4D and 2D Evaporating Dilatonic Black Holes

The picture of S-wave scatering from a 4D extremal dilatonic black hole is examined. Classically, a small matter shock wave will form a non-extremal black hole. In the "throat region" the r-t geometry is exactly that of a collapsing 2D black hole. The 4D Hawking radiation (in this classical background) gives the 2D Hawking radiation exactly in the throat region. Inclusion of the back-reaction changes this picture: the 4D solution can then be matched to the 2D one only if the Hawking radiation is very small and only at the beginning of the radiation. We give that 4D solution. When the total radiating energy approaches the energy carried by the shock wave, the 4D picture breaks down. This happens even before an apparent horizon is formed, which suggests that the 4D semi-classical solution is quite different from the 2D one.Comment: 18 pages, BRX-TH-34

Relaxed spanners for directed disk graphs

Let $(V,\delta)$ be a finite metric space, where $V$ is a set of $n$ points and $\delta$ is a distance function defined for these points. Assume that $(V,\delta)$ has a constant doubling dimension $d$ and assume that each point $p\in V$ has a disk of radius $r(p)$ around it. The disk graph that corresponds to $V$ and $r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices are the points of $V$ and whose edge set includes a directed edge from $p$ to $q$ if $\delta(p,q)\leq r(p)$. In \cite{PeRo08} we presented an algorithm for constructing a (1+\eps)-spanner of size O(n/\eps^d \log M), where $M$ is the maximal radius $r(p)$. The current paper presents two results. The first shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of $M$. The second result shows that by slightly relaxing the requirements and allowing a small perturbation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph I(V,E,r_{1+\eps}), where r_{1+\eps}(p) = (1+\eps)\cdot r(p) for every $p\in V$, then it is possible to get a (1+\eps)-spanner of size O(n/\eps^d) for $I(V,E,r)$. Our algorithm is simple and can be implemented efficiently

Performance Guarantees of the Thresholding Algorithm for the Co-Sparse Analysis Model

The co-sparse analysis model for signals assumes that the signal of interest can be multiplied by an analysis dictionary \Omega, leading to a sparse outcome. This model stands as an interesting alternative to the more classical synthesis based sparse representation model. In this work we propose a theoretical study of the performance guarantee of the thresholding algorithm for the pursuit problem in the presence of noise. Our analysis reveals two significant properties of \Omega, which govern the pursuit performance: The first is the degree of linear dependencies between sets of rows in \Omega, depicted by the co-sparsity level. The second property, termed the Restricted Orthogonal Projection Property (ROPP), is the level of independence between such dependent sets and other rows in \Omega. We show how these dictionary properties are meaningful and useful, both in the theoretical bounds derived, and in a series of experiments that are shown to align well with the theoretical prediction.Comment: submitted to IEEE Trans. on Information Theor