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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 93 (No. 10) |
| Halaman | : | 775–780 |
| Abstrak | : | We analyze the wavelengths of a bright colored line on an optical disk illuminated by light rays at large angles with respect to the normal to the plane of the disk. The reflected rays constructively interfere, allowing only those with specific resonant wavelengths to be seen. We observe that, in certain circumstances, the color of the line noticeably varies along its length. To account for this phenomenon, we develop an equation for the wavelength that results in constructive interference at a given reflection point and validate it through a set of experiments. |
Efficient and Accurate Algorithm for the Full Modal Green's Kernel of the Scalar Wave Equation in Helioseismology
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | |
| Abstrak | : | In this work, we provide an algorithm to compute efficiently and accurately the full outgoing modal Green's kernel for the scalar wave equation in local helioseismology under spherical symmetry. Due to the high computational cost of a full Green's function, current helioseismic studies rely on single-source computations. However, a more realistic modelization of the helioseismic products (cross-covariance and power spectrum) requires the full Green's kernel. In the classical approach, the Dirac source is discretized and one simulation gives the Green's function on a line. Here, we propose a two-step algorithm which, with two simulations, provides the full kernel on the domain. Moreover, our method is more accurate, as the singularity of the solution due to the Dirac source is described exactly. In addition, it is coupled with the exact Dirichlet-to-Neumann boundary condition, providing optimal accuracy in approximating the outgoing Green's kernel, which we demonstrate in our experiments. In addition, we show that high-frequency approximations of the nonlocal radiation boundary conditions can represent accurately the helioseismic products. |
Predator-Prey Interactions with Hunger Structure
| Pengarang | : | Paulo Amorim |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2631-2656 |
| Abstrak | : | We present, analyze, and simulate a model for predator-prey interaction structured by hunger. The model consists of a nonlocal transport equation for the predator, coupled to an ordinary differential equation for the prey. We deduce a system of 3 ODEs for some integral quantities of the transport equation, which generalizes some classical Lotka--Volterra systems. By taking an asymptotic regime of fast hunger variation, we find that this system provides new interpretations of classical Holling-type functional responses. We also show that, with a small extension, the system can reproduce predator-prey role reversal bistability results observed in the literature. We next establish a well-posedness result for the nonlocal transport equation. Finally, we show that in the basin of attraction of the nontrivial equilibrium, the asymptotic behavior of the original coupled PDE-ODE system is completely described by solutions of the ODE system. |
Persistence and Propagation of a PDE and Discrete-Time Map Hybrid Animal Movement Model With Habitat Shift Driven by Climate Change
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2608-2630 |
| Abstrak | : | Persistence or extinction of moving animal species is a fundamental question in spatial ecology. This paper focuses on the impact of habitat shift driven by climate change on the persistence and propagation of a population with birth pulse. We first present a class of impulsive reaction-diffusion models with heterogeneous nonlinear reaction in high-dimensional space and study their threshold dynamics. We provide the persistence criterion of the system in bounded domains, and prove the existence, uniqueness, and global attraction of a positive steady state. Then we extend the results from bounded domains to the whole space. Our results indicate how the speed of the shifting habitat edge and impulsive reproduction (or harvesting) rate determine the persistence and extinction of the population. Numerical simulations are presented to illustrate the theoretical results. |
Transport Phenomena in Biological Field Effect Transistors
| Pengarang | : | Ryan M. Evans |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2586-2607 |
| Abstrak | : | A mathematical model for simulating biological field effect transistor (Bio-FET) experiments is introduced. It takes the form of a nonlinear equation that describes evolution of reacting species concentration at the boundary coupled to a diffusion equation. Using analytic techniques, this coupled system of equations is reduced to a singular integrodifferential equation (IDE). A numerical approximation of this equation is developed that achieves greater than first-order accuracy in time and greater than second-order accuracy in space, despite the presence of a singular temporal convolution kernel and a discontinuous boundary condition. The mathematical model was validated using Bio-FET data, and stochastic regression was employed to separate signal from noise. Results show that our IDE provides a robust way of estimating important parameters such as diffusion coefficients and kinetic rate constants. |
Convolutional Neural Networks in Phase Space and Inverse Problems
| Pengarang | : | Gunther Uhlmann |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2560-2585 |
| Abstrak | : | We study inverse problems consisting of determining medium properties using the responses to probing waves from the machine learning point of view. Based on the analysis of propagation of waves and their nonlinear interactions, we construct a deep convolutional neural network to reconstruct the coefficients of nonlinear wave equations that model the medium properties. Furthermore, for given approximation accuracy, we obtain the depth and number of units of the network and their quantitative dependence on the complexity of the medium. |
Stability for the Acoustic Inverse Source Problem in Inhomogeneous Media
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2547-2559 |
| Abstrak | : | In this paper, we show for the first time the stability of the inverse source problem for the three-dimensional Helmholtz equation in an inhomogeneous background medium. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The analysis employs scattering theory to obtain the holomorphic domain and an upper bound for the resolvent of the elliptic operator. |
Classifying Local Anisotropy Formed by Rigid Molecules: Symmetries and Tensors
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2518-2546 |
| Abstrak | : | We consider an infinitesimal volume where there are many rigid molecules of the same kind and discuss the description and classification of the local anisotropy in this volume by tensors. First, we examine the symmetry of a rigid molecule, which is described by a point group in ??????????(3). For each point group in ??????????(3), we find the tensors invariant under the rotations in the group. These tensors shall be symmetric and traceless. We write down the explicit expressions. The order parameters to describe the local anisotropy are then chosen as some of the invariant tensors averaged about the density function. Next, we discuss the classification of local anisotropy by the symmetry of the whole infinitesimal volume. This mesoscopic symmetry can be recognized by the value of the order parameter tensors in the sense of maximum entropy state. For some sets of order parameter tensors involving different molecular symmetries, we give the classification of mesoscopic symmetries, in which the threefold, fourfold, and polyhedral symmetries are examined. |
Stochastic Peierls--Nabarro Model for Dislocations in High Entropy Alloys
| Pengarang | : | Tianpeng Jiang |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2496-2517 |
| Abstrak | : | High entropy alloys (HEAs) are single phase crystals that consist of random solid solutions of multiple elements in approximately equal proportions. This class of novel materials have exhibited superb mechanical properties, such as high strength combined with other desired features. The strength of crystalline materials is associated with the motion of dislocations. In this paper, we derive a stochastic continuum model based on the Peierls--Nabarro framework for interlayer dislocations in a bilayer HEA from an atomistic model that incorporates the atomic level randomness. We use asymptotic analysis and limit theorem in the convergence from the atomistic model to the continuum model. The total energy in the continuum model consists of a stochastic elastic energy in the two layers, and a stochastic misfit energy that accounts for the interlayer nonlinear interaction. The obtained continuum model can be considered as a stochastic generalization of the classical, deterministic Peierls--Nabarro model for the dislocation core and related properties. This derivation also validates the stochastic model adopted by Zhang et al. [Acta Mater., 166 (2019), pp. 424--434]. |
Amplitude Blowup in Radial Isentropic Euler Flow
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2472-2495 |
| Abstrak | : | We show that the compressible Euler system for isentropic gas flow admits unbounded solutions. The examples are radial flows of similarity type and describe a spherically symmetric and continuous wave moving toward the origin. At time of focusing, both the density and the velocity become unbounded at the origin. This is followed by an expanding shock wave which slows down as it interacts with the incoming flow. While unbounded radial Euler flows have been known since the work of Guderley [Luftfahrtforschung, 19 (1942), pp. 302--311], those are at the borderline of the regime covered by the Euler model: The upstream pressure field vanishes identically (either because of vanishing temperature or vanishing density there). In contrast, the solutions we build exhibit an everywhere strictly positive pressure field, demonstrating that the geometric effect of wave focusing is strong enough on its own to drive the primary flow variables to infinity. |