Jorge P. Zubelli

Image Reconstruction of the Interior of Bodies That Diffuse Radiation

A method for reconstructing images from projections is described. The unique aspect of the procedure is that the reconstruction of the internal structure can be carried out for objects that diffuse the incident radiation. The method may be used with photons, phonons, neutrons, and many other kinds of radiation. The procedure has applications to medical imaging, industrial imaging, and geophysical imaging.

Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV

We study a certain family of Schrödinger operators whose eigenfunctions ϕ(χ, λ) satisfy a differential equation in the spectral parameter λ of the formB(λ,∂λ)ϕ=Θ(x)ϕ. We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class ofbispectral potentials. This extends and complements a result of Duistermaat and Grünbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy.

On the inverse problem for a size-structured population model

We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such a question as an inverse problem for an integro-differential equation posed on the half line. We develop firstly a regular dependence theory for the solution in terms of the coefficients and, secondly, a novel regularization technique for tackling this inverse problem which takes into account the specific nature of the equation. Our results also rely on generalized relative entropy estimates and related Poincare inequalities.

Numerical Solution of an Inverse Problem in Size-Structured Population Dynamics

We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We propose a new regularization technique based on a filtering approach. We prove convergence of the algorithm and validate the theoretical results by implementing numerical simulations, based on classical techniques. We compare the results for direct and inverse problems, for the filtering method and for the quasi-reversibility method proposed in Perthame and Zubelli (2007 Inverse Problems 23 1037).

Differential equations in the spectral parameter for matrix differential operators

We use matrix Darboux transformation to generate a class of matrix differential operators L that have the following property: There exist a family ψ(x,k) of eigenfunctions of L also satisfying a differential equation in the spectral parameter k of the form B(k,∂k)ψ = Θ(x)ψ, whereB(∂k, k is a diffrential and Θ is a non-constant function of x. We also obtain Nth-order scalar operators which have this property.

A Structured Population Model of Cell Differentiation

We introduce and analyze several aspects of a new model for cell differentiation. It assumes that differentiation of progenitor cells is a continuous process. From the mathematical point of view, it is based on partial differential equations of transport type. Specifically, it consists of a structured population equation with a nonlinear feedback loop. This models the signaling process due to cytokines, which regulate the differentiation and proliferation process. We compare the continuous model to its discrete counterpart, a multi-compartmental model of a discrete collection of cell subpopulations recently proposed by Marciniak-Czochra et al. [17] to investigate the dynamics of the hematopoietic system. We obtain uniform bounds for the solutions, characterize steady state solutions, and analyze their linearized stability. We show how persistence or extinction might occur according to values of parameters that characterize the stem cells self-renewal. We also perform numerical simulations and discuss the qualitative behavior of the continuous model vis a vis the discrete one.

Rational solutions of nonlinear evolution equations, vertex operators, and bispectrality

Abstract We prove that if q ( x , t 2 , …, t m ) and r ( x , t 2 , …, t m ) are certain rational solutions of the AKNS hierarchy, then there are eigenfunctions ψ ( x , k ) of the AKNS/ZS operator L= ∂ x −q r −∂ x satisfying a differential equation in the spectral parameter k of the form B ( k , ∂ k ) ψ = Θ ( x ) ψ , where B ( ∂ k , k ) is a matrix differential operator, independent of x , and Θ is a nonconstant function of x . We also discuss the relation between this result and a similar one for the rational solutions of the Schrodinger operator with potentials in the manifold of rational solutions of the KdV hierarchy.

Global Stability for a Class of Virus Models with Cytotoxic T Lymphocyte Immune Response and Antigenic Variation

We study the global stability of a class of models for in-vivo virus dynamics that take into account the Cytotoxic T Lymphocyte immune response and display antigenic variation. This class includes a number of models that have been extensively used to model HIV dynamics. We show that models in this class are globally asymptotically stable, under mild hypothesis, by using appropriate Lyapunov functions. We also characterise the stable equilibrium points for the entire biologically relevant parameter range. As a by-product, we are able to determine what is the diversity of the persistent strains.

Tangent Graeffe iteration

Summary. Graeffe iteration was the choice algorithm for solving univariate polynomials in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modern digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after N iterations and the running cost. The main ideas from which this algorithm is built are: classical Graeffe iteration and Newton Diagrams, changes of scale (renormalization), and replacement of a difference technique by a differentiation one. The algorithm was implemented successfully and a number of numerical experiments are displayed.

Diffuse tomography: computational aspects of the isotropic case

In the context of diffuse tomography, the authors study some computational aspects of two algorithms for the reconstruction of the absorption parameters of a two-dimensional square lattice from a set of external measurements when the movement of the probing particles is isotropic.