765 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
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
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How to Measure the Average and Peak Age of Information in Real Networks?
The Age of information (AoI) was proposed in the literature to quantify the freshness of information. The majority of the work done in this area has theoretically evaluated AoI and its Peak (PAoI). In this paper, a method for obtaining the value of AoI and PAoI from experiments is proposed. We conducted an experiment emulating an M/M/1 queue and used the proposed method to evaluate AoI and PAoI. The values were compared to the expressions presented previously in the literature. Our results show that the proposed method is accurate for the M/M/1 queue. A statistical test was conducted to confirm the reliability of this conclusion
Interaction of Lactobacillus acidophilus NCFM grown on different carbohydrates with human intestinal epithelial cells: Adhesion Properties and roles of S-layer Proteins
Detection of Local Wall Stiffness Drop in Steel-Lined Pressure Tunnels and Shafts of Hydroelectric Power Plants Using Steep Pressure Wave Excitation and Wavelet Decomposition
A new monitoring approach for detecting, locating, and quantifying structurally weak reaches of steel-lined pressure tunnels and shafts is presented. These reaches arise from local deterioration of the backfill concrete and the rock mass surrounding the liner. The change of wave speed generated by the weakening of the radial-liner supports creates reflection boundaries for the incident pressure waves. The monitoring approach is based on the generation of transient pressure with a steep wave front and the analysis of the reflected pressure signals using the fast Fourier transform and wavelet decomposition methods. Laboratory experiments have been carried out to validate the monitoring technique. The multilayer system (steel-concrete-rock) of the pressurized shafts and tunnels is modeled by a one-layer system of the test pipe. This latter was divided into several reaches having different wall stiffnesses. Different longitudinal placements of a steel, aluminum, and PVC pipe reach were tested to validate the identification method of the weak section. DOI: 10.1061/(ASCE)HY.1943-7900.0000478. (C) 2012 American Society of Civil Engineers
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