Two-term Expansion Research Articles

An L 2 Comparison Between the Bootstrap and the Empirical Edgeworth Expansion

The Bootstrap estimate for studentized statistics is more accurate than both the normal approximation and the two-term empirical Edgeworth expansion. In this article, it will be shown that the three-term empirical Edgeworth expansion for studentized statistics compares well with the bootstrap. It is also shown that the three-term Edgeworth expansion is superior to the bootstrap in some cases, using more efficient estimators than sample moments in the Edgeworth expansion, such as using maximum likelihood estimators in the one-parameter exponential family.

Asymptotics for the Expected Lifetime of Brownian Motion on Thin Domains in ℝ n

We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in ℝn, and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime. The approach is similar to that which we used previously to study the eigenvalues of the Dirichlet Laplacian and consists of scaling the domain in one direction and deriving the corresponding asymptotic expansions as the scaling parameter goes to zero. Apart from being dominated by the one-dimensional Brownian motion along the direction of the scaling, we also see that the symmetry of the perturbation plays a role in the expansion.

Two-term Edgeworth expansions of the distributions of fit indexes under fixed alternatives in covariance structure models

Asymptotic expansions of the distributions of thirteen fit indexes used in covariance structure analysis in practice are obtained. The fit indexes include the usual log likelihood ratio statistic for a posited model and the functions of this statistic and the corresponding statistic of the so-called baseline model of uncorrelated observed variables. The results are derived by the two-term Edgeworth expansion under fixed alternatives for possibly nonnormally distributed data. A numerical example using a misspecified factor analysis model is shown to see the behavior of the asymptotic results in finite samples.

Multi-displacement continuum model for discrete systems

A first-order multi-displacement microstructure continuum model is introduced to represent a discrete diatomic lattice system. This model is developed based on a two-term Taylor series expansion of the local displacement of the lattice. It is found that the multi-displacement continuum model obtained by keeping two terms in the Taylor series yields, in general, a better representation of the lattice system than the effective modulus model. However, this microstructure continuum model cannot characterize the negative group velocity of an optical mode of harmonic wave motion in the diatomic lattice. To capture the negative group velocity, a higher-order multi-displacement continuum model is necessary.

Correlating data: examples from turbulent wall layers

Flow fields in fluid mechanics often contain regions where different physical events occur. The relative size of the regions changes as a parameter is varied. Correlating field data, either physical or DNS calculation experiments, in these situations can be aided by using ideas from matched asymptotic expansions from applied mathematics. A second situation is when two slightly different processes occur in the same spatial region. For this case, a two-term asymptotic expansion is needed. This article discusses how composite expansions and the common part matching behavior are useful in correlating data.

Optimizing the principal eigenvalue of the Laplacian in a sphere with interior traps

The method of matched asymptotic expansions is used to calculate a two-term asymptotic expansion for the principal eigenvalue λ ( ε ) of the Laplacian in a three-dimensional domain Ω with a reflecting boundary that contains N interior traps of asymptotically small radii. In the limit of small trap radii ε → 0 , this principal eigenvalue is inversely proportional to the average mean first passage time (MFPT), defined as the expected time required for a Brownian particle undergoing free diffusion, and with a uniformly distributed initial starting point in Ω , to be captured by one of the localized traps. The coefficient of the second-order term in the asymptotic expansion of λ ( ε ) is found to depend on the spatial locations of the traps inside the domain, together with the Neumann Green’s function for the Laplacian. For a spherical domain, where this Green’s function is known analytically, the discrete variational problem of maximizing the coefficient of the second-order term in the expansion of λ ( ε ) , or correspondingly minimizing the average MFPT, is studied numerically by global optimization methods for N ≤ 20 traps. Moreover, the effect on the average MFPT of the fragmentation of the trap set is shown to be rather significant for a fixed trap volume fraction when N is not too large. Finally, the method of matched asymptotic expansions is used to calculate the splitting probability in a three-dimensional domain, defined as the probability of reaching a specific target trap from an initial source point before reaching any of the other traps. Some examples of the asymptotic theory for the calculation of splitting probabilities are given for a spherical domain.

Scaling and correlation of vorticity fluctuations in turbulent channels

Data from direct numerical simulations have been correlated to examine the effects of Reynolds number on vorticity fluctuation profiles. Three different scalings are found: two in the inner region and the third in the outer region. In the inner region a two-term asymptotic expansion is required to represent the mean square vorticity profiles; ⟨ωω⟩#=⟨ωω⟩0#+⟨ωω⟩1+u∗/U0. The first scaling ⟨ωω⟩0#=⟨ωω⟩0/(u∗3U0/ν2) is for inactive motions that do not generate Reynolds shear stress. It applies to the streamwise and spanwise components. However, the zero-order term is zero for the normal vorticity component. The scrubbing of the inactive motions over the wall generates vorticity, which is a maximum at the wall, and diffuses out to about y+=50 before it decays. The fluctuating viscous wall shear stress is due entirely to this motion, and as a result the stress ratio (rms/mean) τ′/τ0=CU0/u∗ depends on the Reynolds number. The second scaling ⟨ωω⟩1+=⟨ωω⟩1/(u∗4/ν2), the same scaling as the Reynolds shear stress, is active motions. All three components participate in these motions, which are zero at the wall, rise to a peak at about y+=13–20, and fall to zero at about y+=400. This is consistent with the fact that the Reynolds shear stress is identically zero at the wall. The third scaling correlates data in the outer region using the Kolmogorov time scale ⟨ωω⟩/(u∗3/hν). Matching between the inner and outer regions has a common part (the equivalent of the log law) of ∼C/y+ or ∼C/Y for all three components. All components decrease further and become isotropic as the centerline is approached. Scaling with the Kolmogorov scale implies that dissipation is the important activity. The correlated data are fitted to simplified equations that can be formed into composite expansions to predict profiles at any Reynolds number.

Effective ionization coefficients and transport parameters in binary and ultradilute SF6–Ar mixtures using Boltzmann equation analysis

The effective ionization coefficients and transport parameters such as electron mean energy drift velocity and transverse diffusion coefficient in binary and ultradilute SF6–Ar gas mixtures have been calculated for density reduced electric field strength E/N values from 10 to 400 Td. These calculations have been performed by using the two-term spherical harmonic expansion to obtain the numerical solution of the Boltzmann transport equation based on the finite element method under steady-state Townsend condition. In order to confirm the model and code developed in this study, the Reid ramp model has been used as a benchmark test and then effective ionization coefficients and transport parameters have been evaluated for SF6 contents of 1%, 10%, 25%, 50%, 70% and 100% in the binary mixture. Finally SF6 contents in the ultradilute mixtures of 0.1%, 0.3%, 0.5% and 0.7% are taken into account with the evaluated effective ionizations and transport parameters of electron mean energy, drift velocity and transverse diffusion coefficients.

Circumferential Slot on a Charged Cylindrical Shell

An approximate solution to Laplace's equation in an axisymmetric geometry is constructed in terms of a pair of two simple functions for the potential in an aperture cut in an infinite conducting tube. The two halves of the tube are held at opposite potentials. Upon differentiation, one aperture potential function provides for a constant axial electric field, and the other recovers the expected edge-condition behavior at the knife-edge of a charged conductor. Continuity of potential and one simple integration, in the spirit of the generalized Ritz-Galerkin method, furnishes integral expressions for the required constants in the expansion. These integrals are evaluated using a combination of analytic regularization and quadrature. The validity of the two-term expansion is verified by comparison with an independent moment method calculation.

Fourier–Legendre expansion of the one-electron density matrix of ground-state two-electron atoms

The density matrix rho(r,r(')) of a spherically symmetric system can be expanded as a Fourier-Legendre series of Legendre polynomials P(l)(cos theta=rr(')rr(')). Application is here made to harmonically trapped electron pairs (i.e., Moshinsky's and Hooke's atoms), for which exact wavefunctions are known, and to the helium atom, using a near-exact wavefunction. In the present approach, generic closed form expressions are derived for the series coefficients of rho(r,r(')). The series expansions are shown to converge rapidly in each case, with respect to both the electron number and the kinetic energy. In practice, a two-term expansion accounts for most of the correlation effects, so that the correlated density matrices of the atoms at issue are essentially a linear functions of P(l)(cos theta)=cos theta. For example, in the case of Hooke's atom, a two-term expansion takes in 99.9% of the electrons and 99.6% of the kinetic energy. The correlated density matrices obtained are finally compared to their determinantal counterparts, using a simplified representation of the density matrix rho(r,r(')), suggested by the Legendre expansion. Interestingly, two-particle correlation is shown to impact the angular delocalization of each electron, in the one-particle space spanned by the r and r(') variables.

Higher Order Asymptotic Cumulants of Studentized Estimators in Covariance Structures

In covariance structure analysis, the Studentized pivotal statistic of a parameter estimator is often used since the statistic is asymptotically normally distributed with mean zero and unit variance. For more accurate asymptotic distribution, the first and third asymptotic cumulants can be used to have the single-term Edgeworth, Cornish-Fisher, and Hall type asymptotic expansions. In this paper, the higher order asymptotic variance and the fourth asymptotic cumulant of the statistic are obtained under nonnormality when the partial derivatives of a parameter estimator with respect to sample variances and covariances up to the third order and the moments of the associated observed variables up to the eighth order are available. The result can be used to have the two-term Edgeworth expansion. Simulations are performed to see the accuracy of the asymptotic results in finite samples.

Ab initio study of the torsional potential energy surfaces of N2O3 and N2O4: Origin of the torsional barriers

Intrinsic reaction coordinate (IRC) torsional potentials were calculated for N(2)O(4) and N(2)O(3) based on optimized B3LYP/aug-cc-pVDZ geometries of the respective 90 degrees -twisted saddle points. These potentials were refined by obtaining CCSD(T)aug-cc-pVXZ energies [in the complete basis set (CBS) limit] of points along the IRC. A comparison is made between these ab initio potentials and an analytical form based on a two-term cosine expansion in terms of the N-N dihedral angle. The shapes of these two potential curves are in close agreement. The torsional barriers in N(2)O(4) and N(2)O(3) obtained from the CCSD(T)/CBS//B3LYP/aug-cc-pVDZ calculations are 2333 and 1704 cm(-1), respectively. For N(2)O(4) the torsion fundamental frequency from the IRC potential is 87.06 cm(-1), which is in good agreement with the experimentally reported value of 81.73 cm(-1). However, in the case of N(2)O(3) the torsional frequency found from the IRC potential, 144 cm(-1), is considerably larger than the reported experimental values 63-76 cm(-1). Consistent with this discrepancy, the torsional barrier obtained from several different calculations, 1417-1718 cm(-1), is higher than the value of 350 cm(-1) deduced from experimental studies. It is suggested that the assignment of the torsional mode in N(2)O(3) should be reexamined. N(2)O(4) and N(2)O(3) exhibit strong hyperconjugative interactions of in-plane O lone pairs with the central N-N sigma* antibond. Hyperconjugative stabilization is somewhat stronger at the planar geometries because 1,4 interactions of lone pairs on cis O atoms promote delocalization of electrons into the N-N antibond. Calculations therefore suggest that the torsional barriers in these molecules arise principally from a combination of 1,4 interactions and hyperconjugation.

Simulation of electron kinetics in gas discharges

We review the state-of-the-art for the simulation of electron kinetics in gas discharges based on the numerical solution of the Boltzmann equation. The reduction of the six-dimensional Boltzmann equation to a four-dimensional Fokker-Planck equation using two-term spherical harmonics expansion enables efficient and accurate simulation of the electron distribution function in collisional gas discharge plasmas. We illustrate this approach in application to inductively coupled plasmas, capacitively coupled plasmas, and direct current glow discharges. The incorporation of the magnetic field effect into this model is outlined. We also describe recent efforts towards simulating collisionless effect in gas discharge plasma based on Vlasov solvers and outline our views on future development of the numerical models for gas discharge simulations

Steady streaming around a spherical drop displaced from the velocity antinode in an acoustic levitation field

The steady (acoustic) streaming associated with a spherical drop displaced from the velocity antinode of a standing wave is studied. The ratio of the particle size to the acoustic wavelength is treated as small but non-zero, and the solution is developed in the form of a two-term expansion in terms of the corresponding smallness parameter. The drop viscosity is assumed to be much higher than that of the surrounding fluid, which is the case for a drop in a gas medium. There are essentially three distinct regions where the steady streaming flow is analysed: inside the drop (internal circulation), in the Stokes shear-wave layer at the surface on the gas side, and the gas outside the Stokes layer (the outer streaming region). Solutions for the internal circulation and the outer streaming are obtained in the limit of small Reynolds number. Despite the gas-to-liquid viscosity ratio being small, the outer streaming may be dramatically affected by the fact that the sphere is liquid as opposed to solid. The parameter that measures the effect of liquidity is essentially the viscosity ratio divided by the relative (to the particle size) thickness of the Stokes layer. The case of a solid sphere is recovered by letting this parameter go to zero.

Thermal-Runaway Approximation for Ignition Times of Branched-Chain Explosions

An asymptotic analysis for high activation energy of the branching step is developed for predicting ignition times of branched-chain explosions on the basis of a criterion of thermal runaway in homogeneous, isobaric, adiabatic systems. The chemistry includes an initiation step, a branching step, and a recombination step and leads to a nonlinear second-order ordinary differential equation for the temperature as a function of time under adiabatic conditions. One or both of the branching and recombination steps must be exothermic, whereas the initiation step can be endothermic or exothermic. A two-term expansion is derived in a small parameter representing the ratio of the initiation rate to the branching rate, yielding explicit expressions for the ignition time. At leading order, the ignition time is found to be inversely proportional to the net branching rate, the proportionality constant being the logarithm of the ratio of the branching rate to the initiation rate multiplied by energetic and rate parameters associated with branching and recombination. Resulting ignition times are shown to correspond closely with those calculated by numerical integration of the full equations on the basis of a temperature-inflection criterion, except near crossover, where the branching rate equals the recombination rate. Application of the theory to fuel-rich hydrogen‐oxygen systems demonstrates good agreement.

Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models

Fluid models of gas discharges require the input of transport coefficients and rate coefficients that depend on the electron energy distribution function. Such coefficients are usually calculated from collision cross-section data by solving the electron Boltzmann equation (BE). In this paper we present a new user-friendly BE solver developed especially for this purpose, freely available under the name BOLSIG+, which is more general and easier to use than most other BE solvers available. The solver provides steady-state solutions of the BE for electrons in a uniform electric field, using the classical two-term expansion, and is able to account for different growth models, quasi-stationary and oscillating fields, electron–neutral collisions and electron–electron collisions. We show that for the approximations we use, the BE takes the form of a convection-diffusion continuity-equation with a non-local source term in energy space. To solve this equation we use an exponential scheme commonly used for convection-diffusion problems. The calculated electron transport coefficients and rate coefficients are defined so as to ensure maximum consistency with the fluid equations. We discuss how these coefficients are best used in fluid models and illustrate the influence of some essential parameters and approximations.

An asymptotic solution for the surface magnetic field within the paraxial region of a circular cylinder with an impedance boundary condition

It is well-known that the high-frequency asymptotic evaluation of surface fields by the conventional geometrical theory of diffraction (GTD) usually becomes less accurate within the paraxial (close to axial) region of a source excited electrically large circular cylinder. Uniform versions of the GTD based solution for the surface field on a source excited perfect electrically conducting (PEC) circular cylinder were published earlier to yield better accuracy within the paraxial region of the cylinder. However, efficient and sufficiently accurate solutions are needed for the surface field within the paraxial region of a source excited circular cylinder with an impedance boundary condition (IBC). In this work, an alternative approximate asymptotic closed form solution is proposed for the accurate representation of the tangential surface magnetic field within the paraxial region of a tangential magnetic current excited circular cylinder with an IBC. Similar to the treatment for the PEC case, Hankel functions are asymptotically approximated by a two-term Debye expansion within the spectral integral representation of the relevant Green's function pertaining to the IBC case. Although one of the two integrals within the spectral representation is evaluated in an exact fashion, the other integral for which an exact analytical evaluation does not appear to be possible is evaluated asymptotically, unlike the PEC case in which both integrals were evaluated analytically in an exact fashion. Validity of the proposed asymptotic solution is investigated by comparison with the exact eigenfunction solution for the surface magnetic field.

Critical viscosity exponent for classical fluids

A self-consistent mode-coupling calculation of the critical viscosity exponent z(eta) for classical fluids is performed by including the memory effect and the vertex corrections. The incorporation of the memory effect is through a self-consistency procedure that evaluates the order parameter and shear momentum relaxation rates at nonzero frequencies, thereby taking their frequency dependence into account. This approach offers considerable simplification and efficiency in the calculation. The vertex corrections are also demonstrated to have significant effects on the numerical value for the critical viscosity exponent, in contrast to some previous theoretical work which indicated that the vertex corrections tend to cancel out from the final result. By carrying out all of the integrations analytically, we have succeeded in tracing the origin of this discrepancy to an error in earlier work. We provide a thorough treatment of the two-term epsilon expansion, as well as a complete three-dimensional analysis of the fluctuating order-parameter and transverse hydrodynamic modes. The study of the interactions of these modes is carried out to high order so as to arrive at z(eta) = 0.0679+/-0.0007 for comparison with the experimentally observed value, 0.0690+/-0.0006 .

Comparison of kinetic calculation techniques for the analysis of electron swarm transport at low to moderate E/N values

We present a comparison between results for the electron velocity distribution function (evdf), and transport and rate coefficients of an electron swarm obtained under different assumptions for the space and angular dependence of the evdf. Several solution techniques for the Boltzmann equation as well as Monte Carlo simulations have been tested. The comparison is made in neon at a constant and homogeneous reduced electric field in the range 10 Td ≤ E/N ≤ 500 Td taking into account the production of electrons in ionizing collisions. The results show that to obtain an accurate description of the electron swarm we need to take into account the variation in space of the electron density in the representation of the evdf. In what regards the angular dependence on velocity we discuss criteria to estimate the importance of the anisotropy of the evdf for any gas. Depending on the solution technique and on the E/N value, we find good to excellent agreement between the Boltzmann results obtained with a half-range method, a multi-term Legendre expansion, an elliptic approximation and the Monte Carlo results. The accuracy of the transport and rate coefficients obtained with each approach is evaluated and it is found that although the two-term velocity expansion is not sufficiently accurate to be used for cross section fitting, the corresponding rate and transport coefficients can generally be used in discharge modelling.

Fundamental frequency of a circular membrane with a strip of small length

A circular membrane with an arbitrarily placed internal strip of small length is concerned in this article. A two-term asymptotic expansion for the fundamental frequency of the membrane, as the length of the strip approaching to zero, is specified. Comparing it with the one [8] derived for the membrane with an internal circular core, it is found that the position of the internal constraint has more effect than the shape of the internal constraint on the fundamental frequency. The asymptotic approximation is also compared with the numerical data computed by the dual boundary element method [2] for a circular membrane of radius 1 with a radially placed internal strip of length 2c. These two sets of data are in good agreement. The relative error is less than 3 % as c is less than or equal to 0.1, for all positions of the strip. Moreover, the relative error is less than 1 % as c is less than or equal to 0.01.