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This paper is to investigate synchronization theories of a two-group Kuramoto model and a threegroup Kuramoto model. In the settings of these models, every oscillator directly interacts with each other in the same group. In each group, only one oscillator directly interacts with one oscillator in another group. We prove that if the coupling strength is large and the initial configuration of each group is confined to a sector with the arc length less than π, then all oscillators achieve a complete frequency synchronization asymptotically. We emphasize that there is no need to impose any initial condition on the connection between different groups. If, in addition, the natural frequencies in one group are the same, then partial phase synchronization occurs. Moreover, if all natural frequencies are identical, we prove that all oscillators either achieve a complete phase synchronization asymptotically or tend to a bipolar phase-locking state. We also provide several numerical simulations to support the main results. [ABSTRACT FROM AUTHOR]

Evidence shows that resource quality can determine the costs and benefits of the fear effect on consumer dynamics. However, mechanistic modeling and analysis are lacking. This paper formulates a tritrophic level food chain model that integrates both stoichiometric food quality and fear effect. We establish the well-posedness of the model and examine the existence and stability of equilibria. Through extensive numerical simulations, we validate our findings and visually explore the interactive effects of fear and food quality. Our results reveal that the fear effect from predators stabilizes the system. Furthermore, we demonstrate that the fear effect amplifies the influence of food quality on consumers. When food quality is favorable, the fear effect enhances consumer production efficiency, whereas, in the case of poor food quality, the fear effect exacerbates the decline in production efficiency caused by low-nutrient food. [ABSTRACT FROM AUTHOR]

In this paper, we study error diffusion techniques for digital halftoning from the perspective of 1-bit Σ Δ quantization. We introduce a method to generate Σ Δ schemes for two-dimensional signals as a weighted combination of their one-dimensional counterparts and show that various error diffusion schemes proposed in the literature can be represented in this framework via Σ Δ schemes of first order. Under the model of two-dimensional bandlimited signals, which is motivated by a mathematical model of human visual perception, we derive quantitative error bounds for such weighted Σ Δ schemes. We see these bounds as a step towards a mathematical understanding of the good empirical performance of error diffusion, even though they are formulated in the supremum norm, which is known to not fully capture the visual similarity of images. Motivated by the correspondence between existing error diffusion algorithms and first-order Σ Δ schemes, we study the performance of the analogous weighted combinations of second-order Σ Δ schemes and show that they exhibit a superior performance in terms of guaranteed error decay for two-dimensional bandlimited signals. In extensive numerical simulations for real-world images, we demonstrate that with some modifications to enhance stability this superior performance also translates to the problem of digital halftoning. More concretely, we find that certain second-order weighted Σ Δ schemes exhibit competitive performance for digital halftoning of real-world images in terms of the Feature Similarity Index (FSIM), a state-of-the-art measure for image quality assessment. [ABSTRACT FROM AUTHOR]

This paper deals with a receptor-based model which arises from the modeling of interactions between intracellular processes and diffusible signaling factors. We prove the existence of stationary solutions with jump discontinuity by a variational method. Then a singular perturbation problem with discontinuous nonlinearity is studied based on the patching argument and the implicit function theorem, to obtain a solution with a single transition layer. Moreover, we derive a sufficient condition for the stability of stationary solutions with jump discontinuity. In addition, some numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The paper discusses numerical implementations of various inversion schemes for generalized V-line transforms on vector fields introduced in [G. Ambartsoumian, M. J. Latifi, and R. K. Mishra, Inverse Problems, 36 (2020), 104002]. It demonstrates the possibility of efficient recovery of an unknown vector field from five different types of data sets, with and without noise. We examine the performance of the proposed algorithms in a variety of setups, and illustrate our results with numerical simulations on different phantoms. [ABSTRACT FROM AUTHOR]

Social interactions often occur between three or more agents simultaneously. Examining opinion dynamics on hypergraphs allows one to study the effect of such polyadic interactions on the opinions of agents. In this paper, we consider a bounded-confidence model (BCM), in which opinions take continuous values and interacting agents comprise their opinions if they are close enough to each other. We study a density description of a Deffuant--Weisbuch BCM on hypergraphs. We derive a rate equation for the mean-field opinion density as the number of agents becomes infinite, and we prove that this rate equation yields a probability density that converges to noninteracting opinion clusters. Using numerical simulations, we examine bifurcations of the density-based BCM's steady-state opinion clusters and demonstrate that the agent-based BCM converges to the density description of the BCM as the number of agents becomes infinite. [ABSTRACT FROM AUTHOR]

This paper considers the equilibrium positions of n particles in one dimension. Two forces act on the particles; a nonlocal repulsive particle-interaction force and an external force which pushes them to an impenetrable barrier. While the continuum limit as n→∞ is known for a certain class of potentials, numerical simulations show that a discrete boundary layer appears at the impenetrable barrier, i.e. the positions of o(n) particles do not fit to the particle density predicted by the continuum limit. In this paper we establish a first-order Γ-convergence result which guarantees that these o(n) particles converge to a specific continuum boundary-layer profile. [ABSTRACT FROM AUTHOR]

In this paper, the consensus problem of networked multiagent systems (MASs) with time-varying interval uncertain system and output matrices is considered, where the uncertain matrices are bounded by some constant intervals. For this kind of uncertain MAS under a directed graph containing a directed spanning tree, a local interval observer relating only to the bounding information on the uncertainties is introduced to estimate the absolute state of each agent and to implement a robust consensus. Then, a neighborhood framer is designed for this kind of MAS, to estimate the sum of the relative states between each agent and its neighbors, which is used to achieve the desired robust consensus of the systems. Numerical simulations are presented to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we discretize the incompressible Navier-Stokes equations in the framework of finite element exterior calculus. We make use of the Lamb identity to rewrite the equations into a vorticity-velocity-pressure form which fits into the de Rham complex of minimal regularity. We propose a discretization on a large class of finite elements, including arbitrary order polynomial spaces readily available in many libraries. The main advantage of this discretization is that the divergence of the fluid velocity is pointwise zero at the discrete level. This exactness ensures pressure robustness. We focus the analysis on a class of linearized equations for which we prove well-posedness and provide a priori error estimates. The results are validated with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel class of methods for high-order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix φ functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called multirate ExpRB (MERB) methods. They consist of the solution to a sequence of modified "fast" initial-value problems, which may themselves be approximated through subcycling any desired initial-value problem solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous con-vergence analysis of these methods and derive efficient MERB schemes of orders 2 through 6 (the highest-order infinitesimal multirate methods to date). We then present numerical simulations to confirm these theoretical convergence rates and to compare the efficiency of MERB methods against other recently introduced high-order multirate methods. [ABSTRACT FROM AUTHOR]

A coupled system involving a nonlinear scalar PDE and a linear ODE is theoretically investigated. This hypebolic system with relaxation models the propagation of nonlinear waves in a waveguide connected to Helmholtz resonators, this device being an example of a nonlinear acoustic metamaterial. In a previous paper [Sugimoto, J. Fluid. Mech. 1992], it has been shown that this device allows also the propagation of acoustic solitons. In the present paper, the mathematical properties of the coupled system are analysed: formation of singularity in finite time, existence of global smooth solutions for small data, existence of entropy solutions in fractional BV spaces and uniqueness with a single family of entropies. New results are also deduced about weakly coupled systems. Numerical simulations illustrate these findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose a high order numerical scheme to simulate the binary uid{surfactant system by combining the semi-implicit spectral deferred correction (SDC) method and the energy stable linear scheme, in the framework of discontinuous Galerkin (DG) methods. The linear scheme we develop in this paper is decoupled and unconditionally energy stable, which is based on the combination of the convex-concave splitting principle and the invariant energy quadratization approach. However, the scheme is only first order accurate with respect to time, and the SDC method can be employed to iteratively improve the temporal accuracy. Specially, the SDC scheme can be extremely accurate when coupled with an adaptive time stepping strategy. Our numerical scheme leads to a set of decoupled and linear algebraic equations; at each time step, we apply a multigrid solver to solve the equations efficiently. In particular, due to the local property of the DG methods, the resulting algebraic equations can be solved in an explicit way when coupled with the multigrid solver, which is an attractive advantage of the DG method. Various numerical experiments are performed to illustrate the high order accuracy, capability, and efficiency of the proposed methods when solving the binary uid{surfactant system. [ABSTRACT FROM AUTHOR]

In [SIAM J. Sci. Comput., 36 (2014), pp. A693--A713] the authors present a new coarse propagator for the parareal method applied to oscillatory PDEs that exhibit time-scale separation and show, under certain regularity constraints, superlinear convergence which leads to significant parallel speedups over standard parareal methods. The error bound depends on the degree of time-scale separation, ε, and the coarse time step, ƍ, and relies on a bound that holds only in the limit of small. The main result of the present paper is a generalization of this error bound that also holds for finite values of, which can be important for applications in the absence of scale separation. The new error bound is found to depend on an additional parameter, eta, the averaging window used in the nonlinear term of the coarse propagator. The new proof gives insight into how the parareal method can converge even for finite values of . It is also a significant technical advance over the proof presented in [SIAM J. Sci. Comput., 36 (2014), pp. A693--A713]; it requires the introduction of a stiffness regulator function that allows us to control the oscillatory stiffness in the nonlinear term. The new convergence concepts developed in the new proof are confirmed using numerical simulations. [ABSTRACT FROM AUTHOR]

For the predator-prey model with the sigmoid functional response whose denominator has no real roots, the dynamics has already been characterized. Whereas, when the denominator has a real root, the model is singular there, and its dynamics has not been classified. This paper focuses on the case that the denominator has a repeated real root and obtains the next results. The positive equilibrium (if exists) is globally stable provided that it is locally stable. If it is a weak focus, it must be of order one and stable. When the positive equilibrium is unstable, the system has always a limit cycle, which could come from a singular Hopf bifurcation, and undergoes two consecutive canard explosions via relaxation oscillations. In addition, numerical simulations reveal that the curvature of the critical curve at the canard point affects the period of the canard cycle. [ABSTRACT FROM AUTHOR]

The rolling fluid turbine is a bladeless hydraulic device that exhibits peculiar behavior, making it interesting from the aspect of fluid dynamics. The rotor of this turbine has an unusually symmetrical shape and can rotate in two directions--the one it "chooses" seems to be random. Explanation of the underlying hydraulic mechanism is an open problem, which we tackle here. This paper suggests that the hydraulic force on the rotor acts as positive feedback on the motion of the rotor itself. Such an effect can exist since the system is thermodynamically open. Power generation is described in an analytical computation based mostly on the arguments of symmetry. The result is then tested in a numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fundamental problem in modern digital communications known as multiple-input multiple-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite-relaxation-based (SDR-based) algorithms have been proposed to solve the problem in the literature. In this paper, we first show that conventional SDR is generally not tight for the problem. Then, we propose a new and enhanced SDR and show its tightness under an easily checkable condition, which essentially requires the level of the noise to be below a certain threshold. The above results have answered an open question posed by So in [Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10), SIAM, Philadelphia, PA, 2011, pp. 698-711]. Numerical simulation results show that our proposed SDR significantly outperforms the conventional SDR in terms of the relaxation gap. [ABSTRACT FROM AUTHOR]

Oscillators have two main limitations: their synchronization properties are limited (i.e., they have a finite synchronization region) and they have no memory of past interactions (i.e., they return to their intrinsic frequency whenever the entraining signal disappears). We previously proposed a general mechanism to transform an oscillator into an adaptive frequency oscillator which adapts its parameters to learn the frequency of any input signal. The synchronization region then becomes infinite and the oscillator retains the entrainment frequency when the driving signal disappears. While this mechanism has been successfully used in various applications, such as robot control or observer design for active prosthesis, a formal understanding of its properties is still missing. In this paper, we study the adaptation mechanism in the case of strongly coupled phase oscillators and show that nontrivial slow-fast dynamics is at the origin of the adaptation. We show the existence of a layered structure of stable and unstable invariant slow manifolds and demonstrate how the input signal forces the dynamics to jump between these manifolds at regular intervals, leading to exponential convergence of the frequency adaptation. We extend the idea to a network of oscillators with amplitude adaptation and show that the slow invariant manifolds structure persists. Numerical simulations validate our analysis and extend the discussion to more complex cases. [ABSTRACT FROM AUTHOR]

In the last few decades, numerical simulation for nonlinear oscillators has received a great deal of attention, and manyresearchers have been concerned with the design and analysis of numerical methods for solving oscillatory problems. In this paper, from the perspective of the continuous finite element method, we propose and analyze new energy-preserving functionally fitted methods, in particular trigonometrically fitted methods of an arbitrarily high order for solving oscillatory nonlinear Hamiltonian systems with a fixed frequency. To implement these new methods in a widespread way, they are transformed into a class of continuous-stage Runge-Kutta methods. This paper is accompanied by numerical experiments on oscillatory Hamiltonian systems such as the FPU problem and nonlinear Schrödinger equation. The numerical results demonstrate the remarkable accuracy and efficiency of our new methods compared with the existing high-order energy-preserving methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper we undertake a multiscale analysis of nutrient uptake by plant roots by considering different scale relations between the radius of root hairs and the distance between them. We combine the method of formal asymptotic expansions and rigorous derivation of macroscopic equations. The former prompt us to study a distinguished limit (which yields a distinct effective equation), allow us to determine higher-order correctors, and provide motivation for the construction of correctors essential for rigorous derivation of macroscopic equations. The results of our asymptotic analysis are validated by direct comparison with full-geometry numerical simulations. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of boundary stabilization of 1D reaction-diffusion PDEs with a time- and space-varying reaction coefficient. The boundary control design relies on the backstepping approach. The gains of the boundary control are scheduled under two suitable event-triggered mechanisms. More precisely, gains are computed/updated on events according to two state-dependent event-triggering conditions: static-based and dynamic-based conditions, under which the Zeno behavior is avoided and well-posedness and exponential stability of the closed-loop system are guaranteed. Numerical simulations are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide a mathematical and numerical framework for the analysis and design of bubble meta-screens. An acoustic meta-screen is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on acoustic wave propagation. In this paper, periodic subwavelength bubbles mounted on a reective surface (with Dirichlet boundary condition) are considered. It is shown that the structure behaves as an equivalent surface with Neumann boundary condition at the Minnaert resonant frequency which corresponds to a wavelength much greater than the size of the bubbles. An analytical formula for this resonance is derived. Numerical simulations conffirm its accuracy and show how it depends on the ratio between the periodicity of the lattice, the size of the bubble, and the distance from the reective surface. The results of this paper formally explain the superabsorption behavior observed in [V. Leroy et al., Phys. Rev. B, 19 (2015), 02031]. [ABSTRACT FROM AUTHOR]

The binary fluid surfactant phase-field model, coupled with two Cahn--Hilliard equations and Navier--Stokes equations, is a very complex nonlinear system, which poses many challenges to the design of numerical schemes. As far as the author knows, due to the highly nonlinear coupling nature, there is no fully decoupled scheme with second-order accuracy in time for numerical approximation. This paper proposes a novel decoupling approach by introducing a nonlocal auxiliary variable and its associated ODE to deal with the nonlinear coupling terms that satisfy the so-called zero-energy-contribution property. By combining it with other proven effective methods (the projection method of the Navier--Stokes equations and the SAV metho d of linearizing nonlinear potential), we arrive at a fully decoupled, linear, unconditionally energy stable scheme with second-order time accuracy. At each time step, only a few fully decoupled linear elliptic equations with constant coefficients are needed to be solved, which shows the advantages of ease of implementation and efficiency. We also prove the unconditional energy stability rigorously and provide various numerical simulations in two and three dimensions to demonstrate its stability and accuracy, numerically. [ABSTRACT FROM AUTHOR]

This paper introduces a premier and innovative (real-time) multi-scale method for target classification in electrosensing. The intent is that of mimicking the behavior of the weakly electric fish, which is able to retrieve much more information about the target by approaching it. The method is based on a family of transform-invariant shape descriptors computed from generalized polarization tensors (GPTs) reconstructed at multiple scales. The evidence provided by the different descriptors at each scale is fused using Dempster--Shafer theory. Numerical simulations show that the recognition algorithm we propose performs undoubtedly well and yields a robust classification. [ABSTRACT FROM AUTHOR]

This paper deals with the Aw-Rascle-Zhang [A. Aw and M. Rascle, SIAM J. Appl. Math., 60 (2000), pp. 916-938; H. M. Zhang, Transp. Res. B Methodol., 36 (2002), pp. 275-290] model for traffic flow on uni-directional road networks. We construct weak solutions to Riemann problems at the junctions, which conserve the mass and the generalized momentum. In particular, we introduce a new approach to approximate the homogenized pressure through an additional equation for the propagation of a reference pressure. The resulting system of coupled conservation laws is then solved using an appropriate numerical scheme of Godunov type. Numerical simulations show that the proposed approach enables us to approximate the homogenized pressure sufficiently well. The features of the new approach are illustrated through a comparative analysis with other methods proposed in the literature for the Aw-Rascle-Zhang second-order traffic model [2, 31] and the Lighthill-Whitham-Richards model [M. J. Lighthill and G. B. Whitman, Proc. A, 229 (1955), pp. 317-345; P. I. Richards, Oper. Res., 4 (1956), pp. 42-51]. [ABSTRACT FROM AUTHOR]

Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number K, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e., the Navier--Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero along with K, while the second one implies that the GSIS asymptotically preserves the Navier--Stokes limit when K approaches zero. This paper is first dedicated to the rigorous proof of both properties. Second, several numerically challenging cases (especially the two-dimensional thermal edge flow) are used to further demonstrate the accuracy and efficiency of GSIS. [ABSTRACT FROM AUTHOR]

In this paper we develop a two-grid convergence theory for the parallel-in-time scheme known as multigrid reduction in time (MGRIT), as it is implemented in the open-source package [XBraid: Parallel Multigrid in Time, http://llnl.gov/casc/xbraid]. MGRIT is a scalable and multilevel approach to parallel-in-time simulations that nonintrusively uses existing time-stepping schemes, and in a specific two-level setting it is equivalent to the widely known parareal algorithm. The goal of this paper is twofold. First, we present a two-level MGRIT convergence analysis for linear problems where the spatial discretization matrix can be diagonalized, and then apply this analysis to our two basic model problems, the heat equation and the advection equation. One important assumption is that the coarse and fine time-grid propagators can be diagaonalized by the same set of eigenvectors, which is often the case when the same spatial discretization operator is used on the coarse and fine time grids. In many cases, the MGRIT algorithm is guaranteed to converge, and we demonstrate numerically that the theoretically predicted convergence rates are sharp in practice for our model problems. Second, we explore how the convergence of MGRIT compares to the stability of the chosen time-stepping scheme. In particular, we demonstrate that a stable time-stepping scheme does not necessarily imply convergence of MGRIT, although MGRIT with FCF-relaxation always converges for the diffusion dominated problems considered here. [ABSTRACT FROM AUTHOR]

In this paper we consider an ignition-temperature, first-order reaction model of thermo-diffusive combustion that describes dynamics of thick ames arising, for example, in a theory of combustion of hydrogen-oxygen and ethylene-oxygen mixtures. These ames often assume the shape of propagating curved interfaces that can be identified with level sets corresponding to a prescribed ignition temperature. The present paper is concerned with the analysis of such interfaces in two spatial dimensions under a single assumption of their small curvature. We derive a strongly nonlinear evolution equation that governs the dynamics of an interface. This equation relates the normal velocity of the interface to its curvature, the derivatives of the curvature with respect to the arc-length of the interface, and physical parameters of the problem. We study solutions of the evolution equation for various parameter regimes and discuss the ranges of validity of the corresponding simplified models. Our theoretical findings are illustrated and supported by numerical simulations. [ABSTRACT FROM AUTHOR]

The paper is concerned with the optimal harvesting of a marine park, which is described by a parabolic heat equation with Neumann boundary conditions and a nonlinear source term. We consider a cost functional, which is linear with respect to the control; hence the optimal solution can belong to the class of measure-valued control strategies. For each control function, we prove existence and stability estimates for solutions of the parabolic equation. Moreover, we prove the existence of an optimal solution. Finally, some numerical simulations conclude the paper. [ABSTRACT FROM AUTHOR]

In this paper we consider narrow band, active array imaging of weak localized scatterers when only the intensities are recorded at an array with N transducers. We assume that the medium is homogeneous and, hence, wave propagation is fully coherent. This work is an extension of our previous paper [A. Novikov, M. Moscoso, and G. Papanicolaou, SIAM J. Imaging Sci., 8 (2015), pp. 1547{1573], where we showed that using linear combinations of intensity-only measurements, obtained from N² illuminations, imaging of localized scatterers can be carried out efficiently using imaging methods based on the singular value decomposition of the time-reversal matrix. Here we show that the same strategy can be accomplished with only 3N - 2 illuminations, therefore enormously reducing the data acquisition process. Furthermore, we show that in the paraxial regime one can form the images by using six illuminations only. In particular, this paraxial regime includes Fresnel and Fraunhofer diffraction. The key point of this work is that if one controls the illuminations, imaging with intensity only can be easily reduced to an imaging with phases and, therefore, one can apply standard imaging techniques. Detailed numerical simulations illustrate the performance of the proposed imaging strategy with and without data noise. [ABSTRACT FROM AUTHOR]

In this paper we consider the multiscale analysis of a Steklov eigenvalue equation with rapidly oscillating coefficients arising from the modeling of a composite media with a periodic microstructure. There are mainly two new results in the present paper. First, we obtain the convergence rate with ε1/2 for the multiscale asymptotic expansions of the eigenvalues and the eigenfunctions of the Steklov eigenvalue problem. Second, the boundary layer solution is defined. Numerical simulations are then carried out to validate the above theoretical results. [ABSTRACT FROM AUTHOR]

Popular approaches for solving shallow water equations (SWEs) for climate modeling are explicit and semiimplicit methods, and both have certain constraints on the time step size. In this paper, we propose and study a fully implicit method which imposes no limit on the time step size but requires the solution of a large sparse nonlinear system of equations at every time step. The focus of the paper is a parallel, fully coupled, Newton-Krylov-RAS algorithm with a Jacobian matrix explicitly calculated on a weakly nonmatching cubed-sphere mesh. Here, RAS is a restricted additive Schwarz method. We show numerically that with such a preconditioned, nonlinearly implicit method the time step size is no longer constrained by the CFL condition, and we report superlinear speedup of the algorithm on machines with thousands of processors for problems with smooth and nonsmooth solutions. [ABSTRACT FROM AUTHOR]

The goal of this paper is to explain how to derive the classical Rosenzweig--MacArthur model by using a model with two groups of predators in which we can separate the vital dynamic and consumption of prey to describe the behavior of the predators. This will be especially very convenient if we want to add an age or size structure to the predator population. As mentioned by Holling (without mathematical model), we divide the population of predators into the searching and the handling predators. In this article we study some properties of this model and conclude by proving that the model converges to the classical Rosenzweig--MacArthur model by using an appropriate rescaling. This convergence property is observed by using numerical simulations. We also apply this model to the Canadian snowshoe hares and the lynx. [ABSTRACT FROM AUTHOR]

The problem of simulating solid-state dewetting of thin films in three dimensions (3D) by using a sharp-interface approach is considered in this paper. Based on the thermodynamic variation, a speed method is used for calculating the first variation to the total surface energy functional. The speed method shares more advantages than the traditional use of parameterized curves (or surfaces), e.g., it is more intrinsic and its variational structure (related with Cahn-Hoffman ξ-vector) is clearer and more direct. By making use of the first variation, necessary conditions for the equilibrium shape of the solid-state dewetting problem is given, and a kinetic sharp-interface model which includes the surface energy anisotropy is also proposed. This sharp-interface model describes the interface evolution in 3D which occurs through surface diffusion and contact line migration. By solving the proposed model, we perform lots of numerical simulations to investigate the evolution of patterned films, e.g., the evolution of a short cuboid and pinch-off of a long cuboid. Numerical simulations in 3D demonstrate the accuracy and efficacy of the sharp-interface approach to capture many of the complexities observed in solid-state dewetting experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study an agent-based interacting particle system with attractive and singular repulsive forces. We prove the collision avoidance between particles from different groups due to repulsive forces. Moreover, we provide a sufficient condition for the emergence of asymptotic consensus in the same group and separation for different groups. We consider the one-dimensional and multidimensional cases separately since they exhibit different dynamics. Numerical simulations are performed to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on the exponential form of the nonlinear free energy potential is more effective and applicable than the traditional SAV method, which is very popular in constructing energy stable schemes. The first contribution is that the auxiliary variable without square root removes the bounded-frombelow restriction of the nonlinear free energy potential. Then we prove the unconditional energy stability for semidiscrete schemes carefully and rigorously. Another contribution is that we provide a total and explicit discretization of the auxiliary variable combined with the nonlinear term. Such a modification is very efficient for fast calculations. Furthermore, the positivity preserving property of r can be guaranteed, which is very important and reasonable for the models' equivalence. In addition, for complex phase field models with two or more unknown variables and nonlinear terms, we construct a multiple E-SAV (ME-SAV) approach to enhance the applicability of the proposed ESAV approach. A comparative study of classical SAV and E-SAV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy. [ABSTRACT FROM AUTHOR]

The present paper provides exact mathematical expressions for the high-order moments of spiking activity in a recurrently connected network of linear Hawkes processes. It extends previous studies that have explored the case of a (linear) Hawkes network driven by deterministic intensity functions to the case of a stimulation by external inputs (rate functions or spike trains) with arbitrary correlation structure. Our approach describes the spatio-temporal filtering induced by the afferent and recurrent connectivities (with arbitrary synaptic response kernels) using operators acting on the input moments. This algebraic viewpoint provides intuition about how the network ingredients shape the input-output mapping for moments, as well as cumulants. We also show using numerical simulation that our results hold for neurons with refractoriness implemented by self-inhibition, provided the corresponding negative feedback for each neuron only mildly alters its mean firing probability. [ABSTRACT FROM AUTHOR]

In this paper, we consider the evolution of the vortex filament equation (VFE): Xt = Xs\wedge Xss, taking M-sided regular polygons with nonzero torsion as initial data. Using algebraic techniques backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; moreover, the multifractal trajectory of the point X(0, t) is not planar and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as those already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann's nondifferentiable function that are as close to smooth curves as desired when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the self-similar solutions of VFE have finite renormalized energy. [ABSTRACT FROM AUTHOR]

This paper deals with 2D image transformations from a perspective of a 3D heterogeneous shape modeling and computer animation. Shape and image morphing techniques have attracted a lot of attention in artistic design, computer animation, and interactive and streaming applications. We present a novel method for morphing between two topologically arbitrary 2D shapes with sophisticated textures (raster color attributes) using a metamorphosis technique called space-time blending (STB) coupled with space-time transfinite interpolation. The method allows for a smooth transition between source and target objects by generating in-between shapes and associated textures without setting any correspondences between boundary points or features. The method requires no preprocessing and can be applied in 2D animation when position and topology of source and target objects are significantly different. With the conversion of given 2D shapes to signed distance fields, we have detected a number of problems with directly applying STB to them. We propose a set of novel and mathematically substantiated techniques, providing automatic control of the morphing process with STB and an algorithm of applying those techniques in combination. We illustrate our method with applications in 2D animation and interactive applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider the feedback stabilization problem for N -level quantum angular momentum systems undergoing continuous-time measurements. By using stochastic and geometric control tools, we provide sufficient conditions on the feedback control law ensuring almost sure exponential convergence to a predetermined eigenstate of the measurement operator. In order to achieve these results, we establish general features of quantum tra jectories which are of interest by themselves. We illustrate the results by designing a class of feedback control laws satisfying the above-mentioned conditions and finally we demonstrate the effectiveness of our methodology through numerical simulations for three-level quantum angular momentum systems. [ABSTRACT FROM AUTHOR]

When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schr\"odinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated by an arbitrary distance along the edges and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable due to an irreversible drift along the incoming edge toward the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph. [ABSTRACT FROM AUTHOR]

A generalization of the Cucker-Smale model for collective animal behavior is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modeling: (i) stochasticity (imperfections) of individual behavior and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a by-product, a new result regarding the asymptotic behavior of delayed geometric Brownian motion is obtained. In the second part of the paper, results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behavior of the system--namely, that the introduction of an intermediate time delay may facilitate flocking. [ABSTRACT FROM AUTHOR]

Numerical methods that approximate the solution of the Vlasov-Poisson equation by a low-rank representation have been considered recently. These methods can be extremely effective from a computational point of view, but contrary to most semi-Lagrangian or Eulerian Vlasov solvers, they do not conserve mass and momentum, neither globally nor in respecting the corresponding local conservation laws. This can be a significant limitation for intermediate and long time integration. In this paper we propose a numerical algorithm that overcomes some of these difficulties and demonstrate its utility by presenting numerical simulations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the interaction between a two-level system (qubit) and two continuous-mode photons. Two scenarios are investigated: Case 1, how a two-level system changes the pulse shapes of two input photons propagating in a single input channel; and Case 2, how a two-level system responds to two counterpropagating photons, one in each input channel. The steady-state output field states for both cases are derived analytically in both the time and frequency domains. Numerical simulations demonstrate rich and interesting two-photon scattering phenomena induced by a two-level system. [ABSTRACT FROM AUTHOR]

We consider the time-reversal of waves in time-dependent media. The constitutive parameters of a wave equation are assumed constant in time during intervals (0, t \tau)\cup (t+\tau, 2t) for \tau \ll t and time-varying during the time interval (t \tau, t + \tau) to model a quasi-instantaneous time mirror. We show that under appropriate hypotheses, a time-reversed signal is generated by such time-dependent fluctuations. At time 2t, the time-reversed signal is an appropriate differentiation of the original initial condition when the time-varying fluctuations are independent of space. In the case of spatially varying fluctuations, we show that the time-reversed signal enjoys stronger refocusing properties when propagation occurs in a highly heterogeneous environment, as in the case of classical time-reversal. This paper offers additional theoretical justifications to the experimental results obtained in [Bacot et al., Nature Phys., 12 (2016), pp. 972--977] and proposes potential extensions. We present numerical simulations that also corroborate and quantify the theoretical predictions. [ABSTRACT FROM AUTHOR]

We introduce a new second-order particle swarm model on a sphere and study its emergent behaviors. The proposed particle swarm model can be naturally derived from the first-order swarm model on a sphere by adding inertial effect, and as special cases, it can be reduced to other collective particle models, such as the Kuramoto model with inertia and the inertial spin model. In this paper, we provide several sufficient frameworks for the emergent dynamics of the proposed second-order particle swarm model in attractive and repulsive coupling regimes in terms of the system parameters and the initial data. We also perform several numerical simulations to see the effect of couplings and compare them with our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschitz condition, we first show that the SDE is pth moment exponentially stable (for p ? (0, 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size ?t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ? (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems, (P1) and (P2) listed in section 1. [ABSTRACT FROM AUTHOR]

The paper proposes a general framework to analyze control problems for conservation law models on a network. Namely, we consider a general class of junction distribution controls and inflow controls and we establish the compactness in L¹ of a class of flux-traces of solutions. We then derive the existence of solutions for two optimization problems: (I) the maximization of an integral functional depending on the flux-traces of solutions evaluated at points of the incoming and outgoing edges; (II) the minimization of the total variation of the optimal solutions of problem (I). Finally we provide an equivalent variational formulation of the min-max problem (II) and we discuss some numerical simulations for a junction with two incoming and two outgoing edges. [ABSTRACT FROM AUTHOR]

In this paper we discuss discrete conservation laws for diffusion equations over triangular surface meshes from the viewpoint of duality. Conservation laws are very important for us to model physical phenomenon on curved spaces. The key idea of our method is to use the concept of local dual meshes, which are determined by the centroid points of the triangles in the original meshes. The Green formula will provide us a natural way to give a discrete approximation of the Laplace-Beltrami operators on functions over curved regular surfaces. Then, we shall show that discrete conservation laws are fulfilled on the dual meshes for our algorithm. Moreover, our discrete Laplace-Beltrami operators have the local property: it only involves the information on the 1-ring of neighboring vertices. Note that this approach is quite different from the finite element/difference method, used very often in the filed of numerical analysis. Some convergence problems will also be discussed. Numerical simulations are given to support theses results. [ABSTRACT FROM AUTHOR]

Numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. This paper presents the first rigorous PP analysis of conservative schemes with the Lax--Friedrichs (LF) flux for 1D and multidimensional ideal MHD. The significant innovation is the discovery of the theoretical connection between the PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives to the usually expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by the DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality, which is skillfully constructed by technical estimates. Rigorous PP analysis is then presented for finite volume and discontinuous Galerkin schemes with the LF flux on uniform Cartesian meshes. In the 1D case, the PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In the 2D case, we show that the DDF condition is necessary and crucial for achieving the PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves both the positivity and the DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples provided in the supplementary material further confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper we propose a new Eulerian model and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model relies on the use of fractional droplet surface moments and their ability to predict some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. Evaporation is evaluated by using a maximum entropy (ME) reconstruction. The use of fractional moments introduces some theoretical and numerical difficulties. First, relying on a study of the moment space, we extend the ME reconstruction to the case of fractional moments. Then we propose a new high order and robust algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. It involves some negative order fractional moments for which a novel treatment is introduced. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with a minimal number of variables, as well as minimal computational cost, but also provide an additional capacity of coupling with the diffuse interface model and the transport equation of averaged geometrical interface variables, which are essential in order to describe atomization. [ABSTRACT FROM AUTHOR]