Using Determinants for Determining Extreme Values of a Function of One Variable, Two Variables and Three Variables

Finding the maximum and minimum values of a function is essential in high school math. However, Vietnamese high school students have only been taught how to find the extreme values of a function of 1 variable. Seeing the extreme values of a function of 2 and 3 variables is a difficult problem for students. Using the determinants, our aim in this paper is to show the necessary and sufficient conditions for a continuous and differentiable function (1 variable, two variables, and three variables) to reach its maximum over a specified domain. Furthermore, our method can be used to find the extremes of n-variable differentiable functions.

Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

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.

q -Hermite–Hadamard Inequalities for Generalized Exponentially s , m ; η -Preinvex Functions

In this article, we introduce a new extension of classical convexity which is called generalized exponentially s , m ; η -preinvex functions. Also, it is seen that the new definition of generalized exponentially s , m ; η -preinvex functions describes different new classes as special cases. To prove our main results, we derive a new q m κ 2 -integral identity for the twice q m κ 2 -differentiable function. By using this identity, we show essential new results for Hermite–Hadamard-type inequalities for the q m κ 2 -integral by utilizing differentiable exponentially s , m ; η -preinvex functions. The results presented in this article are unification and generalization of the comparable results in the literature.

Asymptotics of $L_p$-error for adaptive approximation of $n$-variable functions by harmonic splines

For a twice continuously differentiable function, defined on $n$-dimensional unit cube, we obtain sharp asymptotics of $L_p$-error for approximation by harmonic splines, and construct the asymptotically optimal sequence of partitions.

New generalized trapezoidal type integral inequalities with applications

Trapezoidal inequalities for functions of diverse nature are useful in numerical computations. The authors have proved an identity for a generalized integral operator via a differentiable function. By applying the established identity, the generalized trapezoidal type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in the recent decades. Various special cases have been identified. Some applications of presented results have been analyzed.

Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

We construct the bivariate form of Bernstein–Schurer operators based on parameter α. We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the help of Peetre’s K-functional of our newly defined operators. Moreover, we define the associated generalized Boolean sum (shortly, GBS) operators and estimate the rate of convergence by means of mixed modulus of smoothness. Finally, the order of approximation for Bögel differentiable function of our GBS operators is presented.