2,543 research outputs found
Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies
Resistance to chemotherapies, particularly to anticancer treatments, is an
increasing medical concern. Among the many mechanisms at work in cancers, one
of the most important is the selection of tumor cells expressing resistance
genes or phenotypes. Motivated by the theory of mutation-selection in adaptive
evolution, we propose a model based on a continuous variable that represents
the expression level of a resistance gene (or genes, yielding a phenotype)
influencing in healthy and tumor cells birth/death rates, effects of
chemotherapies (both cytotoxic and cytostatic) and mutations. We extend
previous work by demonstrating how qualitatively different actions of
chemotherapeutic and cytostatic treatments may induce different levels of
resistance. The mathematical interest of our study is in the formalism of
constrained Hamilton-Jacobi equations in the framework of viscosity solutions.
We derive the long-term temporal dynamics of the fittest traits in the regime
of small mutations. In the context of adaptive cancer management, we also
analyse whether an optimal drug level is better than the maximal tolerated
dose
Investigating Biological Matter with Theoretical Nuclear Physics Methods
The internal dynamics of strongly interacting systems and that of
biomolecules such as proteins display several important analogies, despite the
huge difference in their characteristic energy and length scales. For example,
in all such systems, collective excitations, cooperative transitions and phase
transitions emerge as the result of the interplay of strong correlations with
quantum or thermal fluctuations. In view of such an observation, some
theoretical methods initially developed in the context of theoretical nuclear
physics have been adapted to investigate the dynamics of biomolecules. In this
talk, we review some of our recent studies performed along this direction. In
particular, we discuss how the path integral formulation of the molecular
dynamics allows to overcome some of the long-standing problems and limitations
which emerge when simulating the protein folding dynamics at the atomistic
level of detail.Comment: Prepared for the proceedings of the "XII Meeting on the Problems of
Theoretical Nuclear Physics" (Cortona11
Differential Equations arising from Organising Principles in Biology
This workshop brought together experts in modeling and analysis of organising principles of multiscale biological systems such as cell assemblies, tissues and populations. We focused on questions arising in systems biology and medicine which are related to emergence, function and control of spatial and inter-individual heterogeneity in population dynamics. There were three main areas represented of differential equation models in mathematical biology. The first area involved the mathematical description of structured populations. The second area concerned invasion, pattern formation and collective dynamics. The third area treated the evolution and adaptation of populations, following the Darwinian paradigm. These problems led to differential equations, which frequently are non-trivial extensions of classical problems. The examples included but were not limited to transport-type equations with nonlocal boundary conditions, mixed ODE-reaction-diffusion models, nonlocal diffusion and cross-diffusion problems or kinetic equations
Control theory: history, mathematical achievements and perspectives
These notes are devoted to present some of the mathematical
milestones of Control Theory. To do that, we first overview its origins
and some of the main mathematical achievements. Then, we discuss the
main domains of Sciences and Technologies where Control Theory arises
and applies. This forces us to address modelling issues and to distinguish
between the two main control theoretical approaches, controllability and
optimal control, discussing the advantages and drawbacks of each of
them. In order to give adequate formulations of the main questions,
we have introduced some of the most elementary mathematical material,
avoiding unnecessary technical difficulties and trying to make the paper
accessible to a large class of readers. The subjects we address range
from the basic concepts related to the dynamical systems approach
to (linear and nonlinear) Mathematical Programming and Calculus of
Variations. We also present a simplified version of the outstanding
results by Kalman on the controllability of linear finite dimensional
dynamical systems, Pontryaguin’s maximum principle and the principle
of dynamical programming. Some aspects related to the complexity of
modern control systems, the discrete versus continuous modelling, the
numerical approximation of control problems and its control theoretical
consequences are also discussed. Finally, we describe some of the major
challenging applications in Control Theory for the XXI Century. They
will probably influence strongly the development of this discipline in the
near future.Ministerio de Ciencia y TecnologÃ
Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations
We construct quantum algorithms to compute the solution and/or physical
observables of nonlinear ordinary differential equations (ODEs) and nonlinear
Hamilton-Jacobi equations (HJE) via linear representations or exact mappings
between nonlinear ODEs/HJE and linear partial differential equations (the
Liouville equation and the Koopman-von Neumann equation). The connection
between the linear representations and the original nonlinear system is
established through the Dirac delta function or the level set mechanism. We
compare the quantum linear systems algorithms based methods and the quantum
simulation methods arising from different numerical approximations, including
the finite difference discretisations and the Fourier spectral discretisations
for the two different linear representations, with the result showing that the
quantum simulation methods usually give the best performance in time
complexity. We also propose the Schr\"odinger framework to solve the Liouville
equation for the HJE, since it can be recast as the semiclassical limit of the
Wigner transform of the Schr\"odinger equation. Comparsion between the
Schr\"odinger and the Liouville framework will also be made.Comment: quantum algorithms,linear representations,noninea
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