On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing

We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every direction in the complex plane at the singularity (Stokes directions {\em not} excepted) there exists a nonempty set of solutions whose difference from the ``optimally'' (i.e., near the least term) truncated asymptotic series is of the same (exponentially small) order of magnitude as the least term of the series. There is a family of generalized Borel summation formulas $\mathcal{B}$ which commute with the usual algebraic and analytic operations (addition, multiplication, differentiation, etc). We show that there is exactly one of them, $\mathcal{B}_0$, such that for any formal series solution $\tilde{f}$, $\mathcal{B}_0(\tilde{f})$ differs from the optimal truncation of $\tilde{f}$ by at most the order of the least term of $\tilde{f}$. We show in addition that the Berry (1989) smoothing phenomenon is universal within this class of differential systems. Whenever the terms ``beyond all orders'' {\em change} in crossing a Stokes line, these terms vary smoothly on the Berry scale $\arg(x)\sim |x|^{-1/2}$ and the transition is always given by the error function; under the same conditions we show that Dingle's rule of signs for Stokes transitions holds

Clusters from higher order correlations

Given a set of variables and the correlations among them, we develop a method for finding clustering among the variables. The method takes advantage of information implicit in higher-order (not just pairwise) correlations. The idea is to define a Potts model whose energy is based on the correlations. Each state of this model is a partition of the variables and a Monte Carlo method is used to identify states of lowest energy, those most consistent with the correlations. A set of the 100 or so lowest such partitions is then used to construct a stochastic dynamics (using the adjacency matrix of each partition) whose observable representation gives the clustering. Three examples are studied. For two of them the 3$^\mathrm{rd}$ order correlations are significant for getting the clusters right. The last of these is a toy model of a biological system in which the joint action of several genes or proteins is necessary to accomplish a given process

A search for dark matter with bottom quarks

Despite making up over 80% of the matter in the universe, very little is known about dark matter. Its only well-established property is that it interacts gravitationally, but does not interact with ordinary matter through any of the other known forces. Specific details such as the number of dark matter particles, their quantum properties, and their interactions remain elusive and are only loosely constrained by experiments. In this dissertation I describe a novel search for a particular type of dark matter that couples preferentially to heavy quarks, using LHC proton-proton collisions at ATLAS. With a model-independent framework, comparisons are made to results obtained from other dark matter searches, and new limits are set on various interaction strengths

The Camassa-Holm Equation: Conserved Quantities and the Initial Value Problem

Using a Miura-Gardner-Kruskal type construction, we show that the Camassa-Holm equation has an infinite number of local conserved quantities. We explore the implications of these conserved quantities for global well-posedness.Comment: 8 pages, LaTe

Bi-differential calculus and the KdV equation

A gauged bi-differential calculus over an associative (and not necessarily commutative) algebra A is an N-graded left A-module with two covariant derivatives acting on it which, as a consequence of certain (e.g., nonlinear differential) equations, are flat and anticommute. As a consequence, there is an iterative construction of generalized conserved currents. We associate a gauged bi-differential calculus with the Korteweg-de-Vries equation and use it to compute conserved densities of this equation.Comment: 9 pages, LaTeX, uses amssymb.sty, XXXI Symposium on Mathematical Physics, Torun, May 1999, replaces "A notion of complete integrability in noncommutative geometry and the Korteweg-de-Vries equation

Global solutions in gravity

The method of conformal blocks for construction of global solutions in gravity for a two-dimensional metric having one Killing vector field is described.Comment: 4 pages, 2 figures, minor change