Related papers: Pach's animal problem within the bounding box
In this article, we consider the problems of finding in $d+1$ dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of $d$-dimensional unit-radius…
The classical Pell equation can be extended to the cubic case considering the elements of norm one in $Z[\sqrt[3]{r}]$, which satisfy $x^3 + r y^3 + r^2 z^3 - 3 r x y z = 1$. The solution of the cubic Pell equation is harder than the…
In this paper we consider the generalization of the Cheeger problem which comes by considering the ratio between the perimeter and a certain power of the volume. This generalization has been already sometimes treated, but some of the main…
Isoperimetric robots are large scale, untethered inflatable robots that can undergo large shape changes, but have only been demonstrated in one 3D shape -- an octahedron. These robots consist of independent triangles that can change shape…
In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can…
We present a 3D stylization algorithm that can turn an input shape into the style of a cube while maintaining the content of the original shape. The key insight is that cubic style sculptures can be captured by the as-rigid-as-possible…
We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other…
This paper is concerned with the application of a spacetime structure to a three-dimensional quantum system. There are three components. First, the main part of this paper presents the constraint conditions which build the relation of a…
In this paper we study the problem of augmenting a planar graph such that it becomes 3-regular and remains planar. We show that it is NP-hard to decide whether such an augmentation exists. On the other hand, we give an efficient algorithm…
In this article we introduce the notion of a balloon animal map between broken toric varieties and construct several long exact sequences in cohomology related to them. We give a new proof of the deletion-contraction relation on hypertoric…
A method for converting the geometrical problem of rectangle packing to an algebraic problem of solving a system of polynomial equations is described.
We prove that if $X$ is a real Banach space, with $\dim X\geq 3$, which contains a subspace of codimension 1 which is 1-complemented in $X$ and whose group of isometries is almost transitive then $X$ is isometric to a Hilbert space. This…
In the Colored Bin Packing problem a set of items with varying weights and colors must be packed into bins of uniform weight limit such that no two items of the same color may be packed adjacently within a bin. We solve this problem for the…
In programmable matter, we consider a large number of tiny, primitive computational entities called particles that run distributed algorithms to control global properties of the particle structure. Shape formation problems, where the…
We consider methods for connected reconfigurations by finite automate in the so-called \emph{hybrid} or \emph{Robot-on-Tiles} model of programmable matter, in which a number of simple robots move on and rearrange an arrangement of passive…
A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather…
Framing an issue as a puzzle, problem, or mess is an illustrative approach to characterizing the issue's complexity within organizational theory and systems thinking. We use this approach to characterize the issue of designing collective…
In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peakons. The 3CH model is proven integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system…
A perfect cuboid is formed when an Euler brick whose edges and face diagonals are all integers also has an integer internal diagonal. It is known that if a perfect cuboid exists the internal diagonal is odd. No perfect cuboid has been…
We study entropy numbers and box dimension of (the image of) homogeneous polynomials and holomorphic functions between Banach spaces. First, we see that entropy numbers and box dimensions of subsets of Banach spaces are related. We show…