Related papers: Ranking Unit Squares with Few Visibilities
We study several variations of line segment covering problem with axis-parallel unit squares in $I\!\!R^2$. A set $S$ of $n$ line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at…
Ranking a set of objects from the most dominant one to the least, based on the results of paired comparisons, proves to be useful in many contexts. Using the rankings of teams or individuals players in sports to seed tournaments is an…
We re-examine previous constructions of infinite binary words containing few distinct squares with the goal of finding the "simplest", in a certain sense. We exhibit several new constructions. Rather than using tedious case-based arguments…
Choose a polynomial in three variables with not more than three or four monomials of moderate degree. Take simple coefficients as 1 and -1. Then draw a picture of the solution variety in real three space using a ray-tracing program like…
In this paper, we study arrangements of orthogonal circles, that is, arrangements of circles where every pair of circles must either be disjoint or intersect at a right angle. Using geometric arguments, we show that such arrangements have…
Let $P$ be a simple polygon with $n$ vertices, and let $A$ be a set of $m$ points or line segments inside $P$. We develop data structures that can efficiently count the number of objects from $A$ that are visible to a query point or a query…
We consider the problem of placing k queens on an nxn board so that the total number of attacked squares is as small as possible. In particular, we consider the domain where k is small relative to n and derive nearly tight bounds in this…
This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point…
Given a set of $n\geq 1$ unit disk robots in the Euclidean plane, we consider the fundamental problem of providing mutual visibility to them: the robots must reposition themselves to reach a configuration where they all see each other. This…
In domains like bioinformatics, information retrieval and social network analysis, one can find learning tasks where the goal consists of inferring a ranking of objects, conditioned on a particular target object. We present a general kernel…
A vector space A of matrices is called rank-critical if any vector space that properly contains A has a strictly higher generic rank. I present a sufficient condition for A to be rank-critical, and apply this condition to prove that certain…
Let A, B, C, D be given finite sets of pairs of n-by-n complex matrices. We describe an algorithm to determine, with finitely many computations, whether there is a single unitary matrix U such that each pair of matrices in A is unitarily…
Let $F(x_1,...,x_n)$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever…
We prove that, for all even $n\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \big\lfloor n/6 \big\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$…
The paper suggests a family of paired comparison-based scoring procedures for ranking the participants of a Swiss system chess team tournament. We present the challenges of ranking in Swiss system, the features of individual and team…
A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order $n$ there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality…
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have…
The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…
Given a number of pairwise preferences of items, a common task is to rank all the items. Examples include pairwise movie ratings, New Yorker cartoon caption contests, and many other consumer preferences tasks. What these settings have in…
A vast array of envy-free results have been found for the subdivision of one-dimensional resources, such as the interval $[0,1]$. The goal is to divide the space into $n$ pieces and distribute them among $n$ observers such that each…