For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties. We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.

A modified F-expansion method to find the exact traveling wave solutions of two-component nonlinear partial differential equations (NLPDEs) is discussed. We use this method to construct many new solutions to the nonlinear Whitham-Broer-Kaup system (1+1)-dimensional. The solutions obtained include Jacobi elliptic periodic wave solutions which exactly degenerate to the soliton solutions, triangular periodic wave solutions, exponential solutions and rational solutions under certain limitet conditions. In addition, some figures of partial solutions are provided for direct-viewing analysis. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.

This article presents a formula and a series for approximating 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 superior 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.

Membership function based on probability density functionچندین روش به منظور برآورد تابع چگالی احتمال وجود دارد. از سوی دیگر، در نظریه مجموعه هایفازی یکی از روش های ساختن تابع عضویت بر پایه ی مجموعه داده، روش مبتنی بر تابع چگالی احتمال است.با توجه به روش های متداول در برآورد تابع چگالی، این مسئله می تواند به محاسبه انواع تابع عضویت بر پایهیک مجموعه داده منجر شود. در این مقاله، برخی از این روش ها بیان و با مثال عددی تشریح می شوند..

Soil hydraulic conductivity is a crucial parameter in modeling flow process in soils and in decision making on water management issues. In this research by using 41 soil samples with 11 different soil textures covering sandy, loam, and clay from the UNSODA database, an investigation was conducted on the applicability of specific liquid-vapor interfacial area around the soil particles concept for predicting soil hydraulic conductivity function fitted by van Genuchten model. The results showed that the specific liquid-vapor interfacial area (Aws) meniscus around the soil particles does not monotonically increase as suction increases for all soils. Soil moisture corresponding to the maximum Aws is not equal to residual water content but its value depends on soil characteristic curve. The results showed that between FC and PWP, variable viscosity of water around soil particles did not have a marked effect on unsaturated hydraulic conductivity computations. Variable weight of water in the soil, as compared with constant water weight, had a marked effect on the results.