Density results for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets

We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set $\Omega\subset\mathbb R^n$, $\mathcal{D}(\Omega)$ is dense in $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overline{\Omega}\}$ whenever $\partial\Omega$ has zero Lebesgue measure and $\Omega$ is "thick" (in the sense of Triebel); and (ii) for a $d$-set $\Gamma\subset\mathbb R^n$ ($0<d<n$), $\{u\in H^{s_1}(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}$ is dense in $\{u\in H^{s_2}(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}$ whenever $-\frac{n-d}{2}-m-1<s_{2}\leq s_{1}<-\frac{n-d}{2}-m$ for some $m\in\mathbb N_0$. For (ii), we provide concrete examples, for any $m\in\mathbb N_0$, where density fails when $s_1$ and $s_2$ are on opposite sides of $-\frac{n-d}{2}-m$. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}=\{0\}$ for a given closed set $\Gamma\subset\mathbb R^n$ and $s\in \mathbb R$. They also both arise naturally in the study of boundary integral equation formulations of acoustic wave scattering by fractal screens. We additionally provide analogous results in the more general setting of Besov and Triebel--Lizorkin spaces.Comment: 38 pages, 6 figure

Billiard scattering on rough sets: two-dimensional case

The notion of a rough two-dimensional (convex) body is introduced, and to each rough body there is assigned a measure on T3 describing billiard scattering on the body. The main result is characterization of the set of measures generated by rough bodies. This result can be used to solve various problems of least aerodynamical resistance

Rough sets theory and uncertainty into information system

This article is focused on rough sets approach to expression of uncertainty into information system. We assume that the data are presented in the decision table and that some attribute values are lost. At first the theoretical background is described and after that, computations on real-life data are presented. In computation we wok with uncertainty coming from missing attribute values

On Rough Sets and Hyperlattices

In this paper, we introduce the concepts of upper and lower rough hyper fuzzy ideals (filters) in a hyperlattice and their basic properties are discussed. Let $\theta$ be a hyper congruence relation on $L$. We show that if $\mu$ is a fuzzy subset of $L$, then $\overline{\theta}()=\overline{\theta}()$ and $\overline{\theta}(\mu^*) =\overline{\theta}((\overline{\theta}(\mu))^*)$, where is the least hyper fuzzy ideal of $L$ containing $\mu$ and \mu^*(x) = sup\{\alpha \in [0, 1]: x \in I( \mu_{\alpha} )\}$$ for all $x \in L$. Next, we prove that if $\mu$ is a hyper fuzzy ideal of $L$, then $\mu$ is an upper rough fuzzy ideal. Also, if $\theta$ is a $\wedge-$complete on $L$ and $\mu$ is a hyper fuzzy prime ideal of $L$ such that $\overline{\theta}(\mu)$ is a proper fuzzy subset of $L$, then $\mu$ is an upper rough fuzzy prime ideal. Furthermore, let $\theta$ be a $\vee$-complete congruence relation on $L$. If $\mu$ is a hyper fuzzy ideal, then $\mu$ is a lower rough fuzzy ideal and if $\mu$ is a hyper fuzzy prime ideal such that $\underline{\theta}(\mu)$ is a proper fuzzy subset of $L$, then $\mu$ is a lower rough fuzzy prime ideal

Representative Set of Objects in Rough Sets Based on Galois Connections

This paper introduces a novel definition, called representative set of objects of a decision class, in the framework of decision systems based on rough sets. The idea behind such a notion is to consider subsets of objects that characterize the different classes given by a decision system. Besides the formal definition of representative set of objects of a decision class, we present different mathematical properties of such sets and a relationship with classification tasks based on rough sets. © 2020, Springer Nature Switzerland AG

The reduction subset based on rough sets applied to texture classification

The rough set is a new mathematical approach to imprecision, vagueness and uncertainty. The concept of reduction of the decision table based on the rough sets is very useful for feature selection. The paper describes an application of rough sets method to feature selection and reduction in texture images recognition. The methods applied include continuous data discretization based on Fuzzy c-means and, and rough set method for feature selection and reduction. The trees extractions in the aerial images were applied. The experiments show that the methods presented in this paper are practical and effective.<br /