Terms Of Asymptotic Expansion Research Articles

Space–time structure of weak magnetohydrodynamic turbulence

The two-time energy spectrum of weak magnetohydrodynamic turbulence is found by applying a wave-turbulence closure to the cumulant hierarchy constructed from the dynamical equations. Solutions are facilitated via asymptotic expansions in terms of the small parameter $\varepsilon$ , describing the ratio of time scales corresponding to Alfvénic propagation and nonlinear interactions between counter-propagating Alfvén waves. The strength of nonlinearity at a given spatial scale is further quantified by an integration over all possible delta-correlated modes compliant in a given set of three-wave interactions that are associated with energy flux through the said scale. The wave-turbulence closure for the two-time spectrum uncovers a secularity occurring on a time scale of order $\varepsilon ^{-2}$ , and the asymptotic expansion for the spectrum is reordered in a manner comparable to the one-time case. It is shown that for the regime of stationary turbulence, the two-time energy spectrum exponentially decays on a lagged time scale $(\varepsilon ^2 \gamma _k^s)^{-1}$ in proportion to the strength of the associated three-wave interactions, characterized by nonlinear decorrelation frequency $\gamma _k^s$ . The scaling of the form $k_{\perp } v_0 \chi _0$ exhibited by this frequency is reminiscent of random sweeping by the outer scale with characteristic fluctuation velocity $v_0$ that is modified due to competition with Alfvénic propagation (characterized by $\chi _0$ ) at the said scale. A brief calculation of frequency broadening of the power spectrum due to nonlinear interactions is also presented.

Anderson Localization in the Subwavelength Regime

In this paper, we use recent breakthroughs in the study of coupled subwavelength resonator systems to reveal new insight into the mechanisms responsible for the fundamental features of Anderson localization. The occurrence of strong localization in random media has proved difficult to understand, particularly in physically derived multi-dimensional models and systems with long-range interactions. We show here that the scattering of time-harmonic waves by high-contrast resonators with randomly chosen material parameters reproduces the characteristic features of Anderson localization. In particular, we show that the hybridization of subwavelength resonant modes is responsible for both the repulsion of energy levels as well as the widely observed phase transition, at which point eigenmode symmetries swap and very strong localization is possible. We derive results from first principles, using asymptotic expansions in terms of the material contrast parameter and obtain a characterization of the localized modes in terms of generalized capacitance matrices. This model captures the long-range interactions of the wave-scattering system and provides a concise framework to explain the exotic phenomena that are observed.

Forced convective heat transfer for Stokes flow including viscous dissipation in arbitrary corrugated channels

AbstractThe forced convective heat transfer for Stokes flow, including viscous dissipation in arbitrary corrugated channels, is studied using both an asymptotic method and a numerical solution. The aim is to specify the range of shape parameters for which the validity of the asymptotic approach is ensured, particularly regarding the characteristics of heat transfer. The axial velocity, transversal velocity, pressure, and temperature, for small corrugation's slope compared with unity, are sought as an asymptotic expansion in terms of a parameter that represents the corrugation's slope. The numerical solution is obtained by using ANSYS Fluent solver. Additionally, Python scripting is integrated to automate several parts of the simulations, including the creation of the geometry and a parametric study. Three different types of corrugations are investigated including zigzag, sinusoidal, and arbitrary corrugations defined using a function given by a particular case of the Fourier series. The Nusselt number is calculated to evaluate convective heat transfer. It is found that the asymptotic and numerical solutions for small corrugation's slope, are in good agreement with negligible quantitative differences. However, as the corrugation's slope increases (approaches unity), these quantitative differences increase up to cases where a change in the behavior of the local Nusselt number is observed. The results show that the local Nusselt number decreases in the channel's region with divergent walls due to the decrease in the average velocity. In contrast, it increases in the channel's region with convergent walls due to the increase in the average velocity.

Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior

As the most significant difference from parabolic equations, long-time or short-time behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique determination has been frequently investigated in literature. Unlike all the existing results, in this article we prove the uniqueness of orders and parameters (up to a multiplier for the latter) only by principal terms of asymptotic expansions of solutions near t = 0 at a single spatial point. Moreover, we discover special conditions on unknown initial values or source terms for the coincidence of observation data. As a byproduct, we can even conclude the uniqueness for initial values or source terms by the same data. The proof relies on the asymptotic expansion after taking the Laplace transform and the completeness of generalized eigenfunctions.

Some modifications of Getzler's grading technique

This paper reviews the grading technique developed by Getzler to prove the local index theorem and shows how to adapt it to compute the leading terms of asymptotic expansions of traces of heat kernels in two other situations.

Mathematical analysis of electromagnetic scattering by dielectric nanoparticles with high refractive indices

In this work, we are concerned with the mathematical modeling of the electromagnetic (EM) scattering by arbitrarily shaped non-magnetic nanoparticles with high refractive indices. When illuminated by visible light, such particles can exhibit a very strong isotropic magnetic response, resulting from the coupling of the incident wave with the circular displacement currents of the EM fields. The main aim of this work is to mathematically illustrate this phenomenon. We shall first introduce the EM scattering resolvent and the concept of dielectric subwavelength resonances. Then we derive the a priori estimates for the subwavelength resonances and the associated resonant modes. We also show the existence of resonances and obtain their asymptotic expansions in terms of the small particle size and the high contrast parameter. After that, we investigate the enhancement of the scattering amplitude and the cross sections when the resonances occur. In doing so, we develop a novel multipole radiation framework that directly separates the electric and magnetic multipole moments and allows us to clearly see their orders of magnitude and blow-up rates. We prove that at the dielectric subwavelength resonant frequencies, the nanoparticles with high refractive indices behave like the sum of the electric dipole and the resonant magnetic dipole. Some explicit calculations and numerical experiments are also provided to validate our general results and formulas.

An asymptotic expansion for a Lambert series associated to the symmetric square L-function

Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetric square [Formula: see text]-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function [Formula: see text]. Later, Chakraborty et al. extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series [Formula: see text] where [Formula: see text] is the [Formula: see text]th Fourier coefficient of a Hecke eigenform [Formula: see text] of weight [Formula: see text] over the full modular group.

Asymptotic analysis of long‐term investment with two illiquid and correlated assets

AbstractWe consider a long‐term portfolio choice problem with two illiquid and correlated assets, which is associated with an eigenvalue problem in the form of a variational inequality. The eigenvalue and the free boundaries implied by the variational inequality correspond to the portfolio's optimal long‐term growth rate and the optimal trading strategy, respectively. After proving the existence and uniqueness of viscosity solutions for the eigenvalue problem, we perform an asymptotic expansion in terms of small correlations and obtain semi‐analytical approximations of the free boundaries and the optimal growth rate. Our leading order expansion implies that the free boundaries are orthogonal to each other at four corners and have C1 regularity. We propose an efficient numerical algorithm based on the expansion, which proves to be accurate even for large correlations and transaction costs. Moreover, following the approximate trading strategy, the resulting growth rate is very close to the optimal one.

Acoustic streaming in the cochlea

This presentation examines the fluid motion resulting from airborne and bone conduction excitation in the cochlea. The time-averaged velocity, termed acoustic streaming, driven by the gradient of the Reynolds stress, is of particular interest. The effect of asymmetry in the oval and the round window velocity and its influence pressure gradient across the cochlear partition and acoustic streaming is explored. It is shown that the streaming flow is a function of the reciprocal of the Strouhal, 1/S and the oscillatory Reynolds number, 1/R, have a magnitude much less than one for frequencies of interest. The work presented will focus on the parameter range R/S2 = O(1). The streaming velocity is obtained in terms of asymptotic expansion in terms of the reciprocal of the Strouhal number. Solution of the streaming flow outside the viscous region is obtained using boundary conditions derived from a matched asymptotic analysis.

The closures of test configurations and algebraic singularity types

Given a Kähler manifold X with an ample line bundle L, we consider the metric space of finite energy geodesic rays associated to the Chern class c1(L). We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems.By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge–Ampère energy of a general finite energy ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations.Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge–Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from L1 rays to non-Archimedean rays.

Lambert series associated to Hilbert modular form

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

Computation of the reverse generalized Bessel polynomials and their zeros

It is well known that one of the most relevant applications of the reverse Bessel polynomials θ n ( z ) is filter design. In particular, the poles of the transfer function of a Bessel filter are basically the zeros of θ n ( z ) . In this article we discuss an algorithm to compute the zeros of reverse generalized Bessel polynomials θ n ( z ; a ) . A key ingredient in the algorithm will be a method to compute the polynomials. For this purpose, we analyze the use of recurrence relations and asymptotic expansions in terms of elementary functions to obtain accurate approximations to the polynomials. The performance of all the numerical approximations will be illustrated with examples.

Scattering diagrams from asymptotic analysis on Maurer–Cartan equations

Let $\check{X}_0$ be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration $\check{p}:\check{X}_0 \rightarrow B_0$. We investigate the asymptotic behavior of Maurer-Cartan solutions of the Kodaira-Spencer deformation theory on $\check{X}_0$ by expanding them into Fourier series along fibres of $\check{p}$ over a contractible open subset $U\subset B_0$, following a program set forth by Fukaya in 2005. We prove that semi-classical limits (i.e. leading order terms in asymptotic expansions) of the Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to consistent scattering diagrams, which are tropical combinatorial objects that have played a crucial role in works of Kontsevich-Soibelman and Gross-Siebert on the reconstruction problem in mirror symmetry.

Asymptotic Analysis of Long-Term Investment with Two Illiquid and Correlated Assets

We consider a long-term portfolio choice problem with two illiquid and correlated assets and formulate it as an eigenvalue problem in the form of a variational inequality. The eigenvalue is associated with the portfolio’s optimal long-term growth rate, and the free boundaries implied by the variational inequality correspond to the optimal trading strategy. After proving the existence and uniqueness of viscosity solutions for the eigenvalue problem, we perform an asymptotic expansion in terms of small correlations and obtain semi-analytical approximations of the free boundaries and the optimal growth rate. Our leading order expansion implies that the free boundaries are orthogonal to each other at four corners and have C1 regularity. We propose an efficient numerical algorithm based on the expansion, which proves to be accurate even for large correlations and transaction costs. Moreover, following the approximate trading strategy, the resulting growth rate is very close to the optimal one.

Connection formulae for asymptotics of the fifth Painlevé transcendent on the imaginary axis: I

AbstractLeading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as are found. These asymptotics are parameterized by monodromy data of the associated linear ODE, urn:x-wiley:00222526:media:sapm12323:sapm12323-math-0002The parameterization allows one to derive connection formulae for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulae are also considered.

Asymptotic relations involving 𝑑-orthogonal polynomials

Abstract In this paper, we consider a natural extension in the context of d-orthogonality for asymptotic analysis of orthogonal polynomials. We introduce, for several d-orthogonal polynomials, asymptotic expansions in terms of d-Hermite ones. From these expansions, several limits between d-orthogonal polynomials are obtained.

Local eigenvalue decomposition for embedded Riemannian manifolds

Local Principal Component Analysis can be performed over small domains of an embedded Riemannian manifold in order to relate the covariance analysis of the underlying point set with the local extrinsic and intrinsic curvature. We show that the volume of domains on a submanifold of general codimension, determined by the intersection with higher-dimensional cylinders and balls in the ambient space, have asymptotic expansions in terms of the mean and scalar curvatures. Moreover, we propose a generalization of the classical third fundamental form to general submanifolds and prove that the local eigenvalue decomposition (EVD) of the covariance matrices have asymptotic expansions that contain the curvature information encoded by the traces of this tensor. This proves the general correspondence between the local EVD integral invariants and differential-geometric curvature for arbitrary embedded Riemannian submanifolds, found so far for curves and hypersurfaces only. Thus, we establish a key theoretical bridge, via covariance matrices at scale, for potential applications in manifold learning relating the statistics of point clouds sampled from Riemannian submanifolds to the underlying geometry.

Weak inertial effects on arbitrarily shaped objects in the presence of a wall

Analytic approaches have been previously applied to compute the weak inertial lift on spheres above a plane wall by using an asymptotic expansion in terms of the Reynolds number. By building on these approaches and combining the method of regularized Stokeslets, a numerical method that computes the weak inertial lift on arbitrarily shaped objects in the presence of a plane wall is developed.

Asymptotic analysis of the expected utility maximization problem with respect to perturbations of the numéraire

In an incomplete model, where under an appropriate numéraire, the stock price process is driven by a sigma-bounded semimartingale, we investigate the behavior of the expected utility maximization problem under small perturbations of the numéraire. We establish a quadratic approximation of the value function and a first-order expansion of the terminal wealth. Relying on a description of the base return process in terms of its semimartingale characteristics, we also construct wealth processes and nearly optimal strategies that allow for matching the primal value function up to the second order. We also link perturbations of the numéraire to distortions of the finite-variation part and martingale part of the stock price return and characterize the asymptotic expansions in terms of the risk-tolerance wealth process.

Temperature Distribution to Fluid Transport in a Vertical Porous Tube under the Action of Peristalsis

Peristaltic transport of a Newtonian fluid, with heat transfer, in a vertical porous axisymmetric tube under long wave length approximation is considered. Closed form solution is obtained as an asymptotic expansion in terms of porosity and free convection parameters. Expressions for temperature, coefficient of heat transfer and pressure -flow relationship at the boundary wall of the tube are derived. It is observed that pressure drop increases as amplitude ratio increases. Further, it has been observed that for some specific values of otherparameters under consideration the mean flux significantly increases by about 8 to 10 percent as Grashof number increasesfrom 1 to 2.