Related papers: On dimensions modulo a compact metric ANR and modu…

Repetitiveness in projective and injective resolutions and its influence on homological dimensions are studied. Some variations on the theme of repetitiveness are introduced, and it is shown that the corresponding invariants lead to very…

We study compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces with variable exponents on both bounded and unbounded metric measure spaces. We establish sufficient conditions for compactness, and under additional assumptions, we…

For a compact space $K$ we consider the space $P(K)$, of probability regular Borel measures on $K$, equipped with the $weak^\ast$ topology inherited from $C(K)^\ast$. We discuss possible characterizations of those compact spaces $K$ for…

In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff…

Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\mathrm{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under…

Let $\mathbb K$ be an algebraically closed field of characteristic zero, $\mathbb K[X]$ the polynomial ring in $n$ variables. The vector space $T_n = \mathbb K[X]$ is a $\mathbb K[X]$-module with the action $x_i \cdot v = v_{x_i}'$ for $v…

THEOREM. For every prime $p$ and each $n=2, 3, ... \infty$, there is an action of $G=\prod_{i=1}^{\infty}(Z/ pZ)$ on a two-dimensional compact metric space $X$ with $n$-dimensional orbit space. This theorem was proved in [DW: A.N.…

We begin to study classical dimension theory from the computable analysis (TTE) point of view. For computable metric spaces, several effectivisations of zero-dimensionality are shown to be equivalent. The part of this characterisation that…

We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of the previous works of the authors. This computation allows to determine the…

We construct explicit generators of the K-theory and K-homology of the coordinate algebra of `functions' on quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and…

We give a construction under $CH$ of a non-metrizable compact Hausdorff space $K$ such that any uncountable semi-biorthogonal sequence in $C(K)$ must be of a very specific kind. The space $K$ has many nice properties, such as being…

We derive some necessary conditions on a Riemannian metric $(M, g)$ in four dimensions for it to be locally conformal to K\"ahler. If the conformal curvature is non anti--self--dual, the self--dual Weyl spinor must be of algebraic type $D$…

Fedorchuk's fully closed (continuous) maps and resolutions are applied in constructions of non-metrizable higher-dimensional analogues of Anderson, Choquet, and Cook's continua. Certain theorems on dimension-lowering maps are proved for…

In the literature, the Minkowski-sum and the metric-sum of compact sets are highlighted. While the first is associative, the latter is not. But the major drawback of the Minkowski combination is that, by increasing the number of summands,…

We compute, for a compact set $K\subset\mathbb R^d$, the value of the upper and of the lower $L^q$-dimension of a typical probability measure with support contained in $K$, for any $q\in\mathbb R$. Different definitions of the "dimension"…

One considers Hilbert space valued measures on the Borel sets of a compact metric space. A natural numerical valued integral of vector valued continuous functions with respect to vector valued functions is defined. Using this integral,…

Mean dimension is a topological invariant of dynamical systems, which originates with Mikhail Gromov in 1999 and which was studied with deep applications around 2000 by Elon Lindenstrauss and Benjamin Weiss within the framework of amenable…

We compute projective dimension of translated simple modules in the regular block of the BGG category $\mathcal{O}$ in terms of Kazhdan-Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a…

We investigate integral representation of vector-valued function spaces, i.e., of subspaces $H\subset C(K,E)$, where $K$ is a compact space and $E$ is a (real or complex) Banach space. We point out that there are two possible ways of…

Let $G$ be the adjoint group of a real simple Lie algebra $\mathfrak{g}_0$ equal either $\mathfrak{s}\mathfrak{u}(n,1)$ or $\mathfrak{s}\mathfrak{o}(n,1),$ $K$ its maximal compact subgroup, ${\cal U}(\mathfrak{g})$ the universal enveloping…