Proof the Skewes’ number is not an integer using lattice points and tangent line

Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

An extensive extension of exponentiated exponential distribution using alpha power transformation – statistical properties and applications in engineering science

In this article, an extension of exponentiated exponential distribution is familiarized by adding an extra parameter to the parent distribution using alpha power technique. The new distribution obtained is referred to as Alpha Power Exponentiated Exponential Distribution. Various statistical properties of the proposed distribution like mean, variance, central and non-central moments, reliability functions and entropies have been derived. Two real life data sets have been applied to check the flexibility of the proposed model. The new density model introduced provides the better fit when compared with other related statistical models.

Restructured class of estimators for population mean using an auxiliary variable under simple random sampling scheme

With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coe cients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

Improved strategy for computation of population mean under double stratified sampling framework

The article contains a new technique to estimate the mean of the variate of the interest of the finite population with the help of two auxiliary variates. The technique complies well with the stratified population in which each strata proportion is predicted by taking an initial sample called the first phase sample. When the first phase sample is taken, a second sample is taken from the first sample which is called the second phase sample which is used to estimate the mean of the variate of the interest. In our study, we have considered the population which has two correlated auxiliary variates that pass almost through the origin. In such a situation ratio estimation technique and product estimation technique that provides improved estimates of the mean of the variate of the interest. Our technique considers a ratio-product type exponential estimator of which we have established efficiency theoretically as well as empirically.

A stochastic equivalence approach for an Ornstein-Uhlenbeck process driven power system dynamics

This paper develops a stochastic equivalence approach for an Ornstein-Uhlenbeck process-driven power system. The concept of stochastic equivalence coupled with stochastic differential rule plays the important role to develop the stochastic equivalence approach of this paper. This paper also develops the prediction theory of power system dynamics with the OU process as well.

Vector valued nonuniform nonstationary wavelets and associated MRA on local fields

In this paper we study nonstationary wavelets associated with vector valued nonuniform multiresolution analysis on local fields. By virtue of dimension function a complete characterization of vector valued nonuniform nonstationary wavelets is obtained.

Some fractional derivatives of A-function of multivariable

In the present paper, we study and develop Fractional derivatives of multivariable A – function. We derive two theorems which will act as the key formulas from which can obtain their special cases.

Some new inequalities for generalized convex functions pertaining generalized fractional integral operators and their applications

In this paper, authors establish a new identity for a differentiable function using generic integral operators. By applying it, some new integral inequalities of trapezium, Ostrowski and Simpson type are obtained. Moreover, several special cases have been studied in detail. Finally, many useful applications have been found.

Generalized Lindley-Quasi Xgamma distribution

We obtained a new generalization of Lindley-Quasi Xgamma distribution by adding weight parameter to it through weighting technique and have shown the flexibility of proposed model. Expression for reliability measures, order statistics, Bonferroni curves & indices, Renyi entropy along with some other important properties are derived. Maximum likelihood estimation method is put to use for estimation of unknown parameters of proposed model. Simulation study for checking the performance of maximum likelihood estimates and for model comparison is carried out. Proposed model and its related models are fitted to real life data sets and goodness of fit measure Kolmogorov statistic & p-value, loss of information criteria’s AIC, BIC, AICC & HQIC are computed through R software to check the applicability of proposed model in real life. The significance of weight parameter is also tested by using likelihood ratio test for both randomly generated data as well as real life data.

Applying fractional calculus to analyze final consumption and gross investment influence on GDP

This paper points out the possibility of suitable use of Caputo fractional derivative in regression model. Fitting historical data using a regression model seems to be useful in many fields, among other things, for the short-term prediction of further developments in the state variable. Therefore, it is important to fit the historical data as accurately as possible using the given variables. Using Caputo fractional derivative, this accuracy can be increased in the model described in this paper.