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## Abstract

In this paper, we prove that if *X* is a space with a regular *G*
_{
δ
}-diagonal and *X*
^{2} is star Lindelöf then the cardinality of *X* is at most 2^{c}. We also prove that if *X* is a star Lindelöf space with a symmetric *g*-function such that *g*
^{2}(*n, x*): *n* ∈ *ω*} = {*x*} for each *x* ∈ *X* then the cardinality of *X* is at most 2^{c}. Moreover, we prove that if *X* is a star Lindelöf Hausdorff space satisfying *Hψ*(*X*) = *κ* then *e*(*X*) ^{2κ
}; and if *X* is Hausdorff and *we*(*X*) = *Hψ*(*X*) = *κ*subset of a space then *e*(*X*) ^{
κ
}. Finally, we prove that under *V* = *L* if *X* is a first countable DCCC normal space then *X* has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in *Spaces with property* (*DC*(*ω*
_{1})), *Comment. Math. Univ. Carolin.*, **58(1)** (2017), 131-135.

## Abstract

Fejes Tóth [] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the *square* of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

## Abstract

Let *H*
_{
n
} be the *n*-th harmonic number and let *v*
_{
n
} be its denominator. It is known that *v*
_{
n
} is even for every integer *H*
_{
n
} and prove that for any integer *n*, *v*
_{
n
} = *e*
^{
n(1+o(1))}. In addition, we obtain some results of the logarithmic density of harmonic numbers.

## Abstract

We verify an upper bound of Pach and Tóth from 1997 on the midrange crossing constant. Details of their

## Abstract

Let 0 *< γ*
_{1}
*< γ*
_{2}
*<* ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function *ζ*(*s* + *iγ*
_{
k
}
*h, α*), *h >* 0, with parameter *α* such that the set {log(*m* + *α*): *m* ∈ *γ*
_{
k
}} is applied.

## Abstract

We study certain subgroups of the full group of Hopf algebra automorphisms of twisted tensor biproducts.

## Abstract

In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class *A _{p}
* by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝ

*are obtained.*

^{n}## Abstract

Let *X* be a Hilbert *C**-module over a *C**-algebra *B*. In this paper we introduce two classes of operator algebras on the Hilbert *C**-module *X* called operator algebras with property *B* and *X*. Some of our results generalize the previous results. Also we investigate some properties of these classes of operator algebras.

## Abstract

Let *m* ≠ 0, ±1 and *n* ≥ 2 be integers. The ring of algebraic integers of the pure fields of type *n* = 2, 3,4. It is well known that for *n* = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.

In this paper we explicitly give an integral basis of the field *n*.

## Abstract

Two classes of trigonometric sums about integer powers of secant function are evaluated that are closely related to Jordan's totient function.