Related papers: Construction Techniques for Linear Realizations of…

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far reaching generalization of K\"onig's theorem is only known to be true for $r\leq 3$,…

We present a general approach to the bifurcation analysis of elastic frameworks with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of elastic…

We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair…

The Caccetta-H\"aggkvist conjecture implies that for every integer $k\ge 1$, if $G$ is a bipartite digraph, with $n$ vertices in each part, and every vertex has out-degree more than $n/(k+1)$, then $G$ has a directed cycle of length at most…

We study the following fundamental realization problem of directed acyclic graphs (dags). Given a sequence S:=(a_1,b_1),...,(a_n, b_n) with a_i, b_i in Z_0^+, does there exist a dag (no parallel arcs allowed) with labeled vertex set V:=…

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the…

Offset curves for planar trajectories are interesting in the generation of tool paths for numerically controlled industrial machines and in trajectory planning methods for autonomous driving systems. Theoretical offset curves may exhibit…

There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new results (and problems) concerning families of $t$-intersecting $k$-element multisets of an $n$-set and point out connections to coding theory and classical…

Linear models have found widespread use in statistical investigations. For every linear model there exists a matrix representation for which the ReML (Restricted Maximum Likelihood) can be constructed from the elements of the corresponding…

A (v,k,t) covering design, or covering, is a family of k-subsets, called blocks, chosen from a v-set, such that each t-subset is contained in at least one of the blocks. The number of blocks is the covering's size}, and the minimum size of…

We adapt tools from the algebraic approach to constraint satisfaction problems to answer descriptive set theoretic questions about Borel CSPs. We show that if a structure $\mathcal D$ does not have a Taylor polymorphism, then the…

For given a graph $H$, a graphic sequence $\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\pi$ containing $H$ as a subgraph. In this paper, we characterize the potentially $H$-graphic…

Let $A_{k,t}$ be the matrix that represents the adjacency matrix of the intersection bipartite graph of all subsets of size $t$ of $\{1,2,...,k\}$. We give constructions of large isolation sets in $A_{k,t}$, where, for a large enough $k$,…

Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\delta(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex…

We say that a Hamilton cycle $C=(x_1,\ldots,x_n)$ in a graph $G$ is $k$-symmetric, if the mapping $x_i\mapsto x_{i+n/k}$ for all $i=1,\ldots,n$, where indices are considered modulo $n$, is an automorphism of $G$. In other words, if we lay…

For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a one-realization of $S$ if its feasible set is $S$ and each entry of its chromatic spectrum is either 0 or 1. In this paper, we determine the minimum size of 3-uniform…

We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in C^k when k-1 is a prime power. Using semiregular relative difference sets with parameters…

This paper examines linear binary codes capable of correcting one or more errors. For the single-error-correcting case, it is shown that the Hamming bound is achieved by a constructive method, and an exact expression for the minimal…

For the Kirillov-Poisson structure on the vector space $\g^*$, where $\g$ is a finite-dimensional Lie algebra, it is known at least two canonical deformations quantization of this structure: they are the M. Kontsevich universal formula [K],…

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear…