2,342 research outputs found
Combinatorics of linear iterated function systems with overlaps
Let be points in , and let
be a one-parameter family of similitudes of : where
is our parameter. Then, as is well known, there exists a
unique self-similar attractor satisfying
. Each has
at least one address , i.e.,
.
We show that for sufficiently close to 1, each has different
addresses. If is not too close to 1, then we can still have an
overlap, but there exist 's which have a unique address. However, we
prove that almost every has addresses,
provided contains no holes and at least one proper overlap. We
apply these results to the case of expansions with deleted digits.
Furthermore, we give sharp sufficient conditions for the Open Set Condition
to fail and for the attractor to have no holes.
These results are generalisations of the corresponding one-dimensional
results, however most proofs are different.Comment: Accepted for publication in Nonlinearit
Determination of meteor flux distribution over the celestial sphere
A new method of determination of meteor flux density distribution over the celestial sphere is discussed. The flux density was derived from observations by radar together with measurements of angles of arrival of radio waves reflected from meteor trails. The role of small meteor showers over the sporadic background is shown
Atom trapping with a thin magnetic film
We have created a Rb Bose-Einstein condensate in a magnetic trapping
potential produced by a hard disk platter written with a periodic pattern. Cold
atoms were loaded from an optical dipole trap and then cooled to BEC on the
surface with radiofrequency evaporation. Fragmentation of the atomic cloud due
to imperfections in the magnetic structure was observed at distances closer
than 40 m from the surface. Attempts to use the disk as an atom mirror
showed dispersive effects after reflection.Comment: 4 pages, 5 figure
Higher twists in polarized DIS and the size of the constituent quark
The spontaneous breaking of chiral symmetry implies the presence of a
short-distance scale in the QCD vacuum, which phenomenologically may be
associated with the "size" of the constituent quark, rho ~ 0.3 fm. We discuss
the role of this scale in the matrix elements of the twist-4 and 3 quark-gluon
operators determining the leading power (1/Q^2-) corrections to the moments of
the nucleon spin structure functions. We argue that the flavor-nonsinglet
twist-4 matrix element, f_2^{u - d}, has a sizable negative value of the order
rho^{-2}, due to the presence of sea quarks with virtualities ~ rho^{-2} in the
proton wave function. The twist-3 matrix element, d_2, is not related to the
scale rho^{-2}. Our arguments support the results of previous calculations of
the matrix elements in the instanton vacuum model. We show that this
qualitative picture is in agreement with the phenomenological higher-twist
correction extracted from an NLO QCD fit to the world data on g_1^p and g_1^n,
which include recent data from the Jefferson Lab Hall A and COMPASS
experiments. We comment on the implications of the short-distance scale rho for
quark-hadron duality and the x-dependence of higher-twist contributions.Comment: 8 pages, 4 figure
Computing Garsia Entropy for Bernoulli Convolutions with Algebraic Parameters
We introduce a parameter space containing all algebraic integers
that are not Pisot or Salem numbers, and a sequence of
increasing piecewise continuous function on this parameter space which gives a
lower bound for the Garsia entropy of the Bernoulli convolution .
This allows us to show that for all
with representations in certain open regions of the parameter space.Comment: 21 pages, 2 figures, 5 table
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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