Distance Geometry


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Distance Geometry: Theory, Methods and Applications is the first collection of research surveys dedicated to distance geometry and its applications. The first part of the book discusses theoretical aspects of the Distance Geometry Problem (DGP), where the relation between DGP and other related subjects are also presented. Covered topics include distance matrix theory, Euclidean distance matrix completion, multispherical structure of distance matrices, geometric algebra, algebraic distance geometry theory, visualization of K-dimensional structures in the plane, graph rigidity, and theory of discretizable DGP.
The second part of this volume presents mathematical and computational properties of methods developed to the problems discussed in the first portion, including continuous methods (based on Gaussian and hyperbolic smoothing, difference of convex functions, semidefinite programming, branch-and-bound), discrete methods (based on branch-and-prune, geometric build-up, graph rigidity), and also heuristics methods (based on simulated annealing, genetic algorithms, tabu search, variable neighborhood search).
Applications comprise the third part of the book, which is mainly devoted to the application of DGP to NMR structure calculation. This is an important and strongly multidisciplinary application in biology and biomedicine.


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PART I - Theory

  1. Universal rigidity of bar frameworks in general position: a Euclidean distance matrix approach, by Abdo Y. Alfakih.

  2. Mixed volume and distance geometry techniques for counting Euclidean embeddings of rigid graphs, by Ioannis Z. Emiris, Elias P. Tsigaridas, Antonios Varvitsiotis.

  3. The discretizable molecular distance geometry problem seems easier on proteins, by Leo Liberti, Carlile Lavor, Antonio Mucherino.

  4. Spheres unions and intersections and some of their applications in molecular modeling, by Michel Petitjean.

  5. Is the Distance Geometry Problem in NP?, by Nathanael Beeker, Stéphane Gaubert, Christian Glusa, Leo Liberti.

  6. Solving spatial constraints with generalized distance geometry, by Lu Yang.

  7. A topological interpretation of the walk distances, by Pavel Chebotarev, Michel Deza.

PART II - Methods

  1. Distance geometry methods for protein structure determination, by Zachary Voller, Zhijun Wu.

  2. Solving the discretizable molecular distance geometry problem by multiple realization trees, by Pedro Nucci, Loana Tito Nogueira, Carlile Lavor.

  3. ASAP - An eigenvector synchronization algorithm for the graph realization problem, by Mihai Cucuringu.

  4. Global optimization for atomic cluster distance geometry problems, by Marco Locatelli, Fabio Schoen

  5. Solving molecular distance geometry problems using a continuous optimization approach, by Rodrigo S. Lima and J.M. Martínez.

  6. DC programming approaches for distance geometry problems, by Hoai An Le Thi, Tao Pham Dinh.

  7. Stochastic proximity embedding (SPE): a simple, fast and scalable algorithm for solving the Distance Geometry Problem, by Dimitris K. Agrafiotis, Deepak Bandyopadhyay, Eric Yang.

PART III - Applications to Protein Conformations

  1. Distance geometry for realistic molecular conformations, by Gordon M. Crippen.

  2. Distance geometry in structural biology: new perspectives, by Thérèse E. Malliavin, Antonio Mucherino, Michael Nilges.

  3. Using a distributed SDP approach to solve simulated protein molecular conformation problems, by Xingyuan Fang, Kim-Chuan Toh.

  4. An overview on protein structure determination by NMR. Historical and future perspectives of the use of distance geometry methods, by Fabio C.L. Almeida, Adolfo H. Moraes, Francisco Gomes-Neto.



The editors. From left to right: Antonio Mucherino, Nelson Maculan, Carlile Lavor and Leo Liberti.


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