Approximations to the Normal Distribution Function and An Extended Table for the Mean Range of the Normal Variables

This article presents a formula and a series for approx¬imating the normal distribution function. Over the whole range of the normal variable z, the proposed formula has the greatest absolute error less than 6.5e - 09, and series has a very high accuracy. We examine the accuracy of our proposed formula and series for various values of z’s. In the sense of accuracy, our formula and series are su¬perior to other formulae and series available in the literature. Based on the proposed formula an extended table for the mean range of the normal variables is established

Approximations to the Normal Distribution Function and An Extended Table for the Mean Range of the Normal Variables

This article presents a formula and a series for approx¬imating the normal distribution function. Over the whole range of the normal variable z, the proposed formula has the greatest absolute error less than 6.5e - 09, and series has a very high accuracy. We examine the accuracy of our proposed formula and series for various values of z’s. In the sense of accuracy, our formula and series are su¬perior to other formulae and series available in the literature. Based on the proposed formula an extended table for the mean range of the normal variables is established

Comparison of the h-index for different fields of research using bootstrap methodology

An important disadvantage of the h-index is that typically it cannot take into account the specific field of research of a researcher. Usually sample point estimates of the average and median h-index values for the various fields are reported that are highly variable and dependent of the specific samples and it would be useful to provide confidence intervals of prediction accuracy. In this paper we apply the non-parametric bootstrap technique for constructing confidence intervals for the h-index for different fields of research. In this way no specific assumptions about the distribution of the empirical h-index are required as well as no large samples since that the methodology is based on resampling from the initial sample. The results of the analysis showed important differences between the various fields. The performance of the bootstrap intervals for the mean and median h-index for most fields seems to be rather satisfactory as revealed by the performed simulation

Advantages of the Ilizarov external fixation in the management of intra-articular fractures of the distal tibia

<p>Abstract</p> <p>Background</p> <p>Treatment of distal tibial intra-articular fractures is challenging due to the difficulties in achieving anatomical reduction of the articular surface and the instability which may occur due to ligamentous and soft tissue injury. The purpose of this study is to present an algorithm in the application of external fixation in the management of intra-articular fractures of the distal tibia either from axial compression or from torsional forces.</p> <p>Materials and methods</p> <p>Thirty two patients with intra-articular fractures of the distal tibia have been studied. Based on the mechanism of injury they were divided into two groups. Group I includes 17 fractures due to axial compression and group II 15 fractures due to torsional force. An Ilizarov external fixation was used in 15 patients (11 of group I and 4 of group II). In 17 cases (6 of group I and 11 of group II) a unilateral hinged external fixator was used. In 7 out of 17 fractures of group I an additional fixation of the fibula was performed.</p> <p>Results</p> <p>All fractures were healed. The mean time of removal of the external fixator was 11 weeks for group I and 10 weeks for group II. In group I, 5 patients had radiological osteoarthritic lesions (grade III and IV) but only 2 were symptomatic. Delayed union occurred in 3 patients of group I with fixed fibula. Other complications included one patient of group II with subluxation of the ankle joint after removal of the hinged external fixator, in 2 patients reduction found to be insufficient during the postoperative follow up and were revised and 6 patients had a residual pain. The range of ankle joint motion was larger in group II.</p> <p>Conclusion</p> <p>Intra-articular fractures of the distal tibia due to axial compression are usually complicated with cartilaginous problems and are requiring anatomical reduction of the articular surface. Fractures due to torsional forces are complicated with ankle instability and reduction should be augmented with ligament repair, in order to restore normal movement of talus against the mortise. Both Ilizarov and hinged external fixators are unable to restore ligamentous stability. External fixation is recommended only for fractures of the ankle joint caused by axial compression because it is biomechanically superior and has a lower complication rate.</p

Half‐Cauchy and Power Cauchy Distributions: Ordinary Differential Equations

In this chapter, homogenous ordinary differential equations (ODES) of different orders were obtained for the probability density function, quantile function, survival function inverse survival function, hazard function and reversed hazard functions of half‐Cauchy and power Cauchy distributions. This is possible since the aforementioned probability functions are differentiable. Differentiation and modified product rule were used to obtain the required ordinary differential equations, whose solutions are the respective probability functions. The different conditions necessary for the existence of the ODEs were obtained and it is almost in consistent with the support that defined the various probability functions considered. The parameters that defined each distribution greatly affect the nature of the ODEs obtained. This method provides new ways of classifying and approximating other probability distributions apart from half‐Cauchy and power Cauchy distributions considered in this chapter. In addition, the result of the quantile function can be compared with quantile approximation using the quantile mechanics