Truncated Series Expansion Research Articles

Accurate and efficient reconstruction of discontinuous functions from truncated series expansions

Knowledge of a truncated Fourier series expansion for a discontinuous 2 π 2\pi -periodic function, or a truncated Chebyshev series expansion for a discontinuous nonperiodic function defined on the interval [ − 1 , 1 ] [-1, 1] , is used in this paper to accurately and efficiently reconstruct the corresponding discontinuous function. First an algebraic equation of degree M for the M locations of discontinuities in each period for a periodic function, or in the interval ( − 1 , 1 ) (-1, 1) for a nonperiodic function, is constructed. The M coefficients in that algebraic equation of degree M are obtained by solving a linear algebraic system of equations determined by the coefficients in the known truncated expansion. By solving an additional linear algebraic system for the M jumps of the function at the calculated discontinuity locations, we are able to reconstruct the discontinuous function as a linear combination of step functions and a continuous function.

On the statistical distribution of TLP offsets

The subject of this paper is the determination of the crossing frequency of one of the important design variables in TLP design, namely the horizontal offset of the, platform. With the conceptual design process in mind, an effort has been made to develop approximations for the crossing frequency that are easy to use, and accurate enough for that stage of the design spiral. It has been shown that the first- and second-order horizontal offset of the platform are statistically independent, and that the crossing frequency of the combined offset can be found in terms of a truncated series expansion. The coefficients of the series are found in terms of integrals similar to the ones evaluated when computing the expected value of the second-order force. Some simple approaches to the problem are evaluated, and most of them are found to under-estimate the maximum offset. The crossing frequency of the two horizontal offsets, combined into a radial one, is evaluated, and is shown to take a relatively simple form.

Adaptive Rules for Seminonparametric Estimators That Achieve Asymptotic Normality

In econometrics, seminonparametric (SNP) estimators originated in the consumer demand literature. The Fourier flexible form is a well-known example. The idea is to replace the consumer's indirect utility function with a truncated series expansion and then use a parametric procedure, such as nonlinear multivariate regression, to set a confidence interval on an elasticity. More recently, SNP estimators have been used in nonlinear time series analysis. A truncated Hermite expansion with an ARCH leading term is used as the conditional density of the process. The method of maximum likelihood is used to fit it to data.

On the evaluation of the internal energy and chemical potential of an ideal fermion system

The internal energy and chemical potential of a noninteracting Fermi gas has been evaluated in three ways: numerically (applicable to arbitrary temperatures), through an asymptotic power series expansion (suitable for low temperatures), and by means of standard Padé approximants (appropriate for low and moderate temperatures). The main conclusion of this work is that the Padé approximant, which is constructed from the asymptotic series employing only minimal labor, is far superior to the truncated series expansion.

Prediction Intervals in Nonlinear Regression

AbstractBy treating the nonlinear model as if it were linear in the parameterization θ in the neighbourhood of the least squares estimate θ, we construct two‐sided nominally‐q‐prediction intervals by applying the usual linear model theory. The derivation of the truncated series expansion of the expected coverage of the prediction intervals at a feasible value of the parameter vector is described. The quadratic approximation of the expected coverage is then obtained for a two‐parameter nonlinear model. Finally we show how we may construct the prediction intervals when a certain type of nonlinear transformation of the parameter vector has been applied.

Orthogonal series generalized likelihood ratio test for failure detection and isolation

A new failure detection and isolation algorithm for linear dynamic systems is presented. This algorithm, the Orthogonal Series Generalized Likelihood Ratio (OSGLR) test, is based on the assumption that the failure modes of interest can be represented by truncated series expansions. This assumption leads to a failure detection algorithm with several desirable properties. Computer simulation results are presented for the detection of the failures of actuators and sensors of a C-130 aircraft. The results show that the OSGLR test generally performs as well as the GLR test in terms of time to detect a failure and is more robust to failure mode uncertainty. However, the OSGLR test is also somewhat more sensitive to modeling errors than the GLR test.

A mixed hybrid finite element for three-dimensional elastic crack analysis

A new three-dimensional crack tip element is proposed, which is based on a mixed hybrid stress/displacement model. A truncated series expansion of eigenfunctions for the straight semi-infinite crack is deduced and assumed for the internal stress and displacement fields in the element. The basic approach of constructing these hybrid elements is outlined. Their good capability, efficiency and accuracy for analyzing three-dimensional elastic crack problems are demonstrated by first numerical examples.

Faster 4 × 4 matrix method for uniaxial inhomogeneous media

An exact representation of the transfer matrix for stratified homogeneous uniaxial media is derived. It can be used to calculate optical quantities such as reflectance and transmittance by means of Berreman’s 4 × 4 matrix method, permitting calculations for thick homogeneous slabs such as polarizers in one single step without the commonly used truncated series expansion. When the dielectric tensor of an inhomogeneous medium varies continuously with the normal to the plane of stratification, the medium is divided into thin slabs. The transfer matrix of the whole medium is then obtained by multiplying the transfer matrices of all slabs. Treating each slab as homogeneous gives satisfactory results, as shown in an example of a periodic structure for which an analytic solution is known.

<title>Digital Decoding Of In-Line Holograms</title>

Digitally sampled in-line holograms may be linearly filtered to reconstruct a representation of the original object distribution, thereby decoding the information contained in the hologram. The decoding process is performed by digital computation rather than optically. Substitution of digital for optical decoding has several advantages, including selective suppression of the twin-image artifact, elimination of the far-field requirement, and automation of the data reduction and analysis process. The proposed filter is a truncated series expansion of the inverse of that operator that maps object opacity function to hologram intensity. The first term of the expansion is shown to be equivalent to conventional (optical) reconstruction, with successive terms increasingly sup-pressing the twin image. The algorithm is computationally efficient, requiring only a single fast Fourier transform pair.

An approximate statical solution of the elastoplastic interface for the problem of Galin with a cohesive-frictional material

An approximate statically admissible solution of the elastoplastic interface is described for the plane strain problem of a pressurized circular hole in a plane subject to a non-hydrostatic stress at infinity (Problem of Galin). In contrast to the solution of Galin (Prikl. Mat. Mekh.10, 365–386 (1946)) which applies for the case of a frictionless Tresca material, it is assumed that the material is characterized by a cohesivc-frictional yield strength. The solution of the elasloplastic interface is obtained in the form of a truncated series expansion, for cases where the material has yielded all around the hole. The paper discusses the limiting conditions for which the solution is applicable, and the validity of the solution in regard to an elastoplastic problem.

Spectral algorithms for vector elliptic equations in a spherical gap

Abstract This paper is devoted to the numerical solution of three-dimensional elliptic equations of a vector field in the region contained between two concentric spheres. The method of truncated series expansion in orthogonal functions is considered using Chebyshev polynomials and vector spherical harmonics. Spectral algorithms are described for solving two-point boundary-value problems of the vector modes of the Poisson, Helmholtz, and biharmonic equations supplemented by boundary conditions typical of fluid dynamical applications. The accuracy of the algorithms is illustrated by computing some analytical single-mode examples. The proposed algorithms when combined with efficient transform techniques can be used to solve multi-mode nonlinear problems.

Macro- and Microflux, I: Periodic Lattices

In order to shorten the time of reactor core calculations, the actual core structure is often replaced by a simpler structure, such as a periodic lattice whose neutron flux is determined through some periodic microfluxes and through an overall macroflux. In the framework of the well-known perturbation formalism, it is shown that the macroflux is obtained from a two-group diffusion equation in which the coefficients are determined from transport cross sections and microfluxes. The relationships between microfluxes are given. It is shown that in a finite core the flux is described by an asymptotic and a transient term. A simple problem is solved by means of the presented theory, showing that it is capable of providing a truncated series expansion of the exact results. The theory presented is applied to the evaluation of measurements.

Electronic excited state transport among molecules distributed randomly in a finite volume

A theoretical study of electronic excited state transport among molecules randomly distributed in a finite volume is carried out. Two special cases of the general transport and trapping problem are treated. A truncated series expansion in powers in the chromophore density is used as an approximation for one component systems (i.e., donor–donor transport only). In two component systems of donors and traps, the Förster limit, in which transfer can occur only from donors to traps due to low donor concentrations, is solved exactly for a finite spherical volume. In both cases, the results presented demonstrate that time-dependent observables can be significantly altered in finite volume systems relative to infinite volume systems. These calculations have implications for the interpretation of experiments performed on real finite volume systems, e.g., energy transport among the chromophores of an isolated polymer chain.

Difference equation evaluation of the evolution of radially symmetric ultrasonic fields

A simplified difference equation technique has been developed to calculate the three‐dimensional evolution of radially symmetric complex ultrasonic fields. Localized changes in the magnitude of the field with respect to distance along the axis of propagation are first assumed to be small compared to those with respect to radial displacement. The complex Helmholtz equation in cylindrical coordinates then may be simplified by separating variables and approximating these variables by truncated series expansions. The resulting difference equation determines complex field values at discrete locations in a plane from corresponding adjacent locations in the preceding plane. Stability criteria which limit both the step sizes between consecutive planes and between calculated field values in one plane are discussed. The technique has been applied to the propagation of ultrasonic fields generated by two‐dimensional focusing and non‐focusing uniform and Gaussian profile piezoelectric transducers and theoretical and experimental data are presented. [Work supported in part by NASA and NSF.]

Calculation methods for reacting turbulent flows: A review

The purpose of this review is to describe and appraise components of calculation methods, based on the solution of conservation equations in differential form, for the velocity, temperature and concentration fields in turbulent combusting flows. Particular attention is devoted to the combustion models used within these methods and to gaseous-combustion applications. The differential equations are considered first and the implications of conventional (i.e., unweighted) and density-weighted averaging discussed in the contexts of solution methods and physical interpretation. In general, it is concluded that equations should be solved with dependent variables in density-weighted form and the interpretation of measurements requires special care to distinguish between conventionally averaged and density-weighted properties. Finite difference approximations contained within numerical procedures for solving the equations relevant to two- and three-dimensional recirculating flows, such as are to be found in combustion chambers, are considered briefly. It is concluded that computer storage requirements very often preclude the possibility of reducing the numerical error to entirely negligible proportions everywhere. Considerable care must therefore be taken in both specifying a sufficient number and the distribution of mesh points to be used and also in the interpretation of computational results. Turbulence models are also discussed briefly and deficiencies noted. Since many flows are partly controlled by mechanisms other than diffusion and turbulence transport, these deficiencies are of major importance in a limited range of circumstances which are discussed. The various recent methods proposed to represent reaction in turbulent flames are reviewed in relation to diffusion and premixed flames and to flames in which an element of both is present. The application of laminar flame sheet models, chemical equilibrium assumptions, probability density functions of different forms, and truncated series expansion of reaction rate expressions are considered together with the use of probability density function transport equations and their (Monte Carlo) solution. The appraisal is made in relation to presently available results and future requirements and possibilities.

Radius of convergence of Lie series for some elliptic elements

For equatorial orbits about an oblate body, we show that the Lie series for the elliptic elementse,f,l and $$\varpi$$ diverge when the oblateness exceeds a critical multiple of the transformed eccentricity constant. The use of similar truncated series expansions for such elliptic elements by Brouwer accounts for the first-order errors at low eccentricity in his derived coordinates for an artificial satellite.

Truncation Problems in the Dynamics and Control of Flexible Mechanical Systems

The distributed deformation coordinates of flexible mechanical systems are commonly discretizised by means of a truncated series expansion (Ritz-method). After a short explanation of the governing equations, a mathematical justification of the truncation is given applying singular perturbation theory. The truncated model is compared with the exact, infinite-dimensional model. Easy to evaluate error bounds for the substantial eigenvalues are developped. It is shown under which conditions the control system design basing on the truncated model is admissible. Effects of control spillover are discussed. The results are corroborated by the application to the test model of a flexible manipulator arm.

The Temperature Dependence of the Properties of Electrolyte Solutions. I. A Semi‐Phenomenological Approach to an Electrolyte Theory Including Short Range Forces

AbstractPair distribution functions f,ij(r1, r2) of a symmetrical ionophoric electrolyte compound Y Cz+ Az‐ in solution are determined with the help of a semi‐phenomenological method permitting to avoid a truncated series expansion of the Boltzmann factor exp [ — eo zj ψ i(r) k T] of Coulomb ion‐ion interaction in the vicinity of the i ion and to introduce the mean potential of short range forces. The appropriate chemical potential of the electrolyte component μY(p, T) and the relevant transport equations, especially of electrolyte conductance, are given. The concept of ion association is discussed within the framework of this approach.

Truncated series expansion for the correlation function of permanent and induced depolarised Rayleigh bands

Correlation functions from depolarised Rayleigh bands of molecules with permanent and induced polarisability components are simulated using an expansion in terms of successive memory functions, truncated at an early stage with an exponential decay. The results of Alder et al. for computer argon and Dill et al. for acetone, methyl iodide and benzene up to several kbar of external pressure are simulated closely in the majority of cases, but only some of the oscillations in the experimental results for benzene and methyl iodine are reproduced by the very simple truncated series expansion used here.

Optimum low-order windows for discrete Fourier transform systems

Truncated series expansion is used to obtain discrete-time windows which are optimal in the least-square sense for a given number of terms. The representation suggested in the paper is so rapidly convergent that an excellent approximation to the exact least-square optimum is achieved even if only a few terms are kept. As a consequence, the resulting windows are easy to obtain and economical to implement in practical applications.