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Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator 퐏 ⁢ (i ⁢ ∂ x) = (P 1 ⁢ (i ⁢ ∂ x)) m 1 ⁢ ⋯ ⁢ (P l ⁢ (i ⁢ ∂ x)) m l with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation 퐏 ⁢ (i ⁢ ∂ x) ⁢ u = f in ℝ n . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation. [ABSTRACT FROM AUTHOR]

The asymptotic expansion of the eigenvalue of Sturm-Liouville problem is presented. The problem has a symmetric double well potential that is continuous, symmetrical to both the midpoint and quarter point of the related interval and non-increasing on the quarter interval. [ABSTRACT FROM AUTHOR]

The representability of the Burr distribution as a mixture of Weibull distribution is studied in order to justify its utility for modelling the statistical regularities in extreme values registered in non-stationary flows of informative events. A result of [24] is improved by extending the domain of admissible values of the parameters which provide the representability of the (generalized) Burr distribution as a scale mixture of the Weibull distribution. This result gives an argument in favour of application of the Burr distribution as a model of statistical regularities of extreme values registered within moderate regular time intervals, say, daily (short-term) extremes. In turn, if we are interested in the statistical regularities of the behaviour of the absolute extreme observation over a long period, say, a decade (the long-term extreme), then it can be noted that the daily extreme values form a sample of the Burr-distributed random variables. As is known, the Burr distribution belongs to the domain of max-attraction of the Fréchet distribution. The problem of improving the accuracy of the approximation of the distribution of the absolute extreme by the Fréchet distribution by the construction of an asymptotic expansion for the distribution of the extreme order statistics in the sample of independent identically Burr-distributed random variables is also considered. These results are illustrated by an example of fitting the Burr distribution to the data representing the extreme values of characteristics of the interplanetary magnetic field. [ABSTRACT FROM AUTHOR]

We consider a gas of N weakly interacting bosons in the ground state. Such gases exhibit Bose–Einstein condensation. The binding energy is defined as the energy it takes to remove one particle from the gas. In this article, we prove an asymptotic expansion for the binding energy, and compute the first orders explicitly for the homogeneous gas. Our result addresses in particular a conjecture by Nam (Lett Math Phys 108(1):141–159, 2018), and provides an asymptotic expansion of the ionization energy of bosonic atoms. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic fluctuation of buffer stock level at time t is investigated. Therefore, random walk processes X(t) and Y(t) with two specific barriers have been defined to describe the stochastic fluctuation of the product level. Here X(t) = Y(t) -- a and the parameter a specifies half capacity of the buffer stock warehouse. Next, the one-dimensional distribution of the process X(t) has calculated. Moreover, the ergodicity of the process X(t) has been proven and the exact formula for the characteristic function has been found. Then, the weak convergence theorem has been proven for the standardized process W(t) = X(t)/a, as a → ∞. Additionally, exact and asymptotic expressions for the ergodic moments of the processes X(t) and Y (t) are obtained. [ABSTRACT FROM AUTHOR]

This computational study concerns approximate solutions of singularly perturbed one-dimensional boundary-value problems having perturbed convection and diffusion terms. Such kinds of problems take different stands depending on the perturbation parameters. Typically, when the problem is convection-dominated, classical discretization methods suffer from numerical instability issues. Therefore, standard methods require special treatment in convection dominance. To this end, in this work, the standard Galerkin finite element method (GFEM) is stabilized with the streamlineupwind/Petrov-Galerkin (SUPG) formulation. Beyond that, an asymptotic approach, called the successive complementary expansion method (SCEM), is also proposed. Two test examples are provided to evaluate and compare the proposed methods' performances for various values of the convection and diffusion parameters. [ABSTRACT FROM AUTHOR]

In this paper, based on the discretization method, we construct a new 2-parameter regularly varying discrete distribution generated by Waring-type probability (2-RDWP). Some useful plots are displayed for the model. From the mathematical point of view, to suggest 2-RDWP as a new discrete probability distribution in bioinformatics, some statistical facts such as unimodality, skewness to the right, upward/downward convexity, regular variation at infinity and asymptotically constant slowly varying component are established for the model. We provide the conditions of coincidence of solution for the system of likelihood equations with the maximum likelihood estimators for the unknown parameters. Simulation studies are performed using the Monte Carlo method and Nelder–Mead optimization algorithm to obtain maximum likelihood estimations of the unknown parameters. Asymptotic expansion of the probability function with two terms is considered, and then the moment's existence of integer orders is investigated. Finally, a real count data set is used to show the applicability of the new model compared to other models in bioinformatics. [ABSTRACT FROM AUTHOR]

Let p, q ∈ ℝ with p − q ≽ 0, σ = 1 2 (p + q − 1) and s = 1 2 (1 − p + q) , and let D m (x ; p , q) = D 0 (x ; p , q) + ∑ k = 1 m B 2 k (s) 2 k (x + σ) 2 k , where D 0 (x ; p , q) = ψ (x + p) + ψ (x + q) 2 − ln (x + σ). We establish the asymptotic expansion D 0 (x ; p , q) ∼ − ∑ n = 1 ∞ B 2 n (s) 2 n (x + σ) 2 n a s x → ∞ , where B2n(s) stands for the Bernoulli polynomials. Further, we prove that the functions (− 1) m D m (x ; p , q) and (− 1) m + 1 D m (x ; p , q) are completely monotonic in x on (−σ, ∞) for every m ∈ ℕ0 if and only if p − q ∈ [ 0 , 1 2 ] and p − q = 1, respectively. This not only unifies the two known results but also yields some new results. [ABSTRACT FROM AUTHOR]

Asymptotic expansion is presented for an estimator of the Hurst coefficient of a fractional Brownian motion. We first derive the expansion formula of the principal term of the error of the estimator using a recently developed theory of asymptotic expansion of the distribution of Wiener functionals, and utilize the perturbation method on the obtained formula in order to calculate the expansion of the estimator. We also discuss some second-order modifications of the estimator. Numerical results show that the asymptotic expansion attains higher accuracy than the normal approximation. [ABSTRACT FROM AUTHOR]

The higher-order nonlinear Schrödinger equation with combined nonlinearities is derived by an asymptotic reduction from the (2+4) Korteweg–de Vries model for weakly nonlinear wave packets for the context of interfacial waves in a three-layer symmetric media. Focusing properties and modulation instability effects are discussed for the considered physical context. Instability growth rate, maximum of the increment and the boundaries of the instability interval are derived in terms of three-layer density stratification, their structure on the parameter planes of relative layer depth, carrier wavenumber and envelope amplitude, are considered in detail. [ABSTRACT FROM AUTHOR]

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α ∈ (1 , 2). The error has the asymptotic expansion (d 3 τ 3 - α + d 4 τ 4 - α + d 5 τ 5 - α + ⋯) + (d 2 ∗ τ 4 + d 3 ∗ τ 6 + d 4 ∗ τ 8 + ⋯) at any fixed time t N = T , N ∈ Z + , where d i , i = 3 , 4 , ... and d i ∗ , i = 2 , 3 , ... denote some suitable constants and τ = T / N denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α ∈ (1 , 2) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings. [ABSTRACT FROM AUTHOR]

Multiple testing has been a prominent topic in statistical research. Despite extensive work in this area, controlling false discoveries remains a challenging task, especially when the test statistics exhibit dependence. Various methods have been proposed to estimate the false discovery proportion (FDP) under arbitrary dependencies among the test statistics. One key approach is to transform arbitrary dependence into weak dependence and subsequently establish the strong consistency of FDP and false discovery rate under weak dependence. As a result, FDPs converge to the same asymptotic limit within the framework of weak dependence. However, we have observed that the asymptotic variance of FDP can be significantly influenced by the dependence structure of the test statistics, even when they exhibit only weak dependence. Quantifying this variability is of great practical importance, as it serves as an indicator of the quality of FDP estimation from the data. To the best of our knowledge, there is limited research on this aspect in the literature. In this paper, we aim to fill in this gap by quantifying the variation of FDP, assuming that the test statistics exhibit weak dependence and follow normal distributions. We begin by deriving the asymptotic expansion of the FDP and subsequently investigate how the asymptotic variance of the FDP is influenced by different dependence structures. Based on the insights gained from this study, we recommend that in multiple testing procedures utilizing FDP, reporting both the mean and variance estimates of FDP can provide a more comprehensive assessment of the study's outcomes. [ABSTRACT FROM AUTHOR]

This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O (ε). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O (ε − 1) is considered. The novelty of this article is the case of α < 1 , which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0 , i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε. [ABSTRACT FROM AUTHOR]

We study a system of equations modeling the dynamic tension of a homogeneous round layer of incompressible perfectly rigid-plastic transversely isotropic material obeying the Mises–Hencky criterion. The upper and lower bases are stress-free, the radial velocity is set on the lateral boundary, and the possibility of thickening or thinning of the layer, simulating formation and further development of a neck, is taken into account. Using the method of asymptotic integration, two characteristic tension modes are identified, that is, relations of dimensionless parameters are determined that necessitate taking into account inertial terms. An approximate solution of the problem is constructed when considering the mode associated with the acceleration on the lateral face reaching its critical values. [ABSTRACT FROM AUTHOR]

A formal asymptotic expansion is constructed for the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearities and weak diffusion in the case of several space variables. Under the conditions imposed on the data of the problem, the leading asymptotic term is described by the multidimensional generalized Burgers–Korteweg–de Vries equation. Under certain conditions, the remainder is estimated with respect to the residual. [ABSTRACT FROM AUTHOR]

We consider the model of Mindlin–Timoshenko for a multi-structure composed of an elastic plate surrounded by a thin layer of uniform thickness. From the viewpoint of numerical simulation, the treatment of the behavior of this structure is difficult because of the presence of the thin coating. In order to overcome this difficulty, we use the asymptotic expansion method to identify an approximate model that does not involve the thin layer geometrically but which accounts for its effect through new approximate boundary conditions. These conditions are set on the junction interface between the two sub-structures and depend on the thickness and the physical characteristics of the thin layer. Moreover, we give optimal error estimates between the exact and the approximate solutions of the considered transmission problem, which validate this approximation. [ABSTRACT FROM AUTHOR]

This paper represents the exact solution for the properties of the conducting fluid that is flowing past a rotating sphere immersed in a porous medium. It's a continuation of Sharma and Srivastava (Z Angew Math Mech 103:e202200218, 2022), where the sphere is rotating in the clear fluid region. The solution to the problem was found using Stokes' equation. In this paper the solutions are found using the Brinkman model in terms of special functions- Bessel's function of the first and second kind. In this problem, the drag force experienced by the rotating sphere is also calculated for the less permeability of the porous medium. The model is solved using the asymptotic expansion method for the prescribed boundary conditions. By this method, the model consisting of partial differential equations is reduced into ordinary differential equations – modified Bessel's equations. The results for fluid velocity, stream function, and separation parameters are found in terms of Bessel's functions and these results match with the existing literature in the absence of Reynolds number. The graphs are plotted for the drag force on the rotating sphere and separation parameter. It has been observed that the separation of the particles occurs at the surface of the sphere more often for high permeability and the drag was found to be decreasing for low permeability. Both graphs were plotted for a range of Reynolds numbers. [ABSTRACT FROM AUTHOR]

We study a new class of time-periodic solutions of singularly perturbed systems of reaction–diffusion equations in the case of a fast and a slow equation, which are usually called Tikhonov-type systems. A boundary layer asymptotics of solutions is constructed, the existence of solutions with this asymptotics is proved, and conditions for the Lyapunov asymptotic stability of these solutions treated as solutions of the corresponding initial–boudary value problems are obtained. [ABSTRACT FROM AUTHOR]

The paper provides a precise error estimate for an asymptotic expansion of a certain stochastic control problem related to relative entropy minimization. In particular, it is shown that the expansion error depends on the regularity of functionals on path space. An efficient numerical scheme based on a weak approximation with Monte Carlo simulation is employed to implement the asymptotic expansion in multidimensional settings. Throughout numerical experiments, it is confirmed that the approximation error of the proposed scheme is consistent with the theoretical rate of convergence. [ABSTRACT FROM AUTHOR]

Given the important role an asymptotic expansion of Whittaker functions or Shalika functions plays in the study of L-functions via the Rankin–Selberg method, we first establish an asymptotic expansion of (k, c)-functions for GL n (F) over a p-adic field F via Casselman–Shalika’s argument. As an application, we then show the holomorphy of Ginzburg–Kaplan’s local “tensor product” L-functions for GL c (F) × GL k (F) . In the appendix, following Wallach–Soudry’s argument, we also provide an analogous asymptotic expansion of (k, c)-functions for GL n (F) with F an Archimedean local field. [ABSTRACT FROM AUTHOR]

We present an asymptotic method to compute efficiently second-order telegraph equations with high-frequency extrinsic oscillations. This method is based on asymptotic expansions in inverse powers of the oscillatory parameter. Each term of this asymptotic expansion is the sum of at most two coefficients. Each coefficient is derived either by recursion or by solving a non-oscillatory problem. This leads to method which exhibit improved performance with growing frequency of oscillation. [ABSTRACT FROM AUTHOR]

Infinitesimal bendings for classes of two-dimensional surfaces in $ \mathbb {R}^3 $ R 3 are investigated. The techniques used to construct the bending fields include reduction to solvability of Bers–Vekua type equations and systems of differential equations with periodic coefficients. [ABSTRACT FROM AUTHOR]

The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix Tn(a), whose generating function a is complex-valued and has a power singularity at one point. As a consequence, Tn(a) is non-Hermitian and we know that in this setting, the eigenvalue computation is a nontrivial task for large sizes. First we follow the work of Bogoya, Böttcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion for the eigenvalues. In a second step, we apply matrixless algorithms, in the spirit of the work by Ekström, Furci, Garoni, Serra-Capizzano et al., for computing those eigenvalues. Since the inner and extreme eigenvalues have different asymptotic behaviors, we worked on them independently and combined the results to produce a high precision global numerical and matrixless algorithm. The numerical results are very precise, and the computational cost of the proposed algorithms is independent of the size of the considered matrices for each eigenvalue, which implies a linear cost when the entire spectrum is computed. From the viewpoint of real-world applications, we emphasize that the class under consideration includes the matrices stemming from the numerical approximation of fractional diffusion equations. In the final section a concise discussion on the matter and a few open problems are presented. [ABSTRACT FROM AUTHOR]

We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the stepsize and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low to intermediate frequencies and at low tolerances, where it may use up to 106 times fewer function evaluations. Even in high-frequency regimes, our implementation is on average 10 times faster than other specialized solvers. [ABSTRACT FROM AUTHOR]

Lattice Green's functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order discretization and operate on domains with free-space boundary conditions in all directions. Under these conditions, fast expansion methods are available that enable precomputation of 2-dimensional or 3-dimensional (3D) LGFs in linear time, avoiding the need for brute-force multidimensional quadrature of numerically unstable integrals. Here we focus on higherorder discretizations of the Laplace operator on domains with more general boundary conditions, by (1) providing an algorithm for fast and accurate evaluation of the LGFs associated with highorder dimension-split centered finite differences on unbounded domains, and (2) deriving closed-form expressions for the LGFs associated with both dimension-split and Mehrstellen discretizations on domains with one unbounded dimension. Through numerical experiments we demonstrate that these techniques provide LGF evaluations with near machine-precision accuracy, and that the resulting LGFs allow for numerically consistent solutions to high-order discretizations of the Poisson's equation on fully or partially unbounded 3D domains. [ABSTRACT FROM AUTHOR]

Exact expressions for the relationship between angles of incidence and reflection from a boundary wall in nonlinear dissipative particle models are extremely rare. Here, we study a particle model of a camphor disk floating on water in a low speed limit, for which the model was derived. We begin with a very rough and inaccurate approximation, based in part on a Hamiltonian limit of the model, and then, using symmetry arguments supported by a series of experiments, show that this rough approximation can likely be repaired by introducing a factor dependent upon model parameters, then extract the full parameter dependence of this factor, and finally conjecture a simple and exact asymptotic relationship between the angles of incidence and reflection. There are reasons to believe that this asymptotic relationship may be universal for such models in their corresponding limits. [ABSTRACT FROM AUTHOR]

Approximating the Hadamard finite-part integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the Riemann-Liouville fractional derivative of order α ∈ (1 , 2) and the error is shown to have the asymptotic expansion (d 3 τ 3 - α + d 4 τ 4 - α + d 5 τ 5 - α + ⋯) + (d 2 ∗ τ 4 + d 3 ∗ τ 6 + d 4 ∗ τ 8 + ⋯) at any fixed time, where τ denotes the step size and d l , l = 3 , 4 , ⋯ and d l ∗ , l = 2 , 3 , ⋯ are some suitable constants. Applying the proposed scheme in temporal direction and the central difference scheme in spatial direction, a new finite difference method is developed for approximating the time fractional wave equation. The proposed method is unconditionally stable, convergent with order O (τ 3 - α) , α ∈ (1 , 2) and the error has the asymptotic expansion. Richardson extrapolation is applied to improve the accuracy of the numerical method. The convergence orders are O (τ 4 - α) and O (τ 2 (3 - α)) , α ∈ (1 , 2) , respectively, after first two extrapolations. Numerical examples are presented to show that the numerical results are consistent with the theoretical findings. [ABSTRACT FROM AUTHOR]

Let a,b,c,d ∈ R with a+b = c+d = 2r +1. Then In Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) ~ ∞/Σ/k=1 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 as x→∞, where (δ1,δ2) = (|a−b|,|c−d|) = (1−2θ1,1−2θ2). When 0 ≤ δ2 < δ1 ≤ 1, the function x→ (-1)m [1n Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) - m/Σ/k=1 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 ] for m ∈ N is completely monotonic on (−r,∞). This yields some known and new results. [ABSTRACT FROM AUTHOR]

This work, which can be considered as a small monograph, is devoted to the study of two-and three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of compatibility of asymptotic expansions, the boundary layer method, and the multiscale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very essential weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a problem of optimal control over a finite time interval for a linear system with constant coefficients and a small parameter in the initial data in the class of piecewise continuous controls with smooth geometric constraints. We consider a terminal convex performance index. We substantiate the limit relations as the small parameter tends to zero for the optimal value of the performance index and for the vector generating the optimal control in the problem. We show that the asymptotics of the solution can be of complicated nature. In particular, it may have no expansion in the Poincaré sense in any asymptotic sequence of rational functions of the small parameter or its logarithms. [ABSTRACT FROM AUTHOR]

We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function G (z ; τ). Second, we derive an algorithm for numerically computing the double gamma function and present its complete asymptotic expansion as z → ∞. Third, we derive some new properties, including a new product identity and new representations of the gamma modular forms C (τ) and D (τ). [ABSTRACT FROM AUTHOR]

We consider a metric graph consisting of two edges, one of which has length ε which we send to zero. On this graph we study the resolvent and spectrum of the Laplacian subject to a general vertex condition at the connecting vertex. Despite the singular nature of the perturbation (by a short edge), we find that the resolvent depends analytically on the parameter ε . In contrast, the negative eigenvalues escape to minus infinity at rates that could be fractional, namely, ε 0 , ε - 2 / 3 or ε - 1 . These rates take place when the corresponding eigenfunction localizes, respectively, only on the long edge, on both edges, or only on the short edge. [ABSTRACT FROM AUTHOR]

In the dynamics of linear structures, the impulse response function is of fundamental interest. In some cases one examines the short term response wherein the disturbance is still local and the boundaries have not yet come into play, and for such short-time analysis the geometrical extent of the structure may be taken as unbounded. Here we examine the response of slender beams to angular impulses. The Euler–Bernoulli model, which does not include rotary inertia of cross sections, predicts an unphysical and unbounded initial rotation at the point of application. A finite length Euler–Bernoulli beam, when modeled using finite elements, predicts a mesh-dependent response that shows fast large-amplitude oscillations setting in very quickly. The simplest introduction of rotary inertia yields the Rayleigh beam model, which has more reasonable behavior including a finite wave speed at all frequencies. If a Rayleigh beam is given an impulsive moment at a location away from its boundaries, then the predicted behavior has an instantaneous finite jump in local slope or rotation, followed by smooth evolution of the slope for a finite time interval until reflections arrive from the boundary, causing subsequent slope discontinuities in time. We present a detailed study of the angular impulse response of a simply supported Rayleigh beam, starting with dimensional analysis, followed by modal expansion including all natural frequencies, culminating with an asymptotic formula for the short-time response. The asymptotic formula is obtained by breaking the series solution into two parts to be treated independently term by term, and leads to a polynomial in time. The polynomial matches the response from refined finite element (FE) simulations. [ABSTRACT FROM AUTHOR]

In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results.

We study the dynamics of the flow X solution to the ODE: X ′ (t , x) = b (X (t , x)) with X (0 , x) = x ∈ R d , where b is a regular Z d -periodic vector field in R d. We provide conditions on b to get the fine asymptotic expansion: | X (t , x) − x − t ζ (x) | ≤ M < ∞. To this end, we try to express X (t , x) − x − t ζ (x) as Φ (X (t , x)) − Φ (x) , which yields the desired expansion when Φ is bounded. Then, assuming that the 2D Kolmogorov theorem and some extension for d > 2 hold, we establish several regimes depending on the commensurability of the rotation vectors ζ (x) for which the expansion of X is valid. Moreover, we prove that for any 2D flow with a non vanishing smooth b inducing a unique incommensurable rotation vector ξ , the fine expansion holds in R 2 if, and only if, ξ 1 / ξ 2 is a Diophantine number. The case where ξ is commensurable is also investigated. Finally, several examples illustrate the different results, including the case of a vanishing b which blows up the asymptotic expansion in some direction. In particular, the case of some Euler flows is investigated. [ABSTRACT FROM AUTHOR]

Some well-known results of Grönwall on logarithmic derivative of modified Bessel functions of the first kind concerning exponential bounds are extended to Whittaker functions of the first and second kind M_{\kappa,\mu } and W_{\kappa,\mu }. Moreover, a complete monotonicity result is proved for the logarithmic derivative of the Whittaker function W_{\kappa,\mu }, and some monotonicity results with respect to the parameters and argument are shown for the logarithmic derivative of M_{\kappa,\mu }. The results extend and complement the known results in the literature about modified Bessel functions of the first and second kind. [ABSTRACT FROM AUTHOR]

We consider a mixed boundary value problem for the Poisson equation in a bounded planar domain. Most of the boundary is endowed with the Neumann condition, while the Dirichlet one is imposed on a periodic family of short segments of length e < 1. Consequently, the limit problem, corresponding to e = 0, has mixed boundary conditions with two collision points (where the type of boundary condition changes). The asymptotic representation of the solution to the original, singularly perturbed problem includes boundary layers of two types, namely the exponential one near the segments and the power-law one in the vicinity of the collision points. These boundary layers are solutions of certain problems in the half-strip and the half-plane, respectively. The powerlaw boundary layer causes some surprising phenomena, including the square-root singularities of the main correction asymptotic term at the collision points and their linear depending on the quantity In e. [ABSTRACT FROM AUTHOR]

We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. We perturbed it by a first order differential operator arbitrarily depending on a small multi-dimensional parameter and we study the existence of a limiting (homogenized) operator in the sense of the norm resolvent convergence. The first part of our main results states that the norm resolvent convergence is equivalent to the convergence of the coefficients in the perturbing operator in certain space of multipliers. The second part of our results says that the convergence in the mentioned spaces of multipliers is equivalent to the convergence of certain local mean values over small pieces of the considered domains. These results are supported by series of examples. We also provide a series of ways of generating new non-periodically oscillating perturbations for which our results are applicable. [ABSTRACT FROM AUTHOR]

The present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(a)$$ {T}_n(a) $$ as n$$ n $$ goes to infinity, with a continuous and real‐valued symbol a$$ a $$ having a power singularity of degree γ$$ \gamma $$ with 1<γ<2$$ 1<\gamma <2 $$, at one point. The resulting matrix is dense and its entries decrease slowly to zero when moving away from the main diagonal, we apply the so called simple‐loop (SL) method for constructing and justifying a uniform asymptotic expansion for all the eigenvalues. Note however, that the considered symbol does not fully satisfy the conditions imposed in previous works, but only in a small neighborhood of the singularity point. In the present work: (i) We construct and justify the asymptotic formulas of the SL method for the eigenvalues λj(Tn(a))$$ {\lambda}_j\left({T}_n(a)\right) $$ with j⩾εn$$ j\geqslant \varepsilon n $$, where the eigenvalues are arranged in nondecreasing order and ε$$ \varepsilon $$ is a sufficiently small fixed number. (ii) We show, with the help of numerical calculations, that the obtained formulas give good approximations in the case j<εn$$ j<\varepsilon n $$. (iii) We numerically show that the main term of the asymptotics for eigenvalues with j<εn$$ j<\varepsilon n $$, formally obtained from the formulas of the SL method, coincides with the main term of the asymptotics constructed and justified in the classical works of Widom and Parter. [ABSTRACT FROM AUTHOR]

We study the stationary elastic wave equation with free boundary condition in a locally perturbed half‐space in ℝ3$$ {\mathrm{\mathbb{R}}}&#x0005E;3 $$. Using the stationary phase method, we derive an asymptotic behavior at infinity of the resolvent of the elastic operator uniformly with respect to the direction in 핊+2. Consequently, the body waves and the Rayleigh surface waves appear simultaneously in the expansion. From the far‐field pattern of the expansion, we obtain the scattering amplitude. We also characterize the space of generalized eigenfunctions in terms of Agmon–Hörmander space B∗$$ {\mathcal{B}}&#x0005E;{\ast } $$ and derive their spatial asymptotics. [ABSTRACT FROM AUTHOR]