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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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Stability of MultiComponent Biological Membranes 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. |
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The Eshelby Theorem and Application to the Optimization of an Elastic Patch 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. |
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On a Functional-Differential Equation Arising from a Traffic Flow Model 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. |
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A Unified Variational Formulation for the Parabolic-Elliptic Eddy Current Equations 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. |
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Random Transmission Radii in Greedy Routing Models for Ad Hoc Sensor Networks 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. |
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Controlled Drug Delivery in Cancer Immunotherapy: Stability, Optimization, and Monte Carlo Analysis SIAM J. Appl. Math. 71, pp. 2229-2245 (17 pages) Online Publication Date: December 20, 2011
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A discussion on controlled drug delivery in cancer immunotherapy is presented in this paper. A fifth-order model is adopted to describe the dynamics of the tumor–immune interaction. Natural equilibrium points of this system are sought, and their stability is analyzed. An optimal control problem is stated and solved numerically. Both continuous and discrete controls are treated, and their implications on the therapy protocol are discussed. The robustness of the optimal therapies is assessed a posteriori with a Monte Carlo analysis. This shows that the control policy is effective even when the initial patient conditions are affected by uncertainties. |
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On Spread of Phage Infection of Bacteria in a Petri Dish 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. |
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Analysis of a Nonlocal Model for Spontaneous Cell Polarization 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. |
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Lyapunov Functions and Global Stability for Age-Structured HIV Infection Model SIAM J. Appl. Math. 72, pp. 25-38 (14 pages) Online Publication Date: January 03, 2012
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We study the basic age-structured population model describing the HIV infection process, which is defined by PDEs. The model allows the production rate of viral particles and the death rate of productively infected cells to vary and depend on the infection age. By using the direct Lyapunov method and constructing suitable Lyapunov functions, dynamical properties of the age-structured model without (or with) drug treatment are established. The results show that the global asymptotic stability of the infection-free steady state and the infected steady state depends only on the basic reproductive number determined by the burst size. Further, we establish mathematically that the typical ODE and DDE (delay differential equation) models of HIV infection are equivalent to two special cases of the above PDE models. |
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High Frequency Scattering by a Classically Invisible Body 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 $k_n \to \infty$. |
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Extracting Solitons from Noisy Pulses 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. |
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An Algorithm for Least-Squares Estimation of Nonlinear Parameters SIAM J. Appl. Math. 11, pp. 431-441 (11 pages) Online Publication Date: July 13, 2006
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A New Approach for a Nonlocal, Nonlinear Conservation Law SIAM J. Appl. Math. 72, pp. 464-487 (24 pages) Online Publication Date: February 28, 2012
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We describe an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators we consider do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we examine a nonlocal inviscid Burgers equation, which gives a basic form with which to characterize properties associated with well-posedness, and to examine numerical results for specific cases. We describe the connection to a nonlocal viscous regularization, which mimics the viscous Burgers equation in an appropriate limit. We present numerical results that compare the behavior of the nonlocal Burgers formulation to the standard local case. The developments presented in this paper form the preliminary building blocks upon which to build a theory of nonlocal advection phenomena consistent within the peridynamic theory of continuum mechanics. |
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Analysis of Wetting and Contact Angle Hysteresis on Chemically Patterned Surfaces SIAM J. Appl. Math. 71, pp. 1753-1779 (27 pages) Online Publication Date: September 27, 2011
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Wetting and contact angle hysteresis on chemically patterned surfaces in two dimensions are analyzed from a stationary phase-field model for immiscible two phase fluids. We first study the sharp-interface limit of the model by the method of matched asymptotic expansions. We then justify the results rigorously by the $\Gamma$-convergence theory for the related variational problem and study the properties of the limiting minimizers. The results also provide a clear geometric picture of the equilibrium configuration of the interface. This enables us to explicitly calculate the total surface energy for the two phase systems on chemically patterned surfaces with simple geometries, namely the two phase flow in a channel and the drop spreading. By considering the quasi-static motion of the interface described by the change of volume (or volume fraction), we can follow the change-of-energy landscape which also reveals the mechanism for the stick-slip motion of the interface and contact angle hysteresis on the chemically patterned surfaces. As the interface passes through patterned surfaces, we observe not only stick-slip of the interface and switching of the contact angles but also the hysteresis of contact point and contact angle. Furthermore, as the size of the pattern decreases to zero, the stick-slip becomes weaker but the hysteresis becomes stronger in the sense that one observes either the advancing contact angle or the receding contact angle (when the interface is moving in the opposite direction) without the switching in between. |
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Solving Initial Value Problem by Matching Asymptotic Expansions SIAM J. Appl. Math. 72, pp. 405-416 (12 pages) Online Publication Date: February 16, 2012
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An asymptotic expansion in powers of small dimensionless dispersion coefficient is sought to solve an initial value problem posed for an advection diffusion equation modeling orientation of pulp fibers in a steady fully turbulent flow. The regular expansion is shown to be nonuniform in a small neighborhood of $\phi=0$. Although the highest order derivative with respect to the orientation angle $\phi$ is multiplied by the small parameter, application of matched asymptotic expansions to obtain the inner solution in a small neighborhood of $\phi=0$ matchable with the regular expansion turned out to be unsuccessful. The multiple scales do not lead to the solution either. The problem is solved by matching two asymptotic expansions, one solving the initial value problem in a small neighborhood of the initial point, while another one solves the equation at large distances from the initial point. Thus, this is an example of using the method of matched asymptotic expansions to satisfy the given initial condition by the long-distance approximation of the solution to a nonsingular partial differential equation. |
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A Stochastic Differential Equation SIS Epidemic Model SIAM J. Appl. Math. 71, pp. 876-902 (27 pages) Online Publication Date: June 02, 2011
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In this paper we extend the classical susceptible-infected-susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals $I(t)$. We then prove that this SDE has a unique global positive solution $I(t)$ and establish conditions for extinction and persistence of $I(t)$. We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real-life diseases. |
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Spectral Stability of Deep Two-Dimensional Gravity Water Waves: Repeated Eigenvalues 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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SIAM J. Appl. Math. 71, pp. 854-875 (22 pages) Online Publication Date: June 02, 2011
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We develop a general theory of the swelling kinetics of polymer gels, with the view that a polymer gel is a two-phase fluid. The model we propose is a free boundary problem and can be used to understand both contraction and swelling, including complete dissolving or dehydration of polymeric gels. We show that the equations of motion satisfy a minimum energy dissipation rate principle similar to the Helmholtz minimum dissipation rate principle which holds for a Stokes flow. We also show, using asymptotic analysis and numerical simulation, how the equilibrium swelled state and the swelling rate constant are related to the free energy and rheological properties of the polymer network. |
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Stabilization in a State-Dependent Model of Turning Processes SIAM J. Appl. Math. 72, pp. 1-24 (24 pages) Online Publication Date: January 03, 2012
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We consider a two-degree-of-freedom model for turning processes which involves a system of differential equations with state-dependent delay. Depending on process parameters (e.g., spindle speed, depth of cut) the cutting tool can exhibit unwanted vibrations, resulting in a nonsmooth surface of the workpiece. In this paper we propose a feedback law to stabilize the turning process for a large range of system parameters. The feedback law introduces a generic nonhyperbolic stationary point into the model, which generates the main technical challenge of this work. We establish the stability equivalence between the differential equations with state-dependent delay and a corresponding nonlinear system with the delay fixed at its stationary value. Then we show the stability of that nonlinear system with constant delay by computing its normal form. Finally, we obtain conditions on system parameters which guarantee the stability of the state-dependent delay model at the nonhyperbolic stationary point. |
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A Singularly Perturbed Boundary Value Problem Modelling a Semiconductor Device SIAM J. Appl. Math. 44, pp. 231-256 (26 pages) Online Publication Date: July 12, 2006
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This paper is concerned with the static, one-dimensional modelling of a semiconductor device (namely the $pn$-junction) when a bias is applied. The governing equations are the well-known equations describing carrier transport in a semiconductor which consist of a system of ordinary differential equations subject to boundary conditions imposed at the contacts. Because of the different orders of magnitude of the solution components at the boundaries, we scale the components individually and obtain a singular perturbation problem. We analyse the equilibrium case (zero bias applied) and set up approximate models, posed as singularly perturbed second order equations, by neglecting the hole and electron current densities. This makes sense for small forward bias and for moderate reverse bias. For the full problem we prove an a priori estimate on the number of electron-hole carrier pairs and derive asymptotic expansions (as the perturbation parameter tends to zero) by setting up the reduced system and the boundary layer system. We prove existence theorems for both systems and use the asymptotic expansion to solve the model equations numerically and analyse the dependence of the solutions on the applied bias. |
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