On the complexity of the approximation of nonplanarity parameters for cubic graphs

AbstractLet G=(V,E) be a simple graph. The NON-PLANAR DELETION problem consists in finding a smallest subset E′⊂E such that H=(V,E⧹E′) is a planar graph. The SPLITTING NUMBER problem consists in finding the smallest integer k⩾0, such that a planar graph H can be defined from G by k vertex splitting operations. We establish the Max SNP-hardness of SPLITTING NUMBER and NON-PLANAR DELETION problems for cubic graphs

An evolutionary algorithm for graph planarisation by vertex deletion

A non-planar graph can only be planarised if it is structurally modified. This work presents a new heuristic algorithm that uses vertices deletion to modify a non-planar graph in order to obtain a planar subgraph. The proposed algorithm aims to delete a minimum number of vertices to achieve its goal. The vertex deletion number of a graph G = (V,E) is the smallest integer k ? 0 such that there is an induced planar subgraph of G obtained by the removal of k vertices of G. Considering that the corresponding decision problem is NPcomplete and an approximation algorithm for graph planarisation by vertices deletion does not exist, this work proposes an evolutionary algorithm that uses a constructive heuristic algorithm to planarise a graph. This constructive heuristic has time complexity of O(n+m), where m = |V| and n = |E|, and it is based on the PQ-trees data structure and on the vertex deletion operation. The algorithm performance is verified by means of case studies

07281 Abstracts Collection -- Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs

From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Simulating Self-Assembly of Organosulfur Species on Gold Nanoparticles

This Thesis aims to establish an accurate but computationally effective method for simulating self-assembly of organosulfurs on gold nanoparticles (AuNPs), a process resulting in their functionalisation. A second gold rush is currently rekindling chemists’ interest in the synthesis of novel functionalised AuNPs: these can bear the most chemically diverse functional groups, making them employable in a wide variety of applications, from optoelectronics to catalysis. Some aspects of self-assembly remain experimentally unclear at the mechanistic and electronic levels: achieving its accurate reproduction in silico would indeed represent an important contribution in the synthesis of functionalised AuNPs. This task, however, has so far proven difficult to achieve. In this work, I set out and review four fundamental challenges facing the computational chemist aiming to simulate self-assembly, and describe the strategy chosen to overcome them, using thiols (RSH) as the reference organosulfur. These challenges involve proper reproduction of: I) gold’s relativistic effects and aurophilicity; II) the extensive surface reconstruction occurring upon self-assembly, with formation of RS–Au–SR staples and hydrogen loss; III) the large scale ligands involved in the process and their interactions; and IV) the fluctuating solvent environment in which it occurs. Confined to the AuNP core and RSH headgroups, challenges I and II involve complex electronic properties and entail electronic change, with bonds being cleaved (S–H) and reformed (S–Au, possibly H–H): overcoming them requires explicit simulation of electrons with a QM method (DFT). Challenges III and IV involve the entire RSH-AuNP system, including RSH tails of typically ∼10^2 atoms: QM methods become impracticable at these system sizes, and a less costly classical forcefield treatment (MM) is necessary in this case, at least in part. The work presented here then proceeds towards the stated aim by attempting to resolve each of these challenges I–IV. The eventually devised solution proposes a combination of classical molecular dynamics (MD), followed by the hybrid QM/MM method ONIOM, which allows to combine the ‘best of the QM and MM worlds’ and is well established for other systems. To overcome challenge I, various effective core potentials (ECPs); basis sets; and density functionals are evaluated based on their ability to predict properties and geometries of several pristine AuNPs. These properties and geometries are either derived experimentally, or from high-level ab initio calculations. The chosen QM method PBE/LANL2DZ is then further tested on various systems, assessing its ability (challenge II) to reproduce hydrogen loss and staple formation. Upon proposing to tackle challenge III using ONIOM (with the OPLS-AA forcefield for the MM part), the method’s performance is first compared to that of full QM (PBE/LANL2DZ) in terms of accuracy and efficiency, and in a variety of contexts, including on AuNPs featuring a 38-atom gold core. Once these calculations confirm the considerable time gains afforded by the introduction of ONIOM, I then demonstrate its full applicability in the optimisation of a large, experimentally plausible functionalised AuNP. Finally, I propose to tackle challenge IV by introducing a classical MD simulation stage to precede QM/MM optimisation. As a test, MD is used to generate statistically significant sets of 8-atom AuNPs coated with alkylthiols of different chain lengths, which are then optimised, thereby successfully reproducing the early stages of reconstruction. I then conclude by successfully testing this MD + ONIOM approach on two much larger functionalised AuNPs, having 20-atom gold cores and sixteen or seventeen 64-atom ligands. My Thesis highlights both the strengths and limitations of the ONIOM approach in simulating such a complex process as organosulfur self-assembly on AuNPs. Nonetheless, the chosen MD + ONIOM strategy can indeed reproduce key aspects of self-assembly with increased CPU-efficiency, and, importantly, makes electronically plausible predictions: it therefore represents a viable route for the in silico investigation of this process, and an encouraging fulfilment of my initial aim.Open Acces

Conjectures on exact solution of three - dimensional (3D) simple orthorhombic Ising lattices

We report the conjectures on the three-dimensional (3D) Ising model on simple orthorhombic lattices, together with the details of calculations for a putative exact solution. Two conjectures, an additional rotation in the fourth curled-up dimension and the weight factors on the eigenvectors, are proposed to serve as a boundary condition to deal with the topologic problem of the 3D Ising model. The partition function of the 3D simple orthorhombic Ising model is evaluated by spinor analysis, by employing these conjectures. Based on the validity of the conjectures, the critical temperature of the simple orthorhombic Ising lattices could be determined by the relation of KK* = KK' + KK'' + K'K'' or sinh 2K sinh 2(K' + K'' + K'K''/K) = 1. For a simple cubic Ising lattice, the critical point is putatively determined to locate exactly at the golden ratio xc = exp(-2Kc) = (sq(5) - 1)/2, as derived from K* = 3K or sinh 2K sinh 6K = 1. If the conjectures would be true, the specific heat of the simple orthorhombic Ising system would show a logarithmic singularity at the critical point of the phase transition. The spontaneous magnetization and the spin correlation functions of the simple orthorhombic Ising ferromagnet are derived explicitly. The putative critical exponents derived explicitly for the simple orthorhombic Ising lattices are alpha = 0, beta = 3/8, gamma = 5/4, delta = 13/3, eta = 1/8 and nu = 2/3, showing the universality behavior and satisfying the scaling laws. The cooperative phenomena near the critical point are studied and the results obtained based on the conjectures are compared with those of the approximation methods and the experimental findings. The 3D to 2D crossover phenomenon differs with the 2D to 1D crossover phenomenon and there is a gradual crossover of the exponents from the 3D values to the 2D ones.Comment: 176 pages, 4 figure

A regularization approach for reconstruction and visualization of 3-D data

Esta tesis trata sobre reconstrucción de superficies a partir de imágenes de rango utilizando algunas extensiones de la Regularización de Tikhonov, que produce Splines aplicables a datos en n dimensiones. La idea central es que estos splines se pueden obtener mediante la teoría de regularización, utilizando un equilibrio entre la suavidad y la fidelidad a los datos, por tanto, serán aplicables tanto en la interpolación como en la aproximación de datos exactos o ruidosos. En esta tesis proponemos un enfoque variacional que incluye los datos e información a priori acerca de la solución, dada en forma de funcionales. Solucionamos problemas de optimización que resultan ser una extensión de la teoría de Tikhonov, con el propósito de incluir funcionales con propiedades locales y globales que pueden ser ajustadas mediante parámetros de regularización. El a priori es analizado en términos de las propiedades físicas y geométricas de los funcionales para luego ser agregados a la formulación variacional. Los resultados obtenidos se prueban con datos para reconstrucción de superficies, mostrando notables propiedades de reproducción y aproximación. En particular, utilizamos la reconstrucción de superficies para ilustrar las aplicaciones prácticas, pero nuestro enfoque tiene muchas más aplicaciones. En el centro de nuestra propuesta esta la teoría general de problemas inversos y las aplicaciones de algunas ideas provenientes del análisis funcional. Los splines que obtenemos son combinaciones lineales de las soluciones fundamentales de ciertos operadores en derivadas parciales, frecuentes en la teoría de la elasticidad y no se hace ninguna suposición previa sobre el modelo estadístico de los datos de entrada, de manera que se pueden tomar en términos de una inferencia estadística no paramétrica. Estos splines son implementables en una forma muy estable y se pueden aplicar en problemas de interpolación y suavizado. / Abstract: This thesis is about surface reconstruction from range images using some extensions of Tikhonov regularization that produces splines applicable on n-dimensional data. The central idea is that these splines can be obtained by regularization theory, using a trade-off between fidelity to data and smoothness properties; as a consequence, they are applicable both in interpolation and approximation of exact or noisy data. We propose a variational framework that includes data and a priori information about the solution, given in the form of functionals. We solve optimization problems which are extensions of Tikhonov theory, in order to include functionals with local and global features that can be tuned by regularization parameters. The a priori is thought in terms of geometric and physical properties of functionals and then added to the variational formulation. The results obtained are tested on data for surface reconstruction, showing remarkable reproducing and approximating properties. In this case we use surface reconstruction to illustrate practical applications; nevertheless, our approach has many other applications. In the core of our approach is the general theory of inverse problems and the application of some abstract ideas from functional analysis. The splines obtained are linear combinations of certain fundamental solutions of partial differential operators from elasticity theory and no prior assumption is made on a statistical model for the input data, so it can be thought in terms of nonparametric statistical inference. They are implementable in a very stable form and can be applied for both interpolation and smoothing problems.Doctorad

Plane and simple : using planar subgraphs for efficient algorithms

In this thesis, we showcase how planar subgraphs with special structural properties can be used to fi nd efficient algorithms for two NP-hard problems in combinatorial optimization. In the fi rst part, we develop algorithms for the computation of Tutte paths and show how these special subgraphs can be used to efficiently compute long cycles and other relaxations of Hamiltonicity if we restrict the input to planar graphs. We give an O(n^2) time algorithm for the computation of Tutte paths in circuit graphs and generalize it to the computation of Tutte paths between any two given vertices and a prescribed intermediate edge in 2-connected planar graphs. In the second part, we study the Maximum Planar Subgraph Problem (MPS) and show how dense planar subgraphs can be used to develop new approximation algorithms for this problem. All new algorithms and arguments we present are based on a novel approach that focuses on maximizing the number of triangular faces in the computed subgraph. For this, we define a new optimization problem called Maximum Planar Triangles (MPT). We show that this problem is NP-hard and quantify how good an approximation algorithm for MPT performs as an approximation for MPS. We give a greedy 1/11-approximation algorithm for Mpt and show that the approximation ratio can be improved to 1/6 by using locally optimal triangular cactus subgraphs.In dieser Dissertation zeigen wir, wie planare Teilgraphen mit speziellen Eigenschaften verwendet werden können, um effiziente Algorithmen für zwei NP-schwere Probleme in der kombinatorischen Optimierung zu fi nden. Im ersten Teil entwickeln wir Algorithmen zur Berechnung von Tutte-Wegen und zeigen, wie diese verwendet werden können, um lange Kreise und andere Lockerungen der Hamilton-Charakteristik zu finden, wenn wir uns auf Graphen in der Ebene beschränken. Wir beschreiben zunächst einen O(n^2)-Algorithmus in Circuit-Graphen und verallgemeinern diesen anschließend für die Berechnung von Tutte-Wegen in 2-zusammenhängenden planaren Graphen. Im zweiten Teil untersuchen wir das Maximum Planar Subgraph Problem (MPS) und zeigen, wie besonders dichte planare Teilgraphen verwendet werden können, um neue Approximationsalgorithmen zu entwickeln. Unsere Ergebnisse basieren auf einem neuartigen Ansatz, bei dem die Anzahl der dreieckigen Gebiete im berechneten Teilgraphen maximiert wird. Dazu de finieren wir ein neues Optimierungsproblem namens Maximum Planar Triangles (MPT). Wir zeigen, dass dieses Problem NP-schwer ist und quantifi zieren, wie gut ein Approximationsalgorithmus für MPT als Approximation für MPS funktioniert. Wir geben einen 1/11-Approximationsalgorithmus für MPT und zeigen, wie dies durch die Verwendung von lokal optimaler Kaktus-Teilgraphen auf 1/6 verbessert werden kann

Quantum Fields in Extreme Backgrounds

Quantum field theories behave in interesting and nontrivial ways in the presence of intense electric and/or magnetic fields. Describing such behavior correctly, particularly at finite (nonzero) temperature and density, is of importance for particle physics, nuclear physics, astrophysics, condensed matter physics, and cosmology. Incorporating these conditions as external parameters also provides useful probes into the nonperturbative structure of gauge theories. In this work, formalism for describing matter in a variety of extreme conditions is developed and implemented. We develop several expansions of one-loop finite temperature effects for spinor particles in the presence of magnetic fields, including the effects of confinement, encoded in a nontrivial Polyakov loop. The worldline instanton formalism is extended to the case of finite temperature, which yields a long-sought thermal extension to the celebrated formula of Schwinger for pair production in a constant electric field. The technique is further extended to include the effects of finite density and confinement, as well as some restricted classes of nonabelian electric fields. A persistent source of difficulty in the study of gauge theories at finite density, and/or in the presence of external electric fields, is the so-called sign problem. We advance a novel duality-based approach for lattice simulation of scalar field theories with complex actions, which yields new insights on the old problem of spatial modulations arising in systems with competing interactions. The approach shows promise for simulating scalar theories at finite density and in the presence of external electric fields, and is capable of handling systems in the universality class of the $i\phi^3$ theory, which determines the critical indices of the Lee-Yang edge transition

Boundary value approaches to molecular dynamics simulation

The focus of the research of this dissertation is the mathematical modeling of and use of numerical methods for the study of the dynamics of conformational transitions of biomolecules like proteins and small peptides. While an IV-AA-MDS approach could be considered for this purpose, the focus of this dissertation is a related approach that is called boundary value all-atom molecular dynamics simulation (BV-AA-MDS) in this dissertation. This approach includes the application of a numerical method to seek numerical solutions to two-point boundary value problems (BVP\u27s) for systems of 2nd-order nonlinear ordinary differential equations (ODE\u27s).;In this dissertation, the mathematical framework of AA-MDS, BV-AA-MDS and some numerical methods for BV-AA-MDS---single shooting, multiple shooting, finite differences methods, and stochastic difference equation methods---are described. Important computational limitations of AA-MDS, BV-AA-MDS, and MS for BV-AA-MDS are highlighted and reasons for considering these approaches and methods despite the computational limitations will be provided.;Also, in this dissertation, the application of multiple shooting to BVP\u27s for ODE\u27s corresponding to transitions between two molecular conformations specified by two sets of internal coordinates is proposed. Strategies and issues related to definition of boundary conditions, assignment of initial parameters, and convergence are investigated. Results from the study of transitions between local minima of the potential energy surface of an alanine dipeptide are presented. Implications of the methods and results of this work for application of multiple shooting to the study of conformational transitions in larger systems are discussed.;Defining boundary conditions corresponding to sets of internal coordinates of local minima leads to what is defined to be a full set of 6n boundary conditions, i.e. R = 6n. And, defining parameters of the multiple shooting method as the initial conditions on each subinterval leads to what is defined to be a full set of 6nN parameters. To apply multiple shooting with a full set of parameters to a BVP with a full set of boundary conditions, the number of atoms in the molecule must be limited to avoid excessive computational cost. In this dissertation, for the case of single shooting, an alternate boundary value simulation approach is presented that involves a reduced set of boundary conditions and a reduced set of parameters. We also propose an approach for use a reduced parameter set that is based on an application of principles of normal mode analysis. We provide results from the application of these approaches to the study of transitions between potential energy wells for an alanine dipeptide.;In this dissertation, all-atom distance matrix interpolation (AA-DMI) methods are described. These are methods for generating position trajectories that satisfy certain types of boundary conditions are less computationally demanding than boundary value approaches to AA-MDS, but do provide atomically detailed trajectories. These methods involve an optimization problem with an objective function derived by interpolation of interatomic distances between their values in one conformation and their values in another conformation. Results are presented from the study of conformational transitions of an alanine dipeptide. Future directions of research are discussed. (Abstract shortened by UMI.