5,648 research outputs found
Deterministic Constructions of Binary Measurement Matrices from Finite Geometry
Deterministic constructions of measurement matrices in compressed sensing
(CS) are considered in this paper. The constructions are inspired by the recent
discovery of Dimakis, Smarandache and Vontobel which says that parity-check
matrices of good low-density parity-check (LDPC) codes can be used as
{provably} good measurement matrices for compressed sensing under
-minimization. The performance of the proposed binary measurement
matrices is mainly theoretically analyzed with the help of the analyzing
methods and results from (finite geometry) LDPC codes. Particularly, several
lower bounds of the spark (i.e., the smallest number of columns that are
linearly dependent, which totally characterizes the recovery performance of
-minimization) of general binary matrices and finite geometry matrices
are obtained and they improve the previously known results in most cases.
Simulation results show that the proposed matrices perform comparably to,
sometimes even better than, the corresponding Gaussian random matrices.
Moreover, the proposed matrices are sparse, binary, and most of them have
cyclic or quasi-cyclic structure, which will make the hardware realization
convenient and easy.Comment: 12 pages, 11 figure
Parsing a sequence of qubits
We develop a theoretical framework for frame synchronization, also known as
block synchronization, in the quantum domain which makes it possible to attach
classical and quantum metadata to quantum information over a noisy channel even
when the information source and sink are frame-wise asynchronous. This
eliminates the need of frame synchronization at the hardware level and allows
for parsing qubit sequences during quantum information processing. Our
framework exploits binary constant-weight codes that are self-synchronizing.
Possible applications may include asynchronous quantum communication such as a
self-synchronizing quantum network where one can hop into the channel at any
time, catch the next coming quantum information with a label indicating the
sender, and reply by routing her quantum information with control qubits for
quantum switches all without assuming prior frame synchronization between
users.Comment: 11 pages, 2 figures, 1 table. Final accepted version for publication
in the IEEE Transactions on Information Theor
Half-BPS M2-brane orbifolds
Smooth Freund-Rubin backgrounds of eleven-dimensional supergravity of the
form AdS_4 x X^7 and preserving at least half of the supersymmetry have been
recently classified. Requiring that amount of supersymmetry forces X to be a
spherical space form, whence isometric to the quotient of the round 7-sphere by
a freely-acting finite subgroup of SO(8). The classification is given in terms
of ADE subgroups of the quaternions embedded in SO(8) as the graph of an
automorphism. In this paper we extend this classification by dropping the
requirement that the background be smooth, so that X is now allowed to be an
orbifold of the round 7-sphere. We find that if the background preserves more
than half of the supersymmetry, then it is automatically smooth in accordance
with the homogeneity conjecture, but that there are many half-BPS orbifolds,
most of them new. The classification is now given in terms of pairs of ADE
subgroups of quaternions fibred over the same finite group. We classify such
subgroups and then describe the resulting orbifolds in terms of iterated
quotients. In most cases the resulting orbifold can be described as a sequence
of cyclic quotients.Comment: 51 pages; v3: substantial revision (20% longer): we had missed some
cases, but the paper now includes a check of our results via comparison with
extant classification of finite subgroups of SO(4
A Short Survey of Noncommutative Geometry
We give a survey of selected topics in noncommutative geometry, with some
emphasis on those directly related to physics, including our recent work with
Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at
length two issues. The first is the relevance of the paradigm of geometric
space, based on spectral considerations, which is central in the theory. As a
simple illustration of the spectral formulation of geometry in the ordinary
commutative case, we give a polynomial equation for geometries on the four
dimensional sphere with fixed volume. The equation involves an idempotent e,
playing the role of the instanton, and the Dirac operator D. It expresses the
gamma five matrix as the pairing between the operator theoretic chern
characters of e and D. It is of degree five in the idempotent and four in the
Dirac operator which only appears through its commutant with the idempotent. It
determines both the sphere and all its metrics with fixed volume form.
We also show using the noncommutative analogue of the Polyakov action, how to
obtain the noncommutative metric (in spectral form) on the noncommutative tori
from the formal naive metric. We conclude on some questions related to string
theory.Comment: Invited lecture for JMP 2000, 45
Locally conformal parallel and manifolds
We characterize compact locally conformal parallel (respectively,
) manifolds as fiber bundles over with compact nearly K\"ahler
(respectively, compact nearly parallel ) fiber. A more specific
characterization is provided when the local parallel structures are flat.Comment: References update
On Guichard's nets and Cyclic systems
In the first part, we give a self contained introduction to the theory of
cyclic systems in n-dimensional space which can be considered as immersions
into certain Grassmannians. We show how the (metric) geometries on spaces of
constant curvature arise as subgeometries of Moebius geometry which provides a
slightly new viewpoint. In the second part we characterize Guichard nets which
are given by cyclic systems as being Moebius equivalent to 1-parameter families
of linear Weingarten surfaces. This provides a new method to study families of
parallel Weingarten surfaces in space forms. In particular, analogs of Bonnet's
theorem on parallel constant mean curvature surfaces can be easily obtained in
this setting.Comment: 25 pages, plain Te
A Possible Approach to Inclusion of Space and Time in Frame Fields of Quantum Representations of Real and Complex Numbers
This work is based on the field of reference frames based on quantum
representations of real and complex numbers described in other work. Here frame
domains are expanded to include space and time lattices. Strings of qukits are
described as hybrid systems as they are both mathematical and physical systems.
As mathematical systems they represent numbers. As physical systems in each
frame the strings have a discrete Schrodinger dynamics on the lattices. The
frame field has an iterative structure such that the contents of a stage j
frame have images in a stage j-1 (parent) frame. A discussion of parent frame
images includes the proposal that points of stage j frame lattices have images
as hybrid systems in parent frames. The resulting association of energy with
images of lattice point locations, as hybrid systems states, is discussed.
Representations and images of other physical systems in the different frames
are also described.Comment: Paper has been greatly revised and shortened to 26 pages, 2 figures,
per referees comment
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