The Dirichlet Problem for Harmonic Functions on Compact Sets

For any compact set $K\subset \mathbb{R}^n$ we develop the theory of Jensen measures and subharmonic peak points, which form the set $\mathcal{O}_K$, to study the Dirichlet problem on $K$. Initially we consider the space $h(K)$ of functions on $K$ which can be uniformly approximated by functions harmonic in a neighborhood of $K$ as possible solutions. As in the classical theory, our Theorem 8.1 shows $C(\mathcal{O}_K)\cong h(K)$ for compact sets with $\mathcal{O}_K$ closed. However, in general a continuous solution cannot be expected even for continuous data on \rO_K as illustrated by Theorem 8.1. Consequently, we show that the solution can be found in a class of finely harmonic functions. Moreover by Theorem 8.7, in complete analogy with the classical situation, this class is isometrically isomorphic to $C_b(\mathcal{O}_K)$ for all compact sets $K$.Comment: There have been a large number of changes made from the first version. They mostly consists of shortening the article and supplying additional reference

Arithmetic based fractals associated with Pascal's triangle

Our goal is to study Pascal-Sierpinski gaskets, which are certain fractal sets defined in terms of divisibility of entries in Pascal's triangle. The principal tool is a "carry rule" for the addition of the base-q representation of coordinates of points in the unit square. In the case that q = p is prime, we connect the carry rule to the power of p appearing in the prime factorization of binomialcoefficients. We use the carry rule to define a family of fractal subsets Bqr of the unit square, and we show that when q = p is prime, Bqr coincides with the Pascal-Sierpinski gasket corresponding to N = pr . We go on to describe Bqr as the limit of an iterated function system of "partial similarities", and we determine its Hausdorff dimension. We consider also the corresponding fractal sets in higher-dimensional Euclidean space

Complementarity of representations in quantum mechanics

We show that Bohr's principle of complementarity between position and momentum descriptions can be formulated rigorously as a claim about the existence of representations of the CCRs. In particular, in any representation where the position operator has eigenstates, there is no momentum operator, and vice versa. Equivalently, if there are nonzero projections corresponding to sharp position values, all spectral projections of the momentum operator map onto the zero element.Comment: 14 pages, LaTe