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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]

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]

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]

The automated removal of layers by scarfing to repair carbon fibre reinforced plastics (CFRP) has been investigated with vacuum suction blasting (VSB) showing promising results. However, its removal mechanism and particle behaviour were not yet fully understood. The removal leads to cavities on the surface and the central area is less removed after longer static blasting times. Further theoretical considerations and experiments of measured forces and velocities of the VSB particles can explain these effects to optimize the large-scale removal. Former studies on abrasive blasting used simplified approaches with incompressible Bernoulli equations taking the same velocities of particles and carrier medium. For VSB, the air is much faster than the particles and more complex equations are derived in this study. Additionally, it is taken into account, that the impinging forces act against the vacuum forces. To specify the power and removal mechanism of VSB for large scale removal, forces and velocities of the blasting particles are studied. Calculations by compressible Bernoulli equations, measurements of blasting particle velocities and impact forces give a good approximation. The air velocities and forces are calculated and compared with the measurements. Forces are measured by a piezo-electric sensor. Particle velocities are measured with a Particle Image Velocimetry (PIV) system tracking the abrasive particle movement through a transparent outer nozzle. The influence of different blasting parameters on the forces and velocities are identified. Results correspond to the material removal from former experiments and indicate the favourable process parameter set. Despite the impinging forces are in general low, they lead together with the particle velocity to the large-scale removal rate. The measured particle directions explain the removal behaviour of this VSB process. The measured velocities match to those well-known from grinding with bonded grinding tools. Article highlights: Force and velocity measurements of vacuum suction blasting process PIV measurements of blasting particles Forces calculated and measured by piezo-electric sensor. [ABSTRACT FROM AUTHOR]

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, 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]

For a sequence of continuous, monotone functions f1,…,fn:I→R (I is an interval) we define the mapping M:In→In as a Cartesian product of quasi-arithmetic means generated by fj-s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of In. We will prove that whenever all fj-s are C2 with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit VarMk+1(v)(VarMk(v))2 will be calculated in a nondegenerated case. [ABSTRACT FROM AUTHOR]

We consider surface area approximations by Lagrange and Crouzeix-Raviart interpolations on triangulations. For Lagrange interpolation, we give an alternative proof for Young's classical result that claims the areas of inscribed polygonal surfaces converge to the area of the original surface under the maximum angle condition on the triangulation. For Crouzeix-Raviart interpolation we show that the approximated surface areas converge to the area of the original surface without any geometric conditions on the triangulation. [ABSTRACT FROM AUTHOR]

In this study, we prove equality for twice partially differentiable mappings involving the double generalized fractional integral. Using the established identity, we offer some Simpson's type inequalities for partially differentiable co-ordinated convex functions in a rectangle from the plane R 2 . [ABSTRACT FROM AUTHOR]

Archimedes’ well known theorem on the area of a parabolic segment says that this area is 4/3 of the area of a certain inscribed triangle. In this paper we generalize this theorem to the n-dimensional euclidean space, n ≥ 3. It appears that the ratio of the volume of an n-dimensional solid bounded by an (n − 1)-dimensional hyper-paraboloid and an (n − 1)-dimensional hyperplane and the volume of a certain inscribed cone (we analogously repeat Archimedes’ procedure) depends only on the dimension of the euclidean space and it equals to 2n/(n +1).

We show that the Euclidean ball has the smallest volume among sublevel sets of nonnegative forms of bounded Bombieri norm as well as among sublevel sets of sum of squares forms whose Gram matrix has bounded Frobenius or nuclear (or, more generally, p-Schatten) norm. These volume-minimizing properties of the Euclidean ball with respect to its representation (as a sublevel set of a form of fixed even degree) complement its numerous intrinsic geometric properties. We also provide a probabilistic interpretation of the results. [ABSTRACT FROM AUTHOR]

In this note, a vector space of real-analytic functions on the plane is explicitly constructed such that all its nonzero functions are non-integrable but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space is dense in the space of all real continuous functions on the plane endowed with the compact-open topology. [ABSTRACT FROM AUTHOR]

The aim of this note is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body K in Euclidean n-space, defined as the volume of the union of K and one of its translates, and the volume of K and a translate of a homothetic copy of K, respectively, as functions of the translation vector. In particular, we prove that the convex hull function of the body K does not determine K. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horváth and Lángi about translative constant volume property of convex bodies. We give a short proof of some theorems of Jerónimo-Castro about the homothetic convex hull function, and prove a homothetic variant of the translative constant volume property conjecture for 3-dimensional convex polyhedra. We also apply our results to describe the properties of the illumination bodies of convex bodies. [ABSTRACT FROM AUTHOR]

A geometric measure for the entanglement of a unit length tensor T ∈ (C n ) ⊗ k is given by − 2 log 2 ⁡ ∥ T ∥ σ , where ∥ ⋅ ∥ σ denotes the spectral norm. A simple induction gives an upper bound of (k − 1) log 2 ⁡ (n) for the entanglement. We show the existence of tensors with entanglement larger than k log 2 ⁡ (n) − log 2 ⁡ (k) − o (log 2 ⁡ (k)). Friedland and Kemp (Proc. AMS, 2018) have similar results in the case of symmetric tensors. Our techniques give improvements in this case. [ABSTRACT FROM AUTHOR]

The purpose of this article is to provide an explicit formula for the bounds of integration of the regular simplex centred at the origin. Furthermore, this article rigorously proves that these integration bounds recover the volume of the regular simplex. To the authors' knowledge, this is the first time that such integration bounds have been explicitly formulated in the mainstream literature. In addition, the derivation of the simplex volume using integration bounds appears to be new. As a result, this article has the potential to serve as a useful resource for a broad audience, including teachers of mathematics, engineers, and recreational geometers. [ABSTRACT FROM AUTHOR]

In this study, we prove an identity for twice partially differentiable mappings involving the double generalized fractional integral and some parameters. By using this established identity, we offer some generalized inequalities for differentiable co-ordinated convex functions with a rectangle in the plane R 2 . Furthermore, by special choice of parameters in our main results, we obtain several well-known inequalities such as the Ostrowski inequality, trapezoidal inequality, and the Simpson inequality for Riemann and Riemann–Liouville fractional integrals. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Opial type inequalities for Riemann-Stieltjes integrals of functions with two variables. The obtained inequalities generalize those previously demonstrated (see [2])

We minimise the Canham–Helfrich energy in the class of closed immersions with prescribed genus, surface area, and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false under varifold convergence by a counter example by Große-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of R 3 . [ABSTRACT FROM AUTHOR]

We address the following generalization $$\mathbf {P}$$ of the Löwner-John ellipsoid problem. Given a (non necessarily convex) compact set $$\mathbf {K}\subset \mathbb {R}^n$$ and an even integer $$d\in \mathbb {N}$$ , find an homogeneous polynomial $$g$$ of degree $$d$$ such that $$\mathbf {K}\subset \mathbf {G}:=\{\mathbf {x}:g(\mathbf {x})\le 1\}$$ and $$\mathbf {G}$$ has minimum volume among all such sets. We show that $$\mathbf {P}$$ is a convex optimization problem even if neither $$\mathbf {K}$$ nor $$\mathbf {G}$$ are convex! We next show that $$\mathbf {P}$$ has a unique optimal solution and a characterization with at most $${n+d-1\atopwithdelims ()d}$$ contacts points in $$\mathbf {K}\cap \mathbf {G}$$ is also provided. This is the analogue for $$d>2$$ of Löwner-John's theorem in the quadratic case $$d=2$$ , but importantly, we neither require the set $$\mathbf {K}$$ nor the sublevel set $$\mathbf {G}$$ to be convex. More generally, there is also an homogeneous polynomial $$g$$ of even degree $$d$$ and a point $$\mathbf {a}\in \mathbb {R}^n$$ such that $$\mathbf {K}\subset \mathbf {G}_\mathbf {a}:=\{\mathbf {x}:g(\mathbf {x}-\mathbf {a})\le 1\}$$ and $$\mathbf {G}_\mathbf {a}$$ has minimum volume among all such sets (but uniqueness is not guaranteed). Finally, we also outline a numerical scheme to approximate as closely as desired the optimal value and an optimal solution. It consists of solving a hierarchy of convex optimization problems with strictly convex objective function and Linear Matrix Inequality constraints. [ABSTRACT FROM AUTHOR]

We discuss the error analysis of linear interpolation on triangular elements. We claim that the circumradius condition is more essential than the well-known maximum angle condition for convergence of the finite element method, especially for the linear Lagrange finite element. Numerical experiments show that this condition is the best possible. We also point out that the circumradius condition is closely related to the definition of surface area. [ABSTRACT FROM AUTHOR]

In this note we investigate the behavior of the volume that the convex hull of two congruent and intersecting simplices in Euclidean $$n$$-space can have. We prove some useful equalities and inequalities on this volume. For the regular simplex we determine the maximal possible volume for the case when the two simplices are related to each other via reflection at a hyperplane intersecting them. [ABSTRACT FROM AUTHOR]