Métrique sur le fibré unitaire tangent au plan hyperbolique

RÉSUMÉ: Toute variété différentiable $M$ admet une métrique dite métrique riemannienne.\\ En définissant $\mathbb{H}=\lbrace z\in\mathbb{C}: Im(z)>0\rbrace$, on peut munir de $\mathbb{H}$ d'une métrique riemannienne $ds^{2}=\frac{dzd\bar{z}}{(Im(z))^{2}}=\frac{dx^{2}+dy^{2}}{y^{2}}$.\\ Muni de cette métrique, $\mathbb{H}$ est une variété riemannienne à la quelle on associe le fibré tangent, $T\mathbb{H}$ ainsi que le fibré unitaire tangent, $T^{1}\mathbb{H}$. Les éléments de $T^{1}\mathbb{H}$ peuvent être exprimés, de façon bijective, en termes des éléments du groupe PSL(2,$\mathbb{R}$) dont l'action sur $T^{1}\mathbb{H}$ est transitive et libre.\\ La métrique définie sur $M$ (en particulier sur $M=\mathbb{H}$) permet de définir sur $TM$ (en particulier sur $T^{1}\mathbb{H}$) une métrique connue sous le nom de métrique de Sasaki.

Problems in number theory and hyperbolic geometry

In the first part of this thesis we generalize a theorem of Kiming and Olsson concerning the existence of Ramanujan-type congruences for a class of eta quotients. Specifically, we consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 (mod ℓ). We use this last result to answer a question of H.C. Chan. In the second part of this thesis [S2] we explore a natural analog of D. Calegari’s result that there are no hyperbolic once-punctured torus bundles over S^1 with trace field having a real place. We prove a contrasting theorem showing the existence of several infinite families of pairs (−χ, p) such that there exist hyperbolic surface bundles over S^1 with trace field of having a real place and with fiber having p punctures and Euler characteristic χ. This supports our conjecture that with finitely many known exceptions there exist such examples for each pair ( −χ, p).

On the critical point structure of eigenfunctions belonging to the first nonzero eigenvalue of a genus two closed hyperbolic surface

We develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds

"Not one word of it made any sense" : Hyperbolic Synecdoche in the British National Corpus

A distinct metonymic pattern was discovered in the course of conducting a corpus-based study of figurative uses of WORD. The pattern involved examples such as Not one word of it made any sense and I agree with every word. It was labelled ‘hyperbolic synecdoche’, defined as a case in which a lexeme which typically refers to part of an entity (a) is used to stand for the whole entity and (b) is described with reference to the end point on a scale. Specifically, the speaker/writer selects the perspective of a lower-level unit (such as word for ‘utterance’), which is quantified as NOTHING or ALL, thus forming a subset of ‘extreme case formulations’. Hyperbolic synecdoche was found to exhibit a restricted range of lexicogrammatical patterns involving word, with the negated NOTHING patterns being considerably more common than the ALL patterns. The phenomenon was shown to be common in metonymic uses in general, constituting one-fifth of all cases of metonymy in word. The examples of hyperbolic synecdoche were found not to be covered by the oftquoted ‘abbreviation’ rationale for metonymy; instead, they represent a more roundabout way of expression. It is shown that other cases of hyperbolic synecdoche exist outside of word and the domain of communication (such as ‘time’ and ‘money’).

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles SrM of a Riemannian manifold (M,g) of dim ≥ 3, assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I^G and symplectic structure ω^G on the manifold TM , generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel-Whitney characteristic classes of the manifolds TM and SrM.

Mathematical models for describing the shape of the in vitro unstretched human crystalline lens

We developed orthogonal least-squares techniques for fitting crystalline lens shapes, and used the bootstrap method to determine uncertainties associated with the estimated vertex radii of curvature and asphericities of five different models. Three existing models were investigated including one that uses two separate conics for the anterior and posterior surfaces, and two whole lens models based on a modulated hyperbolic cosine function and on a generalized conic function. Two new models were proposed including one that uses two interdependent conics and a polynomial based whole lens model. The models were used to describe the in vitro shape for a data set of twenty human lenses with ages 7–82 years. The two-conic-surface model (7 mm zone diameter) and the interdependent surfaces model had significantly lower merit functions than the other three models for the data set, indicating that most likely they can describe human lens shape over a wide age range better than the other models (although with the two-conic-surfaces model being unable to describe the lens equatorial region). Considerable differences were found between some models regarding estimates of radii of curvature and surface asphericities. The hyperbolic cosine model and the new polynomial based whole lens model had the best precision in determining the radii of curvature and surface asphericities across the five considered models. Most models found significant increase in anterior, but not posterior, radius of curvature with age. Most models found a wide scatter of asphericities, but with the asphericities usually being positive and not significantly related to age. As the interdependent surfaces model had lower merit function than three whole lens models, there is further scope to develop an accurate model of the complete shape of human lenses of all ages. The results highlight the continued difficulty in selecting an appropriate model for the crystalline lens shape.

Bulk compressive properties of the heel fat pad during walking: a pilot investigation in plantar heel pain

Background: Altered mechanical properties of the heel pad have been implicated in the development of plantar heel pain. However, the in vivo properties of the heel pad during gait remain largely unexplored in this cohort. The aim of the current study was to characterise the bulk compressive properties of the heel pad in individuals with and without plantar heel pain while walking. ---------- Methods: The sagittal thickness and axial compressive strain of the heel pad were estimated in vivo from dynamic lateral foot radiographs acquired from nine subjects with unilateral plantar heel pain and an equivalent number of matched controls, while walking at their preferred speed. Compressive stress was derived from simultaneously acquired plantar pressure data. Principal viscoelastic parameters of the heel pad, including peak strain, secant modulus and energy dissipation (hysteresis), were estimated from subsequent stress–strain curves.---------- Findings: There was no significant difference in loaded and unloaded heel pad thickness, peak stress, peak strain, or secant and tangent modulus in subjects with and without heel pain. However, the fat pad of symptomatic feet had a significantly lower energy dissipation ratio (0.55 ± 0.17 vs. 0.69 ± 0.08) when compared to asymptomatic feet (P < .05).---------- Interpretation: Plantar heel pain is characterised by reduced energy dissipation ratio of the heel pad when measured in vivo and under physiologically relevant strain rates.

Axisymmetric shell structures for multi-use

Shell structures find use in many fields of engineering, notably structural, mechanical, aerospace and nuclear-reactor disciplines. Axisymmetric shell structures are used as dome type of roofs, hyperbolic cooling towers, silos for storage of grain, oil and industrial chemicals and water tanks. Despite their thin walls, strength is derived due to the curvature. The generally high strength-to-weight ratio of the shell form, combined with its inherent stiffness, has formed the basis of this vast application. With the advent in computation technology, the finite element method and optimisation techniques, structural engineers have extremely versatile tools for the optimum design of such structures. Optimisation of shell structures can result not only in improved designs, but also in a large saving of material. The finite element method being a general numerical procedure that could be used to treat any shell problem to any desired degree of accuracy, requires several runs in order to obtain a complete picture of the effect of one parameter on the shell structure. This redesign I re-analysis cycle has been achieved via structural optimisation in the present research, and MSC/NASTRAN (a commercially available finite element code) has been used in this context for volume optimisation of axisymmetric shell structures under axisymmetric and non-axisymmetric loading conditions. The parametric study of different axisymmetric shell structures has revealed that the hyperbolic shape is the most economical solution of shells of revolution. To establish this, axisymmetric loading; self-weight and hydrostatic pressure, and non-axisymmetric loading; wind pressure and earthquake dynamic forces have been modelled on graphical pre and post processor (PATRAN) and analysis has been performed on two finite element codes (ABAQUS and NASTRAN), numerical model verification studies are performed, and optimum material volume required in the walls of cylindrical, conical, parabolic and hyperbolic forms of axisymmetric shell structures are evaluated and reviewed. Free vibration and transient earthquake analysis of hyperbolic shells have been performed once it was established that hyperbolic shape is the most economical under all possible loading conditions. Effect of important parameters of hyperbolic shell structures; shell wall thickness, height and curvature, have been evaluated and empirical relationships have been developed to estimate an approximate value of the lowest (first) natural frequency of vibration. The outcome of this thesis has been the generation of new research information on performance characteristics of axisymmetric shell structures that will facilitate improved designs of shells with better choice of shapes and enhanced levels of economy and performance. Key words; Axisymmetric shell structures, Finite element analysis, Volume Optimisation_ Free vibration_ Transient response.

Velocity-jump models with crowding effects

Velocity jump processes are discrete random walk models that have many applications including the study of biological and ecological collective motion. In particular, velocity jump models are often used to represent a type of persistent motion, known as a “run and tumble”, which is exhibited by some isolated bacteria cells. All previous velocity jump processes are non-interacting, which means that crowding effects and agent-to-agent interactions are neglected. By neglecting these agent-to-agent interactions, traditional velocity jump models are only applicable to very dilute systems. Our work is motivated by the fact that many applications in cell biology, such as wound healing, cancer invasion and development, often involve tissues that are densely packed with cells where cell-to-cell contact and crowding effects can be important. To describe these kinds of high cell density problems using a velocity jump process we introduce three different classes of crowding interactions into a one-dimensional model. Simulation data and averaging arguments lead to a suite of continuum descriptions of the interacting velocity jump processes. We show that the resulting systems of hyperbolic partial differential equations predict the mean behavior of the stochastic simulations very well.

Spin

“Spin” borrows idioms and metaphors from sports commentary and squeezes them into a single emotional rollercoaster. Accompanied by a driving soundtrack, text appears and disappears one word at a time. As the work progresses, multiple words fade in and out at the same time, filling the screen and testing our ability to read and assimilate these well-worn phrases. On the one hand, the work mimes some of what we enjoy about sport – its ability to take us to another place, to incite passion and emotion, and to enable us to share in common experiences, goals and desires. On the other hand, it plays up the hyperbolic language often associated with sports broadcasting. The very language that helps take us to another place, incite passion and make us feel part of something bigger than ourselves, is pushed to its extreme and starts to burst at the seams. This work was commissioned for “Kick Off: contemporary video art program” at Metricon Stadium, Gold Coast, and supported by Project Services, Department of Public Works, Queensland Government.

Travelling waves for a velocity–jump model of cell migration and proliferation

Cell invasion, characterised by moving fronts of cells, is an essential aspect of development, repair and disease. Typically, mathematical models of cell invasion are based on the Fisher–Kolmogorov equation. These traditional parabolic models can not be used to represent experimental measurements of individual cell velocities within the invading population since they imply that information propagates with infinite speed. To overcome this limitation we study combined cell motility and proliferation based on a velocity–jump process where information propagates with finite speed. The model treats the total population of cells as two interacting subpopulations: a subpopulation of left–moving cells, $L(x,t)$, and a subpopulation of right–moving cells, $R(x,t)$. This leads to a system of hyperbolic partial differential equations that includes a turning rate, $\Lambda \ge 0$, describing the rate at which individuals in the population change direction of movement. We present exact travelling wave solutions of the system of partial differential equations for the special case where $\Lambda = 0$ and in the limit that $\Lambda \to \infty$. For intermediate turning rates, $0 < \Lambda < \infty$, we analyse the travelling waves using the phase plane and we demonstrate a transition from smooth monotone travelling waves to smooth nonmonotone travelling waves as $\Lambda$ decreases through a critical value $\Lambda_{crit}$. We conclude by providing a qualitative comparison between the travelling wave solutions of our model and experimental observations of cell invasion. This comparison indicates that the small $\Lambda$ limit produces results that are consistent with experimental observations.

Characteristics of police-reported road traffic crashes in Ethiopia over a six year period

Ethiopia has one of Africa’s fastest growing non-oil producing economies and an increasing level of motorisation (AfDB, OECD, UNDP, & UNECA, 2012). This rapidly increasing mobility has created some unique road safety concerns; however there is scant published information and related commentary (United Nations Economic Commission for Africa, 2009). The objective of this paper is to quantify police-reported traffic crashes in Ethiopia and characterise the existing state of road safety. Six years (July 2005 - June 2011) of police-reported crash data were analysed, consisting of 12,140 fatal and 29,454 injury crashes on the country’s road network. The 12,140 fatal crashes involved 1,070 drivers, 5,702 passengers, and 7,770 pedestrians, totalling 14,542 fatalities, an average of 1.2 road user fatalities per crash. An important and glaring trend that emerges is that more than half of the fatalities in Ethiopia involve pedestrians. The majority of the crashes occur during daytime hours, involve males, and involve persons in the 18-50 age group—Ethiopia’s active workforce. Crashes frequently occur in mid blocks or roadways. The predominant collision between motor vehicles and pedestrians was a rollover on a road tangent section. Failing to observe the priority of pedestrians and speeding were the major causes of crashes attributed by police. Trucks and minibus taxis were involved in the majority of crashes, while automobiles (small vehicles) were less involved in crashes relative to other vehicle types, partially because small vehicles tend to be driven fewer kilometres per annum. These data illustrate and justify a high priority to identify and implement effective programs, policies, and countermeasures focused on reducing pedestrian crashes.

Effect of overload cycles on thin steel roof claddings during cyclonic winds

During an investigation on thin steel roof claddings under simulated cyclonic wind loading, it was found that trapezoidal roof claddings behaved quite differently to corrugated (arc and tangent type) roof claddings due to the presence of overload cycles. The overload cycles caused a reduction in fatigue life for corrugated roofing whereas the reverse occurred for trapezoidal roofing. This contrasting behavior of the two crest-fixed roof claddings was investigated using small scale roofing models instead of the commonly used large scale two-span roof claddings. It was found that overload cycles formed a weaker locally dimpled mechanism around the fastener holes of corrugated roofing and thus accelerated the fatigue-caused pull-through failure. In contrast, a stronger deformed shape was formed in trapezoidal roofing which delayed the pull-through failure. Both laboratory testing and finite element analysis of small scale models were used to study the contrasting behavior of roof claddings.