Unbounded Region Research Articles

Enhanced photon squeezing in two-photon Dicke model

We explore the phenomena of quadrature squeezing of photons in the two-photon Dicke model under the mean-field approximation in the thermodynamic limit. An effective squeezing of the photonic quadratures beyond the standard quantum limit is observed in the superradiant phase. The magnitude of squeezing is enhanced, and it persists longer as one moves away from the phase boundary and approaches the boundary between the superradiant phase and the unbounded region. This unbounded region is encountered where the coupling strength is of the same order of magnitude as one of the fundamental frequencies of the system. We find that the squeezing is prolonged near the said boundary for varying transition energy of the qubits. Furthermore, we explore the critical behavior of photon squeezing near the unbounded region.

Believing in Infinity: Exploring Students’ Conceptions of Improper Integrals

We discuss student conceptions of improper integrals and infinity in the context of a second-semester calculus course (in a three-course sequence). Our observations stem from a sequence of activities used in an online course over a three-day period. Throughout the enactment of these activities, students are challenged to develop conceptions of improper integrals that are compatible with the conventions of the discipline, and they are forced to wrestle with their underlying beliefs about infinity, limits, and areas of unbounded regions.

On the behaviour of derivative of algebraic polynomials in the regions with cusps

ABSTRACT In this paper, we study the behavior of derivatives of algebraic polynomials in bounded and unbounded regions of the complex plane. At the same time, both interior and exterior cusp points are allowed on the boundary of such regions. Bernstein-Walsh-type estimates are obtained for derivatives of algebraic polynomials in the specified region with corners for exterior points, as well as Markoff-type estimates for closure of the region. As a result, estimates are found for the derivatives of algebraic polynomials in the whole complex plane. It is also shown that the inequality for the closed region is exact in order for the given region.

An Enhanced Strategy for Adaptive Output-Feedback Control of Uncertain Nonlinear Systems

We are concerned with global practical tracking for uncertain nonlinear systems with intrinsic dependence on unmeasured states. Typically, we aim to remove the polynomial restrictions on system growth rates in relevant works, admitting the rates to be of arbitrary function-of-output type. Note that such highly nonlinear rates could alter so drastically in magnitude that no polynomial can upper bound them on any unbounded region, and would undermine the design rationale and the analysis route that are followed in polynomial rate scenarios. This entails refining/integrating state-of-the-art techniques and developing new ones, from design to analysis. Concretely, to circumvent the need for knowledge on polynomials, dual dynamic high gains are devised ingeniously which particularly incorporate certain highly nonlinear components to counteract the arbitrary function-of-output rates and other inherent nonlinearities and uncertainties. But unfavorable impacts of the components would also arise putting the effectiveness of the high gains at risk, which necessitates noticeable changes in the choice of vital variables/functions. In addition, to provide tractable observer error dynamics for control design and analysis, a nonlinear high-gain observer is pursued, which is filter based and of reduced order, following the design methodologies in relevant works. In terms of performance analysis, practical tracking suffers extra time variants and additive uncertainties which together with the highly nonlinear rates would carry over to stability analysis inevitably and in turn ruin the existing analysis routes. Hence, a new analysis route is paved to verify the anticipated performance. Notably, by unfolding an important implication, the performance verification boils down largely to the boundedness of the high gains. This makes plain and intelligible the verification, although sophisticated integration of multiple composite Lyapunov functions is still required. An exception, i.e., asymptotic stabilization, is presented to demonstrate its close connection with practical tracking.

Complete Graph Drawings up to Triangle Mutations

The main result of the paper can be stated in the following way: a complete graph drawing in the sphere, where two edges have at most one common point, which is either a crossing or a common endpoint, and no three edges share a crossing, is determined by the circular ordering of edges at each vertex, or equivalently by the set of pairs of edges that cross, up to homeomorphism and a sequence of triangle mutations. A triangle mutation (or switch, or flip) consists in passing an edge across the intersection of two other edges, assuming the three edges cross each other and the region delimited by the three edges has an empty intersection with the drawing. Equivalently, the result holds for a drawing in the plane assuming the drawing is given with a pair of edges indicating where the unbounded region is. The proof is constructive, based on an inductive algorithm that adds vertices and their incident edges one by one (actually yielding an extra property for the sequence of triangle mutations). This result generalizes Ringel’s theorem on uniform pseudoline arrangements (or rank 3 uniform oriented matroids) to complete graph drawings. We also apply this result to plane projections (or visualizations) of a geometric spatial complete graph, in terms of the rank 4 uniform oriented matroid defined by its vertices. Independently, we prove that, for a complete graph drawing, the set of pairs of edges that cross determine, by first order logic formulas, the circular ordering of edges at each vertex, as well as further information.

Orbital Stability of Internal Waves

This paper studies the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler equations, with the density of each fluid being constant but distinct. A diverse collection of small-amplitude solitary wave solutions for this system have been constructed by several authors in the case of strong surface tension (as measured by the Bond number) and slightly subcritical Froude number. We prove that all of these waves are (conditionally) orbitally stable in the natural energy space. Moreover, the trivial solution is shown to be conditionally stable when the Bond and Froude numbers lie in a certain unbounded parameter region. For the near critical surface tension regime, we prove that one can infer conditional orbital stability or orbital instability of small-amplitude traveling waves solutions to the full Euler system from considerations of a dispersive PDE model equation. These results are obtained by reformulating the problem as an infinite-dimensional Hamiltonian system, then applying a version of the Grillakis–Shatah–Strauss method recently introduced in Varholm et al. (Commun Pure Appl Math 73:2634–2684, 2020). A key part of the analysis consists of computing the spectrum of the linearized augmented Hamiltonian at a shear flow or small-amplitude wave. For this, we generalize an idea used by Mielke (R Soc Lond Philos Trans Ser A Math Phys Eng Sci 360:2337–2358, 2002) to treat capillary-gravity water waves beneath vacuum.

Numerical approximation to semi-linear stiff neutral equations via implicit–explicit general linear methods

Among the initial value problems of semi-linear neutral equations, there are a class of so-called stiff problems, where the classical explicit methods do not work as the methods have only bounded stability regions, which confine the computational stepsize to be excessively small and thus leads to an unsuccessful calculation. For resolving this difficult issue, ones turned to develop the implicit methods with unbounded stability regions to solve the stiff problems. Nevertheless, it is well-known that the implementation of an implicit method needs a large computational cost. In order to improve the computational efficiency, in Refs. Tan and Zhang (2018, 2020), the authors adopted the implicit–explicit (IMEX) splitting technique to derive the extended IMEX one-leg methods and IMEX Runge–Kutta methods, respectively. Unfortunately, these two methods have the serious order barrier. So far, for stiff neutral equations (SNEs), no IMEX method with order more than two has been found. To improve the computational accuracy and efficiency of IMEX methods, in the present paper, we construct a class of extended implicit–explicit general linear (EIEGL) methods for solving semi-linear SNEs. Under some suitable conditions, an EIEGL method is proved to be stable and convergent of order p whenever the underlying implicit–explicit general linear (IEGL) method has order p and stage order p. With applications to several concrete problems of SNEs, the computational accuracy of EIEGL methods are further illustrated, where we also verify that the convergence order of EIEGL methods can exceed two, namely, third- and fourth-order EIEGL methods can be obtained. Moreover, based on a numerical comparison with the extended implicit general linear (EIGL) methods, the advantage of EIEGL methods in computational efficiency is shown.

Non-linear periodic long waves based on Boussinesq equation for shallow water waves: A coupled FEM modeling

A coupled mathematical model is developed based on the non-linear Boussinesq equation (BE) for shallow water waves to investigate the influence of the periodic non – linear long waves inside the irregular domain. A coupled numerical model based on the solution of the BE in a bounded region with the incorporation of the effects of viscous dissipation, dispersion, convective non-linearities, and variable bathymetry is obtained by using Galerkin Finite Element Method (GFEM), and an analytical approach based on Fourier-Bessel series solution for the scattered waves in unbounded region. The current coupled numerical model is validated by comparing the simulation results with existing studies. Moreover, the accuracy of the current scheme is also examined using convergence analysis for the rectangular domain. The present numerical model is applied on T and TT–shaped harbors to investigate the role of geometry to induce the amplification. In addition, the current approach implemented on the realistic Paradip port, Odisha, India, and Pohang New Harbor (PNH), South Korea at six record stations for practical implication. Therefore, the present numerical model can be utilized to construct/redesign an artificial port in the coastal region including variable bathymetry considering the effect of partial reflection, dissipation, and convective non – linearities.

Discrete hybrid Izhikevich neuron model: Nodal and network behaviours considering electromagnetic flux coupling

We analyse the dynamics of the improved discretised version of the well known Izhikevich neuron model under the action of external electromagnetic field. It is found that the improved three dimensional IZH map shows rich dynamics. With the variation of the electromagnetic field, period-doubling route to chaos in a repeating fashion is observed from the bifurcation diagram. Even the forward and backward continuation bifurcation diagram which do not completely overlap suggests that there is multistability in the system. The phenomenon of bistability (coexistence of periodic and chaotic attractors) is observed. The presence of periodic and chaotic attractor is aided by the maximal Lyapunov exponent diagram. The Lyapunov phase diagram of electromagnetic field and synapses current shows a large parameter region of chaotic and periodic behaviours with the presence of unbounded regions as well. The IZH map shows a plethora of spiking and bursting patterns such as mixed-mode patterns, tonic spiking, phasic spiking, steady spikes, regular spikes, spike bursting, periodic bursting, phasic bursting, chaotic firing etc with the variation of electromagnetic coupling strength and the synapses current. We also investigate the presence of chimera states in a ring-star, ring, star networks of IZH map neurons. Chimera states are found in the case of ring-star and ring network while synchronised clusters were found in the case of star network and are aided by the spatiotemporal plots, space-time plot, recurrence plots. The rich dynamics shown by the discretised IZH map makes it a promising research model to study about neurodynamics.

Open-pit slope design using a DtN-FEM: Parameter space exploration

Given the sustained mineral-deposits ore-grade decrease, it becomes necessary to reach greater depths when extracting ore by open-pit mining. Steeper slope angles are thus likely to be required, leading to geomechanical instabilities. In order to determine excavation stability, mathematical modelling and numerical simulation are often used to compute the rock-mass stress-state, to which some stability criterion needs to be added. A problem with this approach is that the volume surrounding the excavation has no clear borders and in practice it might be regarded as an unbounded region. Then, it is necessary to use advanced methods capable of dealing efficiently with this difficulty. In this work, a DtN-FEM procedure is applied to calculate displacements and stresses in open-pit slopes under geostatic stress conditions. This procedure was previously devised by the authors to numerically treat this kind of problems where the surrounding domain is semi-infinite. Its efficiency makes possible to simulate, in a short amount of time, multiple open-pit slope configurations. Therefore, the method potentiality for open-pit slope design is investigated. A regular open-pit slope geometry is assumed, parameterised by the overall-slope and bench-face angles. Multiple geometrically admissible slopes are explored and their stability is assessed by using the computed stress-field and the Mohr–Coulomb failure criterion. Regions of stability and instability are thus explored in the parametric space, opening the way for a new and flexible designing tool for open-pit slopes and related problems.

A Seamless, Extended DG Approach for Advection–Diffusion Problems on Unbounded Domains

We propose and analyze a seamless extended Discontinuous Galerkin (DG) discretization of advection–diffusion equations on semi-infinite domains. The semi-infinite half line is split into a finite subdomain where the model uses a standard polynomial basis, and a semi-unbounded subdomain where scaled Laguerre functions are employed as basis and test functions. Numerical fluxes enable the coupling at the interface between the two subdomains in the same way as standard single domain DG interelement fluxes. A novel linear analysis on the extended DG model yields unconditional stability with respect to the Péclet number. Errors due to the use of different sets of basis functions on different portions of the domain are negligible, as highlighted in numerical experiments with the linear advection–diffusion and viscous Burgers’ equations. With an added damping term on the semi-infinite subdomain, the extended framework is able to efficiently simulate absorbing boundary conditions without additional conditions at the interface. A few modes in the semi-infinite subdomain are found to suffice to deal with outgoing single wave and wave train signals more accurately than standard approaches at a given computational cost, thus providing an appealing model for fluid flow simulations in unbounded regions.

Moored ship motion under the resonance conditions with breakwaters: A coupled numerical approach

A hybrid element model has been formulated to understand the wave interaction with moored ship motion using mild slope equation (MSE) including the influence of breakwaters and wave absorbers, under resonance conditions for variable bathymetry. In a hybrid element formulation, a combined approach with the finite element method (FEM) in an interior domain (port region) including moored ship region and an analytical approximation in the unbounded region (open sea) is utilized to determine the wave response of moored ship motion. Especially, in this approach, a FEM and analytical approximation are firstly utilized to solve MSE with the consideration of permeable breakwaters for moored ship motion under the resonance conditions. The non-uniform discretization pattern enhanced the numerical accuracy of the coupled numerical approach and is verified by using convergence analysis. Further, the simulation results are compared with existing studies to validate the applicability of the present numerical model. Wave absorbers and permeable breakwaters are employed to determine the incident wave response over moored ship motion inside the realistic Paradip port, Odisha, India. In addition, the efficacy of permeable breakwaters and wave absorbers on moored ship motion is investigated. It is shown that breakwaters and wave absorbers at appropriate locations play a significant role to induce/reduce the resonance inside the port. Therefore, the current numerical model provides the countermeasures tactics such as installing breakwaters, wave absorbers to mitigate the resonance in arbitrarily shaped ports for safe moorings.

Nitrogen ion-implanted resistive regions for edge termination of vertical Ga2O3 rectifiers

N+ ion implantation to form resistive regions for edge termination at the anode contact periphery was used to maximize the breakdown voltage (VB) of vertical geometry β-Ga2O3 rectifiers. The design was examined via simulations using the Florida object oriented device and process Technology Computer-Aided Design (TCAD) simulator. The configuration of the implanted region was investigated with these simulations and then implemented experimentally. Significant increases of ∼200%–250% in VB were achieved for 50–1000 μm diameter rectifiers with an unbounded resistive region and an implantation depth of ∼0.5 μm. The on-state resistance and on/off ratios of the rectifiers were essentially unchanged by the addition of the implanted regions. An optimized implanted edge termination structure maximizes the breakdown voltage with no associated increase in device resistance in vertical geometry Ga2O3 rectifiers.

Analysis and application of an overlapped FEM-BEM for wave propagation in unbounded and heterogeneous media

An overlapped continuous model framework, for the Helmholtz wave propagation problem in unbounded regions comprising bounded heterogeneous media, was recently introduced and analyzed by the authors (2020) [10]. The continuous Helmholtz system incorporates a radiation condition (RC) and our equivalent hybrid framework facilitates application of widely used finite element methods (FEM) and boundary element methods (BEM), and the resulting discrete systems retain the RC exactly. The FEM and BEM discretizations, respectively, applied to the designed interior heterogeneous and exterior homogeneous media Helmholtz systems include the FEM and BEM solutions matching in artificial interface domains, and allow for computations of the exact ansatz based far-fields. In this article we present rigorous numerical analysis of a discrete two-dimensional FEM-BEM overlapped coupling implementation of the algorithm. We also demonstrate the efficiency of our discrete FEM-BEM framework and analysis using numerical experiments, including applications to non-convex heterogeneous multiple particle Janus configurations. Simulations of the far-field induced differential scattering cross sections (DSCS) of heterogeneous configurations and orientation-averaged (OA) counterparts are important for several applications, including inverse wave problems. Our robust FEM-BEM framework facilitates computations of such quantities of interest, without boundedness or homogeneity or shape restrictions on the wave propagation model.

The Local and Parallel Finite Element Scheme for Electric Structure Eigenvalue Problems

In this paper, an efficient multiscale finite element method via local defect-correction technique is developed. This method is used to solve the Schrödinger eigenvalue problem with three-dimensional domain. First, this paper considers a three-dimensional bounded spherical region, which is the truncation of a three-dimensional unbounded region. Using polar coordinate transformation, we successfully transform the three-dimensional problem into a series of one-dimensional eigenvalue problems. These one-dimensional eigenvalue problems also bring singularity. Second, using local refinement technique, we establish a new multiscale finite element discretization method. The scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. Finally, the error estimates of eigenvalues and eigenfunctions are also proved. Numerical examples show that our numerical method can significantly improve the accuracy of eigenvalues.

On some inequalities for derivatives of algebraic polynomials in unbounded regions with angles

In this work we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with interior zero and exterior non zero angles.

The implementation of the boundary element method to the Helmholtz equation of acoustics

Introduction: To solve the Helmholtz equation is important for the branches of engineering that require the simulation of wave phenomenon. Numerical methods allow effectiveness’ enhancing of the related computations. Methods: To find a numerical solution of the Helmholtz equation one may apply the boundary element method. Only the surface mesh constructed for the boundary of the three-dimensional domain of interest must be supplied to make the computations possible. This method’s trait makes it possible toconduct numerical experiments in the regions which are external in relation to some Euclidian three-dimensional subdomain bounded in the three-dimensional space. The later also provides the opportunity of not using additional geometric techniques to consider the infinitely distant boundary. However, it’s only possible to use the boundary element methods either for the homogeneous domains or for the domains composed out of adjacent homogeneous subdomains. Results: The implementation of the boundary elementmethod was committed in the program complex named Quasar. The discrepancy between the analytic solution approximation and the numerical results computed through the boundary element method for internal and external boundary value problems was analyzed. The results computed via the finite element method for the model boundary value problems are also provided for the purpose of the comparative analysis done between these two approaches. Practical relevance: The method gives an opportunityto solve the Helmholtz equation in an unbounded region which is a significant advantage over the numerical methods requiring the volume discretization of computational domains in general and over the finite element method in particular. Discussion: It is planned to make a coupling of the two methods for the purpose of providing the opportunity to conduct the computations in the complex regions with unbounded homogeneous subdomain and subdomains with substantial inhomogeneity inside.

Further developments on boundary-field equation methods for nonlinear transmission problems

This paper is concerned with new insights on the application of the boundary-field equation approach, which refers basically to the combined use of finite element and boundary integral equation methods, to numerically solve nonlinear exterior transmission problems in 2D and 3D. As a model, we consider a nonlinear second-order elliptic equation in divergence form holding in an annular domain coupled with the Laplace equation in the corresponding unbounded exterior region, together with transmission conditions on the interface and a suitable radiation condition at infinity. We first extend the classical Johnson & Nédélec coupling procedure, which makes use of only one boundary integral equation, to our nonlinear model without assuming any restrictive smoothness requirement on the boundary but only Lipschitz-continuity. Next, we extend the applicability of a recently introduced modification of the Costabel & Han coupling method, which employs the two boundary integral equations arising from the Green representation formula, to our nonlinear model as well. This new boundary-field equation method is based on the introduction of both Cauchy data on the boundary as independent unknowns. Primal and dual-mixed variational formulations are analyzed for each extension described above, and suitable hypotheses on the nonlinear coefficient of the elliptic equation allow to establish well-posedness of the corresponding continuous and discrete schemes by using monotone operator and nonlinear Babuška-Brezzi theories. Finally, a priori error estimates are established.

Eliminating synchronization of coupled neurons adaptively by using feedback coupling with heterogeneous delays.

In this paper, we present an adaptive scheme involving heterogeneous delay interactions to suppress synchronization in a large population of oscillators. We analytically investigate the incoherent state stability regions for several specific kinds of distributions for delays. Interestingly, we find that, among the distributions that we discuss, the exponential distribution may offer great convenience to the performance of our adaptive scheme because this distribution renders an unbounded stability region. Moreover, we demonstrate our scheme in the realization of synchronization elimination in some representative, realistic neuronal networks, which makes it possible to deepen the understanding and even refine the existing techniques of deep brain stimulation in the treatment of some synchronization-induced mental disorders.

Host relatedness and landscape connectivity shape pathogen spread in the puma, a large secretive carnivore

Urban expansion can fundamentally alter wildlife movement and gene flow, but how urbanization alters pathogen spread is poorly understood. Here, we combine high resolution host and viral genomic data with landscape variables to examine the context of viral spread in puma (Puma concolor) from two contrasting regions: one bounded by the wildland urban interface (WUI) and one unbounded with minimal anthropogenic development (UB). We found landscape variables and host gene flow explained significant amounts of variation of feline immunodeficiency virus (FIV) spread in the WUI, but not in the unbounded region. The most important predictors of viral spread also differed; host spatial proximity, host relatedness, and mountain ranges played a role in FIV spread in the WUI, whereas roads might have facilitated viral spread in the unbounded region. Our research demonstrates how anthropogenic landscapes can alter pathogen spread, providing a more nuanced understanding of host-pathogen relationships to inform disease ecology in free-ranging species.