Dissertations / Theses: 'Mathematics. Metric spaces. Mappings (Mathematics)'

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Niyitegeka, Jean Marie Vianney. "Generalizations of some fixed point theorems in banach and metric spaces." Thesis, Nelson Mandela Metropolitan University, 2015. http://hdl.handle.net/10948/5265.

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A fixed point of a mapping is an element in the domain of the mapping that is mapped into itself by the mapping. The study of fixed points has been a field of interests to mathematicians since the discovery of the Banach contraction theorem, i.e. if is a complete metric space and is a contraction mapping (i.e. there exists such that for all ), then has a unique fixed point. The Banach contraction theorem has found many applications in pure and applied mathematics. Due to fixed point theory being a mixture of analysis, geometry, algebra and topology, its applications to other fields such as physics, economics, game theory, chemistry, engineering and many others has become vital. The theory is nowadays a very active field of research in which many new theorems are published, some of them applied and many others generalized. Motivated by all of this, we give an exposition of some generalizations of fixed point theorems in metric fixed point theory, which is a branch of fixed point theory about results of fixed points of mappings between metric spaces, where certain properties of the mappings involved need not be preserved under equivalent metrics. For instance, the contractive property of mappings between metric spaces need not be preserved under equivalent metrics. Since metric fixed point theory is wide, we limit ourselves to fixed point theorems for self and non-self-mappings on Banach and metric spaces. We also take a look at some open problems on this topic of study. At the end of the dissertation, we suggest our own problems for future research.

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Ruth, Harry Leonard Jr. "Conformal densities and deformations of uniform loewner metric spaces." Cincinnati, Ohio : University of Cincinnati, 2008. http://www.ohiolink.edu/etd/view.cgi?ucin1210203872.

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Thesis (Ph.D.)--University of Cincinnati, 2008.
Committee/Advisors: David Herron PhD (Committee Chair), David Minda PhD (Committee Member), Nageswari Shanmugalingam PhD (Committee Member). Title from electronic thesis title page (viewed Sep.3, 2008). Keywords: conformal density; uniform spaces; Loewner; quasisymmetry; quasiconofrmal. Includes abstract. Includes bibliographical references.

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Stofile, Simfumene. "Fixed points of single-valued and multi-valued mappings with applications." Thesis, Rhodes University, 2013. http://hdl.handle.net/10962/d1002960.

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The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.

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Medwid, Mark Edward. "Rigidity of Quasiconformal Maps on Carnot Groups." Bowling Green State University / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1497620176117104.

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Hume, David S. "Embeddings of infinite groups into Banach spaces." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:e38f58ec-484c-4088-bb44-1556bc647cde.

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In this thesis we build on the theory concerning the metric geometry of relatively hyperbolic and mapping class groups, especially with respect to the difficulty of embedding such groups into Banach spaces. In Chapter 3 (joint with Alessandro Sisto) we construct simple embeddings of closed graph manifold groups into a product of three metric trees, answering positively a conjecture of Smirnov concerning the Assouad-Nagata dimension of such spaces. Consequently, we obtain optimal embeddings of such spaces into certain Banach spaces. The ideas here have been extended to other closed three-manifolds and to higher dimensional analogues of graph manifolds. In Chapter 4 we give an explicit method of embedding relatively hyperbolic groups into certain Banach spaces, which yields optimal bounds on the compression exponent of such groups relative to their peripheral subgroups. From this we deduce that the fundamental group of every closed three-manifold has Hilbert compression exponent one. In Chapter 5 we prove that relatively hyperbolic spaces with a tree-graded quasi-isometry representative can be characterised by a relative version of Manning's bottleneck property. This applies to the Bestvina-Bromberg-Fujiwara quasi-trees of spaces, yielding an embedding of each mapping class group of a closed surface into a finite product of simplicial trees. From this we obtain explicit embeddings of mapping class groups into certain Banach spaces and deduce that these groups have finite Assouad-Nagata dimension. It also applies to relatively hyperbolic groups, proving that such groups have finite Assouad-Nagata dimension if and only if each peripheral subgroup does.

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Gonzalez, Villasanti Hugo Jose. "Stability of Input/Output Dynamical Systems on Metric Spaces: Theory and Applications." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu155558269238935.

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Razafindrakoto, Ando Desire. "Hyperconvex metric spaces." Thesis, Stellenbosch : University of Stellenbosch, 2010. http://hdl.handle.net/10019.1/4106.

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Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.
ENGLISH ABSTRACT: One of the early results that we encounter in Analysis is that every metric space admits a completion, that is a complete metric space in which it can be densely embedded. We present in this work a new construction which appears to be more general and yet has nice properties. These spaces subsequently called hyperconvex spaces allow one to extend nonexpansive mappings, that is mappings that do not increase distances, disregarding the properties of the spaces in which they are defined. In particular, theorems of Hahn-Banach type can be deduced for normed spaces and some subsidiary results such as fixed point theorems can be observed. Our main purpose is to look at the structures of this new type of “completion”. We will see in particular that the class of hyperconvex spaces is as large as that of complete metric spaces.
AFRIKAANSE OPSOMMING: Een van die eerste resultate wat in die Analise teegekom word is dat enige metriese ruimte ’n vervollediging het, oftewel dat daar ’n volledige metriese ruimte bestaan waarin die betrokke metriese ruimte dig bevat word. In hierdie werkstuk beskryf ons sogenaamde hiperkonvekse ruimtes. Dit gee ’n konstruksie wat blyk om meer algemeen te wees, maar steeds gunstige eienskappe het. Hiermee kan nie-uitbreidende, oftewel afbeeldings wat nie afstande rek nie, uitgebrei word sodanig dat die eienskappe van die ruimte waarop dit gedefinieer is nie ’n rol speel nie. In die besonder kan stellings van die Hahn- Banach-tipe afgelei word vir genormeerde ruimtes en sekere addisionele ressultate ondere vastepuntstellings kan bewys word. Ons hoofdoel is om hiperkonvekse ruimtes te ondersoek. In die besonder toon ons aan dat die klas van alle hiperkonvekse ruimtes net so groot soos die klas van alle metriese ruimtes is.

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Otafudu, Olivier Olela. "Convexity in quasi-metric spaces." Doctoral thesis, University of Cape Town, 2012. http://hdl.handle.net/11427/10950.

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Includes abstract.
Includes bibliographical references.
The principal aim of this thesis is to investigate the existence of an injective hull in the categories of T-quasi-metric spaces and of T-ultra-quasi-metric spaces with nonexpansive maps.

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Jeganathan, P. "Fixed points for nonexpansive mappings in Banach spaces." Master's thesis, University of Cape Town, 1991. http://hdl.handle.net/11427/17067.

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Sarantopoulos, I. C. "Polynomials and multilinear mappings in Banach spaces." Thesis, Brunel University, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.376057.

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Eriksson-Bique, Sylvester David. "Quantitative Embeddability and Connectivity in Metric Spaces." Thesis, New York University, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10261097.

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This thesis studies three analytic and quantitative questions on doubling metric (measure) spaces. These results are largely independent and will be presented in separate chapters.

The first question concerns representing metric spaces arising from complete Riemannian manifolds in Euclidean space. More precisely, we find bi-Lipschitz embeddings ƒ for subsets A of complete Riemannian manifolds M of dimension n, where N could depend on a bound on the curvature and diameter of A. The main difficulty here is to control the distortion of such embeddings in terms of the curvature of the manifold. In constructing the embeddings, we will study the collapsing theory of manifolds in detail and at multiple scales. Similar techniques give embeddings for subsets of complete Riemannian orbifolds and quotient metric spaces.

The second part of the thesis answers a question about finding quantitative and weak conditions that ensure large families of rectifiable curves connecting pairs of points. These families of rectifiable curves are quantified in terms of Poincaré inequalities. We identify a new quantitative connectivity condition in terms of curve fragments, which is equivalent to possessing a Poincaré inequality with some exponent. The connectivity condition arises naturally in three different contexts, and we present methods to find Poincaré inequalities for the spaces involved. In particular, we prove such inequalities for spaces with weak curvature bounds and thus resolve a question of Tapio Rajala.

In the final part of the thesis we study the local geometry of spaces admitting differentiation of Lipschitz functions with certain Banach space targets. The main result shows that such spaces can be characterized in terms of Poincaré inequalities and doubling conditions. In fact, such spaces can be covered by countably many pieces, each of which is an isometric subset of a doubling metric measure space admitting a Poincaré inequality. In proving this, we will find a new way to use hyperbolic fillings to enlarge certain sub-sets into spaces admitting Poincaré inequalities.

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Lee, Seunghwan Han. "Probabilistic reasoning on metric spaces." [Bloomington, Ind.] : Indiana University, 2009. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3380096.

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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics and Cognitive Science, 2009.
Title from PDF t.p. (viewed on Jul 19, 2010). Source: Dissertation Abstracts International, Volume: 70-12, Section: B, page: 7604. Adviser: Lawrence S. Moss.

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Farnana, Zohra. "The Double Obstacle Problem on Metric Spaces." Doctoral thesis, Linköpings universitet, Tillämpad matematik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-51588.

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In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We minimize the p-energy integral among all functions which have prescribed boundary values and lie between two given obstacles. This is a generalization of the Dirichlet problem for p-harmonic functions, in which case the obstacles are —∞ and ∞. We show the existence and uniqueness of solutions, and their continuity when the obstacles are continuous. Moreover we show that the continuous solution is p-harmonic in the open set where it does not touch the continuous obstacles. If the obstacles are not continuous, but satisfy a Wiener type regularity condition, we prove that the solution is still continuous. The Hölder continuity for solutions is shown, when the obstacles are Hölder continuous. Boundary regularity of the solutions is also studied. Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles and the boundary values vary and show the convergence of the solutions. We also consider generalized solutions for insoluble obstacle problems, using the convergence results. Moreover we show that for soluble obstacle problems the generalized solution coincides, locally, with the standard solution. Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω.

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Mushaandja, Zechariah. "A quasi-pseudometrizability problem for ordered metric spaces." Doctoral thesis, University of Cape Town, 2009. http://hdl.handle.net/11427/4914.

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Includes abstract.
Includes bibliographical references (leaves 83-88).
In this dissertation we obtain several results in the setting of ordered topological spaces related to the Hanai-Morita-Stone Theorem. The latter says that if f is a closed continuous map of a metric space X onto a topological space Y then the following statements are equivalent: (i) Y satisfies the first countability axiom; (ii) For each y 2 Y, f−1{y} has a compact boundary in X; (iii) Y is metrizable. A partial analogue of the above theorem for ordered topological spaces is herein obtained.

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Birch, Thomas. "Algorithmic randomness on computable metric spaces and hyperspaces." Master's thesis, University of Cape Town, 2012. http://hdl.handle.net/11427/22093.

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In this text we shall be focusing on generalizing Martin-Löf randomness to computable metric spaces with arbitrary measure (for examples of this type of generalization see Gács [14], Rojas and Hoyrup [15]. The aim of this generalization is to define algorithmic randomness on the hyperspace of non-empty compact subsets of a computable metric space, the study of which was first proposed by Barmpalias et al. [16] at the University of Florida in their work on the random closed subsets of the Cantor space. Much work has been done in the study of random sets with authors such as Diamondstone and Kjos-Hanssen [17] continuing the Florida approach, whilst others such as Axon [18] and Cenzer and Broadhead [19] have been studying the use of capacities to define hyperspace measures for use in randomness tests. Lastly in section 6.4 we shall be looking at the work done by Hertling and Weihrauch [13] on universal randomness tests in effective topological measure spaces and relate their results to randomness on computable metric measure spaces and in particular to the randomness of compact sets in the hyperspace of non-empty compact subsets of computable metric spaces.

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Stares, Ian S. "Extension of functions and generalised metric spaces." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386678.

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Al-Nayef, Anwar Ali Bayer, and mikewood@deakin edu au. "Semi-hyperbolic mappings in Banach spaces." Deakin University. School of Computing and Mathematics, 1997. http://tux.lib.deakin.edu.au./adt-VDU/public/adt-VDU20051208.110247.

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The definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations. Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation. It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations. Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is chaotic.

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Putwain, Rosemary Johanna. "Partial translation algebras for certain discrete metric spaces." Thesis, University of Southampton, 2010. https://eprints.soton.ac.uk/170227/.

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The notion of a partial translation algebra was introduced by Brodzki, Niblo and Wright in [11] to provide an analogue of the reduced group C*-algebra for metric spaces. Such an algebra is constructed from a partial translation structure, a structure which any bounded geometry uniformly discrete metric space admits; we prove that these structures restrict to subspaces and are preserved by uniform bijections, leading to a new proof of an existing theorem. We examine a number of examples of partial translation structures and the algebras they give rise to in detail, in particular studying cases where two different algebras may be associated with the same metric space. We introduce the notion of a map between partial translation structures and use this to describe when a map of metric spaces gives rise to a homomorphism of related partial translation algebras. Using this homomorphism, we construct a C*-algebra extension for subspaces of groups, which we employ to compute K-theory for the algebra arising from a particular subspace of the integers. We also examine a way to form a groupoid from a partial translation structure, and prove that in the case of a discrete group the associated C*-algebra is the same as the reduced group C*-algebra. In addition to this we present several subsidiary results relating to partial translations and cotranslations and the operators these give rise to.

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Färm, David. "Upper gradients and Sobolev spaces on metric spaces." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5816.

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The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.

All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.

Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts.

This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.

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Howroyd, John David. "On the theory of Hausdorff measures in metric spaces." Thesis, University College London (University of London), 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.283290.

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Shao, Chuang Gao Su. "Urysohn ultrametric spaces and isometry groups." [Denton, Tex.] : University of North Texas, 2009. http://digital.library.unt.edu/permalink/meta-dc-9918.

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Farnana, Zohra. "The Double Obstacle Problem on Metric Spaces." Licentiate thesis, Linköping : Linköpings universitet, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-10621.

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Stover, Derrick D. "Continuous Mappings and Some New Classes of Spaces." View abstract, 2009. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:3371579.

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Lau, Chi-hin, and 劉智軒. "Holomorphic maps from rational homogeneous spaces onto projective manifolds." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2003. http://hub.hku.hk/bib/B3124550X.

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RUTH, HARRY LEONARD JR. "Conformal Densities and Deformations of Uniform Loewner Metric Spaces." University of Cincinnati / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1210203872.

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CAMFIELD, CHRISTOPHER SCOTT. "Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces." University of Cincinnati / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1211551579.

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Tran, Anh Tuyet. "1p spaces." CSUSB ScholarWorks, 2002. https://scholarworks.lib.csusb.edu/etd-project/2238.

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Chowdhury, Samir. "Metric and Topological Approaches to Network Data Analysis." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114.

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Lopez, Marcos D. "Discrete Approximations of Metric Measure Spaces with Controlled Geometry." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439305529.

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Duda, Jakub. "Aspects of delta-convexity /." free to MU campus, to others for purchase, 2003. http://wwwlib.umi.com/cr/mo/fullcit?p3115539.

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Shao, Chuang. "Urysohn ultrametric spaces and isometry groups." Thesis, University of North Texas, 2009. https://digital.library.unt.edu/ark:/67531/metadc9918/.

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In this dissertation we study a special sub-collection of Polish metric spaces: complete separable ultrametric spaces. Polish metric spaces have been studied for quite a long while, and a lot of results have been obtained. Motivated by some of earlier research, we work on the following two main parts in this dissertation. In the first part, we show the existence of Urysohn Polish R-ultrametric spaces, for an arbitrary countable set R of non-negative numbers, including 0. Then we give point-by-point construction of a countable R-ultra-Urysohn space. We also obtain a complete characterization for the set R which corresponding to a R-Urysohn metric space. From this characterization we conclude that there exist R-Urysohn spaces for a wide family of countable R. Moreover, we determine the complexity of the classification of all Polish ultrametric spaces. In the second part, we investigate the isometry groups of Polish ultrametric spaces. We prove that isometry group of an Urysohn Polish R-ultrametric space is universal among isometry groups of Polish R-ultrametric spaces. We completely characterize the isometry groups of finite ultrametric spaces and the isometry groups of countable compact ultrametric spaces. Moreover, we give some necessary conditions for finite groups to be isomorphic to some isometry groups of finite ultrametric spaces.

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Chang, Cheng. "The Relative Complexity of Various Classification Problems among Compact Metric Spaces." Thesis, University of North Texas, 2016. https://digital.library.unt.edu/ark:/67531/metadc849626/.

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In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.

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Li, Xining. "Preservation of bounded geometry under transformations metric spaces." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439309722.

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Estep, Dewey. "Prime End Boundaries of Domains in Metric Spaces and the Dirichlet Problem." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439295199.

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Thompson, Scotty L. "Comparing Topological Spaces Using New Approaches to Cleavability." View abstract, 2009. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:3372574.

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Lesser, Alice. "Optimal and Hereditarily Optimal Realizations of Metric Spaces." Doctoral thesis, Uppsala University, Department of Mathematics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8297.

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This PhD thesis, consisting of an introduction, four papers, and some supplementary results, studies the problem of finding an optimal realization of a given finite metric space: a weighted graph which preserves the metric's distances and has minimal total edge weight. This problem is known to be NP-hard, and solutions are not necessarily unique.

It has been conjectured that extremally weighted optimal realizations may be found as subgraphs of the hereditarily optimal realization Γ_d, a graph which in general has a higher total edge weight than the optimal realization but has the advantages of being unique, and possible to construct explicitly via the tight span of the metric.

In Paper I, we prove that the graph Γ_d is equivalent to the 1-skeleton of the tight span precisely when the metric considered is totally split-decomposable. For the subset of totally split-decomposable metrics known as consistent metrics this implies that Γ_d is isomorphic to the easily constructed Buneman graph.

In Paper II, we show that for any metric on at most five points, any optimal realization can be found as a subgraph of Γ_d.

In Paper III we provide a series of counterexamples; metrics for which there exist extremally weighted optimal realizations which are not subgraphs of Γ_d. However, for these examples there also exists at least one optimal realization which is a subgraph.

Finally, Paper IV examines a weakened conjecture suggested by the above counterexamples: can we always find some optimal realization as a subgraph in Γ_d? Defining extremal optimal realizations as those having the maximum possible number of shortest paths, we prove that any embedding of the vertices of an extremal optimal realization into Γ_d is injective. Moreover, we prove that this weakened conjecture holds for the subset of consistent metrics which have a 2-dimensional tight span

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Silwal, Sharad Deep. "Harnack's inequality in spaces of homogeneous type." Diss., Kansas State University, 2012. http://hdl.handle.net/2097/14189.

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Doctor of Philosophy
Department of Mathematics
Diego Maldonado
Originally introduced in 1961 by Carl Gustav Axel Harnack [36] in the context of harmonic functions in R[superscript]2, the so-called Harnack inequality has since been established for solutions to a wide variety of different partial differential equations (PDEs) by mathematicians at different times of its historical development. Among them, Moser's iterative scheme [47-49] and Krylov-Safonov's probabilistic method [43, 44] stand out as pioneering theories, both in terms of their originality and their impact on the study of regularity of solutions to PDEs. Caffarelli's work [12] in 1989 greatly simplified Krylov-Safonov's theory and established Harnack's inequality in the context of fully non-linear elliptic PDEs. In this scenario, Caffarelli and Gutierrez's study of the linearized Monge-Ampere equation [15, 16] in 2002-2003 served as a motivation for axiomatizations of Krylov-Safonov-Caffarelli theory [3, 25, 57]. The main work in this dissertation is a new axiomatization of Krylov-Safonov-Caffarelli theory. Our axiomatic approach to Harnack's inequality in spaces of homogeneous type has some distinctive features. It sheds more light onto the role of the so-called critical density property, a property which is at the heart of the techniques developed by Krylov and Safonov. Our structural assumptions become more natural, and thus, our theory better suited, in the context of variational PDEs. We base our method on the theory of Muckenhoupt's A[subscript]p weights. The dissertation also gives an application of our axiomatic approach to Harnack's inequality in the context of infinite graphs. We provide an alternate proof of Harnack's inequality for harmonic functions on graphs originally proved in [21].

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Kaczynski, Tomasz. "Topological transversality of condensing set-valued maps." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=73995.

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39

Zhang, Tan. "Manifolds with indefinite metrics whose skew-symmetric curvature operator has constant eigenvalues /." view abstract or download file of text, 2000. http://wwwlib.umi.com/cr/uoregon/fullcit?p9978605.

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Thesis (Ph. D.)--University of Oregon, 2000.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 123-128). Also available for download via the World Wide Web; free to University of Oregon users.

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40

Zhang, Tan 1969. "Manifolds with indefinite metrics whose skew-symmetric curvature operator has constant eigenvalues." Thesis, University of Oregon, 2000. http://hdl.handle.net/1794/150.

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Adviser: Peter B. Gilkey. ix, 128 leaves
A print copy of this title is available through the UO Libraries under the call number: MATH QA613 .Z43 2000
Relative to a non-degenerate metric of signature (p, q), an algebraic curvature tensor is said to be IP if the associated skew-symmetric curvature operator R(π) has constant eigenvalues and if the kernel of R(π) has constant dimension on the Grassmanian of non-degenerate oriented 2-planes. A pseudo-Riemannian manifold with a non-degenerate indefinite metric of signature (p, q) is said to be IP if the curvature tensor of the Levi-Civita connection is IP at every point; the eigenvalues are permitted to vary with the point. In the Riemannian setting (p, q) = (0, m), the work of Gilkey, Leahy, and Sadofsky and the work of Ivanov and Petrova have classified the IP metrics and IP algebraic curvature tensors if the dimension is at least 4 and if the dimension is not 7. We use techniques from algebraic topology and from differential geometry to extend some of their results to the Lorentzian setting (p, q) = (1, m – 1) and to the setting of metrics of signature (p, q) = (2, m – 2).

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41

Shchur, Vladimir. "Quasi-isometries between hyperbolic metric spaces, quantitative aspects." Phd thesis, Université Paris Sud - Paris XI, 2013. http://tel.archives-ouvertes.fr/tel-00867709.

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In this thesis we discuss possible ways to give quantitative measurement for two spaces not being quasi-isometric. From this quantitative point of view, we reconsider the definition of quasi-isometries and propose a notion of ''quasi-isometric distortion growth'' between two metric spaces. We revise our article [32] where an optimal upper-bound for Morse Lemma is given, together with the dual variant which we call Anti-Morse Lemma, and their applications.Next, we focus on lower bounds on quasi-isometric distortion growth for hyperbolic metric spaces. In this class, $L^p$-cohomology spaces provides useful quasi-isometry invariants and Poincaré constants of balls are their quantitative incarnation. We study how Poincaré constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincaré constants for locally homogeneous metrics of the form $dt^2+\sum_ie^{2\mu_it}dx_i^2$, and give a lower bound on quasi-isometric distortion growth among such spaces.This allows us to give examples of different quasi-isometric distortion growths, including a sublinear one (logarithmic).

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42

Siebert, Kitzeln B. "A modern presentation of "dimension and outer measure"." Columbus, Ohio : Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1211395297.

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43

Peske, Wendy Ann. "A topological approach to nonlinear analysis." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2779.

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A topological approach to nonlinear analysis allows for strikingly beautiful proofs and simplified calculations. This topological approach employs many of the ideas of continuous topology, including convergence, compactness, metrization, complete metric spaces, uniform spaces and function spaces. This thesis illustrates using the topological approach in proving the Cauchy-Peano Existence theorem. The topological proof utilizes the ideas of complete metric spaces, Ascoli-Arzela theorem, topological properties in Euclidean n-space and normed linear spaces, and the extension of Brouwer's fixed point theorem to Schauder's fixed point theorem, and Picard's theorem.

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44

Silva, Camila Tolin Santos da. "Descobrindo a Topologia : um compêndio de fundamentos teóricos e atividades lúdicas para auxiliar na formalização de conceitos topológicos no ensino básico /." Universidade Estadual Paulista (UNESP), 2018. http://hdl.handle.net/11449/155883.

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Submitted by CAMILA TOLIN SANTOS DA SILVA (camilatolin@yahoo.com.br) on 2018-08-30T19:02:42Z No. of bitstreams: 1 Camila Tolin - Dissertação mestrado versão final.pdf: 53772690 bytes, checksum: e820d5f4112aab671ce28d3bd47eab57 (MD5)
Rejected by Elza Mitiko Sato null (elzasato@ibilce.unesp.br), reason: Solicitamos que realize correções na submissão seguindo as orientações abaixo: Problema 01) É necessário que o arquivo contendo a sua dissertação esteja no formato PDF (Portable Document Format) e não esteja protegido . Problema 02) A ficha catalográfica deve ser corrigida pois nela consta: ”Instituto de Biociências, Letras e Ciências Exatas”, Presidente Prudente, o nome do Câmpus de Presidente Prudente é Faculdade de Ciências e Tecnologia. Problema 03) Solicito que corrija também na descrição na natureza da pesquisa na folha de rosto e aprovação: Dissertação apresentada como parte dos requisitos para obtenção do título de Mestre, junto ao programa de Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, da Faculdade de Ciências e Tecnologia da Universidade Estadual Paulista Júlio de Mesquita Filho , Câmpus de Presidente Prudente. Problema 04) O arquivo contém 12(doze) páginas em branco (pags. 04, 06, 08, 10, 12, 14, 24, 42, 52, 82, 98 e 138) as mesmas devem retiradas pois o arquivo não pode conter páginas em branco. e deve ser corrigido também na ficha catalográfica o número de folhas. Lembramos que o arquivo depositado no repositório deve ser igual ao impresso, o rigor com o padrão da Universidade se deve ao fato de que o seu trabalho passará a ser visível mundialmente. Agradecemos a compreensão. on 2018-08-31T13:19:11Z (GMT)
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A topologia é um ramo da matemática, sutilmente entrelaçado com a Geometria, de aplicação em diversas áreas do conhecimento, cuja conceituação foi apresentada de forma expressiva nas escolas durante as décadas de 60 e 70, com o movimento educacional conhecido como Matemática Moderna. Através das mudanças curriculares, muitos temas abordados no ensino fundamental e médio foram reestruturados dentro de um conjunto de parâmetros para a organização curricular da base nacional comum, os PCN's, que normatizam a base do ensino e orientam que a matemática deve ser apresentada para o desenvolvimento de habilidades inerentes à resolução de problemas, aquisição de linguagem simbólica, modelagem e interpretação de situações cotidianas, argumentação e aplicação em situações da vida real. Portanto, esse trabalho foi elaborado com o objetivo de fornecer suporte para o ensino da topologia no ensino básico, através da compilação de fatos históricos, formalização de definições básicas de caráter introdutório como continuidade, espaços métricos, espaços topológicos, entre outros, apresentação de atividades que poderão ser trabalhadas conjuntamente com o ensino da geometria, que de forma lúdica e intuitiva, ajudarão a alicerçar a base para um futuro aprofundamento desses conceitos, auxiliando no desenvolvimento do pensamento topológico.
Topology is a branch of mathematics, subtly intertwined with geometry, of application in several areas of knowledge, whose conceptualization was presented expressively in schools during the 60s and 70s, with the educational movement known as Modern Mathematics. Through the curricular changes, many topics addressed in elementary and secondary education have been restructured within a set of parameters for the curriculum organization of the common national base, the PCNs, that normalize the base of the teaching and guide that the mathematics must be presented for the development of inherent abilities to solve problems, acquisition of symbolic language, modeling and interpretation of everyday situations, argumentation and application in real life situations. Thus, this work was developed with the purpose of providing support for the teaching of topology in basic education, through the compilation of historical facts, formalization of basic de nitions of introductory character such as continuity, metric spaces, topological spaces, among others, presentation of activities which can be worked together with the teaching geometry, which in a playful and intuitive way, will help to lay the foundation for a future deepening of these concepts, aiding in the development of topological thinking.

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45

Chen, Li. "Quasi transformées de Riesz, espaces de Hardy et estimations sous-gaussiennes du noyau de la chaleur." Phd thesis, Université Paris Sud - Paris XI, 2014. http://tel.archives-ouvertes.fr/tel-01001868.

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Dans cette thèse nous étudions les transformées de Riesz et les espaces de Hardy associés à un opérateur sur un espace métrique mesuré. Ces deux sujets sont en lien avec des estimations du noyau de la chaleur associé à cet opérateur. Dans les Chapitres 1, 2 et 4, on étudie les transformées quasi de Riesz sur les variétés riemannienne et sur les graphes. Dans le Chapitre 1, on prouve que les quasi transformées de Riesz sont bornées dans Lp pour 1

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46

Bettinelli, Jérémie. "Limite d'échelle de cartes aléatoires en genre quelconque." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00638065.

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Au cours de ce travail, nous nous intéressons aux limites d'échelle de deux classes de cartes. Dans un premier temps, nous regardons les quadrangulations biparties de genre strictement positif g fixé et, dans un second temps, les quadrangulations planaires à bord dont la longueur du bord est de l'ordre de la racine carrée du nombre de faces. Nous voyons ces objets comme des espaces métriques, en munissant leurs ensembles de sommets de la distance de graphe, convenablement renormalisée. Nous montrons qu'une carte prise uniformément parmi les cartes ayant n faces dans l'une de ces deux classes tend en loi, au moins à extraction près, vers un espace métrique limite aléatoire lorsque n tend vers l'infini. Cette convergence s'entend au sens de la topologie de Gromov--Hausdorff. On dispose de plus des informations suivantes sur l'espace limite que l'on obtient. Dans le premier cas, c'est presque sûrement un espace de dimension de Hausdorff 4 homéomorphe à la surface de genre g. Dans le second cas, c'est presque sûrement un espace de dimension 4 avec une frontière de dimension 2, homéomorphe au disque unité de R^2. Nous montrons en outre que, dans le second cas, si la longueur du bord est un petit~o de la racine carrée du nombre de faces, on obtient la même limite que pour les quadrangulations sans bord, c'est-à-dire la carte brownienne, et l'extraction n'est plus requise.

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47

Triestino, Michele. "La dynamique des difféomorphismes du cercle selon le point de vue de la mesure." Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2014. http://tel.archives-ouvertes.fr/tel-01065468.

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Les travaux de ma thèse s'articulent en trois parties distinctes.Dans la première partie j'étudie les mesures de Malliavin-Shavguldize sur les difféomorphismes du cercle et de l'intervalle. Il s'agit de mesures de type " Haar " pour ces groupes de dimension infinie : elles furent introduites il a une vingtaine d'années pour permettre une étude de leur théorie des représentations. Un premier chapitre est dédié à recueillir les résultats présents dans la littérature et et les représenter dans une forme plus étendue, avec un regard particulier sur les propriétés de quasi-invariance de ces mesures. Ensuite j'étudie de problèmes de nature plus dynamique : quelle est la dynamique qu'on doit s'attendre d'un difféomorphisme choisi uniformément par rapport à une mesure de Malliavin-Shavguldize ? Je démontre en particulier qu'il y a une forte présence des difféomorphismes de type Morse-Smale.La partie suivante vient de mon premier travail publié, obtenu en collaboration avec Andrés Navas. Inspirés d'un théorème récent de Avila et Kocsard sur l'unicité des distributions invariantes par un difféomorphisme lisse minimal du cercle, nous analysons le même problème en régularité faible, avec des argument plus géométriques.La dernière partie est constituée des résultats récemment obtenus avec Mikhail Khristoforov et Victor Kleptsyn. Nous abordons les problèmes reliés à la gravité quantique de Liouville en étudiant des espaces auto-similaires qui sont la limite de graphes finis. Nous démontrons qu'il est possible de trouver des distances aléatoires non-triviales sur ces espaces qui sont compatibles avec la structure auto-similaire.

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48

Lins, Brian C. "Asymptotic behavior and Denjoy-Wolff theorems for Hilbert metric nonexpansive maps." 2007. http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.16723.

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49

Saradhi, P. V. S. P. "Sets of periods of continuous self maps on some metric spaces." Thesis, 1997. http://hdl.handle.net/2009/1097.

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50

Balakumar, G. P. "Rigidity And Regularity Of Holomorphic Mappings." Thesis, 2012. http://etd.iisc.ernet.in/handle/2005/2447.

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We deal with two themes that are illustrative of the rigidity and regularity of holomorphic mappings. The first one concerns the regularity of continuous CR mappings between smooth pseudo convex, finite type hypersurfaces which is a well studied subject for it is linked with the problem of studying the boundary behaviour of proper holomorphic mappings between domains bounded by such hypersurfaces. More specifically, we study the regularity of Lipschitz CR mappings from an h-extendible(or semi-regular) hypersurface in Cn .Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudo convex domains is also proved. The second theme dealt with, is the classification upto biholomorphic equivalence of model domains with abelian automorphism group in C3 .It is shown that every model domain i.e.,a hyperbolic rigid polynomial domainin C3 of finite type, with abelian automorphism group is equivalent to a domain that is balanced with respect to some weight.

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Dissertations / Theses on the topic 'Mathematics. Metric spaces. Mappings (Mathematics)'

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