The generalized Taylor expansion including a secret auxiliary parameter h which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of h can’t be understood in the frame of the homotopy analysis method. This is a serious shortcoming of Liao’s method. We solve the problem. Through a detailed study of a simple ...

We describe a new method for analyzing, classifying, and evaluating filters that can be applied to interpolation filters as well as to arbitrary derivative filters of any order. Our analysis is based on the Taylor series expansion of the convolution sum. Our analysis shows the need and derives the method for the normalization of derivative filter weights. Under certain minimal restrictions of t...

In this paper, a direct synthesis approach to fractional order controller design Is investigated. The proposed algorithm makes use of Taylor series of both desired closed-loop and actual closed-loop transfer function which is truncated to the first five terms. FOPID Controller parameters are synthesized in order to match the closed-loop response of the plant to the desired closed-loop response....

It is known that stability properties of delay-differential equations are not preserved by Taylor series expansion of the delayed term. Still, this technique is often used to approximate delayed systems by ordinary differential equations in different engineering and biological applications. In this brief, it is demonstrated through some simple second-order scalar systems that low-order Taylor s...

The major aim of this article is modeling of nonlinear systems with friction structure that, thismethod is essentially extended based on taylore expansion polynomial. So in this study, thetaylore expansion was extended in the generalized form for the differential equations of the statespaceform. The proposed structure is based on multi independent variables taylore extended.According to the pro...

In the field of Geometric Function Theory, one can not deny the importance of analytic and univalent functions. The characteristics of these functions including their taylor series expansion, their coefficients in these representations as well as their associated functional inequalities have always attracted the researchers. In particular, Fekete-Szegö inequality is one of such vastly studied a...

We describe a formula for the Taylor series expansion of the Gaussian kernel around the origin of Rn × R.

For the continuous-time risk model with compound Poisson input, the ((nite-horizont) joint probability P (t; X x; Y y) of ruin time , surplus X just before ruin and deecit Y at ruin time is considered as a function of the arrival rate of claims. It is expanded into a Taylor series at = 0. A certain extension of a corresponding result for innnite-horizont joint probabilities, which previously ha...

Molecular dynamics computer simulation methods are very important for understanding mechanisms of chemical, physical, and biological processes. The reliability of molecular dynamics simulations strongly depends on the integration schemes used in the simulations. In this work, we developed new rigid body integration schemes for molecular dynamics simulations. Our approach is based on a numerical...

Most analytical models include variables that have randomness or uncertainties associated with them. These models span many scientific and engineering disciplines. Analyzing these models without incorporating the uncertainties may provide misleading results. In this paper, several methods for evaluating uncertainty are summarized. One of these methods is illustrated using the Lotus 1-2-3 spread...