Related papers: Optimal shapes of compact strings
We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the…
Using computed x-ray tomography we determine the three dimensional (3d) structure of binary hard sphere mixtures as a function of composition and size ratio of the particles, q. Using a recently introduced four-point correlation function we…
We study a class of rearrangement problems under a novel pick-n-swap prehensile manipulation model, in which a robotic manipulator, capable of carrying an item and making item swaps, is tasked to sort items stored in lattices of variable…
This work studies rearrangement problems involving the sorting of robots or objects in stack-like containers, which can be accessed only from one side. Two scenarios are considered: one where every robot or object needs to reach a…
Many standard structural quantities, such as order parameters and correlation functions, exist for common condensed matter systems, such as spherical and rod-like particles. However, these structural quantities are often insufficient for…
Structural quantities such as order parameters and correlation functions are often employed to gain insight into the physical behavior and properties of condensed matter systems. While standard quantities for characterizing structure exist,…
Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for…
Particle packing problems have fascinated people since the dawn of civilization, and continue to intrigue mathematicians and scientists. Resurgent interest has been spurred by the recent proof of Kepler's conjecture: the face-centered cubic…
The prediction of the three-dimensional native structure of proteins from the knowledge of their amino acid sequence, known as the protein folding problem, is one of the most important yet unsolved issues of modern science. Since the…
The search for optimal configurations of pointsets, the most notable examples being the problems of Kepler and Thompson, have an extremely rich history with diverse applications in physics, chemistry, communication theory, and scientific…
We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex…
Thomson problem is a classical problem in physics to study how $n$ number of charged particles distribute themselves on the surface of a sphere of $k$ dimensions. When $k=2$, i.e. a 2-sphere (a circle), the particles appear at equally…
Combinatorial optimization can be described as the problem of finding a feasible subset that maximizes a objective function. The paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is…
The Bin Packing Problem involves efficiently packing items into a limited number of bins without exceeding their capacity. In this paper, we try to answer a specific question in this field. Mathematically the combinatorial optimization…
Given a set of vectors $\F=\{f_1,\dots,f_m\}$ in a Hilbert space $\HH$, and given a family $\CC$ of closed subspaces of $\HH$, the {\it subspace clustering problem} consists in finding a union of subspaces in $\CC$ that best approximates…
Computing an optimal chain of fragments is a classical problem in string algorithms, with important applications in computational biology. There exist two efficient dynamic programming algorithms solving this problem, based on different…
We introduce geometric consideration into the theory of formal languages. We aim to shed light on our understanding of global patterns that occur on infinite strings. We utilise methods of geometric group theory. Our emphasis is on large…
Nearly orthogonal lattices were formally defined in [4], where their applications to image compression were also discussed. The idea of ``near orthogonality" in $2$-dimensions goes back to the work of Gauss. In this paper, we focus on…
Many discrete optimization problems amount to selecting a feasible set of edges of least weight. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets.…
In preference modelling, it is essential to determine the number of questions and their arrangements to ask from the decision maker. We focus on incomplete pairwise comparison matrices, and provide the optimal filling in patterns, which…