Analysis of a time-dependent problem of mixed migration and population dynamics

In this work, we consider a system of differential equations modeling the dynamics of some populations of preys and predators, moving in space according to rapidly oscillating time-dependent transport terms, and interacting with each other through a Lotka-Volterra term. These two contributions naturally induce two separated time-scales in the problem. A generalized center manifold theorem is derived to handle the situation where the linear terms are depending on the fast time in a periodic way. The resulting equations are then amenable to averaging methods. As a product of these combined techniques, one obtains an autonomous differential system in reduced dimension whose dynamics can be analyzed in a much simpler way as compared to original equations. Strikingly enough, this system is of Lotka-Volterra form with modified coefficients. Besides, a higher order perturbation analysis allows to show that the oscillations on the original model destabilize the cycles of the averaged Volterra system in a way that can be explicitely computed

Semiclassical resolvent estimates for Schroedinger operators with Coulomb singularities

Consider the Schroedinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator's resolvent at a positive energy E are bounded by O(1/h) if and only if the associated Hamilton flow is non-trapping at energy E. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.Comment: 39 pages, no figures, corrected versio

Second order averaging for the nonlinear Schroedinger equation with strongly anisotropic potential

International audienceWe consider the three dimensional Gross-Pitaevskii equation (GPE) describing a Bose-Einstein Condensate (BEC) which is highly confi ned in vertical z direction. The highly confi ned potential induces high oscillations in time. If the confi nement in the z direction is a harmonic trap (which is widely used in physical experiments), the very special structure of the spectrum of the confi nement operator will imply that the oscillations are periodic in time. Based on this observation, it can be proved that the GPE can be averaged out with an error of order of epsilon, which is the typical period of the oscillations. In this article, we construct a more accurate averaged model, which approximates the GPE up to errors of order epsilon squared. Then, expansions of this model over the eigenfunctions (modes) of the vertical Hamiltonian Hz are given in convenience of numerical application. Effi cient numerical methods are constructed for solving the GPE with cylindrical symmetry in 3D and the approximation model with radial symmetry in 2D, and numerical results are presented for various kinds of initial data

Mammographic texture synthesis: second-generation clustered lumpy backgrounds using a genetic algorithm

Synthetic yet realistic images are valuable for many applications in visual sciences and medical imaging. Typically, investigators develop algorithms and adjust their parameters to generate images that are visually similar to real images. In this study, we used a genetic algorithm and an objective, statistical similarity measure to optimize a particular texture generation algorithm, the clustered lumpy backgrounds (CLB) technique, and synthesize images mimicking real mammograms textures. We combined this approach with psychophysical experiments involving the judgment of radiologists, who were asked to qualify the visual realism of the images. Both objective and psychophysical approaches show that the optimized versions are significantly more realistic than the previous CLB model. Anatomical structures are well reproduced, and arbitrary large databases of mammographic texture with visual and statistical realism can be generated. Potential applications include detection experiments, where large amounts of statistically traceable yet realistic images are needed. (C) 2008 Optical Society of America

Masses detection in breast tomosynthesis and digital mammography: a model observer study

In this study, we adapt and apply model observers within the framework of realistic detection tasks in breast tomosynthesis (BT). We use images consisting of realistic masses digitally embedded in real patient anatomical backgrounds, and we adapt specific model observers that have been previously applied to digital mammography (DM). We design alternative forced-choice experiments (AFC) studies for DM and BT tasks in the signal known exactly but variable (SKEV) framework. We compare performance of various linear model observers (non-prewhitening matched filter with an eye filter, and several channelized Hotelling observers (CHO) against human. A good agreement in performance between human and model observers can be obtained when an appropriate internal noise level is adopted. Models achieve the same detection performance across BT and DM with about three times less projected signal intensity in BT than in DM (humans: 3.8), due to the anatomical noise reduction in BT. We suggest that, in the future, model observers can potentially be used as an objective tool for automating the optimization of BT acquisition parameters or reconstruction algorithms, or narrowing a wide span of possible parameter combinations, without requiring human observers studies