Exact results for a tunnel-coupled pair of trapped Bose-Einstein condensates

A model describing coherent quantum tunnelling between two trapped Bose-Einstein condensates is discussed. It is not well known that the model admits an exact solution, obtained some time ago, with the energy spectrum derived through the algebraic Bethe ansatz. An asymptotic analysis of the Bethe ansatz equations leads us to explicit expressions for the energies of the ground and the first excited states in the limit of weak tunnelling and all energies for strong tunnelling. The results are used to extract the asymptotic limits of the quantum fluctuations of the boson number difference between the two Bose-Einstein condensates and to characterize the degree of coherence in the system.

Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.

Spectral properties of the Laplacian of multiplex networks.

One of the more challenging tasks in the understanding of dynamical properties of models on top of complex networks is to capture the precise role of multiplex topologies. In a recent paper, Gómez et al. [ Phys. Rev. Lett. 110 028701 (2013)], some of the authors proposed a framework for the study of diffusion processes in such networks. Here, we extend the previous framework to deal with general configurations in several layers of networks and analyze the behavior of the spectrum of the Laplacian of the full multiplex. We derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them. Capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues. This spectrum is used to gain insight into physical phenomena on top of multiplex, specifically, diffusion processes and synchronizability.

Modèles de flammelette en combustion turbulente avec extinction et réallumage : étude asymptotique et numérique, estimation d’erreur a posteriori et modélisation adaptative

On s’intéresse ici aux erreurs de modélisation liées à l’usage de modèles de flammelette sous-maille en combustion turbulente non prémélangée. Le but de cette thèse est de développer une stratégie d’estimation d’erreur a posteriori pour déterminer le meilleur modèle parmi une hiérarchie, à un coût numérique similaire à l’utilisation de ces mêmes modèles. Dans un premier temps, une stratégie faisant appel à un estimateur basé sur les résidus pondérés est développée et testée sur un système d’équations d’advection-diffusion-réaction. Dans un deuxième temps, on teste la méthodologie d’estimation d’erreur sur un autre système d’équations, où des effets d’extinction et de réallumage sont ajoutés. Lorsqu’il n’y a pas d’advection, une analyse asymptotique rigoureuse montre l’existence de plusieurs régimes de combustion déjà observés dans les simulations numériques. Nous obtenons une approximation des paramètres de réallumage et d’extinction avec la courbe en «S», un graphe de la température maximale de la flamme en fonction du nombre de Damköhler, composée de trois branches et d’une double courbure. En ajoutant des effets advectifs, on obtient également une courbe en «S» correspondant aux régimes de combustion déjà identifiés. Nous comparons les erreurs de modélisation liées aux approximations asymptotiques dans les deux régimes stables et établissons une nouvelle hiérarchie des modèles en fonction du régime de combustion. Ces erreurs sont comparées aux estimations données par la stratégie d’estimation d’erreur. Si un seul régime stable de combustion existe, l’estimateur d’erreur l’identifie correctement ; si plus d’un régime est possible, on obtient une fac˛on systématique de choisir un régime. Pour les régimes où plus d’un modèle est approprié, la hiérarchie prédite par l’estimateur est correcte.

Stability of preconditioned finite volume schemes at low Mach numbers

In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step Dt does not satisfy the requirement to be O(M2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Dt=O(M), M to 0, which results from the well-known CFL-condition. We present a comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M-2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. Thereby, we present statements for both the standard preconditioner used by Guillard and Viozat [4] and the more general one due to Turkel [21]. The theoretical results are after wards confirmed by numerical experiments.

Numerical Simulation of Flows at Low Mach Numbers with Heat Sources

This work is concerned with finite volume methods for flows at low mach numbers which are under buoyancy and heat sources. As a particular application, fires in car tunnels will be considered. To extend the scheme for compressible flow into the low Mach number regime, a preconditioning technique is used and a stability result on this is proven. The source terms for gravity and heat are incorporated using operator splitting and the resulting method is analyzed.

Mathematical model for low density lipoprotein (LDL) endocytosis by hepatocytes

Individuals with elevated levels of plasma low density lipoprotein (LDL) cholesterol (LDL-C) are considered to be at risk of developing coronary heart disease. LDL particles are removed from the blood by a process known as receptor-mediated endocytosis, which occurs mainly in the liver. A series of classical experiments delineated the major steps in the endocytotic process; apolipoprotein B-100 present on LDL particles binds to a specific receptor (LDL receptor, LDL-R) in specialized areas of the cell surface called clathrin-coated pits. The pit comprising the LDL-LDL-R complex is internalized forming a cytoplasmic endosome. Fusion of the endosome with a lysosome leads to degradation of the LDL into its constituent parts (that is, cholesterol, fatty acids, and amino acids), which are released for reuse by the cell, or are excreted. In this paper, we formulate a mathematical model of LDL endocytosis, consisting of a system of ordinary differential equations. We validate our model against existing in vitro experimental data, and we use it to explore differences in system behavior when a single bolus of extracellular LDL is supplied to cells, compared to when a continuous supply of LDL particles is available. Whereas the former situation is common to in vitro experimental systems, the latter better reflects the in vivo situation. We use asymptotic analysis and numerical simulations to study the longtime behavior of model solutions. The implications of model-derived insights for experimental design are discussed.

Bode's maximum available feedback and phase margin

'Maximum Available Feedback' is Bode's term for the highest possible loop gain over a given bandwidth, with specified stability margins, in a single loop feedback system. His work using asymptotic analysis allowed Bode to develop a methodology for achieving this. However, the actual system performance differs from that specified, due to the use of asymptotic approximations, and the author[2] has described how, for instance, the actual phase margin is often much lower than required when the bandwidth is high, and proposed novel modifications to the asymptotes to address the issue. This paper gives some new analysis of such systems, showing that the method also contravenes Bode's definition of phase margin, and shows how the author's modifications can be used for different amounts of bandwidth.

Asymptotics for Gegenbauer-Sobolev orthogonal polynomials associated with non-coherent pairs of measures

Inner products of the type < f, g >(S) = < f, g >psi(0) + < f', g'>psi(1), where one of the measures psi(0) or psi(1) is the measure associated with the Gegenbauer polynomials, are usually referred to as Gegenbauer-Sobolev inner products. This paper deals with some asymptotic relations for the orthogonal polynomials with respect to a class of Gegenbauer-Sobolev inner products. The inner products are such that the associated pairs of symmetric measures (psi(0), psi(1)) are not within the concept of symmetrically coherent pairs of measures.

Two-dimensional integrable generalization of the Camassa-Holm equation

In this letter we discuss the (2 + 1)-dimensional generalization of the Camassa-Holm equation. We require that this generalization be, at the same time, integrable and physically derivable under the same asymptotic analysis as the original Camassa-Holm equation. First, we find the equation in a perturbative calculation in shallow-water theory. We then demonstrate its integrability and find several particular solutions describing (2 + 1) solitary-wave like solutions. © 1999 Published by Elsevier Science B.V. All rights reserved.

Optimal Boussinesq model for shallow-water waves interacting with a microstructure

In this paper, we consider the propagation of water waves in a long-wave asymptotic regime, when the bottom topography is periodic on a short length scale. We perform a multiscale asymptotic analysis of the full potential theory model and of a family of reduced Boussinesq systems parametrized by a free parameter that is the depth at which the velocity is evaluated. We obtain explicit expressions for the coefficients of the resulting effective Korteweg-de Vries (KdV) equations. We show that it is possible to choose the free parameter of the reduced model so as to match the KdV limits of the full and reduced models. Hence the reduced model is optimal regarding the embedded linear weakly dispersive and weakly nonlinear characteristics of the underlying physical problem, which has a microstructure. We also discuss the impact of the rough bottom on the effective wave propagation. In particular, nonlinearity is enhanced and we can distinguish two regimes depending on the period of the bottom where the dispersion is either enhanced or reduced compared to the flat bottom case. © 2007 The American Physical Society.

Ciclos limites e a equação de van der Pol

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Modelagem e imageamento 2.5D no domínio do tempo através de diferenças finitas

A modelagem 2.5D consiste em simular a propagação do campo de ondas em 3D em meios com simetria de translação em uma direção. Nesta tese esta abordagem é formulada para meios elásticos e anisotrópicos com classe de simetria arbitrária e a geometria de aquisição não precisa coincidir com um plano de simetria do meio. A migração por reversão no tempo do campo de ondas é formulada e implementada através de diferenças finitas 2.5D. Para reduzir os efeitos de retro-espalhamento e melhorar a recuperação da amplitude dos eventos migrados, propomos uma nova condição de imagem para migração reversa no tempo baseada na análise assintótica da condição de imagem clássica por correlação cruzada. Experimentos numéricos indicam que a migração reversa no tempo 2.5D com a nova condição de imagem proposta, melhora a resolução da imagem em relação à migração reversa no tempo 2D e reduz acentuadamente os ruídos causados por retro-espalhamento.

Differential orthogonality: Laguerre and Hermite cases with applications

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Variations in Structure Explain the Viscometric Behavior of AOT Microemulsions at Low Water/AOT Molar Ratios

The viscosity of AOT/water/decane water-in-oil microemulsions exhibits a well-known maximum as a function of water/AOT molar ratio, which is usually attributed to increased attractions among nearly spherical droplets. The maximum can be removed by adding salt or by changing the oil to CCl4. Systematic small-angle X-ray scattering (SAXS) measurements have been used to monitor the structure of the microemulsion droplets in the composition regime where the maximum appears. On increasing the droplet concentration, the scattering intensity is found to scale with the inverse of the wavevector, a behavior which is consistent with cylindrical structures. The inverse wavevector scaling is not observed when the molar ratio is changed, moving the system away from the value corresponding to the viscosity maximum. It is also not present in the scattering from systems containing enough added salt to essentially eliminate the viscosity maximum. An asymptotic analysis of the SAXS data, complemented by some quantitative modeling, is consistent with cylindrical growth of droplets as their concentration is increased. Such elongated structures are familiar from related AOT systems in which the sodium counterion has been exchanged for a divalent one. However, the results of this study suggest that the formation of non-spherical aggregates at low molar ratios is an intrinsic property of AOT.