Related papers: Asymptotics of $d$-Dimensional Visibility
A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…
Suppose a d-dimensional lattice cube of size n^d is colored in several colors so that no face of its triangulation (subdivision of the standard partition into n^d small cubes) is colored in m+2 colors. Then one color is used at least…
It is well-known that the $d$-dimensional hypercube contains a Hamilton cycle for $d\ge 2$. In this paper we address the analogous problem in the $3$-uniform cube hypergraph, a $3$-uniform analogue of the hypercube: for simple parity…
In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in discrete quantum systems. It is known that for dimensions $d$ that are powers of prime numbers, there exists a set of up to $d+1$ bases that form an MUB set.…
In this paper we are proving the following fact. Let P be an arbitrary simple polygon, and let S be an arbitrary set of 15 points inside P. Then there exists a subset T of S that is not "visually discernible", that is, T is not equal to the…
We show that in a complex d-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have manifold applications…
We study $n$-dimensional matrices with $\{0,1\}$-entries ($n$-cubes) such that all their $2$-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized…
We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of…
In a given hypercube, draw grid lines parallel to the edges, and consider all hypercuboids (or hypercubes) whose edges are lying on the grid lines or the boundary. We find the limit of the value of the ratio of the arithmetic mean of the…
We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if…
We establish that almost every positive integer $n$ is the sum of four cubes, two of which are at most $n^{\theta}$, as long as $\theta\geq192/869$. An asymptotic formula for the number of such representations is established when…
In this paper we show that the number of distinct distances determined by a set of $n$ points on a constant-degree two-dimensional algebraic variety $V$ (i.e., a surface) in $\mathbb R^3$ is at least $\Omega\left(n^{7/9}/{\rm polylog}…
Given $n$ pairwise openly disjoint triangles in 3-space, their vertical depth relation may contain cycles. We show that, for any $\varepsilon>0$, the triangles can be cut into $O(n^{3/2+\varepsilon})$ connected semi-algebraic pieces, whose…
The dimension of the visible part of self-affine sets, that satisfy domination and a projection condition, is being studied. The main result is that the assouad dimension of the visible part equals to 1 for all directions outside the set of…
We determine the density of the largest sum-free subset of the lattice cube $\{1, 2, \dots, n\}^d$ for $d = 3$ and $d = 4$. This solves a conjecture of Cameron and Aydinian in dimensions $3$ and $4$.
For a point set of $n$ elements in the $d$-dimensional unit cube and a class of test sets we are interested in the largest volume of a test set which does not contain any point. For all natural numbers $n$, $d$ and under the assumption of a…
Higher-dimensional spaces are ubiquitous in applications of mathematics. Yet, as we live in a three-dimensional space, visualizing, say, a four-dimensional space is challenging. We introduce a novel method of interactive visualization of…
In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition,…
A visibility representation of a graph $G$ is an assignment of the vertices of $G$ to geometric objects such that vertices are adjacent if and only if their corresponding objects are "visible" each other, that is, there is an uninterrupted…
Counter-propagating light fields have the ability to create self-organized one-dimensional optically bound arrays of microscopic particles, where the light fields adapt to the particle locations and vice versa. We develop a theoretical…