Balanced Allocation on Graphs: A Random Walk Approach

In this paper we propose algorithms for allocating $n$ sequential balls into $n$ bins that are interconnected as a $d$-regular $n$-vertex graph $G$, where $d\ge3$ can be any integer.Let $l$ be a given positive integer. In each round $t$, $1\le t\le n$, ball $t$ picks a node of $G$ uniformly at random and performs a non-backtracking random walk of length $l$ from the chosen node.Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that $G$ has a sufficiently large girth and $d=\omega(\log n)$. Then we establish an upper bound for the maximum number of balls at any bin after allocating $n$ balls by the algorithm, called {\it maximum load}, in terms of $l$ with high probability. We also show that the upper bound is at most an $O(\log\log n)$ factor above the lower bound that is proved for the algorithm. In particular, we show that if we set $l=\lfloor(\log n)^{\frac{1+\epsilon}{2}}\rfloor$, for every constant $\epsilon\in (0, 1)$, and $G$ has girth at least $\omega(l)$, then the maximum load attained by the algorithm is bounded by $O(1/\epsilon)$ with high probability.Finally, we slightly modify the algorithm to have similar results for balanced allocation on $d$-regular graph with $d\in[3, O(\log n)]$ and sufficiently large girth

A fluorophore attached to nicotinic acetylcholine receptor beta M2 detects productive binding of agonist to the alpha delta site

To study conformational transitions at the muscle nicotinic acetylcholine (ACh) receptor (nAChR), a rhodamine fluorophore was tethered to a Cys side chain introduced at the beta-19' position in the M2 region of the nAChR expressed in Xenopus oocytes. This procedure led to only minor changes in receptor function. During agonist application, fluorescence increased by (Delta-F/F) approximate to 10%, and the emission peak shifted to lower wavelengths, indicating a more hydrophobic environment for the fluorophore. The dose-response relations for Delta-F agreed well with those for epibatidine-induced currents, but were shifted approximate to 100-fold to the left of those for ACh-induced currents. Because (i) epibatidine binds more tightly to the alpha-gamma-binding site than to the alpha-delta site and (ii) ACh binds with reverse-site selectivity, these data suggest that Delta-F monitors an event linked to binding specifically at the alpha-delta-subunit interface. In experiments with flash-applied agonists, the earliest detectable Delta-F occurs within milliseconds, i.e., during activation. At low [ACh] (less than or equal to 10 muM), a phase of Delta-F occurs with the same time constant as desensitization, presumably monitoring an increased population of agonist-bound receptors. However, recovery from Delta-F is complete before the slowest phase of recovery from desensitization (time constant approximate to 250 s), showing that one or more desensitized states have fluorescence like that of the resting channel. That conformational transitions at the alpha-delta-binding site are not tightly coupled to channel activation suggests that sequential rather than fully concerted transitions occur during receptor gating. Thus, time-resolved fluorescence changes provide a powerful probe of nAChR conformational changes

Noise sensitivity of an atomic velocity sensor

We use Bloch oscillations to accelerate coherently Rubidium atoms. The variation of the velocity induced by this acceleration is an integer number times the recoil velocity due to the absorption of one photon. The measurement of the velocity variation is achieved using two velocity selective Raman pi-pulses: the first pulse transfers atoms from the hyperfine state 5S1/2 |F=2, mF=0> to 5S1/2, |F=1, mF = 0> into a narrow velocity class. After the acceleration of this selected atomic slice, we apply the second Raman pulse to bring the resonant atoms back to the initial state 5S1/2, |F=2, mF = 0>. The populations in (F=1 and F=2) are measured separately by using a one-dimensional time-of-flight technique. To plot the final velocity distribution we repeat this procedure by scanning the Raman beam frequency of the second pulse. This two pi-pulses system constitutes then a velocity sensor. Any noise in the relative phase shift of the Raman beams induces an error in the measured velocity. In this paper we present a theoretical and an experimental analysis of this velocity sensor, which take into account the phase fluctuations during the Raman pulses

Dipole and Bloch oscillations of cold atoms in a parabolic lattice

The paper studies the dynamics of a Bose-Einstein condensate loaded into a 1D parabolic optical lattice, and excited by a sudden shift of the lattice center. Depending on the magnitude of the initial shift, the condensate undergoes either dipole or Bloch oscillations. The effects of dephasing and of atom-atom interactions on these oscillations are discussed.Comment: 3 pages, to appear in proceeding of LPHYS'05 conference (July 4-8, 2005, Kyoto, Japan

Loop structure of the lowest Bloch band for a Bose-Einstein condensate

We investigate analytically and numerically Bloch waves for a Bose--Einstein condensate in a sinusoidal external potential. At low densities the dependence of the energy on the quasimomentum is similar to that for a single particle, but at densities greater than a critical one the lowest band becomes triple-valued near the boundary of the first Brillouin zone and develops the structure characteristic of the swallow-tail catastrophe. We comment on the experimental consequences of this behavior.Comment: 4 pages, 7 figure

Scaling property of the critical hopping parameters for the Bose-Hubbard model

Recently precise results for the boundary between the Mott insulator phase and the superfluid phase of the homogeneous Bose-Hubbard model have become available for arbitrary integer filling factor g and any lattice dimension d > 1. We use these data for demonstrating that the critical hopping parameters obey a scaling relationship which allows one to map results for different g onto each other. Unexpectedly, the mean-field result captures the dependence of the exact critical parameters on the filling factor almost fully. We also present an approximation formula which describes the critical parameters for d > 1 and any g with high accuracy.Comment: 5 pages, 5 figures. to appear in EPJ

Periodically-dressed Bose-Einstein condensates: a superfluid with an anisotropic and variable critical velocity

Two intersecting laser beams can produce a spatially-periodic coupling between two components of an atomic gas and thereby modify the dispersion relation of the gas according to a dressed-state formalism. Properties of a Bose-Einstein condensate of such a gas are strongly affected by this modification. A Bogoliubov transformation is presented which accounts for interparticle interactions to obtain the quasiparticle excitation spectrum in such a condensate. The Landau critical velocity is found to be anisotropic and can be widely tuned by varying properties of the dressing laser beams.Comment: 5 pages, 4 figure