135 research outputs found
Finite Fields: Theory and Applications
Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation
Eisenstein series and automorphic representations
We provide an introduction to the theory of Eisenstein series and automorphic
forms on real simple Lie groups G, emphasising the role of representation
theory. It is useful to take a slightly wider view and define all objects over
the (rational) adeles A, thereby also paving the way for connections to number
theory, representation theory and the Langlands program. Most of the results we
present are already scattered throughout the mathematics literature but our
exposition collects them together and is driven by examples. Many interesting
aspects of these functions are hidden in their Fourier coefficients with
respect to unipotent subgroups and a large part of our focus is to explain and
derive general theorems on these Fourier expansions. Specifically, we give
complete proofs of the Langlands constant term formula for Eisenstein series on
adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic
spherical Whittaker function associated to unramified automorphic
representations of G(Q_p). In addition, we explain how the classical theory of
Hecke operators fits into the modern theory of automorphic representations of
adelic groups, thereby providing a connection with some key elements in the
Langlands program, such as the Langlands dual group LG and automorphic
L-functions. Somewhat surprisingly, all these results have natural
interpretations as encoding physical effects in string theory. We therefore
also introduce some basic concepts of string theory, aimed toward
mathematicians, emphasising the role of automorphic forms. In particular, we
provide a detailed treatment of supersymmetry constraints on string amplitudes
which enforce differential equations of the same type that are satisfied by
automorphic forms. Our treatise concludes with a detailed list of interesting
open questions and pointers to additional topics which go beyond the scope of
this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with
highlighted applications to string theory. v2: 375 pages. Substantially
extended and small correction
Introduction to Modern Canonical Quantum General Relativity
This is an introduction to the by now fifteen years old research field of
canonical quantum general relativity, sometimes called "loop quantum gravity".
The term "modern" in the title refers to the fact that the quantum theory is
based on formulating classical general relativity as a theory of connections
rather than metrics as compared to in original version due to Arnowitt, Deser
and Misner. Canonical quantum general relativity is an attempt to define a
mathematically rigorous, non-perturbative, background independent theory of
Lorentzian quantum gravity in four spacetime dimensions in the continuum. The
approach is minimal in that one simply analyzes the logical consequences of
combining the principles of general relativity with the principles of quantum
mechanics. The requirement to preserve background independence has lead to new,
fascinating mathematical structures which one does not see in perturbative
approaches, e.g. a fundamental discreteness of spacetime seems to be a
prediction of the theory providing a first substantial evidence for a theory in
which the gravitational field acts as a natural UV cut-off. An effort has been
made to provide a self-contained exposition of a restricted amount of material
at the appropriate level of rigour which at the same time is accessible to
graduate students with only basic knowledge of general relativity and quantum
field theory on Minkowski space.Comment: 301 pages, Latex; based in part on the author's Habilitation Thesis
"Mathematische Formulierung der Quanten-Einstein-Gleichungen", University of
Potsdam, Potsdam, Germany, January 2000; submitted to the on-line journal
Living Reviews; subject to being updated on at least a bi-annual basi
Vacuum Energy in Modern Cosmology: an analysis of quantum field theory in curved spaces and its application to cosmological spacetimes
The last decades have witnessed an unprecedented advancement in our knowledge
of the large scale universe. In particular, increasingly accurate cosmological
observations have allowed us to discover a form of "dark energy", which
presently dominates the expansion of the universe. On the other hand,
fundamental problems in the standard cosmological model point towards the
possibility of a primordial inflationary period. Both these expansion phases
have in common the fact that they should be governed by forms of energy with
properties much similar to those of vacuum energy of classical or quantum
fields. In the meanwhile, quantum field theory in curved spaces (QFTCS) has
proved a rich framework to analyze phenomena of a quantum nature in regimes
where spacetime curvature is relevant, but not too extreme; particularly, it
yields novel insights on the structure and dynamics of quantum vacuum. In this
dissertation, we make a thorough exposition of the fundamentals of QFTCS and
present some of its applications in cosmological spacetimes. Particular
attention is given to the construction of an empirical notion of particles
through an idealized model of particle detectors, and to the phenomenon of
particle creation in expanding FLRW spacetimes. Further, we develop the
procedure of adiabatic renormalization, and use it to compute the renormalized
stress tensor in these spacetimes. For a noninteracting scalar field in de
Sitter spaces, we find that it takes the form of a cosmological constant,
although a quantitatively self-consistent value with the background expansion
can only be found at Planckian densities. We also present a construction of a
simple inflationary model, driven by a self-interacting classical scalar field,
and show how the quantized fluctuations of this field could give rise to a
nearly scale-invariant power spectrum, like the one that is currently observed
in the CMB.Comment: 246 page
Soft Processing Techniques for Quantum Key Distribution Applications
This thesis deals with soft-information based information reconciliation and data sifting for
Quantum Key Distribution (QKD). A novel composite channel model for QKD is identified, which
includes both a hard output quantum channel and a soft output classic channel. The Log-Likelihood
Ratios, - also called soft-metrics - derived from the two channels are jointly processed at the receiver,
exploiting capacity achieving soft-metric based iteratively decoded block codes. The performance
of the proposed mixed-soft-metric algorithms are studied via simulations as a function of the system
parameters.
The core ideas of the thesis are employing Forward Error Correction (FEC) coding as opposed to
two-way communication for information reconciliation in QKD schemes, exploiting all the available
information for data processing at the receiver including information available from the quantum
channel, since optimized use of this information can lead to significant performance improvement,
and providing a security versus secret-key rate trade-off to the end-user within the context of QKD
systems
Sparse graph codes for compression, sensing, and secrecy
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from student PDF version of thesis.Includes bibliographical references (p. 201-212).Sparse graph codes were first introduced by Gallager over 40 years ago. Over the last two decades, such codes have been the subject of intense research, and capacity approaching sparse graph codes with low complexity encoding and decoding algorithms have been designed for many channels. Motivated by the success of sparse graph codes for channel coding, we explore the use of sparse graph codes for four other problems related to compression, sensing, and security. First, we construct locally encodable and decodable source codes for a simple class of sources. Local encodability refers to the property that when the original source data changes slightly, the compression produced by the source code can be updated easily. Local decodability refers to the property that a single source symbol can be recovered without having to decode the entire source block. Second, we analyze a simple message-passing algorithm for compressed sensing recovery, and show that our algorithm provides a nontrivial f1/f1 guarantee. We also show that very sparse matrices and matrices whose entries must be either 0 or 1 have poor performance with respect to the restricted isometry property for the f2 norm. Third, we analyze the performance of a special class of sparse graph codes, LDPC codes, for the problem of quantizing a uniformly random bit string under Hamming distortion. We show that LDPC codes can come arbitrarily close to the rate-distortion bound using an optimal quantizer. This is a special case of a general result showing a duality between lossy source coding and channel coding-if we ignore computational complexity, then good channel codes are automatically good lossy source codes. We also prove a lower bound on the average degree of vertices in an LDPC code as a function of the gap to the rate-distortion bound. Finally, we construct efficient, capacity-achieving codes for the wiretap channel, a model of communication that allows one to provide information-theoretic, rather than computational, security guarantees. Our main results include the introduction of a new security critertion which is an information-theoretic analog of semantic security, the construction of capacity-achieving codes possessing strong security with nearly linear time encoding and decoding algorithms for any degraded wiretap channel, and the construction of capacity-achieving codes possessing semantic security with linear time encoding and decoding algorithms for erasure wiretap channels. Our analysis relies on a relatively small set of tools. One tool is density evolution, a powerful method for analyzing the behavior of message-passing algorithms on long, random sparse graph codes. Another concept we use extensively is the notion of an expander graph. Expander graphs have powerful properties that allow us to prove adversarial, rather than probabilistic, guarantees for message-passing algorithms. Expander graphs are also useful in the context of the wiretap channel because they provide a method for constructing randomness extractors. Finally, we use several well-known isoperimetric inequalities (Harper's inequality, Azuma's inequality, and the Gaussian Isoperimetric inequality) in our analysis of the duality between lossy source coding and channel coding.by Venkat Bala Chandar.Ph.D
Proceedings of the 35th WIC Symposium on Information Theory in the Benelux and the 4th joint WIC/IEEE Symposium on Information Theory and Signal Processing in the Benelux, Eindhoven, the Netherlands May 12-13, 2014
Compressive sensing (CS) as an approach for data acquisition has recently received much attention. In CS, the signal recovery problem from the observed data requires the solution of a sparse vector from an underdetermined system of equations. The underlying sparse signal recovery problem is quite general with many applications and is the focus of this talk. The main emphasis will be on Bayesian approaches for sparse signal recovery. We will examine sparse priors such as the super-Gaussian and student-t priors and appropriate MAP estimation methods. In particular, re-weighted l2 and re-weighted l1 methods developed to solve the optimization problem will be discussed. The talk will also examine a hierarchical Bayesian framework and then study in detail an empirical Bayesian method, the Sparse Bayesian Learning (SBL) method. If time permits, we will also discuss Bayesian methods for sparse recovery problems with structure; Intra-vector correlation in the context of the block sparse model and inter-vector correlation in the context of the multiple measurement vector problem
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