24 research outputs found
Fourier spectra of measures associated with algorithmically random Brownian motion
In this paper we study the behaviour at infinity of the Fourier transform of
Radon measures supported by the images of fractal sets under an algorithmically
random Brownian motion. We show that, under some computability conditions on
these sets, the Fourier transform of the associated measures have, relative to
the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity.
The argument relies heavily on a direct characterisation, due to Asarin and
Pokrovskii, of algorithmically random Brownian motion in terms of the prefix
free Kolmogorov complexity of finite binary sequences. The study also
necessitates a closer look at the potential theory over fractals from a
computable point of view.Comment: 24 page
Noise Covariance Properties in Dual-Tree Wavelet Decompositions
Dual-tree wavelet decompositions have recently gained much popularity, mainly
due to their ability to provide an accurate directional analysis of images
combined with a reduced redundancy. When the decomposition of a random process
is performed -- which occurs in particular when an additive noise is corrupting
the signal to be analyzed -- it is useful to characterize the statistical
properties of the dual-tree wavelet coefficients of this process. As dual-tree
decompositions constitute overcomplete frame expansions, correlation structures
are introduced among the coefficients, even when a white noise is analyzed. In
this paper, we show that it is possible to provide an accurate description of
the covariance properties of the dual-tree coefficients of a wide-sense
stationary process. The expressions of the (cross-)covariance sequences of the
coefficients are derived in the one and two-dimensional cases. Asymptotic
results are also provided, allowing to predict the behaviour of the
second-order moments for large lag values or at coarse resolution. In addition,
the cross-correlations between the primal and dual wavelets, which play a
primary role in our theoretical analysis, are calculated for a number of
classical wavelet families. Simulation results are finally provided to validate
these results
Theory of structure and thermodynamic function of liquid ⁴He (Review Article)
A new method of calculation of the density matrix of a many-boson system is proposed. The calculation of thermodynamic and structure functions at finite temperatures based on density matrix of Bose liquid is made. The structure factor of liquid ⁴He at T = 0 K is used as an input information for numerical calculation instead of the interatomic potential. We found a good agreement of the calculated quantities with experimental data
Hilbert's Tenth Problem in Coq (Extended Version)
We formalise the undecidability of solvability of Diophantine equations, i.e.
polynomial equations over natural numbers, in Coq's constructive type theory.
To do so, we give the first full mechanisation of the
Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively
enumerable problem -- in our case by a Minsky machine -- is Diophantine. We
obtain an elegant and comprehensible proof by using a synthetic approach to
computability and by introducing Conway's FRACTRAN language as intermediate
layer. Additionally, we prove the reverse direction and show that every
Diophantine relation is recognisable by -recursive functions and give a
certified compiler from -recursive functions to Minsky machines.Comment: submitted to LMC
Nonlinear spectral calculus and super-expanders
Nonlinear spectral gaps with respect to uniformly convex normed spaces are
shown to satisfy a spectral calculus inequality that establishes their decay
along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to
behave sub-multiplicatively under zigzag products. These results yield a
combinatorial construction of super-expanders, i.e., a sequence of 3-regular
graphs that does not admit a coarse embedding into any uniformly convex normed
space.Comment: Typos fixed based on referee comments. Some of the results of this
paper were announced in arXiv:0910.2041. The corresponding parts of
arXiv:0910.2041 are subsumed by the current pape
Comparison of metric spectral gaps
Let be an by symmetric stochastic matrix. For
and a metric space , let be the
infimum over those for which every
satisfy
Thus measures the magnitude of the {\em nonlinear spectral
gap} of the matrix with respect to the kernel . We study pairs of metric spaces and for which
there exists such that for every symmetric stochastic with
. When is linear a complete geometric
characterization is obtained.
Our estimates on nonlinear spectral gaps yield new embeddability results as
well as new nonembeddability results. For example, it is shown that if and then for every there exist
such that {equation}\label{eq:p factor} \forall\, i,j\in
\{1,...,n\},\quad \|x_i-x_j\|_2\lesssim p\|f_i-f_j\|_p, {equation} and
This statement is impossible for , and the asymptotic dependence
on in \eqref{eq:p factor} is sharp. We also obtain the best known lower
bound on the distortion of Ramanujan graphs, improving over the work of
Matou\v{s}ek. Links to Bourgain--Milman--Wolfson type and a conjectural
nonlinear Maurey--Pisier theorem are studied.Comment: Clarifying remarks added, definition of p(n,d) modified, typos fixed,
references adde
Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse
The Birkhoff conjecture says that the boundary of a strictly convex
integrable billiard table is necessarily an ellipse. In this article, we
consider a stronger notion of integrability, namely, integrability close to the
boundary, and prove a local version of this conjecture: a small perturbation of
almost every ellipse that preserves integrability near the boundary, is itself
an ellipse. We apply this result to study local spectral rigidity of ellipses
using the connection between the wave trace of the Laplacian and the dynamics
near the boundary and establish rigidity for almost all of them.Comment: 69 pages, 5 figure