The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile

Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $Y_{ij}^{n}=\frac{\sigma(i/N,j/n)}{\sqrt{n}} X_{ij}^{n}$, the $X_{ij}^{n}$ being centered i.i.d. and $\sigma:[0,1]^2 \to (0,\infty)$ being a continuous function called a variance profile. Consider now a deterministic $N\times n$ matrix $\Lambda_n=(\Lambda_{ij}^{n})$ whose non diagonal elements are zero. Denote by $\Sigma_n$ the non-centered matrix $Y_n + \Lambda_n$. Then under the assumption that $\lim_{n\to \infty} \frac Nn =c>0$ and $$\frac{1}{N} \sum_{i=1}^{N} \delta_{(\frac{i}{N}, (\Lambda_{ii}^n)^2)} \xrightarrow[n\to \infty]{} H(dx,d\lambda),$$ where $H$ is a probability measure, it is proven that the empirical distribution of the eigenvalues of $\Sigma_n \Sigma_n^T$ converges almost surely in distribution to a non random probability measure. This measure is characterized in terms of its Stieltjes transform, which is obtained with the help of an auxiliary system of equations. This kind of results is of interest in the field of wireless communication.Comment: 25 pages, revised version. Assumption (A2) has been relaxe

The empirical eigenvalue distribution of a Gram matrix: From independence to stationarity

Consider a $N\times n$ random matrix $Z_n=(Z^n_{j_1 j_2})$ where the individual entries are a realization of a properly rescaled stationary gaussian random field. The purpose of this article is to study the limiting empirical distribution of the eigenvalues of Gram random matrices such as $Z_n Z_n ^*$ and $(Z_n +A_n)(Z_n +A_n)^*$ where $A_n$ is a deterministic matrix with appropriate assumptions in the case where $n\to \infty$ and $\frac Nn \to c \in (0,\infty)$. The proof relies on related results for matrices with independent but not identically distributed entries and substantially differs from related works in the literature (Boutet de Monvel et al., Girko, etc.).Comment: 15 page

A sharp-front moving boundary model for malignant invasion

We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells in both populations undergo diffusive migration and logistic proliferation. The interface between the two populations moves according to a two-phase Stefan condition. Unlike many reaction-diffusion models of malignant invasion, the moving boundary model explicitly describes the motion of the sharp front between the cancer and surrounding tissues without needing to introduce degenerate nonlinear diffusion. Numerical simulations suggest the model gives rise to very interesting travelling wave solutions that move with speed $c$, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative $x$-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed $c \ge c^* > 0$, the moving boundary model leads to travelling wave solutions with $|c| < c^{**}$. We interpret these travelling wave solutions in the phase plane and show that they are associated with several features of the classical Fisher-Kolmogorov phase plane that are often disregarded as being nonphysical. We show, numerically, that the phase plane analysis compares well with long time solutions from the full partial differential equation model as well as providing accurate perturbation approximations for the shape of the travelling waves.Comment: 48 pages, 21 figure

On the capacity achieving covariance matrix for Rician MIMO channels: an asymptotic approach

The capacity-achieving input covariance matrices for coherent block-fading correlated MIMO Rician channels are determined. In this case, no closed-form expressions for the eigenvectors of the optimum input covariance matrix are available. An approximation of the average mutual information is evaluated in this paper in the asymptotic regime where the number of transmit and receive antennas converge to $+\infty$. New results related to the accuracy of the corresponding large system approximation are provided. An attractive optimization algorithm of this approximation is proposed and we establish that it yields an effective way to compute the capacity achieving covariance matrix for the average mutual information. Finally, numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information, while being much more computationally attractive.Comment: 56 pp. Extended version of the published article in IEEE Inf. Th. (march 2010) with more proof

Invading and receding sharp-fronted travelling waves

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade into vacant regions, are routinely studied using partial differential equation (PDE) models based upon the classical Fisher--KPP model. While the Fisher--KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often--overlooked limitation of the Fisher--KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work we study the \textit{Fisher--Stefan} model, which is a generalisation of the Fisher--KPP model obtained by reformulating the Fisher--KPP model as a moving boundary problem. The nondimensional Fisher--Stefan model involves just one single parameter, $\kappa$, which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, $c$. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher--Stefan model for both slowly invading and slowly receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between $c$ and $\kappa$ so that commonly--reported experimental estimates of $c$ can be used to provide estimates of the unknown parameter $\kappa$. Interestingly, when we reinterpret the Fisher--KPP model as a moving boundary problem, many disregarded features of the classical Fisher--KPP phase plane take on a new interpretation since travelling waves solutions with $c < 2$ are not normally considered. This means that our analysis of the Fisher--Stefan model has both practical value and an inherent mathematical value.Comment: 47 pages, 13 figure

Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation

A family of travelling wave solutions to the Fisher-KPP equation with speeds $c=\pm 5/\sqrt{6}$ can be expressed exactly using Weierstrass elliptic functions. The well-known solution for $c=5/\sqrt{6}$, which decays to zero in the far-field, is exceptional in the sense that it can be written simply in terms of an exponential function. This solution has the property that the phase-plane trajectory is a heteroclinic orbit beginning at a saddle point and ends at the origin. For $c=-5/\sqrt{6}$, there is also a trajectory that begins at the saddle point, but this solution is normally disregarded as being unphysical as it blows up for finite $z$. We reinterpret this special trajectory as an exact sharp-fronted travelling solution to a \textit{Fisher-Stefan} type moving boundary problem, where the population is receding from, instead of advancing into, an empty space. By simulating the full moving boundary problem numerically, we demonstrate how time-dependent solutions evolve to this exact travelling solution for large time. The relevance of such receding travelling waves to mathematical models for cell migration and cell proliferation is also discussed

Importance of collection in gene set enrichment analysis of drug response in cancer cell lines

Gene set enrichment analysis (GSEA) associates gene sets and phenotypes, its use is predicated on the choice of a pre-defined collection of sets. The defacto standard implementation of GSEA provides seven collections yet there are no guidelines for the choice of collections and the impact of such choice, if any, is unknown. Here we compare each of the standard gene set collections in the context of a large dataset of drug response in human cancer cell lines. We define and test a new collection based on gene co-expression in cancer cell lines to compare the performance of the standard collections to an externally derived cell line based collection. The results show that GSEA findings vary significantly depending on the collection chosen for analysis. Henceforth, collections should be carefully selected and reported in studies that leverage GSEA

Topical Peroxisome Proliferator Activated Receptor Activators Accelerate Postnatal Stratum Corneum Acidification

Previous studies have shown that pH declines from between 6 and 7 at birth to adult levels (pH 5.0–5.5) over 5–6 days in neonatal rat stratum corneum (SC). As a result, at birth, neonatal epidermis displays decreased permeability barrier homeostasis and SC integrity, improving days 5–6. We determined here whether peroxisome proliferator-activated receptor (PPAR) activators accelerate postnatal SC acidification. Topical treatment with two different PPARα activators, clofibrate and WY14643, accelerated the postnatal decline in SC surface pH, whereas treatment with PPARγ activators did not and a PPARβ/δ activator had only a modest effect. Treatment with clofibrate significantly accelerated normalization of barrier function. The morphological basis for the improvement in barrier function in PPARα-treated animals includes accelerated secretion of lamellar bodies and enhanced, postsecretory processing of secreted lamellar body contents into mature lamellar membranes. Activity of β-glucocerebrosidase increased after PPARα-activator treatment. PPARα activator also improved SC integrity, which correlated with an increase in corneodesmosome density and increased desmoglein-1 content, with a decline in serine protease activity. Topical treatment of newborn animals with a PPARα activator increased secretory phospholipase A2 activity, which likely accounts for accelerated SC acidification. Thus, PPARα activators accelerate neonatal SC acidification, in parallel with improved permeability homeostasis and SC integrity/cohesion. Hence, PPARα activators might be useful to prevent or treat certain common neonatal dermatoses