Asymptotic Calculations Research Articles

Does accounting for gene‐environment (G×E) interaction increase the power to detect the effect of a gene in a multifactorial disease?

Despite tremendous efforts, few genes involved in the susceptibility for complex disorders have been identified. One explanation is that these disorders are a result of an interaction between genes and environment, and under such conditions, it may be difficult to measure the true genetic effect without accounting for the interaction. Umbach and Weinberg ([2000] Am. J. Hum. Genet. 66:251-261) proposed an association test which looks at the joint effects of genotype and environment, using case-parent trios. In this study, we explore under which conditions accounting for GxE interaction enhances one's ability to detect the role of genetic factors in complex diseases. Using asymptotic power calculations, we investigate the power to detect the gene effect over varying exposure frequencies and different scenarios of GxE interaction. We show that for a given sample size, interaction scenario, and allele frequency, the actual gain in power while accounting for the interaction depends on the magnitude of the exposure frequency: the largest gains are seen for relatively low exposure frequencies. Moreover, a loss of power can be observed when the exposure is frequent and/or the exposure effect is strong. If we consider a gene with a disease allele frequency of 0.2, with no effect in the absence of exposure, an exposure with a 10-fold increase risk and a GxE relative risk of 2, then when the exposure frequency is 0.1, accounting for GxE interaction increases the power to detect the gene effect in 200 trios by 10%; alternatively, when the exposure frequency is 0.9, it decreases the power by 15%.

The ‘Indian wire trick’ via parametric excitation: a comparison between theory and experiment

A seemingly paradoxical experiment is described whereby a length of wire is stabilized upside down by vertical periodic oscillation of its support. The experimental results reveal an upper and a lower bound on the excitation frequency for stability. The results of recent theories are presented and used to explain the essential details of the observations. The theory relies on a novel phenomenon of so–called resonancetongue interaction. The result is verified via asymptotic calculations based on a one–dimensional rod model and numerical results on a spatially discretized system of links. This gravity–defying effect has potential application to the stabilization of other spatially extended systems via parametric excitation.

On blest's measure of rank correlation

AbstractBlest (2000) proposed a new nonparametric measure of correlation between two random variables. His coefficient, which is dissymmetric in its arguments, emphasizes discrepancies observed among the first ranks in the orderings induced by the variables. The authors derive the limiting distribution of Blest's index and suggest symmetric variants whose merits as statistics for testing independence are explored using asymptotic relative efficiency calculations and Monte Carlo simulations.

The van der Waals interactions of concentric nanotubes

Concentric fullerene tubes (nanotubes) are stabilized by a competition between the short-range kinetic-energy repulsion and the long-range van der Waals binding. At the relevant binding distances (3–3.5 Å or 6–7 Bohr radii) the traditional first-principle density-functional theory calculations cannot account for these combined interactions. An accurate quantum-physics account of the multi-walled nanotube structure and dynamics must therefore also include a determination of the intertube van der Waals forces. We use a successful model of the electrodynamical response of the nanotube electron gas to provide such a combined description in which we reflect the important self-consistent screening effects arising within the nanotube electron gases. Our description differs significantly from traditional asymptotic calculations of the van der Waals interactions and offers an approximate determination at the relevant intertube binding distances.

Evaluating the Results of Genomewide Linkage Scans of Complex Traits by Locus Counting

The evaluation of results from primary genomewide linkage scans of complex human traits remains an area of importance and considerable debate. Apart from the usual assessment of statistical significance by use of asymptotic and empirical calculations, an additional means of evaluation--based on counting the number of distinct regions showing evidence of linkage--is possible. We have explored the characteristics of such a locus-counting method over a range of experimental conditions typically encountered during genomewide scans for complex trait loci. Under the null hypothesis, factors that have an impact on the informativeness of the data--such as map density, availability of parental data, and completeness of genotyping--are seen to markedly influence the number of regions of excess allele sharing and the empirically derived genomewide significance of the associated LOD score thresholds. In some circumstances, the expected number of regions is less than one-quarter of that predicted under the assumption of a dense map and complete extraction of inheritance information. We have applied this method to a previously analyzed data set--the Warren 2 genome scan for type 2-diabetes susceptibility--and demonstrate that more regions showing evidence for linkage were observed in the primary genome scan than would be expected by chance, across the whole range of LOD scores, even though no single linkage result achieved empirical genomewide statistical significance. Locus counting may be useful in assessing the results from genome scans for complex traits in general, especially because relatively few scans generate evidence for linkage reaching genomewide significance by dense-map criteria. By taking account of the effects of reduced data informativeness on the expected number of regions showing evidence for linkage, a more meaningful, and less conservative, evaluation of the results from such linkage studies is possible.

On the Critical Solutions in Coating and Rimming Flow on a Uniformly Rotating Horizontal Cylinder

We use a combination of analytical and numerical techniques to re-examine the question posed by Moffatt [Journal de Mae#169;canique 16 (1977) 651-673] of determining the critical weights of fluid that can be maintained per unit length in a steady, smoothly varying, two-dimensional film on either the outside ('coating flow') or the inside ('rimming flow') of a rotating horizontal cylinder. We use a pseudospectral method to obtain highly accurate numerical solutions for steady Stokes flow on a cylinder and hence to calculate the critical weights. In particular, these calculations reveal that the behaviour of the critical solutions in the thin-film limit 0 (where is the aspect ratio of the film) in an inner region near the horizontal on the ascending side of the cylinder (where Moffatt's leading-order outer solution has a corner) are not captured by naive outer asymptotic solutions in integer powers of . Motivated by these numerical results we obtain the uniformly valid critical asymptotic solutions in the thin-film limit to sufficient accuracy to enable us to calculate the critical fluxes and weights to accuracies o(4/3 (log )-3) and o(4/3 (log )-2) relative to Moffatt's leading-order values, respectively. We find that our asymptotic solutions for the critical weights are in good agreement with the numerically calculated results over a wide range of values of . In particular, our numerical and asymptotic calculations show that, even in the absence of surface-tension effects, the corner predicted by Moffatt's leading-order outer solution never actually occurs. In practice the higher-order terms obtained in the present work dominate the formally lower-order term that can be obtained straightforwardly without a detailed knowledge of the solution in the inner region, and so these higher-order terms must be included in order to obtain accurate corrections to Moffatt's leading-order value of the critical weight. In particular, in practice the critical weights in both coating and rimming flow always exceed Moffatt's value.

The viscous damping of capillary-gravity waves in a brimful circular cylinder

The natural frequency and damping rate of surface capillary-gravity waves in a circular cylinder with pinned-end edge conditions and clean surfaces depend on the inverse gravitational Reynolds number C, the Bond number B, the aspect ratio of the cylinder Λ, and the radial and azimuthal wave numbers of each excited mode. A semianalytical method valid for arbitrary values of C, B, and Λ is used. It is shown that the damping rate has a weak dependence on the Bond number and also that the oscillations become overdamped for large values of C or low values of Λ. The results are compared with recent asymptotic calculations for small viscosity (which include viscous dissipation in the boundary layers and in the bulk), showing that the approximations yield good results if C≲10−3 and Λ≳0.1, for all values of B. The results are also compared with experimental measurements (which include values of C larger than 10−3), showing a very good agreement.

State-Space Based Approach to Particle Creation in Spatially Uniform Electric Fields

Our formalism, described recently in (C. E. Dolby and S. F. Gull, Ann. Phys.293 (2001), 189–214.) is applied to the study of particle creation in spatially uniform electric fields, concentrating on the cases of a time-invariant electric field and a so-called adiabatic electric field. Several problems are resolved by incorporating the Bogoliubov coefficient approach and the tunnelling approach into a single consistent, gauge invariant formulation. The value of a time-dependent particle interpretation is demonstrated by presenting a coherent account of the time-development of the particle creation process, in which the particles are created with small momentum (in the frame of the electric field) and are then accelerated by the electric field to make up the bulge of created particles predicted by asymptotic calculations [2, 3]. An initial state comprising one particle is also considered, and its evolution is described as being the sum of two contributions: the sea of current produced by the evolved vacuum and the extra current arising from the initial particle state.

Motion of a nonrectilinear fiber in a viscous fluid flow

The plane problem of dynamic interaction of a laminar viscous fluid flow and an inextensible pliable fiber of finite length is solved using the perturbation method. Two types of rheological two‐dimensional flows — pure shear and simple shear — are considered. Formulas are obtained for the evolution of the tensile force and the shape of the fiber. Results of asymptotic and numerical calculations are compared.

Axisymmetric pulse recycling and motion in bulk semiconductors.

The Kroemer model for the Gunn effect in a circular geometry (Corbino disks) has been numerically solved. The results have been interpreted by means of asymptotic calculations. Above a certain onset dc voltage bias, axisymmetric pulses of the electric field are periodically shed by an inner circular cathode. These pulses decay as they move towards the outer anode, which they may not reach. As a pulse advances, the external current increases continuously until a new pulse is generated. Then the current abruptly decreases, in agreement with existing experimental results. Depending on the bias, more complex patterns with multiple pulse shedding are possible.

Comparative study of air-core and coaxial Bragg fibers: single-mode transmission and dispersion characteristics

Using an asymptotic formalism we developed in an earlier paper, we compare the dispersion properties of the air-core Bragg fiber with those of the coaxial Bragg fiber. In particular we are interested in the way the inner core of the coaxial fiber influence the dispersion relation. It is shown that, given appropriate structural parameters, large single-mode frequency windows with a zero-dispersion point can be achieved for the TM mode in coaxial fibers. We provide an intuitive interpretation based on perturbation analysis and the results of our asymptotic calculations are confirmed by Finite Difference Time Domain (FDTD) simulations.

Flow past a sphere undergoing unsteady rectilinear motion and unsteady drag at small Reynolds number

The flow induced by a sphere which undergoes unsteady motion in a Newtonian fluid at small Reynolds number is considered at distances large compared with sphere radius a. Previous solutions of the unsteady Oseen equations (Ockendon 1968; Lovalenti & Brady 1993b) for rectilinear motion are refined. Three-dimensional Fourier transforms of the disturbance field are integrated over Fourier space to derive new concise equations for the velocity field and history force in terms of single history integrals.Various slip-velocity profiles are classified by the ratio A of the particle relative displacement, z′p(t′) − z′p(τ′), to the diffusion length, l′D = 2[v(t′ − τ′)]1/2, where v is the kinematic viscosity of the fluid. Most previous studies are concerned with large-displacement motions for which the ratio is large in the long-time limit. It is shown using asymptotic calculations that the flow at any point at large distance z past a sphere for arbitrary large-displacement and non-reversing motion is the same as the steady-state laminar wake if z is expressed in terms of the time elapsed since the particle was at that point in the laboratory frame. The point source solution for the remainder of the far flow is also valid for the unsteady case.A start-up motion with slip velocity V′p = γ′(t′)−1/2, t′ > 0, is investigated for which A is finite. A self-similar solution for the flow field is obtained in terms of space coordinates scaled by the diffusion length, u′ = auss(η)/t′ where η = r′/2(vt′)1/2. The unsteady Oseen correction to the drag is inversely proportional to time.When A is small in the long-time limit (a small-displacement motion) the flow field also depends on the space coordinates in terms of η. The distribution of the streamwise velocity uz is symmetrical in z.

Nonlinear dielectric and magnetic relaxation after a sudden change of applied field

A dilute suspension of particles with permanent electric or magnetic dipole moment is subjected to an applied electric or magnetic field which suddenly changes its value. The relaxation of polarization or magnetization to the new equilibrium value is calculated from the solution of the Smoluchowski equation for the orientational distribution function. It is assumed that the new field has the same or opposite direction as the old one. For reversal of a strong field, comparison is made with an asymptotic calculation of van Kampen and with the solution of an approximate relaxation equation proposed by Martsenyuk et al.

Unmasking the Trend Sought by Jonckheere‐Type Tests for Right‐Censored Data

Medical and epidemiological studies often involve groups of subjects associated with increasing levels of exposure to a risk factor. Survival of the groups is expected to follow the same order as the level of exposure. Formal tests for this trend fall into the regression framework if one knows what function of exposure to use as a covariate. When unknown, a linear function of exposure level is often used. Jonckheere‐type tests for trend have generated continued interest largely because they do not require specification of a covariate. This paper shows that the Jonckheere‐type test statistics are special cases of a generalized linear rank statistic with time‐dependent covariates which unfortunately depend on the initial group sizes and censoring distributions. Using asymptotic relative efficiency calculations, the Jonckheere tests are compared to standard linear rank tests based on a linear covariate over a spectrum of shapes for the true trend.

FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES

We describe a robust correction to Black-Scholes American derivatives prices that accounts for uncertain and changing market volatility. It exploits the tendency of volatility to cluster, or fast mean-reversion, and is simply calibrated from the observed implied volatility skew. The two-dimensional free-boundary problem for the derivative pricing function under a stochastic volatility model is reduced to a one-dimensional free-boundary problem (the Black-Scholes price) plus the solution of a fixed boundary-value problem. The formal asymptotic calculation that achieves this is presented here. We discuss numerical implementation and analyze the effect of the volatility skew.

Upper branch nonstationary modes of the boundary layer due to a rotating disk

A treatment of asymptotic calculation of upper branch nonstationary instability modes is undertaken in the boundary layer flow due to a rotating disk. A numerical spectral solution of the eigenvalue problem shows good agreement with the results of a rational asymptotic approach, based on the extension of the multideck theory of [1].

Penalized likelihood inference in extreme value analyses

Models for extreme values are usually based on detailed asymptotic argument, for which strong ergodic assumptions such as stationarity, or prescribed perturbations from stationarity, are required. In most applications of extreme value modelling such assumptions are not satisfied, but the type of departure from stationarity is either unknown or complex, making asymptotic calculations unfeasible. This has led to various approaches in which standard extreme value models are used as building blocks for conditional or local behaviour of processes, with more general statistical techniques being used at the modelling stage to handle the non-stationarity. This paper presents another approach in this direction based on penalized likelihood. There are some advantages to this particular approach: the method has a simple interpretation; computations for estimation are relatively straightforward using standard algorithms; and a simple reinterpretation of the model enables broader inferences, such as confidence intervals, to be obtained using MCMC methodology. Methodological details together with applications to both athletics and environmental data are given.

Asymptotic analysis and finite element calculation of a rubber notch contacting with a rigid wedge

In this paper, the strain energy function given by Knowles-Sternberg in 1973 is used, and the contact problem of a rubber notch with a rigid wedge is analyzed. The basic equations of the deformation field near the notch corner are derived and solved. An analytical solution is obtained for the expanding sector while the numerical solution is given for two shrinking sectors. A special interesting case is that a half rubber-like plane contacts with a rigid wedge, for which the completely analytical solutions are obtained for both expanding sector and shrinking sectors. The analysis of this paper is also valid for the contact problem of a rubber wedge with a rigid wedge. To verify the analytical results, a finite element program of nonlinear elasticity is made with which the same problem is calculated, and the results are consistent with the analytical results very well.

Asymptotic analysis and finite element calculation of a rubber wedge under tension

The problem of a cone under tension of a concentrated load at its tip is investigated by adopting the constitutive equation of rubber-like materials given by Knowles and Sternberg (1973). The problem was treated as axial-symmetry case, and large deformation was taken into account. The asymptotic solution to the stress-strain field near the apex of the cone is obtained and solved analytically. By means of the finite element method, the stress-strain field is also calculated. The numerical results are consistent well with that of the analysis.

Microwave-Frequency Modulation of the Gasdynamic Parameters of Plasma by Laser Radiation

Numerical and asymptotic calculations are performed of the gasdynamic parameters of laser-induced plasma under the effect of high-power laser radiation with microwave-frequency modulation of its intensity. The results of these calculations are compared for the amplitudes of oscillation of these parameters. It is demonstrated that the values of the parameters may be found using the averaging method and represented in the form of the sum of two parts, of which one part is the solution in the absence of modulation, and the other part describes small-scale oscillations about this solution.