893 research outputs found
Multidimensional continued fractions and a Minkowski function
The Minkowski Question Mark function can be characterized as the unique
homeomorphism of the real unit interval that conjugates the Farey map with the
tent map. We construct an n-dimensional analogue of the Minkowski function as
the only homeomorphism of an n-simplex that conjugates the piecewise-fractional
map associated to the Monkemeyer continued fraction algorithm with an
appropriate tent map.Comment: 17 pages, 3 figures. Revised version according to the referee's
suggestions. Proof of Lemma 2.3 more detailed, other minor modifications. To
appear in Monatshefte fur Mathemati
Channel Capacity under Sub-Nyquist Nonuniform Sampling
This paper investigates the effect of sub-Nyquist sampling upon the capacity
of an analog channel. The channel is assumed to be a linear time-invariant
Gaussian channel, where perfect channel knowledge is available at both the
transmitter and the receiver. We consider a general class of right-invertible
time-preserving sampling methods which include irregular nonuniform sampling,
and characterize in closed form the channel capacity achievable by this class
of sampling methods, under a sampling rate and power constraint. Our results
indicate that the optimal sampling structures extract out the set of
frequencies that exhibits the highest signal-to-noise ratio among all spectral
sets of measure equal to the sampling rate. This can be attained through
filterbank sampling with uniform sampling at each branch with possibly
different rates, or through a single branch of modulation and filtering
followed by uniform sampling. These results reveal that for a large class of
channels, employing irregular nonuniform sampling sets, while typically
complicated to realize, does not provide capacity gain over uniform sampling
sets with appropriate preprocessing. Our findings demonstrate that aliasing or
scrambling of spectral components does not provide capacity gain, which is in
contrast to the benefits obtained from random mixing in spectrum-blind
compressive sampling schemes.Comment: accepted to IEEE Transactions on Information Theory, 201
The quantum cat map on the modular discretization of extremal black hole horizons
Based on our recent work on the discretization of the radial AdS geometry
of extremal BH horizons,we present a toy model for the chaotic unitary
evolution of infalling single particle wave packets.
We construct explicitly the eigenstates and eigenvalues for the single
particle dynamics for an observer falling into the BH horizon, with time
evolution operator the quantum Arnol'd cat map (QACM).
Using these results we investigate the validity of the eigenstate
thermalization hypothesis (ETH), as well as that of the fast scrambling time
bound (STB).
We find that the QACM, while possessing a linear spectrum, has eigenstates,
which are random and satisfy the assumptions of the ETH.
We also find that the thermalization of infalling wave packets in this
particular model is exponentially fast, thereby saturating the STB, under the
constraint that the finite dimension of the single--particle Hilbert space
takes values in the set of Fibonacci integers.Comment: 28 pages LaTeX2e, 8 jpeg figures. Clarified certain issues pertaining
to the relation between mixing time and scrambling time; enhanced discussion
of the Eigenstate Thermalization Hypothesis; revised figures and updated
references. Typos correcte
Modular discretization of the AdS2/CFT1 Holography
We propose a finite discretization for the black hole geometry and dynamics.
We realize our proposal, in the case of extremal black holes, for which the
radial and temporal near horizon geometry is known to be
AdS. We implement its discretization by
replacing the set of real numbers with the set of integers modulo
, with AdS going over to the finite geometry
AdS.
We model the dynamics of the microscopic degrees of freedom by generalized
Arnol'd cat maps, , which are isometries of the
geometry at both the classical and quantum levels.
These exhibit well studied properties of strong arithmetic chaos, dynamical
entropy, nonlocality and factorization in the cutoff discretization , which
are crucial for fast quantum information processing.
We construct, finally, a new kind of unitary and holographic correspondence,
for AdS/CFT, via coherent states of both the bulk and boundary
geometries.Comment: 33 pages LaTeX2e, 1 JPEG figure. Typos corrected, references added.
Clarification of several points in the abstract and the tex
Improving the efficiency of the LDPC code-based McEliece cryptosystem through irregular codes
We consider the framework of the McEliece cryptosystem based on LDPC codes,
which is a promising post-quantum alternative to classical public key
cryptosystems. The use of LDPC codes in this context allows to achieve good
security levels with very compact keys, which is an important advantage over
the classical McEliece cryptosystem based on Goppa codes. However, only regular
LDPC codes have been considered up to now, while some further improvement can
be achieved by using irregular LDPC codes, which are known to achieve better
error correction performance than regular LDPC codes. This is shown in this
paper, for the first time at our knowledge. The possible use of irregular
transformation matrices is also investigated, which further increases the
efficiency of the system, especially in regard to the public key size.Comment: 6 pages, 3 figures, presented at ISCC 201
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