Approximation methods

In the present work, an approximation method was used to determine both the crystallite size and microstrain from XRD profile of TiSiN thin film deposited on high speed steel substrates. The estimated crystallite size obtained via this approximation method was in good agreement with the resulting microstructure observation using the scanning electron microscope (SEM). The approximation method was used to determine microstrain, and its corresponding compressive stress was related to the result of scratch adhesion measurement of the TiSiN thin film. Comparison of crystallite size and microstrain were investigated using different definitions of line broadening, β. The approximation method was found to be useful in cases when crystallite size and microstrain contributed in the line broadening simultaneously. This research demonstrated the reliability of using the approximation method in determining the resulting crystallite size and microstrain from the XRD line broadening analysis in the TiSiN thin films. © 2011 Elsevier Ltd. All rights reserved.

http://www.sciencedirect.com/science/article/pii/S0042207X11003745
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The aim of this paper is to employ fractional order proportional integral derivative (FO-PID) controller and integer order PID controller to control the position of the levitated object in a magnetic levitation system (MLS), which is inherently nonlinear and unstable system. The proposal is to deploy discrete optimal pole-zero approximation method for realization of digital fractional order controller. An approach of phase shaping by slope cancellation of asymptotic phase plots for zeros and poles within given bandwidth is explored. The controller parameters are tuned using dynamic particle swarm optimization (dPSO) technique. Effectiveness of the proposed control scheme is verified by simulation and experimental results. The performance of realized digital FO-PID controller has been compared with that of the integer order PID controllers. It is observed that effort required in fractional order control is smaller as compared with its integer counterpart for obtaining the same system performance.

This paper provides an introduction to least-squares collocation, a process already well established in statistical geodesy for prediction, filtering and modelling.  Variance/covariance propagation laws are shown to have hitherto-unnoticed applications including prediction and filtering formulae usually obtained by least-squares criteria.  An example from coordinate transformations is given in which collocation is shown to be more accurate than traditional least-squares modelling.

For three-parameter datum transformations to be applied rigorously, geodetic coordinates on the first ellipsoid need to be converted to Cartesian coordinates before application of the shifts, then converted to geodetic coordinates on the second ellipsoid. The Standard Molodensky method of datum transformation is more direct but is inexact. It also fails to reproduce the original coordinates when applied forward and back. However, this paper shows a pattern of proportionality between the misclosures and the errors in the forward approximations. This gives rise to a new method of computing the transformations, best described as " Standard Molodensky in two stages with applied misclosures " (SMITSWAM). The method is shown to be more than 1600 times more accurate than Standard Molodensky, coming close to the accuracy of the rigorous approach. SMITSWAM is also shown to be around 48% faster than the traditional form of the rigorous method which uses iteration.

The first derivative of a real-valued function may be approximated at a certain point by the derivative of a polynomial collocating with the function at this point and a number of other distinct points.  The particular points which minimise the magnification of any rounding errors in the function values for any fixed point of truncation error are shown to be closely related to the turning points of a related Chebyshev polynomial.

Neutrosophic concept is known undirected graph theory to involve with complex logistic networks, not clearly given and unpredictable real life situations, where fuzzy logic malfunctions to model. The transportation objective is to ship all logistic nodes in the network. The logistic network mostly experiences in stable condition, but for some edges found to be volatile. The weight of these erratic edges may vary at random (bridgelifting/bascule, ad hoc accident on road, traffic condition) In this article, we propose an approximation algorithm for solving minimum spanning tree (MST) of an undirected neutrosophic graphs (UNG), in which the edge weights represent neutrosophic values. The approximation upon the balanced score calculation is introduced for all known configurations in alternative MST. As the result, we further compute decisive threshold value for the weak weights amid minimum cost pre-computation. If the threshold triggers then the proper MST can direct the decision and avoid post-computation. The proposed algorithm is also related to other existing approaches and a numerical analysis is presented.

This paper considers the problem of calculating the vertical gravitational attraction at an arbitrary point of a rectangular mass with uniform density.  A rigorous approach is needed for a large mass which is very close to that point.  Treating the mass as an infinite summation of point masses requires the computation of a triple integral.  However, the problem can be solved by treating the mass as a summation of cylindrical strips.  A subroutine and an algorithm are proposed.

In the production of geographic data covering regions of the world, a typical project requirement is a set of matrices totalling 100 million heights interpolated from contour data.  An investigation on a sample area using the DAP shows a 10 to 1 advantage in speed, even when based on a method originally designed for a serial computer.  The use of the DAP to manipulate and re-arrange data with inconvenient structures is also demonstrated.