Abstract:
We consider the optimal covering of fractal sets in a two-dimensional space using ellipses which become increasingly anisotropic as their size is reduced. If the semi-minor axis is \epsilon and the semi-major axis is \delta, we set \delta=\epsilon^\alpha, where 0<\alpha<1 is an exponent characterising the anisotropy of the covers. For point set fractals, in most cases we find that the number of points N which can be covered by an ellipse centred on any given point has expectation value < N > ~ \epsilon^\beta, where \beta is a generalised dimension. We investigate the function \beta(\alpha) numerically for various sets, showing that it may be different for sets which have the same fractal dimension.

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:
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:
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:
A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. upper H-null, directed P-null and P-null sets are defined likewise, with other fractal dimensions in place of Hausdorff dimension. We investigate these sets and show that in 2^\omega{} they coincide, respectively, with strongly null, meager-additive, T' and null-additive sets. Some consequences: A subset of 2^\omega{} is meager-additive if and only if it is E-additive; if f:2^\omega->2^\omega{} is continuous and X is meager-additive, then so is f(X), and likewise for null-additive and T'-sets.

Abstract:
In this paper two zero-dimensional compact sets with equal topological and fractal dimensions but embedded in Euclidean space by different ways are under study. Diffraction of plane electromagnetic wave propagated and reflected by fractal surfaces is considered for each of these compact sets placed in vacuum. It is obtained, that the embedding of compact influences on characteristics of wave in final state. Thus, the embedding of Cantor set in Euclidean space is additional property of a fractal which can be important both for applications of fractal electrodynamics and for physics of strong interactions.

Abstract:
In the space and the time with a fractional dimensions the Lorents transformations fulfill only as a good approach and become exact only when dimensions are integer. So the principle of relativity (it is exact when dimensions are integer) may be treated also as a good approximation and may remain valid (but modified) in case of small fractional corrections to integer dimensions of time and space. In this paper presented the gravitation field theory in the fractal time and space (based on the fractal theory of time and space developed by author early). In the theory are taken into account the alteration of Lorents transformations for case including $v=c$ and are described the real gravitational fields with spin equal 2 in the fractal time defined on the Riemann or Minkowski measure carrier. In the theory introduced the new "quasi-spin", given four equations for gravitational fields (with different "quasi spins" and real and imaginary energies). For integer dimensions the theory coincide with Einstein GR or Logunov- Mestvirichvili gravitation theory.

Abstract:
We have performed a detailed analysis of the fast multipole method (FMM) in the adaptive case, in which the depth of the FMM tree is non-uniform. Previous works in this area have focused mostly on special types of adaptive distributions, for example when points accumulate on a 2D manifold or accumulate around a few points in space. Instead, we considered a more general situation in which fractal sets, e.g., Cantor sets and generalizations, are used to create adaptive sets of points. Such sets are characterized by their dimension, a number between 0 and 3. We introduced a mathematical framework to define a converging sequence of octrees, and based on that, demonstrated how to increase $N \to \infty$. A new complexity analysis for the adaptive FMM is introduced. It is shown that the ${\cal{O}}(N)$ complexity is achievable for any distribution of particles, when a modified adaptive FMM is exploited. We analyzed how the FMM performs for fractal point distributions, and how optimal parameters can be picked, e.g., the criterion used to stop the subdivision of an FMM cell. A new subdividing double-threshold method is introduced, and better performance demonstrated. Parameters in the FMM are modeled as a function of particle distribution dimension, and the optimal values are obtained. A three dimensional kernel independent black box adaptive FMM is implemented and used for all calculations.

Abstract:
We study the existence of traces of Besov spaces on fractal $h$-sets $\Gamma$ with the special focus laid on necessary assumptions implying this existence, or, in other words, present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of [Br4] and a continuation of the recent paper [Ca2]. Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that -- depending on the function space and the set $\Gamma$ -- there occurs an alternative: either the trace on $\Gamma$ exists, or smooth functions compactly supported outside $\Gamma$ are dense in the space. This notion was introduced by Triebel in [Tr7] for the special case of $d$-sets.