Asymptotic Approximation Research Articles

A convergent and asymptotic Laplace method for integrals

Watson’s lemma and Laplace’s method provide asymptotic expansions of Laplace integrals F(z)≔∫0∞e−zf(t)g(t)dt for large values of the parameter z. They are useful tools in the asymptotic approximation of special functions that have a Laplace integral representation. But in most of the important examples of special functions, the asymptotic expansion derived by means of Watson’s lemma or Laplace’s method is not convergent. A modification of Watson’s lemma was introduced in [Nielsen, 1906] where, by the use of inverse factorial series, a new asymptotic as well as convergent expansion of F(z), for the particular case f(t)=t, was derived. In this paper we go some steps further and investigate a modification of the Laplace’s method for F(z), with a general phase function f(t), to derive asymptotic expansions of F(z) that are also convergent, accompanied by error bounds. An analysis of the remainder of this new expansion shows that it is convergent under a mild condition for the functions f(t) and g(t), namely, these functions must be analytic in certain starlike complex regions that contain the positive axis [0,∞). In many practical situations (in many examples of special functions), the singularities of f(t) and g(t) are off this region and then this method provides asymptotic expansions that are also convergent. We illustrate this modification of the Laplace’s method with the parabolic cylinder function U(a,z), providing an asymptotic expansions of this function for large z that is also convergent.

Approximation algorithms for a virtual machine allocation problem with finite types

We study the virtual machine allocation (VMA) problem with physical machines (PMs) of identical configuration and virtual machines (VMs) of a constant number of types. Only CPU and memory (MEM) resources are considered, so it is a special two-dimensional vector bin packing (2-DVBP) problem. Applying the best fit heuristic, we develop a fast online algorithm, which uses 209OPT(L)+1 PMs at most, where OPT(L) is the optimal value. According to the limited types of VMs, the progress of combining VMs and dividing the PMs' capacity contributes to a linear-time offline algorithm that uses 109OPT(L)+2 PMs. It outperforms the state-of-the-art (1.405+ϵ)-asymptotic approximation algorithm for the general 2-DVBP problem. These ideas can be extended to develop fast and practical algorithms.

Asymptotic analysis of in-plane dynamic problems for elastic media with rigid clusters of small inclusions.

We present formal asymptotic approximations of fields representing the in-plane dynamic response of elastic solids containing clusters of closely interacting small rigid inclusions. For finite densely perforated bodies, the asymptotic scheme is developed to approximate the eigenfrequencies and the associated eigenmodes of the elastic medium with clamped boundaries. The asymptotic algorithm is also adapted to address the scattering of in-plane waves in infinite elastic media containing dense clusters. The results are accompanied by numerical simulations that illustrate the accuracy of the asymptotic approach.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)’.

Oviposition Model for a Southern Population of Mountain Pine Beetle.

We develop a predictive oviposition model for a southern population of mountain pine beetle (MPB) using a previously developed rate curve, incorporating variation in both oviposition rate and fecundity. We also introduce a method for determining the time delay before oviposition. The model describes the probability of oviposition for a season of MPB attacks using hourly phloem temperature and adult MPB attack data. We also develop an asymptotic approximation of MPB oviposition that is much less computationally taxing. The detailed oviposition model and its asymptotic approximation are compared with other ovipositional models for MPB; the predictive capacity of each model is evaluated using previously published observations.

Endless Process of Bifurcations in Delay Differential Equations

The paper is devoted to studying the singularly perturbed equations and systems with one or multiple delays. In focus is a bifurcation effect that, as the small parameter present in the system tends to zero, an endless process of forward and backward bifurcations repetition occur. To describe this effect and give its analytical background, we use the analog of the normal form, the quasinormal form, a nonlinear partial differential equation that regularly depends on a small parameter. The solutions of the quasinormal form give the main part of the asymptotic approximation of the solution of the differential equation with delay. Also effects of endless process of bifurcations is demonstrated in detail in two examples. In the first example, we consider second order equation with large delay, and in the second example — first order equation with two delays.

Strategies for Determining the Cascade Rate in MHD Turbulence: Isotropy, Anisotropy, and Spacecraft Sampling

Exact laws for evaluating cascade rates, tracing back to the Kolmogorov “4/5” law, have been extended to many systems of interest including magnetohydrodynamics (MHD), and compressible flows of the magnetofluid and ordinary fluid types. It is understood that implementations may be limited by the quantity of available data and by the lack of turbulence symmetry. Assessment of the accuracy and feasibility of such third-order (or Yaglom) relations is most effectively accomplished by examining the von Kármán–Howarth equation in increment form, a framework from which the third-order laws are derived as asymptotic approximations. Using this approach, we examine the context of third-order laws for incompressible MHD in some detail. The simplest versions rely on the assumption of isotropy and the presence of a well-defined inertial range, while related procedures generalize the same idea to arbitrary rotational symmetries. Conditions for obtaining correct and accurate values of the dissipation rate from these laws based on several sampling and fitting strategies are investigated using results from simulations. The questions we address are of particular relevance to sampling of solar wind turbulence by one or more spacecraft.

On eigenfunctions of Hill's equation with symmetric double well potential

Throughout this paper the asymptotic approximations for eigen- functions of eigenvalue problems associated with Hill’s equation satisfying periodic and semi-periodic boundary conditions are derived when the potential is symmetric double well. These approximations are used to determine the Green’s functions of the related problems. Then, the obtained results are adapted to the Whittaker-Hill equation which has the symmetric double well potential and is widely investigated in the literature.

Asymptotic approximation of the likelihood of stationary determinantal point processes

AbstractContinuous determinantal point processes (DPPs) are a class of repulsive point processes on with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behavior when the observation window grows toward . This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on and compare favorably to common alternative estimation methods for continuous DPPs.

Feasibility Analysis Upon Optimal Pollutant Degradation via Compartmental Modeling

Due to a lack of plausible kinetic modeling for contaminant attenuation, this study first proposed a “bi-exponential disposition” model to demonstrate the feasibility of wide-ranging applications for the biotic and abiotic degradation of pollutant(s). As a consequence of under-determined systems of bisphenol A (BPA) degradation via advanced oxidation processes (AOPs), a previous study proposed asymptotic approximation singular perturbation for kinetic modeling. The present study extended this model to provide the key performance indicator (KPI); namely, the area under the time course (AUC) of pollutant concentrations from time zero to the endpoint (i.e., AUC0→tf), quantitatively revealing the most promising strategy for pollutant (bio)degradation. Compared to the typical KPI (percentage of pollutant removal), AUC better illustrated the overall efficiency. Compartmental modeling predicted maximal pollutant mitigation through optimal schemes of operation for global optimization. The feasibility of adopting AUC for system prediction was confirmed in cases of azo and anthraquinone dye biodecolorization and acetaminophen (APAP), glyphosate, and bisphenol A (BPA) degradation. Regarding azo dye biodecolorization, the AUC and SDRmax indicated the need to consider cell concentrations. Compartment kinetics could be used for the serial acclimation of anthraquinone dye removal. Moreover, compartmental assessment upon glyphosate and acetaminophen abiotic degradation was also feasible for further applications. To minimize the AUC for optimal degradation of A⇌k3k1B→k2C, the maximal forward rate constants k1 and k2 and minimal backward rate constant k3 should be satisfied simultaneously. Thus, this AUC approach might be broadened to demonstrate overall optimization via Pontryagin’s maximum principle.

Non-local effects on travelling waves arising in a moving-boundary reaction–diffusion model

We examine travelling wave solutions of the partial differential equation u t = u xx + u(1 − u∗ϕ) on a moving domain x ⩽ L(t), where u∗ϕ is the spatial convolution of the population density with a kernel ϕ(y). We provide asymptotic approximations of the resulting travelling waves in various asymptotic limits of the wavespeed, the non-local interaction strength, and the moving boundary condition. Crucially, we find that when the moving boundary has a weak interactive strength with the population density flux, there can be two different travelling wave solutions that move at the same wavespeed.

Anomaly Detection for a Large Number of Streams: A Permutation-Based Higher Criticism Approach

Anomaly detection when observing a large number of data streams is essential in a variety of applications, ranging from epidemiological studies to monitoring of complex systems. High-dimensional scenarios are usually tackled with scan-statistics and related methods, requiring stringent modeling assumptions for proper calibration. In this work we take a nonparametric stance, and propose a permutation-based variant of the higher criticism statistic not requiring knowledge of the null distribution. This results in an exact test in finite samples which is asymptotically optimal in the wide class of exponential models. We demonstrate the power loss in finite samples is minimal with respect to the oracle test. Furthermore, since the proposed statistic does not rely on asymptotic approximations it typically performs better than popular variants of higher criticism that rely on such approximations. We include recommendations such that the test can be readily applied in practice, and demonstrate its applicability in monitoring the content uniformity of an active ingredient for a batch-produced drug product. Supplementary materials for this article are available online.

The generalized Gamma Matrix Function via Jordan Canonical Form and its Approximations

The generalized Gamma matrix function via Jordan canonical form are provided. A more general case of Beta matrix functions with two positive stable matrices are also obtained. Asymptotic approximations are derived for the Gamma matrix function with two positive stable matrices.

Contact stress induced by the rotation of a rigid bolt bonded to an elastic body

Torsional deformation of an elastic half-space containing a vertical cylindrical cavity caused by a rotation of a bonded rigid bolt with a long shank is studied in this article by employing Weber integral transforms and integral equations. In the limiting case of the infinite shank, we suggest two different techniques reducing the problem to the regular integral equations of the second kind, which permit efficient solutions in distinct ranges of geometric parameters and allow us to derive a highly accurate approximate analytic solution of the problem for any values of the parameters. Almost exact approximations are suggested for the stress concentration at the edge of the bolt head. In the case of a bolt with a long finite shank, relationships between the torque and the angle of twist of the bolt are derived by utilizing solutions obtained for the bolt with the infinite shank as asymptotic approximations. The effect of the geometric parameters is investigated providing insight into the behavior of the bolt and contact stress in dependence on the bolt head radius and the length of the shank.

Description of covalent bond in terms of generalized charges: potential and dissociation energy of homonuclear compound

The theory of generalized charges is an asymptotic approximation of quantum mechanics for interatomic forces. It is based on the model of multicomponent electron gas, which extends the Thomas–Fermi model of inhomogeneous electron gas to pair electronic states. The present research takes in consideration the participation of the field of generalized charges in interatomic bonds. Herein, the definition of generalized charges (GC) by means of the overlap integral of the wave functions of valence electrons is given, and properties of GC are found out. Equations are derived for the potential and length of a homopolar bond, and dependence of these parameters on the generalized charges is shown. The dissociation energy of a diatomic molecule is expressed in terms of the nuclear charges, the function and multiplicity of the covalent bond, the angular momentum, and the vibrational energy of the binding electrons. The parametrization rules are substantiated and an a priori calculation of the dissociation energy of diatomic homonuclear molecules is carried out against the electronic configuration of the bond electrons and the nuclear charge. The calculation results for homonuclear compounds of the elements of the first four rows of the periodic table are shown to be of satisfactory accuracy.

Orbital stability of smooth solitary waves for the Degasperis-Procesi equation

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space.

Non-parametric tests for serial dependence in time series based on asymptotic implementations of ordinal-pattern statistics.

Ordinal patterns can be used to construct non-parametric hypothesis tests that aim to discover (possibly non-linear) serial dependence in a real-valued time series. We derive the asymptotic distribution of the vector of sample frequencies of ordinal patterns and that of various corresponding tests statistics such that the targeted tests for serial dependence are easily implemented based on asymptotic approximations. Simulations are used to check the finite-sample performance of these tests as well as their power properties with respect to various alternative scenarios. The application and interpretation of the tests in practice are illustrated by an environmental data example.

A unified framework for linear thermo-visco-elastic wave propagation including the effects of stress-relaxation

We present a unified framework for the study of wave propagation in homogeneous linear thermo-visco-elastic (TVE) continua, starting from conservation laws. In free-space such media admit two thermo-compressional modes and a shear mode. We provide asymptotic approximations to the corresponding wavenumbers which facilitate the understanding of dispersion of these modes, and consider common solids and fluids as well as soft materials where creep compliance and stress relaxation are important. We further illustrate how commonly used simpler acoustic/elastic dissipative theories can be derived via particular limits of this framework. Consequently, our framework allows us to: (i) simultaneously model interfaces involving both fluids and solids and (ii) easily quantify the influence of thermal or viscous losses in a given configuration of interest. As an example, the general framework is appliedto the canonical problem of scattering from an interface between two TVE half spaces in perfect contact. To illustrate, we provide results for fluid–solid interfaces involving air, water, steel and rubber, paying particular attention to the effects of stress relaxation.

Asymptotic approximations for the phase space Schrödinger equation

We consider semi-classical time evolution for the phase space Schrödinger equation and present two methods of constructing short time asymptotic solutions. The first method consists of constructing a semi-classical phase space propagator in terms of semi-classical Gaussian wave packets on the basis of the anisotropic Gaussian approximation, related to the nearby orbit approximation, which is identified with the phase space representation of the Littlejohn semi-classical propagator. By means of the semi-classical propagator, we derive an asymptotic solution for configuration space WKB initial data. The second method consists of constructing a phase space narrow beam asymptotic solution, following the complex WKB theory developed by Maslov, on the basis of a canonical system in double phase space, analogous to the Berezin–Shubin–Marinov Hamilton–Jacobi and transport equations. We illustrate the methods for sub-quadratic potentials in .

On the soliton resolution and the asymptotic stability of N-soliton solution for the Wadati-Konno-Ichikawa equation with finite density initial data in space-time solitonic regions

In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data in space-time solitonic regions,iqt+(q1+|q|2)xx=0,q(x,0)=q0(x),limx→±∞⁡q0(x)=q±,q0(x)−q±∈H4,4(R). By analyzing the Lax spectrum problem, the solution of the original problem is convert to the solution of the corresponding Riemann-Hilbert problem for the first time with the initial boundary value condition. By developing the ∂¯-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution q(x,t) in soliton region of space-time, −(yΦ02t)=z0 for any z0∈R, and give bounds for the error decaying as |t|→∞. Based on the resulting asymptotic behavior, the asymptotic approximation of the WKI equation is characterized with the soliton term confirmed by N(I)-soliton on discrete spectrum and the t−12 leading order term on continuous spectrum with residual error up to O(t−34). Our results also confirm the soliton resolution conjecture and the asymptotic stability of N-soliton solution for the WKI equation with finite density initial data.

Meso-scale method of asymptotic analysis of elastic vibrations in periodic and non-periodic multi-structures

Summary The method of meso-scale asymptotic approximations has proved to be very effective for the analysis of models of solids containing large clusters of defects, such as small inclusions or voids. Here, we present a new avenue where the method is extended to elastic multi-structures. Geometrically, a multi-structure makes a step up in the context of overall dimensions, compared to the dimensions of its individual constituents. The main mathematical challenge comes from the analysis of the junction regions assigned to the multi-structure itself. Attention is given to problems of vibration and on the coupling of vibration modes corresponding to displacements of different orientations. The method is demonstrated through the dynamic analysis of infinite or finite multi-scale asymmetric flexural systems consisting of a heavy beam connected to a non-periodic array of massless flexural resonators within some interval. In modelling the interaction between the beam and the resonators, we derive a vectorial system of partial differential equations through which the axial and flexural motions of the heavy beam are coupled. The solution of these equations is written explicitly in terms of Green’s functions having intensities determined from a linear algebraic system. The influence of the resonators on the heavy beam is investigated within the framework of scattering and eigenvalue problems. For large collections of resonators, dynamic homogenization approximations for the medium within the location of the resonant array are derived, leading to (i) the classical Rayleigh beam for symmetric systems and (ii) a generalized Rayleigh beam for asymmetric structures that support flexural–longitudinal wave coupling. Independent numerical simulations are also presented that demonstrate the accuracy of the analytical results.