Asymptotic Calculations Research Articles

Nonlinear eigenvalue problems

This paper presents an asymptotic study of the differential equation y′(x) = cos [πxy(x)] subject to the initial condition y(0) = a. While this differential equation is nonlinear, the solutions to the initial-value problem bear a striking resemblance to solutions to the linear time-independent Schrödinger eigenvalue problem. As x increases from 0, y(x) oscillates and thus resembles a quantum wave function in a classically allowed region. At a critical value x = xcrit, where xcrit depends on a, the solution y(x) undergoes a transition; the oscillations abruptly cease and y(x) decays to 0 monotonically as x → ∞. This transition resembles the transition in a wave function at a turning point as one enters the classically forbidden region. Furthermore, the initial condition a falls into discrete classes; in the nth class of initial conditions an − 1 < a < an (n = 1, 2, 3, …), y(x) exhibits exactly n maxima in the oscillatory region. The boundaries an of these classes are the analogues of quantum-mechanical eigenvalues. An asymptotic calculation of an for large n is analogous to a high-energy semiclassical (WKB) calculation of eigenvalues in quantum mechanics. The principal result of this paper is that as n → ∞, , where A = 25/6. Numerical analysis reveals that the first Painlevé transcendent has an eigenvalue structure that is quite similar to that of the equation y′(x) = cos [πxy(x)] and that the nth eigenvalue grows with n like a constant times n3/5 as n → ∞. Finally, it is noted that the constant A is numerically very close to the lower bound on the power-series constant P in the theory of complex variables, which is associated with the asymptotic behavior of zeros of partial sums of Taylor series.

Remarks on asymptotic efficient estimation for regression effects in stationary and nonstationary models for panel count data

In a panel count data setup, repeated counts of an individual are assumed to be influenced by the individual’s random effect. Consequently, conditional on the random effect, the repeated responses of the individual are assumed to be serially correlated. Under the assumption that the random effects of the individuals follow a normal distribution, Jowaheer and Sutradhar (Statist. Probab. Letters 79 (2009) 1928–1934) have demonstrated that the generalized quasi-likelihood (GQL) estimation approach produces more efficient estimates than the so-called generalized method of moments (GMM) approach for both regression effects and the variance component of the normal random effects. For the cases where the distribution of the random effects is unknown, there exist two estimation approaches, namely the conditional maximum likelihood (CML) and instrumental variables based GMM (IVGMM) approaches, for the estimation of the regression effects. The purpose of this paper is to examine the asymptotic efficiency performances of the CML and IVGMM approaches as compared to the GQL approach for the regression estimation. When the covariates are stationary, that is, time independent, it is, however, known that the CML and IVGMM approaches are useless for the regression estimation, whereas the GQL approach does not encounter any such limitations. For the general case, that is, when the covariates are time dependent, the IVGMM approach appears to be computationally expensive and hence it is not included in efficiency comparison. Between the CML and GQL approaches, it is found through exact asymptotic variance calculations that the GQL approach is asymptotically more efficient than the CML approach in estimating the regression effects. This makes the GQL as a unified efficient approach irrespective of the cases whether the panel count data are stationary or nonstationary.

Variational Approach to Percolation Threshold of Nanocomposites Considering Clustering Effect

AbstractTo mathematically model the phenomenon of complete dispersion, a crucial step is taken in a recent paper on isotropic formulation of the homogenized Eshelby’s tensor, leading to derivation of the ellipsoidal bound. In this paper, the writers show that in the same variational framework, the effect of clustering can be mathematically modeled as anisotropy of the homogenized Eshelby’s tensor. The degree of clustering or dispersion is accordingly represented by the aspect ratio of the ellipsoidal shape associated with the anisotropic Eshelby’s tensor. The asymptotic results and calculation demonstrate that such a new anisotropic formulation consistently leads to an increase of the percolation threshold due to the clustering effect.

Nonlinear resonance interaction of waves on a jet in a radial electric field

Nonlinear analytic asymptotic second-order calculations show that the motion of a charged jet relative to a material medium results in the appearance of rise-time periodic wave motions of the interface (Kelvin-Helmholtz instabilities), as well as an internal nonlinear resonance interaction of waves, whose parameters (the intensity and the characteristic time of interaction) depend on the physical parameters of the system: the electric charge density at the jet, its velocity relative the medium, the mass density, the values of the wavenumbers of the interacting waves, and the value of the surface tension coefficient of the boundary.

On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood

The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead to biased estimates for the latent correlations, and the univariate marginal parameters that are not regression coefficients. We implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi-Monte Carlo methods. Asymptotic and small-sample efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate normal copula-based models. An illustrative example is given to show the use of our simulated likelihood method.

Destructive impact of imperfect beam collimation in extraordinary optical transmission

We investigate the difference between analytic predictions, numerical simulations, and experiments measuring the transmission of energy through subwavelength, periodically arranged holes in a metal film. At normal incidence, theory predicts a sharp transmission minimum when the wavelength is equal to the periodicity, and sharp transmission maxima at one or more nearby wavelengths. In experiments, the sharpest maximum from the theory is not observed, while the others appear less sharp. In numerical simulations using commercial electromagnetic field solvers, we find that the sharpest maximum appears and approaches our predictions as the computational resources are increased. To determine possible origins of the destruction of the sharp maximum, we incorporate additional features in our model. Incorporating imperfect conductivity and imperfect periodicity in our model leaves the sharp maximum intact. Imperfect collimation, on the other hand, incorporated into the model causes the destruction of the sharp maximum as happens in experiments. We provide analytic support through an asymptotic calculation for both the existence of the sharp maximum and the destructive impact of imperfect collimation.

Internal nonlinear resonance on a charged jet

Nonlinear analytical asymptotic calculations of the second order of smallness show that the motion of a charged jet in a material medium generates periodic wave motions of the jet-medium interface (Kelvin-Helmholtz instabilities), which grow in time. In addition, the motion of the jet gives rise to nonlinear internal resonance interaction of waves. The parameters of this interaction (intensity and characteristic time) depend on the physical parameters of the system: electric charge density of the jet, its velocity in the medium, mass density, wavenumbers of interacting waves, and the interfacial tension coefficient.

Impact of Multiple Matched Controls on Design Sensitivity in Observational Studies

In an observational study, one treated subject may be matched for observed covariates to either one or several untreated controls. The common motivation for using several controls rather than one is to increase the power of a test of no effect under the doubtful assumption that matching for observed covariates suffices to remove bias from nonrandom treatment assignment. Does the choice between one or several matched controls affect the sensitivity of conclusions to violations of this doubtful assumption? With continuous responses, it is known that reducing the heterogeneity of matched pair differences reduces sensitivity to unmeasured biases, but increasing the sample size has a highly circumscribed effect on sensitivity to bias. Is the use of several controls rather than one analogous to a reduction in heterogeneity or to an increase in sample size? The issue is examined for Huber's m-statistics, including the t-test, the examination having three components: an example, asymptotic calculations using design sensitivity, and a simulation. Use of multiple controls with continuous responses yields a nontrivial reduction in sensitivity to unmeasured biases. An example looks at lead and cadmium in the blood of smokers from the 2008 National Health and Nutrition Examination Survey. A by-product of the discussion is a new result giving the design sensitivity for the permutation distribution of m-statistics.

Asymptotic calculation of wave period statistics in non-Gaussian mixed sea

This article concerns the calculation of the wave period probability densities in non-Gaussian mixed sea states. The calculations are carried out by incorporating a second order nonlinear wave model into an asymptotic analysis method which is a novel approach to the calculation of wave period probability densities. Since all of the calculations are performed in the probability domain, the approach avoids long time-domain simulations. The accuracy and efficiency of the asymptotic analysis method for calculating the wave period probability densities are validated by comparing the results predicted using the method with those predicted by using the Monte-Carlo simulation (MCS) method.

Unsteady feeding and optimal strokes of model ciliates

AbstractThe flow field created by swimming micro-organisms not only enables their locomotion but also leads to advective transport of nutrients. In this paper we address analytically and computationally the link between unsteady feeding and unsteady swimming on a model micro-organism, the spherical squirmer, actuating the fluid in a time-periodic manner. We start by performing asymptotic calculations at low Péclet number ($\mathit{Pe}$) on the advection–diffusion problem for the nutrients. We show that the mean rate of feeding as well as its fluctuations in time depend only on the swimming modes of the squirmer up to order ${\mathit{Pe}}^{3/ 2} $, even when no swimming occurs on average, while the influence of non-swimming modes comes in only at order ${\mathit{Pe}}^{2} $. We also show that generically we expect a phase delay between feeding and swimming of $1/ 8\mathrm{th} $ of a period. Numerical computations for illustrative strokes at finite $\mathit{Pe}$ confirm quantitatively our analytical results linking swimming and feeding. We finally derive, and use, an adjoint-based optimization algorithm to determine the optimal unsteady strokes maximizing feeding rate for a fixed energy budget. The overall optimal feeder is always the optimal steady swimmer. Within the set of time-periodic strokes, the optimal feeding strokes are found to be equivalent to those optimizing periodic swimming for all values of the Péclet number, and correspond to a regularization of the overall steady optimal.

The impact of covariance misspecification in multivariate Gaussian mixtures on estimation and inference: an application to longitudinal modeling.

Multivariate Gaussian mixtures are a class of models that provide a flexible parametric approach for the representation of heterogeneous multivariate outcomes. When the outcome is a vector of repeated measurements taken on the same subject, there is often inherent dependence between observations. However, a common covariance assumption is conditional independence-that is, given the mixture component label, the outcomes for subjects are independent. In this paper, we study, through asymptotic bias calculations and simulation, the impact of covariance misspecification in multivariate Gaussian mixtures. Although maximum likelihood estimators of regression and mixing probability parameters are not consistent under misspecification, they have little asymptotic bias when mixture components are well separated or if the assumed correlation is close to the truth even when the covariance is misspecified. We also present a robust standard error estimator and show that it outperforms conventional estimators in simulations and can indicate that the model is misspecified. Body mass index data from a national longitudinal study are used to demonstrate the effects of misspecification on potential inferences made in practice.

Asymptotic calculation of the dynamics of self-sustained detonations in condensed phase explosives

AbstractWe use the weak-curvature, slow-time asymptotics of detonation shock dynamics (DSD) to calculate an intrinsic relation between the normal acceleration, the normal velocity and the curvature of a lead detonation shock for self-sustained detonation waves in condensed phase explosives. The formulation uses the compressible Euler equations for an explosive that is described by a general equation of state with multiple reaction progress variables. The results extend an earlier asymptotic theory for a polytropic equation of state and a single-step reaction rate model discussed by Kasimov (Theory of instability and nonlinear evolution of self-sustained detonation waves. PhD thesis, University of Illinois Urbana-Champaign, Urbana, Illinois) and by Kasimov &amp; Stewart (Phys. Fluids, vol. 16, 2004, pp. 3566–3578). The asymptotic relation is used to study the dynamics of ignition events in solid explosive PBX-9501 and in porous PETN powders. In the case of porous or powdered explosives, two composition variables are used to represent the extent of exothermic chemical reaction and endothermic compaction. Predictions of the asymptotic formulation are compared against those of alternative DSD calculations and against shock-fitted direct numerical simulations of the reactive Euler equations.

Testing the fit of the logistic model for matched case-control studies

With numerous statistical packages being easily available to conduct the logistic regression analysis, assessment for the goodness-of-fit in the logistic case-control studies becomes more important in practice. While various methods for model checking in conventional case-control studies have been proposed in the literature, methods for checking model adequacy with matched case-control data get relatively less attention. In this study, we propose an omnibus goodness-of-fit test to assess adequacy of the conditional logistic model for matched case-control data. The proposed test can be either constructed based on the discrepancy between two moment estimations or derived to be a score-type test under a general random-effects model. Computation of the proposed test is quite simple in which it does not need to partition the covariate space or to estimate p-value of the test via simulations. The asymptotic null distribution and power calculation of the test are derived under a sequence of alternatives. Empirical type I error rates and powers of the test are performed by simulation studies. An example has been used to illustrate the proposed method as well.

Derivative Moments for Characteristic Polynomials from the CUE

We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity. The latter asymptotic calculation allows us to prove a long-standing conjecture from random matrix theory.

Propagating two-dimensional magnetic droplets

Propagating, solitary magnetic wave solutions of the Landau–Lifshitz equation with uniaxial, easy-axis anisotropy in thin (two-dimensional) magnetic films are investigated. These localized, nontopological wave structures, parametrized by their precessional frequency and propagation speed, extend the stationary, coherently precessing “magnon droplet” to the moving frame, a non-trivial generalization due to the lack of Galilean invariance. Propagating droplets move on a spin wave background with a nonlinear droplet dispersion relation that yields a limited range of allowable droplet speeds and frequencies. An iterative numerical technique is used to compute the propagating droplet’s structure and properties. The results agree with previous asymptotic calculations in the weakly nonlinear regime. Furthermore, an analytical criterion for the droplet’s orbital stability is confirmed. Time-dependent numerical simulations further verify the propagating droplet’s robustness to perturbation when its frequency and speed lie within the allowable range.

Nonlinear analysis of governing laws of realization of the wave motion on the surface of a charged jet moving with respect to a material medium

On the base of analytic asymptotic calculations which are quadratic with respect to the ratio of the wave amplitude and the jet radius it is shown that the presence of a tangential jump in the velocity field on the jet surface leads to generation of a periodic wave motion on the interface between the media and has the destabilizing effect for both axisymmetric and bending and bending-deformation waves. It is found that there is a degenerate internal nonlinear resonance interaction between waves on the jet surface. This interaction may be of six different types in which the energy can be transferred between the interacting waves including waves of different symmetry. In the last case the energy is transferred from waves determining the initial deformation to axisymmetric waves.

Asymptotic Calculation of the Intensity of Millimeter Wave Propagation over an Undulating Surface Using the Diffraction Integral with a High-Degree Phase Function

Asymptotic Calculation of the Intensity of Millimeter Wave Propagation over an Undulating Surface Using the Diffraction Integral with a High-Degree Phase Function

The equiconvergence of the eigenfunction expansion for a singular version of one-dimensional Schrodinger operator with explosive factor

This paper is devoted to prove the equiconvergence formula of the eigenfunction expansion for some version of Schrodinger operator with explosive factor. The analysis relies on asymptotic calculation and complex integration. The paper is of great interest for the community working in the area.

A fiducial approach to multiple comparisons

Comparing treatment means from populations that follow independent normal distributions is a common statistical problem. Many frequentist solutions exist to test for significant differences amongst the treatment means. A different approach would be to determine how likely it is that particular means are grouped as equal. We developed a fiducial framework for this situation. Our method provides fiducial probabilities that any number of means are equal based on the data and the assumed normal distributions. This methodology was developed when there is constant and non-constant variance across populations. Simulations suggest that our method selects the correct grouping of means at a relatively high rate for small sample sizes and asymptotic calculations demonstrate good properties. Additionally, we have demonstrated the flexibility in the methods ability to calculate the fiducial probability for any number of equal means. This was done by analyzing a simulated data set and a data set measuring the nitrogen levels of red clover plants that were inoculated with different treatments.

Weighted scores method for regression models with dependent data

There are copula-based statistical models in the literature for regression with dependent data such as clustered and longitudinal overdispersed counts, for which parameter estimation and inference are straightforward. For situations where the main interest is in the regression and other univariate parameters and not the dependence, we propose a "weighted scores method", which is based on weighting score functions of the univariate margins. The weight matrices are obtained initially fitting a discretized multivariate normal distribution, which admits a wide range of dependence. The general methodology is applied to negative binomial regression models. Asymptotic and small-sample efficiency calculations show that our method is robust and nearly as efficient as maximum likelihood for fully specified copula models. An illustrative example is given to show the use of our weighted scores method to analyze utilization of health care based on family characteristics.