Related papers: A one-dimensional planar Besicovitch-type set
A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the…
Let $\Gamma$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\Gamma$ on the circle is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle.
In this article, we give a geometric description for any invertible operator on a finite dimensional inner--product space. With the aid of such a description, we are able to decompose any given conformal transformation as a product of…
In a series of papers the authors associated to an $L^2$-acyclic group $\Gamma$ an invariant $\mathcal{P}(\Gamma)$ that is a formal difference of polytopes in the vector space $H_1(\Gamma;\Bbb{R})$. This invariant is in particular defined…
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In this paper we show that the Minkowski…
Given an open, bounded, planar set $\Omega$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac12,+\infty)\rightarrow\mathcal{P}((0,|\Omega|])$ associating to each $p$ the set of…
The paper examines machines of the type of the $\Gamma$-spaces of Segal which describe homotopy structures on topological spaces. The main result of the paper shows that for any such machine one can find an algebraic theory characterizing…
Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this…
It is studied the following problem: for a given function $f$ what kind of may be a set of all rotations $\gamma$ for which $\int f$ is not differentiable with respect to $\gamma$-rotation of a given basis $B$? In particular, for…
We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $\Gamma$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there…
Let $P$ be a set of $n$ points on the plane in general position. We say that a set $\Gamma$ of convex polygons with vertices in $P$ is a convex decomposition of $P$ if: Union of all elements in $\Gamma$ is the convex hull of $P,$ every…
The set \[ \Gamma {\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2 \] has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface…
We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the…
We validate the Timoshenko beam model as an approximation of the linear-elasticity model of a three-dimensional beam-like body. Our validation is achieved within the framework of $\Gamma$-convergence theory, in two steps: firstly, we…
The $ k $-configuration space $ B_k\Gamma $ of a topological space $ \Gamma $ is the space of sets of $ k $ distinct points in $ \Gamma $. In this paper, we consider the case where $ \Gamma $ is a graph of circumference at most $1$. We show…
This note presents the definition of a proposed generalization of the conchoid at the plane. Known conchoids, such as the Nicomedes and the Lima\c{c}on of Pascal are part of this set. Following the definition, one can generate other…
A linear isometry $R$ of $\mathbb{R}^d$ is called a similarity isometry of a lattice $\Gamma \subseteq \mathbb{R}^d$ if there exists a positive real number $\beta$ such that $\beta R\Gamma$ is a sublattice of (finite index in) $\Gamma$. The…
We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of…
A family of relativistic geometric models is defined as a generalization of the actual anti-de Sitter (1+1) model of the relativistic harmonic oscillator. It is shown that all these models lead to the usual harmonic oscillator in the…
We construct a self-similar local regular Dirichlet form on the Sierpi\'nski gasket using $\Gamma$-convergence of stable-like non-local closed forms. As a continuation of a recent paper by Grigor'yan and the author, we give the first…