2,308 research outputs found
Supercritical holes for the doubling map
For a map and an open connected set ( a hole) we
define to be the set of points in whose -orbit avoids
. We say that a hole is supercritical if (i) for any hole such
that the set is either empty or contains
only fixed points of ; (ii) for any hole such that \barH\subset H_0
the Hausdorff dimension of is positive.
The purpose of this note to completely characterize all supercritical holes
for the doubling map .Comment: This is a new version, where a full characterization of supercritical
holes for the doubling map is obtaine
Super Landau Models on Odd Cosets
We construct d=1 sigma models of the Wess-Zumino type on the SU(n|1)/U(n)
fermionic cosets. Such models can be regarded as a particular supersymmetric
extension (with a target space supersymmetry) of the classical Landau model,
when a charged particle possesses only fermionic coordinates. We consider both
classical and quantum models, and prove the unitarity of the quantum model by
introducing the metric operator on the Hilbert space of the quantum states,
such that all their norms become positive-definite. It is remarkable that the
quantum n=2 model exhibits hidden SU(2|2) symmetry. We also discuss the planar
limit of these models. The Hilbert space in the planar n=2 case is shown to
carry SU(2|2) symmetry which is different from that of the SU(2|1)/U(1) model.Comment: 1 + 33 pages, some typos correcte
Rosenblatt's first theorem and frugality of deep learning
First Rosenblatt's theorem about omnipotence of shallow networks states that
elementary perceptrons can solve any classification problem if there are no
discrepancies in the training set. Minsky and Papert considered elementary
perceptrons with restrictions on the neural inputs: a bounded number of
connections or a relatively small diameter of the receptive field for each
neuron at the hidden layer. They proved that under these constraints, an
elementary perceptron cannot solve some problems, such as the connectivity of
input images or the parity of pixels in them. In this note, we demonstrated
first Rosenblatt's theorem at work, showed how an elementary perceptron can
solve a version of the travel maze problem, and analysed the complexity of that
solution. We constructed also a deep network algorithm for the same problem. It
is much more efficient. The shallow network uses an exponentially large number
of neurons on the hidden layer (Rosenblatt's -elements), whereas for the
deep network the second order polynomial complexity is sufficient. We
demonstrated that for the same complex problem deep network can be much smaller
and reveal a heuristic behind this effect
Possibility of local pair existence in optimally doped SmFeAsO(1-x) in pseudogap regime
We report the analysis of pseudogap Delta* derived from resistivity
experiments in FeAs-based superconductor SmFeAsO(0.85), having a critical
temperature T_c = 55 K. Rather specific dependence Delta*(T) with two
representative temperatures followed by a minimum at about 120 K was observed.
Below T_s = 147 K, corresponding to the structural transition in SmFeAsO,
Delta*(T) decreases linearly down to the temperature T_AFM = 133 K. This last
peculiarity can likely be attributed to the antiferromagnetic (AFM) ordering of
Fe spins. It is believed that the found behavior can be explained in terms of
Machida, Nokura, and Matsubara (MNM) theory developed for the AFM
superconductors.Comment: 5 pages, 2 figure
Influence of Rb, Cs and Ba on Superconductivity of Magnesium Diboride
Magnesium diboride has been thermally treated in the presence of Rb, Cs, and
Ba. Magnetic susceptibility shows onsets of superconductivity in the resulting
samples at 52K (Rb), 58K (Cs) and 45K (Ba). Room-temperature 11B NMR indicates
to cubic symmetry of the electric field gradient at boron site for the samples
reacted with Rb and Cs, in contrast to the axial symmetry in the initial MgB2
and in the sample treated with Ba.Comment: 3 pages (twocolumn), 2 figure
Golden gaskets: variations on the Sierpi\'nski sieve
We consider the iterated function systems (IFSs) that consist of three
general similitudes in the plane with centres at three non-collinear points,
and with a common contraction factor \la\in(0,1).
As is well known, for \la=1/2 the invariant set, \S_\la, is a fractal
called the Sierpi\'nski sieve, and for \la<1/2 it is also a fractal. Our goal
is to study \S_\la for this IFS for 1/2<\la<2/3, i.e., when there are
"overlaps" in \S_\la as well as "holes". In this introductory paper we show
that despite the overlaps (i.e., the Open Set Condition breaking down
completely), the attractor can still be a totally self-similar fractal,
although this happens only for a very special family of algebraic \la's
(so-called "multinacci numbers"). We evaluate \dim_H(\S_\la) for these
special values by showing that \S_\la is essentially the attractor for an
infinite IFS which does satisfy the Open Set Condition. We also show that the
set of points in the attractor with a unique ``address'' is self-similar, and
compute its dimension.
For ``non-multinacci'' values of \la we show that if \la is close to 2/3,
then \S_\la has a nonempty interior and that if \la<1/\sqrt{3} then \S_\la$
has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of
the model in question.Comment: 27 pages, 10 figure
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