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We consider the diffusion asymptotics of a coupled model arising in radiative transfer in a unit disk in R 2 with one-speed velocity. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled a nonhomogeneous heat equation satisfied by the material temperature. For the well-prepared data, we prove the existence and the nonequilibrium diffusion limit of solutions to the initial boundary problem for the coupled model. [ABSTRACT FROM AUTHOR]

In this paper, we consider spin-diffusion Landau–Lifshitz–Gilbert equations (SDLLG), which consist of the time-dependent Landau–Lifshitz–Gilbert (LLG) equation coupled with a time-dependent diffusion equation for the electron spin accumulation. The model takes into account the diffusion process of the spin accumulation in the magnetization dynamics of ferromagnetic multilayers. We prove that in the limit of long times, the system reduces to simpler equations in which the LLG equation is coupled to a nonlinear and nonlocal steady-state equation, referred to as SLLG. As a by-product, the existence of global weak solutions to the SLLG equation is obtained. Moreover, we prove weak-strong uniqueness of solutions of SLLG, i.e., all weak solutions coincide with the (unique) strong solution as long as the latter exists in time. The results provide a solid mathematical ground to the qualitative behavior originally predicted by Zhang , Levy , and Fert in [44] in ferromagnetic multilayers. [ABSTRACT FROM AUTHOR]

Numerous traits under migration–selection balance are shown to exhibit complex patterns of genetic architecture with large variance in effect sizes. However, the conditions under which such genetic architectures are stable have yet to be investigated, because studying the influence of a large number of small allelic effects on the maintenance of spatial polymorphism is mathematically challenging, due to the high complexity of the systems that arise. In particular, in the most simple case of a haploid population in a two-patch environment, while it is known from population genetics that polymorphism at a single major-effect locus is stable in the symmetric case, there exist no analytical predictions on how this polymorphism holds when a polygenic background also contributes to the trait. Here we propose to answer this question by introducing a new eco-evo methodology that allows us to take into account the combined contributions of a major-effect locus and of a quantitative background resulting from small-effect loci, where inheritance is encoded according to an extension to the infinitesimal model. In a regime of small variance contributed by the quantitative loci, we justify that traits are concentrated around the major alleles, according to a normal distribution, using new convex analysis arguments. This allows a reduction in the complexity of the system using a separation of time scales approach. We predict an undocumented phenomenon of loss of polymorphism at the major-effect locus despite strong selection for local adaptation, because the quantitative background slowly disrupts the rapidly established polymorphism at the major-effect locus, which is confirmed by individual-based simulations. Our study highlights how segregation of a quantitative background can greatly impact the dynamics of major-effect loci by provoking migrational meltdowns. We also provide a comprehensive toolbox designed to describe how to apply our method to more complex population genetic models. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the solutions to an initial boundary value problem of the Hunter-Saxton equation in the half space when the viscosity tends to zero. By means of the asymptotic analysis with multiple scales, we first formally derive the equations for boundary layer profiles. Next, we study the well-posedness of the equations for the boundary layer profiles by using the compactness argument. Moreover, we construct an accurate approximate solution and use the energy method to obtain the convergence results of the vanishing viscosity limit. [ABSTRACT FROM AUTHOR]

In this paper we identify a set of two-dimensional variational inequalities that model the displacement of a linearly elastic elliptic membrane shell subjected to a confinement condition, expressing that all the points of the admissible deformed configurations remain in a given half-space. [ABSTRACT FROM AUTHOR]

Plasmon resonance is the resonant oscillation of conduction electrons at the interface between negative and positive permittivity material stimulated by incident light, which forms the fundamental basis of many cutting-edge industrial applications. We are concerned with the quantitative theoretical understanding of this peculiar resonance phenomenon. It is known that the occurrence of plasmon resonance as well as its quantitative behaviours critically depend on the geometry of the material structure, the corresponding material parameters and the operating wave frequency, which are delicately coupled together. In this paper, we study the plasmon resonance for a 2D nanorod structure, which presents an anisotropic geometry and arises in the transverse electromagnetic scattering. We present delicate spectral and asymptotic analysis to establish the accurate resonant conditions as well as sharply characterize the quantitative behaviours of the resonant field. [ABSTRACT FROM AUTHOR]

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials { − Δ u n = W n (x) u n p n , u n > 0 , in Ω , u n = 0 , on ∂ Ω , ∫ Ω p n W n (x) u n p n d x ≤ C , where Ω is a smooth bounded domain in R 2 , W n (x) ≥ 0 are bounded functions with zeros in Ω, and p n → ∞ as n → ∞. A typical example is W n (x) = | x | 2 α with 0 ∈ Ω , i.e. the equation turns to be the well-known Hénon equation. The asymptotic behavior for α = 0 has been well studied in the literature. While for α > 0 , the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case α > 0 and prove a quantization property (suppose 0 is a concentration point) p n | x | 2 α u n (x) p n − 1 + t → 8 π e t 2 ∑ i = 1 k δ a i + 8 π (1 + α) e t 2 c t δ 0 , t = 0 , 1 , 2 , for some k ≥ 0 , a i ∈ Ω ∖ { 0 } and some c ≥ 1. Moreover, for α ∉ N , we show that the blow up must be simple, i.e. c = 1. As applications, we also obtain the complete asymptotic behavior of ground state solutions for the Hénon equation. [ABSTRACT FROM AUTHOR]

We consider a problem of approximation of d -variate functions defined on R d which belong to the Hilbert space with tensor product-type reproducing Gaussian kernel with constant shape parameter. Within worst case setting, we investigate the growth of the information complexity as d → ∞. The asymptotics are obtained for the case of fixed error threshold and for the case when it goes to zero as d → ∞. [ABSTRACT FROM AUTHOR]

In high performance computing environments, we observe an ongoing increase in the available number of cores. For example, the current TOP500 list reveals that nine clusters have more than 1 million cores. This development calls for re-emphasizing performance (scalability) analysis and speedup laws as suggested in the literature (e.g., Amdahl's law and Gustafson's law), with a focus on asymptotic performance. Understanding speedup and efficiency issues of algorithmic parallelism is useful for several purposes, including the optimization of system operations, temporal predictions on the execution of a program, the analysis of asymptotic properties, and the determination of speedup bounds. However, the literature is fragmented and shows a large diversity and heterogeneity of speedup models and laws. These phenomena make it challenging to obtain an overview of the models and their relationships, to identify the determinants of performance in a given algorithmic and computational context, and, finally, to determine the applicability of performance models and laws to a particular parallel computing setting. In this work, I provide a generic speedup (and thus also efficiency) model for homogeneous computing environments. My approach generalizes many prominent models suggested in the literature and allows showing that they can be considered special cases of a unifying approach. The genericity of the unifying speedup model is achieved through parameterization. Considering combinations of parameter ranges, I identify six different asymptotic speedup cases and eight different asymptotic efficiency cases. Jointly applying these speedup and efficiency cases, I derive eleven scalability cases, from which I build a scalability typology. Researchers can draw upon my suggested typology to classify their speedup model and to determine the asymptotic behavior when the number of parallel processing units increases. Also, the description of two computational experiments demonstrates the practical application of the model and the typology. In addition, my results may be used and extended in future research to address various extensions of my setting. • We develop a generic speedup and efficiency model for computational parallelization. • The unifying model generalizes many prominent models suggested in the literature. • Asymptotic analysis extends existing speedup laws. • Asymptotic analysis allows explaining sublinear, linear and superlinear speedup. • Based upon asymptotic speedup and efficiency analyses, we build a scalability typology. [ABSTRACT FROM AUTHOR]

We discuss conditions for Lipschitz and C 1 regularity of solutions for a uniformly elliptic equation in divergence form. We focus on coefficients having the form that was introduced by Gilbarg & Serrin. In particular, we find cases where Lipschitz or C 1 regularity holds but the coefficients are not Dini continuous, or do not even have Dini mean oscillation. The form of the coefficients also enables us to obtain specific conditions and examples for which there exists a weak solution that is not Lipschitz continuous. [ABSTRACT FROM AUTHOR]

We consider the equation − Δ u = | x | α | u | p − 1 u for any α ≥ 0 , either in R 2 or in the unit ball B of R 2 centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior as p → + ∞ of all the radial solutions to these problems and we show that there is no uniform a priori bound for nodal solutions under Neumann or Dirichlet boundary conditions. This contrasts with the existence of uniform bounds for positive solutions, as shown in [32] for α = 0 and Dirichlet boundary conditions. [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]

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 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 a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Ω ε in the plane R 2. Here Ω ε is Ω ∪ ω ε ∪ Γ , where Ω is a fixed bounded domain with boundary Γ, ω ε is a curvilinear band of width O (ε) , and Γ = Ω ‾ ∩ ω ‾ ε. The density and stiffness constants are of order ε − m − t and ε − t respectively in this band, while they are of order 1 in Ω; t ≥ 1 , m > 2 , and ε is a small positive parameter. We address the asymptotic behavior, as ε → 0 , for the eigenvalues and the corresponding eigenfunctions. In particular, we show certain localization effects for eigenfunctions associated with low frequencies. This is deeply involved with the extrema of the curvature of Γ. [ABSTRACT FROM AUTHOR]

The effects of spatial heterogeneity on population dynamics have been studied extensively. However, the effects of temporal periodicity on the dynamics of general periodic-parabolic reaction-diffusion systems remain largely unexplored. As a first attempt to understand such effects, we analyze the asymptotic behavior of the principal eigenvalue for linear cooperative periodic-parabolic systems with small diffusion rates. As an application, we show that if a cooperative system of periodic ordinary differential equations has a unique positive periodic solution which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann or regular oblique derivative boundary condition also has a unique positive periodic solution which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. The role of temporal periodicity, spatial heterogeneity and their combined effects with diffusion will be studied in subsequent papers for further understanding and illustration. [ABSTRACT FROM AUTHOR]

The paper treats a nonlinear eigenvalue problem that describes propagation of a transverse-magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides. The dielectric's permittivity is characterised by an arbitrary monotonically increasing self-action nonlinearity. The full set of guided modes is described by eigenvalues of the corresponding (nonlinear) Maxwell operator with appropriate boundary conditions. We give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions. Several results about existence of the eigenvalues are proved, their distribution and asymptotic are found; zeros of the eigenfunctions and their location are also determined; criterion of periodicity for the eigenfunctions is found, comparison theorem is derived, etc. It is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions. This results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes. [ABSTRACT FROM AUTHOR]

• A multi-DOF unstable mechanical system coupled to a set of NESs is studied. • We analyzed the model using a multiple-scale approach. • The analysis allows to predict the steady-state response regimes. • The method is finally validated numerically. A general method to predict the steady-state regimes of a multi-degree-of-freedom unstable vibrating system (the primary system) coupled to several nonlinear energy sinks (NESs) is proposed. The method has three main steps. The first step consists in the diagonalization of the primary underline linear system using the so-called biorthogonal transformation. Within the assumption of a primary system with only one unstable mode the dynamics of the diagonalized system is reduced ignoring the stable modes and keeping only the unstable mode. The complexification method is applied in the second step with the aim of obtaining the slow-flow of the reduced system. Then, the third step is an asymptotic analysis of the slow-flow based geometric singular perturbation theory. The analysis shows that the critical manifold of the system can be reduced to a one dimensional parametric curve evolving in a multidimensional space. The shape and the stability properties of the critical manifold and the stability properties of the fixed points of the slow-flow provide an analytical tool to predict the nature of the possible steady-state regimes of the system. Finally, two examples are considered to evaluate the effectiveness and advancement of the proposed method. The method is first applied to the prediction of the mitigation limit of a breaking system subject to friction-induced vibrations coupled to two NESs, and next an airfoil model undergoing an aeroelastic instability coupled to a NESs setup (from one to four) is discussed. Theoretical results are compared, for validation purposes, to direct numerical integration of the system. The comparisons show good agreement. [ABSTRACT FROM AUTHOR]

Abstract The Fourier transform, F , on R N (N ≥ 3) transforms the Cauchy problem for the strongly damped wave equation u t t − Δ u t − Δ u = 0 to an ordinary differential equation in time. We let u (t , x) be the solution of the problem given by the Fourier transform, and ν (t , ξ) be the asymptotic profile of F (u) (t , ξ) = u ˆ (t , ξ) found by Ikehata in the paper Asymptotic profiles for wave equations with strong damping (2014). In this paper we study the asymptotic expansions of the squared L 2 -norms of u (t , x) , u ˆ (t , ξ) − ν (t , ξ) , and ν (t , ξ) as t → ∞. With suitable initial data u (0 , x) and u t (0 , x) , we establish the rate of decay of the squared L 2 -norms of u (t , x) and ν (t , ξ) as t → ∞. By noting the cancellation of leading terms of their respective expansions, we conclude that the rate of convergence between u ˆ (t , ξ) and ν (t , ξ) in the L 2 -norm occurs quickly relative to their individual behaviors. This observation is similar to the diffusion phenomenon, which has been well studied. [ABSTRACT FROM AUTHOR]

Abstract We study approximation properties of centred additive random fields Y d , d ∈ N. The average case approximation complexity n Y d (ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Y d , with relative 2-average error not exceeding a given threshold ε ∈ (0 , 1). We investigate the growth of n Y d (ε) for arbitrary fixed ε ∈ (0 , 1) and d → ∞. Under natural assumptions we obtain general results concerning asymptotics of n Y d (ε). We apply our results to additive random fields with marginal random processes corresponding to the Korobov kernels. [ABSTRACT FROM AUTHOR]

Abstract In-host mutation of a cross-species infectious disease to a form that is transmissible between humans has resulted with devastating global pandemics in the past. We use simple mathematical models to describe this process with the aim to better understand the emergence of an epidemic resulting from such a mutation and the extent of measures that are needed to control it. The feared outbreak of a human–human transmissible form of avian influenza leading to a global epidemic is the paradigm for this study. We extend the SIR approach to derive a deterministic and a stochastic formulation to describe the evolution of two classes of susceptible and infected states and a removed state, leading to a system of ordinary differential equations and a stochastic equivalent based on a Markov process. For the deterministic model, the contrasting timescale of the mutation process and disease infectiousness is exploited in two limits using asymptotic analysis in order to determine, in terms of the model parameters, necessary conditions for an epidemic to take place and timescales for the onset of the epidemic, the size and duration of the epidemic and the maximum level of the infected individuals at one time. Furthermore, the basic reproduction number R 0 is determined from asymptotic analysis of a distinguished limit. Comparisons between the deterministic and stochastic model demonstrate that stochasticity has little effect on most aspects of an epidemic, but does have significant impact on its onset particularly for smaller populations and lower mutation rates for representatively large populations. The deterministic model is extended to investigate a range of quarantine and vaccination programmes, whereby in the two asymptotic limits analysed, quantitative estimates on the outcomes and effectiveness of these control measures are established. [ABSTRACT FROM AUTHOR]

Abstract The paper presents experimental and theoretical studies of wave phenomena accompanying transient edge waves (EW) excitation by a piezoelectric transducer attached at the edge of an elastic plate. Theoretical investigations are conducted on the basis of the three-dimensional elastodynamic theory. The boundary value problem describing non-stationary wave motion is solved with the use of integral transforms and the modal expansion technique. The asymptotic analysis with the wavenumber considered as a small or large parameter is applied to the three-dimensional problem. In the first case (long-wave vibrations) the approximate formulae for the fundamental EW's pole and corresponding residue are derived. These relations take into account the influence of the transverse shear load arising because of coupling between long-wave integral and Saint-Venant's boundary layer. In the second case (short-wave vibrations) an infinite series of poles corresponding to higher order EW is revealed. Results of numerical investigations of EW dispersion properties and waveforms are presented and used for an analysis of experimental data acquired with the use of Laser Doppler vibrometry. It is shown that the contribution of EW is predominant in the wave-field excited by the load under consideration. A good agreement between theoretical predictions and measurements is demonstrated. For the calculation of transient wave-field three different models of actuator-plate interaction are developed: in the low-frequency range the interaction is considered as a static one, while the lower and higher eigenfrequencies of the actuator are taken into account in the medium- and high-frequency ranges respectively. For the low-frequency range, an explicit analytical formula is derived for calculation of EW contribution into the transient wave-field. In the high-frequency range, the excitation of the first higher order EW is predicted by the numerical solution and observed experimentally. Graphical abstract Image 1 Highlights • Edge waves excited by a transducer at the edge of a plate are studied. • Contribution of edge waves is predominant in the transient wave-field at the edge. • An explicit analytical formula for calculation of edge wave contribution is derived. • Laser vibrometry measurements verify theoretically predicted properties of edge waves. • Excitation of a higher order edge wave is observed in the experiments. [ABSTRACT FROM AUTHOR]

Highlights • The study demonstrates how parameter inference can be used to motivate asymptotic reduction of a mathematical model to derive a simpler description with identifiable parameters. • Reduced models of the gibberellin biosynthesis pathway are derived and shown to fit experimental data. Parameter estimates and their confidence intervals are calculated for the reduced models. • The analysis reveals the limiting steps in the biosynthesis of the plant hormone gibberellin (GA). Abstract Developing effective strategies to use models in conjunction with experimental data is essential to understand the dynamics of biological regulatory networks. In this study, we demonstrate how combining parameter estimation with asymptotic analysis can reveal the key features of a network and lead to simplified models that capture the observed network dynamics. Our approach involves fitting the model to experimental data and using the profile likelihood to identify small parameters and cases where model dynamics are insensitive to changing particular individual parameters. Such parameter diagnostics provide understanding of the dominant features of the model and motivate asymptotic model reductions to derive simpler models in terms of identifiable parameter groupings. We focus on the particular example of biosynthesis of the plant hormone gibberellin (GA), which controls plant growth and has been mutated in many current crop varieties. This pathway comprises two parallel series of enzyme-substrate reactions, which have previously been modelled using the law of mass action (Middleton et al., 2012). Considering the GA20ox-mediated steps, we analyse the identifiability of the model parameters using published experimental data; the analysis reveals the ratio between enzyme and GA levels to be small and motivates us to perform a quasi-steady state analysis to derive a reduced model. Fitting the parameters in the reduced model reveals additional features of the pathway and motivates further asymptotic analysis which produces a hierarchy of reduced models. Calculating the Akaike information criterion and parameter confidence intervals enables us to select a parsimonious model with identifiable parameters. As well as demonstrating the benefits of combining parameter estimation and asymptotic analysis, the analysis shows how GA biosynthesis is limited by the final GA20ox-mediated steps in the pathway and generates a simple mathematical description of this part of the GA biosynthesis pathway. [ABSTRACT FROM AUTHOR]

In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator Δ. Concretely, we treat the generalized Charlier weights in the framework of Δ -Sobolev orthogonality. We obtain an asymptotic expansion for these orthogonal polynomials where the falling factorial polynomials play an important role. [ABSTRACT FROM AUTHOR]

Let Ω be a curvilinear polygon and Q Ω γ be the Laplacian in L 2 ( Ω ) , Q Ω γ ψ = − Δ ψ , with the Robin boundary condition ∂ ν ψ = γ ψ , where ∂ ν is the outer normal derivative and γ > 0 . We are interested in the behavior of the eigenvalues of Q Ω γ as γ becomes large. We prove that the asymptotics of the first eigenvalues of Q γ Ω is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with ∂Ω. In the particular case of a polygon with straight edges the first eigenpairs are exponentially close to those of the model operators. Finally, we prove a Weyl asymptotics for the eigenvalue counting function of Q Ω γ for a threshold depending on γ , and show that the leading term is the same as for smooth domains. [ABSTRACT FROM AUTHOR]

We undertake a symmetry analysis of a sequence of evolution equations of orders n = 3 , 4 , 5 , 6 , … which includes at lower orders partial differential equations having important applications. For n = 3 , 4 and 5 the equations in this sequence are the Korteweg–de Vries equation, the dissipative Kuramoto–Sivashinsky equation and the so-called Korteweg–de Vries equation of fifth order. We give a detailed discussion of both classical and nonclassical symmetries for this sequence of equations. It is in this latter case, using an approach based on the compatibility of the members of this sequence of equations with a first order differential equation and in the case where the infinitesimal τ = 0 , that we make our main new insights. Further results are also given for two of the obtained reductions of this sequence of evolution equations to ordinary differential equations. In addition, a generalization of the approach to reductions based on compatibility is also considered, and is found to provide much promise for future work. [ABSTRACT FROM AUTHOR]

The mevalonate pathway is normally found in eukaryotes, and allows for the production of isoprenoids, a useful class of organic compounds. This pathway has been successfully introduced to Escherichia coli , enabling a biosynthetic production route for many isoprenoids. In this paper, we develop and solve a mathematical model for the concentration of metabolites in the mevalonate pathway over time, accounting for the loss of acetyl-CoA to other metabolic pathways. Additionally, we successfully test our theoretical predictions experimentally by introducing part of the pathway into Cupriavidus necator . In our model, we exploit the natural separation of time scales as well as of metabolite concentrations to make significant asymptotic progress in understanding the system. We confirm that our asymptotic results agree well with numerical simulations, the former enabling us to predict the most important reactions to increase isopentenyl diphosphate production whilst minimizing the levels of HMG-CoA, which inhibits cell growth. Thus, our mathematical model allows us to recommend the upregulation of certain combinations of enzymes to improve production through the mevalonate pathway. [ABSTRACT FROM AUTHOR]

We investigate eigenfunctions of the Neumann Laplacian in a bounded domain Ω ⊂ R d , where a cuspidal singularity is caused by a cavity consisting of two touching balls, or discs in the planar case. We prove that the eigenfunctions with all of their derivatives are bounded in Ω ‾ , if the dimension d equals 2, but in dimension d ≥ 3 their gradients have a strong singularity O ( | x − O | − α ) , α ∈ ( 0 , 2 − 2 ] at the point of tangency O . Our study is based on dimension reduction and other asymptotic procedures, as well as the Kondratiev theory applied to the limit differential equation in the punctured hyperplane R d − 1 ∖ O . We also discuss other shapes producing thinning gaps between touching cavities. [ABSTRACT FROM AUTHOR]

We investigate the spectrum of three-dimensional Schrödinger operators with δ -interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues below the threshold of the essential spectrum. We focus on spectral properties for sharp cones, that is when the cone aperture goes to zero, and we describe the asymptotic behavior of the eigenvalues and of the eigenvalue counting function. A part of the results are given in terms of numerical constants appearing as solutions of transcendental equations involving modified Bessel functions. [ABSTRACT FROM AUTHOR]

Immunotherapies exploit the immune system to target and kill cancer cells, while sparing healthy tissue. Antibody therapies, an important class of immunotherapies, involve the binding to specific antigens on the surface of the tumour cells of antibodies that activate natural killer (NK) cells to kill the tumour cells. Preclinical assessment of molecules that may cause antibody-dependent cellular cytotoxicity (ADCC) involves co-culturing cancer cells, NK cells and antibody in vitro for several hours and measuring subsequent levels of tumour cell lysis. Here we develop a mathematical model of such an in vitro ADCC assay, formulated as a system of time-dependent ordinary differential equations and in which NK cells kill cancer cells at a rate which depends on the amount of antibody bound to each cancer cell. Numerical simulations generated using experimentally-based parameter estimates reveal that the system evolves on two timescales: a fast timescale on which antibodies bind to receptors on the surface of the tumour cells, and NK cells form complexes with the cancer cells, and a longer time-scale on which the NK cells kill the cancer cells. We construct approximate model solutions on each timescale, and show that they are in good agreement with numerical simulations of the full system. Our results show how the processes involved in ADCC change as the initial concentration of antibody and NK-cancer cell ratio are varied. We use these results to explain what information about the tumour cell kill rate can be extracted from the cytotoxicity assays. [ABSTRACT FROM AUTHOR]

We consider the semilinear Lane–Emden problem: ( E p ) { − Δ u = | u | p − 1 u in B u = 0 on ∂ B where B is the unit ball of R N , N ≥ 3 , centered at the origin and 1 < p < p S , p S = N + 2 N − 2 . We prove that for any radial solution u p of ( E p ) with m nodal domains its Morse index m ( u p ) is given by the formula m ( u p ) = m + N ( m − 1 ) if p is sufficiently close to p S . [ABSTRACT FROM AUTHOR]

We give in this paper an uniqueness result of the lubricated Elrod Adams model in stationary and transitory cases and in one dimensional space. We give also an asymptotic behaviour in time. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic expansions of the amplitude of the solution of the Helmholtz equation. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator. This work uses first and second order approximations of this operator to derive new asymptotic expressions of the normal derivative of the total field. The resulting expansions can be used to appropriately choose the ansatz in the design of high-frequency numerical solvers, such as those based on integral equations, in order to produce more accurate approximation of the solutions around the shadow and the deep shadow regions than the ones based on the usual ansatz. [ABSTRACT FROM AUTHOR]

We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory. [ABSTRACT FROM AUTHOR]

We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation − Δ u = u n + 2 n − 2 with isolated boundary singularities, where n ≥ 3 is the dimension. [ABSTRACT FROM AUTHOR]

We establish the existence of an open set of initial conditions through which pass solutions without singularities to Newton's gravitational equations in R 3 on a semi-infinite interval in forward time, for which every pair of particles separates like A t , A > 0 , as t → ∞ . The solutions are constructable as series with rapid uniform convergence and their asymptotic behavior to any order is prescribed. We show that this family of solutions depends on 6 N parameters subject to certain constraints. [ABSTRACT FROM AUTHOR]

The group of genetically mediated diseases, known collectively as retinitis pigmentosa (RP), cause retinal degeneration and, hence, loss of vision. The most common inherited retinal degeneration, RP is currently untreatable. The retina detects light using cells known as photoreceptors, of which there are two types: rods and cones. In RP, genetic mutations cause patches of photoreceptors to degenerate and typically directly affect either rods or cones, but not both. During disease progression, degenerate patches spread and the unaffected photoreceptor type also begins to degenerate. The cause underlying these phenomena is currently unknown. The oxygen toxicity hypothesis proposes that secondary photoreceptor loss is due to hyperoxia (toxically high oxygen levels), which results from the decrease in oxygen uptake following the initial loss of photoreceptors. In this paper, we construct mathematical models, formulated as 1D systems of partial differential equations, to investigate this hypothesis. Using a combination of numerical simulations, asymptotic analysis and travelling wave analysis, we find that degeneration may spread due to hyperoxia, and generate spatio-temporal patterns of degeneration similar to those seen in vivo . We determine the conditions under which a degenerate patch will spread and show that the wave speed of degeneration is a monotone decreasing function of the local photoreceptor density. Lastly, the effects of treatment with antioxidants and trophic factors, and of capillary loss, upon the dynamics of photoreceptor loss and recovery are considered. [ABSTRACT FROM AUTHOR]

In this paper, we study a double-barrier option with a stochastic volatility model whose volatility is driven by a fast mean-reverting process, where the option's payoff is extinguished as the underlying asset crosses one of two barriers. By using an asymptotic analysis and Mellin transform techniques, we derive semi-analytic option pricing formulas with the sum of a leading-order term and a correction-order term, and then the accuracy of the first approximation price of the double-barrier option is verified by using Monte Carlo simulation. Moreover, we analyze the impact of stochastic volatility on the double-barrier option prices. Finally, we demonstrate that our results enhance the existing double-barrier option price structures in view of flexibility and applicability through the market price of volatility risk. [ABSTRACT FROM AUTHOR]

We present a minimal model of solvent evaporation and absorption in thin films consisting of a volatile solvent and non-volatile solutes. An asymptotic analysis yields expressions that facilitate the extraction of physically significant model parameters from experimental data, namely the mass transfer coefficient and composition-dependent diffusivity. The model can be used to predict the dynamics of drying and film formation, as well as sorption/desorption, over a wide range of experimental conditions. A state diagram is used to understand the experimental conditions that lead to the formation of a solute-rich layer, or “skin”, at the evaporating surface during drying. In the case of solvent absorption, the model captures the existence of a saturation front that propagates from the film surface towards the substrate. The theoretical results are found to be in excellent agreement with data produced from dynamic vapour sorption experiments of ternary mixtures comprising an aluminium salt, glycerol, and water. Moreover, the model should be generally applicable to a variety of practical contexts, from paints and coatings, to personal care, packaging, and electronics. [ABSTRACT FROM AUTHOR]

We establish a homogenization result for a fully nonlinear degenerate parabolic system with critical growth arising from the heat and moisture flow through a partially saturated porous media. Existence of a global weak solution of the mesoscale problem is proven by means of a semidiscretization in time, a priori estimates and passing to the limit from discrete approximations. After that, porous material exhibiting periodic spatial oscillations is considered and the two-scale convergence (as the oscillation period vanishes) to a corresponding homogenized problem is rigorously proven. [ABSTRACT FROM AUTHOR]

The identification of a single global model for a stochastic dynamical system operating under various conditions is considered. Each operating condition is assumed to have a pseudo-static effect on the dynamics and be characterized by a single measurable scheduling variable. Identification is accomplished within a recently introduced Functionally Pooled (FP) framework, which offers a number of advantages over Linear Parameter Varying (LPV) identification techniques. The focus of the work is on the extension of the framework to include the important FP-ARMAX model case. Compared to their simpler FP–ARX counterparts, FP–ARMAX models are much more general and offer improved flexibility in describing various types of stochastic noise, but at the same time lead to a more complicated, non-quadratic, estimation problem. Prediction Error (PE), Maximum Likelihood (ML), and multi-stage estimation methods are postulated, and the PE estimator optimality, in terms of consistency and asymptotic efficiency, is analytically established. The postulated estimators are numerically assessed via Monte Carlo experiments, while the effectiveness of the approach and its superiority over its FP–ARX counterpart are demonstrated via an application case study pertaining to simulated railway vehicle suspension dynamics under various mass loading conditions. [ABSTRACT FROM AUTHOR]

Three asymptotic limits exist for the Euler equations at low Mach number - purely convective, purely acoustic, and mixed convective-acoustic. Standard collocated density-based numerical schemes for compressible flow are known to fail at low Mach number due to the incorrect asymptotic scaling of the artificial diffusion. Previous studies of this class of schemes have shown a variety of behaviours across the different limits and proposed guidelines for the design of low-Mach schemes. However, these studies have primarily focused on specific discretisations and/or only the convective limit. In this paper, we review the low-Mach behaviour using the modified equations - the continuous Euler equations augmented with artificial diffusion terms - which are representative of a wide range of schemes in this class. By considering both convective and acoustic effects, we show that three diffusion scalings naturally arise. Single- and multiple-scale asymptotic analysis of these scalings shows that many of the important low-Mach features of this class of schemes can be reproduced in a straightforward manner in the continuous setting. As an example, we show that many existing low-Mach Roe-type finite-volume schemes match one of these three scalings. Our analysis corroborates previous analysis of these schemes, and we are able to refine previous guidelines on the design of low-Mach schemes by including both convective and acoustic effects. Discrete analysis and numerical examples demonstrate the behaviour of minimal Roe-type schemes with each of the three scalings for convective, acoustic, and mixed flows. • Asymptotic analysis of numerical schemes at low Mach number. • Consider the convective, acoustic, and mixed convective-acoustic low Mach limits. • Derivation of required asymptotic scaling of artificial diffusion at each limit. • Use of modified equations applies to finite-volume or finite-difference schemes. • Application to Roe schemes shows excellent agreement with previous literature. [ABSTRACT FROM AUTHOR]

The nervous system is today recognized to play an important role in the development of cancer. Indeed, neurons extend long processes (axons) that grow and infiltrate tumors in order to regulate the progression of the disease in a positive or negative way, depending on the type of neuron considered. Mathematical modeling of this biological process allows to formalize the nerve–tumor interactions and to test hypotheses in silico to better understand this phenomenon. In this work, we introduce a system of differential equations modeling the progression of pancreatic ductal adenocarcinoma (PDAC) coupled with associated changes in axonal innervation. The study of the asymptotic behavior of the model confirms the experimental observations that PDAC development is correlated with the type and densities of axons in the tissue. We study then the identifiability and the sensitivity of the model parameters. The identifiability analysis informs on the adequacy between the parameters of the model and the experimental data and the sensitivity analysis on the most contributing factors on the development of cancer. It leads to significant insights on the main neural checkpoints and mechanisms controlling the progression of pancreatic cancer. Finally, we give an example of a simulation of the effects of partial or complete denervation that sheds lights on complex correlation between the healthy, pre-cancerous and cancerous cell densities and axons with opposite functions. • Dynamical system modeling the pancreatic cancer progression and the axons regulation • The in silico denervations are consistent with the in vivo observations • The autonomic axons slow the PDAC progression at early stages • The sensory axons have a pro-tumoral effect especially on the proliferation process [ABSTRACT FROM AUTHOR]

The local eigenvalue statistics of large random matrices near a hard edge transitioning into a soft edge are described by the Bessel process associated with a large parameter α. For this point process, we obtain (1) exponential moment asymptotics, up to and including the constant term, (2) asymptotics for the expectation and variance of the counting function, (3) several central limit theorems and (4) a global rigidity upper bound. [ABSTRACT FROM AUTHOR]