Abstract

We introduce new star selection principles defined by neighbourhoods and stars which are weaker versions of the of Menger, Rothberger and Hurewicz properties; in particular the properties introduced are between strong star versions and star versions of the corresponding properties defined in [12]. Some properties of these neighbourhood star selection principles are proved and some examples are given.

Keywords: Selection principles, (strongly) star-Rothberger; (strongly) star-Menger; (strongly) star-Hurewicz; neighbourhood star-Rothberger; neighbourhood star-Menger; neighbourhood star-Hurewicz; $\omega$-cover; $\gamma$-cover.

MSC: 54D20

Abstract

New dimension functions $\Cal{G}$-dim and $\Cal{R}$-dim, where $\Cal{G}$ is a class of finite simplicial complexes and $\Cal{R}$ is a class of $ANR$-compacta, are introduced. Their definitions are based on the theorem on partitions and on the theorem on inessential mappings to cubes, respectively. If $\Cal{R}$ is a class of compact polyhedra, then for its arbitrary triangulation $\tau$, we have ${\Cal{R}}_\tau\text{-dim}\,X={\Cal{R}}\text{-dim}\,X$ for an arbitrary normal space $X$. To investigate the dimension function $\Cal{R}$-dim we apply results of extension theory. Internal properties of this dimension function are similar to those of the Lebesgue dimension. The following inequality $\Cal{R}\text{-dim}\,X\leq\tx{\rm dim}\,X$ holds for an arbitrary class $\Cal{R}$. We discuss the following Question: When $\Cal{R}$-$\text{\rm dim}\,X<\infty\Rightarrow\text{\rm dim}\,X<\infty$?

Keywords: Dimension; simplicial complex; $ANR$-compactum; extension theory.

MSC: 54F45, 55M10

Abstract

In the present note we give a full quantitative version of a theorem of Floater dealing with the asymptotic behaviour of differentiated Bernstein polynomials. While Floater's result is a generalization of the classical Voronovskaya theorem, ours generalizes a hardly known quantitative version of this theorem due to Videnski\uı, among others.

Keywords: Voronovskaya theorem; Bernstein operators; simultaneous approximation; $k$-th Kantorovich modification; convex operators.

MSC: 41A10, 41A17, 41A25, 41A28, 41A36

Abstract

The paper considers the existence of the maximal possible hyperplane partition of a continuous probability Borel measure in $\Bbb{R}^{4}$. The emphases is on the use of the equivariant ideal valued index theory of Fadell and Husseini. The presented result is the tightest positive solution to one of the oldest and most relentless partition problems posed by B. Grünbaum~[12].

Keywords: Partition of measures; Fadell-Husseini index theory.

MSC: 52A37, 55N91, 55M35

Abstract

Let $\mu$ be a proper Borel probability measure on the sphere $S^{2}$ in $\Bbb{R}^{3}$. It was conjectured that for every triple of rational numbers $(q_{1},q_{2},q_{3})$ with the property $q_{1}+q_{2}+q_{3}=\tfrac{1}{2}$, there exist three planes in $\Bbb{R}^{3}$ intersecting along the common line through the origin such that the six angular sectors on the sphere determined by those planes have respectively $q_{1}$, $q_{2}$, $q_{3}$, $q_{1}$, $q_{2}$, $q_{3}$ amount of the measure $\mu$. In this paper we give an exact and explicitly realized algorithm which, for every triple $(q_{1},q_{2},q_{3})$ of the form $q_{2}=q_{3}$, establishes whether there exists a configuration of three planes splitting the measure in the required proportion.

Keywords: Partition of measures; $k$-fans; equivariant obstruction theory.

MSC: 52A37, 55S35, 55M35

Abstract

In this note we give a uniqueness theorem for solutions $(u,\pi)$ to the Navier-Stokes Cauchy problem, assuming that $u$ belongs to $L^\infty((0,T)\times\Bbb R^n)$ and $(1+|x|)^{-n-1}\pi\in L^1(0,T;L^1(\R^n))$, $n\geq2$. The interest to our theorem is motivated by the fact that a possible pressure field $\widetilde \pi$, belonging to $L^1(0,T;\text{\rm{BMO}})$, satisfies in a suitable sense our assumption on the pressure, and by the fact that the proof is very simple.

Keywords: Uniqueness; Weak solutions; Navier-Stokes equations.

MSC: 35Q30, 76D05, 76N10

Abstract

The transfinite inductive dimensions of a space by a normal bases introduced by S. D. Iliadis are studied. These dimensions generalize both classical large transfinite inductive dimension and relative large transfinite inductive dimensions. The main theorems of dimension theory (sum theorem, subset theorem, product theorem) are proved.

Keywords: Dimension-like function; Normal base.

MSC: 54B99, 54C25