<p> Several test statistics, which are known, can be used for testing for exponentiality. A new test statistic TE is proposed. TE is based on a censored sample and is similar to Tiku's T statistic for testing for normality. The distribution of TE tends to normality with increasing sample size. Besides, TE is easy to compute and is both origin and scale invariant. The power of TE for non-exponential distributions is comparable with Shapiro & Wilk statistic W-exponential. </p> / Thesis / Master of Science (MSc)

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The convergence properties of the permanent exponential interpolation series f(Z) = 1^Zf(0) + (2^Z - 1^Z)Δf(0) + (3^Z - 2.2^Z + 1^Z/2!)Δ(Δ - 1)f(0) + … have been investigated. Using the following notation U_n(Z) = ∑ⁿ_k=0 (-1)^k(ⁿ_k)(n - i + 1)^Z, Δ⁽ⁿ⁾ f(0) = Δ(Δ-1)…(Δ - n + 1)f(0), the series can be written more compactly as f(Z) = ∑^∞₀ U_n(Z)/n!Δ⁽ⁿ⁾ f(0). It is shown that Δ⁽ⁿ⁾ f(0) can be represented as Δ⁽ⁿ⁾ f(0) = M_n(f) = 1/2πi ∫_Γ (e^ω - 1)⁽ⁿ⁾ F(ω)dω, where F(ω) is the Borel transform of f(Z) and Γ encloses the convex hull of the singularities of F(ω). It is further shown that the series ∑^∞₀ U_n(Z)/n! (e^ω - 1)⁽ⁿ⁾ forms a uniformly convergent Gregory-Newton series, convergent to e^Zω in any bounded region in the strip |I(ω)| < π/2. The Polya representation of an entire function of exponential type is then formed, and the method of kernel expansion (R. P. Boas, and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, 1964) yields the desired result. This result is summed up in the following: Theorem Any entire function of exponential type such that the convex hull of the set of singularities of its Borel transform lies in the strip |I(ω)| < π/2. admits the convergent exponential interpolation series expansion f(Z) = ∑^∞_n=0 U_n(Z)/n!Δ⁽ⁿ⁾ f(0) for all Z. / M.S.

This thesis discusses a development of the exponential and logarithmic functions.

Thesis (Ph. D.)--Illinois State University, 2005. / Title from title page screen, viewed on April 13, 2007. Dissertation Committee: Norma C. Presmeg (chair), Beverly S. Rich, Nerida F. Ellerton, Sherry L. Meier. Includes bibliographical references (leaves 207-217) and abstract. Also available in print.

This study utilized historical population statistics data and related reports to investigate the population distribution and transition trend of Kaohsiung Main Urban Area. There were four purposes of this study; firstly to understand the current population distribution pattern of Kaohsiung Main Urban Area, secondly to investigate the change trend of Kaohsiung Main Urban Area in the past 36 years, thirdly using Kaohsiung Main Urban Area population data to verify the population density model and to examine the correspondence with the current society, and fourthly to explore the impact of the establishment of Kaohsiung Mass Rapid Transit System and Taiwan High Speed Rail to the transition and distribution of population in Kaohsiung Main Urban Area. This study employed the population statistical database from year 1972 to 2007, and divided the Kaohsiung metropolitan area into 32 sub-divisions. The population data was re-organized accordingly in order to perform the systematic analyses and exploration in making the understanding of the current population status and transition process. Furthermore, employing the population data of Kaohsiung metropolitan area, the validation of negative exponential model and quadratic exponential model, the population model proposed by Clark and Newling respectively 58 year ago, were performed to verify the applicability of the theories to modern society. In addition, this study also performed the exploration of population transition and distribution in Kaohsiung metropolitan area during the construction period of Kaohsiung Mass Rapid Transit System and Taiwan High Speed Rail and could be of reference to urban planning and development in the future. This study found that both Clark and Newling¡¦s population model still has high interpretation capability after 58 years later. The Kaohsiung metropolitan area population distribution model and trend forecast proposed by Ho in 1991 is still true. However, because of the slow down in overall development trend of Taiwan, the transition trend of Kaohsiung metropolitan also mitigated. Besides, due to the trend mitigation, the volcano peak phenomenon which was mentioned in the literature review will be postponed and expected to happen in year 2018.

The theory of partial differential equations constitutes today one of the most important topics of scientific understanding. A standard approach for solving a time-dependent partial differential equation consists in discretizing the spatial variables by finite differences or finite elements. This results in a huge system of (stiff) ordinary differential equations that has to be integrated in time. Exponential integrators constitute an interesting class of numerical methods for the time integration of stiff systems of differential equations. Their efficient implementation heavily relies on the fast computation of the action of certain matrix functions; among those, the matrix exponential is the most prominent one. In this manuscript, we go through the steps that led to the development of backward error accurate routines for computing the action of the matrix exponential.

Doctor of Philosophy / Department of Mathematics / Christopher G. Pinner

International Telemetering Conference Proceedings / October 30-November 02, 1995 / Riviera Hotel, Las Vegas, Nevada / The national telemetry ranges are being pushed to provide higher data rate telemetry services by users with increasingly complex test procedure for increasingly complex weapon systems. At the same time they are having trouble obtaining more spectrum in which to provide these higher rates because of the demand for spectrum in SHF range from various mobile/cellular Personal Communications Services (PCS) as well as congress’s desire to auction spectrum and to transfer as much spectrum as possible to commercial uses. In light of these pressures the industry is in need of a modulation standard that will out perform the existing PCM/FM standard. The motivation for the present review and analysis of the performance of various coded/uncoded modulation schemes arises from this issue. Comparison of the performance of these schemes will be utilized in the following work to find a suitable solution to the existing problem.

In this dissertation, we study iterative methods for computing eigenvalues and exponentials of large matrices. These types of computational problems arise in a large number of applications, including mathematical models in economics, physical and biological processes. Although numerical methods for computing eigenvalues and matrix exponentials have been well studied in the literature, there is a lack of analysis in inexact iterative methods for eigenvalue computation and certain variants of the Krylov subspace methods for approximating the matrix exponentials. In this work, we proposed an inexact inverse subspace iteration method that generalizes the inexact inverse iteration for computing multiple and clustered eigenvalues of a generalized eigenvalue problem. Compared with other methods, the inexact inverse subspace iteration method is generally more robust. Convergence analysis showed that the linear convergence rate of the exact case is preserved. The second part of the work is to present an inverse Lanczos method to approximate the product of a matrix exponential and a vector. This is proposed to allow use of larger time step in a time-propagation scheme for solving linear initial value problems. Error analysis is given for the inverse Lanczos method, the standard Lanczos method as well as the shift-and-invert Lanczos method. The analysis demonstrates different behaviors of these variants and helps in choosing which variant to use in practice.