Homogenization of the one-dimensional wave equation

We present a method for two-scale model derivation of the periodic homogenization of the one-dimensional wave equation in a bounded domain. It allows for analyzing the oscillations occurring on both microscopic and macroscopic scales. The novelty reported here is on the asymptotic behavior of high frequency waves and especially on the boundary conditions of the homogenized equation. Numerical simulations are reported

Asymptotic analysis of pollution filtration through thin random fissures between two porous media

We describe the asymptotic behaviour of a filtration problem from a contaminated porous medium to a non-contaminated porous medium through thin vertical fissures of fixed height h>0, of random thinness of order {\epsilon} and which are $\epsilon$-periodically distributed. We compute the limit velocity of the flow and the limit flux of pollutant at the interfaces between the two porous media and the intermediate one

Locally periodic unfolding method and two-scale convergence on surfaces of locally periodic microstructures

In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of non-periodic microstructures, especially to derive macroscopic equations for problems posed in domains with perforations distributed non-periodically. Using the methods of locally periodic two-scale convergence (l-t-s) on oscillating surfaces and the locally periodic (l-p) boundary unfolding operator, we are able to analyze differential equations defined on boundaries of non-periodic microstructures and consider non-homogeneous Neumann conditions on the boundaries of perforations, distributed non-periodically

Homogenization of Biomechanical Models for Plant Tissues

In this paper homogenization of a mathematical model for plant tissue biomechanics is presented. The microscopic model constitutes a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equations for fluid flow inside the cells. The chemical process in cells and the elastic properties of cell walls and middle lamella are coupled because elastic moduli depend on densities involved in chemical reactions, whereas chemical reactions depend on mechanical stresses. Using homogenization techniques we derive rigorously a macroscopic model for plant biomechanics. To pass to the limit in the nonlinear reaction terms, which depend on elastic strain, we prove the strong two-scale convergence of the displacement gradient and velocity field

Long Term Stabilization of Expanding Aortic Aneurysms by a Short Course of Cyclosporine A through Transforming Growth Factor-Beta Induction

Abdominal aortic aneurysms (AAAs) expand as a consequence of extracellular matrix destruction, and vascular smooth muscle cell (VSMC) depletion. Transforming growth factor (TGF)-beta 1 overexpression stabilizes expanding AAAs in rat. Cyclosporine A (CsA) promotes tissue accumulation and induces TGF -beta1 and, could thereby exert beneficial effects on AAA remodelling and expansion. In this study, we assessed whether a short administration of CsA could durably stabilize AAAs through TGF-beta induction. We showed that CsA induced TGF-beta1 and decreased MMP-9 expression dose-dependently in fragments of human AAAs in vitro, and in animal models of AAA in vivo. CsA prevented AAA formation at 14 days in the rat elastase (diameter increase: CsA: 131.9±44.2%; vehicle: 225.9±57.0%, P = 0.003) and calcium chloride mouse models (diameters: CsA: 0.72±0.14 mm; vehicle: 1.10±0.11 mm, P = .008), preserved elastic fiber network and VSMC content, and decreased inflammation. A seven day administration of CsA stabilized formed AAAs in rats seven weeks after drug withdrawal (diameter increase: CsA: 14.2±15.1%; vehicle: 45.2±13.7%, P = .017), down-regulated wall inflammation, and increased αSMA-positive cell content. Co-administration of a blocking anti-TGF-beta antibody abrogated CsA impact on inflammation, αSMA-positive cell accumulation and diameter control in expanding AAAs. Our study demonstrates that pharmacological induction of TGF-beta1 by a short course of CsA administration represents a new approach to induce aneurysm stabilization by shifting the degradation/repair balance towards healing

The Role of Mesotocin on Social Bonding in Pinyon Jays

The neuropeptide oxytocin influences mammalian social bonding by facilitating the building and maintenance of parental, sexual, and same‐sex social relationships. However, we do not know whether the function of the avian homologue mesotocin is evolutionarily conserved across birds. While it does influence avian prosocial behavior, mesotocin\u27s role in avian social bonding remains unclear. Here, we investigated whether mesotocin regulates the formation and maintenance of same‐sex social bonding in pinyon jays (Gymnorhinus cyanocephalus), a member of the crow family. We formed squads of four individually housed birds. In the first, “pair‐formation” phase of the experiment, we repeatedly placed pairs of birds from within the squad together in a cage for short periods of time. Prior to entering the cage, we intranasally administered one of three hormone solutions to both members of the pair: mesotocin, oxytocin antagonist, or saline. Pairs received repeated sessions with administration of the same hormone. In the second, “pair‐maintenance” phase of the experiment, all four members of the squad were placed together in a large cage, and no hormones were administered. For both phases, we measured the physical proximity between pairs as our proxy for social bonding. We found that, compared with saline, administering mesotocin or oxytocin antagonist did not result in different proximities in either the pair‐formation or pair‐maintenance phase of the experiment. Therefore, at the dosages and time frames used here, exogenously introduced mesotocin did not influence same‐sex social bond formation or maintenance. Like oxytocin in mammals, mesotocin regulates avian prosocial behavior; however, unlike oxytocin, we do not have evidence that mesotocin regulates social bonds in birds

Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation

We consider the nonlinear Schr\"{o}dinger equation $(-\Delta +V(x))u = \Gamma(x) |u|^{p-1}u$, $x\in \R^n$ with $V(x) = V_1(x) \chi_{\{x_1>0\}}(x)+V_2(x) \chi_{\{x_1<0\}}(x)$ and $\Gamma(x) = \Gamma_1(x) \chi_{\{x_1>0\}}(x)+\Gamma_2(x) \chi_{\{x_1<0\}}(x)$ and with $V_1, V_2, \Gamma_1, \Gamma_2$ periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state $H^1$ solutions (surface gap soliton ground states) for $0<\min \sigma(-\Delta +V)$. Using a concentration compactness argument, we provide an abstract criterion for the existence based on ground state energies of each periodic problem (with $V\equiv V_1, \Gamma\equiv \Gamma_1$ and $V\equiv V_2, \Gamma\equiv \Gamma_2$) as well as a more practical criterion based on ground states themselves. Examples of interfaces satisfying these criteria are provided. In 1D it is shown that, surprisingly, the criteria can be reduced to conditions on the linear Bloch waves of the operators $-\tfrac{d^2}{dx^2} +V_1(x)$ and $-\tfrac{d^2}{dx^2} +V_2(x)$.Comment: definition of ground and bound states added, assumption (H2) weakened (sign changing nonlinearity is now allowed); 33 pages, 4 figure

On the commutability of homogenization and linearization in finite elasticity

We study non-convex elastic energy functionals associated to (spatially) periodic, frame indifferent energy densities with a single non-degenerate energy well at SO(n). Under the assumption that the energy density admits a quadratic Taylor expansion at identity, we prove that the Gamma-limits associated to homogenization and linearization commute. Moreover, we show that the homogenized energy density, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the quadratic term associated to the linearization of the initial energy density

Asymptotics of Eigenvalues and Eigenfunctions for the Laplace Operator in a Domain with Oscillating Boundary. Multiple Eigenvalue Case

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics