Related papers: On the general dyadic grids in $\mathbb{R}^d$
A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is…
The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive…
A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were…
A collection of pairwise mutually unbiased bases (in short: MUB) in d>1 dimensions may consist of at most d+1 bases. Such "complete" collections are known to exists in C^d when d is a power of a prime. However, in general little is known…
In this paper, we propose an end-to-end graph learning framework, namely Iterative Deep Graph Learning (IDGL), for jointly and iteratively learning graph structure and graph embedding. The key rationale of IDGL is to learn a better graph…
The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by K\H{o}nig, $d$ colors suffice if the graph is bipartite. We investigate…
The general position number ${\rm gp}(G)$ of a graph $G$ is the cardinality of a largest set of vertices $S$ such that no element of $S$ lies on a geodesic between two other elements of $S$. The complementary prism $G\overline{G}$ of $G$ is…
Many real-world machine learning problems involve inherently hierarchical data, yet traditional approaches rely on Euclidean metrics that fail to capture the discrete, branching nature of hierarchical relationships. We present a theoretical…
Reider's Theorem on the very ampleness of adjoint linear series on a complex projective algebraic surface is extended in two new directions. First, Reider-type inequalities are shown to imply nefness of linear series of the form dH - E on…
One of the defining features of complex networks is the connectivity properties that we observe emerging from local interactions. Recently, hypergraphs have emerged as a versatile tool to model networks with non-dyadic, higher-order…
If we pick $n$ random points uniformly in $[0,1]^d$ and connect each point to its $k-$nearest neighbors, then it is well known that there exists a giant connected component with high probability. We prove that in $[0,1]^d$ it suffices to…
Recently, Okounkov, Lazarsfeld and Mustata, and Kaveh and Khovanskii have shown that the growth of a graded linear series on a projective variety over an algebraically closed field is asymptotic to a polynomial. We give a complete…
Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite…
This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is…
In granular computing, fuzzy sets can be approximated by granularly representable sets that are as close as possible to the original fuzzy set w.r.t. a given closeness measure. Such sets are called granular approximations. In this article,…
We develop the theory of the additive dimension ${\rm dim} (A)$, i.e. the size of a maximal dissociated subset of a set $A$. It was shown that the additive dimension is closely connected with the growth of higher sumsets $nA$ of our set…
Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set, PDDS, in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and…
We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of…
This article proves hypersurfaces of degree d in projective n-space are "rationally simply-connected" if $d^2 \leq n$. In a forthcoming paper, de Jong and I prove a slightly weaker result when $d^2 \leq n+1$.
Two adjoint functors can be seen as generalisations of the two functions within a Galois connection. If instead the adjoints are not generalised from functions, but from relations, then analogously the object of study becomes a more general…