Hans Volkmer

Riesz Bases of Solutions of Sturm-Liouville Equations

This article concerns the stability of orthogonal bases of solutions of Sturm-Liouville equations with different types of initial conditions. The investigation is based on the stability of Riesz bases of cosines and sines in the Hibert space L2[0,π].

On the regularity of wavelets

The regularity index alpha /sub N/ of the scaling functions /sub N/ phi , N=2, 3, . . . of multiresolution analysis introduced by I. Daubechies (1988) is investigated. It is shown that 0.51 >

Invariants of weak equivalence in primitive matrices

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one dimensional inverse limit spaces.

Asymptotic regularity of compactly supported wavelets

We study the asymptotic regularity of orthonormal bases of compactly supported wavelets when their support width tends to infinity. We construct sequences of wavelets whose ratio of regularity to support width is greater than that in previously known examples.

On multiparameter theory

Abstract In this paper we point out a mistake in the paper “Abstract Multiparameter Theory I” of P. J. Browne. We give new proofs for some of the results of P. J. Browne.

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.

FOUR REMARKS ON EIGENVALUES OF LAMÉ'S EQUATION

Four properties of eigenvalues of Lames differential equation are studied: interlacing of eigenvalues, behavior as k→1, width of stability bands and expansion in powers of k2.

Eigenfunction expansions for a fundamental solution of Laplace's equation on R 3 in parabolic and elliptic cylinder coordinates

A fundamental solution of Laplace’s equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functionsJ0(kr) orK0(kr),r 2 = (x−x0) 2 + (y−y0) 2 , in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that K0(kr) is a fundamental solution and J0(kr) is the Riemann function of partial differential equations on the Euclidean plane.

An idealized model of the one-dimensional carbon dioxide rectifier effect

The net flux of carbon dioxide (CO2) from the land surface into the atmospheric boundary layer has a diurnal cycle. Drawdown of CO2 occurs during daytime photosynthesis, and return of CO2 to the atmosphere occurs during night. Even when the net diurnal-average surface flux vanishes, the diurnal-average profile of atmospheric CO2 mixing ratio is usually not vertically uniform. This is because of the diurnal rectifier effect, by which atmospheric vertical transport and the surface flux conspire to produce a surplus of CO2 near the ground and a deficit aloft. This paper constructs an idealized, 1-D, eddy-diffusivity model of the rectifier effect and provides an analytic series solution. When non-dimensionalized, the intensity of the rectifier effect is related solely to a single ‘rectifier parameter’. Given this model’s governing equation and boundary conditions, we prove that the existence of the rectifier effect is related to the correlation of CO2 gradient and transport, and also to the day–night symmetry of transport. The rectifier-induced near-surface CO2 surplus ought to be included in inverse calculations that use near-surface CO2 mixing ratio to infer land–surface sources and sinks of carbon. Such inverse modeling is facilitated by our model’s simplicity. To illustrate, we use a 1-D inverse calculation to infer the amplitude of diurnal CO2 surface flux.

Generalized ellipsoidal and sphero-conal harmonics.

Classical ellipsoidal and sphero-conal harmonics are polynomial solutions of the Laplace equation that can be expressed in terms of Lame polynomials. Generalized ellipsoidal and sphero-conal harmonics are polynomial solutions of the more general Dunkl equation that can be expressed in terms of Stieltjes polynomials. Nivens formula connec- ting ellipsoidal and sphero-conal harmonics is generalized. Moreover, generalized ellipsoidal harmonics are applied to solve the Dirichlet problem for Dunkls equation on ellipsoids.