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 $\Theta$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\cap_{s<1/2} H^s(\Theta)$. 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 $T\subset{\mathbb R}^n$ be a fixed set. By a scaled copy of $T$ around $x\in{\mathbb R}^n$ we mean a set of the form $x+rT$ for some $r>0$. 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 $T$ around every point of a set of given size? We will consider the cases when $T$ is circle or sphere centered at the origin, Cantor set in ${\mathbb R}$, the boundary of a square centered at the origin, or more generally the $k$-skeleton ($0\le k<n$) of an $n$-dimensional cube centered at the origin or the $k$-skeleton of a more general polytope of ${\mathbb R}^n$. 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

Growth in solvable subgroups of GL_r(Z/pZ)

Let $K=Z/pZ$ and let $A$ 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

Efficient Quantum Tensor Product Expanders and k-designs

Quantum expanders are a quantum analogue of expanders, and k-tensor product expanders are a generalisation to graphs that randomise k correlated walkers. Here we give an efficient construction of constant-degree, constant-gap quantum k-tensor product expanders. The key ingredients are an efficient classical tensor product expander and the quantum Fourier transform. Our construction works whenever k=O(n/log n), where n is the number of qubits. An immediate corollary of this result is an efficient construction of an approximate unitary k-design, which is a quantum analogue of an approximate k-wise independent function, on n qubits for any k=O(n/log n). Previously, no efficient constructions were known for k>2, while state designs, of which unitary designs are a generalisation, were constructed efficiently in [Ambainis, Emerson 2007].Comment: 16 pages, typo in references fixe

Genericity of Fr\'echet smooth spaces

If a separable Banach space contains an isometric copy of every separable reflexive Fr\'echet smooth Banach space, then it contains an isometric copy of every separable Banach space. The same conclusion holds if we consider separable Banach spaces with Fr\'echet smooth dual space. This improves a result of G. Godefroy and N. J. Kalton.Comment: 34 page

Enhanced soliton transport in quasi-periodic lattices with short-range aperiodicity

We study linear transmission and nonlinear soliton transport through quasi-periodic structures, which profiles are described by multiple modulation frequencies. We show that resonant scattering at mixed-frequency resonances limits transmission efficiency of localized wave packets, leading to radiation and possible trapping of solitons. We obtain an explicit analytical expression for optimal quasi-periodic lattice profiles, where additional aperiodic modulations suppress mixed-frequency resonances, resulting in dramatic enhancement of soliton mobility. Our results can be applied to the design of photonic waveguide structures, and arrays of magnetic micro-traps for atomic Bose-Einstein condensates.Comment: 4 pages, 4 figure

Quasi-periodic solutions of completely resonant forced wave equations

We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.Comment: 25 pages, 1 figur