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In this paper, we obtain the N th-order rational solutions for the defocusing non-local nonlinear Schrödinger equation by the Darboux transformation and some limit technique. Then, via an improved asymptotic analysis method relying on the balance between different algebraic terms, we derive the explicit expressions of all asymptotic solitons of the rational solutions with the order 1≤N≤4. It turns out that the asymptotic solitons are localized in the straight lines or algebraic curves, and the exact solutions approach the curved asymptotic solitons with a slower rate than the straight ones. Moreover, we find that all the rational solutions exhibit just five different types of soliton interactions, and the interacting solitons are divided into two halves with each having the same amplitudes. Particularly for the curved asymptotic solitons, there may exist a slight difference for their velocities between at t and −t with certain parametric conditions. In addition, we reveal that the soliton interactions in the rational solutions with N≥2 are stronger than those in the exponential and exponential-and-rational solutions. [ABSTRACT FROM AUTHOR]

Distortion risk measure, defined by an integral of a distorted tail probability, has been widely used in behavioral economics and risk management as an alternative to expected utility. The sensitivity of the distortion risk measure is a functional of certain distribution sensitivities. We propose a new sensitivity estimator for the distortion risk measure that uses generalized likelihood ratio estimators for distribution sensitivities as input and establish a central limit theorem for the new estimator. The proposed estimator can handle discontinuous sample paths and distortion functions. [ABSTRACT FROM AUTHOR]

We consider a family of linearly viscoelastic shells with thickness 2 ε , all having the same middle surface S = θ (ω ¯) ⊂ I R 3 , where ω ⊂ I R 2 is a bounded and connected open set with a Lipschitz-continuous boundary γ and θ ∈ C 3 (ω ¯ ; I R 3) . The shells are clamped on a portion of their lateral face, whose middle line is θ (γ 0) , where γ 0 is a non-empty portion of γ . The aim of this work is to show that the viscoelastic Koiter's model is the most accurate two-dimensional approach in order to solve the displacements problem of a viscoelastic shell. Furthermore, the solution of the Koiter's model, ξ K ε = (ξ K , i ε) , is in H 1 (0 , T ; V K (ω)) , with ξ K , i ε : [ 0 , T ] × ω ¯ → R the covariant components of the displacements field ξ K , i ε a i of the points of the middle surface S and where V K (ω) : = { η = (η i) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω) ; η i = ∂ ν η 3 = 0 in γ 0 } , with ∂ ν denoting the outer normal derivative along γ . Under the same assumptions as for the viscoelastic elliptic membranes problem, we show that the displacement field, ξ K , i ε a i , converges to ξ i a i (the solution of the two-dimensional problem for a viscoelastic elliptic membrane) in H 1 (0 , T ; H 1 (ω)) for the tangential components, and in H 1 (0 , T ; L 2 (ω)) for the normal component, as ε → 0 . Under the same assumptions as in the viscoelastic flexural shell problem, we show that the displacement field, ξ K , i ε a i , converges to ξ i a i (the solution of the two-dimensional problem for a viscoelastic flexural shell) in H 1 (0 , T ; H 1 (ω)) for the tangential components, and in H 1 (0 , T ; H 2 (ω)) for the normal component, as ε → 0 . Also, we obtain analogous results assuming the same assumptions as in the viscoelastic generalized membranes problem. Therefore, we justify the two-dimensional viscoelastic model of Koiter for all kind of viscoelastic shells. [ABSTRACT FROM AUTHOR]

Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin‐Bona‐Mahony (BBM) equation ut+uux=uxxt are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg‐de Vries equation ut+uux+uxxx=0. The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two‐phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two‐phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self‐similar solution of the BBM equation whose limit as t→∞ is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation iqt(x,t)+qxx(x,t)+2q2(x,t)q¯(−x,t)=0,x∈R,t>0, with a step‐like boundary values: q(x,t)→0 as x→−∞ and q(x,t)→A as x→∞ for all t≥0, where A>0 is a constant. In a recent paper, we presented the long‐time asymptotics of the solution q(x,t) of this problem along the rays x/t=C≠0, where C is a constant. In the present paper, we extend the asymptotics into a region that is asymptotically closer to the ray x=0 than any of these rays. We specify a one‐parameter family of wedges in the x,t‐plane, with curved boundaries, characterized by qualitatively different asymptotic behavior of q(x,t), and present the main asymptotic terms for each wedge. Particularly, for wedges within x<0, we show that the solution decays as tplnt with p<0 depending on the wedge. For wedges within x>0, we show that the asymptotics has an oscillatory nature, with the phase functions specific for each wedge and depending on a slow variable parameterizing the wedges. [ABSTRACT FROM AUTHOR]

We present a nonoverlapping domain decomposition method for steady flow in high contrast heterogeneous media modelled by an elliptic equation with coefficients that have very large amplitude variations on a small spatial scale. The linear system of equations resulting from matching the solution trace and the fluxes through the boundary of the subdomains is ill-conditioned, especially for fine meshes needed to capture the rapid variations of the solution. Our main contribution is to show with analysis and numerical simulations how to use an asymptotic approximation of the Dirichlet to Neumann map of the sub-domain problems to obtain a preconditioner for an efficient domain decomposition algorithm. [ABSTRACT FROM AUTHOR]

We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and the need for adapted methods in the presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provides valuable insights to tackle plasmonic problems and general high aspect ratio particles. For this problem, we find that the boundary integral operator is nearly singular due to the collapsing geometry from an ellipse to a line segment. We show that this nearly singular behavior leads to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this nearly singular behavior and this parity into account, computed solutions incur a large error. To address these challenges, we introduce a new method called quadrature by parity asymptotic expansions (QPAX) that effectively and efficiently addresses these issues. We first develop QPAX to solve the Dirichlet problem for Laplace's equation in a high aspect ratio ellipse. Then, we extend QPAX for scattering by a sound-hard, high aspect ratio ellipse. We demonstrate the effectiveness of QPAX through several numerical examples. [ABSTRACT FROM AUTHOR]

Since optical links are easily blocked by obstacles in the environment, they have been considered difficult to directly perform wireless communications. However, reconfigurable intelligent surfaces (RISs), as a new type of digital coding meta-materials, can significantly improve the coverage of optical communication by establishing new links. We propose a controllable multi-branch wireless optical communication system based on optical RISs. By setting up multiple optical RISs in the environment, multiple artificial channels are built to improve the system performance and to reduce the outage probability. In this paper, we investigate three factors affecting channel coefficients, including beam jitter, RIS jitter, and obstruction probability. We derive closed-form probability density function of channel coefficients, asymptotic average bit-error rate (BER), and the outage probability for systems with multiple branches. Our analysis reveals that the probability density function contains an impulse function, which causes irreducible error rate and outage probability floors. Based on our numerical results, the BER and outage probability floor of the multi-branch system are significantly reduced compared with the single direct path system. Therefore, the optical RIS assisted multi-branch wireless communication is a promising solution against obstacles of optical channel. [ABSTRACT FROM AUTHOR]

Our first objective is to identify two-dimensional equations that model the displacement of a linearly elastic flexural shell subjected to the action of an external heat source. To this end, we embed the shell into a family of linearly elastic flexural shells, all sharing the same middle surface θ (ω ‾) , where ω is a domain in R 2 and θ : ω ‾ → E 3 is a smooth enough immersion and whose thickness 2 ε > 0 is considered as a "small" parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as ε approaches zero, the corresponding "limit" two-dimensional variational problem. Our second objective is to identify and justify a set of two-dimensional equations that are meant to approximate the original three-dimensional model in the case where the shell under consideration is either an elliptic membrane shell or a flexural shell. [ABSTRACT FROM AUTHOR]

We use an individual-based model and its associated kinetic equation to study the generation of long jumps in the motion of E. coli. These models relate the run-and-tumble process to the intracellular reaction where the intrinsic noise plays a central role. Compared with previous work in Perthame et al. (Z Angew Math Phys 69(3):1–15, 2018), in which the parametric assumptions are mainly targeted for mathematical convenience but not well-suited for numerical simulations or comparison with experimental results, our current paper makes use of biologically meaningful pathways and tumbling kernels. The main contribution of this current work is bridging the gap between the theoretical results and experimentally available data. Some particular forms of how the tumbling frequency depends on the internal variable are proposed. Moreover, we propose two individual-based models, one for the tumbling frequency and the other for the receptor activity, and perform numerical simulations. Power-law decay of the run length distribution, which corresponds to Lévy-type motions, is observed in our numerical results. The particular decay rate agrees quantitatively with the analytical result. [ABSTRACT FROM AUTHOR]

Atherosclerotic plaques develop over a long time and can cause heart attacks and strokes. There are no simple mathematical models that capture the different timescales of rapid macrophage and lipid dynamics and slow plaque growth. We propose a simple ODE model for lipid dynamics that includes macrophage preference for ingesting apoptotic material and modified low-density lipoproteins (modLDL) over ingesting necrotic material. We use multiple timescale analysis to show that if the necrosis rate is small then the necrotic core in the model plaque may continue to develop slowly even when the lipid levels in plaque macrophages, apoptotic material and modLDL appear to have reached equilibrium. We use the model to explore the effect of macrophage emigration, apoptotic cell necrosis, total rate of macrophage phagocytosis and modLDL influx into the plaque on plaque lipid accumulation. [ABSTRACT FROM AUTHOR]

Let (X t) t ≥ 0 be the overdamped Langevin process on R d , i.e. the solution of the stochastic differential equation d X t = − ∇ f (X t) d t + h d B t. Let Ω ⊂ R d be a bounded domain. In this work, when X 0 = x ∈ Ω , we derive new sharp asymptotic equivalents (with optimal error terms) in the limit h → 0 of the mean exit time from Ω of the process (X t) t ≥ 0 (which is the solution of (− h 2 Δ + ∇ f · ∇) w = 1 in Ω and w = 0 on ∂ Ω ), when the function f : Ω ¯ → R has critical points on ∂ Ω. Such a setting is the one considered in many cases in molecular dynamics simulations. This problem has been extensively studied in the literature but such a setting has never been treated. The proof, mainly based on techniques from partial differential equations, uses recent spectral results from [Le Peutrec and Nectoux, Anal. PDE, 2020] and its starting point is a formula from the potential theory. We also provide new sharp leveling results on the mean exit time from Ω. [ABSTRACT FROM AUTHOR]

Parabolic cylinder functions are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain very sharp bounds for the ratio Φn(x)=U(n−1,x)/U(n,x) and the double ratio Φn(x)/Φn+1(x) in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for U(n,z)/U(n,y) are also made available. The bounds are very sharp as x→±∞ and n→+∞, and this simultaneous sharpness in three different directions explains their remarkable global accuracy. Upper and lower elementary bounds are obtained which are able to produce several digits of accuracy for moderately large |x| and/or n. [ABSTRACT FROM AUTHOR]

We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, exp(nsz), over the interval [−1,1], where s∈C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, s∈iR, due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as n→∞ have recently been studied for s∈iR, and our main goal is to extend these results to all s in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter s is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter s approaches a breaking curve, by considering double scaling limits as s approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points s=±2 or some other points on the breaking curve. [ABSTRACT FROM AUTHOR]

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of N -fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal P T -symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals. [ABSTRACT FROM AUTHOR]

In this paper, we consider a model of linear thermoelasticity for an elastic threedimensional body with high thermal conductivity covered by a thin layer on one of its faces. We show that Ventcel's boundary conditions may be obtained, as the thickness of the rigid body goes to zero. [ABSTRACT FROM AUTHOR]

In this paper, by considering the average mean squared error (AMSE) of channel estimation, we primarily obtain the closed-from expressions of the probability density function (PDF) and cumulative distribution function of AMSE for the least squares (LS)/minimum mean squared error (MMSE) estimation method as the line-of-sight (LOS) component is known, where the asymptotic analysis is executed in Rayleigh fading and strong LOS conditions. Secondly, the closed-form expressions for the expectation of AMSE (Expamse) and variance of AMSE (Varamse) are acquired, where Varamse is inversely proportional to the number of antennas (M). As M becomes infinite, the PDF of AMSE at Expamse has an order of root M. When the pilot power decreases with M in a power law, the LS case keeps deteriorating while the MMSE case converges to a constant which basically depends on the Rician K-factor. Next, the spectral efficiency is investigated by considering AMSE. When Expamse accelerates, the spectral efficiency of the LS method keeps dropping and that of the MMSE method firstly is degraded and then is improved to a constant except Rayleigh fading. Finally, all results are validated via simulations. [ABSTRACT FROM AUTHOR]

The one‐dimensional radial vibration model of piezoelectric disks has been widely used to determine the relevant material coefficients from admittance measurements. However, the one‐dimensional model assumes infinitely thin disks, and therefore cannot predict their axial displacements. We extend the one‐dimensional model by performing an asymptotic analysis of the axisymmetric radial vibration of thin disks. The asymptotic expansions include the asymptotic axial displacement and the second‐order corrections to the admittance and the radial displacement in the one‐dimensional model. We verify the asymptotic expansions and the one‐dimensional model with the Chebyshev tau method. In the one‐dimensional model, the frequencies of the maximum admittance fn in the first and second radial modes are accurate to 1% for Pz27 disks with thickness‐to‐diameter ratios of 0.15 and 0.065, respectively. For a general piezoelectric disk in the forced vibration, the error of fn in the one‐dimensional model can be estimated from the second‐order correction of the asymptotic resonance frequency in the free vibration. [ABSTRACT FROM AUTHOR]

We develop novel numerical methods and perturbation approaches to determine the mean first passage time (MFPT) for a Brownian particle to be captured by either small stationary or mobile traps inside a bounded 2-D confining domain. Of particular interest is to identify optimal arrangements of small absorbing traps that minimize the average MFPT. Although the MFPT, and the associated optimal trap arrangement problem, has been well-studied for disk-shaped domains, there are very few analytical or numerical results available for general star-shaped domains or for thin domains with large aspect ratio. Analytical progress is challenging owing to the need to determine the Neumann Green's function for the Laplacian, while the numerical challenge results from a lack of easy-to-use and fast numerical tools for first computing the MFPT and then optimizing over a class of trap configurations. In this direction, and for the stationary trap problem, we develop a simple embedded numerical method, based on the closest point method (CPM), to perform MFPT simulations on elliptical and star-shaped domains. For periodic mobile trap problems, we develop a robust CPM to compute the average MFPT. Optimal trap arrangements are identified numerically through either a refined discrete sampling approach or from a particle-swarm optimization procedure. To confirm some of the numerical findings, novel perturbation approaches are developed to approximate the average MFPT and identify optimal trap configurations for a class of near-disk confining domains or for an arbitrary thin domain of large aspect ratio. [ABSTRACT FROM AUTHOR]

We consider a characteristic initial value problem, with initial data given on a future null cone, for the Einstein (massless) scalar field system with a positive cosmological constant, in Bondi coordinates. We prove that, for small data, this system has a unique global classical solution which is causally geodesically complete to the future and decays polynomially in radius and exponentially in Bondi time, approaching the de Sitter solution. [ABSTRACT FROM AUTHOR]

We provide an asymptotic analysis of a fractional Fisher-KPP type equation in periodic non-connected media with Dirichlet conditions outside the domain. After showing the existence and uniqueness of a non-trivial bounded stationary state n + , we prove that it invades the unstable state zero exponentially fast in time. [ABSTRACT FROM AUTHOR]

This article studies the cardinality of perfect multi deletion binary codes. The lower bound for any perfect deletion code with fixed code length and number of deletions, and the asymptotic achievability of Levenshtein’s upper bound are shown. [ABSTRACT FROM AUTHOR]

In this article, we investigate community detection in networks in the presence of node covariates. In many instances, covariates and networks individually only give a partial view of the cluster structure. One needs to jointly infer the full cluster structure by considering both. In statistics, an emerging body of work has been focused on combining information from both the edges in the network and the node covariates to infer community memberships. However, so far the theoretical guarantees have been established in the dense regime, where the network can lead to perfect clustering under a broad parameter regime, and hence the role of covariates is often not clear. In this article, we examine sparse networks in conjunction with finite dimensional sub-Gaussian mixtures as covariates under moderate separation conditions. In this setting each individual source can only cluster a nonvanishing fraction of nodes correctly. We propose a simple optimization framework which improves clustering accuracy when the two sources carry partial information about the cluster memberships, and hence perform poorly on their own. Our optimization problem can be solved by scalable convex optimization algorithms. With a variety of simulated and real data examples, we show that the proposed method outperforms other existing methodology. for this article are available online. [ABSTRACT FROM AUTHOR]

We prove norm-resolvent and spectral convergence in L² of solutions to the Neumann Poisson problem --Δε= f on a domain Ωε perforated by Dirichlet holes and shrinking to a 1- dimensional interval. The limit u satisfies an equation of the type --u" + μu = f on the interval (0, 1), where μ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighborhood and the vertex neighborhood is chosen correctly, the constant μ will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation. [ABSTRACT FROM AUTHOR]

This paper studies the channel estimation performance in massive multiple-input multiple-output (MIMO) systems with Ricean fading through considering the relative channel estimation error (RCEE) metric. The closed-form expressions of the probability distribution function (PDF) and cumulative distribution function of RCEE are derived, according to which the PDFs of RCEE with Rayleigh fading and pure line-of-sight (LoS) propagation are obtained as special cases. Through analyzing RCEE, it notes that as the number of base-station antennas M becomes unbounded, RCEE approaches a given constant. Moreover, the asymptotic analysis proves that, in high M-regime, the increasing rate of the RCEE scales in the order of √M for Rayleigh fading and pure LoS propagation, while the scaling order with general Ricean fading is capped by √M. All the above mentioned results are verified through Monte-Carlo simulations. [ABSTRACT FROM AUTHOR]

A two-order asymptotic solution for sharp V-notch tip mechanics field in elasto-viscoplastic solids with power-law form under mode I loading is developed. The stress exponents and angular distributions for various notch angles and viscoplastic exponents for the second terms are reported. The stress exponent of the first order is singular and is non-singular except for low creep exponent with small notch opening angle. Compared with finite element results from several V-notched specimens, it is found that the two-order solution can have a better estimate for those cases with small notch opening angles, e.g., α = 30° under n = 5, than that of one-order solution regardless of viscoplastic extent, specimen type and loading level. One-order term solution is demonstrated to be sufficient to characterize the full field of sharp V-notch specimen for those conditions that elastic term is considered into higher order term, e.g., α = 60° under n = 5, in all specimens under various viscoplasticity extents. [ABSTRACT FROM AUTHOR]

This work is concerned with the problem of identifying the shape, size and location of a small embedded tumor from measured temperature on the skin surface. The temperature distribution in the biological tissue is governed by the Pennes model equation. The proposed approach is based on the Kohn–Vogelius formulation and the topological sensitivity analysis method. The ill-posed geometric inverse problem is reformulated as a topology optimization. The temperature field perturbation, caused by the presence of a small anomaly, is analyzed and estimated. A topological asymptotic formula, describing the variation of the considered Kohn–Vogelius type functional with respect to the presence of a small anomaly is derived. [ABSTRACT FROM AUTHOR]

New expressions are derived for the Green function (GF) for diffraction and radiation of waves by a two-dimensional (2D) body in finite water depth. The finite depth GF is expressed as a sum of singularities, the infinite depth GF and smoothly-varying integrals that are convergent for all parameter values. The infinite depth component is given explicitly, making it very fast to compute. Explicit expressions are derived for the limiting cases of zero and infinite frequency, for both finite and infinite water depth. The low frequency limit of the 2D GF is inconsistent with the zero-frequency 2D GF, with the real part tending to infinity when the water depth is infinite and the imaginary part tending to infinity in finite water depth. The inconsistencies with the zero frequency GF differ between the 2D and 3D cases. These inconsistencies lead to differences between the low-frequency behaviour of the added mass and damping of an oscillating body and the values at zero frequency. It is shown that these differences can be inferred directly from the behaviour of the GF at low frequencies. • Finite depth Green function (GF) expressed as infinite depth GF plus perturbations. • Explicit expression derived for infinite depth Green function. • Explicit expressions derived for zero and infinite frequency GF in finite depth. • Inconsistencies in low frequency added mass and damping related to behaviour of GF. [ABSTRACT FROM AUTHOR]

In this paper, we consider the monodromy and, in particular, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. The isomonodromy sets of accessory parameters for the Heun class equations are described by Taylor or Laurent coefficients of the corresponding Painlevé functions, or the associated tau functions, at the positions of the critical values. As an application of these results, we derive some asymptotic approximations for the isomonodromy sets of accessory parameters in the Heun class equations, including the confluent Heun equation, the doubly‐confluent Heun equation, and the reduced biconfluent Heun equation. [ABSTRACT FROM AUTHOR]

A recent three-dimensional (3D) model that revisited earlier theoretical work for longitudinal electromagnetic stirring in the continuous casting of steel blooms is analyzed further to explore how the bloom width interacts with the pole pitch of the stirrer to affect the magnetic flux density. Whereas the first work indicated the presence of a boundary layer in the steel near the interface with the stirrer, with all three components of the magnetic flux density vector being coupled to each other, in the analysis presented here we find that the component along the direction of the travelling wave decouples from those in the other two directions and can even be determined analytically in the form of a series solution. Moreover, it is found that the remaining two components can be found via a two-dimensional computation, but that it is not possible in general to determine these components without taking into account the surrounding air. The validity of the asymptotically reduced model solution is confirmed by comparing it with the results of 3D numerical computations. Moreover, the asymptotic approach provides a way to compute the time-averaged Lorentz force components that requires two orders of magnitude less computational time than the fully 3D approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior as 𝜀 → 0 + of solutions u 𝜀 to the nonlocal stationary Fisher-KPP type equation 1 𝜀 m ∫ ℝ N J 𝜀 (x − y) (u 𝜀 (y) − u 𝜀 (x)) d y + u 𝜀 (x) (a (x) − u 𝜀 (x)) = 0  in  ℝ N , where 𝜀 > 0 and 0 ≤ m < 2. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution u 𝜀 and that u 𝜀 → a + as 𝜀 → 0 + where a + = max { 0 , a }. This generalizes the previously known results and answers an open question raised by Berestycki et al. Our method of proof is also of independent interest as it shows how to reduce this nonlocal problem to a local one. The sharpness of our assumptions is also briefly discussed. [ABSTRACT FROM AUTHOR]

The Cahn–Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. Various dynamic boundary conditions have already been introduced in the literature to model interactions of the materials with the boundary more precisely. To take long-range interactions into account, we propose a new model consisting of a nonlocal Cahn–Hilliard equation with a nonlocal dynamic boundary condition comprising an additional boundary penalization term. We rigorously derive our model as the gradient flow of a nonlocal free energy with respect to a suitable inner product of order H − 1 containing both bulk and surface contributions. In the main model, the chemical potentials are coupled by a Robin type boundary condition depending on a specific relaxation parameter. We prove weak and strong well-posedness of this system, and we investigate the singular limits attained when this relaxation parameter tends to zero or infinity. [ABSTRACT FROM AUTHOR]

We consider singularly perturbed convection–diffusion equations on one‐dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling conditions at inner vertices are derived that guarantee conservation of mass and dissipation of a mathematical energy which allows us to prove stability and well‐posedness. For single intervals and appropriately specified initial conditions, it is well‐known that the solutions of the convection–diffusion problem converge to that of the transport problem with order O(ϵ) in the L∞(L2)‐norm with diffusion ϵ → 0. In this paper, we prove a corresponding result for problems on one‐dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers. [ABSTRACT FROM AUTHOR]

This work addresses the mathematical analysis – by means of asymptotic analysis – of a penalisation strategy for the full discretisation of elastic wave propagation problems in quasi-incompressible media that has been recently developed by the authors. We provide a convergence analysis of the solution to the continuous version of the penalised problem towards its formal limit when the penalisation parameter tends to infinity. Moreover, as a fundamental intermediate step we provide an asymptotic analysis of the convergence of solutions to quasi-incompressible problems towards solutions to purely incompressible problems when the incompressibility parameter tends to infinity. Finally, we further detail the regularity assumptions required to guarantee that the mentioned convergence holds. [ABSTRACT FROM AUTHOR]

Accurate asymptotic solutions are presented for axisymmetric deformation of thin layers constrained by either two rigid plates or two rigid spheres. Those solutions are developed using Saint-Venant's principle and the layer thinness as the only assumptions. The solutions are valid in the entire range of Poisson's ratios, and allow one to distinguish among compressible, nearly incompressible, and incompressible layers. That classification involves both material and geometric parameters. [ABSTRACT FROM AUTHOR]

We study a resource allocation setting where m discrete items are to be divided among n agents with additive utilities, and the agents' utilities for individual items are drawn at random from a probability distribution. Since common fairness notions like envy-freeness and proportionality cannot always be satisfied in this setting, an important question is when allocations satisfying these notions exist. In this paper, we close several gaps in the line of work on asymptotic fair division. First, we prove that the classical round-robin algorithm is likely to produce an envy-free allocation provided that m = (n log n= log log n), matching the lower bound from prior work. We then show that a proportional allocation exists with high probability as long as m = n, while an allocation satisfying envy-freeness up to any item (EFX) is likely to be present for any relation between m and n. Finally, we consider a related setting where each agent is assigned exactly one item and the remaining items are left unassigned, and show that the transition from nonexistence to existence with respect to envy-free assignments occurs at m = en. [ABSTRACT FROM AUTHOR]

We study the time-domain acoustic scattering problem by a cluster of small holes (i.e., sound-soft obstacles). Based on the retarded boundary integral equation method, we derive the asymptotic expansion of the scattered field as the size of the holes goes to zero. Under certain geometrical constraints on the size and the minimum distance between the holes, we show that the scattered field is approximated by a linear combination of point-sources where the weights are given by the capacitance of each hole and the causal signals (of these point-sources) can be computed by solving a retarded in time linear algebraic system. A rigorous justification of the asymptotic expansion and the unique solvability of the linear algebraic system is shown under natural conditions on the cluster of holes. As an application of the asymptotic expansion, we derive, in the limit case when the holes are densely distributed and occupy a bounded domain, the equivalent effective acoustic medium (an equivalent mass density characterized by the capacitance of the holes) that generates, approximately, the same scattered field as the cluster of holes. Conversely, given a locally variable, smooth, and positive mass density, satisfying a certain subharmonicity condition, we can design a perforated material with holes, having appropriate capacitances, that generates approximately the same acoustic field as the acoustic medium modeled by the given mass density (and constant speed of propagation). Finally, we numerically verify the asymptotic expansions by comparing the asymptotic approximations with the numerical solutions of the scattered fields via the finite element method. [ABSTRACT FROM AUTHOR]

We consider the scattering of time-harmonic electromagnetic waves by a penetrable thin tubular scattering object in three-dimensional free space. We establish an asymptotic representation formula for the scattered wave away from the thin tubular scatterer as the radius of its cross-section tends to zero. The shape, the relative electric permittivity, and the relative magnetic permeability of the scattering object enter this asymptotic representation formula by means of the center curve of the thin tubular scatterer and two electric and magnetic polarization tensors. We give an explicit characterization of these two three-dimensional polarization tensors in terms of the center curve and of the two two-dimensional polarization tensors for the cross-section of the scattering object. As an application we demonstrate how this formula may be used to approximate the residual and the shape derivative in an iterative reconstruction algorithm for an inverse scattering problem with thin tubular scattering objects. We present numerical results to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

The Lagrange-multiplier technique is a multiscale numerical scheme designed to solve evolution problems containing some stiff transport terms. Such multiscale problems arise very often in kinetic models for the description of thermonuclear plasma dynamics. The particularity of this scheme is that it permits one to capture (without having to refine the discretization) even the asymptotic limit when the small parameter ε describing the stiffness of the problem goes to zero. This property is called asymptotic-preserving and has been validated numerically in previous works of the author. In the present work, the missing mathematical study of this Lagrange-multiplier approach is performed, with special emphasis on the asymptotic limit when ε → 0. [ABSTRACT FROM AUTHOR]

We make precise some results on the cloaking of displacement fields in linear elasticity. In the spirit of transformation media theory, the transformed governing equations in Cosserat and Willis frameworks are shown to be equivalent to certain high-contrast small defect problems for the usual Navier equations. We discuss near-cloaking for elasticity systems via a regularized transform and perform numerical experiments to illustrate our near-cloaking results. We also study the sharpness of the estimates from [H. Ammari, H. Kang, K. Kim, and H. Lee, J. Differential Equations, 254 (2013), pp. 4446-4464], wherein the convergence of the solutions to the transmission problems is investigated, when the Lamé parameters in the inclusion tend to extreme values. Both soft and hard inclusion limits are studied and we also touch upon the finite frequency case. Finally, we propose an approximate isotropic cloak algorithm for a symmetrized Cosserat cloak. [ABSTRACT FROM AUTHOR]

The outage performance of multiple-input multiple-output (MIMO) technique has received intense attention in order to ensure the ultra-reliability requirement for broadcasting systems. In this article, the outage probability at high signal-to-noise ratio (SNR) is asymptotically studied for MIMO channels to thoroughly investigate the transmission reliability. To fully capture the spatial correlation effects, the MIMO fading channel matrix is modelled according to three types of Kronecker correlation structure, i.e., independent, semi-correlated and full-correlated Rayleigh MIMO channels. The outage probabilities under all three Kronecker models are expressed as weighted sum representations of generalized Fox’s H functions. The tractable results empower the asymptotic outage analyses, which are conducted not only to reveal helpful insights into understanding the behavior of fading effects, but also to offer useful design guideline for MIMO configurations. Particularly, the asymptotic outage probability is proved to be a monotonically increasing and convex function of the transmission rate. In the absence of the channel state information (CSI), the transmitter tends to equally allocate the total transmit power among its antennas to enhance the reception reliability especially in high SNR regime. In the end, the analytical results are validated through extensive numerical experiments. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for the existence of weak solutions to quasilinear elliptic Dirichlet problem driven by the A-Laplace operator in a bounded domain Ω. The techniques, based on a variant of the symmetric mountain pass theorem, exploit variational methods. We also provide information about the asymptotic behavior of the solutions as a suitable parameter goes to 0 + . In this case, we point out the existence of a blow-up phenomenon. The analysis developed in this paper extends and complements various qualitative and asymptotic properties for some cases described by homogeneous differential operators. [ABSTRACT FROM AUTHOR]

We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability δ of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any δ > 0 , with enhanced diffusivity (≫ O (δ 2)) in the small δ regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form δ ξ n with ξ n 's being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero δ limit), with diffusivity between O ( 1 | log δ | ) and O ( 1 log | log δ | ). Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as O (log − 2 δ). [ABSTRACT FROM AUTHOR]

The Petrowsky type equation y t t ε + ε y x x x x ε − y x x ε = 0 , ε > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order ε occurring at the extremities, these boundary controls get singular as ε goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y ε and derive a rigorous second order asymptotic expansion of the control of minimal weighted L 2 -norm, with respect to the parameter ε. The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J.-L. Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any ε small enough. Numerical experiments support our study. [ABSTRACT FROM AUTHOR]

The standard engineer's model for heat transfer between the fluid flowing through the pipe and the exterior medium neglects the effects of the pipe's wall. The goal of this paper is to prove that they are not always negligible. Comparing the ratio between diffusivities of the fluid and the wall with the wall's thickness, using rigorous asymptotic analysis, we find five different models for effective description of the heat exchange process. [ABSTRACT FROM AUTHOR]

Efflux pumps are a mechanism of intrinsic and evolved resistance in bacteria. If an efflux pump can expel an antibiotic so that its concentration within the cell is below a killing threshold the bacteria are resistant to the antibiotic. Efflux pumps may be specific or they may pump various different substances. This is why many efflux pumps confer multi drug resistance (MDR). In particular over expression of the AcrAB−TolC efflux pump system confers MDR in both Salmonella and Escherichia coli. We consider the complex gene regulation network that controls expression of genes central to controlling the efflux associated genes acrAB and acrEF in Salmonella. We present the first mathematical model of this gene regulatory network in the form of a system of ordinary differential equations. Using a time dependent asymptotic analysis, we examine in detail the behaviour of the efflux system on various different timescales. Asymptotic approximations of the steady states provide an analytical comparison of targets for efflux inhibition. [ABSTRACT FROM AUTHOR]

We present a numerical algorithm for an accurate and efficient computation of the convolution of the frequency domain layered media Green's function with a given density function. Instead of compressing the convolution matrix directly as in the classical fast multipole, fast direct solver, and H -matrix algorithms, this new algorithm considers a translated form of the original matrix so existing blocks from the highly optimized free-space fast multipole method can be easily adapted to the layered media Green's function. An asymptotic analysis is performed on the Sommerfeld integrals to provide an estimate of the decay rate in the new "multipole" and "local" expansions. To avoid the highly oscillatory integrand in the original integral representations when the source and target are close to each other, mathematically equivalent alternative direction integral representations are introduced. The convergence of the new expansions and quadrature rules for the original and alternative direction representations are numerically validated. [ABSTRACT FROM AUTHOR]

We consider the propagation of time-harmonic acoustic waves in a device made of three unbounded channels connected by thin slits. The wavenumber is chosen such that only one mode can propagate. The main goal of this work is to present a device which can serve as an energy distributor. More precisely, the geometry is first designed so that for an incident wave coming from one channel, the energy is almost completely transmitted in the two other channels. Additionally, adjusting slightly two geometrical parameters, we can control the ratio of energy transmitted in the two channels. The approach is based on asymptotic analysis for thin slits around resonance lengths. We also provide numerical results to illustrate the theory. [ABSTRACT FROM AUTHOR]

Simulations of classical pattern-forming reaction–diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator–inhibitor systems such as the two-component Gierer–Meinhardt (GM) model, one can consider the singular limit Da ≪ Dh, where Da and Dh are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction–transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate α between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in Caenorhabditis elegans. We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit α → ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for α → ∞ with a graphical construction for finite α. [ABSTRACT FROM AUTHOR]