8,960 research outputs found
Vector bundles on curves and generalized theta functions: recent results and open problems
Riemann surface carries a natural line bundle, the determinant bundle. The
space of sections of this line bundle (or its multiples) constitutes a natural
non-abelian generalization of the spaces of theta functions on the Jacobian.
There has been much progress in the last few years towards a better
understanding of these spaces, including a rigorous proof of the celebrated
Verlinde formula which gives their dimension. This survey paper tries to
explain what is now known and what remains open.Comment: 15 pages, Plain Te
Decision Making for Rapid Information Acquisition in the Reconnaissance of Random Fields
Research into several aspects of robot-enabled reconnaissance of random
fields is reported. The work has two major components: the underlying theory of
information acquisition in the exploration of unknown fields and the results of
experiments on how humans use sensor-equipped robots to perform a simulated
reconnaissance exercise.
The theoretical framework reported herein extends work on robotic exploration
that has been reported by ourselves and others. Several new figures of merit
for evaluating exploration strategies are proposed and compared. Using concepts
from differential topology and information theory, we develop the theoretical
foundation of search strategies aimed at rapid discovery of topological
features (locations of critical points and critical level sets) of a priori
unknown differentiable random fields. The theory enables study of efficient
reconnaissance strategies in which the tradeoff between speed and accuracy can
be understood. The proposed approach to rapid discovery of topological features
has led in a natural way to to the creation of parsimonious reconnaissance
routines that do not rely on any prior knowledge of the environment. The design
of topology-guided search protocols uses a mathematical framework that
quantifies the relationship between what is discovered and what remains to be
discovered. The quantification rests on an information theory inspired model
whose properties allow us to treat search as a problem in optimal information
acquisition. A central theme in this approach is that "conservative" and
"aggressive" search strategies can be precisely defined, and search decisions
regarding "exploration" vs. "exploitation" choices are informed by the rate at
which the information metric is changing.Comment: 34 pages, 20 figure
The Yangian Symmetry of the Hubbard Models with Variable Range Hopping
We present two pairs of Y() Yangian symmetries for the trigonometric
and hyperbolic versions of the Hubbard model with non-nearest-neighbour
hopping. In both cases the Yangians are mutually commuting, hence can be
combined into a Y()Y() Yangian. Their mutual commutativity
is of dynamical origin. The known Yangians of the Haldane-Shastry spin chain
and the nearest neighbour Hubbard model are contained as limiting cases of our
new representations.Comment: 10 pages, Late
Real-valued, time-periodic localized weak solutions for a semilinear wave equation with periodic potentials
We consider the semilinear wave equation for three different classes (P1), (P2), (P3) of periodic potentials
. (P1) consists of periodically extended delta-distributions, (P2) of
periodic step potentials and (P3) contains certain periodic potentials V,q\in
H^r_{\per}(\R) for . Among other assumptions we suppose that
for some and . In each class we can find
suitable potentials that give rise to a critical exponent such that
for both in the "+" and the "-" case we can use variational
methods to prove existence of time-periodic real-valued solutions that are
localized in the space direction. The potentials are constructed explicitely in
class (P1) and (P2) and are found by a recent result from inverse spectral
theory in class (P3). The critical exponent depends on the regularity
of . Our result builds upon a Fourier expansion of the solution and a
detailed analysis of the spectrum of the wave operator. In fact, it turns out
that by a careful choice of the potentials and the spatial and temporal
periods, the spectrum of the wave operator
(considered on suitable space of time-periodic functions) is bounded away from
. This allows to find weak solutions as critical points of a functional on a
suitable Hilbert space and to apply tools for strongly indefinite variational
problems
Large sets of consecutive Maass forms and fluctuations in the Weyl remainder
We explore an algorithm which systematically finds all discrete eigenvalues
of an analytic eigenvalue problem. The algorithm is more simple and elementary
as could be expected before. It consists of Hejhal's identity, linearisation,
and Turing bounds. Using the algorithm, we compute more than one hundredsixty
thousand consecutive eigenvalues of the Laplacian on the modular surface, and
investigate the asymptotic and statistic properties of the fluctuations in the
Weyl remainder. We summarize the findings in two conjectures. One is on the
maximum size of the Weyl remainder, and the other is on the distribution of a
suitably scaled version of the Weyl remainder.Comment: A version with higher resolution figures can be downloaded from
http://www.maths.bris.ac.uk/~mahlt/research/T2012a.pd
From quantum stochastic differential equations to Gisin-Percival state diffusion
Starting from the quantum stochastic differential equations of Hudson and
Parthasarathy (Comm. Math. Phys. 93, 301 (1984)) and exploiting the
Wiener-Ito-Segal isomorphism between the Boson Fock reservoir space
and
the Hilbert space , where is the Wiener probability measure of
a complex -dimensional vector-valued standard Brownian motion
, we derive a non-linear stochastic Schrodinger
equation describing a classical diffusion of states of a quantum system, driven
by the Brownian motion . Changing this Brownian motion by an
appropriate Girsanov transformation, we arrive at the Gisin-Percival state
diffusion equation (J. Phys. A, 167, 315 (1992)). This approach also yields an
explicit solution of the Gisin-Percival equation, in terms of the
Hudson-Parthasarathy unitary process and a radomized Weyl displacement process.
Irreversible dynamics of system density operators described by the well-known
Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by
coarse-graining over the Gisin-Percival quantum state trajectories.Comment: 28 pages, one pdf figure. An error in the multiplying factor in Eq.
(102) corrected. To appear in Journal of Mathematical Physic
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