Different Orders Of Approximation Research Articles

A MATHEMATICAL THEORY FOR VIBRATIONAL LEVELS ASSOCIATED WITH HYDROGEN BONDS II: THE NON-SYMMETRIC CASE

We propose an alternative to the usual time-independent Born–Oppenheimer approximation that is specifically designed to describe molecules with non-symmetrical hydrogen bonds. In our approach, the masses of the hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. As a result, the different vibrational modes appear at different orders of approximation. Although we develop a general theory, our analysis is motivated by an examination of the FHCl- ion. We describe our results for it in detail. We prove the existence of quasimodes and quasienergies for the nuclear vibrational and rotational motion to arbitrary order in the Born–Oppenheimer parameter ∊. When the electronic motion is also included, we provide simple formulas for the quasienergies up to order ∊3 that compare well with experiment and numerical results.

A Mathematical Theory for Vibrational Levels Associated with Hydrogen Bonds

We describe recent work in which we propose an alternative to the usual time independent Born–Oppenheimer approximation that is specifically designed to describe molecules with Hydrogen bonds. In our approach, the masses of the Hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. Consequently, anharmonic effects play a role in the leading order calculations of vibrational levels for symmetric molecules. For non-symmetrical molecules, the different vibrational modes appear at different orders of approximation.

Accurate, Efficient Quantification of Uncertainty for Flow in Heterogeneous Reservoirs Using the KLME Approach

Summary Accurate modeling of flow in oil/gas reservoirs requires a detailed description of reservoir properties such as permeability and porosity. However, such reservoirs are inherently heterogeneous and exhibit a high degree of spatial variability in medium properties. Significant spatial heterogeneity and a limited number of measurements lead to uncertainty in characterization of reservoir properties and thus to uncertainty in predicting flow in the reservoirs. As a result, the equations that govern flow in such reservoirs are treated as stochastic partial differential equations. The current industrial practice is to tackle the problem of uncertainty quantification by Monte Carlo simulations (MCS). This entails generating a large number of equally likely random realizations of the reservoir fields with parameter statistics derived from sampling, solving deterministic flow equations for each realization, and post-processing the results over all realizations to obtain sample moments of the solution. This approach has the advantages of applying to a broad range of both linear and nonlinear flow problems, but it has a number of potential drawbacks. To properly resolve high-frequency space-time fluctuations in random parameters, it is necessary to employ fine numerical grids in space-time. Therefore, the computational effort for each realization is usually large, especially for large-scale reservoirs. As a result, a detailed assessment of the uncertainty associated with flow-performance predictions is rarely performed. In this work, we develop an accurate yet efficient approach for solving flow problems in heterogeneous reservoirs. We do so by obtaining higher-order solutions of the prediction and the associated uncertainty of reservoir flow quantities using the moment-equation approach based on Karhunen-Loéve decomposition (KLME). The KLME approach is developed on the basis of the Karhunen-Loéve (KL) decomposition, polynomial expansion, and perturbation methods. We conduct MCS and compare these results against different orders of approximations from the KLME method. The 3D computational examples demonstrate that this KLME method is computationally more efficient than both Monte Carlo simulations and the conventional moment-equation method. The KLME approach allows us to evaluate higher-order terms that are needed for highly heterogeneous reservoirs. In addition, like the Monte Carlo method, the KLME approach can be implemented with existing simulators in a straightforward manner, and they are inherently parallel. The efficiency of the KLME method makes it possible to simulate fluid flow in large-scale heterogeneous reservoirs.

Time domain surface impedance concept for low frequency electromagnetic problems—Part II: Application to transient skin and proximity effect problems in cylindrical conductors

A time domain boundary element formulation employing the surface impedance boundary conditions (SIBCs) is developed for the 3-dimensional transient eddy current problem of cylindrical conductors. SIBCs of different orders of approximation are implemented using the perturbation technique in the small parameter proportional to the ratio of the skin depth and characteristic size of the conductor cross-section. The formulation consists of a set of time domain surface integral equations that have identical left-hand sides and can be solved using the same program procedure. The number of equations is determined by the order of approximation of the SIBC, namely: solutions in the perfect electrical conductor (PEC) limit (lowest order) and in the so-called Rytov approximation (highest order) are given by one and four equations, respectively. It is demonstrated that each equation admits separation of variables into space and time components, a property that significantly reduces computational costs compared with traditional time domain formulations that require the integral equations to be solved at each time step. For the purpose of validation, a test problem is solved by the proposed formulation and by the ‘original’ BEM based on the time-dependent fundamental solution. Conditions of applicability are discussed and the effect of such factors as the shape of the incident current pulse and proximity effect are considered.

Microscopic quantum dynamics approach to the dilute condensed Bose gas

We derive quantum evolution equations for the dynamics of dilute condensed Bose gases. The approach contains, at different orders of approximation, for cases close to equilibrium, the Gross-Pitaevskii equation and the first-order Hartree-Fock-Bogoliubov theory. The proposed approach is also suited for the description of the dynamics of condensed gases that are far away from equilibrium. As an example the scattering of two Bose condensates is discussed.

Systematic weakly nonlinear analysis of interfacial instabilities in Hele-Shaw flows.

We develop a systematic method to derive all orders of mode couplings in a weakly nonlinear approach to the dynamics of the interface between two immiscible viscous fluids in a Hele-Shaw cell. The method is completely general: it applies to arbitrary geometry and driving. Here we apply it to the channel geometry driven by gravity and pressure. The finite radius of convergence of the mode-coupling expansion is found. Calculation up to third-order couplings is done, which is necessary to account for the time-dependent Saffman-Taylor finger solution and the case of zero viscosity contrast. The explicit results provide relevant analytical information about the role that the viscosity contrast and the surface tension play in the dynamics of the system. We finally check the quantitative validity of different orders of approximation and a resummation scheme against a physically relevant, exact time-dependent solution. The agreement between the low-order approximations and the exact solution is excellent within the radius of convergence, and is even reasonably good beyond this radius.

THE PROPERTIES OF A HADRON

Using the analogy of a well-known mathematical formalism, an analytical form of the probability density of a hadron as a quarkonium system has been suggested which corresponds to that obtained from the statistical model. The aforesaid prescription also offers the necessary framework to incorporate different orders of approximations in it. Unlike all existing theoretical approaches, the model reveals the fractal characteristics of a hadron, like the fractal dimension DF, correlation dimension DCand the anomalous dimension DA. The direct physical bearings of these dimensional quantities on the physical parameters related to the recent experimental data have also been discussed.

A general form of color difference formula based on color discrimination ellipsoid parameters

AbstractA general color difference formula has been derived based on the parameters of color discrimination ellipsoids in the CIE 1976 L*a*b* color space. By using different orders of approximation, the general formula resembles the basic forms of the current formulae. The method described in this article suggests a framework for modifying the CIE 1976 L*a*b* color difference formula.

Propagation of electromagnetic waves in inhomogeneous plasmas

The reflection and transmission coefficients for an electromagnetic beam propagating in an inhomogeneous plasma are calculated analytically using the Magnus approximation in different physical configurations. The theoretical predictions for such coefficients are expressed in simple analytical form, and are compared with the exact results obtained by numerical solution of the wave propagation equations, using the Berreman 4 × 4 matrix method. It is shown that the theoretical approach is able to reproduce the correct results for reflection and transmission coefficients over a wide range of physical parameters. The accuracy of the theoretical analysis, at different orders of approximation, is also discussed.

Approximation Theory for Linear-Quadratic-Gaussian Control of Flexible Structures

This paper presents approximation theory for the linear-quadratic-Gaussian optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite-dimensional compensators that approximate closely the optimal compensator, which is infinite-dimensional. Design of the optimal compensator separates into an optimal linear-quadratic control problem and a dual optimal state estimation problem; the solution to each problem lies in the solution to an infinite-dimensional Riccati operator equation. The approximation scheme in the paper approximates the infinite-dimensional LQG problem with a sequence of finite-dimensional LQG problems defined for a sequence of finite-dimensional, usually finite-element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite-dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite-dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite-dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite-dimensional Riccati equations to the solutions of the infinite-dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.

Anisotropic Plates With Various Openings Under Uniform Loading or Pure Bending

In the case when an anisotropic plate contains a triangular, oval, or square opening, the only solution available in the literature is an approximate solution for orthotropic plates with openings, which was obtained by Lekhnitskii using the complex variable formulation. Solutions for any kind of anisotropic plates with various openings are presented in this paper by applying the Stroh formalism and using the technique of conformal mapping. Unlike the former results, which have different orders of approximation for different openings, the solutions presented here have only one simple unified expression for various openings such as the ellipse, circle, crack, triangle, oval, and square. Two special loading conditions are considered: one is uniform loading, the other is pure bending. Through the use of identities developed in the literature, the hoop stress along the opening boundary is obtained in real form. The results show that the effect of anisotropy on the stress concentration is totally determined through the fundamental elasticity matrices N1 and N3 introduced by Stroh.

Representation of the approximational error term and sharp estimates for some local splines

In the present paper we study local splines of defect 1 on a uniform net with step approximating functions and their derivatives of arbitrary order. Namely, splines of the first, third, and fifth degrees realizing different orders of approximation of f(s) are considered, including splines realizing the maximally possible orders O(~2), O(~), O(~6), respectively, for the minimal number of net values of the function f operated on (so-called splines of minimal model, SMM). In addition, the simplest splines of arbitrary degree exactly reproducing the derivatives ps+1(s)(t) of polynomials of degree s + 1 are considered, and also SMM of arbitrary degree r exactly reproducing Pr(t). In the paper we find explicit expressions for the error terms arising in the approximation by the splines listed above of functions and their derivatives of arbitrary order. The expressions obtained let us find sharp estimates for the error of approximation on the corresponding classes of functions. These results are based on the asymptotic expansions of the error terms in powers of ~ established in [I, 2] and also on one of the results of [3].

Pole and zero estimation in linear circuits

A unified method for fast, accurate estimation of dominant and nondominant poles and zeros of linear active circuit transfer functions is presented. The basis for both pole and zero estimates is an efficiently computed time-constant matrix. The major computation in the estimation process involves only the solution of linear equations, which are independent of capacitor values. This feature permits the implementation of fast real-time dominant pole-zero design based on capacitor variation. Formulas with concomitant applicability criteria are developed for estimating the four most dominant poles and zeros of a transfer function. A broad range of circuit examples is included to demonstrate the power and relative accuracy of different orders of approximation through fourth order.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Dean equations extended to a helical pipe flow

In this paper the Dean (1928) equations are extended to the case of a helical pipe flow, and it is shown that they depend not only on the Dean number K but also on a new parameter λ/[Rscr ] where λ is the ratio of the torsion τ to the curvature κ of the pipe axis and [Rscr ] the Reynolds number referred in the usual way to the pipe radius a and to the equivalent maximum speed in a straight pipe under the same axial pressure gradient. The fact that the torsion has no first-order effect on the flow is confirmed, but it is shown that this is peculiar to a circular cross-section. In the case of an elliptical cross-section there is a first-order effect of the torsion on the secondary flow, and in the limit λ/[Rscr ] → ∞ (twisted pipes, provided only with torsion), the first-order ‘displacement’ effect of the walls on the secondary flow, analysed in detail by Choi (1988), is recovered.Different systems of coordinates and different orders of approximations have recently been adopted in the study of the flow in a helical pipe. Thus comparisons between the equations and the results presented in different reports are in some cases difficult and uneasy. In this paper the extended Dean equations for a helical pipe flow recently derived by Kao (1987) are converted to a simpler form by introducing an appropriate modified stream function, and their equivalence with the present set of equations is recovered. Finally, the first-order equivalence of this set of equations with the equations obtained by Murata et al. (1981) is discussed.

Systematic analytical solutions for minority-carrier transport in semiconductors with position-dependent composition, with application to heavily doped silicon

For quasi-neutral regions of semiconductor devices with position-dependent composition, we have derived expressions for the position dependence of the excess minority-carrier density and for relevant recombination currents. To make the development concrete, we study nonuniformly and heavily doped emitter regions of silicon p-n junction devices. The expressions developed differ from those previously advanced in that they are in the form of a multiple integral series, yielding, by truncation, many different orders of approximation. Correspondences exist between some of the different orders of approximation and various solutions previously obtained. All of the mechanisms relating to hole and electron transport in position-dependent heavily doped semiconductors are accounted for in the new expressions. These mechanisms include bandgap narrowing, majority-carrier degeneracy, Auger recombination lifetime, etc. To assess the accuracy of the various orders of approximation, we compare their predictions with a numerical solution. We determine that the simplest approximation, which contains only one term of the integral series, is accurate to within about 5 percent of the numerical solution for thin emitters (∼0.2 µm) provided the surface recombination velocity is less than 105cm/s. It thus applies directly to bipolar transistors with polysilicon contacts, and to surface-passivated solar cells. This new solution, which we call the zeroth-order or quasi-neutral quasi-equilibrium approximation, is simpler than solutions previously put forward. If the emitter junction is deep or the contact is ohmic, the higher-order approximations provide whatever accuracy is needed. We separate the emitter recombination current into charge and a characteristic time, enabling the calculation of the contribution of emitter to capacitance and to delay in the time domain or phase shift in the frequency domain.

Comments on relativistic two-body interaction in atoms

The relativistic two-body interaction in many-electron atoms is investigated in detail. The authors show that the transferred energy contribution which appears in the expression of the two-body interaction is significant at the present level of experimental accuracy. They also show that the Coulomb and Feynman gauge expressions for the relativistic two-body interactions are equivalent in the on-shell energy case for a local potential, but the same terms appear in different orders of approximation in the two gauges. In addition, a critical comparison between the MCDF-EAL and MCDF-OL methods is made from this point of view.

Phase diagrams of random Ising models : simple and systematic successive approximations

Abstract 2014 We present a simple and systematic method to calculate phase diagrams of random Ising models. It applies to a wide class of systems in which, however, randomness has to be described by discrete random varia-bles. We calculated, at different orders of approximation, various quantities like critical temperatures and perco- lation thresholds in good agreement with exact known results. J. Physique 44 (1983) 1143-1147 OCTOBRE 1983, : Classification Physics Abstracts 64.60 Most papers dealing with random Ising systems make use either of real space renormalization group methods [1] or the replica trick combined with themean-field approximation [2]. This paper presents a much less sophisticated method due to one of us [3] which is, conceptually, as simple as an ordinary mean- field approximation and has the advantage of leading to systematic successive approximations which, even at the lowest order, give, from a numerical point of view, better results than the mean-field

Measurement of Velocities in Solitary Waves

The water surface profiles and corresponding water particle velocities of several solitary waves have been obtained. The measurements of the horizontal and vertical velocity components are conducted using a two-dimensional laser-Doppler velocimeter (LDV). Results are presented for three different wave height-to-depth ratios: ϵ = 0.11, 0.19 and 0.29. The experimental results are compared with existing theories which follow different orders of approximation, and the theories are found to agree well with the experiments. It has been convincingly demonstrated that the LDV measurement technique offers an unmatched advantage to determine the water particle velocities under water waves, especially in the region above the still water level.

Dual-order parallelogram finite elements for the axisymmetric vector Poisson equation

A new family of parallelogram finite elements, in which the order of approximation can be separately specified for the two pairs of opposing sides, is described. These elements are uniquely useful in problems involving long, narrow subregions and successfully complement high-order triangular elements for a wide range of applications. In addition to their use for narrow subregions they effectively fill up large, open spaces (a traditional use of rectangular elements) and uniquely allow different orders of approximation to be achieved in different regions without sacrificing the conformity of the solution. The assembly of these elements at run-time is accomplished without costly numerical integration by the use of pre-calculated universal element matrices. The element matrices are calculated exactly for one-dimensional elements through sixth order and are less expensive to generate and smaller than the corresponding matrices for triangular elements. Element matrices are given for approximation orders up to three and a sample problem is solved.

Radiative transfer in atmospheres with spherical symmetry—IV. the non-conservative problem

Non-conservative transfer problems are conceptually more complicated than the conservative ones because the source function introduces a new scale height. This problem is specially important in spherical geometry, where, due to the curvature of the layers, one must also take into account the scale height of the opacity. With the introduction of the auxiliary variables μ c and μ r , it has been possible to obtain very simple linear closure relations for the μ moment system; μ c is the cosine of the critical zenithal angle ϑ c from which the intensity peak appears; μ r is an intermediate value of μ between μ c and 1 which permits an explanation of the geometrical dilution and leads easily to a peaking linear relation. With these closure relations, we have solved the moment system for different opacity laws and source functions which represent many cases of astrophysical interest. Likewise, we have compared the quality of our results for different orders of approximation.