Linear Geometry: flats, ranks, regularity, parallelity

In this thesis we present two geometric theories designed to extend general relativity. It can be seen as one of the aims of such theories to model the observed accelerating expansion of the universe as a gravitational phenomenon, or to provide a mathematical structure for the formulation of quantum field theories on curved spacetimes and quantum gravity. This thesis splits into two parts: In the first part we consider multimetric gravity theories containing N > 1 standard model copies which interact only gravitationally and repel each other in the Newtonian limit. The dynamics of each of the standard model copies is governed by its own metric tensor. We show that the antisymmetric case, in which the mutual repulsion between the different matter sectors is of equal strength compared to the attractive gravitational force within each sector, is prohibited by a no-go theorem for N = 2. We further show that this theorem does not hold for N > 2 by explicitly constructing an antisymmetric multimetric repulsive gravity theory. We then examine several properties of this theory. Most notably, we derive a simple cosmological model and show that the accelerating expansion of the late universe can indeed be explained by the mutual repulsion between the different matter sectors. We further present a simple model for structure formation and show that our model leads to the formation of filament-like structures and voids. Finally, we show that multimetric repulsive gravity is compatible with high-precision solar system data using the parametrized post-Newtonian formalism. In the second part of the thesis we propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the differentiable manifold structure of classical spacetime. In this picture we demonstrate that classical spacetime emerges as a finite-dimensional manifold through the topological identification of all quantum points with identical position expectation value. We speculate on the possible relevance of this geometry to quantum field theory and gravity.

Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Nevertheless we still miss a model in which networks develop an emergent complex geometry. Here we show that a single two parameter network model, the growing geometrical network, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit planar random geometries with non-trivial modularity. Finally we find that these properties of the geometrical growing networks are present in a large set of real networks describing biological, social and technological systems.

Nonlocal density functional study of mixed copper–cobalt clusters

A case study of a project for the geometric documentation of a roman mausoleum from the 4th century is used as the overarching element in order to put forward a reflection on some of the steps for the process of documentation itself, as well as on the utility of the outcomes. The main issues considered are: (1) documentation project, (2) three-dimensional modeling, (3) use of different kinds of two-dimensional and three-dimensional representations, (4) Augmented Reality and (5) archive and dissemination of information.

Arrangements of planar curves are fundamental structures in computational geometry. We describe the recent developments in the arrangement package of Cgal, the Computational Geometry Algorithms Library, making it easier to use, to extend and to adapt to a variety of applications. This improved flexibility of the code does not come at the expense of efficiency as we mainly use generic-programming techniques, which make dexterous use of the compilation process. To the contrary, we expedited key operations as we demonstrate by experiments.

Lower envelopes are fundamental structures in computational geometry, which have many applications, such as computing general Voronoi diagrams and performing hidden surface removal in computer graphics. We present a generic, robust and efficient implementation of the divide-and-conquer algorithm for computing the envelopes of surfaces in IR3. To the best of our knowledge, this is the first exact implementation that computes envelopes in three-dimensional space. Our implementation is based on CGAL (the Computational Geometry Algorithms Library) and is designated as a CGAL package. The separation of topology and geometry in our solution allows for the reuse of the algorithm with different families of surfaces, provided that a small set of geometric objects and operations on them is supplied. We used our algorithm to compute the lower and upper envelope for several types of surfaces. Exact arithmetic is typically slower than floating-point arithmetic, especially when higher order surfaces are involved. Since our implementation follows the exact geometric computation paradigm, we minimize the number of geometric operations, and by that significantly improve the performance of the algorithm in practice. Our experiments show interesting phenomena in the behavior of the divide-and-conquer algorithm and the combinatorics of lower envelopes of random surfaces.

Geometric spanner is a fundamental structure in computational geometry and plays an important role in many geometric networks design applications. In this paper, we consider a generalization of the classical geometric spanner problem (called segment spanner): Given a set S of disjoint 2-D segments, find a spanning network G with minimum size so that for any pair of points in S, there exists a path in G with length no more than t times their Euclidean distance. Based on a number of interesting techniques (such as weakly dominating set, strongly dominating set, and interval cover), we present an efficient algorithm to construct the segment spanner. Our approach first identifies a set of Steiner points in S, then construct a point spanner for them. Our algorithm runs in O(|Q| + n2 log n) time, where Q is the set of Steiner points. We show that Q is an O(1)-approximation in terms of its size when S is relatively "well" separated by a constant. For arbitrary rectilinear segments under L1 distance, the approximation ratio improves to 2.

The well-separated pair decomposition (WSPD) is a fundamental structure in computational geometry. Given a set P of n points in d-dimensional space and a positive separation parameter s, an s-WSPD is a concise representation of all the O(n2) pairs of P requiring only O(sdn) storage. The WSPD has numerous applications in spatial data processing, such as computing spanner graphs, minimum spanning trees, shortest-path oracles, and statistics on interpoint distances. We consider the problem of maintaining a WSPD when points are inserted to or deleted from P. Worst-case arguments suggest that the addition or deletion of a single point could result in the generation (or removal) up to Ω(sd) pairs, which can be unacceptably high in many applications. Fortunately, the actual number of well separated pairs can be significantly smaller in practice, particularly when the points are well clustered. This suggests the importance of being able to respond to insertions and deletions in a manner that is output sensitive, that is, whose running time depends on the actual number of pairs that have been added or removed. We present the first output-sensitive algorithms for maintaining a WSPD of a point set under insertion and deletion. We show that our algorithms are nearly optimal, in the sense that these operations can be performed in time that is roughly equal to the number of changes to the WSPD.

A plant cell division algorithm based on cell biomechanics and ellipse-fitting.

The European rail network includes a large number of masonry arch bridges for which the actual safety level is uncertain due to the rise in traffic demand and the material deterioration that have occurred since they have been built. For this study, a representative sample of 50 Italian large-span railway masonry arch bridges dating to the 19th and 20th centuries is selected. The sample includes single-span and multi-span bridges, shallow and deep arches, viaducts with slender and stocky piers, for which the geometric properties are taken from original design drawings and calculation reports. Despite such variability, the main structural characteristics and construction methods recur and comply with the design criteria suggested by historical treatises. The load-carrying capacity of the bridges and the corresponding safety factor with respect to rail traffic loads are assessed through a fiber beam based modeling approach. Safety levels are related to the shape of the arch, the height of the piers and the number of spans. Different constitutive assumptions are considered and the results of yield-design based methods are compared with those accounting for limited ductility of masonry in compression. The influence of uncertainties in the knowledge of geometric and mechanical parameters is also evaluated.

Plan drawings are graphical documents critical to the documentation of architectural features at historic sites. These drawings include important geometric information such as the location, shape, and size of architectural features, which, for decaying or collapsed structures, may be the only existing records of the intact structure. This paper discusses an algorithm that estimates the geometry and semantic interpretation of architectural structures from a plan drawing. The estimated values are used to automatically generate a 3D structure using the estimated semantic labels of structural elements in the plan drawing. We demonstrate the utility of this approach by parsing several plan drawings of medieval castles and fortresses and generating 3D reconstructions of these structures and detail typical circumstances that prevent the system from generating a valid reconstruction. Since the 3D model is derived from plan drawings where the architectural contour is well-defined, the approach automatically provides near-pixel level accuracy at all locations which is very difficult and time-consuming to guarantee when manually constructing 3D models from the same drawing. Hence, these automatically-produced models can provide unprecedented accuracy to the in-situ remains not feasible with conventional manual model-building techniques. While this article represents initial work on this topic with limited scope (castles/fortresses), we envision that subsequent enhancements to this method will be a valuable tool for efficiently generating accurate 3D models for many different historic structures.

Local Geometry and Elasticity in Compact Chromatin Structure

Spider silk, as a natural material, shows exceptional performance in its properties. The combination of the superior properties of spider silk and the geometry of spider structures make the spider web very resilient. A spider web structure can be considered as a cable-like structure with inappreciable torsional, bending and shear rigidities. An investigation emphasising on natural frequencies and corresponding mode shapes with and without the consideration of geometric nonlinearity is presented in this paper. This study is the world’s first discovery of large amplitude free vibration behaviours of spider web structures. Large deformable finite element 3D models of spider web structures have been developed and validated. By using the energy method, the variational model of the spider web structures have been established to further extend the finite element model, consisting of the strain energy due to axial deformation, kinetic energy due to the spider web movement and the virtual work caused by the self-weight per unit unstretched length. The emphasis of this study is placed on the linear and geometric nonlinear behaviour of the spider web structures considering different structural patterns and material properties. To determine the large-amplitude free vibrational behaviours, a series of pretension load is applied to the first step in Abaqus to initiate the nonlinear strain-displacement relationships enabling a precursor to free vibrations. The parametric studies stemming from structural patterns (the number of radial and capture threads), elastic modulus, density, and inertia moment have been highlighted. The insight will help engineers and scientists to adapt the concept of spider webs, their geometric properties, and damage patterns for the design of any structural membranes, preventing any failure from dynamic resonances and nonlinear phenomena.

The two independent shear moduli, Gxz and Gyz, of a novel cellular core material (NorCore) have been calculated and their effects on the fundamental natural frequency of sandwich plates investigated. In order to characterize the dynamic properties of this novel material, an analysis of the geometry of a typical structural cell has been presented. Based upon the introduction of a typical representative volume element for the core material, the cellular structure of the material has been modeled using triangular finite elements. A series of numerical calculations have been performed in order to examine the variation of the shear moduli as functions of the geometrical properties of the core material. The shear moduli have been nondimensionalized are plotted against the web angle θ. In conjunction with a 3-D isoparametric element, an off-set quadrilateral finite element approach, that account for the effects of transverse deformation is used to compute the fundamental natural frequencies of the sandwich plates made by NorCore and FRP faces. The results show that the unique geometry of the cellular structure of NorCore has significant effects on the shear moduli and fundamental natural frequencies.

A series of graphene-based nanomolecule devices are constructed by connecting the graphene nanodot to two Au electrodes through different bond length between the electrodes and molecules. The geometric structure and electronic properties are studied by using density functional theory calculations. Basing on the optimized structure, we calculate the quantum conductance of the system by using the Green's function method. We find that the geometry structures of the molecule and the transport properties are sensitive to the bond length d Au-H . The plane of carbon atoms increasingly bends with the decrease of the d Au-H . The I SD-VSD curves have the same threshold value under different d Au-H .