Background: Measures of central tendency are one of the foundational concepts of statistics, with the most commonly used measures being mean, median, and mode. While these are all very simple to calculate when data conform to a unimodal symmetric distribution, either discrete or continuous, measures of central tendency are more challenging to calculate for data distributed asymmetrically. There is a gap in the current statistical literature on computing median and mode for most skewed unimodal continuous distributions. For example, for a standardized normal distribution, mean, median, and mode are all equal to 0. The mean, median, and mode are all equal to each other. For a more general normal distribution, the mode and median are still equal to the mean. Unfortunately, the mean is highly affected by extreme values. If the distribution is skewed either positively or negatively, the mean is pulled in the direction of the skew; however, the median and mode are more robust statistics and are not pulled as far as the mean. The traditional response is to provide an estimate of the median and mode as current methodological approaches are limited in determining their exact value once the mean is pulled away. Methods: The purpose of this study is to test a new statistical method, utilizing the first order and second order partial derivatives in Taylor series expansion, for approximating the median and mode of skewed unimodal continuous distributions. Specifically, to compute the approximated mode, the first order derivatives of the sum of the first three terms in the Taylor series expansion is set to zero and then the equation is solved to find the unknown. To compute the approximated median, the integration from negative infinity to the median is set to be one half and then the equation is solved for the median. Finally, to evaluate the accuracy of our derived formulae for computing the mode and median of the skewed unimodal continuous distributions, simulation study will be conducted with respect to skew normal distributions, skew t-distributions, skew exponential distributions, and others, with various parameters. Conclusions: The potential of this study may have a great impact on the advancement of current central tendency measurement, the gold standard used in public health and social science research. The study may answer an important question concerning the precision of median and mode estimates for skewed unimodal continuous distributions of data. If this method proves to be an accurate approximation of the median and mode, then it should become the method of choice when measures of central tendency are required.

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In this study, novel Ray Based finite difference method for Time Domain electromagnetics(RBTD) has been developed. Instead of solving Maxwell&rsquo / s hyperbolic partial differential equations directly, Geometrical Optics tools (wavefronts, rays) and Taylor series have been utilized. Discontinuities of electromagnetic fields lie on wavefronts and propagate along rays. They are transported in the computational domain by transport equations which are ordinary differential equations. Then time dependent field solutions at a point are constructed by using Taylor series expansion in time whose coefficients are these transported distincontinuties. RBTD utilizes grid structure conforming to wave fronts and rays and treats all electromagnetic problems, regardless of their dimensions, as one dimensional problem along the rays. Hence CFL stability condition is implemented always at one dimensional eqaulity case on the ray. Accuracy of RBTD depends on the accuracy of grid generation and numerical solution of transport equations. Simulations for isotropic medium (homogeneous/inhomogeneous) have been conducted. Basic electromagnetic phenomena such as propagation, reflection and refraction have been implemented. Simulation results prove that RBTD eliminates numerical dispersion inherent to FDTD and is promising to be a novel method for computational electromagnetics.

Submitted by Leonardo Cavalcante (leo.ocavalcante@gmail.com) on 2018-05-02T16:28:27Z No. of bitstreams: 1 Arquivototal.pdf: 6277340 bytes, checksum: 0dbd4fb516303e62d7475fa9335d391b (MD5) / Made available in DSpace on 2018-05-02T16:28:27Z (GMT). No. of bitstreams: 1 Arquivototal.pdf: 6277340 bytes, checksum: 0dbd4fb516303e62d7475fa9335d391b (MD5) Previous issue date: 2017-08-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work,we present two important concepts: Taylor Polynomialand Taylor Series. We discus show theTaylor Polynomial can be used toapproximate the value of Analytic function sin the neighbor hoodo fagiven point, an destimate the precision of the approximation obtained. Subsequently,we study the possibility oflocallyre- presenting functions through a power system,called theTaylor Serie. We concludeby presenting some application sof the result sobtained. / Neste trabalho,abordamos dois conceitos importantes:o Polinômiode Taylor e a Série de Taylor. Apresentamos como o Polinômio de Taylor pode ser usado para aproximar o valor de funções analíticas na vizinhança de um ponto determinado e esti- mamos a precisão da aproximação obtida.Posteriormente,estudamos a possibilidade de representar,localmente,funções através de uma serie de potências,chamadas série de Taylor Finalizamos apresentando algumas aplicações dos resultados obtidos.

The objective of this project is to get familiar with the numerical solution of partial differential equations. This solution will be implemented by using a grid refinement based on the aposteriory error estimation.

This paper is concerned with the problem of computing theapproximate D-optimal design for polynomial regression with weight function w(x)>0 on the design interval I=[m_0-a,m_0+a]. It is shown that if w'(x)/w(x) is a rational function on I and a is close to zero, then the problem of constructing D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix including a finite number of auxiliary unknown constants, which can be approximated by a Taylor expansion. We provide a recursive algorithm to compute Taylor expansion of these constants. Moreover, the D-optimal interior support points are the zeros of a polynomial which has coefficients that can be computed from a linear system.

This thesis deals with the design system for multiple integrals for diferential expression with space variables. Today, integration is one of engineering problems. Reader is acquainted with different method of integration, then with numerican integration and Taylor series. The practical aim of this work is to design software and hardware system of numerican integration multiple integrals.

This term project describes transformation of system of diferential equations into polynomial form. Such transformed systems of diferential equations can be subsequently solved using Taylor series. This method enables computing of initial problem's numeric solution using dynamical order selection in order to achieve required accuracy. The work mathematically proves, that transformed systems of diferential equations have the same solution as the original systems. This transformation can be used for all mathematic functions commonly used in technical applications. The work also focuses on optimization of given problem and implements it in programme taylor. This progamme enables user to solve given diferential equations with chosen parameters.

This thesis deals with continuous system simulation. The systems can be described by system of differential equations or block diagram. Differential equations are usually solved by numerical methods that are integrated into simulation software such as Matlab, Maple or TKSL. Taylor series method has been used for numerical solutions of differential equations. The presented method has been proved to be both very accurate and fast and also procesed in parallel systems. The aim of the thesis is to design, implement and compare a few versions of the parallel system.

Uncertainty analysis is an important part of system design. The formula for error propagation through a system model that is most-often cited in literature is based on a first-order Taylor series. This formula makes several important assumptions and has several important limitations that are often ignored. This thesis explores these assumptions and addresses two of the major limitations. First, the results obtained from propagating error through nonlinear systems can be wrong by one or more orders of magnitude, due to the linearization inherent in a first-order Taylor series. This thesis presents a method for overcoming that inaccuracy that is capable of achieving fourth-order accuracy without significant additional computational cost. Second, system designers using a Taylor series to propagate error typically only propagate a mean and variance and ignore all higher-order statistics. Consequently, a Gaussian output distribution must be assumed, which often does not reflect reality. This thesis presents a proof that nonlinear systems do not produce Gaussian output distributions, even when inputs are Gaussian. A second-order Taylor series is then used to propagate both skewness and kurtosis through a system model. This allows the system designer to obtain a fully-described non-Gaussian output distribution. The benefits of having a fully-described output distribution are demonstrated using the examples of both a flat rolling metalworking process and the propeller component of a solar-powered unmanned aerial vehicle.