In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

I will present the mean curvature flow in Euclidean spaces and the Lagrangian mean curvature flow. We will first study the mean curvature evolution of submanifolds in Euclidean spaces, with an emphasis on the case of hypersurfaces. Along the way we will demonstrate the basic techniques in the study of geometric flows in general (for example, various maximum principles and the treatment of singularities). After that we will move on to the study of Lagrangian mean curvature flows. We will make the relevant definitions and prove the fundamental result that the Lagrangian condition is preserved along the mean curvature flow in Kähler-Einstein manifolds, which started the extensive, and still ongoing, research on Lagrangian mean curvature flows. We will also define special Lagrangian submanifolds as calibrated submanifolds in Calabi-Yau manifolds. Finally, we will study the mean curvature flow of conormal bundles as submanifolds of C^n. Using some tools developed recently, we will show that if a surface has strictly negative curvatures, then away from the zero section, the Lagrangian mean curvature flow starting from a conormal bundle does not develop Type I singularities.

The research described in this thesis aimed to evaluate the intrinsic DNA curvature ofthe region of the TP53 tumour suppressor gene that codes for the sequence-specific DNA-binding domain of the p53 protein, a key protein that protects the cell from chemical insultsand tumourogenesis. There have been no previous attempts to experimentally investigate theintrinsic DNA curvature within TP53 or its relation to the functional or structural properties ofthe gene, such as DNA repair and nucleosomal architecture. The present study usedtheoretical models of TP53 in concert with an atomic force microscopy based experimentalinvestigation of TP53 DNA molecules to analyse intrinsic DNA curvature within the gene. Thiswas achieved by developing a novel software platform for the atomic force microscopy basedinvestigation of DNA curvature, named ADIPAS. Dinucleotide wedge models of DNA curvaturewere used to model TP53 in order to investigate the relationship between intrinsic DNAcurvature and the structure and function of the gene. ADIPAS was applied to atomic forcemicroscopy images of TP53 DNA molecules immobilised on a mica surface in order toexperimentally measure intrinsic DNA curvature. The experimental findings were compared totheoretical models of intrinsic curvature in TP53. The resulting intrinsic curvature profilesshowed that exons exhibited significantly lower intrinsic DNA curvature than introns withinTP53, this was also shown to be true for regions of slow DNA repair. This indicated that DNAcurvature may play a role in TP53 as a controlling factor for nucleosomal architecture tofacilitate open chromatin and active DNA transcription. The evolutionary selection for intrinsiccurvature may have played a role in the development of exons with low intrinsic DNAcurvature. Low intrinsic curvature in exon position has also been implicated in the reducedefficiency of DNA repair in a number of cancer specific mutation hotspots.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Dado um espaÃo-tempo M─n+1 = I x à Fn Robertson-Walker generalizado onde à à a funÃÃo warping que verifica uma certa condiÃÃo de convexidade, vamos classificar hipersuperfÃcies tipo-espaÃo fortemente estÃveis com curvatura mÃdia constante. Mais precisamente, vamos mostrar que, considerando x : Mn→ M─n+1 uma hipersuperfÃcie tipo-espaÃo fortemente estÃvel, fechada imersa em M─n+1 com curvatura mÃdia constante H, se a funÃÃo warping à satisfaz Ãâ ≥ max {H Ãâ, 0} ao longo de M, entÃo Mn à maximal ou uma folha tipo-espaÃo Mto={to} x F, para algum to Є I. / Give a generalized M─n+1 = I xà Fn Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given x : Mn → M─n+1 a closed, strongly stable spacelike hypersurfaces of M─n+1 with constant mean curvature H, if the warping function à satisfying à ≥ max {HÃ', 0} along M, is either maximal or a spacelike slice Mto = {to} x F, for some to Є I.

No description available.

This thesis presents a theory of multi-scale, curvature and torsion based shape representation for planar and space curves. The theory presented has been developed to satisfy various criteria considered useful for evaluating shape representation methods in computer vision. The criteria are: invariance, uniqueness, stability, efficiency, ease of implementation and computation of shape properties. The regular representation for planar curves is referred to as the curvature scale space image and the regular representation for space curves is referred to as the torsion scale space image. Two variants of the regular representations, referred to as the renormalized and resampled curvature and torsion scale space images, have also been proposed. A number of experiments have been carried out on the representations which show that they are very stable under severe noise conditions and very useful for tasks which call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation. Planar or space curves are described at varying levels of detail by convolving their parametric representations with Gaussian functions of varying standard deviations. The curvature or torsion of each such curve is then computed using mathematical equations which express curvature and torsion in terms of the convolutions of derivatives of Gaussian functions and parametric representations of the input curves. Curvature or torsion zero-crossing points of those curves are then located and combined to form one of the representations mentioned above. The process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve. This thesis contains a number of theorems about evolution and arc length evolution of planar and space curves along with their proofs. Some of these theorems demonstrate that evolution and arc length evolution do not change the physical interpretation of curves as object boundaries and others are in fact statements on the global properties of planar and space curves during evolution and arc length evolution and their representations. Other theoretical results shed light on the local behavior of planar and space curves just before and just after the formation of a cusp point during evolution and arc length evolution. Together these results provide a sound theoretical foundation for the representation methods proposed in this thesis. / Science, Faculty of / Computer Science, Department of / Graduate

This dissertation discusses the time and place that Brownian motion on a Riemannian manifold first exit a normal ball of small radius. A general procedure is given for computing asymptotic expansions of joint moments of the first exit time and place random variables as the radius of the geodesic ball decreases to zero. The asymptotic expansion of the joint Laplace transform of exit time and spherical harmonics of exit position is derived for a ball of small radius. A generalized Pizetti’s formula is used to expand the solution of the related partial differential equations. These expansions are represented in terms of curvature in the manifold. Asymptotic Independence Conditions (AIC) and Asymptotic Uncorrelated Conditions (AUC) are defined for the joint distributions of exit time and place. Computations using the methods developed in this work demonstrate that AIC and AUC produce the same curvature conditions up to a certain level of asymptotics. It is conjectured that AUC implies AIC. Further, a generalized method is given for computing the Laplace transform, and therefore the moments of the exit time. This work is related to and also extends the work of M. Liao and H. R. Hughes in stochastic geometric analysis.

The present study deals with the use of fiberglass reinforced polymer bars (GFRP) as a replacement for the common steel of a reinforced concrete circular pile, in order to avoid the corrosion of durability of reinforcing bars and thus improve them. The comparative analysis was carried out between a pile reinforced with GFRP and another with steel, where the ductility was evaluated by obtaining moment-curvature diagram. As a result, said idealized moment-curvature diagrams and ductility indices are presented, concluding the ductility of the section reinforced with GFRP in 20% more than that of steel.

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science, November 2012. / Hominoids practice a diverse array of locomotor behavior, from obligate terrestrial bipedalism to arboreal suspensory behavior, which is reflected in the variable morphology found in their foot bones. That hominin foot bones reflect locomotor behavior is also clear, but the forms of locomotor behaviors to be inferred are less clear. Pressure plate studies indicate that the center of pressure tends to move medially in the human foot during the last half of stance phase of bipedal gaits, while it tends to remain relatively more lateral in the bonobo and chimpanzee foot during the last half of stance phase. Here is presented a comparison of metatarsals of Homo sapiens[n=22] and two species of Pan (Pan paniscus [n=15] Pan troglodytes schweinfurthii [n=22]in order to explore the relationship between Homo and Pan metatarsal morphology and foot function. Specifically, this dissertation addresses whether cortical thickness is associated with the position of maximum change in geometry on the plantar surface in metatarsals.