arXiv subjects

Sinisa Vrecica

Publications and source records attributed to Sinisa Vrecica.

6 recordsLinked to original sources

arXiv2013

Measurable patterns, necklaces, and sets indiscernible by measure

In some recent papers the classical `splitting necklace theorem' is linked in an interesting way with a geometric `pattern avoidance problem'. We explore the topological constraints on the existence of a (relaxed) measurable coloring of R^d such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Lason, we show that for every collection μ_1,...,μ_{2d-1} of 2d-1 continuous finite measures on R^d, there exist two nontrivial axis-aligned d-dimensional cuboids (rectangular parallelepipeds) C_1 and C_2 such that μ_i(C_1)=μ_i(C_2) for each i=1,...,2d-1. We also show by examples that the bound 2d-1 cannot be improved in general. These results are steps in the direction of studying general topological obstructions for the existence of non-repetitive colorings of measurable spaces.

math.CO↗

arXiv2010

Topological obstructions to totally skew embeddings

Following Ghomi and Tabachnikov we study topological obstructions to totally skew embeddings of a smooth manifold M in Euclidean spaces. This problem is naturally related to the question of estimating the geometric dimension of the stable normal bundle of the configuration space F_2(M) of ordered pairs of distinct points in M. We demonstrate that in a number of interesting cases the lower bounds obtained by this method are quite accurate and very close to the best known general upper bound. We also provide some evidence for the conjecture that each n-dimensional, compact smooth manifold M^n (n>1), admits a totally skew embedding in the Euclidean space of dimension N = 4n-2alpha(n)+1 where alpha(n)=number of non-zero digits in the binary representation of n. This is a revised version of the paper (accepted for publication in A.M.S. Transactions).

math.AT↗

arXiv2008

Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem

The cyclohedron (Bott-Taubes polytope) arises both as the polyhedral realization of the poset of all cyclic bracketings of a circular word and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S^1. The "polygonal pegs problem" asks whether every simple, closed curve in the plane or in the higher dimensional space admits an inscribed polygon of a given shape. We develop a new approach to the polygonal pegs problem based on the Fulton-MacPherson (Axelrod-Singer, Kontsevich) compactification of the configuration space of (cyclically) ordered n-element subsets in S^1. Among the results obtained by this method are proofs of Grunbaum's conjecture about affine regular hexagons inscribed in smooth Jordan curves and a new proof of the conjecture of Hadwiger about inscribed parallelograms in smooth, simple, closed curves in the 3-space (originally established by Victor Makeev).

math.CO↗

arXiv2007

Cycle-free chessboard complexes and symmetric homology of algebras

Chessboard complexes and their relatives have been one of important recurring themes of topological combinatorics. Closely related ``cycle-free chessboard complexes'' have been recently introduced by Ault and Fiedorowicz as a tool for computing symmetric analogues of the cyclic homology of algebras. We study connectivity properties of these complexes and prove a result that confirms a strengthened conjecture of Ault and Fiedorowicz.

math.AT↗

arXiv2007

Equipartitions of a Mass in Boxes

The aim of this paper is to provide the sufficient condition for a mass distribution in R^d to admit an equipartition with a collection of hyperplanes some of which are parallel. The results extend the previously obtained results for the equipartitions with non-parallel hyperplanes.

math.CO↗

arXiv2003

Topology and Combinatorics of Partitions of Masses by Hyperplanes

One of our result is that 5 measurable sets in $R^8$ always admit an equipartition by 2 hyperplanes. This is an instance of a general equipartition problem (formulated by B. Gr{\" u}nbaum and H. Hadwiger) which can be reduced to the question of (non)existence of a $W_k$-equivariant map where $W_k$ is the group of symmetries of a $k$-cube. We show that the computation of relevant cohomology/bordism obstruction classes often reduces to the question of enumerating the classes of immersed curves in $\mathbb{R}^2$ with a prescribed type and number of intersections with the coordinate axes, which in turn leads to a problem of enumerating classes of cyclic signed $AB$-words.

math.CO↗