Related papers: The maximum diameter of 2-dimensional simplicial c…
Many well-studied problems in extremal combinatorics deal with the maximum possible size of a family of objects in which every pair of objects satisfies a given restriction. One problem of this type was recently raised by Alon, Gujgiczer,…
A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational…
A method based on the envelope theory is presented to compute approximate solutions for $N$-body Hamiltonians with identical particles in $D$ dimensions ($D\ge 2$). In some favorable cases, the approximate eigenvalues can be analytically…
We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of "super…
We examine the Maximum Independent Set Problem in an undirected graph. The main result is that this problem can be considered as the solving the same problem in a subclass of the weighted normal twin-orthogonal graphs. The problem is…
We study the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box in $\Z^d$. We focus on the cases when an edge between $x$ and $y$ is added with probability decaying with the Euclidean…
We present a new formulation of the maximum clique problem of a graph in complex space. We start observing that the adjacency matrix A of a graph can always be written in the form A = B B where B is a complex, symmetric matrix formed by…
Given a square-free monomial ideal $I$, we define a simplicial complex labeled by the generators of $I^2$ which supports a free resolution of $I^2$. As a consequence, we obtain (sharp) upper bounds on the Betti numbers of the second power…
Let $K$ be a finite simplicial complex. We prove that the normalized expected Betti numbers of a random subcomplex in its $d$-th barycentric subdivision $\text{Sd}^d (K)$ converge to universal limits as $d$ grows to $+ \infty$. In…
We study a specific convex maximization problem in n-dimensional space. The conjectured solution is proved to be a vertex of the polyhedral feasible region, but only a partial proof of local maximality is known. Integer sequences with…
Let $P$ be a set of $n$ points in $d$-dimensions. The simplicial depth, $\sigma_P(q)$ of a point $q$ is the number of $d$-simplices with vertices in $P$ that contain $q$ in their convex hulls. The simplicial depth is a notion of data depth…
In this paper we describe the intersection between the balls of maximal symplectic packings of $\P^2$. This analysis shows the existence of singular points for maximal packings of $\P^2$ by more than three equal balls. It also yields a…
We investigate the giant component formed via high-dimensional paths in the multi-parameter random simplicial complex (MRSC) model. For a $d$-dimensional simplicial complex, we define $d$-dimensional connectivity through incidence between…
When can we compute the diameter of a graph in quasi linear time? We address this question for the class of {\em split graphs}, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that…
Using transversality and a dimension reduction argument, a result of A. Bezdek and W. Kuperberg is applied to polycylinders $\mathbb{D}^2\times \mathbb{R}^n$, showing that the optimal packing density is $\pi/\sqrt{12}$ in any dimension.
The maximal contact symmetry dimensions for scalar ODEs of order $\ge 4$ and vector ODEs of order $\ge 3$ are well known. Using a Cartan-geometric approach, we determine for these ODEs the next largest realizable (submaximal) symmetry…
We consider closed simplicial and cubical $n$-complexes in terms of link of their $(n-2)$-faces. Especially, we consider the case, when this link has size 3 or 4, i.e., every $(n-2)$-face is contained in 3 or 4 $n$-faces. Such simplicial…
We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time…
We prove that for every $d\geq 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for…
Motivated by potential applications in network theory, engineering and computer science, we study $r$-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of…