Asymptotic Expansion Research Articles

Uniform Asymptotic Expansions for the Zeros of Bessel Functions

Uniform Asymptotic Expansions for the Zeros of Bessel Functions

Finite-Time Nonconvex Optimization Using Time-Varying Dynamical Systems

In this paper, we study the finite-time convergence of the time-varying dynamical systems for solving convex and nonconvex optimization problems in different scenarios. We first show the asymptotic convergence of the trajectories of dynamical systems while only requiring convexity of the objective function. Under the Kurdyka–Łojasiewicz (KL) exponent of the objective function, we establish the finite-time convergence of the trajectories to the optima from any initial point. Making use of the Moreau envelope, we adapt our finite-time convergent algorithm to solve weakly convex nonsmooth optimization problems. In addition, we unify and extend the contemporary results on the KL exponent of the Moreau envelope of weakly convex functions. A dynamical system is also introduced to find a fixed point of a nonexpansive operator in finite time and fixed time under additional regularity properties. We then apply it to address the composite optimization problems with finite-time and fixed-time convergence.

Modelling collective invasion with reaction–diffusion equations: When does domain curvature matter?

Real-world cellular invasion processes often take place in curved geometries. Such problems are frequently simplified in models to neglect the curved geometry in favour of computational simplicity, yet doing so risks inaccuracies in any model-based predictions. To quantify the conditions under which neglecting a curved geometry is justifiable, we explore the dynamics of a system of reaction–diffusion equations (RDEs) on a two-dimensional annular geometry analytically. Defining ϵ as the ratio of the annulus thickness δ and radius r0 we derive, through an asymptotic expansion, the conditions under which it is appropriate to ignore the domain curvature for a general system of reaction–diffusion equations. To highlight the consequences of these results, we simulate solutions to the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) model, a paradigm nonlinear RDE typically used to model spatial invasion, on an annular geometry. Thus, we quantify the size of the deviation from an analogous simulation on the rectangle, and how this deviation changes across the width of the annulus. We further characterise the nature of the solutions through numerical simulations for different values of r0 and δ. Our results provide insight into when it is appropriate to neglect the domain curvature in studying travelling wave behaviour in RDEs.

The Asymptotic Expansion of the Spacetime Metric at the Event Horizon

AbstractHawking’s local rigidity theorem, proven in the smooth setting by Alexakis-Ionescu-Klainerman, says that the event horizon of any stationary non-extremal black hole is a non-degenerate Killing horizon. In this paper, we prove that the full asymptotic expansion of any smooth vacuum metric at a non-degenerate Killing horizon is determined by the geometry of the horizon. This gives a new perspective on the black hole uniqueness conjecture. In spacetime dimension 4, we also prove an existence theorem: Given any non-degenerate horizon geometry, Einstein’s vacuum equations can be solved to infinite order at the horizon in a unique way (up to isometry). The latter is a gauge invariant version of Moncrief’s classical existence result, without any restriction on the topology of the horizon. In the real analytic setting, the asymptotic expansion is shown to converge and we get well-posedness of this characteristic Cauchy problem.

Globalization of convergence of the constrained piecewise Levenberg–Marquardt method

We develop linesearch algorithms intended for globalization of convergence of the piecewise Levenberg–Marquardt method for constrained piecewise smooth equation. Conditions ensuring global convergence properties and asymptotic superlinear convergence rate are proposed. The peculiarities of the global convergence results in the piecewise smooth case are discussed and illustrated by examples. We also provide numerical results for unconstrained and constrained reformulations of nonlinear complementarity problems, comparing the performance of the globalized piecewise Levenberg–Marquardt algorithm with some relevant alternatives.

Cooperative Load Transportation with Quadrotors using Adaptive RISE Control

The problem of transporting cable-suspended payloads using two quadrotors is analyzed in this work. The system kinematics are treated using a virtual structure formation controller, which generates the acceleration references commanded to the aerial vehicles. The disturbances caused by the payload are treated as unmodeled disturbances and a novel robust integral of the sign of the error (RISE) controller is proposed, guaranteeing the asymptotic convergence of the tracking errors. A model reference adaptive control is also incorporated into the RISE controller, combining the advantages of adaptive and robust control. The proposal is validated by numerous experiments using two quadrotors to transport a bar-shaped payload in adverse conditions. The results allow the conclusion that the proposed system is capable of performing transportation and orientation tasks with the payload, subject to translational accelerations up to 3.5 m/s2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document}.

Asymptotic Expansions for the Radii of Starlikeness of Normalized q-Bessel Functions

In this paper we study the asymptotic behavior of the radii of starlikeness of the normalized Jackson’s second and third q-Bessel functions, focusing on their large orders. To achieve this, we use the Rayleigh sums of positive zeros of both q-Bessel functions, and determine the coefficients of the asymptotic expansions. We derive complete asymptotic expansions for these radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums of positive zeros of q-Bessel functions, Kvitsinsky’s results on spectral zeta functions for q-Bessel functions and asymptotic inversion. Moreover, we derive bounds for the radius of starlikeness of q-Bessel functions by using potential polynomials.

Dragging of the particle spin and spin–spin coupling effect on its periapsis shift

Abstract The periapsis shift (PS) of spinning test particles in the equatorial plane of an arbitrary stationary and axisymmetric spacetime is studied using the post-Newtonian method. The result is expressed as a half-integer power series of $M/p$, where $M$ is the spacetime mass and $p$ is the semilatus rectum. The coefficients of the series are polynomials of the particle spin, the asymptotic expansion coefficients of the metric functions, and the eccentricity of the orbit. The particle spin is shown to have a similar effect as the Lense-Thirring (LT) effect on the PS, and both of them appear from the $(M/p)^{-3/2}$ order in the PS. The coupling between the spacetime and particle spins will increase (or decrease) the PS if they are parallel (or antiparallel). For Jupiter and Saturn rotating around the Sun and exceptionally designed satellites around Mercury and Moon, and the space-based gravitational wave observatories LISA and TaiJi around the Sun, the particle spin effect can be comparable to or even larger than the LT one in size. The PS in other spacetime studied are is not distinguishable from that in the Kerr spacetime to the $(M/p)^{-3/2}$ order.

Laminar boundary layers in three‐dimensional MHD natural convective flow past a semi‐infinite vertical plate with variable sinusoidal suction

AbstractThe primary goal of this study is to investigate how the diffusion‐thermo and thermo‐diffusion affect a three‐dimensional hydromagnetic natural convective flow of a radiating fluid past a uniformly moving isothermal and isosolutal semi‐infinite vertical plate subjected to a sinusoidal suction velocity. The asymptotic series expansion method is adopted to solve the equations governing the flow model and to find the analytical solutions. The effect of variable suction on the flow and transport characteristics is mainly highlighted in the study. Both the diffusion‐thermo and thermo‐diffusion effects improve the fluid velocity.

Transient Dynamics in Fast–Slow Predator–Prey Model with Monod–Haldane Functional Response and Asymptotic Behaviors

This paper studies the transient dynamics in a fast–slow ecological model with a Monod–Haldane functional response. Geometric singular perturbation theory is utilized to examine the transient dynamics generated by intrinsic multiple time scales. An asymptotic series solution is derived for the normally hyperbolic sub-manifold in the slow flow. The paper proves the existence of relaxation oscillations near fold points and the existence of a smooth family of canard cycles arising from folded singularities. Furthermore, it investigates the resulting canard explosion and transient dynamics numerically. By incorporating the Monod–Haldane functional response, which represents the inhibitory effect on predation at high prey population levels, this study illustrates the occurrence of transient dynamics under this mechanism.

Two-Term Asymptotics of the Exchange Energy of the Electron Gas on Symmetric Polytopes in the High-Density Limit

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.

Stabilization of fractional nonlinear systems with disturbances via sliding mode control

AbstractIn this article, the sliding mode control (SMC) of fractional nonlinear systems (FNSs) with disturbances is studied. First, a useful fractional power‐rate inequality for is extended to a more general form . Based on the generalized inequality, the method of a modified fractional integral SMC is completed, in which the parameter of the symbolic function is extended to and . This increases the degree of freedom of the sliding mode surface (SMS). Then, the SMC of FNSs is studied for the cases of known and unknown disturbances. In the case of known disturbance, it is proved by the generalized inequality and quadratic Lyapunov function method that the state of the FNSs can converge asymptotically to zero on the SMS. For the case of unknown disturbance, a fractional disturbance observer is used to estimate the disturbance by introducing auxiliary variables . The disturbance estimation error of the proposed FNSs is proved to be bounded. An equivalent global control law is obtained and asymptotic convergence of the FNSs under the action of the controller is proved. Finally, two numerical simulation examples are given to verify the feasibility of the method proposed in this article.

Asymptotic expansion for the Fourier coefficients associated with the inverse of the modular discriminant function $$\Delta $$

Asymptotic expansion for the Fourier coefficients associated with the inverse of the modular discriminant function $$\Delta $$

Nanoparticle uptake by a semi-permeable, spherical cell from an external planar diffusive field. II. Numerical study of temporal and spatial development validated using FEM

In this paper, we present a mathematical study of particle diffusion inside and outside a spherical biological cell that has been exposed on one side to a propagating planar diffusive front. The media inside and outside the spherical cell are differentiated by their respective diffusion constants. A closed form, large-time, asymptotic solution is derived by the combined means of Laplace transform, separation of variables, and asymptotic series development. The solution process is assisted by means of an effective far-field boundary condition, which is instrumental in resolving the conflict of planar and spherical geometries. The focus of the paper is on a numerical comparison to determine the accuracy of the asymptotic solution relative to a fully numerical solution obtained using the finite element method. The asymptotic solution is shown to be highly effective in capturing the dynamic behaviour of the system, both internal and external to the cell, under a range of diffusive conditions.

Two types of series expansions valid at strong coupling

It is known that perturbative expansions in powers of the coupling in quantum mechanics (QM) and quantum field theory (QFT) are asymptotic series. This can be useful at weak coupling but fails at strong coupling. In this work, we present two types of series expansions valid at strong coupling. We apply the series to a basic integral as well as a QM path integral containing a quadratic and quartic term with coupling constant λ. The first series is the usual asymptotic one, where the quartic interaction is expanded in powers of λ. The second series is an expansion of the quadratic part where the interaction is left alone. This yields an absolutely convergent series in inverse powers of λ valid at strong coupling. For the basic integral, we revisit the first series and identify what makes it diverge even though the original integral is finite. We fix the problem and obtain, remarkably, a series in powers of the coupling which is absolutely convergent and valid at strong coupling. We explain how this series avoids Dyson’s argument on convergence. We then consider the QM path integral (discretized with time interval divided into N equal segments). As before, the second series is absolutely convergent and we obtain analytical expressions in inverse powers of λ for the nth order terms by taking functional derivatives of generalized hypergeometric functions. The expressions are functions of N and we work them out explicitly up to third order. The general procedure has been implemented in a Mathematica program that generates the expressions at any order n. We present numerical results at strong coupling for different values of N starting at N = 2. The series matches the exact numerical value for a given N (up to a certain accuracy). The continuum is formally reached when N → ∞ but in practice this can be reached at small N.

Image-Based Visual Servoing of Manipulators With Unknown Depth: A Recurrent Neural Network Approach.

The image-based visual servoing (IBVS) of manipulators is important for intelligent manipulation using visual feedbacks. While the traditional IBVS methods for manipulators require the knowledge of the depth information in the interaction matrix, in this article, we propose a novel IBVS method for manipulators without depth estimation by leveraging the property of the associated image Jacobian. Because of a novel transformation, the IBVS problem is converted into a convex optimization problem subject to the kinematic constraint, joint constraints, and other constraints that are not explicitly related to the depth information. The problem is then solved by developing a recurrent neural network of global asymptotic convergence, and a dynamic neural control law without depth estimation emerges for the IBVS of manipulators. The theoretical guarantee and simulation results are provided to show the efficacy of the proposed method.

Variable Selection in Semi-Functional Partially Linear Regression Models with Time Series Data

This article investigates a variable selection method in semi-functional partially linear regression (SFPLR) models for strong α-mixing functional time series data. We construct penalized least squares estimators for unknown parameters and unknown link functions in our models. Under some regularity assumptions, we establish the asymptotic convergence rate and asymptotic distribution for the proposed estimators. Furthermore, we make a comparison of our variable selection method with the oracle method without variable selection in simulation studies and an electricity consumption data analysis. Simulation experiments and real data analysis results indicate that the variable selection method performs well at extracting the primary information and reducing dimensionality.

Population Dynamics of Nile Tilapia (Oreochromis niloticus, Linnaeus.1758) from Khashm El-Girba Reservoir, Atbara River, Eastern Sudan

This study was conducted to assess the population parameters of Nile tilapia, Oreochromis niloticus, in the Khashm El-Girba Dam Reservoir in Eastern Sudan. Fish samples were collected from December 2019 to November 2020. Length-frequency data were collected from 1,671 specimens ranging from 5.1 to 34.2 cm in total length (TL). Analysis of the population parameters was determined using the ELEFAN I routine and the FiSAT II software computer program revealed that the estimated the Von Bertalanffy growth parameters were as follows: asymptotic length (L∞ = 36.75 cm), growth rate (K = 0.650 year -1) and age at length zero (t0 = -0.629) and the growth performance index (Φ') was calculated at 2.943. The instantaneous total mortality (Z), natural mortality (M), and fishing mortality (F) rates were determined using the length-converted catch curve and empirical models. The values obtained were Z = 2.38 year -1, M = 1.22 year -1, and F = 1.16 year -1, respectively. The exploitation rate (E) of the Nile tilapia was calculated at 0.49 year -1. The size at first capture (Lc) was estimated at 15.64 cm, which is slightly greater than the size at first maturity (Lm = 14.5 cm) and longevity (Tmax) was found to be 4.62 years. These findings indicate that the stock of the Nile tilapia population in Khashm El-Girba reservoir is overexploited, and that monitoring and management practices need to be employed to maintain the stocks at the optimum exploitation level.

Analytic NNLO QCD corrections to top quark pair production in electron-positron collisions

We present the analytic total cross section of top quark pair production in electron-positron annihilation at next-to-next-to-leading order (NNLO) in Quantum Chromodynamics (QCD). By utilizing the optical theorem, the NNLO corrections are related to the imaginary parts of three-loop self-energy Feynman diagrams, of which the master integrals are calculated with canonical differential equations. The analytic results for the NNLO corrections are expressed in terms of multiple polylogarithms as well as elliptic functions. We discuss the asymptotic expansions near the threshold and in the high energy limit in detail. Numerical results are provided for the total cross section of top quark pair production at future lepton colliders.

Asymptotics of the partition function of the perturbed Gross–Witten–Wadia unitary matrix model

AbstractWe consider the asymptotics of the partition function of the extended Gross–Witten–Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a ‐function sequence of the Painlevé equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta‐function and the Barnes ‐function. A third‐order phase transition in the leading terms of the asymptotic expansions is also observed.