Terms Of Asymptotic Expansion Research Articles

Reconstruction of missing data using iterative harmonic expansion

In the cosmic microwave background or galaxy density maps, missing fluctuations in masked regions can be reconstructed from fluctuations in the surrounding unmasked regions if the original fluctuations are sufficiently smooth. One reconstruction method involves applying a harmonic expansion iteratively to fluctuations in the unmasked region. In this paper, we discuss how well this reconstruction method can recover the original fluctuations depending on the prior of fluctuations and property of the masked region. The reconstruction method is formulated with an asymptotic expansion in terms of the size of mask for a fixed iteration number. The reconstruction accuracy depends on the mask size, the spectrum of the underlying density fluctuations, the scales of the fluctuations to be reconstructed and the number of iterations. For Gaussian fluctuations with the Harrison--Zel'dovich spectrum, the reconstruction method provides more accurate restoration than naive methods based on brute--forth matrix inversion or the singular value decomposition. We also demonstrate that an isotropic non-Gaussian prior does not change the results but an anisotropic non-Gaussian prior can yield a higher reconstruction accuracy compared to the Gaussian prior case.

A Higher Order Asymptotic Expansion of the Krawtchouk Polynomials

The paper extends a classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide a uniform asymptotic expansion in terms of Hermite polynomials and obtain explicit expressions for a few first terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation. Bibliography: 10 titles.

A Photoelastic Study for Multiparametric Analysis of the Near Crack Tip Stress Field Under Mixed Mode Loading

The study is aimed at theoretical, experimental and computational determination of the coefficients in crack tip asymptotic expansions for a wide class of specimens under mixed mode loading conditions. Multiparametric presentation of the stress filed near the crack tips for a wide class of specimens is given. Theoretical, experimental and computational results obtained in this research show that the isochromatic fringes in the vicinity of the crack tip require to keep the higher order stress terms in the asymptotic expansion of the stress field around the crack tip since the contribution of the higher order stress terms (besides the stress intensity factors and the T-stress) is not negligible in the crack tip stress field. One can see that the higher order terms of the asymptotic expansion are important when the stress distribution has to be known also farther from the crack tip and it is necessary to extend the domain of validity of the Williams solution. It is shown that at large distances from the crack tips the effect of the higher order terms of the Williams series expansion becomes more considerable. The knowledge of more terms of the stress asymptotic expansions will allow us to approximate the stress field near the crack tips with high accuracy.

Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter

By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in the first part of this paper for the Lam\'e and Mathieu functions with a large real parameter. These approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid in their respective real open intervals. In all cases explicit bounds are supplied for the error terms associated with the approximations. Approximations are also obtained for the large order behaviour for the respective eigenvalues. We restrict ourselves to a two term uniform approximation. Theoretically more terms in these approximations could be computed, but the coefficients would be very complicated. In the second part of this paper we use a simplified method to obtain uniform asymptotic expansions for these functions. The coefficients are just polynomials and satisfy simple recurrence relations. The price to pay is that these asymptotic expansions hold only in a shrinking interval as their respective parameters become large; this interval however encapsulates all the interesting oscillatory behaviour of the functions. This simplified method also gives many terms in asymptotic expansions for these eigenvalues, derived simultaneously with the coefficients in the function expansions. We provide rigorous realistic error bounds for the function expansions when truncated and order estimates for the error when the eigenvalue expansions are truncated. With this paper we confirm that many of the formal results in the literature are correct.

Asymptotic expansions and numerical simulations of I-V relations via a steady state Poisson-Nernst-Planck system

A steady state Poisson-Nernst-Planck (PNP) system is studied both analytically and numerically with particular attention on I-V relations of ion channels. Assuming the dielectric constant $\varepsilon$ is small, the PNP system can be viewed as a singularly perturbed system. Due to the special structures of the zeroth order inner and outer systems, one is able to derive more explicit expressions of higher order terms in asymptotic expansions. For the case of zero permanent charge, under the assumption of electro-neutrality at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion species is that the third order correction is \textit{cubic} in $V$, and, furthermore (Theorem \ref{3rd order}), up to the third order, the cubic I-V relation has \textit{three distinct real roots} (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Three numerical experiments are conducted to check the cubic-like feature of the I-V curve, study the boundary value effect on the I-V relation and investigate the permanent charge effect on the I-V curve, respectively.

Models of discretized moduli spaces, cohomological field theories, and Gaussian means

We prove combinatorially the explicit relation between genus filtrated s -loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces M g , s disc given by N g , s ( P 1 , … , P s ) for ( P 1 , … , P s ) ∈ Z + s . This generating function therefore enjoys the topological recursion, and we prove that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition. We use another Givental-type decomposition obtained for this model by the second authors in 1995 in terms of special times related to the discretization of moduli spaces thus representing its asymptotic expansion terms (and therefore those of the Gaussian means) as finite sums over graphs weighted by lower-order monomials in times thus giving another proof of (quasi)polynomiality of the discrete volumes. As an application, we find the coefficients in the first subleading order for M g , 1 in two ways: by using the refined Harer–Zagier recursion and by exploiting the above Givental-type transformation. We put forward the conjecture that the above graph expansions can be used for probing the reduction structure of the Deligne–Mumford compactification M ¯ g , s of moduli spaces of punctured Riemann surfaces.

Eigenvibrations of thick cascade junctions with `very heavy' concentrated masses

We study small-parameter asymptotics of eigenelements of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses. There are five qualitatively different cases in the asymptotic behaviour of eigenvalues and eigenfunctions as the small parameter tends to zero (`light', `intermediate', `slightly heavy', `intermediate heavy' and `very heavy' concentrated masses). We study the influence of concentrated masses on the asymptotics of eigenvibrations in the last two cases. We construct the leading terms of asymptotic expansions for eigenfunctions and eigenvalues and rigorously justify them by appropriate asymptotic estimates. We also find new types of high-frequency eigenvibrations.

Asymptotic form of the matrix elements of the direct collision integral in the Boltzmann equation

The asymptotic form is investigated for large indices of the matrix elements of the collision integral in the Boltzmann equation, which are the expansion coefficients of the collision integral in spherical Hermitian polynomials in the isotropic case for power interaction potentials. It is established that the principal term of the asymptotic expansion can be presented as the product of the power function of one of the indices by the homogeneous function of the ratios of the indices with the degree of homogeneity determined by the exponent in the interaction potential. The principal term of asymptotic expansion, which is the same for all ratios of the matrix element indices, is obtained for the matrix elements of the integral of direct collisions.

Convergent and asymptotic expansions of the Pearcey integral

We consider the Pearcey integral P(x,y) for large values of |x|, x,y∈C. We can find in the literature several convergent or asymptotic expansions in terms of elementary and special functions, with different levels of complexity. Most of them are based on analytic, in particular asymptotic, techniques applied to the integral definition of P(x,y). In this paper we consider a different method: the iterative technique used for differential equations in López, 2012. Using this technique in a differential equation satisfied by P(x,y) we obtain a new convergent expansion analytically simple that is valid for any complex x and y and has an asymptotic property when |x|→∞ uniformly for y in bounded sets. The accuracy of the approximation is illustrated with some numerical experiments and compared with other expansions given in the literature.

Stochastic Perturbations of Periodic Orbits with Sliding

Vector fields that are discontinuous on codimension-one surfaces are known as Filippov systems and can have attracting periodic orbits involving segments that are contained on a discontinuity surface of the vector field. In this paper we consider the addition of small noise to a general Filippov system and study the resulting stochastic dynamics near such a periodic orbit. Since a straight-forward asymptotic expansion in terms of the noise amplitude is not possible due to the presence of discontinuity surfaces, in order to quantitatively determine the basic statistical properties of the dynamics, we treat different parts of the periodic orbit separately. Dynamics distant from discontinuity surfaces is analyzed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analyzed through stochastic averaging methods. The influence of noise on points at which the periodic orbit escapes a discontinuity surface is determined by zooming into the transition point. We combine the results to quantitatively determine the effect of noise on the oscillation time for a three-dimensional canonical model of relay control. For some parameter values of this model, small noise induces a significantly large reduction in the average oscillation time. By interpreting our results geometrically, we are able to identify four features of the relay control system that contribute to this phenomenon.

Quantifying nuisance parameter effects via decompositions of asymptotic refinements for likelihood-based statistics

Accurate inference on a scalar interest parameter in the presence of a nuisance parameter may be obtained using an adjusted version of the signed root likelihood ratio statistic, in particular Barndorff-Nielsen’s R∗ statistic. The adjustment made by this statistic may be decomposed into a sum of two terms, interpreted as correcting respectively for the possible effect of nuisance parameters and the deviation from standard normality of the signed root likelihood ratio statistic itself. We show that the adjustment terms are determined to second-order in the sample size by their means. Explicit expressions are obtained for the leading terms in asymptotic expansions of these means. These are easily calculated, allowing a simple way of quantifying and interpreting the respective effects of the two adjustments, in particular of the effect of a high dimensional nuisance parameter. Illustrations are given for a number of examples, which provide theoretical insight to the effect of nuisance parameters on parametric inference. The analysis provides a decomposition of the mean of the signed root statistic involving two terms: the first has the property of taking the same value whether there are no nuisance parameters or whether there is an orthogonal nuisance parameter, while the second is zero when there are no nuisance parameters. Similar decompositions are discussed for the Bartlett correction factor of the likelihood ratio statistic, and for other asymptotically standard normal pivots.

Novel Heavy-Traffic Regimes for Large-Scale Service Systems

We introduce a family of heavy-traffic regimes for large-scale service systems, presenting a range of scalings that include both moderate and extreme heavy traffic, as compared to classical heavy traffic. The heavy-traffic regimes can be translated into capacity sizing rules that lead to economies-of-scales, so that the system utilization approaches 100% while congestion remains limited. We obtain heavy-traffic approximations for stationary performance measures in terms of asymptotic expansions, using a nonstandard saddle point method, tailored to the specific form of integral expressions for the performance measures, in combination with the heavy-traffic regimes.

Discrete Bartlett-Type Transformed Statistics Using a Discontinuous Term of Asymptotic Expansion

Discrete Bartlett-Type Transformed Statistics Using a Discontinuous Term of Asymptotic Expansion

Thermal influence on charge carrier transport in solar cells based on GaAs PN junctions

The electron and hole one-dimensional transport in a solar cell based on a Gallium Arsenide (GaAs) PN junction and its dependency with electron and lattice temperatures are studied here. Electrons and heat transport are treated on an equal footing, and a cell operating at high temperatures using concentrators is considered. The equations of a two-temperature hydrodynamic model are written in terms of asymptotic expansions for the dependent variables with the electron Reynolds number as a perturbation parameter. The dependency of the electron and hole densities through the junction with the temperature is analyzed solving the steady-state model at low Reynolds numbers. Lattice temperature distribution throughout the device is obtained considering the change of kinetic energy of electrons due to interactions with the lattice and heat absorbed from sunlight. In terms of performance, higher values of power output are obtained with low lattice temperature and hot energy carriers. This modeling contributes to improve the design of heat exchange devices and thermal management strategies in photovoltaic technologies.

A singular element for reissner plate bending problem with V-shaped notches

A novel singular finite element is presented to study Reissner plate bending problem with V-shaped notches. Firstly, the expressions of the first four order asymptotic displacement field around the tip of V-shaped notch are derived systematically using William’s eigenfunction series expansion method. Subsequently, the expressions are used to construct a novel displacement mode singular finite element, which can well depict the characteristic of singular stress field around the tip of V-shaped notch having arbitrary opening angle. Combining with conventional finite elements, the novel element can be applied to solve Reissner plate bending problem with V-shaped notches, and mode I, mode II and mode III stress intensity factors can all be determined directly by the coefficients of the asymptotic expansion terms. Finally, four numerical examples are performed to demonstrate the performance of the present method, and the influences of the size and number of export nodes of the singular element on the calculation of bending stress intensity factors are also discussed. Numerical results show that the singular element method is a practical and effective numerical technique for obtaining directly and synchronously mode I, mode II and mode III stress intensity factors without other numerical techniques, such as extrapolation.

Trace Estimates for Relativistic Stable Processes

In this paper, we study the asymptotic behavior, as the time t goes to zero, of the trace of the semigroup of a killed relativistic α-stable process in bounded C 1,1 open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of t of the trace with an error bound of order t 2/α t −d/α for C 1,1 open sets and of order t 1/α t −d/α for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders t −d/α and t (2−d)/α for C 1,1 open sets, and, when α ∈ (0, 1), between the orders t −d/α and t (1−d)/α for Lipschitz open sets.

The perturbation of electromagnetic fields at distances that are large compared with the object's size

Abstract We rigorously derive the leading-order terms in asymptotic expansions for the scattered electric and magnetic fields in the presence of a small object at distances that are large compared with its size. Our expansions hold for fixed wavenumber when the scatterer is a (lossy) homogeneous dielectric object with constant material parameters or a perfect conductor. We also derive the corresponding leading-order terms in expansions for the fields for a low-frequency problem when the scatterer is a non-lossy homogeneous dielectric object with constant material parameters or a perfect conductor. In each case, we express our results in terms of polarization tensors.

Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis

We consider the uniform asymptotics of polynomials orthogonal on (0, ∞) with respect to the exponential weight w(x) = x α e −Q(x) ,w here α> − 1a ndQ(x) is a polynomial with positive leading coefficient. In this paper, we have obtained two types of asymptotic expansions in terms of Laguerre polynomials and elementary functions for z in different overlapping regions, respectively. These two regions together cover the whole complex plane. Our approach is based on the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295-368).

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ϵ‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ϵ → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ϵ → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.

Coulomb-distorted plane wave: Partial wave expansion and asymptotic forms

Partial wave expansion of the Coulomb-distorted plane wave is determined and studied. Dominant and sub-dominant asymptotic expansion terms are given and leading order three-dimensional asymptotic form is derived. The generalized hypergeometric function 2F2(a, a; a + l + 1, a − l; z) is expressed with the help of confluent hypergeometric functions and the asymptotic expansion of 2F2(a, a; a + l + 1, a − l; z) is simplified.