On Singularity Formation of a Nonlinear Nonlocal System

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove the global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.Comment: 28 pages, 9 figure

Pathwise Performance of Debt Based Policies for Wireless Networks with Hard Delay Constraints

Hou et al have introduced a framework to serve clients over wireless channels when there are hard deadline constraints along with a minimum delivery ratio for each client's flow. Policies based on "debt," called maximum debt first policies (MDF) were introduced, and shown to be throughput optimal. By "throughput optimality" it is meant that if there exists a policy that fulfils a set of clients with a given vector of delivery ratios and a vector of channel reliabilities, then the MDF policy will also fulfill them. The debt of a user is the difference between the number of packets that should have been delivered so as to meet the delivery ratio and the number of packets that have been delivered for that client. The maximum debt first (MDF) prioritizes the clients in decreasing order of debts at the beginning of every period. Note that a throughput optimal policy only guarantees that \begin{small} \liminf_{T \to \infty} \frac{1}{T}\sum_{t=1}^{T} \mathbbm{1}\{\{client n$'s packet is delivered in frame$t} \} \geq q_{i} \end{small}, where the right hand side is the required delivery ratio for client $i$. Thus, it only guarantees that the debts of each user are $o(T)$, and can be otherwise arbitrarily large. This raises the interesting question about what is the growth rate of the debts under the MDF policy. We show the optimality of MDF policy in the case when the channel reliabilities of all users are same, and obtain performance bounds for the general case. For the performance bound we obtain the almost sure bounds on $\limsup_{t\to\infty}\frac{d_{i}(t)}{\phi(t)}$ for all $i$, where $\phi(t) = \sqrt{2t\log\log t}$

Self-Diffusion in 2D Dusty Plasma Liquids: Numerical Simulation Results

We perform Brownian dynamics simulations for studying the self-diffusion in two-dimensional (2D) dusty plasma liquids, in terms of both mean-square displacement and velocity autocorrelation function (VAF). Super-diffusion of charged dust particles has been observed to be most significant at infinitely small damping rate $\gamma$ for intermediate coupling strength, where the long-time asymptotic behavior of VAF is found to be the product of $t^{-1}$ and $\exp{(-\gamma t)}$. The former represents the prediction of early theories in 2D simple liquids and the latter the VAF of a free Brownian particle. This leads to a smooth transition from super-diffusion to normal diffusion, and then to sub-diffusion with an increase of the damping rate. These results well explain the seemingly contradictory scattered in recent classical molecular dynamics simulations and experiments of dusty plasmas.Comment: 10 pages 5 figures, accepted by PR

No association of CTLA-4 polymorphisms with susceptibility to Behcet disease

Background: Cytotoxic T lymphocyte-associated antigen 4 (CTLA-4) is a key negative regulator of T lymphocytes and has been shown to be associated with a number of autoimmune diseases. The present study was performed to assess the association between CTLA-4 polymorphisms and Behcet disease (BD) in Chinese patients. Methods: Two hundred and twenty-eight BD patients and 207 controls were analysed for four single nucleotide polymorphisms (SNPs) (21661A/G, 2318C/T, + 49G/A and CT60G/A) in the CTLA-4 gene by PCR-restriction fragment length polymorphism (RFLP) analysis. The association between SNP +49A/G and BD in Chinese population as well as other ethnic groups was analysed by meta-analysis. Results: No association could be detected between CTLA-4 SNPs or haplotypes and BD. Also, no association was observed between CTLA-4 polymorphisms and BD subgroups, stratified by clinical features. A meta-analysis showed that there was no heterogeneity between studies (p = 0.60, I-2 = 0%) and that CTLA-4 SNP + 49 was not associated with BD (overall effect: Z = 0.26, p = 0.79). Conclusion: This study and a meta-analysis failed to demonstrate any association between the tested CTLA-4 polymorphisms and B

The non-linear evolution of bispectrum from the scale-free N-body simulation

We have accurately measured the bispectrum for four scale-free models of structure formation with the spectral index $n=1$, 0, -1, and -2. The measurement is based on a new method that can effectively eliminate the alias and numerical artifacts, and reliably extend the analysis into the strongly non-linear regime. The work makes use of a set of state-of-the art N-body simulations that have significantly increased the resolution range compared with the previous studies on the subject. With these measured results, we demonstrated that the measured bispectrum depends on the shape and size of $k$-triangle even in the strongly nonlinear regime. It increases with wavenumber and decreases with the spectral index. These results are in contrast with the hypothesis that the reduced bispectrum is a constant in the strongly non-linear regime. We also show that the fitting formula of Scoccimarro & Frieman (1999) does not describe our simulation results well (with a typical error about 40 percent). In the end, we present a new fitting formula for the reduced bispectrum that is valid for $-2 \leq n \leq 0$ with a typical error of 10 percent only.Comment: 33 pages, including 1 table, 14 figures, accepted by Ap