32,782 research outputs found
Extension of information geometry for modelling non-statistical systems
In this dissertation, an abstract formalism extending information geometry is
introduced. This framework encompasses a broad range of modelling problems,
including possible applications in machine learning and in the information
theoretical foundations of quantum theory. Its purely geometrical foundations
make no use of probability theory and very little assumptions about the data or
the models are made. Starting only from a divergence function, a Riemannian
geometrical structure consisting of a metric tensor and an affine connection is
constructed and its properties are investigated. Also the relation to
information geometry and in particular the geometry of exponential families of
probability distributions is elucidated. It turns out this geometrical
framework offers a straightforward way to determine whether or not a
parametrised family of distributions can be written in exponential form. Apart
from the main theoretical chapter, the dissertation also contains a chapter of
examples illustrating the application of the formalism and its geometric
properties, a brief introduction to differential geometry and a historical
overview of the development of information geometry.Comment: PhD thesis, University of Antwerp, Advisors: Prof. dr. Jan Naudts and
Prof. dr. Jacques Tempere, December 2014, 108 page
Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
We develop exterior calculus approaches for partial differential equations on
radial manifolds. We introduce numerical methods that approximate with spectral
accuracy the exterior derivative , Hodge star , and their
compositions. To achieve discretizations with high precision and symmetry, we
develop hyperinterpolation methods based on spherical harmonics and Lebedev
quadrature. We perform convergence studies of our numerical exterior derivative
operator and Hodge star operator
showing each converge spectrally to and . We show how the
numerical operators can be naturally composed to formulate general numerical
approximations for solving differential equations on manifolds. We present
results for the Laplace-Beltrami equations demonstrating our approach.Comment: 22 pages, 13 figure
GPU-accelerated discontinuous Galerkin methods on hybrid meshes
We present a time-explicit discontinuous Galerkin (DG) solver for the
time-domain acoustic wave equation on hybrid meshes containing vertex-mapped
hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable
formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto
(Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions
for hybrid meshes are derived by bounding the spectral radius of the DG
operator using order-dependent constants in trace and Markov inequalities.
Computational efficiency is achieved under a combination of element-specific
kernels (including new quadrature-free operators for the pyramid), multi-rate
timestepping, and acceleration using Graphics Processing Units.Comment: Submitted to CMAM
Hadron Structure Functions in a Chiral Quark Model: Regularization, Scaling and Sum Rules
We provide a consistent regularization procedure for calculating hadron
structure functions in a chiral quark model. The structure functions are
extracted from the absorptive part of the forward Compton amplitude in the
Bjorken limit. Since this amplitude is obtained as a time-ordered correlation
function its regularization is consistently determined from the regularization
of the bosonized action. We find that the Pauli-Villars regularization scheme
is most suitable because it preserves both the anomaly structure of QCD and the
leading scaling behavior of hadron structure functions in the Bjorken limit. We
show that this procedure yields the correct pion structure function. In order
to render the sum rules of the regularized polarized nucleon structure
functions consistent with their corresponding axial charges we find it
mandatory to further specify the regularization procedure. This specification
goes beyond the double subtraction scheme commonly employed when studying
static hadron properties in this model. In particular the present approach
serves to determine the regularization prescription for structure functions
whose leading moments are not given by matrix elements of local operators. In
this regard we conclude somewhat surprisingly that in this model the Gottfried
sum rule does not undergo regularization.Comment: 42 pages LaTex, 5 figures included via epsfi
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