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 Wenhao Gui Journal of Quality and Reliability Engineering , 2013, DOI: 10.1155/2013/219473 Abstract: We investigate the statistical inferences and applications of the half exponential power distribution for the first time. The proposed model defined on the nonnegative reals extends the half normal distribution and is more flexible. The characterizations and properties involving moments and some measures based on moments of this distribution are derived. The inference aspects using methods of moment and maximum likelihood are presented. We also study the performance of the estimators using the Monte Carlo simulation. Finally, we illustrate it with two real applications. 1. Introduction The well-known exponential power (EP) distribution or the generalized normal distribution has the following density function: where is the shape parameter. This family consists of a wide range of symmetric distributions and allows continuous variation from normality to nonnormality. It includes the normal distribution as the special case when and the Laplace distribution when . Nadarajah [1] provided a comprehensive treatment of its mathematical properties. Its tails can be more platykurtic ( ) or more leptokurtic ( ) than the normal distribution ( ). The distribution has been widely used in the Bayes analysis and robustness studies (see Box and Tiao [2], Genc [3], Goodman and Kotz [4], and Tiao and Lund [5].) On the other hand, since the most popular models used to describe the lifetime process are defined on nonnegative measurements, which motivate us to take a positive truncation in the model (1) and develop a half exponential power (HEP) distribution. As far as we know, this model has not been previously studied although, we believe, it plays an important role in data analysis. The resulting nonnegative half exponential power distribution generalizes the half normal (HN) distribution, and it is more flexible. In our work, we aim to investigate the statistical features of the nonnegative model and apply them to fit the lifetime data. The rest of this paper is organized as follows: in Section 2, we present the new distribution and study its properties. Section 3 discusses the inference, moments, and maximum likelihood estimation for the parameters. In Section 4, we discuss a useful technique, a half normal plot with a simulated envelope, to assess the model adequacy. Simulation studies are performed in Section 5. Section 6 gives two illustrative examples and reports the results. Section 7 concludes our work. 2. The Half Exponential Power Distribution 2.1. The Density and Hazard Function Definition 1. A random variable has a half exponential power slash distribution if
 Grzegorz Bancerek Formalized Mathematics , 2009, DOI: 10.2478/v10037-009-0032-8 Abstract: An epsilon number is a transfinite number which is a fixed point of an exponential map: ω = . The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, &uarr2;). Namely, the ordinal indexing of epsilon numbers is defined as follows: and for limit ordinal λ: Tetration stabilizes at ω: Every ordinal number α can be uniquely written as where κ is a natural number, n1, n2, …, nk are positive integers, and β1 > β2 > … > βκ are ordinal numbers (βκ = 0). This decomposition of α is called the Cantor Normal Form of α.
 Radu Pantilie Mathematics , 2013, Abstract: We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere, with nonnegative normal bundles.
 Physics , 2006, DOI: 10.1088/1126-6708/2007/02/040 Abstract: It is proved that the Laurent expansion of the following Gauss hypergeometric functions, 2F1(I1+a*epsilon, I2+b*ep; I3+c*epsilon;z), 2F1(I1+a*epsilon, I2+b*epsilon;I3+1/2+c*epsilon;z), 2F1(I1+1/2+a*epsilon, I2+b*epsilon; I3+c*epsilon;z), 2F1(I1+1/2+a*epsilon, I2+b*epsilon; I3+1/2+c*epsilon;z), 2F1(I1+1/2+a*epsilon,I2+1/2+b*epsilon; I3+1/2+c*epsilon;z), where I1,I2,I3 are an arbitrary integer nonnegative numbers, a,b,c are an arbitrary numbers and epsilon is an arbitrary small parameters, are expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with polynomial coefficients. An efficient algorithm for the calculation of the higher-order coefficients of Laurent expansion is constructed. Some particular cases of Gauss hypergeometric functions are also discussed.
 Simin Feng Physics , 2011, DOI: 10.1103/PhysRevLett.108.193904 Abstract: Contrary to conventional wisdom that light bends away from the normal at the interface when it passes from high to low refractive index media, here we demonstrate an exotic phenomenon that the direction of electromagnetic power bends towards the normal when light is incident from arbitrary high refractive index medium to \epsilon-near-zero metamaterial. Moreover, the direction of the transmitted beam is close to the normal for all angles of incidence. In other words, the electromagnetic power coming from different directions in air or arbitrary high refractive index medium can be redirected to the direction almost parallel to the normal upon entering the \epsilon-near-zero metamaterial. This phenomenon is counterintuitive to the behavior described by conventional Snell's law and resulted from the interplay between \epsilon-near-zero and material loss. This property has potential applications in communications to increase acceptance angle and energy delivery without using optical lenses and mechanical gimbals.
 M. Yu. Kalmykov Physics , 2006, DOI: 10.1088/1126-6708/2006/04/056 Abstract: The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hypergeometric functions. The epsilon-expansion of functions of one of this type (type F in our classification) demands the introduction of new functions related to generalizations of elliptic functions. For the five other types of functions the higher-order epsilon-expansion up to functions of weight 4 are constructed. The result of the expansion is expressible in terms of Nielsen polylogarithms only. The reductions and epsilon-expansion of q-loop off-shell propagator diagrams with one massive line and q massless lines and q-loop bubble with two-massive lines and q-1 massless lines are considered. The code (Mathematica/FORM) is available via the www at this URL http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html
 Benjamin Lopez-Garcia;Devata Venkata Bhyrava Murthy;Alonso Corona-Chavez PIER , 2011, DOI: 10.2528/PIER10100707 Abstract: A new design of microwave band pass filter design is presented using metamaterial-inspired Epsilon Near Zero (ENZ) and Mu Near Zero (MNZ) behaviors. These filters are based on waveguide technology. The proposed structure allows us to reduce the number of tunnels normally used for passband filter design by reducing its size. It is also incorporated the half mode concept to the tunnels leading a greater miniaturization. Two Chebyshev filters with two and four-poles were designed, fabricated and measured showing good agreement between simulated and experimental results.
 Physics , 2003, Abstract: Following the discovery of superconductivity in epsilon-iron, subsequent experiments hinted at non-Fermi liquid behaviour of the normal phase and sensitive dependence of the superconducting state on disorder, both signatures of unconventional pairing. We report further resistive measurements under pressure of samples of iron from multiple sources. The normal state resistivity of epsilon-iron varied as rho_0+AT^{5/3} at low temperature over the entire superconducting pressure domain. The superconductivity could be destroyed by mechanical work, and was restored by annealing, demonstrating sensitivity to the residual resistivity rho_0. There is a strong correlation between the rho_0 and A coefficients and the superconducting critical temperature T_c. Within the partial resistive transition there was a significant current dependence, with V(I)=a(I-I_0)+bI^2, with a >> b, possibly indicating flux-flow resistivity, even in the absence of an externally applied magnetic field.
 Journal of Probability and Statistics , 2013, DOI: 10.1155/2013/491628 Abstract: We study some mathematical properties of the beta generalized half-normal distribution recently proposed by Pescim et al. (2010). This model is quite flexible for analyzing positive real data since it contains as special models the half-normal, exponentiated half-normal, and generalized half-normal distributions. We provide a useful power series for the quantile function. Some new explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability, and entropy. We demonstrate that the density function of the beta generalized half-normal order statistics can be expressed as a mixture of generalized half-normal densities. We obtain two closed-form expressions for their moments and other statistical measures. The method of maximum likelihood is used to estimate the model parameters censored data. The beta generalized half-normal model is modified to cope with long-term survivors may be present in the data. The usefulness of this distribution is illustrated in the analysis of four real data sets. 1. Introduction Cooray and Ananda [1] pioneered the generalized half-normal (GHN) distribution with shape parameter and scale parameter defined by the cumulative distribution function (cdf) where the standard normal cdf and the error function are given by Following an idea due to Eugene et al. [2], Pescim et al. [3] proposed the beta generalized half-normal (BGHN) distribution, which seems to be superior over the GHN model for some applications. The justification for the practicability of this model is based on the fatigue crack growth under variable stress or cyclic load. In this paper, we study several mathematical properties of the BGHN model with the hope that it will attract wider applications in reliability, engineering and in other areas of research. The four-parameter BGHN cdf is defined from (1) by (for ), where is the beta function, is the incomplete beta function ratio, and and are two additional shape parameters. The probability density function (pdf) and the hazard rate function (hrf) corresponding to (3) are respectively. Hereafter, a random variable with pdf (4) is denoted by . Pescim et al. [3] demonstrated that the cdf and pdf of can be expressed as infinite power series of the GHN cumulative distribution. Here, all expansions in power series are around the point zero. If is a real noninteger, we can expand the binomial term in (3) to obtain where . The pdf corresponding to (6) can be expressed as If is an integer, (7) provides the BGHN density function as an infinite power series of the GHN cumulative distribution.
 A. Pich Physics , 2000, DOI: 10.1016/S0920-5632(00)01113-0 Abstract: The Standard Model prediction for epsilon'/epsilon is updated, taking into account the most recent theoretical developments. The final numerical value, epsilon'/\epsilon = (17\pm 6) x 10^{-4}, is in good agreement with present measurements.
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