In this paper, a new homological dimension of modules, copresented dimension, is defined. We study some basic properties of this homological dimension. Some ring extensions are considered, too. For instance, we prove that if $Sgeq R$ is a finite normalizing extension and $S_R$ is a projective module, then for each right $S$-module $M_S$, the copresented dimension of $M_S$ does not exceed the c...

The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization exact categories triangulated categories. Our first aim is to provide an extension Auslander's formula: for some $\mathcal{C}$, there exists localization sequence $\operatorname{def}\mathcal{C} \to \mod\mathcal{C} \operatorname{lex}\mathcal{C}$, where $\operatorname{lex}\mathcal{C}$ den...

We consider completely invariant subsets A of weakly expanding piecewise monotonic transformations T on [0, 1]. It is shown that the upper box dimension of A is bounded by the minimum tA of all parameters t for which a t-conformal measure with support A exists. In particular, this implies equality of box dimension and Hausdorff dimension of A.

Three aspects of texture are considered by the fractal geometry: Fractal Dimension (FD), Lacunarity and Succolarity. Fractal Dimension has been well studied; a great number of approaches have been presented to extract it from images. It can be computed from black-white to multi-band image. There are many approaches also, from the simple Box-Dimension to the most complex Hausdorff Dimension. The...

In this paper we present some new properties of the metric dimension defined by Bouligand in 1928 and prove the following new projection theorem: Let dimb(A − A) denote the Bouligand dimension of the set A − A of differences between elements of A. Given any compact set A ⊆ R such that dimb(A−A) < m, then almost every orthogonal projection P of A of rank m is injective on A and P |A has Lipschit...

The paper deals with an algorithm for Hausdorff dimension estimation based on box-counting dimension calculation. The main goal of the paper is to propose a new approach to box-counting dimension calculation with less computational demands. Streszczenie. W artykule opisano algorytm estymacji wymiaru Hausdorffa oparty o wyznaczanie wymiaru pudełkowego. Głównym celem pracy jest zaproponowanie now...