Structure in the 3D Galaxy Distribution: I. Methods and Example Results

Three methods for detecting and characterizing structure in point data, such as that generated by redshift surveys, are described: classification using self-organizing maps, segmentation using Bayesian blocks, and density estimation using adaptive kernels. The first two methods are new, and allow detection and characterization of structures of arbitrary shape and at a wide range of spatial scales. These methods should elucidate not only clusters, but also the more distributed, wide-ranging filaments and sheets, and further allow the possibility of detecting and characterizing an even broader class of shapes. The methods are demonstrated and compared in application to three data sets: a carefully selected volume-limited sample from the Sloan Digital Sky Survey redshift data, a similarly selected sample from the Millennium Simulation, and a set of points independently drawn from a uniform probability distribution -- a so-called Poisson distribution. We demonstrate a few of the many ways in which these methods elucidate large scale structure in the distribution of galaxies in the nearby Universe.Comment: Re-posted after referee corrections along with partially re-written introduction. 80 pages, 31 figures, ApJ in Press. For full sized figures please download from: http://astrophysics.arc.nasa.gov/~mway/lss1.pd

Combinatorial and Hodge Laplacians: Similarity and Difference

As key subjects in spectral geometry and combinatorial graph theory respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data and in their realization of diffusion and minimization of harmonic measures. It is believed that they also both associate with vector calculus, through the gradient, curl, and divergence, as argued in the popular usage of "Hodge Laplacians on graphs" in the literature. Nevertheless, these Laplacians are intrinsically different in their domains of definitions and applicability to specific data formats, hindering any in-depth comparison of the two approaches. To facilitate the comparison and bridge the gap between the combinatorial Laplacian and Hodge Laplacian for the discretization of continuous manifolds with boundary, we further introduce Boundary-Induced Graph (BIG) Laplacians using tools from Discrete Exterior Calculus (DEC). BIG Laplacians are defined on discrete domains with appropriate boundary conditions to characterize the topology and shape of data. The similarities and differences of the combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are then examined. Through an Eulerian representation of 3D domains as level-set functions on regular grids, we show experimentally the conditions for the convergence of BIG Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes.Comment: 26 page

Defect-induced fracture topologies in Al₂O₃ ceramic-graphene nanocomposites

Models of ceramic-graphene nanocomposites are used to study how the manufacturing process-dependent arrangement of reduced graphene oxide (rGO) inclusions governs nano-crack network development. The work builds upon recent studies of such composites where a novel combinatorial approach was used to investigate the effect of rGO arrangements on electrical conductivity and porosity. This approach considers explicitly the discrete structure of the composite and represents it as a collection of entities of different dimensions - grains, grain boundaries, triple junctions, and quadruple points. Here, the combinatorial approach is developed further by considering the effects of rGO agglomerations, stress concentrators and adhesion energies on intergranular cracking. The results show that the fracture networks can be effectively controlled by the local ordering of rGO inclusions to allow for a concurrent increase in the strength and conductivity of the ceramic composites. It is shown that the ratio of local stress concentrators related to rGO inclusions and cracks is the most significant factor affecting the nano-crack network topology. The local spatial arrangement of rGO inclusions becomes an effective tool for controlling nano-crack network topology only when this ratio approaches one. It is anticipated that these results will inform future design of toughness-enhanced composites

Evolutionary games between epithelial cells: the impact of population structure and tissue dynamics on the success of cooperation

Cooperation is usually understood as a social phenomenon. However, it also occurs on the cellular level. A number of key mutations associated with malignancy can be considered cooperative, as they rely on the production of diffusible growth factors to confer a fitness benefit. Evolutionary game theory provides a framework for modelling the evolutionary dynamics of these cooperative mutations. This thesis uses evolutionary game theory to examine the evolutionary dynamics of cooperation within epithelial cells, which are the origin point of most cancers. In particular, we consider how the structure and dynamics of an epithelium affect cooperative success. We use the Voronoi tessellation model to represent an epithelium. This allows us much greater flexibility, compared to evolutionary graph theory models, to explore realistic dynamics for population updating. Initially, we consider a model where death and division are spatially decoupled. We analyse pairwise social dilemma games, focussing on the additive prisoner’s dilemma, and multiplayer public goods games. We calculate fixation probabilities, and conditions for cooperative success, by simulation, as well as deriving quasi-analytic results. Comparing with results for graph structured populations with spatially coupled birth and death, or well-mixed populations, we find that in general cooperation is promoted by local game play, but global competition for offspring. We then introduce a more realistic model of population updating, whereby death and division are spatially coupled as a consequence of contact inhibition. The strength of this coupling is positively correlated with the strength of contact inhibition. However, the extent to which strong spatial coupling inhibits cooperation depends on mechanical properties of the tissue

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

Mathematical modeling of Lynch syndrome carcinogenesis

Cancer is one of the leading causes of disease-related death worldwide. In recent years, large amounts of data on cancer genetics and molecular characteristics have become available and accumulated with increasing speed. However, the current understanding of cancer as a disease is still limited by the lack of suitable models that allow interpreting these data in proper ways. Thus, the highly interdisciplinary research field of mathematical oncology has evolved to use mathematics, modeling, and simulations to study cancer with the overall goal to improve clinical patient care. This dissertation aims at developing mathematical models and tools for different spatial scales of cancer development at the example of colorectal cancer in Lynch syndrome, the most common inherited colorectal cancer predisposition syndrome. We derive model-driven approaches for carcinogenesis at the DNA, cell, and crypt level, as well as data-driven methods for cancer-immune interactions at the DNA level and for the evaluation of diagnostic procedures at the Lynch syndrome population level. The developed models present an important step toward an improved understanding of hereditary cancer as a disease aiming at rapid implementation into clinical management guidelines and into the development of novel, innovative approaches for prevention and treatment

Characterisation of Disordered Structures

In this thesis I will look at how large, complex structures can be interpreted and evaluated using an information theoretic approach. The work specifically investigates techniques to understand disordered materials. It explains a novel framework using statistical methods to investigate structural information of very large data sets. This framework facilitates understanding of complex structures through the quantification of information and disorder. Large scale structures including granular media and amorphous atomic systems can also be processed. The need to deal with larger complex structures has been driven by new methods used to characterise amorphous materials, such as atomic scale tomography. In addition, computers are allowing for the creation of larger and larger data sets for researchers to analyse, requiring new techniques for storing and understanding information. As it has become possible to analyse large complex systems there has been a corresponding increase in attempts to scientifically understand these systems. New, man-made, complex systems have emerged such as the stock market and on-line networks. This has boosted interest in their interpretation, with the hopes they can be more easily manipulated or controlled. Crystallography has been applied to great effect in biology, having been used to discover the structure of DNA and develop new drugs (UNESCO,2013). However it only describes crystal structure, which can be a drawback as a large majority of matter is amorphous. As such it is hoped that interpreting and understanding disorder may lead to similar breakthroughs in disordered materials. Entropic measures such as the mutual information and Kullback Leibler Divergence are used to investigate the nature of structural information and its impact on the system. I examine how this information propagates in a system, and how it could quantify the amount of organisation in a system that is structurally disordered. The methodology introduced in this thesis extracts useful information from large data sets to allow for a quantification of disorder. The calculated entropy for amorphous packings is generally less than 1 bit with Mutual information between 0 and 0.1 bits. The results verify direct correlation between Mutual Information and the correlation coefficient using various techniques. The Mutual information shows most information is obtained where sphere density is highest, following a similar trend to that of the Radial distribution function, and generally increasing for higher packing fractions. Evidence of the Random Close Packed (RCP) and Random Loose Packed (RLP) limits in two dimensions is shown, as well as evidence of both phases in time-lapsed 3D packings. The Kullback Leibler Divergence is also explored as a relative measure of disorder. This is achieved by calculating redundant information in packings so that areas of low and high order can be shown. Results present colour maps displaying relative information in random disk packings from which motifs can be identified. For higher packing fractions distinct borders form for areas of low and high information, particularly where crystallisation has occurred. Again, these results show an increase in information for more densely packed structures, as expected, with a Kullback Leibler divergence of between 0 and 1 bits. Finally I introduce the concept of self-referential order which provides a way to quantify structural organisation in non-crystalline materials, by referencing part of the system in a similar way to a unit cell. This allows a step forward in understanding and characterising disorder, helping to develop a framework to encode amorphous structures in an efficient way. These results show increasing information for higher packing fractions as well as further evidence of RLP and RCP limits around packing fractions of 0.54 and 0.64 respectively

Discrete model for discontinuous dynamic recrystallisation applied to grain structure evolution inside adiabatic shear bands

Discontinuous dynamic recrystallisation (DDRX) is a well-known phenomenon playing a significant role in the high-temperature processing of metals, including industrial forming and severe plastic deformations. The ongoing discussion on the Zener–Hollomon (Z–H) parameter as a descriptor of materials’ propensity to DDRX and a measure of microstructure homogeneity leaves more questions than answers and prevents its practical application. Most of the existing DDRX models are continuous, and so the geometry and topology of real grain microstructures cannot be considered. The present study uses a fully discrete representation of polycrystalline aluminium alloys as 2D/3D Voronoi space tessellations corresponding to EBSD maps. Such tessellations are geometric realisations of combinatorial structures referred to as polytopal cell complexes. Combining discrete models with FEM LS-Dyna simulations of shock-wave propagation in AA1050 and AW5083 aluminium alloys makes it possible to estimate for the first time the contribution of DDRX to the final material microstructure inside adiabatic shear bands. It is shown that the increase of the initial fraction of high-angle grain boundaries, caused by preliminary deformation, significantly increases the spatial homogeneity and decreases the clustering of DDRX grains. The obtained results contradict the conventional assumption that the microstructures obtained by severe plastic deformation under quasi-static and dynamic deformation conditions are similar due to the similar value of the Z–H parameter: competition between the two recrystallisation mechanisms leads to almost unpredictable final grain structures inside share bands that require further comprehensive experimental studies. This agrees with experimental evidence for high material sensitivity to the Z–H parameter