On Weighted Simpson’s 38 Rule

Numerical approximations of definite integrals and related error estimations can be made using Simpson’s rules (inequalities). There are two well-known rules: Simpson’s 13 rule or Simpson’s quadrature formula and Simpson’s 38 rule or Simpson’s second formula. The aim of the present paper is to extend several inequalities that hold for Simpson’s 13 rule to Simpson’s 38 rule. More precisely, we prove a weighted version of Simpson’s second type inequality and some Simpson’s second type inequalities for Lipschitzian, bounded variations, convex functions and the functions that belong to Lq. Some applications of the second type Simpson’s inequalities relate to approximations of special means and Simpson’s 38 formula, and moments of random variables are made.

New-type Hoeffding’s inequalities and application in tail bounds

It is well known that Hoeffding's inequality has a lot of applications in the signal and information processing fields. How to improve Hoeffding's inequality and find the refinements of its applications have always attracted much attentions. An improvement of Hoeffding inequality was recently given by Hertz [<a href="#1">1</a>]. Eventhough such an improvement is not so big, it still can be used to update many known results with original Hoeffding's inequality, especially for Hoeffding-Azuma inequality for martingales. However, the results in original Hoeffding's inequality and its refined version by Hertz only considered the first order moment of random variables. In this paper, we present a new type of Hoeffding's inequalities, where the high order moments of random variables are taken into account. It can get some considerable improvements in the tail bounds evaluation compared with the known results. It is expected that the developed new type Hoeffding's inequalities could get more interesting applications in some related fields that use Hoeffding's results.

On Some New Weighted Inequalities for Differentiable Exponentially Convex and Exponentially Quasi-Convex Functions with Applications

In this article, we aim to establish several inequalities for differentiable exponentially convex and exponentially quasi-convex mapping, which are connected with the famous Hermite–Hadamard (HH) integral inequality. Moreover, we have provided applications of our findings to error estimations in numerical analysis and higher moments of random variables.

Application of the Method of Galerkin to Non Linear Problem Stochastic Heat Conduction One-Dimensional

The Galerkin method is applied in the non-linear stochastic diffusion problem. The uncertainty is present in the coefficients of diffusion equation. The uncertainty is modeled by random variables. The chaos polynomials is used to approximate the stochastic behavior of the problem. The approximate solutions obtained through Galerkin method are compared with Monte Carlo simulation in terms of the statistical moments of random variables generated by the random field solution.