Related papers: A solution to the Pyjama Problem

Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph is the smallest number of…

Let P be a set of n points in R^3. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present two randomized algorithms for computing a 2-center of P. The first algorithm runs in O(n^3 log^5…

We consider harmonic maps from Minkowski space into the three sphere. We are especially interested in solutions which are asymptotically constant, i.e. converge to the same value in all directions of spatial infinity. Physical 3-space can…

We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter. Because of…

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the 1-dimensional curve bounding a disk. For any nonnegative integer, a circle is called $n$-enclosing if…

In this note, we give a closed formula for the partition function of the dimer model living on a (2 x n) strip of squares or hexagons on the torus for arbitrary even n. The result is derived in two ways, by using a Potts model like…

Let $\Gamma$ be an arrangement of Jordan curves in the plane, i.e., simple closed curves in the plane. For any curve $\gamma \in \Gamma$, we denote the bounded region enclosed by $\gamma$ as $\tilde{\gamma}$. We say that $\Gamma$ is…

The string propagation equations in axisymmetric spacetimes are exactly solved by quadratures for a planetoid Ansatz. This is a straight non-oscillating string, radially disposed, which rotates uniformly around the symmetry axis of the…

Spherical caps play a crucial role in establishing a criterion for the existence of solutions to the Yamabe problem on a compact Riemannian manifold with boundary, similar to the role played by the standard sphere in the problem on a closed…

A (multi)set of segments in the plane may form a TSP tour, a matching, a tree, or any multigraph. If two segments cross, then we can reduce the total length with the following flip operation. We remove a pair of crossing segments, and…

In mathematics, the classical Plateau problem consists of finding the surface of least area that spans a given rigid boundary curve. A physical realization of the problem is obtained by dipping a stiff wire frame of some given shape in…

We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can…

Toy Stars are gas masses where the compressibility is treated without approximations but gravity is replaced by a force which, for any pair of masses, is along their line of centres and proportional to their separation. They provide an…

Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k=1,2,..n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a…

Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a…

Hadwiger's covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an…

A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viwed as the converse of Tarski's plank problem (Bang's theorem): Is it true that if the total width of a collection of planks…

Consider a problem where we are given a bipartite graph H with vertices arranged on two horizontal lines in the plane, such that the two sets of vertices placed on the two lines form a bipartition of H. We additionally require that H admits…

We consider the periodic Manhattan lattice with alternating orientations going north-south and east-west. Place obstructions on vertices independently with probability $0<p<1$. A particle is moving on the edges with unit speed following the…

A generalization of Rip\`a's square spiral solution for the $n \times n \times \cdots \times n$ Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the $k$-dimensional $n_1 \times n_2 \times \cdots \times n_k$…