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 [1].
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 question as an inverse problem for an integro-differential equation posed on the half line. We develop firstly a regular dependency 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 rely also on generalized relative entropy estimates and related Poincaré inequalities.
This manuscript describes harmonic generation in semiconductor superlattices, starting from a nonequilibrium Green's functions input to relaxation rate-type analytical approximations for the Boltzmann equation in which imperfections in the structure lead to asymmetric current flow and scattering processes under forward and reverse bias. The resulting current-voltage curves and the predicted consequences on harmonic generation, notably the development of even harmonics, are in good agreement with experiments. Significant output for frequencies close to 1 THz (7th harmonic) at room temperature, after excitation by a 141-GHz input signal, demonstrate the potential of superlattice devices for gigahertz to terahertz applications.
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.
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. in 2009 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
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