Variability in rainfall dispersion in Madhya Pradesh

An attempt has been made to examine distribution and dispersion in rainfall variability in Madhya Pradesh by applying Gamma distribution probability model, The spatial and regional distribution of shape and scale parameters of the Gamma distribution have been examined, Periods of water surpluses and deficiencies have been identified by comparing the probability rainfall with the water requirement. Regression equations have been developed to find probabilitistic rainfall from the mean rainfall. Agronomic practices have been evaluated for efficient utilization of water resources for crop planning.

Bootstrapping Bloch bands

Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.

“The Crooked Smile of TCR”: Banks’ Solvency and Restructuring Costs in the European Banking Industry

In this study, we analyze the probability of bank failure, the expected losses, and the costs of bank restructuring with the application of a lognormal distribution probability function for three categories of European banks, that is, small, medium, and large, over the post-crisis period from 2012 to 2016. Our goal was to determine whether the total capital ratio (TCR) properly reflects banks’ solvency under stress conditions. We identified a phenomenon that one can call the “crooked smile of TCR”. Medium-sized banks with relatively high TCRs performed poorly in stress tests; however, the probability of bank failure increases slightly with the size of the bank, while the TCR decreases. We claim that the focus on capital adequacy measures is not sufficient to achieve the goal of improving banks’ stability and reducing their restructuring costs. Our results are of special importance for medium-sized banks, as these banks are not regularly subjected to publicly available stress tests.

A family of Gamma-generated distributions: Statistical properties and applications

In this paper, we concentrate on the statistical properties of Gamma-X family of distributions. A special case of this family is the Gamma-Weibull distribution. Therefore, the statistical properties of Gamma-Weibull distribution as a sub-model of Gamma-X family are discussed such as moments, variance, skewness, kurtosis and Rényi entropy. Also, the parameters of the Gamma-Weibull distribution are estimated by the method of maximum likelihood. Some sub-models of the Gamma-X are investigated, including the cumulative distribution, probability density, survival and hazard functions. The Monte Carlo simulation study is conducted to assess the performances of these estimators. Finally, the adequacy of Gamma-Weibull distribution in data modeling is verified by the two clinical real data sets. Mathematics Subject Classification: 62E99; 62E15

Multifractal Characteristics Analysis Based on Slope Distribution Probability in the Yellow River Basin, China

Multifractal theory provides a reliable method for the scientific quantification of the geomorphological features of basins. However, most of the existing research has investigated small and medium-sized basins rather than complex and large basins. In this study, the Yellow River Basin and its sub-basins were selected as the research areas, and the generalized fractal dimension and multifractal spectrum were computed and analyzed with a multifractal technique based on the slope distribution probability. The results showed that the Yellow River Basin and its sub-basins exhibit clear multifractal characteristics, which indicates that the multifractal theory can be applied well to the analysis of large-scale basin geomorphological features. We also concluded that the region with the most uneven terrain is the Yellow River Downstream Basin with the “overhanging river”, followed by the Weihe River Basin, the Yellow River Mainstream Basin, and the Fenhe River Basin. Multifractal analysis can reflect the geomorphological feature information of the basins comprehensively with the generalized fractal dimension and the multifractal spectrum. There is a strong correlation between some common topographic parameters and multifractal parameters, and the correlation coefficients between them are greater than 0.8. The results provide a scientific basis for analyzing the geomorphic characteristics of large-scale basins and for the further research of the morphogenesis of the forms.