Related papers: Optimal shapes of compact strings
We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids, and certain smoothly-shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we…
Lattices and periodic point sets are well known objects from discrete geometry. They are also used in crystallography as one of the models of atomic structure of periodic crystals. In this paper we study the embedding properties of spaces…
We consider the lattice model for an ideal-linear polymer chain to mimic the conformations of the semi-flexible homo-polymer chain. The polymer chain is assumed to confine in the fairly small area, such that the flexible chain conformations…
In this thesis we study string compactifications on manifolds equipped with a $G$-structure, placing a special emphasis on the interplay between geometry and physics. We follow two complementary approaches. In the first part of the thesis…
Hard-particle packings have served as useful starting points to study the structure of diverse systems such as liquids, living cells, granular media, glasses, and amorphous solids. Howard Reiss has played a major role in helping to…
Lattice-model simulations and experiments of some small proteins suggest that folding is essentially controlled by a few conserved contacts. Residues of these conserved contacts form the minimum set of native contacts needed to ensure…
The task of finding the optimal compression of a polyline with straight-line segments and arcs is performed in many applications, such as polyline compression, noise filtering, and feature recognition. Optimal compression algorithms find…
The optimal binning is the optimal discretization of a variable into bins given a discrete or continuous numeric target. We present a rigorous and extensible mathematical programming formulation for solving the optimal binning problem for a…
We consider sets of trace-normalized non-negative operators in Hilbert-Schmidt balls that maximize their mutual Hilbert-Schmidt distance; these are optimal arrangements in the sets of purity-limited classical or quantum states on a…
Finding correspondences between shapes is a fundamental problem in computer vision and graphics, which is relevant for many applications, including 3D reconstruction, object tracking, and style transfer. The vast majority of correspondence…
The shape of the cross section of a dense fiber bundle is related to the symmetry of its molecular packing. However, this statement might be belied by type I collagen fibrils which have a rounded section of high symmetry while structural…
The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or…
Patchy particles have proven to be a prominent model for studying the self-assembly behavior of various systems, ranging from finite clusters to bulk crystal assemblies, and from synthetic colloidal particles to viruses. The patchy particle…
A fast and reliable geometry optimization algorithm is presented that optimizes atomic positions and lattice vectors simultaneously. Using a series of benchmarks, it is shown that the method presented in this paper outperforms in most cases…
We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every…
We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous…
Max-cut, clustering, and many other partitioning problems that are of significant importance to machine learning and other scientific fields are NP-hard, a reality that has motivated researchers to develop a wealth of approximation…
We consider random perfect matchings on a general class of contracting bipartite graphs by letting certain edge weights be 0 on the contracting square-hexagon lattice in a periodic way. We obtain a deterministic limit shape in the scaling…
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…