain continuity, ω-chain completeness, and the least fixed point property for ω-chain continuous functions. Some of the relations between completeness and fixed point properties in partially ordered sets are stated and it is briefly shown how the question basic to the dissertation arises.

In the second section, two examples are given showing that a partially ordered set need not be ω-chain complete to have the least fixed point property for ω-chain continuous functions.

Retracts are discussed in section 3, where it is seen that they are not sufficient …

Three Counterexamples Concerning Ω-Chain Continuous Functions And Fixed-Point Properties, Joe Mashburn

Mathematics Faculty Publications

A partially ordered set is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.

Numerical Solution Of A Quadratic Matrix Equation, George J. Davis

Mathematics & Statistics ETDs

This paper is concerned with the efficient numerical solution of the matrix equation AX2 + BX + C =0, where A,B,C and X are all square matrices. Such a matrix X is called a solvent. This matrix equation is very closely related to the problem of finding scalars lambda and nonzero vectors x such that (lambda2A + lambdaB + C)x=0. The latter equation represents a quadratic eigenvalue problem with each lambda and x called an eigenvalue and eigenvector, respectively. Such equations have many important physical applications which we survey.

By presenting an algorithm to calculate solvents, we show how the …

Double Stochastic Integrals, Tsung-Dow Huang

Mathematics & Statistics ETDs

In this paper we study the double stochastic integral or stochastic quadratic form

Q =integreal(integral(phi(x,y)Z(dx)Z(dy) ))

with respect to a Gaussian random spectral measure Z. Letting F be the corresponding spectral distribution function, the kernel function is required to satisfy a) the function phi(x,x) is integrable respect to F and b) phi(x,y) is square integrable with respect to FxF. A definition of Q is given in terms of a carefully constructed sequence of step functions converging to phi. Iterated integral formulae, a discussion of the stochastic contribution to Q due to the diagonal values of phi, and an improvement of …

Stochastic Approximation Procedures For Mixing Stochastic Processes, Katherine Campbell

Mathematics & Statistics ETDs

Stochastic approximation methods for estimating the parameters of a stationary autoregressive process of finite order are investigated. The emphasis is on robust methods, and the non-linear scoring functions associated with such methods require the development of new techniques for establishing convergence. A mixing condition falling between the traditional strong and uniform mixing conditions is investigated in detail, and used to establish almost sure and mean square convergence of the proposed algorithms when the underlying process satisfies this condition. A short Monte Carlo study verifies the desirable properties of the robust algorithm in the presence of heavy-tailed innovations.

Estimation Of The Acceleration Function, Le Than Chap

Mathematics & Statistics ETDs

Let x1, x2,…, xm be a random sample of m failure times under normal conditions with the underlying distribution function F(x) and Y1, Y2,…, Yn be a random sample on n failure times under accelerated conditions with the underlying distribution function G(x):

G(x)=f(F(x))

Where f(x) is the acceleration function.

In this dissertation we propose several estimators for the acceleration function f(x) and investigate their properties.