David L. Ragozin

Harvest policies and nonmarket valuation in a predator — prey system

Abstract Although prey may not have commercial value, their economic value can be ascertained in a predator-prey model if the predator has a harvest value. The economic optimal (recovery) path of the predator and prey are carefully described when growth is quadratic in the predator (prey) and linear in prey (predator). Parameter values, in part, resembling Pacific halibut are used to provide numerical illustrations.

Error bounds for derivative estimates based on spline smoothing of exact or noisy data

Abstract Estimates are found for the L 2 error in approximating the j th derivative of a given smooth function f by the corresponding derivative of the 2 m th order smoothing spline based on an n -point sample from the function. The results cover both the case of an exact sample from f and the case when the sample is subject to some random noise. In the noisy case, the estimates are for the expected value of the approximation error. These bounds show that, even in the presence of noise, the derivatives of the smoothing splines of order less than m can be expected to converge to those of f as the number of (uniform) sample points increases, and the smoothing parameter approaches zero at a rate appropriately related to m , n , and the order of differentiability of f.

Zonal measure algebras on isotropy irreducible homogeneous spaces

This paper analyzes the convolution algebra M(K\GK) of zonal measures on a Lie group G, with compact subgroup K, primarily for the case when M(K\GK) is commutative and GK is isotropy irreducible. A basic result for such (G, K) is that the convolution of dim GK continuous (on GK) zonal measures is absolutely continuous. Using this, the spectrum (maximal ideal space) of M(K\GK) is determined and shown to be in 1-1 correspondence with the bounded Borel spherical functions. Also, certain asymptotic results for the continuous spherical functions are derived. For the special case when G is compact, all the idempotents in M(K\GK) are determined.

A Simple Approach to the Variational Theory for Interpolation on Spheres

In this paper we consider the problem of developing a variational theory for interpolation by radial basis functions on spheres. The interpolants have the property that they minimise the value of a certain semi-norm, which we construct explicitly. We then go on to investigate forms of the interpolant which are suitable for computation. Our main aim is to derive error bounds for interpolation from scattered data sets, which we do in the final section of the paper.

Energies, group-invariant kernels and numerical integration on compact manifolds

The purpose of this paper is to derive quadrature estimates on compact, homogeneous manifolds embedded in Euclidean spaces, via energy functionals associated with a class of group-invariant kernels which are generalizations of zonal kernels on the spheres or radial kernels in euclidean spaces. Our results apply, in particular, to weighted Riesz kernels defined on spheres and certain projective spaces. Our energy functionals describe both uniform and perturbed uniform distribution of quadrature point sets.

Radial basis interpolation on homogeneous manifolds: convergence rates

Pointwise error estimates for approximation on compact homogeneous manifolds using radial kernels are presented. For a

A Class of Bernstein Polynomials that Satisfy Descartes' Rule of Signs Exactly

{\mathcal C}^{2r}

Limits of periodic smoothing splines

positive definite kernel κ the pointwise error at x for interpolation by translates of κ goes to 0 like ρr, where ρ is the density of the interpolating set on a fixed neighbourhood of x. Tangent space techniques are used to lift the problem from the manifold to Euclidean space, where methods for proving such error estimates are well established.

Non-smooth Wavelets: Graphing functions unbounded on every interval

Let Ta = b where \(a = \left\{ {{{a}_{i}}} \right\}_{{i = 0}}^{n}\)and \(b = \left\{ {{{b}_{i}}} \right\}_{{i = 0}}^{n}\) are the coefficients of a polynomial in the power and Bernstein bases respectively, and T is the transformation matrix between the bases. If USV T is the singular value decomposition of T, it is shown that the Bernstein polynomial p r (x) whose coefficients are given by column r of the left singular matrixU,that is, \(b = \left\{ {{{u}_{{ir}}}} \right\}_{{i = 0}}^{n}\), satisfies Descartes’ rule of signs exactly because the number of sign changes of the coefficients b i is exactly equal to the number of roots of p r (x) in the interval [0,1]. This result provides a new interpretation of polynomial basis conversion because U also determines the numerical condition of the basis transformation equation Ta = b. This connection is established by showing that T is a totally non—negative matrix and TT T is an oscillation matrix. Examples that illustrate the theoretical results are presented.

Limits of generalized periodic D-splines

The 2m — 1st degree 1-periodic smoothing splines for a fixed data vector y possess a limit as m approaches infinity if and only if the smoothing parameter approaches zero essentially as t2m. This limit, when it exists, is the least squares projection of the (proximal) trigonometric interpolant onto the trigonometric polynomials of degree at most [12πt], with respect to the (semi-) inner product given by evaluation at the data points.