Order Correction Terms Research Articles

Boundary Crossing Probabilities for Locally Poisson Processes

We derive large deviation approximations to boundary crossing probabilities for a class of point processes which can be approximated locally as Poisson processes. In the special case of empirical processes, we are able to obtain second order correction terms. The methods are applied to Kolmogorov-Smirnov testing, where we are able to obtain accurate approximations to the significance level when the null hypothesis is an exponential family with unknown nuisance parameters.

CMOS stabilised DC power source

A new approach for realising a DC power source is reported. Its function is to stabilise power in the presence of resistive load variations. Only linear circuit blocks are required, no analogue multiplier is needed. This is achieved by making the power stabilisation around the nominal operating point and keeping only the first order correction terms in the compensation equation. The error introduction by this approach is smaller than 0.2% for a 10% load variation which corresponds to a stabilisation factor of 50. The circuit was integrated as a driver for the resistor heating element of an air flow sensor co-integrated on the same chip. >

Cylindrical bending of thick plates

Sixth and twelfth order plate theory solutions for cylindrical bending plate problems, efficiently obtained by a preliminary reduction of the twelfth order theory for general cylindrical bending to a boundary value problem for a fourth order ODE, demonstrate the adequacy of these theories for the interior of thick plates with a thickness-to-span ratio λ up to ½ (at least). In contrast. Kirchhoff's classical thin plate theory is not adequate for values of the same ratio greater than 1/10. Comparison with the corresponding interior solutions of 3-D elastostatics shows that the twelfth order theory captures the (numerically small) first order correction term in the thicknessto-span parameter ε not possible by lower order plate theories. While it is a qualitatively important feature, this additional term associated with the effect of transverse normal stress does not significantly improve the plate theory approximation quantitatively for Isotropic plates.

Renormalization and the Effective Field Theory Programme

Quantum field theory (QFT) poses the following well-known problem. Calculations of values for certain quantities begin with a first-order approximation to which are added higher order corrections. In many cases, the calculation of even the very first higher order correction term yields infinite values. These infinite values are due to divergences in the theory. These divergences can be traced to their source in the standard fundamental assumptions defining QFT. No matter which version of the formalism we adopt, we are trying to incorporate into the mathematics certain fundamental physical constraints on the nature of the vacuum state, on the states that can be occupied by systems, on the comeasurability of observables, and so on. The resulting formalism contains divergences.

Theory of dispersive relaxation in amorphous semiconductors

A theoretical derivation is given to the empirical formula δn( t) = δn(0) K( v c t) −1+ α exp [−( t/ τ eff ) α ] for dispersive relaxation of photoconductivity and photoluminescence in amorphous semiconductors of which the power law t −1+ α also forms a limiting case. Limits of validity of the formula have been brought out and higher order correction terms have been obtained. At higher excitations or longer times saturation effect becomes important and a quasi-equilibrium between free and trapped charges can be assumed. This gives t −1/1− α decay for the later part of the transient.

Low-energy pion photoproduction from p-shell nuclei

Low-energy pion photoproduction off 6Li, 10B, and 14N has been reinvestigated in a DWIA framework that includes a number of improvements neglected in previous analyses. The production operator is based on Feynman diagrams and includes correction terms of order p2/M2 and higher. An s-channel delta resonance term is included with both longitudinal and transverse electromagnetic couplings. Rather than using on harmonic oscillator for the nucleon orbitals we employ Woods–Saxon wave functions that have been adjusted to fit electron-scattering form factors and single-particle binding energies. Furthermore, we include the Coulomb potential without approximation in our momentum-space approach. Using more realistic wave functions and including corrections to the production amplitude that have been neglected before leads to considerable improvement in the case of 10B and 14N when compared with existing data. The Coulomb effects are shown to change the cross section by about 30% close to threshold but are negligible at higher energies.

A Perturbation Theory of Zener Transitions in Dissipative Driven Systems11The project supported by Chinese Science Foundation for young scientists through grant No. 0187703 and Chinese Science Foundation through grant No. 1870744.

The effect of inelastic interactions on the Zener transitions in dissipative driven systems has been investigated by conventional perturbation theory in adiabatic approximation. Two canonical transformations are used to simplify the model Hamiltonian. In zeroth order approximation, we reproduce the previous results. Moreover, our theory can be used to calculate easily the higher order correction terms that would be difficult to obtain in the previous theory.

Probability density and exceedance rate functions of locally Gaussian turbulence

A locally Gaussian model of turbulence velocities is postulated which consists of the superposition of a slowly varying strictly Gaussian component representing slow temporal changes in the mean wind speed and a more rapidly varying locally Gaussian turbulence component which possesses a temporally fluctuating local variance. Series expansions of the probability density and exceedance rate functions of the turbulence velocity model, based on Taylor's series, are derived. The first terms in these expansions are familiar stationary Gaussian process approximations and the first non-zero correction terms represent first order corrections to these Gaussian process approximations. Since the functional forms of these first order correction terms are fixed and unique, corrections to the familiar stationary Gaussian process approximations of probability density and exceedance rate functions are predicted to have fixed functional forms for fractionally small variance fluctuations. Comparisons of the resulting two-term approximations with measured probability density and exceedance rate functions of atmospheric turbulence velocity records show encouraging agreement, thereby confirming the consistency of the measured records with the locally Gaussian model, which attributes deviations from stationary Gaussian behavior to temporal fluctuations in the local variance or squared intensity of locally Gaussian turbulence velocities. The probability density expansion is shown to be identical with the expansion provided by the Gram-Charlier series of type A through the term containing the Hermite polynomial of degree six, even though its derivation is fundamentally different from conventional derivations of the Gram-Charlier series. Explicit formulas are derived for computing all required expansion coefficients from measured turbulence records.

Single server with linear state-dependent mean rate

A birth-death queueing system with asingle server, first-come first-served discipline, Poisson arrivals and mean service rate which depends linearly on the number of customers in the system, is considered. Explicit expressions are derived for the equilibrium densities of the sojourn and waiting times. Simple approximations to the densities, including the first order correction terms, are obtained in a heavy-traffic situation.

The effective viscosity of suspensions and emulsions of spherical particles

We study the high frequency effective viscosity of suspensions of spherical particles and of emulsions of spherical fluid droplets. Assuming that in the bulk of the dispersion the distribution of particles on average is uniform and isotropic we derive an exact expression for the effective viscosity. We evaluate the expression in two-body approximation and show that for hard spheres with stick boundary conditions this leads to a divergence of the effective viscosity at a volume fraction of about thirty-six percent. This rather low value indicates the necessity of calculating higher order correction terms.

Conditioned Response-Time Distribution for a Large Closed Processor-Sharing System with Multiple Classes in Very Heavy Usage

A large closed processor-sharing system with multiple job-classes is considered. The system consists of a bank of terminals grouped according to job-classes, where the class structure allows for distinctions in the user’s behavior in the terminal, and in the service requirements from the central processing unit. Our earlier results for the single-class system in very heavy usage are extended to the multiple-class systems. For large systems, asymptotic approximations to the distribution of the equilibrium response time, conditioned on the required service time $\tau $, are derived. The first order correction term to the leading normal approximation, which leads to a skewness of the distribution, is shown to have the same structure as the single-class system. Equations are obtained for determining the asymptotic approximations to the mean, variance, and skewness of the response time as functions of the required service time $\tau $.

Gap structures in the phonon spectrum of a linear chain with varying masses

The phonon spectrum of a one-dimensional lattice with masses varying incommensurately with the underlying lattice is studied by a discrete version of a multiple-scale perturbation theory. From the resonances in the higher order correction terms we determine the location of the gaps in the spectrum. These resonances also provide a gap labeling in terms of two integers. Approximate analytic expressions are found predicting the behavior of the integrated density of states close to the two widest gaps and the exponential growth rate of the displacements within a gap. In the continuum limit the equation of motion of the discrete model reduces to the Mathieu equation.

Vibrating beam accelerometer with velocity change output

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Characteristic functions and the aberrations of symmetric optical systems II Addition of aberrations

We describe a method for the calculation of the eikonal coefficients of a symmetric optical system composed of two symmetric subsystems. We consider coefficients of second to tenth order in the direction cosines. The system eikonal is seen to consist of an additive part and correction terms of sixth and higher order. We discuss the interpretation and the magnitude of the correction terms.

The uncertainty in absolute values of surface tension of water

The large discrepancies among the values of surface tension (γ a w ) of water reported in the literature prompted a careful experimental determination of γ a w using the Wilhelmy plate method. The extent of wetting of both smooth and roughened Wilhelmy plates was examined by an autoradiographic technique which indicated that the smooth plates are not wetted completely. The effect of the roughness of the Wilhelmy plate on the determination of the surface tension of water was also investigated. Contrary to a recent investigation [Colloids Surfaces, 6 (1983) 221], our results show that it is necessary to account for a peripheral correction term in order to obtain accurate values of γ a w . We obtained 72.94 ± 0.03 and 72.13 ± 0.04 mN m −1 for γ a w at 20 and 25 °C, respectively, values which are slightly higher than the so-called standard values.

Analysis of the Parallel-Disk Sample Holder for Dielectric Permittivity Measurement

The parallel-disk sample holder used to measure the conductivity and the dielectric permittivity of dissipative materials is analyzed in terms of quasistatic electromagnetic theory. It is shown that the commonly used static solution is not accurate enough to interpret data at high frequencies. Interpretation of the data is improved when higher order correction terms are added to the zeroth-order static solution.

Acoustic radiation force on a particle in a temperature gradient

A general expression is derived for the acoustic radiation force on a small spherical particle of radius R in a standing wave field in a temperature gradient.The case of a particle inside a long tube chamber with a temperature gradient along the axis of symmetry is examined in more detail. The analysis is considerably simplified by the introduction of the mass flux density potential ψ. Assuming kR≪1, and neglecting convection, acoustic streaming, heat conduction, and viscosity effects, an expression is obtained for the force that consists of a ‘‘local’’ version of Gor’kov’s result as well as correction terms of order β/k, where β∼(1/T)(dT/dz).

The orbital current for the 0 gr+→1 + M1 excitations in the even-even f [formula omitted] shell nuclei

The role of orbital current for the 0 gr +→1 + M1 excitations in the even-even f 7 2 shell nuclei is studied in the full f 7 2 n+f 7 2 n−1 (f 5 2 p 3 2 p 1 2 ) 1 configuration space. Most of the orbital contribution is concentrated in the lowest 1 + state. The first order correction term enhances appreciably the orbital-to-spin ratios compared with the pure f 7 2 model. The observed orbital-to-spin ratio for the 0 gr +→1 1 + M1 transition in 46Ti is well reproduced.

The uncertainty in absolute values of surface tension of water

The large discrepancies among the values of surface tension (⇐ a/w) of water reported in the literature prompted a careful experimental determination of ⇐ a/w using the Wilhelmy plate method. The extent of wetting of both smooth and roughened Wilhelmy plates was examined by an autoradiographic technique which indicated that the smooth plates are not wetted completely. The effect of the roughness of the Wilhelmy plate on the determination of the surface tension of water was also investigated. Contrary to a recent investigation [Colloids Surfaces, 6 (1983) 221], our results show that it is necessary to account for a peripheral correction term in order to obtain accurate values of ⇐ a/w. We obtained 72.94±0.03 and 72.13±0.04 mN m −1 for ⇐ a/w at 20 and 25°C, respectively, values which are slightly higher than the so-called standard values.

Boundary behavior of diffusion approximations to Markov jump processes

We show that diffusion approximations, including modified diffusion approximations, can be problematic since the proper choice of local boundary conditions (if any exist) is not obvious. For a class of Markov processes in one dimension, we show that to leading order it is proper to use a diffusion (Fokker-Planck) approximation to compute mean exit times with a simple absorbing boundary condition. However, this is only true for the leading term in the asymptotic expansion of the mean exit time. Higher order correction terms do not, in general, satisfy simple absorbing boundary conditions. In addition, the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary, and leads to an increasing relative error. By introducing a boundary layer, we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time. We illustrate these considerations with a number of examples, including a jump process which leads to Kramers' diffusion model. This example represents an extension to a multivariate process.