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In this paper, utilizing an identity given by Yıldız and Sarıkaya in (Yildiz and Sarikaya in Int. J. Anal. Appl. 13(1):64–69, 2017), we establish some weighted Ostrowski type inequalities for co-ordinated convex functions in a rectangle from the plane R 2 . Moreover, as special cases of our main results, we give some weighted Hermite–Hadamard type inequalities. The results given in this paper provide generalizations of some result established in earlier works. [ABSTRACT FROM AUTHOR]

In this paper, operational matrices of two-dimensional hybrid of block-pulse functions and Legendre polynomials are employed to approximate the solutions of two-dimensional nonlinear Volterra integral equations of the first kind (2DNVIEF). To this aim, we first convert 2DNVIEF to the two-dimensional linear Volterra integral equations of the second kind (2DLVIES). Then the operational matrices of integration, together with the product operational matrix, can be applied to reduce the 2DLVIES to a system of algebraic equations. The convergence of the proposed method is studied, and an error estimation under the L 2 -norm is provided. Finally, some numerical examples are prepared to clarify the accuracy and efficiency of the numerical method especially in dealing with functions that are not continuous in the whole interval. [ABSTRACT FROM AUTHOR]

An investigation into families of integrals containing powers of the arctanh and log functions will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals. In another special case we prove that the corresponding log-tanh integral can be represented as a linear combination of the product of zeta functions and the Dirichlet eta function. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Hermite–Hadamard–Fejér type inclusions for the product of two co-ordinated convex interval-valued functions. These inclusions are generalizations of some results given in earlier works. [ABSTRACT FROM AUTHOR]

This paper deals with some trapezoid and mid-point type inequalities on closed balls in R 3 . Three kinds of functions are considered: convex, Lipschitz, and bounded functions. The spherical coordinates are used to obtain sharp inequalities. Also a reverse result is given for the right-hand side of Hermite–Hadamard's inequality obtained on closed balls in R 3 . [ABSTRACT FROM AUTHOR]

In the paper we investigate the continuity properties of the mapping $${\Phi}$$ which sends any non-empty compact connected hv-convex planar set K to the associated generalized conic function f. The function f measures the average taxicab distance of the points in the plane from the focal set K by integration. The main area of applications is geometric tomography because f involves the coordinate X-rays' information as second order partial derivatives (Nagy and Vincze, J Approx Theory 164: 371-390, ). We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the supremum-norm and the L-norm provided that we restrict the domain to the collection of non-empty compact connected hv-convex planar sets contained in a fixed box (reference set) with parallel sides to the coordinate axes. We also have that $${\Phi^{-1}}$$ is upper semi-continuous as a set-valued mapping. The upper semi-continuity establishes an approximating process in the sense that if f is close to f then L must be close to an element $${K^\prime}$$ such that $${f_{K}=f_{K^\prime}}$$ . Therefore K and $${K^\prime}$$ have the same coordinate X-rays almost everywhere. Lower semi-continuity is usually related to the existence of continuous selections. If a set-valued mapping is both upper and lower semi-continuous at a point of its domain it is called continuous. The last section of the paper is devoted to the case of non-empty compact convex planar sets. We show that the class of convex bodies that are determined by their coordinate X-rays coincides with the family of convex bodies K for which f is a point of lower semi-continuity for $${\Phi^{-1}}$$ . [ABSTRACT FROM AUTHOR]

This paper presents some critical perspectives regarding pedagogical approaches to the method of reversing the order of integration in double integrals from prevailing educational literature on multivariable calculus. First, we question the message found in popular textbooks that the traditional process of reversing the order of integration is necessary when solving well-known problems. Second, we illustrate that the method of integration by parts can be directly applied to many of the classic pedagogical problems in the literature concerning double integrals, without taking the well-worn steps associated with reversing the order of integration. Third, we examine the benefits and limitations of such a method. In our conclusion, we advocate for integration by parts to be a part of the pedagogical conversation in the learning and teaching of double integral methods; and call for more debate around its use in the learning and teaching of other areas of mathematics. Finally, we emphasize the need for critical approaches in the pedagogy of mathematics more broadly. [ABSTRACT FROM PUBLISHER]

In this paper, the following generalized multiple fractional part integrals:andare considered for complex numbers, positive integersand positive numbera, wheredenotes the fractional part ofu. Furthermore,andcan be transformed into the calculation of definite fractional part integraland, which overcomes the shortcomings of low speed and poor accuracy of the multiple case. Moreover, some identities and recursive formulas of the above integrals are obtained. [ABSTRACT FROM PUBLISHER]

In this paper, the following multiple fractional part integralsandare calculated for non-negative integersand positive integern, wheredenotes the fractional part ofuandIt is proved thathas a closed form for non-negative integer. We also prove that(and) andcan be expressed as linear combinations of the Riemann zeta function, logarithmic function and some binomial coefficients. In particular,has a closed form for. Moreover, some identities and recursive formulas of the above integrals are obtained. [ABSTRACT FROM PUBLISHER]

In this paper, we establish Luzin's condition (N) for mappings in certain Sobolev-Orlicz spaces with certain moduli of continuity. Further, given a mapping in these Sobolev-Orlicz spaces, we give bounds on the size of the exceptional set where Luzin's condition (N) may fail. If a mapping violates Luzin's condition (N), we show that there is a Cantor set of measure zero that is mapped to a set of positive measure. [ABSTRACT FROM AUTHOR]