306 research outputs found
The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile
Consider a random matrix where the entries are
given by , the
being centered i.i.d. and being a
continuous function called a variance profile. Consider now a deterministic
matrix whose non diagonal elements
are zero. Denote by the non-centered matrix . Then
under the assumption that and where is a probability measure, it is proven
that the empirical distribution of the eigenvalues of
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 random matrix 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 and where is a deterministic matrix with appropriate
assumptions in the case where and .
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 , and the model supports both malignant invasion and malignant
retreat, where the travelling wave can move in either the positive or negative
-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher
models where travelling waves move with a minimum wave speed ,
the moving boundary model leads to travelling wave solutions with . 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 . 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, , which relates the shape of the density front at the moving
boundary to the speed of the associated travelling wave, . 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 and
so that commonly--reported experimental estimates of can be used
to provide estimates of the unknown parameter . 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 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
can be expressed exactly using Weierstrass elliptic
functions. The well-known solution for , 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 , 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 . 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
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