C. Van

Alexander G. Ramm, C. Van

In this paper the theory is developed for creating a material in which the heat is transmitted along a given line. This gives a possibility to transfer information using heat signals. This seems to be a novel idea. The technical part of the theory is the construction of the potential $q(x)$. This potential describes the heat equation $u_t = Δu - q(x)u$ in the limiting medium which is obtained after the small impedance particles are distributed in a given domain. A numerical method is also established to construct numerically such a potential.

C. Van

We consider the 3D inverse scattering problem with non-over-determined scattering data. The data are the scattering amplitude $A(β, α_0, k)$ for all $β\in S_β^2$, where $S_β^2$ is an open subset of the unit sphere $S^2$ in $\mathbb{R}^3$, $α_0 \in S^2$ is fixed, and for all $k \in (a,b), 0 \leq a < b$. The basic uniqueness theorem for this problem belongs to Ramm \cite{R603}. In this paper, a numerical method is given for solving this problem and the numerical results are presented.

A. G. Ramm, C. Van

Suppose the data consist of a set $S$ of points $x_j, 1 \leq j \leq J$, distributed in a bounded domain $D \subset R^N$, where $N$ and $J$ are large numbers. In this paper an algorithm is proposed for checking whether there exists a manifold $\mathbb{M}$ of low dimension near which many of the points of $S$ lie and finding such $\mathbb{M}$ if it exists. There are many dimension reduction algorithms, both linear and non-linear. Our algorithm is simple to implement and has some advantages compared with the known algorithms. If there is a manifold of low dimension near which most of the data points lie, the proposed algorithm will find it. Some numerical results are presented illustrating the algorithm and analyzing its performance compared to the classical PCA (principal component analysis) and Isomap.