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2007

Volume 67, Issue 6, pp. 1523-1807


$P$ Matrix Properties, Injectivity, and Stability in Chemical Reaction Systems

Murad Banaji, Pete Donnell, and Stephen Baigent

SIAM J. Appl. Math. 67, pp. 1523-1547 (25 pages) | Cited 10 times

Online Publication Date: September 07, 2007

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In this paper we examine matrices which arise naturally as Jacobians in chemical dynamics. We are particularly interested in when these Jacobians are $P$ matrices (up to a sign change), ensuring certain bounds on their eigenvalues, precluding certain behavior such as multiple equilibria, and sometimes implying stability. We first explore reaction systems and derive results which provide a deep connection between system structure and the $P$ matrix property. We then examine a class of systems consisting of reactions coupled to an external rate-dependent negative feedback process and characterize conditions which ensure that the $P$ matrix property survives the negative feedback. The techniques presented are applied to examples published in the mathematical and biological literature.

Weak Lacunae of Electromagnetic Waves in Dilute Plasma

S. V. Tsynkov

SIAM J. Appl. Math. 67, pp. 1548-1581 (34 pages) | Cited 3 times

Online Publication Date: September 12, 2007

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The propagation of waves is said to be diffusionless, and the corresponding governing PDE (or system) is said to satisfy Huygens' principle if the waves due to compactly supported sources have sharp aft fronts. The areas of no disturbance behind the aft fronts are called lacunae. Diffusionless propagation of waves is rare, whereas its opposite—diffusive propagation accompanied by aftereffects—is common. Nonetheless, lacunae can still be observed in a number of important applications, including the Maxwell equations in vacuum or in dielectrics with static response. In the framework of these applications, lacunae can be efficiently exploited for the numerical simulation of unsteady waves, and considerable progress has been made toward the development of lacunae-based methods for computational electromagnetism. Maxwell equations in vacuum are Huygens' because they reduce to a set of d'Alembert equations. Besides d'Alembert equations, there are no other scalar Huygens' equations in the standard $3+1$-dimensional Minkowski space-time. In terms of physics, this means that the mechanisms of dissipation and dispersion destroy the lacunae. In fact, all conventional low-frequency electromagnetic models, such as metals with Ohm conductivity, semiconductors, and magnetohydrodynamic media, are diffusive. An important case of the propagation of high-frequency electromagnetic waves in plasma is governed by the Klein–Gordon equation. It does not reduce to the d'Alembert equation either, and therefore the corresponding propagation is diffusive as well. However, one can still identify “weak lacunae” in the solutions of the Klein–Gordon equation, with the aft fronts that can be clearly observed, although they may not be as sharp as in the pure Huygens' case. Moreover, one can show that the “depth” of a weak lacuna is controlled by the dimensionless ratio of the Langmuir frequency to the primary carrying frequency of the waves.

Active Control of Sound for Composite Regions

A. W. Peterson and S. V. Tsynkov

SIAM J. Appl. Math. 67, pp. 1582-1609 (28 pages) | Cited 5 times

Online Publication Date: September 12, 2007

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We present a methodology for the active control of time-harmonic wave fields, e.g., acoustic disturbances, in composite regions. This methodology extends our previous approach developed for the case of arcwise connected regions. The overall objective is to eliminate the effect of all outside field sources on a given domain of interest, i.e., to shield this domain. In this context, active shielding means introducing additional field sources, called active controls, that generate the annihilating signal and cancel out the unwanted component of the field. As such, the problem of active shielding can be interpreted as a special inverse source problem for the governing differential equation or system. For a composite domain, not only do the controls prevent interference from all exterior sources, but they can also enforce a predetermined communication pattern between the individual subdomains (as many as desired). In other words, they either allow the subdomains to communicate freely with one another or otherwise have them shielded from their peers. In the paper, we obtain a general solution for the composite active shielding problem and show that it reduces to solving a collection of auxiliary problems for arcwise connected domains. The general solution is constructed in two stages. Namely, if a particular subdomain is not allowed to hear another subdomain, then the supplementary controls are employed first. They communicate the required data prior to building the final set of controls. The general solution can be obtained with only the knowledge of the acoustic signals propagating through the boundaries of the subdomains. No knowledge of the field sources is required, nor is any knowledge of the properties of the medium needed.

On the Uplink of a Cellular System with Imperfect Power Control and Multiple Services

John A. Morrison and Phil Whiting

SIAM J. Appl. Math. 67, pp. 1610-1632 (23 pages)

Online Publication Date: September 12, 2007

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We analyze the reverse link of a single wireless cell in which mobile phones simultaneously transmit to the base station using code division multiple access (CDMA). The mobiles are transmitting data which is delay intolerant, so that scheduling cannot be employed. There is a finite number of data classes, and the users transmit either data or a lower rate synchronizing signal. To overcome the near-far problem, received power control is used, which has a log-normal error. For each class there is an outage probability that the user's signal-to-noise ratio (SNR) will be met (when active and when idle). Refinements of the central limit theorem are used to determine the number of users of each class that can be supported, i.e., the capacity. The approximation can also be used to determine the minimal target powers necessary to meet the outage requirements. Comparison with simulation shows these approximations to be accurate.

Cavitation on Deformable Glacier Beds

Christian Schoof

SIAM J. Appl. Math. 67, pp. 1633-1653 (21 pages) | Cited 5 times

Online Publication Date: September 14, 2007

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The formation of water-filled cavities at the interface between a glacier and its bed can significantly affect the drainage of meltwater along the base of a glacier, which in turn is one of the most important controls on glacier sliding. In this paper, we analyze a mathematical model for cavity formation on deformable glacier beds. By contrast with the case of rigid glacier beds, the cavities described here are the result of an interfacial instability in coupled ice-sediment flow. This instability causes bumps on the ice-sediment interface to grow until normal stress in the lee of bed bumps drops to the local porewater pressure, at which point the ice begins to lose contact with the surface of the sediment. We extend the basic instability model to cover the case of cavity formation, and analyze the corresponding traveling wave problem. This takes the form of a viscous contact problem in which the obstacle on the boundary—the traveling bed bump caused by the initial instability—must be determined as part of the solution. A classical complex variable method allows the traveling wave problem to be cast as an eigenvalue problem which is straightforward to solve numerically. Our results show that solutions for different wavelengths can be obtained from an apparently unique solution to a scaled problem, and that the amplitude of traveling waves increases with wavelength, while their speed decreases with wavelength.

Attractors in Confined Source Problems for Coupled Nonlinear Diffusion

D. V. Strunin

SIAM J. Appl. Math. 67, pp. 1654-1674 (21 pages)

Online Publication Date: September 26, 2007

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In processes driven by nonlinear diffusion, a signal from a concentrated source is confined in a finite region. Such solutions can be sought in the form of power series in a spatial coordinate. We use this approach in problems involving coupled agents. To test the method, we consider a single equation with (a) linear and (b) quadratic diffusivity in order to recover the known results. The original set of PDEs is converted into a dynamical system with respect to the time-dependent series coefficients. As an application we consider an expansion of a free turbulent jet. Some example trajectories from the respective dynamical system are presented. The structure of the system hints at the existence of an attracting center manifold. The attractor is explicitly found for a reduced version of the system.

Numerical Investigation of Cavitation in Multidimensional Compressible Flows

Kristen J. DeVault, Pierre A. Gremaud, and Helge Kristian Jenssen

SIAM J. Appl. Math. 67, pp. 1675-1692 (18 pages) | Cited 1 time

Online Publication Date: September 26, 2007

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The compressible Navier–Stokes equations for an ideal polytropic gas are considered in ${R}^n$, $n = 2,3$. The question of possible vacuum formation, an open theoretical problem, is investigated numerically using highly accurate computational methods. The flow is assumed to be symmetric about the origin with a purely radial velocity field. The numerical results indicate that there are weak solutions to the Navier–Stokes system in two and three space dimensions, which display formation of vacuum when the initial data are discontinuous and sufficiently large. The initial density is constant, while the initial velocity field is symmetric, points radially away from the origin, and belongs to $H^s_{loc}$ for all $s < n/2$. In addition, in the one-dimensional case, the numerical solutions are in agreement with known theoretical results.

Hopf Bifurcation in Differential Equations with Delay for Tumor–Immune System Competition Model

Radouane Yafia

SIAM J. Appl. Math. 67, pp. 1693-1703 (11 pages) | Cited 1 time

Online Publication Date: September 26, 2007

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This paper deals with the qualitative analysis of the solutions to a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a system of differential equations with one delay. It is shown that the dynamics depends crucially on the time delay parameter. By using the time delay as a parameter of bifurcation, the analysis is focused on the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state. The obtained results depict the oscillations, given by simulations (see [M. Galach, Int. J. Appl. Math. Comput. Sci., 13 (2003), pp. 395–406]), which are observed in reality (see [D. Kirschner and J. C. Panetta, J. Math. Biol., 37 (1998), pp. 235–252]). It is suggested to examine by laboratory experiments how to employ these results for control of tumor growth.

A Stochastic Model and Associated Fokker–Planck Equation for the Fiber Lay-Down Process in Nonwoven Production Processes

T. Götz, A. Klar, N. Marheineke, and R. Wegener

SIAM J. Appl. Math. 67, pp. 1704-1717 (14 pages) | Cited 5 times

Online Publication Date: September 26, 2007

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In this paper we present and investigate a stochastic model and its associated Fokker–Planck equation for the lay-down of fibers on a conveyor belt in the production process of nonwoven materials. The model is based on a stochastic differential equation taking into account the motion of the fiber under the influence of turbulence. A reformulation as a stochastic Hamiltonian system and an application of the stochastic averaging theorem lead to further simplifications of the model. Finally, the model is used to compute the distribution of functionals of the process that are important for the quality assessment of industrial fabrics.

Pulse Propagation and Time Reversal in Random Waveguides

Josselin Garnier and George Papanicolaou

SIAM J. Appl. Math. 67, pp. 1718-1739 (22 pages) | Cited 3 times

Online Publication Date: October 05, 2007

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Mode coupling in a random waveguide can be analyzed with asymptotic analysis based on separation of scales when the propagation distance is large compared to the size of the random inhomogeneities, which have small variance, and when the wavelength is comparable to the scale of the inhomogeneities. In this paper we study the asymptotic form of the joint distribution of the mode amplitudes at different frequencies. We derive a deterministic system of transport equations that describe the evolution of mode powers. This result is applied to the computations of pulse spreading in a random waveguide. It is also applied to the analysis of time reversal in a random waveguide. We show that randomness enhances spatial refocusing and that diffraction-limited focal spots can be obtained even with small-size time-reversal mirrors. The refocused field is statistically stable for broadband pulses in general. We show here that it is also stable for narrowband pulses, provided that the time-reversal mirror is large enough.

The Dynamics and Interaction of Quantized Vortices in the Ginzburg–Landau–Schrödinger Equation

Yanzhi Zhang, Weizhu Bao, and Qiang Du

SIAM J. Appl. Math. 67, pp. 1740-1775 (36 pages) | Cited 3 times

Online Publication Date: October 05, 2007

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The dynamic laws of quantized vortex interactions in the Ginzburg–Landau–Schrödinger equation (GLSE) are analytically and numerically studied. A review of the reduced dynamic laws governing the motion of vortex centers in the GLSE is provided. The reduced dynamic laws are solved analytically for some special initial data. By directly simulating the GLSE with an efficient and accurate numerical method proposed recently in [Y. Zhang, W. Bao, and Q. Du, Numerical simulation of vortex dynamics in Ginzburg–Landau–Schrödinger equation, European J. Appl. Math., to appear], we can qualitatively and quantitatively compare quantized vortex interaction patterns of the GLSE with those from the reduced dynamic laws. Some conclusive findings are obtained, and discussions on numerical and theoretical results are made to provide further understanding of vortex interactions in the GLSE. Finally, the vortex motion under an inhomogeneous potential in the GLSE is also studied.

Small- and Waiting-Time Behavior of the Thin-Film Equation

James F. Blowey, John R. King, and Stephen Langdon

SIAM J. Appl. Math. 67, pp. 1776-1807 (32 pages)

Online Publication Date: October 05, 2007

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We consider the small-time behavior of interfaces of zero contact angle solutions to the thin-film equation. For a certain class of initial data, through asymptotic analyses, we deduce a wide variety of behavior for the free boundary point. These are supported by extensive numerical simulations.
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