Representation of noncommutative phase space

The representations of the algebra of coordinates and momenta of noncommutative phase space are given. We study, as an example, the harmonic oscillator in noncommutative space of any dimension. Finally the map of Sch$\ddot{o}$dinger equation from noncommutative space to commutative space is obtained.Comment: Latex file, 10 page

Unequal Error Protected JPEG 2000 Broadcast Scheme with Progressive Fountain Codes

This paper proposes a novel scheme, based on progressive fountain codes, for broadcasting JPEG 2000 multimedia. In such a broadcast scheme, progressive resolution levels of images/video have been unequally protected when transmitted using the proposed progressive fountain codes. With progressive fountain codes applied in the broadcast scheme, the resolutions of images (JPEG 2000) or videos (MJPEG 2000) received by different users can be automatically adaptive to their channel qualities, i.e. the users with good channel qualities are possible to receive the high resolution images/vedio while the users with bad channel qualities may receive low resolution images/vedio. Finally, the performance of the proposed scheme is evaluated with the MJPEG 2000 broadcast prototype

Rateless Codes with Progressive Recovery for Layered Multimedia Delivery

This paper proposes a novel approach, based on unequal error protection, to enhance rateless codes with progressive recovery for layered multimedia delivery. With a parallel encoding structure, the proposed Progressive Rateless codes (PRC) assign unequal redundancy to each layer in accordance with their importance. Each output symbol contains information from all layers, and thus the stream layers can be recovered progressively at the expected received ratios of output symbols. Furthermore, the dependency between layers is naturally considered. The performance of the PRC is evaluated and compared with some related UEP approaches. Results show that our PRC approach provides better recovery performance with lower overhead both theoretically and numerically

A Cautionary Note on Generalized Linear Models for Covariance of Unbalanced Longitudinal Data

Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positive-definiteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positive-definiteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations