Related papers: A note on the intersection property for flat boxes…

We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more…

In this paper we study the regularity of embeddings of finite--dimensional subsets of Banach spaces into Euclidean spaces. In 1999, Hunt and Kaloshin [Nonlinearity 12 1263-1275] introduced the thickness exponent and proved an embedding…

The effects of boundary conditions of the fields for the compactified space directions on the supersymmetric theories are discussed. The boundary conditions can be taken to be periodic up to the degrees of freedom of localized $U(1)_{R}$…

For any natural number $d$ and positive number $\varepsilon$, we present a point set in the $d$-dimensional unit cube $[0,1]^d$ that intersects every axis-aligned box of volume greater than $\varepsilon$. These point sets are very easy to…

The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…

Geometric embeddings have recently received attention for their natural ability to represent transitive asymmetric relations via containment. Box embeddings, where objects are represented by n-dimensional hyperrectangles, are a particularly…

We explore spacetime torsion in a two-dimensional setting, wherein it corresponds to a vector field. Without invoking field equations of a particular gravitational theory, we develop visualization techniques for such torsion fields,…

Qualitatively, a no-dimensional Helly-type theorem says that if every small subfamily of convex sets has a common point in a bounded region, then suitable neighborhoods of all the sets in the whole family have a common point. Quantitative…

The Casimir effect for rectangular boxes has been studied for several decades. But there are still some points unclear. Recently, there are new developments related to this topic, including the demonstration of the equivalence of the…

We study the relationship between the sizes of two sets $B, S\subset\mathbb{R}^2$ when $B$ contains either the whole boundary, or the four vertices, of a square with axes-parallel sides and center in every point of $S$, where size refers to…

We investigate $\lambda$-Hilbert transform, $\lambda$-Possion integral and conjugate $\lambda$-Poisson integral on the atomic Hardy space in the Dunkl setting and establish a new version of Paley type inequality which extends the results in…

In this paper we study generalizations of classical results on intersection patterns of set systems in $\mathbb{R}^d$, such as the fractional Helly theorem or the $(p,q)$-theorem, in the setting of arbitrary triangulable spaces with a…

We introduce a model structure on the category of graphs, which is Quillen equivalent to the category of $\mathbb{Z}_2$-spaces. A weak equivalence is a graph homomorphism which induces a $\mathbb{Z}_2$-homotopy equivalence between their box…

We prove that for a topological space X with the property that $H_p(U)=0$ for $p\geq d$ and every open subset $U$ of $X$, a finite family of open sets in $X$ has nonempty intersection if for any subfamily of size $j$, $1\leq j \leq d+1$,…

We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $\mu$ is a Borel finite measure on $X$…

Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times...\times [a_k,b_k]$. The {\it boxicity} of $G$,…

We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional…

We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex…

Two axis-aligned boxes in $\mathbb{R}^d$ are \emph{$k$-neighborly} if their intersection has dimension at least $d-k$ and at most $d-1$. The maximum number of pairwise $k$-neighborly boxes in $\mathbb{R}^d$ is denoted by $n(k,d)$. It is…

Planar graphs are the graphs with Dushnik-Miller dimension at most three (W. Schnyder, Planar graphs and poset dimension, Order 5, 323-343, 1989). Consider the intersection graph of interior disjoint axis parallel rectangles in the plane.…