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

Functional maturation of immature β cells: A roadblock for stem cell therapy for type 1 diabetes

Type 1 diabetes mellitus (T1DM) is a chronic autoimmune disease caused by the specific destruction of pancreatic islet β cells and is characterized as the absolute insufficiency of insulin secretion. Current insulin replacement therapy supplies insulin in a non-physiological way and is associated with devastating complications. Experimental islet transplantation therapy has been proven to restore glucose homeostasis in people with severe T1DM. However, it is restricted by many factors such as severe shortage of donor sources, progressive loss of donor cells, high cost, etc. As pluripotent stem cells have the potential to give rise to all cells including islet β cells in the body, stem cell therapy for diabetes has attracted great attention in the academic community and the general public. Transplantation of islet β-like cells differentiated from human pluripotent stem cells (hPSCs) has the potential to be an excellent alternative to islet transplantation. In stem cell therapy, obtaining β cells with complete insulin secretion in vitro is crucial. However, after much research, it has been found that the β-like cells obtained by in vitro differentiation still have many defects, including lack of adult-type glucose stimulated insulin secretion, and multi-hormonal secretion, suggesting that in vitro culture does not allows for obtaining fully mature β-like cells for transplantation. A large number of studies have found that many transcription factors play important roles in the process of transforming immature to mature human islet β cells. Furthermore, PDX1, NKX6.1, SOX9, NGN3, PAX4, etc., are important in inducing hPSC differentiation in vitro. The absent or deficient expression of any of these key factors may lead to the islet development defect in vivo and the failure of stem cells to differentiate into genuine functional β-like cells in vitro. This article reviews β cell maturation in vivo and in vitro and the vital roles of key molecules in this process, in order to explore the current problems in stem cell therapy for diabetes

Evolutionary Dynamics of Fearfulness and Boldness: A Stochastic Simulation Model

A stochastic simulation model is investigated for the evolution of anti-predator behavior in birds. The main goal is to reveal the effects of population size, predation threats, and energy lost per escape on the evolutionary dynamics of fearfulness and boldness. Two pure strategies, fearfulness and boldness, are assumed to have different responses for the predator attacks and nonlethal disturbance. On the other hand, the co-existence mechanism of fearfulness and boldness is also considered. For the effects of total population size, predation threats, and energy lost per escape, our main results show that: (i) the fearful (bold) individuals will be favored in a small (large) population, i.e. in a small (large) population, the fearfulness (boldness) can be considered to be an ESS; (ii) in a population with moderate size, fearfulness would be favored under moderate predator attacks; and (iii) although the total population size is the most important factor for the evolutionary dynamics of both fearful and bold individuals, the small energy lost per escape enables the fearful individuals to have the ability to win the advantage even in a relatively large population. Finally, we show also that the co-existence of fearful and bold individuals is possible when the competitive interactions between individuals are introduced