Euler Transformation Research Articles

Euler Transformation Analysis of Unrecognized Isocenter Shift Errors Induced by Image Guidance

The alignment of visualized features is often used in image guided radiotherapy (IGRT) to drive object registration between the treatment and planning sessions. Rotational adjustments via this process may be neglected or inaccurately performed, prompting incorrect isocenter shifts. This study identifies feature locations that can give rise to significantly sized but undetected isocenter shift errors when the treatment couch is adjusted without rotating about all axes to bring the selected features into apparent alignment.

Lumped-Element Balun for UHF UWB Printed Balanced Antennas

A semi-lumped balun transformer for UHF and ultrawideband (UWB) dipole antennas is presented in this paper. The proposed structure is based on two asymmetric filters that also transform the variable antenna impedance into the desired source impedance. These asymmetric filters make use of a binomial impedance transformer in each filter section. The asymmetric filters (one low pass filter, LPF, and other high pass filter, HPF) allow the balun bandwidth to be increased while the binomial transformer matches the variable balanced dipole impedance. In this way the ripple in the balun response due to the variation of the UWB dipole impedance is reduced. A balun for a UHF and UWB dipole antenna working from 220 to 820 MHz (bandwidth of 4:1) has been achieved with losses lower than 1 dB. These types of baluns are particularly useful in the low microwave frequency band.

A time-averaged model for gas–solids flow in a one-dimensional vertical channel

In this study, we are interested in deriving time-smoothed governing and constitutive equations for gas–solids flow in moderately dense systems where particle–particle collision is the main energy dissipation mechanism. Results obtained from dynamic simulations of a gas–solids flow in a 1D channel are used to show that it is possible to obtain expressions for the time-averaged constitutive relations based on Taylor series expansion. We demonstrate, by comparing with time-averaged transient results, that the 1st term (or laminar) in the series expressions of most non-linear constitutive relations can yield inaccurate quantitative and qualitative results. This means that steady-state models derived by simply removing the partial time derivative from the governing equations are not suitable for gas–solids flows. This study shows that it was necessary to include many terms of the Taylor series expression of non-linear constitutive relations (such as the granular energy dissipation term) due to large-scale oscillations that were computed for all flow variables at all locations in the 1D domain. In some cases, the Taylor series expansion diverged and the Euler transformation was used to improve the convergence of these series. In this moderately dense flow system, turbulence in the gas-phase was found to be just a reaction to turbulence in the solids phase that resulted from the large-scale motion of solids clusters. This resulted in a negative turbulent gas viscosity computed due to the fact that gas (in the horizontal direction) flows only to occupy regions vacated by clusters of solids. The steady-state results obtained using the time-smoothed gas–solids flow model compared well with the time-averaged results obtained using the transient model for all flow variables.

Convergence of the summation formulas constructed by using a symbolic operator approach

This paper deals with the convergence of the summation of power series of the form Σα≤κ(κ)χ κ, where 0 ≤ α ≤ b ≤ ∞, and { (κ)} is a given sequence of numbers with κ ε (α, b) or(t) a differentiable function defined on (a, b). Here, the summation is found by using the symbolic operator approach shown in [1]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Several examples such as the generalized Euler's transformation series will also be given. In addition, we will compare the convergence of the given series transforms.

Modeling of Carrier-based Aircraft Ski Jump Take-off Based on Tensor

A general mathematical model of carrier based aircraft ski jump take-off is derived based on tensor. The carrier, the aircraft body and the movable parts of the landing gears are treated as independent entities. These entities are assembled into a multi-rigid-body system with flexible links. Dynamical equations of each entity are derived on the basis of the Newton law and the Euler transformation. Using the invariance property of the tensor, the dynamical and kinematical equations are converted to tensor forms which are invariant under time dependent coordinate transformations. Then the tensor formed equations are expressed by the matrix operation. Differential equation group of the matrix form is formur lated for the programming. The closure of the model is discussed, and the simulation results are given.

Euler transformation formula for multiple basic hypergeometric series of type A and some applications

A multiple generalization of the Euler transformation formula for basic hypergeometric series 2 φ 1 is derived. It is obtained from the symmetry of the reproducing kernel for Macdonald polynomials by a method of multiple principal specialization. As applications, elementary proofs of the Pfaff–Saalschutz summation formula and the Gauss summation formula for basic hypergeometric series in U( n+1) due to S.C. Milne are given. Some other multiple transformation and summation formulas for very-well-poised 10 φ 9 and 8 φ 7 series, balanced 4 φ 3 series and 3 φ 2 series are also given.

Design Method for the Optimized Acoustic Matching Layers of UT Probes

In this study, we have tried to find the optimized design variables of the matching layer which is important part of thickness mode ultrasonic transducer and finally reach the conclusion that the electrical property of piezo-element must be under consideration when the optimized acoustic impedance is estimated. Proper expression of the effective impedance of front load at free resonant frequency(: /) has been induced by introducing the principle of binomial multilayer transformer and gradient based numerical method is utilized to find the most acceptable value of / . Optimized point of acoustic impedance can be calculated directly from using some simple formula which we propose. We also verify our result by both numerical and experimental method and get a good enhancement especially it concern to the bandwidth of ultrasonic transducer.

A generalization of the continuous Euler transformation and its application to numerical quadrature

The continuous Euler transformation, which was proposed by the present author to accelerate the convergence of integrals of Fourier-type, is an analogue of the well-known Euler transformation for series. In this paper, we propose new continuous transformations including a generalization of the continuous Euler transformation. We show that the generalized continuous Euler transformation is applicable to not only a Fourier-type integral but also an oscillatory integral with an algebraically decaying integrand. We also show some numerical examples using the generalized continuous Euler transformation.

Fabrication of Dual-mode Ultrasonic Transducer using PZT

This study investigates the mechanism of a dual mode probe that generates both of the longitudinal and shear waves simultaneously with a single FZT element. Most of conventional ultrasonic probes are constructed to generate either longitudinal or shear waves. After poling, PZT has the hexagonal 6mm crystal symmetry. All possible crystal cuts are checked to determine appropriate Euler transformation angles for efficient excitation of dual modes. For the selected cut, performance of the dual mode element is analyzed through numerical simulation and experiments. Results of the analysis determine the optimal crystal cut for simultaneous generation of P and S waves of equal strength.

Series solutions for polytropes and the isothermal sphere

The Lane–Emden equation for polytropic index and its limit of the isothermal sphere equation are singular at some negative value of the radius squared. This singularity prevents the real power series solutions about the centre from converging all the way to the outer surface when . However, a simple Euler transformation gives series that do converge all the way to the outer radius. These Euler-transformed series converge significantly faster than the series in the contained mass derived by Roxburgh & Stockman, which are limited to finite radii whenever by a complex conjugate pair of singularities. We construct some compact analytical approximations to the isothermal sphere, and give one for which the density profile is accurate to 0.001 per cent out to the limit of stability against gravothermal collapse.

A continuous Euler transformation and its application to the Fourier transform of a slowly decaying function

The Euler transformation is a linear sequence transformation to accelerate the convergence of an alternating series. The sequence of weights of the transformation is extended to a continuous weight function which can accelerate Fourier-type integrals including Hankel transforms with a slowly convergent integrand. We show that the continuous weight function can also be used to compute the Fourier transform of a slowly decaying function using FFT.

Dirichlet series solution of equations arising in boundary layer theory

The differential equation F′'' + AFF′'+ BF′ 2 = 0, where A and B are arbitrary constants subject to different types of boundary conditions, is considered. This class of equations frequently occurs in boundary-layer theory. The proposed Dirichlet series method, in conjunction with an unconstrained optimization procedure, is found useful in analyzing these problems. The series so generated is analyzed using Euler transformation and Padé approximants.

Computation of irregularly oscillating integrals

A suitable method to compute infinite integrals with oscillatory integrands is to partition the integration interval, for example at the zeros of the integrand. The limit of the resulting sequence of partial sums can then be found by means of extrapolation. This strategy is equally applicable to integrands with a constant period and integrands with an increasingly rapid oscillatory behaviour at infinity. If the phase function of the oscillating factor has a complicated form, its polynomial part can serve as a basis for an asymptotic partition which may be easier to compute. In the case of a nonlinear phase function, the point where the extrapolation process is to be started must be selected carefully, since maxima in the phase function induce steps in the sequence of partial sums. So far, this problem has been neglected in the literature. Almost no off-the-shelf implementations of algorithms suitable for the computation of irregularly oscillating integrals are available. Therefore, this article explores the usefulness of three standard acceleration algorithms that can be applied to a previously computed sequence of partial sums: the Euler transformation, the Δ 2 -process, and the ε-algorithm. They are compared with Sidi's W-transformation that has been devised to evaluate such integrals. The algorithms have been tested with several benchmarks including problems with linear and polynomial phase functions as well as cases appropriate for asymptotic partitioning. Although designed for the latter, the W-transformation is found to have a surprisingly poor performance if the difference between the phase function and its polynomial part is too large, whereas the ε-algorithm yields the best overall results.

A quadratic boundary element implementation in orthotropic elasticity using the real variable approach

This paper revisits the real variable fundamental solution approach to the Boundary Integral Equation (BIE) method in two-dimensional orthotropic elasticity. The numerical implementation was carried out using quadratic isoparametric elements. The strong and weakly singular integrals were directly evaluated using Euler's transformation technique. The limiting process was done in intrinsic coordinates and no separate numerical treatment for strong and weak singular integrals was necessary. For strongly singular integrals a priori interpretation of the Cauchy principal value is not necessary. Two problems from plane stress and strain are presented to demonstrate the numerical efficiency of the approach. Excellent agreement between BEM results and exact solutions was obtained even with relatively coarse mesh discretizations. Copyright © 2000 John Wiley & Sons, Ltd.

Comparison of Design Methods for Binomial Matching Transformers

The binomial transformer is a well known concept in impedance matching. By the use of multiple quarter wavelength sections broadband matching is achieved. Many textbooks handle the problem starting from the approximated theory of small reflections. In this paper, a new and exact derivation for the impedances of the different sections is given. It is shown that not the section coefficients must satisfy a binomial distribution (as commonly used), but a transformation of these coefficients. Furthermore, an alternative formulation starting form the theory of small reflections is given. The results of the different methods are compared. It is shown that the alternative formulation gives the best agreement with the exact method.

Numerical implementation of the integral-transform solution to Lamb's point-load problem

The present work describes a procedure for the numerical evaluation of the classical integral-transform solution of the transient elastodynamic point-load (axisymmetric) Lamb's problem. This solution involves integrals of rapidly oscillatory functions over semi-infinite intervals and inversion of one-sided (time) Laplace transforms. These features introduce difficulties for a numerical treatment and constitute a challenging problem in trying to obtain results for quantities (e.g. displacements) in the interior of the half-space. To deal with the oscillatory integrands, which in addition may take very large values (pseudo-pole behavior) at certain points, we follow the concept of Longman's method but using as accelerator in the summation procedure a modified Epsilon algorithm instead of the standard Euler's transformation. Also, an adaptive procedure using the Gauss 32-point rule is introduced to integrate in the vicinity of the pseudo-pole. The numerical Laplace-transform inversion is based on the robust Fourier-series technique of Dubner/Abate-Crump-Durbin. Extensive results are given for sub-surface displacements, whereas the limit-case results for the surface displacements compare very favorably with previous exact results.

CHARACTERIZING THE STATISTICAL DISTRIBUTION OF ORGANIC CARBON AND EXTRACTABLE PHOSPHORUS AT A REGIONAL SCALE

Greater awareness of potential environmental problems has created the need to monitor total organic carbon (TOC) and extractable phosphorus (P) concentrations at a regional scale. The probability distribution of these soil properties can have a significant effect on the power of statistical tests and the quality of inferences applied to these properties. The objectives of this study were to: (1) evaluate the probability distribution of TOC and extractable P at the regional scale in three Major Land Resource Areas (MLRA), and (2) identify appropriate transformations that will result in a normal distribution. Both TOC and extractable P were non-normally distributed in all three MLRAs. Suggested power transformations did not result in normality, but a natural log and negative binomial transformation did produce distributions that met the assumptions of normality in most cases. Statistical analysis of TOC and extractable P data at the regional scale will need to take into account the non-normal distribution of these properties for accurate and precise estimates.

A Continuous Euler Transformation and its Application to Fourier Transforms of Slowly Decaying Functions

A Continuous Euler Transformation and its Application to Fourier Transforms of Slowly Decaying Functions

Integral symmetries, integral invariants, and monodromy matrices for ordinary differential equations

We consider the transfer and monodromy matrices for the degenerate Heun equation. We use an auxiliary ordinary third-order linear differential equation that is “stable” under the integral Euler transformation. We find the invariant of this transformation and express it via the transfer matrix element.

Error analysis and improvement of FILT

The main errors of FILT (fast inversion of Laplace transform, developed by T. Hosono) are approximation error, truncation error, and cancellation error, which are analyzed theoretically and numerically. The following are clarified: (1) the truncation error is easily controlled by increasing the number of terms used and by use of Euler transformation; (2) the approximation and cancellation errors compete with each other, so that the maximum number of significant digits is decreased to 70% of the mantissa digits; (3) proposed higher-order FILTs give better cancellation errors, and the second-order FILT gives a number of significant digits that is greater than 80% of the mantissa digits. © 1998 Scripta Technica, Electron Comm Jpn Pt 2, 81(8): 10–17, 1998