Volume 34, Issue 4

We introduce a method of differentiation of the length of a variable linear segment with respect to three variable linear segments which generalizes the first variation formula in the three-dimensional Euclidean space. Applying this method, we derive a generalized Gauss tree for a heptagonal pyramid (closed octahedron) by placing two vertices with positive weights at the interior of the convex hull having degree five. Applying the plasticity principle of closed hexahedra, we obtain a plasticity principle of a generalized Gauss tree for a boundary heptagonal pyramid in the three-dimensional Euclidean space.

A classical theorem due to Eneström and Kakeya gives some bounds for the moduli of the zeros of polynomials having a monotone sequence of non-negative (real) coefficients. Nowadays, one can find several modifications and generalizations of this result in the literature. The main subject of the paper is a study of links of coefficients which occur in some of these general results with a view to the recurrence relations fulfilled by systems of orthogonal polynomials on the unit circle. In particular, we discuss the question, if the relevant links are consistent with the recurrence relations or not. This leads to some new insight into the analyzed classes of polynomials.

In this paper, some refinements of Hermite–Hadamard type inequalities for GA-convex functions are obtained. Applications of the obtained results to special means are given.

In this paper, we will prove some new dynamic inequalities on time scales. These inequalities as special cases contain some integral and discrete inequalities due to Hardy and Copson. The main results will be proved by using the Hölder inequality and a simple consequence of Keller's chain rule on time scales.

In this article we introduce the double sequence spaces 2𝒵I(f)${\,}_2\mathcal {Z}^{I}(f)$, 2𝒵0I(f)${\,}_2\mathcal {Z}^{I}_{0}(f)$ and 2𝒵∞I(f)${\,}_2\mathcal {Z}^{I}_{\infty }(f)$ for a modulus function f and study some of the topological and algebraic properties of these spaces.

In this paper, we study vector-valued multiresolution analysis on local fields of positive characteristic p . The associated subspace V 0 of L2(K,ℂN)$L^2(K, \mathbb {C}^N)$ has an orthonormal basis, a collection of translates of the scaling function ϕ of the form {ϕ(t-u(k))}k∈ℙ$\lbrace \varphi (t- u(k))\rbrace _{ k \in \mathbb {P}}$, where ℙ={0,1,2,...}$\mathbb {P}=\lbrace 0, 1, 2,\ldots \rbrace $. We find the necessary and sufficient condition for the existence of associated wavelets.