Related papers: A solution to the Pyjama Problem
We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$.…
A rectangle blanket is a set of non-overlapping axis-aligned rectangles, used to approximately represent the two dimensional image of a shape approximately. The use of a rectangle blanket is a widely considered strategy for speeding-up the…
Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures. For the problem of…
Let $M$ be a 3-manifold. Every knotted (embedded) surface in $M \times \R$ can be moved via an ambient isotopy in such a way that its projection into $M$ is a generic surface. A surface is generic if every point on it is either a regular,…
In this work we consider the problem of finding the minimum-weight loop cover of an undirected graph. This combinatorial optimization problem is called 2-matching and can be seen as a relaxation of the traveling salesman problem since one…
Map vertices of a graph to (not necessarily distinct) points of the plane so that two adjacent vertices are mapped at least a unit distance apart. The plane-width of a graph is the minimum diameter of the image of the vertex set over all…
The classical honeycomb conjecture asserts that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Pappus discusses this problem in his preface to Book V. This paper…
We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of…
We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions…
We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also…
The Orbit Problem consists of determining, given a matrix $A\in \mathbb{R}^{d\times d}$ and vectors $x,y\in \mathbb{R}^d$, whether there exists $n\in \mathbb{N}$ such that $A^n=y$. This problem was shown to be decidable in a seminal work of…
We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Considering a field P of orthogonal projections onto 1-dimensional subspaces, with divergence bounded in L^2, we prove existence and uniqueness for…
We provide a natural answer to Lewis Carroll's pillow problem of what is the probability that a triangle is obtuse, Prob(Obtuse). This arises by straightforward combination of a) Kendall's Theorem - that the space of all triangles is a…
The Plateau's problem seeks to determine a surface of minimal area which spans a given boundary. It is widely studied for its varied mathematical formulations, applications and relevance to physical models such as soap films. We revisit the…
In this paper, on envelopes created by circle families in the plane, all four basic problems (existence problem, representation problem, problem on the number of envelopes, problem on relationships of definitions) are solved.
We study the area ranges where the two possible isoperimetric domains on the infinite cylinder $\mathbb{S}^{1}\times \R$, namely, geodesic disks and cylindrical strips of the form $\mathbb{S}^1\times [0,h]$, satisfy P\'{o}lya's conjecture.…
We prove the existence of classical solutions to the Dirichlet problem for the $\alpha$-translating soliton equation defined in a strip of $\r^2$. We use the Perron method where a family of grim reapers are employed as barriers for solving…
We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently closed curves of length 2) allowing translation and discrete rotations. In particular, we show that the solution is an equilateral triangle…
In this paper, we study the 3D strip packing problem in which we are given a list of 3-dimensional boxes and required to pack all of them into a 3-dimensional strip with length 1 and width 1 and unlimited height to minimize the height used.…
In 2010, Marshall settled the strip conjecture, according to which every polynomial in $\mathbb{R}[x,y]$, nonnegative on the strip $[-1,1]\times\mathbb{R}$, is a sum of squares and of squares times $1-x^2$. We consider affine nonsingular…