-function of an operator: A white noise approach

Let be the canonical framework of white noise analysis over the Gel'fand triple and be the space of continuous linear operators from to . Let be a self-adjoint operator in with spectral representation . In this paper, it is proved that under appropriate conditions upon , there exists a unique linear mapping such that for each . The mapping is then naturally used to define as , where is the Dirac -function. Finally, properties of the mapping are investigated and several results are obtained.

     

The -harmonic forms on rotationally symmetric Riemannian manifolds revisited

We use separation of variables for generalized Dirac operators on rotationally symmetric Riemannian manifolds to recover a theorem of Dodziuk regarding the spaces of -harmonic forms on such manifolds.

     

Telescoping, rational-valued series, and zeta functions

We give an effective procedure for determining whether or not a series telescopes when is a rational function with complex coefficients. We give new examples of series , where is a rational function with integer coefficients, that add up to a rational number. Generalizations of the Euler phi function and the Riemann zeta function are involved. We give an effective procedure for determining which numbers of the form are rational. This procedure is conditional on 3 conjectures, which are shown to be equivalent to conjectures involving the linear independence over the rationals of certain sets of real numbers. For example, one of the conjectures is shown to be equivalent to the well-known conjecture that the set is linearly independent, where is the Riemann zeta function.

Some series of the form , where is a quotient of symmetric polynomials, are shown to be telescoping, as is . Quantum versions of these examples are also given.

     

Decomposition and Group Theoretic Characterization of Pairs of Inverse Relations of the Riordan Type

A new solution to Riordans problem of combinatorial identities classification is presented. An algebgraic characterization of pairs of inverse relations of the Riordan type is given. The use of the integral representation approach for generating new types of combinatorial identities is demonstrated. Supported in part by the National Sciences and Engineering Research Council of Canada on Grant NSERC-108343.Mathematics Subject Classifications (2000) combinatorics, algebra.     

On the Convergence of the Generalized Linear Least Squares Algorithm

This paper considers the issue of parameter estimation for biomedical applications using nonuniformly sampled data. The generalized linear least squares (GLLS) algorithm, first introduced by Feng and Ho (1993), is used in the medical imaging community for removal of bias when the data defining the model are correlated. GLLS provides an efficient iterative linear algorithm for the solution of the non linear parameter estimation problem. This paper presents a theoretical discussion of GLLS and introduces use of both Gauss Newton and an alternating Gauss Newton for solution of the parameter estimation problem in nonlinear form. Numerical examples are presented to contrast the algorithms and emphasize aspects of the theoretical discussion. AMS subject classification (2000) 65F10.R. A. Renaut: This work was partially supported by the Arizona Center for Alzheimer’s Disease Research, by NIH grant EB 2553301 and for the second author by NSF CMG-02223.Received December 2003. Revised November 2004. Communicated by Lars Eldén.     

On the Extrema of Linear Least-Squares Problems

We use the concept of generalized inverses to show that the extrema of the function are minima, A being a rectangular matrix not necessarily of full rank     

On a functional equation generalizing the class of semistable distributions

With ?(p),p≥0 the Laplace-Stieltjes transform of some infinitely divisible probability distribution, we consider the solutions to the functional equation ?(p-e ^?pβΠ _i=1 ^m ?^γi (c _i p) for somem≥1,c _i>0, γ _i >0,i=1., …,m, β ε ®. We supply its complete solutions in terms of semistable distributions (the ones obtained whenm=1). We then show how to obtain these solutions as limit laws (r → ∞) of normalized Poisson sums of iid samples when the Poisson intensity λ(r) grows geometrically withr.     

The Fréchet distance between multivariate normal distributions

The Fréchet distance between two multivariate normal distributions having means μ_X, μ_Y and covariance matrices Σ_X, Σ_Y is shown to be given by $\text{d}{}^2\text{= |μ}{}_{\mathrm{X}}\text{? μ}{}_{\mathrm{Y}}\text{|}{}^2\text{+}\text{tr}\text{(Σ}{}_{\mathrm{X}}\text{+ Σ}{}_{\mathrm{Y}}\text{? 2(Σ}{}_{\mathrm{X}}\text{Σ}{}_{\mathrm{Y}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$. The quantity d₀ given by $\text{d}{}_0{}^2\text{=}\text{tr}\text{(Σ}{}_{\mathrm{X}}\text{+ Σ}{}_{\mathrm{Y}}\text{? 2(Σ}{}_{\mathrm{X}}\text{Σ}{}_{\mathrm{Y}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is a natural metric on the space of real covariance matrices of given order.     

The spectrum of a random 3 × 3 matrix

In this paper, it is shown that the distribution of the spectrum of A + X gives (essentially) a complete set of orthogonal invariants for the (real nonsymmetric) matrix A.     

β-Expectation tolerance regions for a generalized multivariate model with normal error variables

In this paper a procedure of construction of β-expectation tolerance regions in the framework of the structural method of inference has been developed. The procedure has been applied to the generalized multivariate model and the β-expectation tolerance region for this case has been constructed assuming the normal distribution for the error variables of the model.