Let $A$ be a divergence form elliptic operator associated with a quadratic form on $\Omega$ where $\Omega$ is the Euclidean space $\mathbb{R}^n$ or a domain of $\mathbb{R}^n$. Assume that $A$ generates an analytic semigroup $e^{-tA}$ on $L^2(\Omega)$ which has heat kernel bounds of Poisson type, and that the generalised Riesz transform $\nabla A^{-l/2}$ is bounded on $L^2(\Omega)$. We then prove that $\nabla A^{-1/2}$ is of weak type $(l,l)$, hence bounded on $L^p(\Omega)$ for $l \leq p \leq 2$. No specific assumptions are made concerning the Hölder continuity of the coefficients or the smoothness of the boundary of $\Omega$.

We consider a Cellular Neural Network (CNN) with a bias term $z$ in the integer lattice $Z^2$ on the plane $R^2$. We impose a symmetric coupling between nearest neighbors, and also between next-nearest neighbors. Two parameters, $a$ and $c$, are used to describe the weights between such interacting cells. We study patterns that can exist as stable equilibria. In particular, the relationship between mosaic patterns, and the parameter space $(z, a; c)$ can be completely characterized. This, in turn, addresses the so-called "Learning Problem" in CNNs. The complexities of mosaic is also studied.

In joint work with Robin Balean, we have constructed a parameterisation of the space of all solutions of Maxwell's equations in an exterior domain in Minkowski space. The domain has boundary consisting of a null cone and a timelike cylinder, and the construction arises from uniqueness and existences theorems, which we establish within a certain "Lorentz" gauge. The resulting free boundary data are unconstrained.

In the realization of a dynamical system on a computer, all computational processes are of a discretization, where continuum state space is replaced by the finite set of machine arithmetic. When chaos is present, the discretized system often manifests collapsing effects to a fixed point or to short cycles. These phenomena exhibit a statistical structure which can be modelled by random mappings with an absorbing centre. This model gives results which are very much in line with computational experiments and there appears to be a type of universality summarized by an Arcsine Law. The effects are discussed with special reference to the family of mappings $f_e(x) = 1 - |1 - 2x|^l, x \in [0, 1], 1 \lt l \leq 2$. Computer experiments display close agreement with the predictions of the model.

It is known that matrix-free numerical implementations for solving stiff ordinary differential equations (ODEs) can be considerably more effective than implementations which rely on direct linear algebra techniques to solve the implicit equations governing the stage values. In this paper it will be shown how fully implicit, high order Runge-Kutta methods can be efficiently implemented in a matrix-free, parallel environment. The advantage of this is that no new parallel algorithms need be developed and that existing sequential methods that are adpated using these techniques need have no special structure (such as singly implicitness). This is demonstrated by the conversion of an existing Radau IIA method (RADAU5) to a matrix-free implementation using a dynamically pre-conditioned GMRES algorithm to solve the appropriate linear systems. Numerical results are presented for an implementation on a shared memory SGI Power Challenge and show the efficacy of this approach.

This paper concerns with the balancing theory for the system governed by diffusion partial differential equation, which is refereed here as infinite dimensional system or distributed-parameter system. Based on the eigenvalue-eigenfunction structure of Laplacian differential operator, the approximate controllability and initial observability are constructed. In order to perform the balanced realization for infinite dimensional system, a brief review on finite dimensional balancing is presented, and more intuitive meaning of balanced realization is then obtained. After defining the Hankel operator of the infinite dimensional system, we compute the Hankel singular value and construct the energy balancing. And then, the model reduction problem is discussed. In the sequel, numerical simulation of the balanced model reduction for one-dimensional heat equation is conducted.

This review discusses work in progress and related earlier studies by many authors. We have attempted to place our results in their broadcontext beginning with the $L^2$ index theorem of Atiyah and Singer, subsequent extensions and the motivation for our results and conjectures. The geometric setting is the analysis of $L^2$ invariants of non-compact covering spaces, several of which are not present (or are trivial) on compact mamfolds. These invariants use the von Neumann algebra of the covering transformation group in an essential way.

This paper first shows that the Cohen-Daubechies-Feauveau (C-D-F) biorthogonal wavelet can be derived from the interpolating wavelet through a lifting process. Its high-pass filter measures the interpolation error of the averaged data. Next, we propose a new wavelet method, called the difference wavelet method, for efficient representation for functions on R. Its analysis part is simply averaging and finite differencing. The corresponding synthesis part involves a finite difference equation, which is solved by the cyclic reduction method to achieve fast transform. The associated wavelets constitute biorthogonal Riesz bases in $L^2(R)$. Their decay and regularity properties are investigated. A comparison study on the efficiency is'sues with C-D-F wavelet method is performed. The comparison includes (i) operation counts. for performing wavelet transform, (ii) the approximate power (i.e. the coefficient in the approximation error estimate), and (iii) the compression ratio (by numerical experiments). The results show that the difference wavelet method is about twice more efficient than the C-D-F wavelet method. This efficiency is due to that the difference wavelets are more regular with just slightly bigger essential support than those of C-D-F wavelets.