In general, a finitely presented group can have very nasty properties, but many of these properties are avoided if the group is assumed to admit a nice action by isometries on a space with a negative curvature property, such as Gromov hyperbolicity. Such groups are surprisingly common: there is a sense in which a random group admits such an action, as do some groups of classical interest, such as fundamental groups of closed Riemannian manifolds with negative sectional curvature. If a group admits an action on a Gromov hyperbolic space then large scale properties of the space give useful invariants of the group. One particularly natural large scale property used in this way is the Gromov boundary. The Gromov boundary of a hyperbolic group is a compact metric space that is, in a sense, approximated by spheres of large radius in the Cayley graph of the group. The technical results contained in this thesis are effective versions of this statement: we see that the presence of a particular topological feature in the boundary of a hyperbolic group is determined by the geometry of balls in the Cayley graph of radius bounded above by some known upper bound, and is therefore algorithmically detectable. Using these technical results one can prove that certain properties of a group can be computed from its presentation. In particular, we show that there are algorithms that, when given a presentation for a one-ended hyperbolic group, compute Bowditch's canonical decomposition of that group and determine whether or not that group is virtually Fuchsian. The final chapter of this thesis studies the problem of detecting Cech cohomological features in boundaries of hyperbolic groups. Epstein asked whether there is an algorithm that computes the Cech cohomology of the boundary of a given hyperbolic group. We answer Epstein's question in the affirmative for a restricted class of hyperbolic groups: those that are fundamental groups of graphs of free groups with cyclic edge groups. We also prove the computability of the Cech cohomology of a space with some similar properties to the boundary of a hyperbolic group: Otal's decomposition space associated to a line pattern in a free group.

Increasing competition and regulation are raising the pressure on manufacturing organisations to innovate their products. Innovation is fraught by significant uncertainty of whole product life cycle costs and this can lead to hesitance in investing which may result in a loss of competitive advantage. Innovative products exist when the minimum information for creating accurate cost models through contemporary forecasting methods does not exist. The scientific research challenge is that there are no forecasting methods available where cost data from only one time period suffices for their application. The aim of this research study was to develop a framework for forecasting cost uncertainty using cost data from only one time period. The developed framework consists of components that prepare minimum information for conversion into a future uncertainty range, forecast a future uncertainty range, and propagate the uncertainty range over time. The uncertainty range is represented as a vector space representing the state space of actual cost variance for 3 to n reasons, the dimensionality of that space is reduced through vector addition and a series of basic operators is applied to the aggregated vector in order to create a future state space of probable cost variance. The framework was validated through three case studies drawn from the United States Department of Defense. The novelty of the framework is found in the use of geometry to increase the amount of insights drawn from the cost data from only one time period and the propagation of cost uncertainty based on the geometric shape of uncertainty ranges. In order to demonstrate its benefits to industry, the framework was implemented at an aerospace manufacturing company for identifying potentially inaccurate cost estimates in early stages of the whole product life cycle.

In this thesis, the critical phenomenon in Berry curvature of nonlinear systems that occurs at phase boundaries is described by using the Bogoliubov excitation of the semiquantal dynamics. Its is shown that when the critical boundary in the parameter space is crossed, the nonlinear geometric phase of the Bogloubov excitations surrounding the elliptic fixed points experiences non-analytic behavior. / 在本論文，我們利用半古典動力學的博戈留波夫激發研究非線性系統的貝里曲率在相邊界上出現的臨界現象。結果顯示，當參數空間中的臨界曲面被越過，環繞橢圓不動點的博戈留波夫激發的非線性幾何相位發生非解析行為。 / Kam, Chon Fai = 非線性系統的貝里曲率 / 甘駿暉. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 49-56). / Abstracts also in Chinese. / Title from PDF title page (viewed on 18, October, 2016). / Kam, Chon Fai = Fei xian xing xi tong de Beili qu lu / Gan Junhui. / Detailed summary in vernacular field only.

Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal without changing the variety is a powerful tool in determining a variety. In general, the quotient and remainder on division of polynomials in more than one variable are not unique. One property of a Groebner basis is that it yields a unique remainder on division. To prove geometric theorems algebraically, we first express the hypotheses and conclusions as polynomials. Then, with the aid of a computer, apply the Groebner Basis Algorithm to determine if the conclusion polynomial(s) vanish on the same variety as the hypotheses.

Features of the infraorbital region, such as infraorbital surface topography, infraorbital surface orientation, and curvature of the zygomaticoalveolar crest, have long played a prominent role in phylogenetic analyses of Homo. However, there is currently considerable debate regarding the phylogenetic reliability of infraorbital characters, as numerous researchers have questioned the degree to which these features are morphologically independent of one another and facial size. These questions largely stem from methodological limitations for accurately quantifying the curvilinear morphology of the infraorbital surface and zygomaticoalveolar crest, which have significantly impeded the ability to discern patterns of infraorbital integration and allometry. In this study, infraorbital surface and zygomaticoalveolar crest morphology are precisely assessed, through geometric morphometric methodologies well-suited for quantifying complex curvilinear structures, in a large sample of fossil (n = 71) and recent Homo (n = 303). Once quantified, measures of infraorbital surface topography, infraorbital surface orientation and zygomaticoalveolar crest curvature are further evaluated for intercorrelation and allometry in order to more fully evaluate the morphological independence of commonly cited infraorbital characters. The results of this study indicate that most aspects of infraorbital surface topography, infraorbital surface orientation and zygomaticoalveolar crest curvature are significantly correlated with facial size across Homo. Moreover, certain aspects of infraorbital shape, such the degree of infraorbital surface depression and the overall curvature of the zygomaticoalveolar crest, appear to show additional, size-independent, intercorrelations, suggesting they form a singular "infraorbital complex." In light of these results, the use of infraorbital characters as separate independent characters in phylogenetic assessments of Homo is called into question, while the importance of facial size in human craniofacial evolution is further highlighted.

Today’s highly endangered orangutan populations of Sumatra and Borneo offer but a glimpse into the taxonomic diversity and vast regional distribution enjoyed by orangutans and their great ape relatives in East Asia over the past 2.5 million years—a time when tropical forest pongine habitats stretched from Java to southern China. In addition to the giant terrestrial ape Gigantopithecus, other great ape genera have been proposed to have existed within this hominid community. The taxonomic diversity of this great ape faunal array is even further complicated when the purported presence of hominins at Early Pleistocene sites older than 1.85 Ma is considered. Highly acidic, the jungle floors of East Asia are notoriously bad at fossil preservation decomposing skeletal and dental evidence quickly. Fortunately, ph-neutral limestone caves have acted to offset these forces. The outcome of this peculiar taphonomy has left us with many teeth, but very little bone. With only unassociated fossil dentition to work with, modern geometric morphometrics offers scientists one of the few cutting-edge tools capable of systematically assessing this material reliably. This dissertation applies modern geometric morphometric statistical analysis to over two thousand fossil hominid teeth (Appendix A) from the Quaternary of southern China and Southeast Asia, which offers unique insight into the taxonomic diversity present in this sole Pleistocene great ape community. This study provides a much clearer understanding of the composition, paleoecology, and regional distribution of Pleistocene great ape communities of East Asia. Concordant with previous research, the main study and pilot study conducted in this dissertation showed Homo sapiens to always be morphologically and statistically distinct from extant and fossil orangutans. In turn, Pongo pygmaeus and Pongo abelii were continuously shown to be distinct from each other as well as from fossil Pongo groups. This investigation refutes hominin assignments for several teeth previously placed within early East Asian hominins (showing them to be orangutans instead) but supports the hominin status of the Jianshi upper third premolar. In combination with a published age of 1.95–2.15 million years (Ma), the hominin assignment reaffirmed here for the Jianshi dentition originally classified as human by Liu, Clarke, & Xing (2010) may offer a challenge to evolutionary models that recognize the 1.85 Ma Dmanisi hominins as the earliest hominins outside of Africa. This fact is often lost on most contemporary scientists due to their preoccupation with the 2.5 Ma Longgupo mandibular fragment, once thought to be a hominin but now assignable to an ape. Like the Jianshi upper third premolar, it is also based on a single specimen (in this case, a mandibular fragment). This dissertation supports the existence of Ciochon’s (2009) “mystery ape”. It refutes Schwartz et al., (1995) multiple Vietnamese Pongo taxa, including the proposed genus “Langsonia,” which is reassigned here to Pongo or the “mystery ape,” while placing Vietnamese fossil orangutans into either Pongo weidenreichi or Pongo devosi. Teeth from the Ralph von Koenigswald collection originally assigned to “Hemanthropus” were also determined to be representative of either the “mystery ape” or Pongo. Indeterminate “hominin” teeth were assignable to either Homo erectus, Homo sapiens, or Pongo only; no evidence was found for any other types of hominin species present in the collections examined for this study.

In this study, the corallite shapes of three species of the scleractinian genus Undaria from the Yague group, Dominican Republic, were examined through a period of time stretching from 6.4 mya to 3.4 mya, a total of 3.0 ma. Corallite shape was measured using 3 dimensional landmarks and manipulated using the well established procedures of geometric morphometrics. Differences in shape and size through time were examined using a variety of tools, including canonical variates analysis, principal components analysis, least squares regression, partial least squares regression, and a variety of evolutionary model fits. Evolutionary model fits were used to test three models against the shape and size variables: general random walk, which models a directional change through time; unbiased random walk, which models random change through time; and stasis, which models stability through time. Stasis is the most common parameter through time, supported in 9 of 15 (60%) of cases, while the unbiased random walk was supported 6 of 15 times. While there was a significant change in one species associated with environmental variables, those variables were also correlated with time and no causal relationship can be reached.

The theory of thermodynamic phase transitions has played a central role both in theoretical physics and in dynamical systems for several decades. One of its fundamental results is the classification of various physical models into equivalence classes with respect to the scaling behavior of solutions near the critical manifold. From that point of view, systems characterized by the same set of critical exponents are equivalent, regardless of how different the original physical models might be. For non-equilibrium phase transitions, the current theoretical framework is much less developed. In particular, an equivalent classification criterion is not available, thus requiring a specific analysis of each model individually. In this thesis, we propose a potential classification method for time-dependent dynamical systems, namely comparing the possible deformations of the original problem, and identifying dynamical systems which share the same deformation space. The specific model on which this procedure is developed is the Kuramoto model for interacting, disordered oscillators. Studied in the mean-field limit by a variety of methods, its associated synchronization phase transition appears as an appropriate model for cooperative phenomena ranging from coupled Josephson junctions to self-ordering patterns in biological and social systems. We investigate the geometric deformation of the dynamical system into the space of univalent maps of the unit disk, related to the Douady-Earle extension and the Denjoy-Wolff theory, and separately the algebraic deformation into the space of nonlinear sigma models for unitary operators. The results indicate that the Kuramoto model is representative for a large class of non-equilibrium synchronization models, with a rich phase-space diagram.

This paper describes an intuitive way of defining geometry design variables for solving structural topology optimization problems using a genetic algorithm (GA). The geometry representation scheme works by defining a skeleton that represents the underlying topology/connectivity of the continuum structure. As the effectiveness of any GA is highly dependent on the chromosome encoding of the design variables, the encoding used here is a directed graph which reflects this underlying topology so that the genetic crossover and mutation operators of the GA can recombine and preserve any desirable geometric characteristics through succeeding generations of the evolutionary process. The overall optimization procedure is tested by solving a simulated topology optimization problem in which a 'target' geometry is pre-defined with the aim of having the design solutions converge towards this target shape. The procedure is also applied to design a straight-line compliant mechanism : a large displacement flexural structure that generates a vertical straight line path at some point when given a horizontal straight line input displacement at another point. / Singapore-MIT Alliance (SMA)

We consider the problem of moving a three dimensional solid object among polyhedral obstacles. The traditional formulation of configuration space for this problem uses three translational parameters and three angles (typically Euler angles), and the constraints between the object and obstacles involve transcendental functions. We show that a quaternion representation of rotation yields constraints which are purely algebraic in a higher-dimensional space. By simple manipulation, the constraints may be projected down into a six dimensional space with no increase in complexity. Using this formulation, we derive an efficient exact intersection test for an object which is translating and rotating among obstacles.