37,186 research outputs found
Some properties of q-ary functions based on spectral analysis
In this paper, we generalize some existing results on Boolean
functions to the -ary functions defined over \BBZ_q, where
is an integer, and obtain some new characterization of
-ary functions based on spectral analysis. We provide a
relationship between Walsh-Hadamard spectra of two -ary functions
and (for a prime) and their derivative . We
provide a relationship between the Walsh-Hadamard spectra and the
decompositions of any two -ary functions. Further, we investigate
a relationship between the Walsh-Hadamard spectra and the
autocorrelation of any two -ary functions
Glauber dynamics on trees:Boundary conditions and mixing time
We give the first comprehensive analysis of the effect of boundary conditions
on the mixing time of the Glauber dynamics in the so-called Bethe
approximation. Specifically, we show that spectral gap and the log-Sobolev
constant of the Glauber dynamics for the Ising model on an n-vertex regular
tree with plus-boundary are bounded below by a constant independent of n at all
temperatures and all external fields. This implies that the mixing time is
O(log n) (in contrast to the free boundary case, where it is not bounded by any
fixed polynomial at low temperatures). In addition, our methods yield simpler
proofs and stronger results for the spectral gap and log-Sobolev constant in
the regime where there are multiple phases but the mixing time is insensitive
to the boundary condition. Our techniques also apply to a much wider class of
models, including those with hard-core constraints like the antiferromagnetic
Potts model at zero temperature (proper colorings) and the hard--core lattice
gas (independent sets)
On Locally Dyadic Stationary Processes
We introduce the concept of local dyadic stationarity, to account for
non-stationary time series, within the framework of Walsh-Fourier analysis. We
define and study the time varying dyadic ARMA models (tvDARMA). It is proven
that the general tvDARMA process can be approximated locally by either a tvDMA
and a tvDAR process.Comment: 27 pages, 2 figure
Codes as fractals and noncommutative spaces
We consider the CSS algorithm relating self-orthogonal classical linear codes
to q-ary quantum stabilizer codes and we show that to such a pair of a
classical and a quantum code one can associate geometric spaces constructed
using methods from noncommutative geometry, arising from rational
noncommutative tori and finite abelian group actions on Cuntz algebras and
fractals associated to the classical codes.Comment: 18 pages LaTeX, one png figur
Approximation of L\"owdin Orthogonalization to a Spectrally Efficient Orthogonal Overlapping PPM Design for UWB Impulse Radio
In this paper we consider the design of spectrally efficient time-limited
pulses for ultrawideband (UWB) systems using an overlapping pulse position
modulation scheme. For this we investigate an orthogonalization method, which
was developed in 1950 by Per-Olov L\"owdin. Our objective is to obtain a set of
N orthogonal (L\"owdin) pulses, which remain time-limited and spectrally
efficient for UWB systems, from a set of N equidistant translates of a
time-limited optimal spectral designed UWB pulse. We derive an approximate
L\"owdin orthogonalization (ALO) by using circulant approximations for the Gram
matrix to obtain a practical filter implementation. We show that the centered
ALO and L\"owdin pulses converge pointwise to the same Nyquist pulse as N tends
to infinity. The set of translates of the Nyquist pulse forms an orthonormal
basis or the shift-invariant space generated by the initial spectral optimal
pulse. The ALO transform provides a closed-form approximation of the L\"owdin
transform, which can be implemented in an analog fashion without the need of
analog to digital conversions. Furthermore, we investigate the interplay
between the optimization and the orthogonalization procedure by using methods
from the theory of shift-invariant spaces. Finally we develop a connection
between our results and wavelet and frame theory.Comment: 33 pages, 11 figures. Accepted for publication 9 Sep 201
Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences
The concept of symbolic sequences play important role in study of complex
systems. In the work we are interested in ultrametric structure of the set of
cyclic sequences naturally arising in theory of dynamical systems. Aimed at
construction of analytic and numerical methods for investigation of clusters we
introduce operator language on the space of symbolic sequences and propose an
approach based on wavelet analysis for study of the cluster hierarchy. The
analytic power of the approach is demonstrated by derivation of a formula for
counting of {\it two-fold de Bruijn sequences}, the extension of the notion of
de Bruijn sequences. Possible advantages of the developed description is also
discussed in context of applied
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