7,130 research outputs found
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
which commute with the usual algebraic and analytic operations (addition,
multiplication, differentiation, etc). We show that there is exactly one of
them, , such that for any formal series solution ,
differs from the optimal truncation of
by at most the order of the least term of . 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
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 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
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