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2012

Volume 72, Issue 2, pp. 489-711


Stability of MultiComponent Biological Membranes

Sefi Givli, Ha Giang, and Kaushik Bhattacharya

SIAM J. Appl. Math. 72, pp. 489-511 (23 pages)

Online Publication Date: March 01, 2012

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Equilibrium equations and stability conditions are derived for a general class of multicomponent biological membranes. The analysis is based on a generalized Helfrich energy that accounts for geometry through the stretch and curvature, the composition, and the interaction between geometry and composition. The use of nonclassical differential operators and related integral theorems in conjunction with appropriate composition and mass conserving variations simplify the derivations. We show that instabilities of multicomponent membranes are significantly different from those in single component membranes, as well as those in systems undergoing spinodal decomposition in flat spaces. This is due to the intricate coupling between composition and shape as well as the nonuniform tension in the membrane. Specifically, critical modes have high frequencies unlike single component vesicles and stability depends on system size unlike in systems undergoing spinodal decomposition in flat space. An important implication is that small perturbations may nucleate localized but very large deformations. We show that the predictions of the analysis are in qualitative agreement with experimental observations.

The Eshelby Theorem and Application to the Optimization of an Elastic Patch

G. Leugering, S. Nazarov, F. Schury, and M. Stingl

SIAM J. Appl. Math. 72, pp. 512-534 (23 pages)

Online Publication Date: March 13, 2012

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We present the analysis for finding optimal locations and rotations of anisotropic material inclusions in a matrix material by using the polarization matrix. We compare different types of cost functionals, in particular local ones, and show their respective differences. We use the Eshelby theorem and the representation of stresses based on the link matrix. As an analytical model reduction technique, this allows for efficient numerical computation which is demonstrated for two selected examples.

Random Transmission Radii in Greedy Routing Models for Ad Hoc Sensor Networks

H. P. Keeler and P. G. Taylor

SIAM J. Appl. Math. 72, pp. 535-557 (23 pages)

Online Publication Date: March 29, 2012

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We study message advancement in ad hoc sensor networks when each node has a randomly distributed transmission radius. The effects of this assumption are outlined and communication considerations are addressed. We introduce communication models and study some specific examples of random transmission models. We derive asymptotic expressions for single hop moments. Based on previous work, we derive multiple integral expressions and evaluate them with quasi-Monte Carlo methods. Theoretical analysis is compared with routing simulations. We discuss the mathematical difficulties faced when analyzing certain models, and give future model extensions and research directions.

A Unified Variational Formulation for the Parabolic-Elliptic Eddy Current Equations

Lilian Arnold and Bastian Harrach

SIAM J. Appl. Math. 72, pp. 558-576 (19 pages)

Online Publication Date: April 04, 2012

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Transient excitation currents generate electromagnetic fields which, in turn, induce electric currents in proximal conductors. For slowly varying fields, this can be described by the eddy current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell's equations. The eddy current equations are of parabolic-elliptic type: In insulating regions, the field instantaneously adapts to the excitation (quasistationary elliptic behavior), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behavior). For fixed conductivity, the equations are well studied. However, little rigorous mathematical results are known for the solution's dependence on the conductivity, in particular for the solution's sensitivity with respect to the equation changing from elliptic to parabolic type. In this work, we derive a new unified variational formulation for the eddy current equations that is uniformly coercive with respect to the conductivity. We then apply our new unified formulation to study the case when the conductivity approaches zero and rigorously linearize the eddy current equations around a non-conducting domain with respect to the introduction of a conducting object.

Extracting Solitons from Noisy Pulses

Jinglai Li and William L. Kath

SIAM J. Appl. Math. 72, pp. 577-593 (17 pages)

Online Publication Date: April 12, 2012

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We describe an iterative method that extracts the underlying soliton from a noisy pulse. The method is formulated as a functional iteration: at each step, the soliton component of the difference between the noisy pulse and the current underlying soliton is determined via soliton perturbation theory; this is then added to the soliton, and the process is repeated. We show that this iteration converges if the perturbation is not too large, and we give the specific types of deviations which most easily cause the iteration to fail to converge. As an example of the method's use, we apply it to obtain improved statistics of the amplitude, phase, frequency, and position of a soliton propagating in an optical fiber in the presence of amplifier noise.

Analysis of a Nonlocal Model for Spontaneous Cell Polarization

Vincent Calvez, Rhoda J. Hawkins, Nicolas Meunier, and Raphael Voituriez

SIAM J. Appl. Math. 72, pp. 594-622 (29 pages)

Online Publication Date: April 12, 2012

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In this work, we investigate the dynamics of a nonlocal model describing spontaneous cell polarization. It consists of a drift-diffusion equation set in the half-space, with the coupling involving the trace value on the boundary. We characterize the following behaviors in the one-dimensional case: solutions are global if the mass is below the critical mass and they blow up in finite time above the critical mass. The higher-dimensional case is also discussed. The results are reminiscent of the classical Keller–Segel system, but critical spaces are different ($L^N$ instead of $L^{N/2}$ due to the coupling on the boundary). In addition, in the one-dimensional case we prove quantitative convergence results using relative entropy techniques. This work is complemented with a more realistic model that takes into account dynamical exchange of molecular content at the boundary. In the one-dimensional case we prove that blow-up is prevented. Furthermore, density converges toward a nontrivial stationary configuration.

On a Functional-Differential Equation Arising from a Traffic Flow Model

Reinhard Illner and Geoffrey McGregor

SIAM J. Appl. Math. 72, pp. 623-645 (23 pages)

Online Publication Date: April 12, 2012

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We provide a derivation in the context of a traffic flow model, as well as both analytical and numerical studies of the functional-differential equation $(z(s)+\alpha)^{2}z'(s)=\beta(z(s+z(s))-z(s)).$ Here, $\alpha$ and $\beta$ are positive parameters, and we are in particular investigating the existence and properties of nonconstant “traveling wave”–type solutions.

High Frequency Scattering by a Classically Invisible Body

E. Lakshtanov, B. D.Sleeman, and B. Vainberg

SIAM J. Appl. Math. 72, pp. 646-669 (24 pages)

Online Publication Date: April 17, 2012

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We consider a polyhedron with zero classical resistance, i.e., a polyhedron invisible to an observer viewing only the paths of geometrical optics rays. The corresponding problem of scattering of plane waves by the polyhedron is studied. The quasi-classical approximation is obtained and justified in the case of impedance boundary conditions with nonzero absorbtion. It is shown that the total momentum transmitted to the obstacle vanishes as the frequency $k$ tends to infinity and that the total cross section oscillates at high frequencies. When the impedance $\lambda_0$ is real (i.e., there is no absorption), it is shown that there exists a sequence of frequencies $k_n$ such that the average of the total cross section over shrinking intervals around $\lambda_0 $ tends to zero as $k_n \to \infty$.

On Spread of Phage Infection of Bacteria in a Petri Dish

Don A. Jones, Hal L. Smith, Horst R. Thieme, and Gergely Röst

SIAM J. Appl. Math. 72, pp. 670-688 (19 pages)

Online Publication Date: April 17, 2012

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A reaction diffusion system with time delay is proposed for virus spread on bacteria immobilized on an agar-coated plate. The delay explicitly accounts for a virus latent period of fixed duration. An interval of possible spreading speeds for virus infection is established, and traveling wave solutions are shown to exist. Linear determinacy of spreading speed breaks down for some parameter values.

Spectral Stability of Deep Two-Dimensional Gravity Water Waves: Repeated Eigenvalues

Benjamin Akers and David P. Nicholls

SIAM J. Appl. Math. 72, pp. 689-711 (23 pages)

Online Publication Date: April 24, 2012

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The spectral stability problem for periodic traveling waves on a two-dimensional fluid of infinite depth is investigated via a perturbative approach, computing the spectrum as a function of the wave amplitude beginning with a flat surface. We generalize our previous results by considering the crucially important situation of eigenvalues with multiplicity greater than one (focusing on the generic case of multiplicity two) in the flat water configuration. We use this extended method of transformed field expansions (which now accounts for the resonant spectrum) to numerically simulate the evolution of the eigenvalues as the wave amplitude is increased. We observe that there are no instabilities that are analytically connected to the flat state: The spectrum loses its analyticity at the Benjamin–Feir threshold. We complement the numerical results with an explicit calculation of the first nonzero correction to the linear spectrum of resonant deep water waves. Two countably infinite families of collisions of eigenvalues with opposite Krein signature which do not lead to instability are presented.
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