A note on eigenvalues of random block Toeplitz matrices with slowly growing bandwidth

This paper can be thought of as a remark of \cite{llw}, where the authors studied the eigenvalue distribution $\mu_{X_N}$ of random block Toeplitz band matrices with given block order $m$. In this note we will give explicit density functions of $\lim\limits_{N\to\infty}\mu_{X_N}$ when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of $m\times m$ Gaussian unitary ensemble (GUE for short). The series $\{\lim\limits_{N\to\infty}\mu_{X_N}|m\in\mathbb{N}\}$ can be seen as a transition from the standard normal distribution to semicircle distribution. We also show a similar relationship between GOE and block Toeplitz band matrices with symmetric blocks.Comment: 6 page

Analysis of an age-structured dengue model with multiple strains and cross immunity

Dengue fever is a typical mosquito-borne infectious disease, and four strains of it are currently found. Clinical medical research has shown that the infected person can provide life-long immunity against the strain after recovering from infection with one strain, but only provide partial and temporary immunity against other strains. On the basis of the complexity of transmission and the diversity of pathogens, in this paper, a multi-strain dengue transmission model with latency age and cross immunity age is proposed. We discuss the well-posedness of this model and give the terms of the basic reproduction number R0 = max{R1, R2} , where Ri is the basic reproduction number of strain i (i = 1, 2). Particularly, we obtain that the model always has a unique diseasefree equilibrium P0 which is locally stable for R0 1, the strain-i dominant equilibrium Pi is locally stable for Rj < R∗ i (i, j = 1, 2, i 6= j). Additionally, the threshold criteria on the uniformly persistence, the existence and global asymptotically stability of coexistence equilibrium are also obtained. Finally, these theoretical results and interesting conclusions are illustrated with some numerical simulations

Genome-Wide Microarrray Analysis Reveals Roles for the REF-1 Family Member HLH-29 in Ferritin Synthesis and Peroxide Stress Response

In Caenorhabditis elegans, the six proteins that make up the REF-1 family have been identified as functional homologs of the Hairy/Enhancer of Split (HES) proteins. These transcription factors act in both Notch dependent and Notch-independent pathways to regulate embryonic events during development; however, their post-embryonic functions are not well defined. As a first step toward understanding how the REF-1 family works together to coordinate post-embryonic events, we used gene expression microarray analysis to identify transcriptional targets of HLH-29 in L4/young adult stage animals. Here we show that HLH-29 targets are genes needed for the regulation of growth and lifespan, including genes required for oxidative stress response and fatty acid metabolism, and the ferritin genes, ftn-1 and ftn-2. We show that HLH-29 regulates ftn-1 expression via promoter sequences upstream of the iron-dependent element that is recognized by the hypoxia inducible factor, HIF-1. Additionally, hlh-29 mutants are more resistant to peroxide stress than wild-type animals and ftn- 1(RNAi) animals, even in the presence of excess iron. Finally we show that HLH-29 acts parallel to DAF-16 but upstream of the microphthalmia transcription factor ortholog, HLH-30, to regulate ftn-1 expression under normal growth conditions