Related papers: Self-projective sets
We introduce a system of light driven microscopic autonomous moving particles that move on a flat surface. The design is simple, yet effective: Micrometer sized objects with wedge shape are produced by photopolymerization, they are covered…
Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over…
In this paper, we study fast escaping set of transcendental semigroup. We discuss some the structure and properties of fast escaping set of transcendental semigroup. We also see how far the classical theory of fast escaping set of…
We investigate dimension-theoretic properties of concentric topological spheres, which are fractal sets emerging both in pure and applied mathematics. We calculate the box dimension and Assouad spectrum of such collections, and use them to…
The attracting set and the inverse limit set are important objects associated to a self-map on a set. We call \emph{stable set} of the self-map the projection of the inverse limit set. It is included in the attracting set, but is not equal…
Self-similarity is a property of fractal structures, a concept introduced by Mandelbrot and one of the fundamental mathematical results of the 20th century. The importance of fractal geometry stems from the fact that these structures were…
Recent advances in deep learning have transformed many fields by introducing generic embedding spaces, capable of achieving great predictive performance with minimal labeling effort. The geology field has not yet met such success. In this…
Fractals are self-repeating patterns which have dimensions given by fractions rather than integers. While the dimension of a system unambiguously defines its properties, a fractional dimensional system can exhibit interesting properties.…
In the present article, topological, metric, and fractal properties of certain sets are investigated. These sets are images of sets whose elements have restrictions on using digits or combinations of digits in own s-adic representations,…
An important result in real algebraic geometry is the projection theorem: every projection of a semialgebraic set is again semialgebraic. This theorem and some of its conclusions lie at the basis of many other results, for example the…
Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 80-s of the last century with the purpose to solve some famous problems in mathematics, including the question raising to von…
The aim of this article is to discuss and clarify the notion of fractality for subgroups of the group of automorphisms of a regular rooted tree. For this purpose we define three types of fractality. We show that they are not equivalent, by…
By a "happy fractal" we mean a metric space with bounded geometry in the sense of a doubling condition and a lot of paths of finite length, so that any pair of points can be connected by a path whose length is less than or equal to a…
We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such…
Our objective is to determine which subsets of $\mathbb{R}^d$ arise as escaping sets of continuous functions from $\mathbb{R}^d$ to itself. We obtain partial answers to this problem, particularly in one dimension, and in the case of open…
Furstenberg, using tools from topological dynamics, defined the notion of a central subset of positive integers, and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-\v{C}ech…
Many astrophysical systems can be approximated as isothermal spheres. In an isothermal sphere, the ``foreground'' objects can act as lenses on ``background'' objects in the same distribution. We study gravitational lensing by a singular…
We study the box dimensions of self-affine sets in $\mathbb{R}^3$ which are generated by a finite collection of generalised permutation matrices. We obtain bounds for the dimensions which hold with very minimal assumptions and give rise to…
Our goal is to derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics. Let A be a small concrete category. We define a functor F, cometic functor,…
Fractals are self-similar recursive structures that have been used in modeling several real world processes. In this work we study how "fractal-like" processes arise in a prediction game where an adversary is generating a sequence of bits…