December 16, 2008Nonlinear Phenomena in Activator-Inhibitor Systems: Stem Cell Self-Organization and Cardiovascular Calcification
Professor Arik YochelisDepartment of Chemical Engineering, Technion & Department of Chemical Physics, Weizmann Institute of Science
AbstractThe self-organization of patterns is fundamental to biological morphogenesis, although its origin and mechanisms are still unclear. One intriguing morphogenetic mechanism was suggested in the pioneering work of Turing. He suggested that chemical ‘morphogens’, interacting as activators or inhibitors, and diffusing through a medium, give rise to a chemical pre-patterning. Recently, several experimental groups indeed suggested a relationship between activator-inhibitor approach and biological self-organization through examples of spatially extended periodic patterns. In my talk, I expand the spatiotemporal theory of morphogenesis beyond Turings' paradigm by uncovering the pattern selection mechanisms in the nonlinear regime. The methods are taken from dynamical systems and pattern formation theory, including local and global bifurcation theory. In particular these allow prediction and creation of a novel isolated localized self-organization in a preparation of vascular-derived mesenchymal stem cells. The biochemical mechanisms of pattern formation suggest therapeutic strategies applicable to bone formation in atherosclerotic lesions in arteries (where it is pathological) and to the regeneration of trabecular bone (recapitulating normal physiological development).
July 20, 2008Essential spectrum of difference operators on periodic graphsProfessor Vladimir Rabinovich, Instituto Politécnico Nacional, Mexico
May 26, 2008The profile of bubbling solutions of the Q-curvature equation Professor Gilbert Weinstein, Department of Mathematics, University of Alabama at Birmingham
AbstractWe study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvatu- re equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions and the standard bubble.
May 5, 2008Well-posedness of nonlinear diffusion models in image processing Dr. Lorina Dascal, Computer Science Department, Technion
AbstractPartial differential equations (PDEs) have led to an entire new field in image processing and computer vision. PDE-based image processing techniques are mainly used for smoothing and restoration purposes. The Beltrami framework for image processing and analysis introduces a non-linear parabolic problem, called in this context the Beltrami flow. This flow is a particular case of a degenerate parabolic equation. We study in the frame- work for functions of bounded variation, the well-posedness of the Beltrami flow. We prove existence and uniqueness of the weak solution in the BV space using lower semi-continuity results for convex functions of measures.The solution is defined via a variational inequality, following Temam's technique. Joint work with Shoshana Kamin and Nir Sochen.
March 31,2008Extensions of smooth functions and Lipschitz selections in jet spacesDr. Pavel Shvartsman, Technion
February 8, 2008Well-posedness of Einstein-Euler equations in asymptotically flat spacetimeDr. Lavi Karp, ORT Braude College, Israel
AbstractThe Einstein-Euler system consists of two coupled evolution equations: one describing the gravitational fields and the other one the evolution of the fluid. In order to determine the system it is necessary to specify a relation between the pressure and the energy density (equation of state). I will discuss how these equations can be written as symmetric hyperbolic systems, the influence of Einstein constraint equations on the initial data in asymptotically flat space-time and some of the technical tools which enable us obtaining a well-posedness under minimal regularity assumptions. This is a joint work with U. Brauer.
January 14, 2008Using Operational Calculus for modeling Water Waves Prof. Yehuda Agnon, Department of Civil and Environmental Engineering, Technion, Haifa, Israel
AbstractThe study of linear and nonlinear diffraction and shoaling of irrotational water waves by uneven bottom topography has received considerable attention. Phenomena such as edge waves, harbour resonance and surf-beats are a few of its manifestations. Spatio-temporal evolution equations are powerful in the study of focusing and of rogue waves. In the case of slow spatial evolution, one can eliminate the vertical coordinate, z. The solution of a three-dimensional (3D) flow field is replaced by a simpler, (approximate) 2D problem. In spatially homogeneous problems (e.g. rectangular basins with an even bottom), the waves are conveniently represented in the wavenumber domain, leading to the formulation of the Zakharov equation. Non-uniformity in space due to non-periodic spatial evolution or to bottom slope, and the study of more general basins, can be accounted for through representation in the frequency, rather than wavenumber, domain. For linear shoaling, fully dispersive models include the mild slope equation (MSE), the extended, and the modified MSE (Chamberlain and Porter, 1995). Most of the work on nonlinear shoaling has been based on the assumption of weak dispersion, either by using Boussinesq type models or by assuming narrow spectra in Schrödinger equation models. This is because the full dispersion operator is an integral operator in space and its pseudo-differential representation involves spatial derivatives to infinite order.Agnon et al. (1993, 1998) and Kaihatu and Kirby (1995) have studied a nonlinear MSE - in which the linear dispersion is exact, but the nonlinear interaction is only accurate for resonant interaction. This, apparently, limited the model's applicability to near resonance quadratic interaction, since it introduced an error in the bound waves that was too large to allow extension of the model to include cubic (four-wave) interaction. Agnon and Sheremet (1997) have derived a stochastic nonlinear MSE. Mei (1999) has derived a nonlinear MSE for long waves induced by short waves. Shemer et al. (2001) have derived a spatial Zakharov equation. Rasmussen and Stiassnie (2001) have derived a discretized spatio-temporal model on an even bottom.Accounting for the dispersive nature of water waves in spatial models complicates the solution and requires consideration of high order derivatives (essentially to any order) of the depth and of the potential. The inherent difficulty of the problem stems from the need to account for non-local interaction. A new model for linear and nonlinear shoaling has been recently derived (Agnon, 1999; Agnon and Pelinovsky, 2001), using operational calculus. It systematically accounts for the effect of terms in high order derivatives of the bathymetry without calculating them. This is done by utilizing the fact that the dynamics of the system is governed by resonant interaction. After deriving a pseudo-differential equation, we expand the nonlocal operator in a small parameter - the detuning from resonance. The result is that the nonlocal operator is approximated by low order differential operators. Here we extend this approach to obtain more accurate nonlinear shoaling .The present work differs from the previous derivations of MSE and its linear and nonlinear extensions: no assumption is made regarding the vertical structure of the wave field (except for the lack of 'global' evanescent modes). This results in a model of very general validity. The derivation provides insight regarding in which regimes the use of existing and new models is appropriate. We study an augmented nonlinear MSE which accurately accounts for the bound waves (subharmonics and superharmonics). This enables the enhancement of the nonlinear MSE to include cubic interaction. Cubic interaction is important in the evolution of ocean waves and is known to play a central role in subharmonic resonance of edge waves. We derive the waves which are induced by the bottom variations, and show that there exists no general MSE-type equation for describing Class II Bragg resonance (i.e. linear in the wave amplitude and quadratic in the bottom variation). We study Class III Bragg resonance (i.e. quadratic in the wave amplitude and linear in the bottom variation) over bathymetries that are not gentle, and derive an extended nonlinear MSE, and a modified nonlinear MSE. The performance of the different models in calculating resonant double-frequency harmonic generation on an oscillatory bottom variation is compared. Spatio-temporal equations are obtained directly from the spatial evolution equations by discretization of the frequency spectrum.