Departamento de Matemática Aplicada
http://hdl.handle.net/10481/31361
Sat, 21 Sep 2019 16:41:39 GMT2019-09-21T16:41:39ZOn the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials
http://hdl.handle.net/10481/56651
On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials
Cañizo Rincón, José Alfredo; Einav, Amit; Lods, Bertrand
In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad’s angular cut-off assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cut-off case and conjecture what we believe to be the right rate of convergence in that case.
http://hdl.handle.net/10481/56651Improved energy methods for nonlocal diffusion problems
http://hdl.handle.net/10481/56086
Improved energy methods for nonlocal diffusion problems
Cañizo Rincón, José Alfredo; Molino Salas, Alexis
We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations:
L
u
(
x
)
:=
∫
R
N
K
(
x
,
y
)
(
u
(
y
)
−
u
(
x
)
)
d
y
,
where
L
acts on a real function
u
defined on
R
N
, and we assume that
K
(
x
,
y
)
is uniformly strictly positive in a neighbourhood of
x
=
y
. The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation
∂
t
u
=
L
u
as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the
L
p
norms of
u
and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases.
http://hdl.handle.net/10481/56086Multiclass classification for skin cancer profiling based on the integration of heterogeneous gene expression series
http://hdl.handle.net/10481/55859
Multiclass classification for skin cancer profiling based on the integration of heterogeneous gene expression series
Gálvez, Juan Manuel; Castillo, Daniel; Herrera Maldonado, Luis Javier; San Román, Belén; Valenzuela Cansino, Olga; Ortuño, Francisco Manuel; Rojas Ruiz, Ignacio
Most of the research studies developed applying microarray technology to the characterization
of different pathological states of any disease may fail in reaching statistically significant
results. This is largely due to the small repertoire of analysed samples, and to the limitation
in the number of states or pathologies usually addressed. Moreover, the influence of potential
deviations on the gene expression quantification is usually disregarded. In spite of the
continuous changes in omic sciences, reflected for instance in the emergence of new Next-
Generation Sequencing-related technologies, the existing availability of a vast amount of
gene expression microarray datasets should be properly exploited. Therefore, this work proposes
a novel methodological approach involving the integration of several heterogeneous
skin cancer series, and a later multiclass classifier design. This approach is thus a way to
provide the clinicians with an intelligent diagnosis support tool based on the use of a robust
set of selected biomarkers, which simultaneously distinguishes among different cancerrelated
skin states. To achieve this, a multi-platform combination of microarray datasets
from Affymetrix and Illumina manufacturers was carried out. This integration is expected to
strengthen the statistical robustness of the study as well as the finding of highly-reliable skin
cancer biomarkers. Specifically, the designed operation pipeline has allowed the identification
of a small subset of 17 differentially expressed genes (DEGs) from which to distinguish
among 7 involved skin states. These genes were obtained from the assessment of a number
of potential batch effects on the gene expression data. The biological interpretation of these
genes was inspected in the specific literature to understand their underlying information in
relation to skin cancer. Finally, in order to assess their possible effectiveness in cancer diagnosis,
a cross-validation Support Vector Machines (SVM)-based classification including feature
ranking was performed. The accuracy attained exceeded the 92% in overall recognition
of the 7 different cancer-related skin states. The proposed integration scheme is expected
to allow the co-integration with other state-of-the-art technologies such as RNA-seq.
http://hdl.handle.net/10481/55859Wearable Intelligent System for the Diagnosis of Cardiac Diseases Working in Real Time and with Low Energy Cost
http://hdl.handle.net/10481/55355
Wearable Intelligent System for the Diagnosis of Cardiac Diseases Working in Real Time and with Low Energy Cost
Valenzuela Cansino, Olga; Prieto Campos, Beatriz; Delgado-Marquez, Elvira; Pomares Cintas, Héctor Emilio; Rojas Ruiz, Ignacio
Heart disease is currently one of the leading causes of death in developed countries. The
electrocardiogram is an important source of information for identifying these conditions, therefore,
becomes necessary to seek an advanced system of diagnosis based on these signals. In this paper
we used samples of electrocardiograms of MIT-related database with ten types of pathologies and
a rate corresponding to normal (healthy patient), which are processed and used for extraction from
its two branches of a wide range of features. Next, various techniques have been applied to feature
selection based on genetic algorithms, principal component analysis and mutual information. To
carry out the task of intelligent classification, 3 different scenarios have been considered. These
techniques allow us to achieve greater efficiency in the classification methods used, namely support
vector machines (SVM) and decision trees (DT) to perform a comparative analysis between them.
Finally, during the development of this contribution, the use of very non-invasive devices (2 channel
ECG) was analyzed, we could practically classify them as wearable, which would not need
interaction by the user, and whose energy consumption is very small to extend the average life of
the user been on it.
http://hdl.handle.net/10481/55355Numerical Approximation using Evolution PDE Variational Splines
http://hdl.handle.net/10481/50935
Numerical Approximation using Evolution PDE Variational Splines
Kouibia Krichi, Abdelouahed; Pasadas Fernández, Miguel; Belhaj, Zakaria
This article deals with a numerical approximation method using an evolutionary partial differential equation
(PDE) by discrete variational splines in a finite element space. To formulate the problem, we need an evolutionary
PDE equation with respect to the time and the position, certain boundary conditions and a set of
approximating points. We show the existence and uniqueness of the solution and we study a computational
method to compute such a solution. Moreover, we established a convergence result with respect to the time
and the position. We provided several numerical and graphic examples of approximation in order to show
the validity and effectiveness of the presented method.
http://hdl.handle.net/10481/50935