Cauchy problem in a product space (that may be particularly involved). For example, consider the heat equation  ∂u ∂t (t, x) = ∆u(t, x), t ≥ 0, x ∈ Ω, ∂w ∂t (t, z) = − ∂w ∂n (t, z), t ≥ 0, z ∈ ∂Ω, w(t, z) = u(t, z), t ≥ 0, z ∈ ∂Ω, where − ∂ ∂n denotes the outer normal derivative. This can be written as U ′(t) = ( ∆ 0 − ∂ ∂n 0 ) U(t), where U = ( u u|∂Ω ) . Observe that a relationship between ...

This paper deals with the coupled procedure of the boundary element method (BEM) and the finite element method (FEM) for the exterior boundary value problems for the Helmholtz equation. A circle is selected as the common boundary on which the integral equation is set up with Fourier expansion. As a result, the exterior problems are transformed into nonlocal boundary value problems in a bounded ...

My lectures at the Minicorsi di Analisi Matematica at Padova in June 2000 are written up in these notes1. They are an updated and extended version of my lectures [37] at Jyväskylä in October 1994. In particular, an account of the exciting recent development of the asymptotic case is included, which is called the ∞-eigenvalue problem. I wish to thank the University of Padova for financial suppor...

We formulate a systematic elegant perturbative scheme for determining the eigenvalues of the Helmholtz equation (∇ + k)ψ = 0 in two dimensions when the normal derivative of ψ vanishes on an irregular closed curve. The unique feature of this method, unlike other perturbation schemes, is that it does not require a separate formalism to treat degeneracies. Degenerate states are handled equally ele...

Let Σ : M → 2 \ {∅} be a set-valued function defined on a Hausdorff compact topological space M and taking values in the normed space (Y, ‖·‖). We deal with the problem of finding the best Chebyshev type approximation of the set-valued function Σ by a singlevalued function g from a given closed convex set V ⊂ C(M, Y ). In an abstract setting this problem is posed as the extremal problem supt∈M ...

We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space and leave the most empty space. In each different dimension and under each different set of restrictions, this question is expected to have a different answer or perhaps no answer at all. As the problem of identifying global minima in most cases appears to be beyon...

Discriminating targets moving against a cluttered background is a huge challenge, let alone detecting a target as small as one or a few pixels and tracking it in flight. In the fly’s visual system, a class of specific neurons, called small target motion detectors (STMDs), have been identified as showing exquisite selectivity for small target motion. Some of the STMDs have also demonstrated dire...

Abstract: This paper is concerned with a shape optimization problem of a viscoelastic body in unilateral dynamic contact with a rigid foundation. The contact with Coulomb friction is assumed to occur at a portion of the boundary of the body. The nonpenetration condition is described in terms of velocities. The thermal deformation is taken into account. Using the material derivative method as we...

A new class of algorithms is described for the analysis of boundaries in a discrete image. The methods are based on well known methods of Gaussian filtering coupled with the use of numerical methods. Instead of the conventional discrete filter and exhaustive evaluation by convolution the filter output is computed only at points of interest. In addition derivative information is gathered at thes...

Synthetic aperture radar (SAR) images are subject to prominent speckle noise, which is generally considered a purely multiplicative noise process. In theory, this multiplicative noise is that the ratio of the standard deviation to the signal value, the “coefficient of variation,” is theoretically constant at every point in a SAR image. Most of the filters for speckle reduction are based on this...