Related papers: Interval projections of self-similar sets
We derive an upper bound for the Assouad dimension of visible parts of self-similar sets generated by iterated function systems with finite rotation groups and satisfying the open set condition. The bound is valid for all visible parts and…
Let $L_1,L_2,\dots,L_K$ be a family of closed subspaces of a Hilbert space $H$, $L_1\cap \dots \cap L_K =\{0\}$; let $P_k$ be the orthogonal projection onto $L_k$. We consider two types of consecutive projections of an element $x_0\in H$:…
The construction of affine interval exchange maps with wandering intervals that are semi-conjugate with a given self-similar interval exchange map is strongly related with the existence of the so called minimal sequences associated with…
We improve the bounds on the number of interval graphs on $n$ vertices. In particular, denoting by $I_n$ the quantity in question, we show that $\log I_n \sim n\log n$ as $n\to \infty$.
We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that…
We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a…
We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than $1$, such that there exist lines for which dimension conservation does not hold and the projection of the measure is…
Most characterizations of interpolating sequences for Bergman spaces include the condition that the sequence be uniformly discrete in the hyperbolic metric. We show that if the notion of interpolation is suitably generalized, two of these…
A large class of problems in sciences and engineering can be formulated as the general problem of constructing random intervals with pre-specified coverage probabilities for the mean. Wee propose a general approach for statistical inference…
Tasks and objects are two predominant ways of specifying distributed problems. A task is specified by an input/output relation, defining for each set of processes that may run concurrently, and each assignment of inputs to the processes in…
We define the similarity boundary of a self-similar set and use it to analyze the properties of self-similar sets in the general setting of any complete metric space. The similarity boundary is an attempt at extending the concept of the…
For an interval finite quiver $Q$, we introduce a class of flat representations. We classify the indecomposable projective objects in the category $\mathrm{rep}(Q)$ of pointwise finite dimensional representations. We show that an object in…
We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement to lattices.…
We study sets $E(\Sigma,q)=\left\{\sum_{i=1}^\infty \sigma_iq^i\colon(\sigma_i)\in\Sigma^{\mathbb N}\right\}$ for a finite set $\Sigma\subset \mathbb R$ and $q\in(0,1)$. Under the assumption $q|\Sigma|=1$ we prove several new equivalent…
A hypergraph $(V,E)$ is called an interval hypergraph if there exists a linear order $l$ on $V$ such that every edge $e\in E$ is an interval w.r.t. $l$; we also assume that $\{j\}\in E$ for every $j\in V$. Our main result is a de…
In this paper we consider fractal percolation random Cantor sets $E$ on the plane constructed with non-homogeneous probabilities. We focus on the case when the probabilities are large enough to guarantee that the almost sure dimension of…
Let $k$ be a field. We describe necessary and sufficient conditions for a $k$-linear abelian category to be a noncommutative projective line, i.e. a noncommutative $\mathbb{P}^{1}$-bundle over a pair of division rings over $k$. As an…
We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point…
We define interval spacing as the difference in the order statistics of data over a gap of some width. We derive its density, expected value, and variance for uniform, exponential, and logistic variates. We show that interval spacing is…
Let $K_1$ and $K_2$ be two one-dimensional homogeneous self-similar sets. Let $f$ be a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the continuous image of $f$ by $$ f_{U}(K_1,K_2)=\{f(x,y):(x,y)\in…