87,301 research outputs found
Minimum degree and density of binary sequences
For with , let denote the infimum density of binary sequences which satisfy the minimum degree condition for all with . We reduce the problem to determine to a combinatorial problem related to the generalized -girth of a graph which is defined as the minimum order of an induced subgraph of of minimum degree at least . Extending results of Kézdy and Markert, and of Bermond and Peyrat, we present a minimum mean cycle formulation which allows to determine for small values of and . For odd values of with , we conjecture and show that this holds for
Spatially Coupled LDPC Codes Constructed from Protographs
In this paper, we construct protograph-based spatially coupled low-density
parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or
uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L,
we obtain a flexible family of code ensembles with varying rates and frame
lengths that can share the same encoding and decoding architecture for
arbitrary L. We demonstrate that the resulting codes combine the best features
of optimized irregular and regular codes in one design: capacity approaching
iterative belief propagation (BP) decoding thresholds and linear growth of
minimum distance with block length. In particular, we show that, for
sufficiently large L, the BP thresholds on both the binary erasure channel
(BEC) and the binary-input additive white Gaussian noise channel (AWGNC)
saturate to a particular value significantly better than the BP decoding
threshold and numerically indistinguishable from the optimal maximum
a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all
variable nodes in the coupled chain have degree greater than two,
asymptotically the error probability converges at least doubly exponentially
with decoding iterations and we obtain sequences of asymptotically good LDPC
codes with fast convergence rates and BP thresholds close to the Shannon limit.
Further, the gap to capacity decreases as the density of the graph increases,
opening up a new way to construct capacity achieving codes on memoryless
binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor
Deterministic Constructions for Large Girth Protograph LDPC Codes
The bit-error threshold of the standard ensemble of Low Density Parity Check
(LDPC) codes is known to be close to capacity, if there is a non-zero fraction
of degree-two bit nodes. However, the degree-two bit nodes preclude the
possibility of a block-error threshold. Interestingly, LDPC codes constructed
using protographs allow the possibility of having both degree-two bit nodes and
a block-error threshold. In this paper, we analyze density evolution for
protograph LDPC codes over the binary erasure channel and show that their
bit-error probability decreases double exponentially with the number of
iterations when the erasure probability is below the bit-error threshold and
long chain of degree-two variable nodes are avoided in the protograph. We
present deterministic constructions of such protograph LDPC codes with girth
logarithmic in blocklength, resulting in an exponential fall in bit-error
probability below the threshold. We provide optimized protographs, whose
block-error thresholds are better than that of the standard ensemble with
minimum bit-node degree three. These protograph LDPC codes are theoretically of
great interest, and have applications, for instance, in coding with strong
secrecy over wiretap channels.Comment: 5 pages, 2 figures; To appear in ISIT 2013; Minor changes in
presentatio
Minimum Description Length Induction, Bayesianism, and Kolmogorov Complexity
The relationship between the Bayesian approach and the minimum description
length approach is established. We sharpen and clarify the general modeling
principles MDL and MML, abstracted as the ideal MDL principle and defined from
Bayes's rule by means of Kolmogorov complexity. The basic condition under which
the ideal principle should be applied is encapsulated as the Fundamental
Inequality, which in broad terms states that the principle is valid when the
data are random, relative to every contemplated hypothesis and also these
hypotheses are random relative to the (universal) prior. Basically, the ideal
principle states that the prior probability associated with the hypothesis
should be given by the algorithmic universal probability, and the sum of the
log universal probability of the model plus the log of the probability of the
data given the model should be minimized. If we restrict the model class to the
finite sets then application of the ideal principle turns into Kolmogorov's
minimal sufficient statistic. In general we show that data compression is
almost always the best strategy, both in hypothesis identification and
prediction.Comment: 35 pages, Latex. Submitted IEEE Trans. Inform. Theor
Maximum mass and universal relations of rotating relativistic hybrid hadron-quark stars
We construct equilibrium models of uniformly and differentially rotating
hybrid hadron-quark stars using equations of state (EOSs) with a first-order
phase transition that gives rise to a third family of compact objects. We find
that the ratio of the maximum possible mass of uniformly rotating
configurations - the supramassive limit - to the Tolman-Oppenheimer-Volkoff
(TOV) limit mass is not EOS-independent, and is between 1.15 and 1.31,in
contrast with the value of 1.20 previously found for hadronic EOSs. Therefore,
some of the constraints placed on the EOS from the observation of the
gravitational wave event GW170817 do not apply to hadron-quark EOSs. However,
the supramassive limit mass for the family of EOSs we treat is consistent with
limits set by GW170817, strengthening the possibility of interpreting GW170817
with a hybrid hadron-quark EOSs. We also find that along constant angular
momentum sequences of uniformly rotating stars, the third family maximum and
minimum mass models satisfy approximate EOS-independent relations, and the
supramassive limit of the third family is approximately 16.5 % larger than the
third family TOV limit. For differentially rotating spheroidal stars, we find
that a lower-limit on the maximum supportable rest mass is 123 % more than the
TOV limit rest mass. Finally, we verify that the recently discovered universal
relations relating angular momentum, rest mass and gravitational mass for
turning-point models hold for hybrid hadron-quark EOSs when uniform rotation is
considered, but have a clear dependence on the degree of differential rotation.Comment: 19 pages, 14 figures, submitted to EPJA Topical Issue "First joint
gravitational wave and electromagnetic observations: Implications for nuclear
and particle physics
Twin Binaries: Studies of Stability, Mass Transfer, and Coalescence
Motivated by suggestions that binaries with almost equal-mass components
("twins") play an important role in the formation of double neutron stars and
may be rather abundant among binaries, we study the stability of synchronized
close and contact binaries with identical components in circular orbits. In
particular, we investigate the dependency of the innermost stable circular
orbit on the core mass, and we study the coalescence of the binary that occurs
at smaller separations. For twin binaries composed of convective main-sequence
stars, subgiants, or giants with low mass cores (M_c <~0.15M, where M is the
mass of a component), a secular instability is reached during the contact
phase, accompanied by a dynamical mass transfer instability at the same or at a
slightly smaller orbital separation. Binaries that come inside this instability
limit transfer mass gradually from one component to the other and then coalesce
quickly as mass is lost through the outer Lagrangian points. For twin giant
binaries with moderate to massive cores (M_c >~0.15M), we find that stable
contact configurations exist at all separations down to the Roche limit, when
mass shedding through the outer Lagrangian points triggers a coalescence of the
envelopes and leaves the cores orbiting in a central tight binary. In addition
to the formation of binary neutron stars, we also discuss the implications of
our results for the production of planetary nebulae with double degenerate
central binaries.Comment: 17 pages, accepted to ApJ, final version includes discussion of
planetary nebulae with central binaries and a new figure about shock heating,
visualizations at http://webpub.allegheny.edu/employee/j/jalombar/movies
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