18,476 research outputs found
On Network Coding Capacity - Matroidal Networks and Network Capacity Regions
One fundamental problem in the field of network coding is to determine the
network coding capacity of networks under various network coding schemes. In
this thesis, we address the problem with two approaches: matroidal networks and
capacity regions.
In our matroidal approach, we prove the converse of the theorem which states
that, if a network is scalar-linearly solvable then it is a matroidal network
associated with a representable matroid over a finite field. As a consequence,
we obtain a correspondence between scalar-linearly solvable networks and
representable matroids over finite fields in the framework of matroidal
networks. We prove a theorem about the scalar-linear solvability of networks
and field characteristics. We provide a method for generating scalar-linearly
solvable networks that are potentially different from the networks that we
already know are scalar-linearly solvable.
In our capacity region approach, we define a multi-dimensional object, called
the network capacity region, associated with networks that is analogous to the
rate regions in information theory. For the network routing capacity region, we
show that the region is a computable rational polytope and provide exact
algorithms and approximation heuristics for computing the region. For the
network linear coding capacity region, we construct a computable rational
polytope, with respect to a given finite field, that inner bounds the linear
coding capacity region and provide exact algorithms and approximation
heuristics for computing the polytope. The exact algorithms and approximation
heuristics we present are not polynomial time schemes and may depend on the
output size.Comment: Master of Engineering Thesis, MIT, September 2010, 70 pages, 10
figure
Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation
Let be a positive integer. In this paper we provide a general theory to
produce full orbit sequences in the affine -dimensional space over a finite
field. For our construction covers the case of the Inversive Congruential
Generators (ICG). In addition, for we show that the sequences produced
using our construction are easier to compute than ICG sequences. Furthermore,
we prove that they have the same discrepancy bounds as the ones constructed
using the ICG.Comment: To appear in Mathematics of Computatio
p-Adic Mathematical Physics
A brief review of some selected topics in p-adic mathematical physics is
presented.Comment: 36 page
Wet paper codes and the dual distance in steganography
In 1998 Crandall introduced a method based on coding theory to secretly embed
a message in a digital support such as an image. Later Fridrich et al. improved
this method to minimize the distortion introduced by the embedding; a process
called wet paper. However, as previously emphasized in the literature, this
method can fail during the embedding step. Here we find sufficient and
necessary conditions to guarantee a successful embedding by studying the dual
distance of a linear code. Since these results are essentially of combinatorial
nature, they can be generalized to systematic codes, a large family containing
all linear codes. We also compute the exact number of solutions and point out
the relationship between wet paper codes and orthogonal arrays
Perturbing an axisymmetric magnetic equilibrium to obtain a quasi-axisymmetric stellarator
It is demonstrated that finite-pressure, approximately quasi-axisymmetric
stellarator equilibria can be directly constructed (without numerical
optimization) via perturbations of given axisymmetric equilibria. The size of
such perturbations is measured in two ways, via the fractional external
rotation and, alternatively, via the relative magnetic field strength, i.e. the
average size of the perturbed magnetic field, divided by the unperturbed field
strength. It is found that significant fractional external rotational transform
can be generated by quasi-axisymmetric perturbations, with a similar value of
the relative field strength, despite the fact that the former scales more
weakly with the perturbation size. High mode number perturbations are
identified as a candidate for generating such transform with local current
distributions. Implications for the development of a general non-perturbative
solver for optimal stellarator equilibria is discussed
Verification of Gyrokinetic codes: theoretical background and applications
In fusion plasmas the strong magnetic field allows the fast gyro-motion to be
systematically removed from the description of the dynamics, resulting in a
considerable model simplification and gain of computational time. Nowadays, the
gyrokinetic (GK) codes play a major role in the understanding of the
development and the saturation of turbulence and in the prediction of the
subsequent transport. Naturally, these codes require thorough verification and
validation.
Here we present a new and generic theoretical framework and specific
numerical applications to test the faithfulness of the implemented models to
theory and to verify the domain of applicability of existing GK codes. For a
sound verification process, the underlying theoretical GK model and the
numerical scheme must be considered at the same time, which has rarely been
done and therefore makes this approach pioneering. At the analytical level, the
main novelty consists in using advanced mathematical tools such as variational
formulation of dynamics for systematization of basic GK code's equations to
access the limits of their applicability. The verification of numerical scheme
is proposed via the benchmark effort.
In this work, specific examples of code verification are presented for two GK
codes: the multi-species electromagnetic ORB5 (PIC) and the radially global
version of GENE (Eulerian). The proposed methodology can be applied to any
existing GK code. We establish a hierarchy of reduced GK Vlasov-Maxwell
equations implemented in the ORB5 and GENE codes using the Lagrangian
variational formulation. At the computational level, detailed verifications of
global electromagnetic test cases developed from the CYCLONE Base Case are
considered, including a parametric -scan covering the transition from
ITG to KBM and the spectral properties at the nominal value.Comment: 16 pages, 2 Figures, APS DPP 2016 invited pape
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