In this article‎, ‎we introduce refinements of Young-type inequalities for scalars‎. ‎We then extend these inequalities to encompass versions based on Hilbert-Schmidt and trace norms‎. ‎The results presented in this paper represent refinements of the findings originally established by Nasiri and Shakoori [6].

The purpose of this paper is to study the bivariate extension of the GENERALIZED Baskakov-KANTOROVICH operators and obtain results on the degree of approximation, Voronovskaja type theorems and their first order derivatives in polynomial weighted spaces. Furthermore, we illustrate the convergence of the bivariate operators to a certain function through graphics using Matlab algorithm. We also discuss the comparison of the convergence of the bivariate GENERALIZED Baskakov KANTOROVICH operators and the bivariate Sz\'{a}sz-KANTOROVICH operators to the function through illustrations using Matlab.

This paper explores an extension of the bivariate Schur-CONSTANT model, introducing an additional parameter to its associated Archimedean copula for greater flexibility. We analyze the dependence properties of the proposed model and illustrate our findings with several examples. Furthermore, provide a likelihood ratio test to compare the performance of the extended Archimedean copula with that of the traditional Archimedean subfamily. Two real-data analysis are also included.

In this article, we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases of these parameters for particular convex compact subsets, we obtain the classical KANTOROVICH operators defined in the 11-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity.

A variety of structural members are expected to safely tolerate torsional moments. These include irregularly-shaped cross-sections (e.g., trapezoidal or triangular sections) in some industries which deserve special considerations for the analysis and design under torsional loading. Therefore, the development of novel methods as alternative approaches seems very necessary, partially because of the deficiency of analytical solutions in treating asymmetric solution domains. Semi-analytical and numerical methods appear as desirable alternatives in most cases. One of the proper tools for dealing with the boundary value problems encountered in the torsional analysis is the vibrational method. The KANTOROVICH semi-analytical method, known as an extension of the Rayleigh-Ritz method, has been proven advantageous among the others, mainly because of relaxing the conventional limitations in selecting the primary function for satisfying the boundary conditions. Therefore, the purpose of the present study is to extend the applicability of the KANTOROVICH method to estimate the warping and stress field of arbitrary trapezoidal sections directly. Finally, the efficiency and accuracy of the present solution is verified against a number of existing analytical and numerical methods. The results indicate high precision and rapid convergence of this semi-analytical method.

The goal of this paper is to calculate the order reduction of the GENERALIZED mKdV equation with CONSTANT-coe, cients (gmKdVcc) and the GENERALIZED mKdV equation with variable-coe, cients (gmKdVvc) using the ,-symmetry method. Moreover we obtain Lagrangian and ,-conservation law of the gmKdVcc equation and the gmKdVvc equation using the variational problem method.