Extreme eigenvalues of sample covariance matrices under generalized elliptical models with applications

We consider the extreme eigenvalues of the sample covariance matrix $Q=YY^*$ under the generalized elliptical model that $Y=\Sigma^{1/2}XD.$ Here $\Sigma$ is a bounded $p \times p$ positive definite deterministic matrix representing the population covariance structure, $X$ is a $p \times n$ random matrix containing either independent columns sampled from the unit sphere in $\mathbb{R}^p$ or i.i.d. centered entries with variance $n^{-1},$ and $D$ is a diagonal random matrix containing i.i.d. entries and independent of $X.$ Such a model finds important applications in statistics and machine learning. In this paper, assuming that $p$ and $n$ are comparably large, we prove that the extreme edge eigenvalues of $Q$ can have several types of distributions depending on $\Sigma$ and $D$ asymptotically. These distributions include: Gumbel, Fr\'echet, Weibull, Tracy-Widom, Gaussian and their mixtures. On the one hand, when the random variables in $D$ have unbounded support, the edge eigenvalues of $Q$ can have either Gumbel or Fr\'echet distribution depending on the tail decay property of $D.$ On the other hand, when the random variables in $D$ have bounded support, under some mild regularity assumptions on $\Sigma,$ the edge eigenvalues of $Q$ can exhibit Weibull, Tracy-Widom, Gaussian or their mixtures. Based on our theoretical results, we consider two important applications. First, we propose some statistics and procedure to detect and estimate the possible spikes for elliptically distributed data. Second, in the context of a factor model, by using the multiplier bootstrap procedure via selecting the weights in $D,$ we propose a new algorithm to infer and estimate the number of factors in the factor model. Numerical simulations also confirm the accuracy and powerfulness of our proposed methods and illustrate better performance compared to some existing methods in the literature.Comment: 90 pages, 6 figures, some typos are correcte

The evolution of BIR domain and its containing proteins

AbstractBIR domain and its containing proteins play critical roles in cell apoptosis and cell division. Here several lines of novelty were revealed based on a comprehensive evolutionary analysis of BIR domains in 11 representative organisms. First, the type II BIR domains in Survivin and Bruce showed more conservation compared with the type I BIR domains in the inhibitors of apoptosis proteins (IAPs). Second, cIAP was derived from a XIAP duplicate and emerged just after the divergence of invertebrates and vertebrates. Third, the three BIR domains of NAIP displayed significantly elevated evolutionary rates compared with the BIR domains in other IAPs

Thermal behaviour of high amylose cornstarch studied by DSC

The thermal behaviour of high amylose cornstarches (80% amylose content) was studied by DSC using high pressure stainless steel pans in the temperature range between 0-350 degrees C. The number of endotherms and the enthalpy of gelatinization were found to depend on moisture content. Up to four endotherms and one exotherm were determined when the moisture content was above 40%. The meaning of each endotherm has been discussed. The enthalpy of gelatinization was calculated based on the summation of all the gelatinization endotherms and found to increase with increasing water content

Investigation of ultra-thin Al₂O₃ film as Cu diffusion barrier on low-k (k=2.5) dielectrics

Ultrathin Al(2)O(3) films were deposited by PEALD as Cu diffusion barrier on low-k (k=2.5) material. The thermal stability and electrical properties of the Cu/low k system with Al(2)O(3) layers with different thickness were studied after annealing. The AES, TEM and EDX results revealed that the ultrathin Al(2)O(3) films are thermally stable and have excellent Cu diffusion barrier performance. The electrical measurements of dielectric breakdown and TDDB tests further confirmed that the ultrathin Al(2)O(3) film is a potential Cu diffusion barrier in the Cu/low-k interconnects system

Starch gelatinization under shearless and shear conditions

This article reviews the development of studying starch gelatinization under shear and shearless conditions, in particular the technologies used to detect the degree of gelatinization. Advantages and disadvantages of each technology were discussed and then some examples were presented to demonstrate their application. A new technology RheoScope, an instrument that can measure viscosity under shear stress and simultaneously observes variation of starch particles using a microscope, was also introduced. It was found the definition of "gelatinization" could be different for different detection technologies. Under shearless condition full gelatinization of starch needs about ratio of water 3/starch 1, while the gelatinization under shear condition requires less water content since shear stress enhances the processing. The number of endotherm and enthalpy of gelatinization depends on amylose/amylopectin, moisture and lipid content

Morphology and properties of thermal/cooling-gel bi-phasic systems based on hydroxypropyl methylcellulose and hydroxypropyl starch

The miscibility between two gels with largely different gelation behaviors is an interesting topic both scientifically and practically. This paper reports a novel bi-phasic system based on two natural polymers, hydroxypropyl methylcellulose (HPMC) which has a thermal gelation behavior, and hydroxypropyl starch (HPS) which has a cooling gelation property. While both biopolymers have the same glucose unit grafted with propylene oxide, and are compatible to a certain degree, they were observed immiscible because of their different gelation behaviors. The immiscibility of these two compatible polymers could result in special structures leading to different blend film properties. Regarding this, the morphology, thermal transition, mechanical properties and oxygen barrier property could be well tailored by the ratio of two biopolymers and the environmental conditions. The knowledge obtained from this work could be useful for understanding other similar systems with desirable structure and properties