Some recent developments in quantization of fractal measures

We give an overview on the quantization problem for fractal measures, including some related results and methods which have been developed in the last decades. Based on the work of Graf and Luschgy, we propose a three-step procedure to estimate the quantization errors. We survey some recent progress, which makes use of this procedure, including the quantization for self-affine measures, Markov-type measures on graph-directed fractals, and product measures on multiscale Moran sets. Several open problems are mentioned.Comment: 13 page

Accelerating universe from gravitational leakage into extra dimensions: confrontation with SNeIa

There is mounting observational evidence that the expansion of our universe is undergoing an acceleration. A dark energy component has usually been invoked as the most feasible mechanism for the acceleration. However, it is desirable to explore alternative possibilities motivated by particle physics before adopting such an untested entity. In this work, we focus our attention on an acceleration mechanism: one arising from gravitational leakage into extra dimensions. We confront this scenario with high-$z$ type Ia supernovae compiled by Tonry et al. (2003) and recent measurements of the X-ray gas mass fractions in clusters of galaxies published by Allen et al. (2002,2003). A combination of the two databases gives at a 99% confidence level that $\Omega_m=0.29^{+0.04}_{-0.02}$, $\Omega_{rc}=0.21^{+0.08}_{-0.08}$, and $\Omega_k=-0.36^{+0.31}_{-0.35}$, indicating a closed universe. We then constrain the model using the test of the turnaround redshift, $z_{q=0}$, at which the universe switches from deceleration to acceleration. We show that, in order to explain that acceleration happened earlier than $z_{q=0} = 0.6$ within the framework of gravitational leakage into extra dimensions, a low matter density, $\Omega_m < 0.27$, or a closed universe is necessary.Comment: 16 pages, 4 figures, accepted for publication in Ap

Thermal Entanglement between Alternate Qubits of a Four-qubit Heisenberg XX Chain in a Magnetic Field

The concurrence of two alternate qubits in a four-qubit Heisenberg XX chain is investigated when a uniform magnetic field B is included. It is found that there is no thermal entanglement between alternate qubits if B is close to zero. Magnetic field can induce entanglement in a certain range both for the antiferromagnetic and ferromagnetic cases. Near zero temperature, the entanglement undergoes two sudden changes with increasing value of the magnetic field B. This is due to the changes in the ground state. This novel property may be used as quantum entanglement switch. The anisotropy in the system can also induce the entanglement between two alternate qubits.Comment: 10 pages, 3 figure

Resampling methods for spatial regression models under a class of stochastic designs

In this paper we consider the problem of bootstrapping a class of spatial regression models when the sampling sites are generated by a (possibly nonuniform) stochastic design and are irregularly spaced. It is shown that the natural extension of the existing block bootstrap methods for grid spatial data does not work for irregularly spaced spatial data under nonuniform stochastic designs. A variant of the blocking mechanism is proposed. It is shown that the proposed block bootstrap method provides a valid approximation to the distribution of a class of M-estimators of the spatial regression parameters. Finite sample properties of the method are investigated through a moderately large simulation study and a real data example is given to illustrate the methodology.Comment: Published at http://dx.doi.org/10.1214/009053606000000551 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cooperative Localization under Limited Connectivity

We report two decentralized multi-agent cooperative localization algorithms in which, to reduce the communication cost, inter-agent state estimate correlations are not maintained but accounted for implicitly. In our first algorithm, to guarantee filter consistency, we account for unknown inter-agent correlations via an upper bound on the joint covariance matrix of the agents. In the second method, we use an optimization framework to estimate the unknown inter-agent cross-covariance matrix. In our algorithms, each agent localizes itself in a global coordinate frame using a local filter driven by local dead reckoning and occasional absolute measurement updates, and opportunistically corrects its pose estimate whenever it can obtain relative measurements with respect to other mobile agents. To process any relative measurement, only the agent taken the measurement and the agent the measurement is taken from need to communicate with each other. Consequently, our algorithms are decentralized algorithms that do not impose restrictive network-wide connectivity condition. Moreover, we make no assumptions about the type of agents or relative measurements. We demonstrate our algorithms in simulation and a robotic~experiment.Comment: 9 pages, 5 figure