Dissertations / Theses on the topic 'Measure metric space'
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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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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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Meizis, Roland [Verfasser], and Anita [Akademischer Betreuer] Winter. "Metric two-level measure spaces : a state space for modeling evolving genealogies in host-parasite systems / Roland Meizis ; Betreuer: Anita Winter." Duisburg, 2019. http://d-nb.info/1191693414/34.
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Malý, Lukáš. "Sobolev-Type Spaces : Properties of Newtonian Functions Based on Quasi-Banach Function Lattices in Metric Spaces." Doctoral thesis, Linköpings universitet, Matematik och tillämpad matematik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-105616.
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This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike
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Enflo, Karin. "Measures of Freedom of Choice." Doctoral thesis, Uppsala universitet, Avdelningen för praktisk filosofi, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-179078.
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This thesis studies the problem of measuring freedom of choice. It analyzes the concept of freedom of choice, discusses conditions that a measure should satisfy, and introduces a new class of measures that uniquely satisfy ten proposed conditions. The study uses a decision-theoretical model to represent situations of choice and a metric space model to represent differences between options. The first part of the thesis analyzes the concept of freedom of choice. Different conceptions of freedom of choice are categorized into evaluative and non-evaluative, as well as preference-dependent and preference-independent kinds. The main focus is on the three conceptions of freedom of choice as cardinality of choice sets, representativeness of the universal set, and diversity of options, as well as the three conceptions of freedom of rational choice, freedom of eligible choice, and freedom of evaluated choice. The second part discusses the conceptions, together with conditions for a measure and a variety of measures proposed in the literature. The discussion mostly focuses on preference-independent conceptions of freedom of choice, in particular the diversity conception. Different conceptions of diversity are discussed, as well as properties that could affect diversity, such as the cardinality of options, the differences between the options, and the distribution of differences between the options. As a result, the diversity conception is accepted as the proper explication of the concept of freedom of choice. In addition, eight conditions for a measure are accepted. The conditions concern domain-insensitivity, strict monotonicity, no-choice situations, dominance of differences, evenness, symmetry, spread of options, and limited function growth. None of the previously proposed measures satisfy all of these conditions. The third part concerns the construction of a ratio-scale measure that satisfies the accepted conditions. Two conditions are added regarding scale-independence and function growth proportional to cardinality. Lastly, it is shown that only one class of measures satisfy all ten conditions, given an additional assumption that the measures should be analytic functions with non-zero partial derivatives with respect to some function of the differences. These measures are introduced as the Ratio root measures.
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Davtyan, Ashot. "Measure generation in the spaces of planes und lines in R^3." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2009. http://nbn-resolving.de/urn:nbn:de:swb:105-7072226.
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Das Ziel der Arbeit besteht darin, einen Beitrag zur Entwicklung der kombinatorischen Integralgeometrie zu leisten. In der Arbeit werden Bewertungen (Valuation) in den Räumen der Geraden und Ebenen im $\R^3$ betrachtet, die von Flagfunktionen abhängen. Unter geeigneten Glattheitsvoraussetzungen an die Flagfunktionen werden notwendige und hinreichende Bedingungen gegeben, die die Fortsetzung der entsprechender Bewertung zu einem signierten Maß sichern. Diese integralgeometrischen Untersuchungen führten zu einer Anzahl von interessanten Ergebnissen, speziell bei der Beschreibung von Metriken im Sinne von Hilberts viertem Problem.
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Jones, Rebekah. "A characterization of quasiconformal maps in terms of sets of finite perimeter." University of Cincinnati / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1560867563841096.
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Malý, Lukáš. "Newtonian Spaces Based on Quasi-Banach Function Lattices." Licentiate thesis, Linköpings universitet, Matematik och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-79166.
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The traditional first-order analysis in Euclidean spaces relies on the Sobolev spaces W1,p(Ω), where Ω ⊂ Rn is open and p ∈ [1, ∞].The Sobolev norm is then defined as the sum of Lp norms of a function and its distributional gradient.We generalize the notion of Sobolev spaces in two different ways. First, the underlying function norm will be replaced by the “norm” of a quasi-Banach function lattice. Second, we will investigate functions defined on an abstract metric measure space and that is why the distributional gradients need to be substituted. The thesis consists of two papers. The first one builds up the elementary theory of Newtonian spaces based on quasi-Banach function lattices. These lattices are complete linear spaces of measurable functions with a topology given by a quasinorm satisfying the lattice property. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces, where the role of weak derivatives is passed on to upper gradients. Tools such asmoduli of curve families and the Sobolev capacity are developed, which allows us to study basic properties of the Newtonian functions.We will see that Newtonian spaces can be equivalently defined using the notion of weak upper gradients, which increases the number of techniques available to study these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are also established. The second paper in the thesis then continues with investigation of properties of Newtonian spaces based on quasi-Banach function lattices. The set of all weak upper gradients of a Newtonian function is of particular interest.We will prove that minimalweak upper gradients exist in this general setting.Assuming that Lebesgue’s differentiation theoremholds for the underlyingmetricmeasure space,wewill find a family of representation formulae. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.
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Alleche, Boualem. "Quelques résultats sur la consonance, les multi-applications, et la séquentialité." Rouen, 1996. http://www.theses.fr/1996ROUES027.
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Cette thèse se subdivise en trois parties, la première traite les hyper-espaces. Apres l'introduction et le développement de certains résultats récents sur la consonance, nous introduisons l'idée originale qui consiste à utiliser les mesures de Radon. Nous démontrons que tout espace consonant est pré-Radon. Nous obtenons que la droite de Sorgenfrey n'est pas consonante, que tout ultra-filtre libre sur les entiers regarde comme un sous-espace du Cantor est non consonant, et nous donnons un espace métrisable séparable héréditairement de Baire non consonant. Cette idée a également inspiré un peu plus tard d'autres mathématiciens pour démontrer la non-consonance de l'ensemble des rationnels. Nous généralisons ensuite un résultat obtenu par M. Arab et J. Calbrix sur la coïncidence de la consonance et l'hyperconsonance. Dans la deuxième partie, nous généralisons le théorème de E. Michael de double sélection multivoque et nous obtenons que tout espace Cech-complet sous-métrisable est sélecteur par rapport aux espaces paracompacts. Nous démontrons que la frontière active d'une multi-application S. C. S. D'un espace compact métrisable dans un espace métrisable est, elle aussi, S. C. S. , et nous trouvons le lien que pour les espaces métrisables séparables co-analytiques, la consonance équivaut à être sélecteur par rapport à l'espace de Cantor. Dans la dernière partie, nous étudions les espaces séquentiels. Nous donnons un théorème sur une certaine classe d'applications quotients et nous obtenons que le produit de n copies du fan séquentiel est séquentiel et d'ordre séquentiel égal à n.
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Belili, Nacereddine. "Problèmes des marges et de transport." Rouen, 1998. http://www.theses.fr/1998ROUES022.
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Cette thèse comprend trois parties, Dans la première partie, on donne une synthèse du théorème de dualité relatif au problème des marges, ses diverses applications comme le théorème de Strassen, la caractérisation de l'ordre stochastique et la représentation des métriques minimales. On donne une preuve du théorème de Goldstein basée sur la représentation de la distance de variation totale. Dans la seconde partie, on considère une suite ( µn de probabilités sur Rd convergeant étroitement vers une probabilité µ absolument continue par rapport à la mesure de Lebesgue. On suppose que µn et µ admettent un moment d'ordre p1. On montre l'existence d'une suite de variables aléatoires {(Xn,X)} à valeurs dans Rd × Rd telles que Xn = n(X), où n: Rd -> Rd est c-cycliquement monotone, Xn converge presque-sûrement vers X et où chaque couple (Xn,X) est c-optimal pour ( µn, µ ). Dans la dernière partie, en collaboration avec H. Heinich, nous donnons des propriétés des probabilités qui vérifient la propriété de transport. En particulier, nous examinons le cas des probabilités fortement diffuses. Nous étudions la relation entre la dérivabilité d'une fonction réelle f et le fait que la probabilité L o f*-1 vérifie la propriété de transport, où L est mesure de Lebesgue et la fonction f*(x) := (x,f(x)).
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Calisti, Matteo. "Differential calculus in metric measure spaces." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21781/.
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L'obbiettivo di questa tesi è la definizione del calcolo differenziale e dell'operatore di Laplace in spazi metrici di misura. Nel primo capitolo vengono introdotte le definizioni e proprietà principali degli spazi metrici di misura mentre nel secondo quelle riguardanti le funzioni lipschitziane e la derivata metrica di curve assolutamente continue. Nel terzo capitolo quindi viene definito il concetto di p-supergradiente debole e di conseguenza la classe di Sobolev S^p. Nel quarto capitolo viene poi studiata la generalizzazione del concetto di differenziale di f applicato al gradiente di g che da luogo a due funzioni che in generale risultano diverse, ma se coincidono lo spazio verrà detto q-infinitesimamente strettamente convesso.
Vengono quindi dimostrate alcune regole della catena per per queste due funzioni attraverso la dualità fra lo spazio S^p e un opportuno spazio di misure dette q-piani test. In particolare mediante l'introduzione del funzionale energia di Cheeger e il suo flusso-gradiente sarà possibile associare un piano di trasporto al gradiente di una funzione in S^p. Nel quinto capitolo viene definito il p-laplaciano e le regole di calcolo provate precedentemente saranno usate per provare quelle per il laplaciano. Verranno poi definiti gli spazi infitesimamente di Hilbert: in questo caso il laplaciano assume un solo valore e risulta linearmente dipendente da g e si dimostra un'identificazione tra differenziali e gradienti. Nell'ultima parte del quinto capitolo infine viene mostrata un'applicazione del calcolo differenziale in spazi metrici di misura al gruppo di Heisenberg, considerandolo uno spazio metrico di misura munito della metrica di Korany e la misura di Lebesgue. Nella prima parte si mostra che il laplaciano metrico coincide con quello subriemanniano. Viene poi considerata nella seconda parte la sottovarietà {x=0} e si dimostra come il laplaciano metrico sia diverso da quello differenziale.
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Capolli, Marco. "Selected Topics in Analysis in Metric Measure Spaces." Doctoral thesis, Università degli studi di Trento, 2021. http://hdl.handle.net/11572/288526.
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The thesis is composed by three sections, each devoted to the study of a specific problem in the setting of PI spaces. The problem analyzed are: a C^m Lusin approximation result for horizontal curves in the Heisenberg group, a limit result in the spirit of Burgain-Brezis-Mironescu for Orlicz-Sobolev spaces in Carnot groups and the differentiability of Lipschitz functions in Laakso spaces.
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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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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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Kopfer, Eva [Verfasser]. "Heat flows on time-dependent metric measure spaces / Eva Kopfer." Bonn : Universitäts- und Landesbibliothek Bonn, 2018. http://d-nb.info/1160594120/34.
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Palmer, Ian Christian. "Riemannian geometry of compact metric spaces." Diss., Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/34744.
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A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the
space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.
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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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Han, Bang-Xian. "Analyse dans les espaces métriques mesurés." Thesis, Paris 9, 2015. http://www.theses.fr/2015PA090014/document.
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Cette thèse traite de plusieurs sujets d'analyse dans les espaces métriques mesurés, en lien avec le transport optimal et des conditions de courbure-dimension. Nous considérons en particulier les équations de continuité dans ces espaces, du point de vue de fonctionnelles continues sur les espaces de Sobolev, et du point de vue de la dualité avec les courbes absolument continues dans l'espace de Wasserstein. Sous une condition de courbure-dimension, mais sans condition de doublement de mesure ou d'inégalité de Poincaré, nous montrons également l'identification des p-gradients faibles. Nous étudions ensuite les espaces de Sobolev sur le produit tordu de l'ensemble des réels et d'un espace métrique mesuré. En particulier, nous montrons la propriété Sobolev-à-Lipschitz sous une certaine condition de courbure-dimension. Enfin, sous une telle condition et dans le cadre d'une théorie non-lisse de Bakry-Emery, nous obtenons une inégalité améliorée de Bochner et proposons une définition du N-tenseur de RicciThis thesis concerns in some topics on calculus in metric measure spaces, in connection with optimal transport theory and curvature-dimension conditions. We study the continuity equations on metric measure spaces, in the viewpoint of continuous functionals on Sobolev spaces, and in the viewpoint of the duality with respect to absolutely continuous curves in the Wasserstein space. We study the Sobolev spaces of warped products of a real line and a metric measure space. We prove the 'Pythagoras theorem' for both cartesian products and warped products, and prove Sobolev-to-Lipschitz property for warped products under a certain curvature-dimension condition. We also prove the identification of p-weak gradients under curvature-dimension condition, without the doubling condition or local Poincaré inequality. At last, using the non-smooth Bakry-Emery theory on metric measure spaces, we obtain a Bochner inequality and propose a definition of N-Ricci tensor
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Carlsson, Niclas. "Markov chains on metric spaces : invariant measures and asymptotic behaviour /." Åbo : Åbo akademi university, 2005. http://catalogue.bnf.fr/ark:/12148/cb400328312.
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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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Suzuki, Kohei. "Convergence of stochastic processes on varying metric spaces." 京都大学 (Kyoto University), 2016. http://hdl.handle.net/2433/215281.
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Tewodrose, David. "Some functional inequalities and spectral properties of metric measure spaces with curvature bounded below." Thesis, Paris Sciences et Lettres (ComUE), 2018. http://www.theses.fr/2018PSLEE076.
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L’objectif de la thèse est de présenter de nouveaux résultats d’analyse sur les espaces métriques mesurés. Nous étendons d’abord à une certaine classe d’espaces avec doublement et Poincaré des inégalités de Sobolev pondérées introduites par V. Minerbe en 2009 dans le cadre des variétés riemanniennes à courbure de Ricci positives. Dans le contexte des espaces RCD(0,N), nous en déduisons une inégalité de Nash pondérée et un contrôle uniforme du noyau de la chaleur pondéré associé. Puis nous démontrons la loi de Weyl sur les espaces RCD(K,N) compactes à l’aide d’un théorème de convergence ponctuelle des noyaux de la chaleur associés à une suite mGH-convergente d’espaces RCD(K,N). Enfin nous abordons dans le contexte RCD(K,N) un théorème de Bérard, Besson et Gallot fournissant, à l’aide du noyau de la chaleur, une famille de plongements asymptotiquement isométriques d’une variété riemannienne fermée dans l’espace de ses fonctions de carré intégrable. Nous introduisons notamment les notions de métrique RCD, de métrique pull-back, et de convergence faible/forte de métriques RCD sur un espace RCD(K,N) compacte, et nous prouvons un résultat de convergence analogue à celui de Bérard, Besson et GallotThe aim of this thesis is to present new results in the analysis of metric measure spaces. We first extend to a certain class of spaces with doubling and Poincaré some weighted Sobolev inequalities introduced by V. Minerbe in 2009 in the context of Riemannian manifolds with non-negative Ricci curvature. In the context of RCD(0,N) spaces, we deduce a weighted Nash inequality and a uniform control of the associated weighted heat kernel. Then we prove Weyl’s law for compact RCD(K,N) spaces thanks to a pointwise convergence theorem for the heat kernels associated with a mGH-convergent sequence of RCD(K,N) spaces. Finally we address in the RCD(K,N) context a theorem from Bérard, Besson and Gallot which provides, by means of the heat kernel, an asymptotically isometric family of embeddings for a closed Riemannian manifold into its space of square integrable functions. We notably introduce the notions of RCD metrics, pull-back metrics, weak/strong convergence of RCD metrics, and we prove a convergence theorem analog to the one of Bérard, Besson and Gallot
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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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Profeta, Angelo [Verfasser]. "Gluing of metric measure spaces and the heat equation with homogeneous Dirichlet boundary values / Angelo Profeta." Bonn : Universitäts- und Landesbibliothek Bonn, 2020. http://d-nb.info/1218301848/34.
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Herán, Andreas [Verfasser], Jens [Akademischer Betreuer] Habermann, and Jens [Gutachter] Habermann. "Existence and Regularity Results for Parabolic Problems on Metric Measure Spaces / Andreas Herán ; Gutachter: Jens Habermann ; Betreuer: Jens Habermann." Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2020. http://d-nb.info/1218785721/34.
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Collins, Michael [Verfasser], Jens [Akademischer Betreuer] Habermann, and Jens [Gutachter] Habermann. "Existence for Variational Solutions to Cauchy-Dirichlet Problems on Metric Measure Spaces / Michael Collins ; Gutachter: Jens Habermann ; Betreuer: Jens Habermann." Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2020. http://d-nb.info/121973683X/34.
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Simmer, Jan [Verfasser], Olaf [Akademischer Betreuer] Post, and Olaf [Gutachter] Post. "Approximation of energy forms on finitely ramified fractals by discrete graphs and related metric measure spaces / Jan Simmer ; Gutachter: Olaf Post ; Betreuer: Olaf Post." Trier : Universität Trier, 2021. http://d-nb.info/1230135057/34.
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Huou, Benoit. "Inégalités isopérimétriques produit pour les élargissements euclidien et uniforme : symétrisation et inégalités fonctionnelles." Thesis, Toulouse 3, 2016. http://www.theses.fr/2016TOU30239/document.
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Le problème isopérimétrique consiste, dans un espace métrique mesuré, à trouver les ensembles qui, à volume fixé, ont la plus petite mesure de surface. Il peut être formulé dans de nombreux cadres (espaces métriques mesurés généraux, variétés riemanniennes à poids, parties de l'espace euclidien...). Deux questions se dégagent de ce problème : - Quels sont les ensembles solutions, c'est-à-dire ayant la plus petite mesure de surface ? (Il faut noter que ces ensembles n'existent pas toujours). - Que vaut la plus petite mesure de surface ? La solution à la deuxième question peut être formulée sous la forme d'une fonction, appelée profil isopérimétrique, qui, à une valeur de volume (pondéré) donnée, associe la plus petite mesure de surface correspondante. La notion de mesure de surface, quant à elle, peut être définie de plusieurs manières (contenu de Minkowski, périmètre géométrique...), toutes dépendant étroitement à la fois de la distance et de la mesure ambiantes. L'objet principal de cette thèse est l'étude du problème isopérimétrique dans des espaces produits, que ce soit pour transférer des inégalités isopérimétriques d'espaces facteurs vers ces produits, ou pour comparer le profil isopérimétrique de l'espace produit à ceux des facteurs. La thèse se découpe en quatre parties : - Étude de l'opération de symétrisation (pour les ensembles) et de réarrangement (pour les fonctions), notions analogues, du point de vue de la théorie de la mesure géométrique et des fonctions à variations bornée. Ces opérations agissent de sorte à ce que n'augmente pas la mesure de surface (pour les ensembles), ou la variation (pour les fonctions). Nous introduisons notamment une nouvelle classe d'espaces modèles, pour lesquels nous obtenons des résultats qualitativement similaires à ceux obtenus pour les espaces modèles classiques : inégalités isopérimétriques transférées aux produits, comparaison d'énergies (pour des fonctionnelles convexes). - Détail d'un argument de minoration du profil isopérimétrique d'un espace métrique produit XxY par une fonction dépendant des profils de X et Y, pour une large classe de distances produits sur XxY. L'étude de ce problème est faite via la minimisation d'une fonctionnelle sur la classe des mesures de Radon. - Étude du problème isopérimétrique dans un espace métrique mesuré produit (le produit d'ordre quelconque du même espace métrique mesuré), muni de la combinaison uniforme de sa distance (élargissement uniforme). Nous donnons un critère pour que tous les profils isopérimétriques (quel que soit l'ordre d'itération du produit) soient minorés par un multiple du minorant du profil isopérimétrique de l'espace originel. Ceci est fait en utilisant notamment des méthodes ayant trait aux inégalités fonctionnelles. Nous appliquons ensuite les résultats aux influences géométriques. - Étude d'inégalités fonctionnelles dites isopérimétriques, permettant d'appréhender le comportement isopérimétrique dans l'espace produit correspondant d'ordre quelconque. Nous résumons l'état des connaissances à propos des inégalités de ce type et proposons une autre méthode qui pourrait aboutir à prouver une telle inégalité dans le cas de mesures réelles particulières, pour lesquelles le problème est ouvertThe isoperimetric problem in a metric measured space consists in finding the sets having minimal boundary measure, with prescribed volume. It can be formulated in various settings (general metric measured spaces, Riemannian manifolds, submanifolds of the Euclidean space, ...). At this point, two questions arise : - What are the optimal sets, namely the sets having smallest boundary measure (it has to be said that they do not always exist) ? - What is the smallest boundary measure ? The solution to the second answer can be expressed by a function called the isoperimetric profile. This function maps a value of (prescribed) measure onto the corresponding smallest boundary measure. As for the precise notion of boundary measure, it can be defined in different ways (Minkowski content, geometric perimeter, ...), all of them closely linked to the ambient distance and measure. The main object of this thesis is the study of the isoperimetric problem in product spaces, in order to transfer isoperimetric inequalities from factor spaces to the product spaces, or to compare their isoperimetric profiles. The thesis is divided into four parts : - Study of the symmetrization operation (for sets) and the rearrangement operation (for functions), analogous notions, from the point of view of Geometric Measure Theory and Bounded Variation functions. These operations cause the boundary measure to decrease (for sets), or the variation (for functions). We introduce a new class of model spaces, for which we obtain similar results to those concerning classic model spaces : transfer of isoperimetric inequalities to the product spaces, energy comparison (for convex functionals). - Detailed proof of an argument of minorization of the isoperimetric profile of a metric measured product space XxY by a function depending on the profiles of X and Y, for a wide class of product distances over XxY. The study of this problem uses the minimization of a functional defined on Radon measures class. - Study of the isoperimetric problem in a metric measured space (n times the same space) equipped with the uniform combination of its distance (uniform enlargement). We give a condition under which every isoperimetric profile (whatever the order of iteration might be) is bounded from below by a quantity which is proportional to the isoperimetric profile of the underlying space. We then apply the result to geometric influences. - Study of isoperimetric functional inequalities, which give information about the isoperimetric behavior of the product spaces. We give an overview of the results about this kind of inequalities, and suggest a method to prove such an inequality in a particular case of real measures for which the problem reamins open
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Ivana, Štajner-Papuga. "Uopštena konvolucija." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2001. https://www.cris.uns.ac.rs/record.jsf?recordId=5987&source=NDLTD&language=en.
Full textAbstract:
U ovoj tezi je definisana uopštena konvolucija koja pripada domenu pseudo-analize i ima veliku primenu u mnogim matematičkim teorijama, npr. u proba-bilističkim metričkim prostorima, PDJ, teorijama odlučivanja, sistema, kontrole i fazi brojeva. Dokazane su bitne osobine ove operacije sa funkcijama. Dokazana je veza izmedju pseudo-konvolucija baziranih na poluprstenima različitih klasa Definisana je (5, C/)-konvolucija bazirana na uslovno distributivnom poluprstenu ([0,1], S, U)).Dat je još jedan vid uopštenja konvolucije baziran na uopštenim pseudo-operacijama.In this thesis the generalized convolution have been defined. This operation with functions has applications in different mathematical theo ries, for example in Probabilistic Metric Spaces, PDE, System and Control Theory, Fuzzy numbers. Some basic properties of this operation has been proved, as well as connection between generalized convolutions based on different classes of semirings. (5, U)-convolution has been defined, as well as convolution based on generalized pseudo-operations.
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Muzellec, Boris. "Leveraging regularization, projections and elliptical distributions in optimal transport." Electronic Thesis or Diss., Institut polytechnique de Paris, 2020. http://www.theses.fr/2020IPPAG009.
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Pouvoir manipuler et de comparer de mesures de probabilité est essentiel pour de nombreuses applications en apprentissage automatique. Le transport optimal (TO) définit des divergences entre distributions fondées sur la géométrie des espaces sous-jacents : partant d'une fonction de coût définie sur l'espace dans lequel elles sont supportées, le TO consiste à trouver un couplage entre les deux mesures qui soit optimal par rapport à ce coût. Par son ancrage géométrique, le TO est particulièrement bien adapté au machine learning, et fait l'objet d'une riche théorie mathématique. En dépit de ces avantages, l'emploi du TO pour les sciences des données a longtemps été limité par les difficultés mathématiques et computationnelles liées au problème d'optimisation sous-jacent. Pour contourner ce problème, une approche consiste à se concentrer sur des cas particuliers admettant des solutions en forme close, ou pouvant se résoudre efficacement. En particulier, le TO entre mesures elliptiques constitue l'un des rares cas pour lesquels le TO admet une forme close, définissant la géométrie de Bures-Wasserstein (BW). Cette thèse s'appuie tout particulièrement sur la géométrie de BW, dans le but de l'utiliser comme outil de base pour des applications en sciences des données. Pour ce faire, nous considérons des situations dans lesquelles la géométrie de BW est tantôt utilisée comme un outil pour l'apprentissage de représentations, étendue à partir de projections sur des sous-espaces, ou régularisée par un terme entropique. Dans une première contribution, la géométrie de BW est utilisée pour définir des plongements sous la forme de distributions elliptiques, étendant la représentation classique sous forme de vecteurs de R^d. Dans une deuxième contribution, nous prouvons l'existence de transports qui extrapolent des applications restreintes à des projections en faible dimension, et montrons que ces plans "sous-espace optimaux" admettent des formes closes dans le cas de mesures gaussiennes. La troisième contribution de cette thèse consiste à obtenir des formes closes pour le transport entropique entre des mesures gaussiennes non-normalisées, qui constituent les premières expressions non triviales pour le transport entropique. Finalement, dans une dernière contribution nous utilisons le transport entropique pour imputer des données manquantes de manière non-paramétrique, tout en préservant les distributions sous-jacentesComparing and matching probability distributions is a crucial in numerous machine learning (ML) algorithms. Optimal transport (OT) defines divergences between distributions that are grounded on geometry: starting from a cost function on the underlying space, OT consists in finding a mapping or coupling between both measures that is optimal with respect to that cost. The fact that OT is deeply grounded in geometry makes it particularly well suited to ML. Further, OT is the object of a rich mathematical theory. Despite those advantages, the applications of OT in data sciences have long been hindered by the mathematical and computational complexities of the underlying optimization problem. To circumvent these issues, one approach consists in focusing on particular cases that admit closed-form solutions or that can be efficiently solved. In particular, OT between elliptical distributions is one of the very few instances for which OT is available in closed form, defining the so-called Bures-Wasserstein (BW) geometry. This thesis builds extensively on the BW geometry, with the aim to use it as basic tool in data science applications. To do so, we consider settings in which it is alternatively employed as a basic tool for representation learning, enhanced using subspace projections, and smoothed further using entropic regularization. In a first contribution, the BW geometry is used to define embeddings as elliptical probability distributions, extending on the classical representation of data as vectors in R^d.In the second contribution, we prove the existence of transportation maps and plans that extrapolate maps restricted to lower-dimensional projections, and show that subspace-optimal plans admit closed forms in the case of Gaussian measures.Our third contribution consists in deriving closed forms for entropic OT between Gaussian measures scaled with a varying total mass, which constitute the first non-trivial closed forms for entropic OT and provide the first continuous test case for the study of entropic OT. Finally, in a last contribution, entropic OT is leveraged to tackle missing data imputation in a non-parametric and distribution-preserving way
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Arnt, Sylvain. "Large scale geometry and isometric affine actions on Banach spaces." Thesis, Orléans, 2014. http://www.theses.fr/2014ORLE2021/document.
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Dans le premier chapitre, nous définissons la notion d’espaces à partitions pondérées qui généralise la structure d’espaces à murs mesurés et qui fournit un cadre géométrique à l’étude des actions isométriques affines sur des espaces de Banach pour les groupes localement compacts à base dénombrable. Dans un premier temps, nous caractérisons les actions isométriques affines propres sur des espaces de Banach en termes d’actions propres par automorphismes sur des espaces à partitions pondérées. Puis, nous nous intéressons aux structures de partitions pondérées naturelles pour les actions de certaines constructions de groupes : somme directe ; produit semi-directe ; produit en couronne et produit libre. Nous établissons ainsi des résultats de stabilité de la propriété PLp par ces constructions. Notamment, nous généralisons un résultat de Cornulier, Stalder et Valette de la façon suivante : le produit en couronne d’un groupe ayant la propriété PLp par un groupe ayant la propriété de Haagerup possède la propriété PLp. Dans le deuxième chapitre, nous nous intéressons aux espaces métriques quasi-médians - une généralisation des espaces hyperboliques à la Gromov et des espaces médians - et à leurs propriétés. Après l’étude de quelques exemples, nous démontrons qu’un espace δ-médian est δ′-médian pour tout δ′ ≥ δ. Ce résultat nous permet par la suite d’établir la stabilité par produit directe et par produit libre d’espaces métriques - notion que nous développons par la même occasion. Le troisième chapitre est consacré à la définition et l’étude d’une distance propre, invariante à gauche et qui engendre la topologie explicite sur les groupes localement compacts, compactement engendrés. Après avoir montré les propriétés précédentes, nous prouvons que cette distance est quasi-isométrique à la distance des mots sur le groupe et que la croissance du volume des boules est contrôlée exponentiellementIn the first chapter, we define the notion of spaces with labelled partitions which generalizes the structure of spaces with measured walls : it provides a geometric setting to study isometric affine actions on Banach spaces of second countable locally compact groups. First, we characterise isometric affine actions on Banach spaces in terms of proper actions by automorphisms on spaces with labelled partitions. Then, we focus on natural structures of labelled partitions for actions of some group constructions : direct sum ; semi-direct product ; wreath product and free product. We establish stability results for property PLp by these constructions. Especially, we generalize a result of Cornulier, Stalder and Valette in the following way : the wreath product of a group having property PLp by a Haagerup group has property PLp. In the second chapter, we focus on the notion of quasi-median metric spaces - a generalization of both Gromov hyperbolic spaces and median spaces - and its properties. After the study of some examples, we show that a δ-median space is δ′-median for all δ′ ≥ δ. This result gives us a way to establish the stability of the quasi-median property by direct product and by free product of metric spaces - notion that we develop at the same time. The third chapter is devoted to the definition and the study of an explicit proper, left-invariant metric which generates the topology on locally compact, compactly generated groups. Having showed these properties, we prove that this metric is quasi-isometric to the word metric and that the volume growth of the balls is exponentially controlled
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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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Chizat, Lénaïc. "Transport optimal de mesures positives : modèles, méthodes numériques, applications." Thesis, Paris Sciences et Lettres (ComUE), 2017. http://www.theses.fr/2017PSLED063/document.
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L'objet de cette thèse est d'étendre le cadre théorique et les méthodes numériques du transport optimal à des objets plus généraux que des mesures de probabilité. En premier lieu, nous définissons des modèles de transport optimal entre mesures positives suivant deux approches, interpolation et couplage de mesures, dont nous montrons l'équivalence. De ces modèles découle une généralisation des métriques de Wasserstein. Dans une seconde partie, nous développons des méthodes numériques pour résoudre les deux formulations et étudions en particulier une nouvelle famille d'algorithmes de "scaling", s'appliquant à une grande variété de problèmes. La troisième partie contient des illustrations ainsi que l'étude théorique et numérique, d'un flot de gradient de type Hele-Shaw dans l'espace des mesures. Pour les mesures à valeurs matricielles, nous proposons aussi un modèle de transport optimal qui permet un bon arbitrage entre fidélité géométrique et efficacité algorithmiqueThis thesis generalizes optimal transport beyond the classical "balanced" setting of probability distributions. We define unbalanced optimal transport models between nonnegative measures, based either on the notion of interpolation or the notion of coupling of measures. We show relationships between these approaches. One of the outcomes of this framework is a generalization of the p-Wasserstein metrics. Secondly, we build numerical methods to solve interpolation and coupling-based models. We study, in particular, a new family of scaling algorithms that generalize Sinkhorn's algorithm. The third part deals with applications. It contains a theoretical and numerical study of a Hele-Shaw type gradient flow in the space of nonnegative measures. It also adresses the case of measures taking values in the cone of positive semi-definite matrices, for which we introduce a model that achieves a balance between geometrical accuracy and algorithmic efficiency
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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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Maitra, Sayantan. "The Space of Metric Measure Spaces." Thesis, 2017. http://etd.iisc.ernet.in/2005/3588.
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This thesis is broadly divided in two parts. In the first part we give a survey of various distances between metric spaces, namely the uniform distance, Lipschitz distance, Hausdor distance and the Gramoz Hausdor distance. Here we talk about only the most basic of their properties and give a few illustrative examples. As we wish to study collections of metric measure spaces, which are triples (X; d; m) consisting of a complete separable metric space (X; d) and a Boral probability measure m on X, there are discussions about some distances between them. Among the three that we discuss, the transportation and distortion distances were introduced by Sturm. The later, denoted by 2, on the space X2 of all metric measure spaces having finite L2-size is the focus of the second part of this thesis.
The second part is an exposition based on the work done by Sturm. Here we prove a number of results on the analytic and geometric properties of (X2; 2). Beginning by noting that (X2; 2) is a non-complete space, we try to understand its completion. Towards this end, the notion of a gauged measure space is useful. These are triples (X; f; m) where X is a Polish space, m a Boral probability measure on X and f a function, also called a gauge, on X X that is symmetric and square integral with respect to the product measure m2. We show that,
Theorem 1. The completion of (X2; 2) consists of all gauged measure spaces where the gauges satisfy triangle inequality almost everywhere. We denote the space of all gauged measure spaces by Y. The space X2 can be embedded in Y and the transportation distance 2 extends easily from X2 to Y. These two spaces turn out to have similar geometric properties.
On both these spaces 2 is a strictly intrinsic metric; i.e. any two members in them can be joined by a shortest path. But more importantly, using a description of the geodesics in these spaces, the following result is proved.
Theorem 2. Both (X2; 2) and (Y; 2) have non-negative curvature in the sense of Alexandrov.
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Lee, Ji Shiang, and 李吉翔. "A note on volume comparison theorem on smooth metric measure space." Thesis, 2015. http://ndltd.ncl.edu.tw/handle/03875326045218760072.
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碩士國立清華大學
數學系
103
Let (Mn,g,e−fdv) be a smooth metric measure space with Bakry-´Emery curvature bounded below, we introduce the volume comparison theorem on such man ifold. If the weighted function is of linear growth or of quadratic growth, we study the volume upper and lower bound estimate of a geodesic ball on M.
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Davtyan, Ashot. "Measure generation in the spaces of planes und lines in R^3." Doctoral thesis, 2001. https://tubaf.qucosa.de/id/qucosa%3A22373.
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Das Ziel der Arbeit besteht darin, einen Beitrag zur Entwicklung der kombinatorischen Integralgeometrie zu leisten. In der Arbeit werden Bewertungen (Valuation) in den Räumen der Geraden und Ebenen im $\R^3$ betrachtet, die von Flagfunktionen abhängen. Unter geeigneten Glattheitsvoraussetzungen an die Flagfunktionen werden notwendige und hinreichende Bedingungen gegeben, die die Fortsetzung der entsprechender Bewertung zu einem signierten Maß sichern. Diese integralgeometrischen Untersuchungen führten zu einer Anzahl von interessanten Ergebnissen, speziell bei der Beschreibung von Metriken im Sinne von Hilberts viertem Problem.
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Ulikowska, Agnieszka. "Structured Population Models in Metric Spaces." Doctoral thesis, 2013. https://depotuw.ceon.pl/handle/item/388.
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The main goal of this thesis is the analysis of a wide class of structured population models in the space of finite, nonnegative Radon measures equipped with the flat metric. This framework allows a unified approach to a variety of problems providing them with basic well-posedness and stability results. The first result is the existence and uniqueness of measure valued solutions to the one-sex structured population model. A nonlinear semigroup is constructed here by means of the operator splitting algorithm. This technique allows to separate the differential operator from the integral one, which leads to a significant simplification of proofs. Concerning stability, the Lipschitz continuity of solutions with respect to the model coefficients is provided. The next analytical result is the well-posedness of the age-structured two-sex population model. Existence and uniqueness of the measure valued solutions is proved by the regularization technique as well as the stability estimates. A brief discussion on a marriage function, which is the main source of the nonlinearity in this model, is carried out and an example of the marriage function fitting into the considered framework is given. The second part of this thesis is devoted to a development of numerical methods for a particular class of one-sex structured population models. The first method is constructed through the splitting technique and corresponds with a current trend basing on a kinetic approach to the population dynamics problems. Separation of a semigroup induced by the transport operator from a semigroup induced by the nonlocal term allows to keep the solution as a sum of Dirac deltas despite of the regularizing character of the nonlocal boundary condition. As the next step, two alternative methods based on different approximations of the boundary condition are analyzed. These are the Escalator Boxcar Train algorithm and its simplification. Convergence of both methods is proved exploiting the concept of semiflows on metric spaces. Last but not least, the rate of convergence for all schemes mentioned above is provided.Celem naukowym niniejszej rozprawy jest analiza matematyczna dynamiki modeli strukturalnych w przestrzeniach metrycznych. Modele strukturalne opisują ewolucję populacji organizmów, zróżnicowanej ze względu na wybrane cechy. Cechy te zależą od modelowanej populacji, mogą być to, między innymi, wiek lub rozmiar osobnika, dojrzałość pojedynczej komórki, stan jej zróżnicowania lub fenotyp. Przestrzenią metryczną, w której analizujemy równania dynamiki populacyjnej jest przestrzeń skończonych, nieujemnych miar Radona z metryką flat. Nasze wyniki dotyczą między innymi istnienia i jednoznaczności miarowych rozwiązań dla szerokiej klasy modeli ze strukturą. W szczególności, rozpatrujemy modele mające zastosowanie w demografii, biologii i epidemiologii. Otrzymane rezultaty gwarantują także stabilność rozwiązań względem współczynników modelu, co bezpośrednio przekłada się na możliwość tworzenia stabilnych schematów numerycznych. Budowa takich schematów, opartych na metodzie cząstek i algorytmie split-up oraz ich zastosowanie do wyżej wymienionych modeli jest istotnym elementem tejże rozprawy
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Bandara, Lashi. "Geometry and the Kato square root problem." Phd thesis, 2013. http://hdl.handle.net/1885/10690.
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The primary focus of this thesis is to consider Kato square root problems for various divergence-form operators on manifolds. This is the study of perturbations of second-order differential operators by bounded, complex, measurable coefficients. In general, such operators are not self-adjoint but uniformly elliptic. The Kato square root problem is then to understand when the square root of such an operator, which exists due to uniform ellipticity, is comparable to its unperturbed counterpart. A remarkably adaptable operator-theoretic framework due to Axelsson, Keith and McIntosh sits in the background of this work. This framework allows us to take a powerful first-order perspective of the problems which we consider in a geometric setting. Through a well established procedure, we reduce these problems to the study of quadratic estimates. Under a set of natural conditions, we prove quadratic estimates for a class of operators on vector bundles over complete measure metric spaces. The first kind of estimates we prove are global, and we establish them on trivial vector bundles when the underlying measure grows at most polynomially. The second kind are local, and there, we allow the vector bundle to be non-trivial but bounded in an appropriate sense. Here, the measure is allowed to grow exponentially. An important consequence of obtaining quadratic estimates on measure metric spaces is that it allows us to consider subelliptic operators on Lie groups. The first-order perspective allows us to reduce the subelliptic problem to a fully elliptic one on a sub-bundle. As a consequence, we are able to solve a homogeneous Kato square root problem for perturbations of subelliptic operators on nilpotent Lie groups. For general Lie groups we solve a similar inhomogeneous problem.
In the situation of complete Riemannian manifolds, we consider uniformly elliptic divergence-form operators arising from connections on vector bundles. Under a set of assumptions, we show that the Kato square root problem can be solved for such operators. As a consequence, we solve this problem on functions under the condition that the Ricci curvature and injectivity radius are bounded. Assuming an additional lower bound for the curvature endomorphism on forms, we solve a similar problem for perturbations of inhomogeneous Hodge-Dirac operators. A theorem for tensors is obtained by additionally assuming boundedness of a second-order Riesz transform. Motivated by the study of these Kato problems, where for technical reasons it is useful to know the density of compactly supported functions in the domains of operators, we study connections and their divergence on a vector bundle. Through a first-order formulation, we show that this density property holds for the domains of these operators if the metric and connection are compatible and the underlying manifold is complete. We also show that compactly supported functions are dense in the second-order Sobolev space on complete manifolds under the sole assumption that the Ricci curvature is bounded below, improving a result that previously required an additional lower bound on the injectivity radius.
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Chen, Yi-Lin, and 陳義麟. "The Function Theory on Complete Smooth Metric Measure Spaces." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/01949315834612010881.
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碩士國立清華大學
數學系
100
In this note, we introduce the gradient estimate on smooth metric measure spaces with nonnegative Bakry-\'Emery Ricci curvature. We also apply such estimate to prove Liouville type theorem and splitting theorem in geometric partial differential equations.
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Dai, Feng-Chih, and 戴夆池. "A Note on Complete Smooth Metric Measure Spaces with Nonnegative Curvature." Thesis, 2018. http://ndltd.ncl.edu.tw/handle/2d7e99.
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Sosa, Garciamarín Gerardo. "On symmetric transformations in metric measured geometry." Doctoral thesis, 2017. https://ul.qucosa.de/id/qucosa%3A16754.
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The central objects of study in this thesis are metric measure spaces. These are metric spaces which are endowed with a reference measure and enriched with basic topological, geometric and measure theoretical properties. The objective of the first part of the work is to characterize metric measure spaces whose symmetry groups admit a differential structure making them Lie groups. The second part is concerned with the analysis of the induced geometry of spaces admitting non-trivial symmetries. More in detail, it is shown that in many cases synthetic notions of Ricci curvature lower bounds are inherited by quotient spaces.
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Divakaran, D. "Compactness Theorems for The Spaces of Distance Measure Spaces and Riemann Surface Laminations." Thesis, 2014. http://hdl.handle.net/2005/3131.
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Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the Gromov-Hausdorff distance, is a theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with the generalised Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple (X, d,µ), where (X, d) forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and µ is a finite Borel measure.
Using this result we prove that the Deligne-Mumford compactification is the completion of the moduli space of Riemann surfaces under the generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov compactness theorem for J-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology, its applicability is limited by the lack of general methods to construct pseudo-holomorphic curves. One hopes that considering a more general class of objects in place of pseudo-holomorphic curves will be useful. Generalising the domain of pseudo-holomorphic curves from Riemann surfaces to Riemann surface laminations is a natural choice. Theorems such as the uniformisation theorem for surface laminations by Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and topological classification of “almost all" leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations, as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation, we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.
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Malý, Lukáš. "Prostory Sobolevova typu na metrických prostorech s mírou." Doctoral thesis, 2014. http://www.nusl.cz/ntk/nusl-342330.
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Title: Sobolev-Type Spaces on Metric Measure Spaces Author: RNDr. Lukáš Malý Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Luboš Pick, CSc., DSc., Department of Mathematical Analysis Abstract: is thesis focuses on function spaces related to rst-order analysis in abstract metric measure spaces. In metric spaces, we can replace distributional gra- dients, whose de nition depends on the linear structure of Rn , by upper gradients that control the functions' behavior along all recti able curves. is gives rise to the so-called Newtonian spaces. e summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid- s. Standard toolbox for the theory is set up in this general setting and Newto- nian spaces are proven complete. Summability of an upper gradient of a function is shown to guarantee the function's absolute continuity on almost all curves. Ex- istence of a unique minimal weak upper gradient is established. Regularization of Newtonian functions via Lipschitz truncations is discussed in doubling Poincaré spaces using weak boundedness of maximal...
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Luckhardt, Daniel. "Benjamini-Schramm Convergence of Normalized Characteristic Numbers of Riemannian Manifolds." Doctoral thesis, 2018. http://hdl.handle.net/21.11130/00-1735-0000-0005-1388-C.
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Amenta, Alex. "Extensions of the theory of tent spaces and applications to boundary value problems." Phd thesis, 2016. http://hdl.handle.net/1885/102564.
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We extend the theory of tent spaces from Euclidean spaces to
various types of metric measure spaces. For doubling spaces we
show that the usual 'global' theory remains valid, and for
'non-uniformly locally doubling' spaces (including R^n with the
Gaussian measure) we establish a satisfactory local theory. In
the doubling context we show that
Hardy–Littlewood–Sobolev-type embeddings hold in the scale of
weighted tent spaces, and in the special case of unbounded
AD-regular metric measure spaces we identify the real
interpolants (the 'Z-spaces') of weighted tent spaces.
Weighted tent spaces and Z-spaces on R^n are used to construct
Hardy–Sobolev and Besov spaces adapted to perturbed Dirac
operators. These spaces play a key role in the classification of
solutions to first-order Cauchy–Riemann systems (or
equivalently, the classification of conormal gradients of
solutions to second-order elliptic systems) within weighted tent
spaces and Z-spaces. We establish this classification, and as a
corollary we obtain a useful characterisation of well-posedness
of Regularity and Neumann problems for second-order
complex-coefficient elliptic systems with boundary data in
Hardy–Sobolev and Besov spaces of order s ∈ (−1, 0).
47
Kuncová, Kristýna. "Neabsolutně konvergentní integrály." Doctoral thesis, 2019. http://www.nusl.cz/ntk/nusl-408083.
Full textAbstract:
Title: Nonabsolutely convergent integrals Author: Krist'yna Kuncov'a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jan Mal'y, DrSc., Department of Mathematical Analysis Abstract: In this thesis we develop the theory of nonabsolutely convergent Hen- stock-Kurzweil type packing integrals in different spaces. In the framework of metric spaces we define the packing integral and the uniformly controlled inte- gral of a function with respect to metric distributions. Applying the theory to the notion of currents we then prove a generalization of the Stokes theorem. In Rn we introduce the packing R and R∗ integrals, which are defined as charges - additive functionals on sets of bounded variation. We provide comparison with miscellaneous types of integrals such as R and R∗ integral in Rn or MCα integral in R. On the real line we then study a scale of integrals based on the so called p-oscillation. We show that our indefinite integrals are a.e. approximately differ- entiable and we give comparison with other nonabsolutely convergent integrals. Keywords: Nonabsolutely convergent integrals, BV sets, Henstock-Kurzweil in- tegral, Divergence theorem, Analysis in metric measure spaces 1