Showing papers in "Quarterly of Applied Mathematics in 2019"
TL;DR: In this article, Ammari et al. provided a mathematical theory for the double-negative refractive index phenomenon in bubble fluids, and proved that the effective mass density and the bulk modulus of the bubble fluid can be negative near the anti-resonance of the two hybridized Minnaert resonances for a single constituent bubble dimer.
Abstract: The aim of this paper is to provide a mathematical theory for understanding the mechanism behind the double-negative refractive index phenomenon in bubbly fluids. The design of double-negative metamaterials generally requires the use of two different kinds of subwavelength resonators, which may limits the applicability of double-negative metamaterials. Herein we rely on media that consists of only a single type of resonant element, and show how to turn the acoustic metamaterial with a single negative effective property obtained in [H. Ammari and H. Zhang, Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency. SIAM J. Math. Anal., 49 (2017), 3252--3276.] into a negative refractive index metamaterial, which refracts waves negatively, hence acting as a superlens. Using bubble dimers made of two identical bubbles, it is proved that both the effective mass density and the bulk modulus of the bubbly fluid can be negative near the anti-resonance of the two hybridized Minnaert resonances for a single constituent bubble dimer. A rigorous justification of the Minnaert resonance hybridization, in the case of a bubble dimer in a homogeneous medium, is established. The acoustic properties of a single bubble dimer are analyzed. Asymptotic formulas for the two hybridized Minnaert resonances are derived. Moreover, it is proved that the bubble dimer can be approximated by a point scatterer with monopole and dipole modes. For an appropriate volume fraction of bubble dimers with certain conditions on their configuration, a double-negative effective medium when the frequency is near the anti-resonance of the hybridized Minnaert resonances can be obtained.
47 citations
TL;DR: In this paper, the existence of traveling waves in a spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattice was studied.
Abstract: We study the existence of traveling waves in a spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. This is a one-dimensional lattice of identical particles connected by alternating nonlinear springs. Following the work of Faver and Wright on the mass dimer, or diatomic, lattice, we find that the lattice equations in the long wave regime are singularly perturbed and apply a method of Beale to produce nanopteron traveling waves with wave speed slightly greater than the lattice's speed of sound. The nanopteron wave profiles are the superposition of an exponentially decaying term (which itself is a small perturbation of a KdV sech2-type soliton) and a periodic term of very small amplitude. Generalizing our work in the diatomic case, we allow the nonlinearity in the spring forces to have the more complicated form "quadratic plus higher order terms." This necessitates the use of composition operators to phrase the long wave problem, and these operators require delicate estimates due to the characteristic superposition of function types from Beale's ansatz. Unlike the diatomic case, the value of the leading order term in the traveling wave profiles alternates between particle sites, so that the spring dimer traveling waves are also "stegotons," in the terminology of LeVeque and Yong. This behavior is absent in the mass dimer and confirms the approximation results of Gaison, Moskow, Wright, and Zhang for the spring dimer.
24 citations
15 citations
TL;DR: In this article, the stationary and traveling wave solutions to a family of degenerate dispersive equations of KdV and NLS-type were analyzed in the $1$ dimension.
Abstract: We analyze the stationary and traveling wave solutions to a family of degenerate dispersive equations of KdV and NLS-type. In stark contrast to the standard soliton solutions for non-degenerate KdV and NLS equations, the degeneracy of the elliptic operators studied here allows for compactly supported steady or traveling states. As we work in $1$ dimension, ODE methods apply, however the models considered have formally conserved Hamiltonian, Mass and Momentum functionals, which allow for variational analysis as well.
14 citations
TL;DR: In this paper, the authors studied the interaction of elementary waves of the Riemann problem with a weak discontinuity for an isothermal no-slip compressible gas-liquid drift flux equation of two-phase flows.
Abstract: In this paper, we study the interaction of elementary waves of the Riemann problem with a weak discontinuity for an isothermal no-slip compressible gas-liquid drift flux equation of two-phase flows. We construct the solution of the Riemann problem in terms of a one parameter family of curves. Using the properties of elementary waves, we prove a necessary and sufficient condition on initial data for which the solution of the Riemann problem consists of a left shock, contact discontinuity, and a right shock. Moreover, we derive the amplitudes of weak discontinuity and discuss the interactions of weak discontinuity with shocks and contact discontinuity. Finally, we carry out some tests to investigate the effect of shock strength and initial data on the jump in shock acceleration and the amplitudes of reflected and transmitted waves.
14 citations
TL;DR: In this article, a new MCMC tool called Attraction-diffusion (AD) is proposed to capture the macroscopic structure of highly non-convex Gibbs densities by measuring the metastability of local modes.
Abstract: An image pattern can be represented by a probability distribution whose density is concentrated on different low-dimensional subspaces in the high-dimensional image space. Such probability densities have an astronomical number of local modes corresponding to typical pattern appearances. Related groups of modes can join to form macroscopic image basins that represent pattern concepts. Recent works use neural networks that capture high-order image statistics to learn Gibbs models capable of synthesizing realistic images of many patterns. However, characterizing a learned probability density to uncover the Hopfield memories of the model, encoded by the structure of the local modes, remains an open challenge. In this work, we present novel computational experiments that map and visualize the local mode structure of Gibbs densities. Efficient mapping requires identifying the global basins without enumerating the countless modes. Inspired by Grenander's jump-diffusion method, we propose a new MCMC tool called Attraction-Diffusion (AD) that can capture the macroscopic structure of highly non-convex densities by measuring metastability of local modes. AD involves altering the target density with a magnetization potential penalizing distance from a known mode and running an MCMC sample of the altered density to measure the stability of the initial chain state. Using a low-dimensional generator network to facilitate exploration, we map image spaces with up to 12,288 dimensions (64 $\times$ 64 pixels in RGB). Our work shows: (1) AD can efficiently map highly non-convex probability densities, (2) metastable regions of pattern probability densities contain coherent groups of images, and (3) the perceptibility of differences between training images influences the metastability of image basins.
13 citations
TL;DR: In this article, the authors considered a sequence of identically distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density and established a new rate of convergence of the Wasserstein distance between the empirical measure of the samples and the true distribution.
Abstract: We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $\\infty-$Wasserstein distance between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slep\\v{c}ev to the case of unbounded density.
12 citations
TL;DR: An inference engine based on energy minimization using an efficient Markov Chain Monte Carlo that uses the ConceptNet in its move proposals to find these structures that describe the image content.
Abstract: The task of interpretation of activities as captured in video extends beyond just the recognition of observed actions and objects. It involves open world reasoning and constructing deep semantic connections that go beyond what is directly observed in the video and annotated in the training data. Prior knowledge plays a big role. Grenander’s canonical pattern theory representation offers an elegant mechanism to capture these semantic connections between what is observed directly in the image and past knowledge in large-scale common sense knowledge bases, such as ConceptNet. We represent interpretations using a connected structure of basic detected (grounded) concepts, such as objects and actions, that are bound by semantics with other background concepts not directly observed, i.e., contextualization cues. Concepts are basic generators and the bonds are defined by the semantic relationships between concepts. Local and global regularity constraints govern these bonds and the overall connection structure. We use an inference engine based on energy minimization using an efficient Markov Chain Monte Carlo that uses the ConceptNet in its move proposals to find these structures that describe the image content. Using four different publicly available large datasets, Charades, Microsoft Visual Description Corpus (MSVD), Breakfast Actions, and CMU Kitchen, we Received March 22, 2018, and, in revised form, October 12, 2018. 2010 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.
11 citations
TL;DR: In this article, the uniqueness of weak solutions to the special relativistic Landau equation was proved under the conditional assumption that the solution satisfies (p^0)^7 F(t,p) \in L^1 ([0,T]; L^\infty] ).
Abstract: We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) \in L^1 ([0,T]; L^\infty)$. The existence of standard weak solutions to the relativistic Landau equation has been shown recently in (arXiv:1806.08720)
11 citations
10 citations
TL;DR: In this paper, the authors considered Riemann problems for a two-phase isothermal flow model of a dispersed phase in a compressible carrier phase, which is a weakly hyperbolic system of conservative partial differential equations.
Abstract: We consider Riemann problems for a two-phase isothermal flow model of a dispersed phase in a compressible carrier phase. It is a weakly hyperbolic system of conservative partial differential equations. This model is the conservation part of a more complete physical model involving phase transitions in case both phases are of the same material. The purpose of this paper is to better understand the mathematical properties of the simplified model. We investigate the characteristic structure of the Riemann problems and construct their exact solutions. Solutions may contain delta shocks or vaporless states. We give examples for initial data corresponding to a system of water bubbles dispersed in liquid water. The analysis is complicated considerably by the fact that a liquid such as water requires an affine equation of state.
TL;DR: In this paper, the first and second boundary value problems for the Stokes system posed in a bounded Lipschitz domain in the context of a Dirichlet condition were reviewed.
Abstract: We review the first and second boundary value problems for the Stokes system posed in a bounded Lipschitz domain in . Particular attention is given to the mixed boundary condition: a Dirichlet cond ...
TL;DR: In this article, it was shown that the renormalization property holds for an electromagnetic field with only a fractional space derivative in some Lebesgue spaces, which is the case for weak solutions to the Vlasov-Maxwell equations.
Abstract: In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{\alpha,p}$, we determine Onsager type exponents $\alpha$ that guarantee the conservation of all entropies. In particular, the Onsager exponent $\alpha$ is smaller than $\alpha = 1/3$ established for fluid models. Entropies conservation is equivalent to the renormalization property, which have been introduced by DiPerna--Lions for studying well-posedness of passive transport equations and collisionless kinetic equations. For smooth solutions renormalization property or entropies conservation are simply the consequence of the chain rule. For weak solutions the use of the chain rule is not always justified. Then arises the question about the minimal regularity needed for weak solutions to guarantee such properties. In the DiPerna--Lions and Bouchut--Ambrosio theories, renormalization property holds under sufficient conditions in terms of the regularity of the advection field, which are roughly speaking an entire derivative in some Lebesgue spaces (DiPerna--Lions) or an entire derivative in the space of measures with finite total variation (Bouchut--Ambrosio). In return there is no smoothness requirement for the advected density, except some natural a priori bounds. Here we show that the renormalization property holds for an electromagnetic field with only a fractional space derivative in some Lebesgue spaces. To compensate this loss of derivative for the electromagnetic field, the distribution function requires an additional smoothness, typically fractional Sobolev differentiability in phase-space. As concerns the conservation of total energy, if the macroscopic kinetic energy is in $L^2$, then total energy is preserved.
TL;DR: In this article, a non-stationary incompressible 3D fluid flow with a nonconstant (temperature dependent) viscosity, subjected to mixed boundary conditions with a given time dependent velocity on a part of the boundary and Tresca's friction law on the other part, is considered.
Abstract: Motivated by extrusion problems, we consider a non-stationary incompress-ible 3D fluid flow with a non-constant (temperature dependent) viscosity, subjected to mixed boundary conditions with a given time dependent velocity on a part of the boundary and Tresca's friction law on the other part. We construct a sequence of approximate solutions by using a regularization of the free boundary condition due to friction combined with a particular penalty method, reminiscent of the " incompressibility limit " of compressible fluids, allowing to get better insights into the links between the fluid velocity and pressure fields. Then we pass to the limit with compactness arguments to obtain a solution to our original problem.
TL;DR: In this paper, a numerical method is developed to study the stability of standing water waves and other time-periodic solutions of the free-surface Euler equations using Floquet theory, and the eigenvalues of the truncated monodromy operator are computed and ordered by the mean wave number of the corresponding eigenfunctions.
Abstract: A numerical method is developed to study the stability of standing water waves and other time-periodic solutions of the free-surface Euler equations using Floquet theory. A Fourier truncation of the monodromy operator is computed by solving the linearized Euler equations about the standing wave with initial conditions ranging over all Fourier modes up to a given wave number. The eigenvalues of the truncated monodromy operator are computed and ordered by the mean wave number of the corresponding eigenfunctions, which we introduce as a method of retaining only accurately computed Floquet multipliers. The mean wave number matches up with analytical results for the zero-amplitude standing wave and is helpful in identifying which Floquet multipliers collide and leave the unit circle to form unstable eigenmodes or rejoin the unit circle to regain stability. For standing waves in deep water, most waves with crest acceleration below $A_c=0.889$ are found to be linearly stable to harmonic perturbations; however, we find several bubbles of instability at lower values of $A_c$ that have not been reported previously in the literature. We also study the stability of several new or recently discovered time-periodic gravity-capillary or gravity waves in deep or shallow water, finding several examples of large-amplitude waves that are stable to harmonic perturbations and others that are not. A new method of matching the Floquet multipliers of two nearby standing waves by solving a linear assignment problem is also proposed to track individual eigenvalues via homotopy from the zero-amplitude state to large-amplitude standing waves.
TL;DR: Aleksandrov et al. as discussed by the authors used the theory of Abelian integral operators and the Riemann-Hilbert problem on a segment to derive the solutions to the model problems and establish the relations between governing systems of integral equations associated with the four models and their solutions.
Abstract: This paper advances the relations by A. Ja. Aleksandrov between axisymmetric and plane strain states to the case of mixed boundary-value problems. We revise two models of a strip-shaped and a circular stamp indented into a half-plane and a half-space, respectively, when the normal and tangential traction components are unknown a priori. Also, two models of a strip-shaped and a penny-shaped interfacial crack are considered. By using the theory of Abelian integral operators and the Riemann-Hilbert problem on a segment we derive the solutions to the model problems and establish the relations between the governing systems of integral equations associated with the four models and their solutions. These relations can be interpreted as mappings between (i) plane and axisymmetric contact problems, (ii) plane and axisymmetric fracture models, (iii) plane contact and fracture problems, and (iv) axisymmetric contact and fracture problems. The mappings enable us to write down the governing systems of integral equations and the solutions to any three models by making use of the governing system and the solution to the fourth problem. The transformations are specified in the scalar cases when there is no friction in the contact zone and when a crack is in a homogeneous elastic medium. By considering the contact frictionless problem of an annulus stamp it is shown that, although an exact solution to the plane strain frictionless contact problem of two stamps is available, a transformation of the plane to the axisymmetric solution in this case is not possible to obtain, and derivation of a closed-form solution to the annulus stamp model is still an open mathematical problem.
TL;DR: In this article, the fundamental equations of wave turbulence theory are rigorously justified for the nonlinear Schrödinger equation on short but nontrivial time scales, and a recent work of the authors in which they obtained the so-called wave kinetic equation, predicted in wave turbulence theories, for the nonsquilibrium Schröding equation is described.
Abstract: The purpose of this note is two-fold: A) We give a brief introduction into the problem of rigorously justifying the fundamental equations of wave turbulence theory (the theory of nonequilibrium statistical mechanics of nonlinear waves), and B) we describe a recent work of the authors in which they obtain the so-called wave kinetic equation, predicted in wave turbulence theory, for the nonlinear Schrödinger equation on short but nontrivial time scales.
TL;DR: In this article, the robustness of nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Planck (KS-FP) equation in the presence of frustrations was studied.
Abstract: We study the robustness in the nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Planck (KS-FP for short) equation in the presence of frustrations. For this, we construct a new unstable mode for the corresponding linear part of the perturbation around the incoherent state, and we show that the nonlinear perturbation stays close to the unstable mode in some small time interval which depends on the initial size of the perturbations. Our instability results improve the previous results on the KS-FP with zero frustration [J. Stat. Phys. 160 (2015), pp. 477–496] by providing a new linear unstable mode and detailed energy estimates.
TL;DR: In this paper, the authors studied the pointwise behavior of the Fokker-Planck Equation with flat confinement in the space and time variables, including large time behavior, initial layer and asymptotic behavior.
Abstract: We study the pointwise (in the space and time variables) behavior of the Fokker-Planck Equation with flat confinement. The solution has very clear description in the $xt-$plane, including large time behavior, initial layer and asymptotic behavior. Moreover, the structure of the solution highly depends on the potential function.
TL;DR: In this article, two sufficient frameworks for the emergent dynamics to the discrete thermodynamic Cucker-Smale (TCS) model are presented in terms of the initial data and system parameters.
Abstract: We present two sufficient frameworks for the emergent dynamics to the discrete thermodynamic Cucker-Smale (TCS) model. Our proposed frameworks are formulated in terms of the initial data and system parameters. The TCS model was first introduced to incorporate the effect of a temperature field in the dynamics of the Cucker-Smale model, and it has been systematically derived from the hydrodynamic model for gas mixture under the spatial homogeneity assumption. The particle model by Cucker and Smale describes the temporal evolution of mechanical observables such as position and velocity, whereas our TCS model governs the dynamics of position, velocity, and temperature of thermodynamic C-S particles. The TCS model conserves the mass, momentum, and energy, and the total entropy is monotonically increasing so that it is consistent with the principle of thermodynamics.
TL;DR: In this article, an iterative coupling scheme based on a weak formulation with nonlinear interior boundary conditions, both for continuous time and with implicit discretization (backward Euler) in time, was proposed.
Abstract: We study the solution of the system of equations describing the dynamical evolution of spontaneous ruptures generated in a prestressed elastic-gravitational deforming body and governed by rate and state friction laws. We propose an iterative coupling scheme based on a weak formulation with nonlinear interior boundary conditions, both for continuous time and with implicit discretization (backward Euler) in time. We regularize the problem by introducing viscosity. This guarantees the convergence of the scheme for solutions of the regularized problems in both cases. We also make precise the conditions on the relevant coefficients for convergence to hold.
TL;DR: It is proved that the xed points of the algorithms under consideration dene lower-bounds on the energy function and the max-marginals of a Markov random eld, independent of initialization.
Abstract: We dene two algorithms for propagating information in classication problems with pairwise relationships. The algorithms involve contraction maps and are related to non-linear diusion and random walks on graphs. The approach is also related to message passing and mean eld methods. The algorithms we describe are guaranteed to converge on graphs with arbitrary topology. Moreover they always converge to a unique xed point, independent of initialization. We prove that the xed points of the algorithms under consideration dene lower-bounds on the energy function and the max-marginals of a Markov random eld. Our theoretical results also illustrate a relationship between message passing algorithms and value iteration for an innite horizon Markov decision process. We illustrate the practical feasibility of our algorithms with preliminary experiments in image restoration and stereo depth estimation.
TL;DR: In this article, the authors derived quantitatively precise estimates of the field enhancement in the narrow region between two inclusions and showed that the field is enhanced by a factor of Ω(n ϵ −1/2 ) where ϵ is the distance between two infinities.
Abstract: A field excited by an emitter can be enhanced due to presence of closely located inclusions. In this paper we consider such field enhancement when inclusions are disks of the same radii, and the emitter is of dipole type and located in the narrow region between two inclusions. We derive quantitatively precise estimates of the field enhancement in the narrow region. The estimates reveal that the field is enhanced by a factor of $\epsilon^{-1/2}$ in most area, where $\epsilon$ is the distance between two inclusions. This factor is the same as that of gradient blow-up when there is a smooth back-ground field, not a field excited by an emitter. The method of deriving estimates shows clearly that enhancement is due to potential gap between two inclusions.
TL;DR: In this article, the authors introduce the concept of hyper-elastic Ricci flow, which is a uniformization of material behavior which follows from application of a parabolic minimum principle.
Abstract: This paper introduces the concept of hyper-elastic Ricci flow. The equation of hyper-elastic Ricci flow amends classical Ricci flow by the addition of the Cauchy stress tensor which itself is derived from the free energy. The main implication of the theory is a uniformization of material behavior which follows from application of a parabolic minimum principle.
TL;DR: In this paper, the authors consider a diffusion process on an evolving surface with a piecewise Lipschitz-continuous boundary from an energetic point of view and derive the generalized diffusion and heat systems on the evolving surface.
Abstract: We consider a diffusion process on an evolving surface with a piecewise Lipschitz-continuous boundary from an energetic point of view. We employ an energetic variational approach with both surface divergence and transport theorems to derive the generalized diffusion and heat systems on the evolving surface. Moreover, we investigate the boundary conditions for the two systems to study the conservation and energy laws of them. As an application, we make a mathematical model for a diffusion process on an evolving double bubble. Especially, this paper is devoted to deriving the representation formula for the unit outer co-normal vector to the boundary of a surface.
TL;DR: In this article, the feasibility of extending the Akaike Information Criterion (AIC) for deterministic systems as a potential model selection criteria for stochastic models is examined.
Abstract: In this paper, we examine the feasibility of extending the Akaike Information Criterion (AIC) for deterministic systems as a potential model selection criteria for stochastic models. We discuss the implementation method for three different classes of stochastic models: continuous time Markov chains (CTMC), stochastic differential equations (SDE), and random differential equations (RDE). The effectiveness and limitations of implementing the AIC for comparison of stochastic models is demonstrated using simulated data from the three types of models and then applied to experimental longitudinal growth data for algae.