826 research outputs found
Expansion in SL_d(Z/qZ), q arbitrary
Let S be a fixed finite symmetric subset of SL_d(Z), and assume that it
generates a Zariski-dense subgroup G. We show that the Cayley graphs of pi_q(G)
with respect to the generating set pi_q(S) form a family of expanders, where
pi_q is the projection map Z->Z/qZ
Random data Cauchy theory for supercritical wave equations II : A global existence result
We prove that the subquartic wave equation on the three dimensional ball
, with Dirichlet boundary conditions admits global strong solutions for
a large set of random supercritical initial data in .
We obtain this result as a consequence of a general random data Cauchy theory
for supercritical wave equations developed in our previous work \cite{BT2} and
invariant measure considerations which allow us to obtain also precise large
time dynamical informations on our solutions
Study of light-assisted collisions between a few cold atoms in a microscopic dipole trap
We study light-assisted collisions in an ensemble containing a small number
(~3) of cold Rb87 atoms trapped in a microscopic dipole trap. Using our ability
to operate with one atom exactly in the trap, we measure the one-body heating
rate associated to a near-resonant laser excitation, and we use this
measurement to extract the two-body loss rate associated to light-assisted
collisions when a few atoms are present in the trap. Our measurements indicate
that the two-body loss rate can reach surprisingly large values beta>10^{-8}
cm^{3}.s^{-1} and varies rapidly with the trap depth and the parameters of the
excitation light.Comment: 6 pages, 7 figure
Evaporative cooling of a small number of atoms in a single-beam microscopic dipole trap
We demonstrate experimentally the evaporative cooling of a few hundred
rubidium 87 atoms in a single-beam microscopic dipole trap. Starting from 800
atoms at a temperature of 125microKelvins, we produce an unpolarized sample of
40 atoms at 110nK, within 3s. The phase-space density at the end of the
evaporation reaches unity, close to quantum degeneracy. The gain in phase-space
density after evaporation is 10^3. We find that the scaling laws used for much
larger numbers of atoms are still valid despite the small number of atoms
involved in the evaporative cooling process. We also compare our results to a
simple kinetic model describing the evaporation process and find good agreement
with the data.Comment: 7 pages, 5 figure
Small union with large set of centers
Let be a fixed set. By a scaled copy of around
we mean a set of the form for some .
In this survey paper we study results about the following type of problems:
How small can a set be if it contains a scaled copy of around every point
of a set of given size? We will consider the cases when is circle or sphere
centered at the origin, Cantor set in , the boundary of a square
centered at the origin, or more generally the -skeleton () of an
-dimensional cube centered at the origin or the -skeleton of a more
general polytope of .
We also study the case when we allow not only scaled copies but also scaled
and rotated copies and also the case when we allow only rotated copies
Expansion in perfect groups
Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an
integer q, denote by Ga_q the subgroup of Ga consisting of the elements that
project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q
with respect to the generating set S form a family of expanders when q ranges
over square-free integers with large prime divisors if and only if the
connected component of the Zariski-closure of Ga is perfect.Comment: 62 pages, no figures, revision based on referee's comments: new ideas
are explained in more details in the introduction, typos corrected, results
and proofs unchange
Sub-Poissonian atom number fluctuations using light-assisted collisions
We investigate experimentally the number statistics of a mesoscopic ensemble
of cold atoms in a microscopic dipole trap loaded from a magneto-optical trap,
and find that the atom number fluctuations are reduced with respect to a
Poisson distribution due to light-assisted two-body collisions. For numbers of
atoms N>2, we measure a reduction factor (Fano factor) of 0.72+/-0.07, which
differs from 1 by more than 4 standard deviations. We analyze this fact by a
general stochastic model describing the competition between the loading of the
trap from a reservoir of cold atoms and multi-atom losses, which leads to a
master equation. Applied to our experimental regime, this model indicates an
asymptotic value of 3/4 for the Fano factor at large N and in steady state. We
thus show that we have reached the ultimate level of reduction in number
fluctuations in our system.Comment: 4 pages, 3 figure
Growth in solvable subgroups of GL_r(Z/pZ)
Let and let be a subset of \GL_r(K) such that is
solvable. We reduce the study of the growth of $A$ under the group operation to
the nilpotent setting. Specifically we prove that either $A$ grows rapidly
(meaning $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), or else there are groups $U_R$
and $S$, with $S/U_R$ nilpotent such that $A_k\cap S$ is large and
$U_R\subseteq A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k
: x_i \in A \cup A^{-1} \cup {1}}$. The implied constants depend only on the
rank $r$ of $\GL_r(K)$.
When combined with recent work by Pyber and Szab\'o, the main result of this
paper implies that it is possible to draw the same conclusions without
supposing that is solvable.Comment: 46 pages. This version includes revisions recommended by an anonymous
referee including, in particular, the statement of a new theorem, Theorem
Scaling of energy spreading in strongly nonlinear disordered lattices
To characterize a destruction of Anderson localization by nonlinearity, we
study the spreading behavior of initially localized states in disordered,
strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized
linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a
phenomenological description by virtue of a nonlinear diffusion equation we
establish a one-parameter scaling relation between the velocity of spreading
and the density, which is confirmed numerically. From this scaling it follows
that for very low densities the spreading slows down compared to the pure power
law.Comment: 4 pages, 4 figure
Observation of suppression of light scattering induced by dipole-dipole interactions in a cold atomic ensemble
We study the emergence of collective scattering in the presence of
dipole-dipole interactions when we illuminate a cold cloud of rubidium atoms
with a near-resonant and weak intensity laser. The size of the atomic sample is
comparable to the wavelength of light. When we gradually increase the atom
number from 1 to 450, we observe a broadening of the line, a small red shift
and, consistently with these, a strong suppression of the scattered light with
respect to the noninteracting atom case. Numerical simulations, which include
the internal atomic level structure, agree with the data.Comment: 5 pages, 5 figure
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