We study sequences of conformal deformations a smooth closed Riemannian manifold dimension $n$, assuming uniform volume bounds and $L^{n/2}$ on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such underlying metric spaces are pre-compact Gromov-Hausdorff topology. Our is based use $A_\infty$-weights from harmonic analysis, provides geometric cont...

The measure contraction property, MCP for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space. We start our investigation from the euclidean case by proving that if a positive Ra...

In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α,β) -metric admits no concurrent vector fields. We also prove that an L-reducible Finsler metric admitting a concurrent vector field reduces to a Landsberg metric.In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α,β) -metric admits no concurrent vector fi...

A multicriterion linear combinatorial problem with a parametric principle of optimality is considered. This principle is defined by a partitioning of partial criteria onto Pareto preference relation groups within each group and the lexicographic preference relation between them. Quasistability of the problem is investigated. This type of stability is a discrete analog of Hausdorff lower semi-co...

We prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley’s theorem: if a countable Hausdorff cat T has a unique complete model of density character λ ≥ ω1, then it has a unique complete model of density character λ for every λ ≥ ω1.

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [1]. If M is a non degenerate multifractal measure with associated metric ρ(x, y) = M([x, y]) and structure function ζ , we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimρH with respect to ρ...

We study random recursive constructions with finite “memory” in complete metric spaces and the Hausdorff dimension of the generated random fractals. With each such construction and any positive number β we associate a linear operator V (β) in a finite dimensional space. We prove that under some conditions on the random construction the Hausdorff dimension of the fractal coincides with the value...

We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work cla...

In the present paper, we investigate structure of metric space M compact spaces considered up to an isometry and endowed with Gromov–Hausdorff in a neighborhood finite space, whose group is trivial. It shown that sufficiently small ball subspace consisting same number points centered at such isometric corresponding ?N norm |(x1, . , xN)| = \( \underset{i}{\max}\left|{x}_i\right| \). Also embedd...