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4D Embryogenesis image analysis using PDE methods of image processing

Paul Bourgine, Róbert Čunderlík, Olga Drblíková-Stašová, Karol Mikula, Mariana Remešíková, Nadine Peyriéras, Barbara Rizzi, Alessandro Sarti (2010)


In this paper, we introduce a set of methods for processing and analyzing long time series of 3D images representing embryo evolution. The images are obtained by in vivo scanning using a confocal microscope where one of the channels represents the cell nuclei and the other one the cell membranes. Our image processing chain consists of three steps: image filtering, object counting (center detection) and segmentation. The corresponding methods are based on numerical solution of nonlinear PDEs, namely...

A decay estimate for a class of hyperbolic pseudo-differential equations

Sandra Lucente, Guido Ziliotti (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider the equation u t i Λ u = 0 , where Λ = λ D x is a first order pseudo-differential operator with real symbol λ ξ . Under a suitable convexity assumption on λ we find the decay properties for u t , x . These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem u t i Λ u = f u , u 0 , x = g x . If f u is a smooth function which satisfies f u u p near u = 0 , and g is small in suitably Sobolev norm, we prove global existence theorems provided p is greater than a critical exponent.

A note on critical times of 2 × 2 quasilinear hyperbolic systems

Ivan Straškraba (1984)

Aplikace matematiky

In this paper the exact formula for the critical time of generating discontinuity (shock wave) in a solution of a 2 × 2 quasilinear hyperbolic system is derived. The applicability of the formula in the engineering praxis is shown on one-dimensional equations of isentropic non-viscous compressible fluid flow.

A note on poroacoustic traveling waves under Darcy's law: Exact solutions

P. M. Jordan, J. K. Fulford (2011)

Applications of Mathematics

A mathematical analysis of poroacoustic traveling wave phenomena is presented. Assuming that the fluid phase satisfies the perfect gas law and that the drag offered by the porous matrix is described by Darcy's law, exact traveling wave solutions (TWS)s, as well as asymptotic/approximate expressions, are derived and examined. In particular, stability issues are addressed, shock and acceleration waves are shown to arise, and special/limiting cases are noted. Lastly, connections to other fields are...

A robust entropy−satisfying finite volume scheme for the isentropic Baer−Nunziato model

Frédéric Coquel, Jean-Marc Hérard, Khaled Saleh, Nicolas Seguin (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing...

A singular perturbation problem in a system of nonlinear Schrödinger equation occurring in Langmuir turbulence

Cédric Galusinski (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The aim of this work is to establish, from a mathematical point of view, the limit α → +∞ in the system i t E + ( . E ) - α 2 × × E = - | E | 2 σ E , where E : 3 3 . This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma Physics. The L2-subcritical σ (that is σ ≤ 2/3) and the H1-subcritical σ (that is σ ≤ 2) are studied. In the physical case σ = 1, the limit is then studied for the H 1 ( 3 ) norm.

A Transmission Strategy for Hyperbolic Internal Waves of Small Width

Olivier Gues, Jeffrey Rauch (2005/2006)

Séminaire Équations aux dérivées partielles

Semilinear hyperbolic problems with source terms piecewise smooth and discontinuous across characteristic surfaces yield similarly piecewise smooth solutions. If the discontinuous source is replaced with a smooth transition layer, the discontinuity of the solution is replaced by a smooth internal layer. In this paper we describe how the layer structure of the solution can be computed from the layer structure of the source in the limit of thin layers. The key idea is to use a transmission problem...

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Mehmet Ersoy, Frédéric Golay, Lyudmyla Yushchenko (2013)

Open Mathematics

We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to...

Air entrainment in transient flows in closed water pipes : A two-layer approach

C. Bourdarias, M. Ersoy, Stéphane Gerbi (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme...

An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment

François Bouchut, Tomás Morales de Luna (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in...

Approche visqueuse de solutions discontinues de systèmes hyperboliques semilinéaires

Franck Sueur (2006)

Annales de l’institut Fourier

On s’intéresse à des systèmes symétriques hyperboliques multidimensionnels en présence d’une semilinéarité. Il est bien connu que ces systèmes admettent des solutions discontinues, régulières de part et d’autre d’une hypersurface lisse caractéristique de multiplicité constante. Une telle solution u 0 étant donnée, on montre que u 0 est limite quand ε 0 de solutions ( u ε ) ε ] 0 , 1 ] du système perturbé par une viscosité de taille ε . La preuve utilise un problème mixte parabolique et des développements de couches limites....

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