The combinatorics of reduced decompositions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.Includes bibliographical references (p. 85-88) and index.This thesis examines several aspects of reduced decompositions in finite Coxeter groups. Effort is primarily concentrated on the symmetric group, although some discussions are subsequently expanded to finite Coxeter groups of types B and D. In the symmetric group, the combined frameworks of permutation patterns and reduced decompositions are used to prove a new characterization of vexillary permutations. This characterization and the methods used yield a variety of new results about the structure of several objects relating to a permutation. These include its commutation classes, the corresponding graph of the classes, the zonotopal tilings of a particular polygon, and a poset defined in terms of these tilings. The class of freely braided permutations behaves particularly well, and its graphs and posets are explicitly determined. The Bruhat order for the symmetric group is examined, and the permutations with boolean principal order ideals are completely characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, it is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed.(cont.) The structure of the intervals and order ideals in this poset is elucidated via patterns, including progress towards understanding the relationship between pattern containment and subintervals in principal order ideals. The final discussions of the thesis are on reduced decompositions in the finite Coxeter groups of types B and D. Reduced decompositions of the longest element in the hyperoctahedral group are studied, and expected values are calculated, expanding on previous work for the symmetric group. These expected values give a quantitative interpretation of the effects of the Coxeter relations on reduced decompositions of the longest element in this group. Finally, the Bruhat order in types B and D is studied, and the elements in these groups with boolean principal order ideals are characterized and enumerated by length.by Bridget Eileen Tenner.Ph.D

Sphere packings revisited

AbstractIn this paper we survey most of the recent and often surprising results on packings of congruent spheres in d-dimensional spaces of constant curvature. The topics discussed are as follows:–Hadwiger numbers of convex bodies and kissing numbers of spheres;–touching numbers of convex bodies;–Newton numbers of convex bodies;–one-sided Hadwiger and kissing numbers;–contact graphs of finite packings and the combinatorial Kepler problem;–isoperimetric problems for Voronoi cells, the strong dodecahedral conjecture and the truncated octahedral conjecture;–the strong Kepler conjecture;–bounds on the density of sphere packings in higher dimensions;–solidity and uniform stability.Each topic is discussed in details along with some of the “most wanted” research problems

Almost finiteness of groupoid actions and Z-stability of C*-algebras associated to tilings

The property of almost finiteness was first introduced by Matui for locally compact Hausdorff totally disconnected étale groupoids with compact unit spaces, and has since been extended by Suzuki to drop the assumption of being totally disconnected. A property of the same name has been defined by Kerr for free actions of discrete groups on compact metric spaces. In this setting, Kerr shows that almost finiteness has direct relevance to the classification programme for simple, separable, unital, nuclear, infinite dimensional C*-algebras, as it implies that the associated crossed product is Z-stable. The motivating example for this thesis comes from the theory of aperiodic tilings. A tiling is a covering of Euclidean space by a collection of sets (called tiles) which overlap only on their boundaries. A tiling is called aperiodic if it does not contain arbitrarily large periodic patterns. Such tilings find physical applications, acting as models for quasicrystals. One may associate a groupoid to certain aperiodic tilings, and the C*-algebras of such groupoids encode information about physical observables in quasicrystalline molecules. In this thesis, we generalise Kerr's notion of almost finiteness of group actions to allow for actions of groupoids. We show that the canonical action of the groupoid associated to any aperiodic, repetitive tiling with finite local complexity on its unit space is almost finite, and we use this to show that the C*-algebra of the tiling is Z-stable. We develop a groupoid version of the Ornstein-Weiss quasitiling machinery, which we use to prove our Z-stability result. Finally, we give a direct proof that tiling C*-algebras are quasidiagonal, which eases the route to classification in the case that the algebra has unique trace

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

Nature's forms are frilly, flexible, and functional

A ubiquitous motif in nature is the self-similar hierarchical buckling of a thin lamina near its margins. This is seen in leaves, flowers, fungi, corals and marine invertebrates. We investigate this morphology from the perspective of non-Euclidean plate theory. We identify a novel type of defect, a branch-point of the normal map, that allows for the generation of such complex wrinkling patterns in thin elastic hyperbolic surfaces, even in the absence of stretching. We argue that branch points are the natural defects in hyperbolic sheets, they carry a topological charge which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating elastic energy. We develop a theory for branch points and investigate their role in determining the mechanical response of hyperbolic sheets to weak external forces. We also develop a discrete differential geometric (DDG) framework for applications to the continuum mechanics of hyperbolic elastic sheets.Comment: 35 pages, 26 figure

Multidisciplinary structural design and optimization for performance, cost, and flexibility

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2005.Includes bibliographical references (p. 155-165).Reducing cost and improving performance are two key factors in structural design. In the aerospace and automotive industries, this is particularly true with respect to design criteria such as strength, stiffness, mass, fatigue resistance, manufacturing cost, and maintenance cost. This design philosophy of reducing cost and improving performance applies to structural components as well as complex structural systems. Design for flexibility is one method of reducing costs and improving performance in these systems. This design methodology allows systems to be modified to respond to changes in desired functionality. A useful tool for this design practice is multi-disciplinary design optimization (MDO). This thesis develops and exercises an MDO framework for exploration of design spaces for structural components, subsystems, and complex systems considering cost, performance, and flexibility. The structural design trade off of sacrificing strength, mass efficiency, manufacturing cost, and other "classical" optimization criteria at the component level for desirable properties such as reconfigurability at higher levels of the structural system hierarchy is explored in three ways in this thesis. First, structural shape optimization is performed at the component level considering structural performance and manufacturing cost. Second, topology optimization is performed for a reconfigurable system of structural elements. Finally, structural design to reduce cost and increase performance is performed for a complex system of structural components. A new concept for modular, reconfigurable spacecraft design is introduced and a design application is presented.by William David Nadir.S.M

Entwicklung und Test von Wechselwirkungspotenzialen für komplexe intermetallische Verbindungen

Complex metallic alloys and quasicrystals show extraordinary physical properties relevant for technological applications, for example hardness at low density. In the study of these systems, atomistic simulation with classical interaction potentials is a very promising tool. Such simulations require classical effective potentials describing the cohesive energy as a function of the atomic coordinates. The quality of the simulation depends crucially on the accuracy with which this potential describes the real interactions. One way to generate physically relevant potentials is the force matching method, where the parameters of a potential are adjusted to optimally reproduce the forces on individual atoms determined from quantum-mechanical calculation. The programme package potfit developed as part of this thesis implements the force matching method efficiently. Potentials are generated for a number of complex metallic alloy systems. A potential for the decagonal basic Ni-rich Al-Co-Ni quasicrystal is used to simulate diffusion processes and melting. In the CaCd6 system built from multishelled clusters, the shape and orientation of the innermost cluster shell is studied. Finally, phonon dispersion in the Mg-Zn system is determined and compared to experiment. The programme potfit is shown to be an effective tool for generating physically justified effective potentials. Potentials created with potfit can greatly improve the understanding of complex metallic alloys through atomistic simulations.Komplexe intermetallische Verbindungen und Quasikristalle zeigen außergewöhnliche physikalische Eigenschaften, wie z.B. Härte bei geringer Dichte. Bei der Untersuchung dieser Systeme sind atomistische Simulationen mit klassischen Wechselwirkungspotenzialen ein wichtiges Werkzeug. Für solche Simulationen benötigt man klassische effektive Potenziale, die die Bindungsenergie als eine Funktion der Atomkoordinaten beschreiben. Die Qualität der Simulation hängt entscheidend von der Genauigkeit ab, mit der diese Potenziale die echten Wechselwirkungen wiedergeben. Eine Möglichkeit, physikalisch relevante Potenziale zu erzeugen, ist die Force-Matching-Methode. Dabei werden die Parameter eines Potenzials so angepasst, dass die mit quantenmechanischen Methoden bestimmten Kräfte auf die einzelnen Atome bestmöglich reproduziert werden. Das als Teil dieser Arbeit entwickelte Programmpaket potfit implementiert die Force-Matching-Methode effizient. Für einige komplexe intermetallische Verbindungen werden Potenziale bestimmt. In dekagonalen Al-Co-Ni-Quasikristallen werden mit Hilfe eines Potenzials Diffusionsprozesse und Schmelzvorgänge simuliert. In der aus mehrschaligen Clustern bestehenden CaCd6-Verbindung wird die Struktur der innersten Clusterschale untersucht. Schließlich wird die Phononendispersion im Mg-Zn-System bestimmt und mit experimentellen Ergebnissen verglichen. Es wird gezeigt, dass das Programm potfit ein effektives Werkzeug zur Erzeugung physikalisch gerechtfertigter Wechselwirkungen ist. Potenziale, die mit potfit erzeugt werden, können zum Verständnis komplexer metallischer Verbindungen durch atomistische Simulationen viel beitragen

Bulletin of the Torrey Botanical Club.

v.30 (1903