136 research outputs found
Recommended from our members
Weak capacity in Ahlfors regular metric spaces
No full textLet be a compact, connected, Ahlfors -regular metric space with Q>1. Using a hyperbolic filling of , we define the notions of the -capacity between certain subsets of and of the weak covering -capacity of path families in . We show comparability results and quasisymmetric invariance. We reprove a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected, Ahlfors -regular metric spaces. Under certain conditions, we identify the Ahlfors regular conformal dimension of with critical exponents arising from weak capacity. Following an approach by Mario Bonk and Bruce Kleiner, we prove a necessary and sufficient condition involving weak capacity for an Ahlfors regular metric space that is topologically to be quasisymmetrically equivalent to
Recommended from our members
Lipschitz Maps in Metric Spaces
No full textIn this dissertation we study Lipschitz and bi-Lipschitz mappings on abstract, non-smooth metric measure spaces. The dissertation consists of two separate parts.The first part considers a well-known class of questions that ask the following: If X and Y are metric measure spaces and f is a Lipschitz mapping from X to Y whose image has positive measure, then must f have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present author) and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors regular topological manifolds. In general, these manifolds need not admit bi-Lipschitz embeddings into any Euclidean space. To prove the result, we use some facts on the Gromov-Hausdorff convergence of manifolds and a topological theorem of Bonk and Kleiner. This also yields a new proof of the uniform rectifiability of some metric manifolds.In the second part, we study the class of ``Lipschitz differentiability spaces'' introduced by Cheeger. These are spaces on which an appropriate version of Rademacher's theorem holds. We show that if an Ahlfors regular Lipschitz differentiability space has a differentiable structure of maximal dimension, then at almost every point all its tangents are uniformly rectifiable. In particular, it admits Euclidean tangents at almost every point. Conversely, we show that if the dimension of the differentiable structure is not extremal, then the space is strongly unrectifiable, in the sense of Ambrosio-Kirchheim. In proving these results, we generalize some results of Cheeger from the setting of doubling spaces with Poincare inequalities (PI spaces) to general doubling Lipschitz differentiability spaces. The starting point is a result of Bate on the local structure of these spaces
Outputs and Inputs of Philippine Commercial Banks
No full textExactly what constitutes bank input and output has not yet been settled. Some argue that deposit liabilities and earning assets are outputs as they represent the capacity of services banks can perform. A number of economists, however, assert that only earning assets can be considered as outputs since inclusion of deposit liabilities fails to distinguish between production in the technical sense and production in the economic sense. To delineate bank outputs and inputs, this study ascertains whether Philippine commercial banks incur positive costs on demand deposits. Utilizing statistical revenue-cost accounting technique, results help to classify bank input and output and to determine the rate of return on the composition of bank portfolio.rate of return, commercial banks, deposit liabilities, revenue-cost accounting model
Recommended from our members
A coarse entropy-rigidity theorem and discrete length-volume inequalities
No full textIn [5], M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-M"obius group actions on Ahlfors n-regular metric spaces with topological dimension n. This led naturally to a rigidity result for quasi-convex geometric actions on CAT(-1)-spaces that can be seen as a metric analog to the ``entropy-rigidity" theorems of U. Hamenst"adt and M. Bourdon. Building on the ideas developed in [5], we establish a rigidity theorem for certain expanding quasi-M"obius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding entropy-rigidity result in the coarse geometric setting.Our analysis of these ``fractal" metric spaces depends heavily on a combinatorial inequality that relates volume to lengths of curves within the space. We extend such inequalities to a broader metric setting and obtain discrete analogs of some results due to W. Derrick. In the process, we shed light on a related question of Y. Burago and V. Zalgaller about pseudometrics on the n-dimensional unit cube
Recommended from our members
Quasiregularly Elliptic Manifolds
No full textThe work in this dissertation is centered around the study of quasiregularly elliptic manifolds. These are manifolds that admit quasiregular maps from Euclidean space. The research of quasiregular maps is motivated by the pursuit of extending theorems from complex analysis and conformal geometry to higher dimensional settings.We first provide a new proof for the Rickman-Picard theorem, which states that a non-constant quasiregular map from Euclidean space to a sphere may omit a bounded number of points depending on the dilatation of the map.We next show that a closed, connected and orientable Riemannian manifold that is quasiregularly elliptic must have bounded dimension of the cohomology independent of the distortion of the map. The bound for the dimension is sharp and proves the Bonk-Heinonen conjecture. A corollary of this theorem answers an open problem posed by Gromov in 1981. He asked whether there exists a simply connected manifold that does not admit a quasiregular mapping from Euclidean space. The result shown gives an affirmative answer to this question.Lastly, we study the behavior of branched covers whose image of their branch set is contained in a simplicial complex. The image of the branch set of a piecewise linear branched cover between piecewise linear manifolds is a simplicial complex. We demonstrate that the reverse implication also holds. A branched cover from a sphere to a sphere with the image of the branch set contained in a codimension two simplicial complex is equivalent up to homeomorphism to a PL mapping. This extends a result by Martio and Srebro in the three dimensional setting
A geometria métrica das aplicações pseudo-Anosov generalizadas
No full textThe article [dC05] introduced the notion of generalized pseudo-Anosov (gpA) maps, which extend the pseudo-Anosov (pA) transformations introduced by Thurston by allowing the presence of an infinite number of singularities, with only a finite number of accumulation points. Similar to the pseudo-Anosov case, the conic-flat structure persists away from the accumulation points, and it is possible to prove [dCH12] that the induced complex structure extends uniquely over these points. In [BdCH21], a continuous family of sphere homeomorphisms was constructed as a quotient of the inverse limit of the tent family of interval endomorphisms. This family includes the unimodal generalized pseudo-Anosov (ugpA) maps as a countable, dense subfamily, whose spheres of definition have well-defined and extensively studied geometric structures. In this work, we examine the family of unimodal generalized pseudo-Anosov maps, exploring their unique geometric and dynamical properties. We show that the associated surfaces exhibit several remarkable characteristics, such as Ahlfors regularity and linear local contractibility, which are crucial in understanding the larger context of quasisymmetric structures on metric surfaces. Consequently, by applying a theorem by Bonk and Kleiner [BK01], we establish that these surfaces are quasisymmetrically equivalent to the topological 2-sphere.O artigo [dC05] introduziu a noção de aplicações pseudo-Anosov generalizadas (gpA), que ampliam as transformações pseudo-Anosov (pA) introduzidas por Thurston, permitindo a presença de um número infinito de singularidades, mas com apenas um número finito de pontos de acumulação. Similarmente ao caso pseudo-Anosov, a estrutura cônico plana persiste fora dos pontos de acumulação, e é possível provar [dCH12] que a estrutura complexa induzida se estende de forma única sobre esses pontos. Em [BdCH21], foi construída uma família contínua de homeomorfismos de esferas como um quociente do limite inverso da família de endomorfismos do intervalo unimodais. Essa família inclui as aplicações pseudo-Anosov generalizadas unimodais (ugpA) como uma subfamília densa e contável, cujas esferas de definição possuem estruturas geométricas bem definidas e amplamente estudadas. Neste trabalho, investigamos a família de aplicações pseudo-Anosov generalizadas unimodais, explorando suas propriedades geométricas e dinâmicas únicas. Demonstramos que as superfícies associadas apresentam características notáveis, como Ahlfors regular e contratibilidade local linear, que são fundamentais para compreender o contexto mais amplo das estruturas quasisimétricas em superfícies métricas. Consequentemente, ao aplicar um teorema de Bonk e Kleiner [BK01], estabelecemos que essas superfícies são quasisimétricamente equivalentes à esfera topológica bidimensional
Lattes Maps and Combinatorial Expansion.
No full textA Lattes map f : C → C is a rational map that is obtained from a finite quotient of
a conformal torus endomorphism. In this thesis, we give a characterization of Lattes
maps by their combinatorial expansion behavior. More specifically, to any Thurston
maps, which are branched covering maps over the 2-sphere with finite post-critical
sets, there are natural cell-decompositions of the 2-sphere induced by the dynamics
following Bonk and Meyer. We show that these cell decompositions give us a natural
Gromov hyperbolic space, and we deduce new necessary and sufficient conditions for
a Thurston map to be topologically conjugate to a Lattes map.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86524/1/qyin_1.pd
Conformal Dimension and the Quasisymmetric Geometry of Metric Spaces.
No full textThe conformal dimension of a metric space measures the optimal dimension of the space under quasisymmetric deformations. We consider metric spaces that are locally connected and have no local cut points, in a quantitative way, and show that such spaces have conformal dimension greater than one.
We then apply this result to hyperbolic groups that do not virtually split over any finite or virtually cyclic subgroup to show that the conformal dimension of the conformal boundary at infinity of such groups is greater than one. This answers a question of Bonk and Kleiner.
One additional result is worth noting: we show that a linearly connected, doubling metric space is connected by quasi-arcs, quantitatively. While this was previously proven by Tukia, our proof is new and much improved.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60878/1/jmmackay_1.pd
Students' experiences and expectations of technologies: an Australian study designed to inform planning and development decisions
No full textThe pace of technological change accompanied by an evolution in social, work-based and study behaviours and norms poses particular challenges for universities as they strive to develop high quality and sustainable technology-rich learning environments. Maintaining currency with the latest advances is resource intensive, hence the costs incurred in upgrading existing and introducing new technologies need to be carefully weighed up against the potential benefits to students. This calls for a multidimensional approach to planning, with the student voice being an important dimension. Three Australian universities have recently completed a project to gain a better understanding of students\u27 experiences and expectations of technologies in everyday life and for study purposes. The LMS and 25 other technologies ranging from established university offerings (email, learning management systems) to freely available social networking technologies (YouTube, Facebook) were surveyed. More than 10,000 students responded. This paper discusses the development of the survey and presents the broad trends that have emerged in relation to the current use of technologies and desired future use of these for learning purposes. The implications of the survey findings for developing institutional infrastructure to engage students and support their learning are highlighted
Recommended from our members
Visual Spheres of Expanding Thurston Maps: Their Weak Tangents and Porous Subsets
No full textWe study certain approximately self-similar metric spaces that arise from expanding Thurston maps called visual spheres. It is known [HP09, Theorem 4.2.11][BM17, Theorem 18.1(ii')]that the quasisymmetry class of a visual sphere of is related to the rationality of . We prove that a visual sphere is indeed approximately self-similar if does not have periodic critical points. This is done by picking a nice visual metric of an expanding Thurston map. Using the nice metric, we study the solenoid of . We put a specific metric on the leaves of and show that the leaves and weak tangents are almost the same thing. We then study the visual spheres of expanding Thurston maps with an emphasis on a quasisymmetric invariant, called the Ahlfors regular conformal dimension. We show that the Ahlfors regular conformal dimension of any weak tangent of a visual sphere is the same as the Ahlfors regular conformal dimension of the visual sphere itself, and that the Ahlfors regular conformal dimension of any weak tangent is attainable if and only if Ahlfors regular conformal dimension of the visual sphere is attainable. We show by an example that the same does not hold for more general metric spaces. Finally, we show that a visual sphere has -thick curve family supported on an -invariant porous subset if has a irreducible -thick -stable multicurve. We give an example to show that when , the condition of a -thick -stable multicurve is sharp
- …
