Showing papers in "Siam Journal on Applied Mathematics in 2001"
TL;DR: A predator-prey system with nonmonotonic functional response is considered and global qualitative and bifurcation analyses are combined to determine the global dynamics of the model.
Abstract: A predator-prey system with nonmonotonic functional response is considered. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The bifurcation a...
458 citations
TL;DR: This work considers traveling front and pulse solutions to a system of integro-differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions.
Abstract: We consider traveling front and pulse solutions to a system of integro-differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension. Our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions. This is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed. Our second goal is to establish the existence of traveling pulse solutions using more rigorous methods. Two techniques are presented. The first, a shooting argument, reduces the problem from finding a specific solution to an integro-differential equation system to finding any solution to an ODE system. The second, a singular perturbation argument, provides a construction of traveling pulse solutions un...
323 citations
TL;DR: The global dynamics are completely determined by the basic reproduction number R0 (p,q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively, which proves that the disease-free equilibrium is globally stable and the disease always dies out.
Abstract: We study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission. The total host population is assumed to have constant density and the incidence term is of the bilinear mass-action form. We prove that the global dynamics are completely determined by the basic reproduction number R0 (p,q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If $R_0(p,q)\le 1,$ the disease-free equilibrium is globally stable and the disease always dies out. If R0 (p,q)>1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the vertical transmission to the basic reproduction number is also analyzed.
283 citations
TL;DR: Two arguments are presented demonstrating the loss of stability of standing pulse solutions through a Hopf bifurcation when more realistic inhibitory dynamics are considered.
Abstract: We consider the existence and stability of standing pulse solutions to a system of integro-differential equations used to describe the activity of synaptically coupled networks of excitatory and inhibitory neurons in a single spatial domain. Assuming an arrangement of synaptic connections described by "lateral inhibition," previous formal arguments have demonstrated the existence of both stable and unstable standing pulses [S. Amari, Biol. Cybern., 27 (1977), pp. 77--87]. These results have formed the basis for several recent hypotheses regarding the generation of sustained activity patterns in prefrontal cortex and other brain regions. Implicit in the lateral inhibition arrangement, however, is the assumption that the dynamics of inhibition are instantaneous. Here we present two arguments demonstrating the loss of stability of standing pulse solutions through a Hopf bifurcation when more realistic inhibitory dynamics are considered. The first argument parallels Amari's formal presentation, while the seco...
167 citations
TL;DR: The variational method is applied to denoise and restore general nonflat image features and Riemannian objects such as metric, distance and Levi--Civita connection play important roles in the models.
Abstract: We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces Familiar examples include the orientation feature (from optical flows or gradient flows) that lives on the unit circle S1 , the alignment feature (from fingerprint waves or certain texture images) that lives on the real projective line $\mathbb{RP}^1$, and the chromaticity feature (from color images) that lives on the unit sphere S2 In this paper, we apply the variational method to denoise and restore general nonflat image features Mathematical models for both continuous image domains and discrete domains (or graphs) are constructed Riemannian objects such as metric, distance and Levi--Civita connection play important roles in the models Computational algorithms are also developed for the resulting nonlinear equations The mathematical framework can be applied to res
145 citations
TL;DR: Using perturbation methods, approximations for the calcium concentration profiles and buffer profiles are derived in three asymptotic limits: an "excess buffer approximation" (EBA), where the mobility of buffer exceeds that of buffer and the fast diffusion of buffer toward the open channel prevents buffer saturation.
Abstract: The "domain" calcium (${\rm Ca}^{\rm 2+}$) concentration near an open ${\rm Ca}^{\rm 2+}$ channel can be modeled as buffered diffusion from a point source. The concentration profiles can be well approximated by hemispherically symmetric steady-state solutions to a system of reaction-diffusion equations. After nondimensionalizing these equations and scaling space so that both reaction terms and the source amplitude are O(1), we identify two dimensionless parameters, ${\varepsilon}_c$ and ${\varepsilon}_b$, that correspond to the diffusion coefficients of dimensionless ${\rm Ca}^{\rm 2+}$ and buffer, respectively.Using perturbation methods, we derive approximations for the ${\rm Ca}^{\rm 2+}$ and buffer profiles in three asymptotic limits: (1) an "excess buffer approximation" (EBA), where the mobility of buffer exceeds that of ${\rm Ca}^{\rm 2+}$ (${\varepsilon}_b \gg {\varepsilon}_c$) and the fast diffusion of buffer toward the ${\rm Ca}^{\rm 2+}$ channel prevents buffer saturation (cf. Neher [ Calcium Ele...
138 citations
TL;DR: The overall bifurcation scenario explains why stable limit cycles and strange attractors with different geometries can coexist.
Abstract: Complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper. First it is shown that the model admits a sequence of pairs of Belyakov bifurcations (codimension-two homoclinic orbits to a critical node). Then fold and period-doubling cycle bifurcation curves associated to each pair of Belyakov points are computed and analyzed. The overall bifurcation scenario explains why stable limit cycles and strange attractors with different geometries can coexist. The analysis is conducted by combining numerical continuation techniques with theoretical arguments.
132 citations
TL;DR: Here it is proved a variational characterization of the front speed for reaction-diffusion-advectionequations in periodically varying heterogeneous media that can be applied to any problem obeying a maximum principle.
Abstract: An important problem in reactive flows is how to estimate the speed of front propagation, especially when inhomogeneities are present. Here we prove a variational characterization of the front speed for reaction-diffusion-advectionequations in periodically varying heterogeneous media. This formulation makes it possible to calculate sharp estimates for the speed explicitly. The method can be applied to any problem obeying a maximum principle. Three examples will be analyzed in detail: a shear flow problem, a problem with rapidly oscillating coefficients, and a discretized diffusion problem. In all cases the effects of the inhomogeneous medium on the speed are discussed in comparison to the homogeneous problem. For the shear flow problem, enhancement of the speed results.
117 citations
TL;DR: A significant but limited class of blurs that can be expressed as convolutions of two-dimensional symmetric Levy "stable" probability density functions is considered, which includes and generalizes Gaussian and Lorentzian distributions.
Abstract: Blind deconvolution seeks to deblur an image without knowing the cause of the blur. Iterative methods are commonly applied to that problem, but the iterative process is slow, uncertain, and often ill-behaved. This paper considers a significant but limited class of blurs that can be expressed as convolutions of two-dimensional symmetric Levy "stable" probability density functions. This class includes and generalizes Gaussian and Lorentzian distributions. For such blurs, methods are developed that can detect the point spread function from one-dimensional Fourier analysis of the blurred image. A separate image deblurring technique uses this detected point spread function to deblur the image. Each of these two steps uses direct noniterative methods and requires interactive tuning of parameters. As a result, blind deblurring of 512 × 512 images can be accomplished in minutes of CPU time on current desktop workstations. Numerous blind experiments on synthetic data show that for a given blurred image, several di...
99 citations
TL;DR: A threshold value is identified for the damping parameter $delta$ such that when $\delta%gt; \delta_{th}$ damping arrests blowup, the solution blows up at the same asymptotic rate as the undamped nonlinear Schrodinger equation.
Abstract: We analyze the effect of damping (absorption) on critical self-focusing. We identify a threshold value $\delta_{th}$ for the damping parameter $\delta$ such that when $\delta%gt; \delta_{th}$ damping arrests blowup. When $\delta < \delta_{th}$, the solution blows up at the same asymptotic rate as the undamped nonlinear Schrodinger equation.
93 citations
TL;DR: This paper analyzes the so-called intrinsic heat equation governing the motion of plane curves obeying a geometric equation and shows how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities.
Abstract: In this paper we study evolution of plane curves satisfying a geometric equation $v= \beta(k,
u)$, where v is the normal velocity and k and $
u$ are the curvature and tangential angle of a plane curve $\Gamma$. We follow the direct approach and we analyze the so-called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation. The intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional $\alpha$. We show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities. From an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when $\beta(k,
u)=\gamma(
u) k^m$, $0 < m\le 2$, and the governing system of equations includes a nontrivial tangential velocity functional.
TL;DR: Chains of N FitzHugh-Nagumo oscillators with a gradient in natural frequencies and strong diffusive coupling are analyzed using geometric theory for singularly perturbed dynamical systems, asymptotic expansions of solutions (rigorously justified), and Lyapunov's method.
Abstract: Chains of N FitzHugh-Nagumo oscillators with a gradient in natural frequencies and strong diffusive coupling are analyzed in this paper. We study the system's dynamics in the limit of infinitely large coupling and then treat the case when the coupling is large but finite as a perturbation of the former case. In the large coupling limit, the 2N -dimensional phase space has an unexpected structure: there is an (N − 1)-dimensional cylinder foliated by periodic orbits with an integral that is constant on each orbit. When the coupling is large but finite, this cylinder becomes an analog of an inertial manifold. The phase trajectories approach the cylinder on the fast time scale and then slowly drift along it toward a unique limit cycle. We analyze these dynamics using geometric theory for singularly perturbed dynamical systems, asymptotic expansions of solutions (rigorously justified), and Lyapunov's method.
TL;DR: It is shown that any regular solution of the Helmholtz equation admits a unique representation by a single layer potential, so that the problem is reduced to the solution of a linear integral equation of the first kind.
Abstract: We consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin. Mathematically this field satisfies the Helmholtz equation. In this paper we consider the model two-dimensional case. We show that any regular solution of this equation admits a unique representation by a single layer potential, so that the problem is reduced to the solution of a linear integral equation of the first kind. We prove uniqueness of reconstruction and obtain a sharp stability estimate. Finally, for two geometries and sources of noise simulating the cabin of the aircraft and two engines, we give results of the numerical solution of this integral equation, comparing regularization by the truncated singular value decomposition and the conjugate gradient method.
TL;DR: A geometrical method is developed to systematically study the singular perturbed problem associated to slowly modulated two-pulse solutions in the pulse-splitting bifurcations and the validity of the quasi-stationary approximation is established.
Abstract: With great sadness, we note the passing away of our mentor and colleague Wiktor Abstract. In this second paper, we develop a geometrical method to systematically study the singular perturbed problem associated to slowly modulated two-pulse solutions. It enables one to see that the characteristics of these solutions are strongly determined by the flow on a slow manifold and, hence, also to identify the saddle-node bifurcations and bifurcations to classical traveling waves in which the solutions constructed in part I are created and annihilated. Moreover, we determine the geometric origin of the critical maximum wave speeds discovered in part I. In this paper, we also study the central role of the slowly varying inhibitor component of the two-pulse solutions in the pulse-splitting bifurcations. Finally, the validity of the quasi-stationary approximation is established here, and we relate the results of both parts of this work to the literature on self-replication.
TL;DR: Theoretical and numerical estimates for the probabilities of nonextinction and convergence to an attractor suggest that random genetic drift is more likely to overcome natural selection in small populations for which pairwise interactions lead to highly variable outcomes, and is less likely to adversaries in large populations with the potential for rapid growth.
Abstract: To understand the relative importance of natural selection and random genetic drift in finite but growing populations, the asymptotic behavior of a class of generalized Polya urns is studied using the method of ordinary differential equation (ODE). Of particular interest is the replicator process: two balls (individuals) are chosen from an urn (the population) at random with replacement and balls of the same colors (strategies) are added or removed according to probabilities that depend only on the colors of the chosen balls. Under the assumption that the expected number of balls being added always exceeds the expected number of balls being removed whenever balls are in the urn, the probability of nonextinction is shown to be positive. On the event of nonextinction, three results are proven: (i) the number of balls increases asymptotically at a linear rate, (ii) the distribution x(n) of strategies at the nth update is a "noisy" Cauchy-Euler approximation to the mean limit ODE of the process, and (iii) the limit set of x(n) is almost surely a connected internally chain recurrent set for the mean limit ODE. Under a stronger set of assumptions, it is shown that for any attractor of the mean limit ODE there is a positive probability that the limit set for x(n) lies in this attractor. Theoretical and numerical estimates for the probabilities of nonextinction and convergence to an attractor suggest that random genetic drift is more likely to overcome natural selection in small populations for which pairwise interactions lead to highly variable outcomes, and is less likely to overcome natural selection in large populations with the potential for rapid growth.
TL;DR: The key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources need to be known.
Abstract: We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaltered those components of the total acoustic field that are deemed friendly. Our key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources need to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, perhaps, in the narrow area of space near the boundary (perimeter) of the domain to be shielded. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are also c...
TL;DR: A chemostat model of n species of microorganisms competing for two perfectly complementary, growth-limiting nutrients is considered and it is shown that every solution converges to one of the single-species or two-species steady states, and hence the dynamics of the model is completely determined.
Abstract: A chemostat model of n species of microorganisms competing for two perfectly complementary, growth-limiting nutrients is considered. Sufficient conditions are given for there to be a single winning species and for two species to coexist, driving the others to extinction. In the case when n=3, it is shown that every solution converges to one of the single-species or two-species steady states, and hence the dynamics of the model is completely determined. The results generalize those of Hsu, Cheng, and Hubbell [SIAM J. Appl. Math., 41 (1981), pp. 422--444] as well as Butler and Wolkowicz [ Math. Biosci., 83 (1987), pp. 1--48] who considered two species.
TL;DR: It is shown that the APEX method can be usefully applied to a wide variety of real blurred images, including astronomical, Landsat, and aerial images, MRI and PET brain scans, and scanning electron microscope images, leading to noticeable improvements in visual quality.
Abstract: The APEX method is an FFT-based direct blind deconvolution technique that can process complex high resolution imagery in seconds or minutes on current desktop platforms. The method is predicated on a restricted class of shift-invariant blurs that can be expressed as finite convolution products of two-dimensional radially symmetric Levy stable probability density functions. This class generalizes Gaussian and Lorentzian densities but excludes defocus and motion blurs. Not all images can be enhanced with the APEX method. However, it is shown that the method can be usefully applied to a wide variety of real blurred images, including astronomical, Landsat, and aerial images, MRI and PET brain scans, and scanning electron microscope images. APEX processing of these images enhances contrast and sharpens structural detail, leading to noticeable improvements in visual quality. The discussion includes a documented example of nonuniqueness, in which distinct point spread functions produce high-quality restorations ...
TL;DR: This four-phase checkerboard encapsulates many limiting and special cases and develops a concise closed-form solution to a basic problem involving four joined quarter planes each of a different resistivity.
Abstract: Two-dimensional periodic rectangular checkerboard media are considered in the situation where mean fluxes are prescribed across the structure. The closed-form solution is obtained in the general case where the checkerboard is constructed using four rectangular cells, each having a different, constant resistivity; this four cell structure then repeats doubly periodically to cover the whole plane. This general solution is then used to calculate the effective properties. Thus this four-phase checkerboard encapsulates many limiting and special cases; as a starting point we develop a concise closed-form solution to a basic problem involving four joined quarter planes each of a different resistivity. Subsequent manipulations yield solutions to problems posed in increasingly convoluted domains while retaining the essentially simple structure found for joined quarter planes.
TL;DR: Singular perturbation methods are used to construct traveling waves for models of thalamic networks and reduce the existence of these waves to simple boundary value problems or algebraic systems which are solved using the computational package AUTO.
Abstract: Singular perturbation methods are used to construct traveling waves for models of thalamic networks. We first study a single layer of mutually inhibitory neurons, each of which has the ability to rebound after hyperpolarization. Two types of waves are constructed: smoothly propagating waves and lurching waves which propagate in a saltatory fashion. We reduce the existence of these waves to simple boundary value problems or algebraic systems which are solved using the computational package AUTO. The resulting calculations are compared to numerical simulations of the network. Finally, some comments are made concerning two-layer networks.
TL;DR: It is proved that generically the dynamics of pulse-coupled integrate-and-fire neurons becomes fully synchronous for any initial conditions if the intrinsic frequencies, the thresholds, and the couplings are not too different.
Abstract: We reconsider the dynamics of pulse-coupled integrate-and-fire neurons analyzed by Mirollo and Strogatz [SIAM J. Appl. Math., 50 (1990), pp. 1645--1662]. Lifting their restriction to identical oscillators, we study the case of different intrinsic frequencies and thresholds of the neurons as well as different but positive couplings. For nonleaky neurons, we prove that generically the dynamics becomes fully synchronous for any initial conditions if the intrinsic frequencies, the thresholds, and the couplings are not too different. For the case of linear evolution functions, this confirms Peskin's conjecture (1975) according to which nearly identical pulse-coupled oscillators, in general, synchronize. It also shows that the requirement of concave evolution functions imposed by Mirollo and Strogatz to ensure global synchronization is not necessary.
TL;DR: An asymptotic formula is established for the displacement vector in terms of the reference Lame coefficients, the location of the imperfections, and their geometry to derive integral boundary formulae that yield to a very effective identification procedure.
Abstract: We consider an isotropic Lame system for an elastic medium consisting of finitely many imperfections of small diameter, embedded in a homogeneous reference medium. The Lame coefficients of the imperfections are different from those of the background medium. First, we establish an asymptotic formula for the displacement vector in terms of the reference Lame coefficients, the location of the imperfections, and their geometry. Second, we use this asymptotic expansion to establish continuous dependence estimates for certain characteristics of the imperfections in terms of the boundary data and to derive integral boundary formulae that yield to a very effective identification procedure.
TL;DR: An ansatz in which the wave speed depends on the underlying lattice as well as on time is introduced to find the traveling wave solutions, which are a product of both the nonlocal diffusion operator and the spinodal effects of the nonlinearity.
Abstract: We consider traveling wave solutions to spatially discrete reaction-diffusion equations with nonlocal variable diffusion and bistable nonlinearities. To find the traveling wave solutions we introduce an ansatz in which the wave speed depends on the underlying lattice as well as on time. For the case of spatially periodic diffusion we obtain analytic solutions for the traveling wave problem using a piecewise linear nonlinearity. The formula for the wave forms is implicitly defined in the general periodic case and we provide an explicit formula for the case of period two diffusion. We present numerical studies for time t = 0 fixed and for the time evolution of the traveling waves. When t = 0 we study the cases of homogeneous, period two, and period four diffusion coefficients using a cubic nonlinearity, and uncover, numerically, a period doubling bifurcation in the wave speed versus detuning parameter relation. For the time evolution case we also discover a detuning parameter dependent bifurcation in observed phenomena, which is a product of both the nonlocal diffusion operator and the spinodal effects of the nonlinearity.
TL;DR: This paper studies resonances in two degrees of freedom, autonomous, Hamiltonian systems, and shows that some of the resonances vanish as lower order resonances due to the presence of a symmetry condition on one of the degrees offreedom.
Abstract: In this paper we study resonances in two degrees of freedom, autonomous, Hamiltonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After giving a sharp estimate of the resonance domain, we investigate this order change of resonance in a rather general potential problem with discrete symmetry and consider as an example the Henon--Heiles family of Hamiltonians. We also study a classical example of a mechanical system with symmetry, the elastic pendulum, which leads to a natural hierarchy of resonances with the 4:1-resonance as the most prominent after the 2:1-resonance and which explains why the 3:1-resonance is neglected.
TL;DR: This paper describes the use of homogenization theory to predict the macroscopic behavior of lung tissue based upon the three dimensional microstructure of respiratory regions, making the simplifying assumption that the microst structure is periodic.
Abstract: High-frequency ventilation is a radical departure from conventional lung ventilation, with frequencies greater than 2Hz, and volumes per breath much smaller than the anatomical dead-space. Its use has been shown to benefit premature infants and patients with severe respiratory distress, but a vital question concerns ventilator-induced damage to the lung tissue, and a clear protocol for the most effective treatment has not been identified. Mathematical modeling can help in understanding the mechanical effects of lung ventilation, and hence in establishing such a protocol. In this paper we describe the use of homogenization theory to predict the macroscopic behavior of lung tissue based upon the three dimensional microstructure of respiratory regions, making the simplifying assumption that the microstructure is periodic. This approach yields equations for macroscopic air flow, pressure, and tissue deformation, with parameters which can be determined from a specification of the tissue microstructure and its ...
TL;DR: It is shown that in a network of two reciprocally connected neurons a single depressing synapse can produce two distinct oscillatory regimes, and that in an intermediate range of $\gbarinh$ values both stable periodic solutions exist simultaneously.
Abstract: Synaptic depression is a common form of short-term plasticity in the central and peripheral nervous systems. We show that in a network of two reciprocally connected neurons a single depressing synapse can produce two distinct oscillatory regimes. These distinct periodic behaviors can be studied by varying the maximal conductance, $\gbarinh$, of the depressing synapse. For small $\gbarinh$, the network has a short-period solution controlled by intrinsic cellular properties. For large $\gbarinh$, the solution has a much longer period and is controlled by properties of the synapse. We show that in an intermediate range of $\gbarinh$ values both stable periodic solutions exist simultaneously. Thus the network can switch oscillatory modes either by changing $\gbarinh$ or, for fixed $\gbarinh$, by changing initial conditions.
TL;DR: A phenomenological theory for phase transition dynamics with memory yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential, and some numerical results indicate that a certain type of two-dimensional relaxation damped oscillation may occur.
Abstract: We present a phenomenological theory for phase transition dynamics with memory which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential. Thereafter, we focus on the implications of our theory in the hyperbolic case, and we derive asymptotically an equation of motion in two dimensions for the interface between two different phases. This equation can be considered as a hyperbolic generalization of the classical flow by mean curvature equation, as well as a generalization of the Born--Infeld equation. We use a crystalline algorithm to study the motion of closed curves for the generalized hyperbolic flow by mean curvature equation our hyperbolic generalization of flow by mean curvature and present some numerical results which indicate that a certain type of two-dimensional relaxation damped oscillation may occur.
TL;DR: The rate of convergence to a stable law is determined for the probability density of the normalized sum of n independent identically distributed random variables, as $narrow \infty$.
Abstract: The rate of convergence to a stable law is determined for the probability density of the normalized sum of n independent identically distributed random variables, as $n\rightarrow \infty$. Methods are given for using these results to fit data to such a law.
TL;DR: A recursive explicit formulation of Taylor's method provides an algorithm leading to symbolic calculations which can be handled by a computer algebra system for microlocally strictly hyperbolic systems.
Abstract: In [ Comm. Pure Appl. Math., 28 (1975), pp. 457--478], M. E. Taylor describes a constructive diagonalization method to investigate the reflection of singularities of the solution to first-order hyperbolic systems. According to further works initiated by Engquist and Majda, this approach proved to be well adapted to the construction of artificial boundary conditions. However, in the case of systems governed by pseudodifferential operators with variable coefficients, Taylor's method involves very elaborate calculations of the symbols of the operators. Hence, a direct approach may be difficult when dealing with high-order conditions. This motivates the first part of this paper, where a recursive explicit formulation of Taylor's method is stated for microlocally strictly hyperbolic systems. Consequently, it provides an algorithm leading to symbolic calculations which can be handled by a computer algebra system. In the second part, an application of the method is investigated for the construction of local radi...
TL;DR: The main focus is on the analysis of the effect of noncommuting limits, an effect which indicates that linear boundary value problems of electromagnetism give formally different results for the long wavelength limits in cases where highly conducting inclusions have refractive indices of different orders of magnitude.
Abstract: We consider formulations for the Helmholtz operator for periodic media containing high contrast inclusions in the limit when the wavelength outside the inclusions tends to infinity. Applications are to problems of electromagnetism. The main focus is on the analysis of the effect of noncommuting limits, an effect which indicates that linear boundary value problems of electromagnetism give formally different results for the long wavelength limits in cases where highly conducting inclusions have refractive indices of different orders of magnitude. Specifically, the effective moduli of the homogenized material will depend on the path used to approach the origin in the coordinate space {wave number, (normalized refractive index of the inclusions)-1}. This mathematical observation gives a physical subtlety which is studied in this paper. The dispersion relation for the lowest frequency (or acoustic mode) is investigated, as are the conditions for existence of an acoustic mode. Cases of both nondispersive and di...