Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Geodesics in Heat

We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required

The Ising Model on a Quenched Ensemble of c = -5 Gravity Graphs

We study with Monte Carlo methods an ensemble of c=-5 gravity graphs, generated by coupling a conformal field theory with central charge c=-5 to two-dimensional quantum gravity. We measure the fractal properties of the ensemble, such as the string susceptibility exponent gamma_s and the intrinsic fractal dimensions d_H. We find gamma_s = -1.5(1) and d_H = 3.36(4), in reasonable agreement with theoretical predictions. In addition, we study the critical behavior of an Ising model on a quenched ensemble of the c=-5 graphs and show that it agrees, within numerical accuracy, with theoretical predictions for the critical behavior of an Ising model coupled dynamically to two-dimensional quantum gravity, provided the total central charge of the matter sector is c=-5. From this we conjecture that the critical behavior of the Ising model is determined solely by the average fractal properties of the graphs, the coupling to the geometry not playing an important role.Comment: 23 pages, Latex, 7 figure

Three-Dimensional Simplicial Gravity and Degenerate Triangulations

I define a model of three-dimensional simplicial gravity using an extended ensemble of triangulations where, in addition to the usual combinatorial triangulations, I allow degenerate triangulations, i.e. triangulations with distinct simplexes defined by the same set of vertexes. I demonstrate, using numerical simulations, that allowing this type of degeneracy substantially reduces the geometric finite-size effects, especially in the crumpled phase of the model, in other respect the phase structure of the model is not affected.Comment: Latex, 19 pages, 10 eps-figur

New error measures and methods for realizing protein graphs from distance data

The interval Distance Geometry Problem (iDGP) consists in finding a realization in $\mathbb{R}^K$ of a simple undirected graph $G=(V,E)$ with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. In this paper, we focus on the application to the conformation of proteins in space, which is a basic step in determining protein function: given interval estimations of some of the inter-atomic distances, find their shape. Among different families of methods for accomplishing this task, we look at mathematical programming based methods, which are well suited for dealing with intervals. The basic question we want to answer is: what is the best such method for the problem? The most meaningful error measure for evaluating solution quality is the coordinate root mean square deviation. We first introduce a new error measure which addresses a particular feature of protein backbones, i.e. many partial reflections also yield acceptable backbones. We then present a set of new and existing quadratic and semidefinite programming formulations of this problem, and a set of new and existing methods for solving these formulations. Finally, we perform a computational evaluation of all the feasible solver$+$formulation combinations according to new and existing error measures, finding that the best methodology is a new heuristic method based on multiplicative weights updates