Abstract:
This paper is a survey, with few proofs, of ideas and notions related to self-similarity of groups, semi-groups and their actions. It attempts to relate these concepts to more familiar ones, such as fractals, self-similar sets, and renormalizable dynamical systems. In particular, it presents a plausible definition of what a "fractal group" should be, and gives many examples of such groups. A particularly interesting class of examples, derived from monodromy groups of iterated branch coverings, or equivalently from Galois groups of iterated polynomials, is presented. This class contains interesting groups from an algebraic point of view (just-non-solvable groups, groups of intermediate growth, branch groups,...), and at the same time the geometry of the group is apparent in that a limit of the group identifies naturally with the Julia set of the covering map. In its survey, the paper discusses finite-state transducers, growth of groups and languages, limit spaces of groups, hyperbolic spaces and groups, dynamical systems, Hecke-type operators, C^*-algebras, random matrices, ergodic theorems and entropy of non-commuting transformations. Self-similar groups appear then as a natural weaving thread through these seemingly different topics.

Abstract:
We show that, almost surely, the Hausdorff dimension $s_0$ of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension $k>s_0$. The result holds for random covering sets with a generating sequence of ball-like sets, and is obtained by investigating orthogonal projections of a class of random Cantor sets.

Abstract:
Intersection of a random fractal or self-affine set with a linear manifold or another fractal set is studied, assuming that one of the sets is in a translational motion with respect to the other. It is shown that the mass of such an intersection is a self-affine function of the relative position of the two sets. The corresponding Hurst exponent h is a function of the scaling exponents of the intersecting sets. A generic expression for h is provided, and its proof is offered for two cases --- intersection of a self-affine curve with a line, and of two fractal sets. The analytical results are tested using Monte-Carlo simulations.

Abstract:
The focus here is on connected fractal sets with topological dimension 1 and a lot of topological activity, and their connections with analysis.

Abstract:
We discuss various questions which arise when one considers the central projection of three dimensional fractal sets (galaxy catalogs) onto the celestial globe. The issues are related to how fractal such projections look. First we show that the lacunarity in the projection can be arbitrarily small. Further characteristics of the projected set---in particular scaling---depend sensitively on how the apparent sizes of galaxies are taken into account. Finally, we discuss the influence of opacity of galaxies. Combining these ideas, seemingly contradictory statements about lacunarity and apparent projections can be reconciled.

Abstract:
Given $n$ points in the plane, a \emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of $n$ points in the plane admits a (possibly self-crossi ng) covering path consisting of $n/2 +O(n/\log{n})$ straight line segments. If the path is required to be noncrossing, we prove that $(1-\eps)n$ straight line segments suffice for a small constant $\eps>0$, and we exhibit $n$-element point sets that require at least $5n/9 -O(1)$ segments in every such path. Further, the analogous question for noncrossing \emph{covering trees} is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for $n$ points in the plane requires $\Omega(n \log{n})$ time in the worst case.

Abstract:
Multi-Granulation Rough Set (MGRS) is a new and interesting topic in rough set theory. In this study, three kinds of Covering Multi-Granulation Fuzzy Rough Sets (CMGFRS) have been first proposed, which extend the covering rough sets from single-granulation to multi-granulation. Firstly, the basic properties of these covering multi-granulation fuzzy rough sets are discussed, it is shown that basic properties of MGRS are special cases of those of CMGFRS. Then, the conditions for two CMGFRS generate identical lower and upper approximations of a set in a covering approximation space are studied. Finally, this study explored the relationship between the three types of CMGFRS.

Abstract:
Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08] proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal upward or downward covering set. For both problems, we raise this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size covering sets are hard or complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischer's result that minimal bidirectional covering sets (i.e., sets that are both minimal upward and minimal downward covering sets) are polynomial-time computable.

Abstract:
Attribute reduction is viewed as an important preprocessing step for pattern recognition and data mining. Most of researches are focused on attribute reduction by using rough sets. Recently, Tsang et al. discussed attribute reduction with covering rough sets in the paper [E. C.C. Tsang, D. Chen, Daniel S. Yeung, Approximations and reducts with covering generalized rough sets, Computers and Mathematics with Applications 56 (2008) 279-289], where an approach based on discernibility matrix was presented to compute all attribute reducts. In this paper, we provide an improved approach by constructing simpler discernibility matrix with covering rough sets, and then proceed to improve some characterizations of attribute reduction provided by Tsang et al. It is proved that the improved discernible matrix is equivalent to the old one, but the computational complexity of discernible matrix is greatly reduced.

Abstract:
Say that a cardinal number $\kappa$ is \emph{small} relative to the space $X$ if $\kappa <\Delta(X)$, where $\Delta(X)$ is the least cardinality of a non-empty open set in $X$. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire $\sigma$-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.