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Wobbling of Pedestrian Bridges
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1537-1557 |
| Abstrak | : | On June 10, 2000, the Millennium Bridge in London opened to the public. As people crossed it, it wobbled, leading to its closure three days later. It reopened after modifications to prevent the wobbling were made. To gain understanding on the physics behind this event, we developed and studied a toy model of a pedestrian-bridge system. Within this model, when the natural frequency of the bridge is the same as the walking frequency of the pedestrian, the lateral oscillations of the bridge and the pedestrian synchronize in antiphase. The model predicts the amplitude of the bridge oscillations. |
Classifying Minimum Energy States for Interacting Particles: Spherical Shells
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1520-1536 |
| Abstrak | : | Particles interacting through long-range attraction and short-range repulsion given by power-laws have been widely used to model physical and biological systems and to predict or explain many of the patterns they display. Apart from rare values of the attractive and repulsive exponents (????,????), the energy minimizing configurations of particles are not explicitly known, although simulations and local stability considerations have led to conjectures with strong evidence over a much wider region of parameters. For dimension ???? ≥2 and for a segment ???? =2 <???? <4 on the mildly repulsive frontier we employ strict convexity to conclude that the energy is uniquely minimized (????∞-locally, up to translation) by a spherical shell. If ???? =1 and ???? =2 <???? −1, we prove that the spherical shell is (i) the unique global energy minimizer and (ii) the unique ????∞-local energy minimizer in the class of even, compactly supported probability measures. In a companion work, we show that, in the mildly repulsive range ???? >???? ≥2, a unimodal threshold 2 <????Δ?????(????) ≤max?{????,4} exists such that equidistribution of particles over a unit diameter regular ????-simplex minimizes the energy if and only if ???? ≥????Δ?????(????) (and minimizes uniquely up to rigid motions if strict inequality holds). For ???? ≥2, the point (????,????) =(2,4) separates these regimes. At this point we show the minimizers all lie on a sphere and are precisely characterized by sharing all first and second moments with the spherical shell. Although the minimizers need not be asymptotically stable, our approach establishes ????????-Lyapunov nonlinear stability of the associated (????2-gradient) aggregation dynamics near the minimizer in both of these adjacent regimes---without reference to linearization. The ????????-Kantorovich--Rubinstein--Wasserstein distance ???????? which quantifies stability is chosen to match the attraction exponent. |
Tatalaksana Status Epileptikus di Instalasi Gawat Darurat
| Pengarang | : | Agung Prasetyo, Bowo Hery Prasetyo |
| Nama Majalah/Jurnal | : | CDK Cermin dunia kedokteran |
| Volume / Edisi | : | 45 (No. 11) |
| Halaman | : | 866-868 |
| Abstrak | : | Status epilepticus is a neurologic emergency with morbidity and mortality that depends on the duration of seizures. The goal of status epilepticus therapy is immediate cessation of seizure activity which will reduce mortality and morbidity. Current guideline recommend the use of benzodiazepines as first-line therapy, phosphenitoin, valproic acid and levetiracetam as second-line therapy and anesthetic agents as third-line therapy. |
Diffraction by a Right-Angled No-Contrast Penetrable Wedge Revisited: A Double Wiener--Hopf Approach
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1495-1519 |
| Abstrak | : | In this paper, we revisit Radlow's innovative approach to diffraction by a penetrable wedge by means of a double Wiener--Hopf technique. We provide a constructive way of obtaining his ansatz and give yet another reason for why his ansatz cannot be the true solution to the diffraction problem at hand. The two-complex-variable Wiener--Hopf equation is reduced to a system of two equations, one of which contains Radlow's ansatz plus some correction term consisting of an explicitly known integral operator applied to a yet unknown function, whereas the other equation, the compatibility equation, governs the behavior of this unknown function. |
Slow Migration of Brine Inclusions in First-Year Sea Ice
| Pengarang | : | Noa Kraitzman |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1470-1494 |
| Abstrak | : | We derive a thermodynamically consistent model for liquid-solid phase change in sea ice by incorporating a phase sensitive scalar to a classical framework of liquid-solid phase change. The entropy of the scalar is taken relative to the liquid molar fraction which induces a chemotactic behavior. This provides a transparent mechanism for the rejection of the scalar under formation of the solid phase. We identify slow varying coordinates, including the scalar density relative to liquid molarity weighted by latent heat, and use multiscale analysis to derive a quasi-equilibrium Stefan-type problem via a sharp interface scaling. The singular limit is underdetermined, and the leading order system is closed by imposing local conservation of the scalar under interface perturbation. The quasi-steady system determines interface motion as balance of curvature, temperature gradient, and scalar density. We resolve this numerically for axisymmetric surfaces and show that the thermal gradients typical of arctic sea ice can have a decisive impact on the mode of pinch-off of cylindrical brine inclusions and on the size distribution of the resultant spherical shapes. The density and distribution of these inclusion sizes is a key component of sea ice albedo which factors into global climate models. |
Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1446-1469 |
| Abstrak | : | We consider the Helmholtz transmission problem with piecewise-constant material coefficients and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur and propose modified boundary integral equations that are not affected by them. |
Soil Searching by an Artificial Root
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1423-1445 |
| Abstrak | : | We model an artificial root which grows in the soil for underground prospecting. Its evolution is described by a controlled system of two integro-partial differential equations: one for the growth of the body and the other for the elongation of the tip. At any given time, the angular velocity of the root is obtained by solving a minimization problem with state constraints. We prove the existence of solutions to the evolution problem, up to the first time where a “breakdown configuration” is reached. Some numerical simulations are performed to test the effectiveness of our feedback control algorithm. |
Boundary Tracing for Laplace's Equation with Conformal Mapping
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1411-1422 |
| Abstrak | : | The conformal mapping technique has long been used to obtain exact solutions to Laplace's equation in two-dimensional domains with awkward geometries. However, a major limitation of the technique is that it is only directly compatible with Dirichlet and zero-flux Neumann boundary conditions. It would be useful to have a means of adapting the technique to handle more general boundary conditions, for example, Robin or nonlinear flux conditions. Boundary tracing is an unconventional method for tackling boundary value problems with generic flux boundary conditions, where one takes a known solution to the field equation and seeks new boundaries satisfying the prescribed boundary condition. In this paper, we adapt boundary tracing for compatibility with conformal mapping to produce a new prescription for studying Laplace's equation coupled with general flux boundary conditions. We illustrate the procedure via two simple examples involving heat transfer. In both cases, we demonstrate how to construct infinite families of nontrivial domains in which the solution to the chosen flux boundary value problem is exactly equal to a selected harmonic function. |
Acoustics of a Partially Partitioned Narrow Slit Connected to a Half-Plane: Case Study for Exponential Quasi-Bound States in the Continuum and their Resonant Excitation
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1387-1410 |
| Abstrak | : | Localized wave oscillations in an open system that do not decay or grow in time, despite their frequency lying within a continuous spectrum of radiation modes carrying energy to or from infinity, are known as bound states in the continuum (BIC). Small perturbations from the typically delicate conditions for BIC almost always result in the waves weakly coupling with the radiation modes, leading to leaky states called quasi-BIC that have a large quality factor. We study the asymptotic nature of this weak coupling in the case of acoustic waves interacting with a rigid substrate featuring a partially partitioned slit---a setup that supports quasi-BIC that exponentially approach BIC as the slit is made increasingly narrow. In that limit, we use the method of matched asymptotic expansions in conjunction with reciprocal relations to study those quasi-BIC and their resonant excitation. In particular, we derive a leading approximation for the exponentially small imaginary part of each wavenumber eigenvalue (inversely proportional to quality factor), which is beyond all orders of the expansion for the wavenumber eigenvalue itself. Furthermore, we derive a leading approximation for the exponentially large amplitudes of the states in the case where they are resonantly excited by a plane wave at oblique incidence. These resonances occur in exponentially narrow wavenumber intervals and are physically manifested in cylindrical-dipolar waves emanating from the slit aperture and exponentially large field enhancements inside the slit. The asymptotic approximations are validated against numerical calculations. |
Analysis of Obstacles Immersed in Viscous Fluids Using Brinkman's Law for Steady Stokes and Navier--Stokes Equations
| Pengarang | : | Jorge Aguayo |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 4) |
| Halaman | : | 1369-1386 |
| Abstrak | : | From the steady Stokes and Navier--Stokes models, a penalization method has been considered by several authors for approximating those fluid equations around obstacles. In this work, we present a justification for using fictitious domains to study obstacles immersed in incompressible viscous fluids through a simplified version of Brinkman's law for porous media. If the scalar function ???? is considered as the inverse of permeability, it is possible to study the singularities of ???? as approximations of obstacles (when ???? tends to ∞) or of the domain corresponding to the fluid (when ???? =0 or is very close to 0). The strong convergence of the solution of the perturbed problem to the solution of the strong problem is studied, also considering error estimates that depend on the penalty parameter, for fluids modeled with both the Stokes and Navier--Stokes equations with inhomogeneous boundary conditions. A numerical experiment is presented that validates this result and allows us to study the application of this perturbed problem simulation of flows and the identification of obstacles. |