A Dialogue of Multipoles: Matched Asymptotic Expansion for Caged Black Holes

No analytic solution is known to date for a black hole in a compact dimension. We develop an analytic perturbation theory where the small parameter is the size of the black hole relative to the size of the compact dimension. We set up a general procedure for an arbitrary order in the perturbation series based on an asymptotic matched expansion between two coordinate patches: the near horizon zone and the asymptotic zone. The procedure is ordinary perturbation expansion in each zone, where additionally some boundary data comes from the other zone, and so the procedure alternates between the zones. It can be viewed as a dialogue of multipoles where the black hole changes its shape (mass multipoles) in response to the field (multipoles) created by its periodic "mirrors", and that in turn changes its field and so on. We present the leading correction to the full metric including the first correction to the area-temperature relation, the leading term for black hole eccentricity and the "Archimedes effect". The next order corrections will appear in a sequel. On the way we determine independently the static perturbations of the Schwarzschild black hole in dimension d>=5, where the system of equations can be reduced to "a master equation" - a single ordinary differential equation. The solutions are hypergeometric functions which in some cases reduce to polynomials.Comment: 47 pages, 12 figures, minor corrections described at the end of the introductio

Fast Searching in Packed Strings

Given strings $P$ and $Q$ the (exact) string matching problem is to find all positions of substrings in $Q$ matching $P$. The classical Knuth-Morris-Pratt algorithm [SIAM J. Comput., 1977] solves the string matching problem in linear time which is optimal if we can only read one character at the time. However, most strings are stored in a computer in a packed representation with several characters in a single word, giving us the opportunity to read multiple characters simultaneously. In this paper we study the worst-case complexity of string matching on strings given in packed representation. Let $m \leq n$ be the lengths $P$ and $Q$, respectively, and let $\sigma$ denote the size of the alphabet. On a standard unit-cost word-RAM with logarithmic word size we present an algorithm using time O\left(\frac{n}{\log_\sigma n} + m + \occ\right). Here \occ is the number of occurrences of $P$ in $Q$. For $m = o(n)$ this improves the $O(n)$ bound of the Knuth-Morris-Pratt algorithm. Furthermore, if $m = O(n/\log_\sigma n)$ our algorithm is optimal since any algorithm must spend at least \Omega(\frac{(n+m)\log \sigma}{\log n} + \occ) = \Omega(\frac{n}{\log_\sigma n} + \occ) time to read the input and report all occurrences. The result is obtained by a novel automaton construction based on the Knuth-Morris-Pratt algorithm combined with a new compact representation of subautomata allowing an optimal tabulation-based simulation.Comment: To appear in Journal of Discrete Algorithms. Special Issue on CPM 200

Purcell Enhancement of Parametric Luminescence: Bright and Broadband Nonlinear Light Emission in Metamaterials

Single-photon and correlated two-photon sources are important elements for optical information systems. Nonlinear downconversion light sources are robust and stable emitters of single photons and entangled photon pairs. However, the rate of downconverted light emission, dictated by the properties of low-symmetry nonlinear crystals, is typically very small, leading to significant constrains in device design and integration. In this paper, we show that the principles for spontaneous emission control (i.e. Purcell effect) of isolated emitters in nanoscale structures, such as metamaterials, can be generalized to describe the enhancement of nonlinear light generation processes such as parametric down conversion. We develop a novel theoretical framework for quantum nonlinear emission in a general anisotropic, dispersive and lossy media. We further find that spontaneous parametric downconversion in media with hyperbolic dispersion is broadband and phase-mismatch-free. We predict a 1000-fold enhancement of the downconverted emission rate with up to 105 photon pairs per second in experimentally realistic nanostructures. Our theoretical formalism and approach to Purcell enhancement of nonlinear optical processes, provides a framework for description of quantum nonlinear optical phenomena in complex nanophotonic structures.Comment: 29 pages, 10 figure

Semiparametric estimation of shifts on compact Lie groups for image registration

In this paper we focus on estimating the deformations that may exist between similar images in the presence of additive noise when a reference template is unknown. The deformations aremodeled as parameters lying in a finite dimensional compact Lie group. A generalmatching criterion based on the Fourier transformand itswell known shift property on compact Lie groups is introduced. M-estimation and semiparametric theory are then used to study the consistency and asymptotic normality of the resulting estimators. As Lie groups are typically nonlinear spaces, our tools rely on statistical estimation for parameters lying in a manifold and take into account the geometrical aspects of the problem. Some simulations are used to illustrate the usefulness of our approach and applications to various areas in image processing are discussed

Pattern matching in compilers

In this thesis we develop tools for effective and flexible pattern matching. We introduce a new pattern matching system called amethyst. Amethyst is not only a generator of parsers of programming languages, but can also serve as an alternative to tools for matching regular expressions. Our framework also produces dynamic parsers. Its intended use is in the context of IDE (accurate syntax highlighting and error detection on the fly). Amethyst offers pattern matching of general data structures. This makes it a useful tool for implementing compiler optimizations such as constant folding, instruction scheduling, and dataflow analysis in general. The parsers produced are essentially top-down parsers. Linear time complexity is obtained by introducing the novel notion of structured grammars and regularized regular expressions. Amethyst uses techniques known from compiler optimizations to produce effective parsers.Comment: master thesi