Gaussian integral by Taylor series and applications

In this paper, we present a solution for a specific Gaussian integral. Introducing a parameter that depends on a n index, we found out a general solution inspired by the Taylor series of a simple function. We demonstrated that this parameter represents the expansion coefficients of this function, a very interesting and new result. We also introduced some Theorems that are proved by mathematical induction. As a test for the solution presented here, we investigated a non-extensive version for the particle number density in Tsallis framework, which enabled us to evaluate the functionality of the method. Besides, solutions for a certain class of the gamma and factorial functions are derived. Moreover, we presented a simple application in fractional calculus. In conclusion, we believe in the relevance of this work because it presents a solution for the Gaussian integral from an unprecedented perspective.

Possible changes of the isotopic composition of ice wedge as a result of self-diffusion

  The subject of this research is the assessment of changes of the isotopic composition of ice wedge as a result of self-diffusion. The author offers a relatively simple method of considering the impact of the process of self-diffusion upon redistribution of paleo-isotopic composition that allows accurately and quickly estimating the changes in the values δ18О and δ2H values in ice wedges at different periods of the quaternary (1000 years, 10,000 years, 100,000 years, and 1,000,000 years). The patterns of diffusion process in physico-chemical systems are described by two differential forms of Fick's diffusion laws, which are transformed into Gaussian integral, using six-digit Chambers charts for calculation of the threshold values δ18О. The article applies the experimentally determined real values of self-diffusion coefficient in the ice – D=(2–10)×10-15 m2/s. The values of the Gaussian integral are calculated for time intervals, at different distances between samples with different concentration; for the period of 1,000 years its values in all cases exceed 0.99. For longer time intervals, these values change significantly. The calculated probabilistic values of changes in the concentration of heavy oxygen isotopes in ice wedge under the impact of self-diffusion at different distances and concentration gradients are noticeable only for the time period of over 100,000 years.  

Numerical Studies on Damage Behavior of Recycled Aggregate Concrete Based on a 3D Model

This paper develops a 3D base force element method (BFEM) based on the potential energy principle. According to the BFEM, the stiffness matrix and node displacement of any eight-node hexahedral element are derived as a uniform expression. Moreover, this expression is explicitly expressed without a Gaussian integral. A 3D random numerical model of recycled aggregate concrete (RAC) is established. The randomness of aggregate was obtained by using the Monte Carlo random method. The effects of the recycled aggregate substitution and adhered mortar percentage on the elastic modulus and compressive strength are explored under uniaxial compression loading. In addition, the failure pattern is also studied. The obtained data show that the 3D BFEM is an efficient method to explore the failure mechanism of heterogeneous materials. The 3D random RAC model is feasible for characterizing the mesostructure of RAC. Both the substitution of recycled aggregate and the percentage of adhering mortar have a non-negligible influence on the mechanical properties of RAC. As the weak points in the specimen, the old interfacial transition zone (ITZ) and adhered mortar are the major factors that lead to the weakened properties of RAC. The first crack always appears in these weak zones, and then, due to the increase and transfer of stress, approximately two-to-three continuous cracks are formed in the 45°direction of the specimen.

New handy and accurate approximation for the Gaussian integrals with applications to science and engineering

In this work, we propose to approximate the Gaussian integral, the error function and the cumulative distribution function by using the power series extender method (PSEM). The approximations proposed in this paper present a high accuracy for the complete domain [–∞,∞]. Furthermore, the approximations are handy and easy computable, avoiding the application of special numerical algorithms. In order to show its high accuracy, three case studies are presented with applications to science and engineering.

Ewald-Based Methods for Gaussian Integral Evaluation: Application to a New Parametrization of GEM*

Our manuscript describes the implementation and performance of Ewald-based approaches for the calculation of Gaussian integrals in the pmemd.gem code as released in the AMBER18 suite, as well as a new parametrization of our density-based Gaussian Electrostatic Model* (GEM*) force field using CCSD(T)/CBS//SAPT2+3/aug-cc-pVTZ data for the fit.