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Törnblom, Arvid. "Measure theory, fractal geometry and their applications on digital sundials." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-435354.
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Childress, Scot Paul. "Quantum measures, arithmetic coils, and generalized fractal strings." Diss., UC access only, 2009. http://proquest.umi.com/pqdweb?index=128&did=1871850181&SrchMode=1&sid=1&Fmt=7&retrieveGroup=0&VType=PQD&VInst=PROD&RQT=309&VName=PQD&TS=1270491013&clientId=48051.
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Thesis (Ph. D.)--University of California, Riverside, 2009.Includes abstract. Includes bibliographical references (leaves 202-204) and index. Issued in print and online. Available via ProQuest Digital Dissertations.
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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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Poosapadi, Arjunan Sridhar, and sridhar arjunan@rmit edu au. "Fractal features of Surface Electromyogram: A new measure for low level muscle activation." RMIT University. Electrical and Computer Engineering, 2009. http://adt.lib.rmit.edu.au/adt/public/adt-VIT20090629.095851.
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Identifying finger and wrist flexion based actions using single channel surface electromyogram have a number of rehabilitation, defence and human computer interface applications. These applications are currently infeasible because of unreliability in classification of sEMG when the level of muscle contraction is low and when there are multiple active muscles. The presence of noise and cross-talk from closely located and simultaneously active muscles is exaggerated when muscles are weakly active such as during maintained wrist and finger flexion. It has been established in literature that surface electromyogram (sEMG) and other such biosignals are fractal signals. Some researchers have determined that fractal dimension (FD) is related to strength of muscle contraction. On careful analysis of fractal properties of sEMG, this research work has established that FD is related to the muscle size and complexity and not to the strength of muscle contraction. The work has also identified a novel feature, maximum fractal length (MFL) of the signal, as a good measure of strength of contraction of the muscle. From the analysis, it is observed that while at high level of contraction, root mean square (RMS) is an indicator of strength of contraction of the muscle, this relationship is not very strong when the muscle contraction is less than 50% maximum voluntary contraction. This work has established that MFL is a more reliable measure of strength of contraction compared to RMS, especially at low levels of contraction. This research work reports the use of fractal properties of sEMG to identify the small changes in strength of muscle contraction and the location of the active muscles. It is observed that fractal dimension (FD) of the signal is related with the properties of the muscle while maximum fractal length (MFL) is related to the strength of contraction of the associated muscle. The results show that classifying MFL and FD of a single channel sEMG from the forearm it is possible to accurately identify a set of finger and wrist flexion based actions even when the muscle activity is very weak. It is proposed that such a system could be used to control a prosthetic hand or for human computer interface.
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Leifsson, Patrik. "Fractal sets and dimensions." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-7320.
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Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.
In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.
A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.
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Simonini, Marina. "Fractal sets and their applications in medicine." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/8763/.
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La geometria euclidea risulta spesso inadeguata a descrivere le forme della natura. I Frattali, oggetti interrotti e irregolari, come indica il nome stesso, sono più adatti a rappresentare la forma frastagliata delle linee costiere o altri elementi naturali. Lo strumento necessario per studiare rigorosamente i frattali sono i teoremi riguardanti la misura di Hausdorff, con i quali possono definirsi gli s-sets, dove s è la dimensione di Hausdorff. Se s non è intero, l'insieme in gioco può riconoscersi come frattale e non presenta tangenti e densità in quasi nessun punto. I frattali più classici, come gli insiemi di Cantor, Koch e Sierpinski, presentano anche la proprietà di auto-similarità e la dimensione di similitudine viene a coincidere con quella di Hausdorff. Una tecnica basata sulla dimensione frattale, detta box-counting, interviene in applicazioni bio-mediche e risulta utile per studiare le placche senili di varie specie di mammiferi tra cui l'uomo o anche per distinguere un melanoma maligno da una diversa lesione della cute.
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Inui, Kanji. "Study of the fractals generated by contractive mappings and their dimensions." Kyoto University, 2020. http://hdl.handle.net/2433/253370.
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Kyoto University (京都大学)0048
新制・課程博士
博士(人間・環境学)
甲第22534号
人博第937号
新制||人||223(附属図書館)
2019||人博||937(吉田南総合図書館)
京都大学大学院人間・環境学研究科共生人間学専攻
(主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史
学位規則第4条第1項該当
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Wang, Nancy. "Fractal Sets: Dynamical, Dimensional and Topological Properties." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-233147.
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Fractals is a relatively new mathematical topic which received thorough treatment only starting with 1960's. Fractals can be observed everywhere in nature and in day-to-day life. To give a few examples, common fractals are the spiral cactus, the romanesco broccoli, human brain and the outline of the Swedish map. Fractal dimension is a dimension which need not take integer values. In fractal geometry, a fractal dimension is a ratio providing an index of the complexity of fractal pattern with regard to how the local geometry changes with the scale at which it is measured. In recent years, fractal analysis is used increasingly in many areas of engineering and technology. Among others, fractal analysis is used in signal and image compression, computer and video design, neuroscience and fractal based cancer modelling and diagnosing. This study consists of two main parts. The first part of the study aims to understand the appearance of an irregular Cantor set generated by the chaotic dynamical system generated by the logistic function on the unit interval [0,1]. In order to understand this irregular Cantor set, we studied the topological properties of the Cantor Middle-thirds set and the generalised Cantor sets, all of which have zero length. The necessity to compare these sets with regard to their size led us to the second part of this paper, namely the dimension studies of fractals. More complex fractals were presented in the second part, three definitions of dimension were introduced. The fractal dimension of the irregular Cantor set generated by the logistic mapping was estimated and we found that the Hausdorff dimension has the widest scope and greatest flexibility in the fractal studies.Fraktaler är ett relativt nytt ämne inom matematik som fick sitt stora genomslag först efter 60-talet. En fraktal är ett självliknande mönster med struktur i alla skalor. Några vardagliga exempel på fraktaler är spiralkaktus, romanescobroccoli, mänskliga hjärnan, blodkärlen och Sveriges fastlandskust. Bråktalsdimension är en typ av dimension där dimensionsindexet tillåts att anta alla icke-negativa reella tal. Inom fraktalgeometri kan dimensionsindexet betraktas som ett komplexitetsindex av mönstret med avseende på hur den lokala geometrin förändras beroende på vilken skala mönstret betraktas i. Under det senaste decenniet har fraktalanalysen använts alltmer flitigt inom tekniska och vetenskapliga tillämpningar. Bland annat har fraktalanalysen använts i signal- och bildkompression, dator- och videoformgivning, neurovetenskap och fraktalbaserad cancerdiagnos. Denna studie består av två huvuddelar. Den första delen fokuserar på att förstår hur en fraktal kan uppstå i ett kaotiskt dynamiskt system. För att vara mer specifik studerades den logistiska funktionen och hur denna ickelinjära avbildning genererar en oregelbunden Cantormängd på intervalet [0,1]. Vidare, för att förstå den oregelbundna Cantormängden studerades Cantormängden (eng. the Cantor Middle-Thirds set) och de generaliserade Cantormängderna, vilka alla har noll längd. För att kunna jämföra de olika Cantormängderna med avseende på storlek, leds denna studie vidare till dimensionsanalys av fraktaler som är huvudämnet i den andra delen av denna studie. Olika topologiska fraktaler presenterades, tre olika definitioner av dimension introducerades, bland annat lådräkningsdimensionen och Hausdorffdimensionen. Slutligen approximerades dimensionen av den oregelbundna Cantormängden med hjälp av Hausdorffdimensionen. Denna studie demonstrerar att Hausdorffdimensionen har större omfattning och mer flexibilitet för fraktalstudier.
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Farkas, Ábel. "Dimension and measure theory of self-similar structures with no separation condition." Thesis, University of St Andrews, 2015. http://hdl.handle.net/10023/7854.
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We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.
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Boore, Graeme C. "Directed graph iterated function systems." Thesis, University of St Andrews, 2011. http://hdl.handle.net/10023/2109.
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This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].
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Haderka, Jan. "Využití fraktální a harmonické analýzy k charakterizaci fyzikálně chemických dějů." Doctoral thesis, Vysoké učení technické v Brně. Fakulta chemická, 2010. http://www.nusl.cz/ntk/nusl-233298.
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Existuje mnoho různých způsobů jak analyzovat disperzní systémy a fyzikálně chemické processy ke kterým v takových systémech dochází. Tato práce byla zaměřena na charakterizaci těchto procesů pomocí metod harmonické fraktální analýzy. Obrazová data sledovaných systémů byly analyzovány pomocí waveletové analýzy. V průběhu práce byly navrženy různé optimalizace samotné analýzy, převážně zaměřené na odstranění manuálních operací během analýzy a tyto optimalizace byly také inkorporovány do softérového vybavení pro Harmonickou Fraktální Analýzu HarFA, který je vyvíjen na Fakultě chemické, VUT Brno.
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Yin, Qinghe. "Fractals and sumsets." Title page, contents and abstract only, 1993. http://web4.library.adelaide.edu.au/theses/09PH/09phy51.pdf.
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Le, Huy. "Numerické metody měření fraktálních dimenzí a fraktálních měr." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2020. http://www.nusl.cz/ntk/nusl-417160.
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Tato diplomová práce se zabývá teorií fraktálů a popisuje patričné potíže při zavedení pojmu fraktál. Dále se v práci navrhuje několik metod, které se použijí na aproximaci fraktálních dimenzí různých množin zobrazených na zařízeních s konečným rozlišením. Tyto metody se otestují na takových množinách, jejichž dimenze známe, a na závěr se výsledky porovnávají.
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Beliaev, Dmitri. "Harmonic measure on random fractals." Doctoral thesis, Stockholm, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-114.
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Alrud, Beng Oscar. "Fractal spectral measures in two dimensions." Doctoral diss., University of Central Florida, 2011. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/4834.
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We study spectral properties for invariant measures associated to affine iterated function systems. We present various conditions under which the existence of a Hadamard pair implies the existence of a spectrum for the fractal measure. This solves a conjecture proposed by Dorin Dutkay and Palle Jorgensen, in several special cases in dimension 2.ID: 030422913; System requirements: World Wide Web browser and PDF reader.; Mode of access: World Wide Web.; Thesis (Ph.D.)--University of Central Florida, 2011.; Includes bibliographical references (p. 75-76).
Ph.D.
Doctorate
Mathematics
Sciences
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McClure, Mark. "Fractal measures on infinite-dimensional sets /." The Ohio State University, 1994. http://rave.ohiolink.edu/etdc/view?acc_num=osu148785391310164.
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Mullins, Edmond N. "Derivation bases, interval functions, and fractal measures /." The Ohio State University, 1996. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487942182325914.
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Chen, Hongjing. "Accuracy of fractal and multifractal measures for signal analysis." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp04/mq23247.pdf.
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Xiao, Yimin. "Fractal Measures and Related Properties of Gaussian Random Fields /." The Ohio State University, 1996. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487935573769563.
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Shmerkin, Pablo. "The structure of overlapping self-affine sets /." Thesis, Connect to this title online; UW restricted, 2006. http://hdl.handle.net/1773/5791.
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Shaw, Donald B. "Classification of transmitter transients using fractal measures and probabilistic neural networks." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp04/mq23494.pdf.
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Dubarry, Blandine. "Comportement asymptotique des systèmes de fonctions itérées et applications aux chaines de Markov d'ordre variable." Thesis, Rennes 1, 2017. http://www.theses.fr/2017REN1S114/document.
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L'objet de cette thèse est l'étude du comportement asymptotique des systèmes de fonctions itérées (IFS). Dans un premier chapitre, nous présenterons les notions liées à l'étude de tels systèmes et nous rappellerons différentes applications possibles des IFS telles que les marches aléatoires sur des graphes ou des pavages apériodiques, les systèmes dynamiques aléatoires, la classification de protéines ou encore les mesures quantiques répétées. Nous nous attarderons sur deux autres applications : les chaînes de Markov d'ordre infini et d'ordre variable. Nous donnerons aussi les principaux résultats de la littérature concernant l'étude des mesures invariantes pour des IFS ainsi que ceux pour le calcul de la dimension de Hausdorff. Le deuxième chapitre sera consacré à l'étude d'une classe d'IFS composés de contractions sur des intervalles réels fermés dont les images se chevauchent au plus en un point et telles que les probabilités de transition sont constantes par morceaux. Nous donnerons un critère pour l'existence et pour l'unicité d'une mesure invariante pour l'IFS ainsi que pour la stabilité asymptotique en termes de bornes sur les probabilités de transition. De plus, quand il existe une unique mesure invariante et sous quelques hypothèses techniques supplémentaires, on peut montrer que la mesure invariante admet une dimension de Hausdorff exacte qui est égale au rapport de l'entropie sur l'exposant de Lyapunov. Ce résultat étend la formule, établie dans la littérature pour des probabilités de transition continues, au cas considéré ici des probabilités de transition constantes par morceaux. Le dernier chapitre de cette thèse est, quant à lui, consacré à un cas particulier d'IFS : les chaînes de Markov de longueur variable (VLMC). On démontrera que sous une condition de non-nullité faible et de continuité pour la distance ultramétrique des probabilités de transitions, elles admettent une unique mesure invariante qui est attractive pour la convergence faibleThe purpose of this thesis is the study of the asymptotic behaviour of iterated function systems (IFS). In a first part, we will introduce the notions related to the study of such systems and we will remind different applications of IFS such as random walks on graphs or aperiodic tilings, random dynamical systems, proteins classification or else $q$-repeated measures. We will focus on two other applications : the chains of infinite order and the variable length Markov chains. We will give the main results in the literature concerning the study of invariant measures for IFS and those for the calculus of the Hausdorff dimension. The second part will be dedicated to the study of a class of iterated function systems (IFSs) with non-overlapping or just-touching contractions on closed real intervals and adapted piecewise constant transition probabilities. We give criteria for the existence and the uniqueness of an invariant probability measure for the IFSs and for the asymptotic stability of the system in terms of bounds of transition probabilities. Additionally, in case there exists a unique invariant measure and under some technical assumptions, we obtain its exact Hausdorff dimension as the ratio of the entropy over the Lyapunov exponent. This result extends the formula, established in the literature for continuous transition probabilities, to the case considered here of piecewise constant probabilities. The last part is dedicated to a special case of IFS : Variable Length Markov Chains (VLMC). We will show that under a weak non-nullness condition and continuity for the ultrametric distance of the transition probabilities, they admit a unique invariant measure which is attractive for the weak convergence
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Eroglu, Kemal Ilgar. "Self-similar sets, projections and arithmetic sums /." Thesis, Connect to this title online; UW restricted, 2007. http://hdl.handle.net/1773/5800.
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Jang, Eric. "Compression of fingerprints based on wavelet packet decomposition and fractal singularity measures." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/mq23354.pdf.
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Dansereau, Richard M. "Progressive image transmission using fractal and wavelet techniques with image complexity measures." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/NQ57505.pdf.
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Fraser, Jonathan M. "Dimension theory and fractal constructions based on self-affine carpets." Thesis, University of St Andrews, 2013. http://hdl.handle.net/10023/3869.
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The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.
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Lebid, Mykola [Verfasser]. "Fractal analysis of singularly continuous measures generated by Cantor series expansions / Mykola Lebid." Bielefeld : Universitätsbibliothek Bielefeld, 2015. http://d-nb.info/1078112460/34.
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Yassierli, Yassierli. "Muscle Fatigue during Isometric and Dynamic Efforts in Shoulder Abduction and Torso Extension: Age Effects and Alternative Electromyographic Measures." Diss., Virginia Tech, 2005. http://hdl.handle.net/10919/29509.
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Aging has been associated with numerous changes in the neuromuscular system. Age effects on muscular performance, however, have been addressed only in limited contexts in earlier research. The present work was conducted primarily to investigate age-related effects on muscle capacity (fatigue and endurance) during isometric and dynamic efforts. This work was also motivated by current theories on muscle fatigue as a potential risk factor for musculoskeletal disorders and recent demographic projections indicating a substantial increase of older adults in the working population. Four main experiments were conducted to investigate development of muscle fatigue during isometric and intermittent efforts in shoulder abduction and torso extension at different contraction levels. Two age groups were involved (n=24 in each), representing the beginning and end of working life. Findings from this study demonstrated that the older group exhibited slower progressions of fatigue, though the age effect was more consistent for the shoulder than the torso muscles. This implied a muscle dependency of the influence of age on fatigue. Several interaction effects of age and effort level were also observed, suggesting that both task and individual factors should be considered simultaneously in job design.
The present investigation also sought to develop alternative electromyography (EMG)-based fatigue parameters for low-level isometric and dynamic contractions, two areas in which improvements are needed in the sensitivity and reliability of existing EMG indices. Several alternative EMG indices were introduced, derived from logarithmic transformation of EMG power spectra, fractal analysis, and parameter estimation based on a Poisson distribution. Potential utility of several of these alternative measures was demonstrated for assessment of muscle fatigue.Ph. D.
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Donzella, Michael A. "The Geometry of Rectifiable and Unrectifiable Sets." Kent State University / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=kent1404332888.
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Reid, James Edward. "Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems." Thesis, University of North Texas, 2017. https://digital.library.unt.edu/ark:/67531/metadc1011825/.
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In the context of fractal geometry, the natural extension of volume in Euclidean space is given by Hausdorff and packing measures. These measures arise naturally in the context of iterated function systems (IFS). For example, if the IFS is finite and conformal, then the Hausdorff and packing dimensions of the limit sets agree and the corresponding Hausdorff and packing measures are positive and finite. Moreover, the map which takes the IFS to its dimension is continuous. Developing on previous work, we show that the map which takes a finite conformal IFS to the numerical value of its packing measure is continuous. In the context of self-similar sets, we introduce the super separation condition. We then combine this condition with known density theorems to get a better handle on finding balls of maximum density. This allows us to extend the work of others and give exact formulas for the numerical value of packing measure for classes of Cantor sets, Sierpinski N-gons, and Sierpinski simplexes.
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Cutter, George Randall. "Spatial, Geostatistical, and Fractal Measures of Seafloor Microtopography Across the Eel River Shelf, off Northern California." W&M ScholarWorks, 1997. https://scholarworks.wm.edu/etd/1539617726.
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Meyer, Daniel. "Melting snowballs /." Thesis, Connect to this title online; UW restricted, 2004. http://hdl.handle.net/1773/5796.
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Berlinkov, Artemi. "Dimensions in Random Constructions." Thesis, University of North Texas, 2002. https://digital.library.unt.edu/ark:/67531/metadc3160/.
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We consider random fractals generated by random recursive constructions, prove
zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.
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Pegon, Paul. "Transport branché et structures fractales." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS444/document.
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Cette thèse est consacrée à l’étude du transport branché, de problèmes variationnels qui y sont liés et de structures fractales qui peuvent y apparaître. Le problème du transport branché consiste à connecter deux mesures de même masse par le biais d’un réseau en minimisant un certain coût, qui sera pour notre étude proportionnel à mLα afin de déplacer une masse m sur une distance L. Plusieurs modèles continus ont été proposés pour formuler le problème, et on s’intéresse plus particulièrement aux deux grands types de modèles statiques : le modèle Lagrangien et le modèle Eulérien, avec une emphase sur le premier. Après avoir posé proprement les bases de ces modèles, on établit rigoureusement leur équivalence en utilisant une décomposition de Smirnov des mesures vectorielles à divergence mesure. On s’intéresse par la suite à un problème d’optimisation de forme lié au transport branché qui consiste à déterminer les ensembles de volume 1 les plus proches de l’origine au sens du transport branché. On démontre l’existence d’une solution, décrite comme un ensemble de sous-niveau de la fonction paysage, désormais standard en transport branché. La régularité Hölder de la fonction paysage, obtenue ici sans hypothèse de régularité a priori sur la solution considérée, permet d’obtenir une borne supérieure sur la dimension de Minkowski de son bord, qui est non-entière et dont on conjecture qu’elle en est la dimension exacte. Des simulations numériques, basées sur une approximation variationnelle à la Modica-Mortola de la fonctionnelle du transport branché, ont été effectuées dans le but d’étayer cette conjecture. Une dernière partie de la thèse se concentre sur la fonction paysage, essentielle à l’étude de problèmes variationnels faisant intervenir le transport branché en ce sens qu’elle apparaît comme une variation première du coût d’irrigation. Le but est d’étendre sa définition et ses propriétés fondamentales au cas d’une source étendue, ce à quoi l’on parvient dans le cas d’un réseau possédant un système fini de racines, par exemple pour des mesures à supports disjoints. On donne une définition satisfaisante de la fonction paysage dans ce cas, qui vérifie en particulier la propriété de variation première et on démontre sa régularité Hölder sous des hypothèses raisonnables sur les mesures à connecterThis thesis is devoted to the study of branched transport, related variational problems and fractal structures that are likely to arise. The branched transport problem consists in connecting two measures of same mass through a network minimizing a certain cost, which in our study will be proportional to mLα in order to move a mass m over a distance L. Several continuous models have been proposed to formulate this problem, and we focus on the two main static models : the Lagrangian and the Eulerian ones, with an emphasis on the first one. After setting properly the bases for these models, we establish rigorously their equivalence using a Smirnov decomposition of vector measures whose divergence is a measure. Secondly, we study a shape optimization problem related to branched transport which consists in finding the sets of unit volume which are closest to the origin in the sense of branched transport. We prove existence of a solution, described as a sublevel set of the landscape function, now standard in branched transport. The Hölder regularity of the landscape function, obtained here without a priori hypotheses on the considered solution, allows us to obtain an upper bound on the Minkowski dimension of its boundary, which is non-integer and which we conjecture to be its exact dimension. Numerical simulations, based on a variational approximation a la Modica-Mortola of the branched transport functional, have been made to support this conjecture. The last part of the thesis focuses on the landscape function, which is essential to the study of variational problems involving branched transport as it appears as a first variation of the irrigation cost. The goal is to extend its definition and fundamental properties to the case of an extended source, which we achieve in the case of networks with finite root systems, for instance if the measures have disjoint supports. We give a satisfying definition of the landscape function in that case, which satisfies the first variation property and we prove its Hölder regularity under reasonable assumptions on the measures we want to connect
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Kabanava, Maryia [Verfasser], Hans [Akademischer Betreuer] Triebel, Martina [Akademischer Betreuer] Zähle, and Fernando Cobos [Akademischer Betreuer] Diaz. "Besov spaces on fractals and tempered Radon measures / Maryia Kabanava. Gutachter: Hans Triebel ; Martina Zähle ; Fernando Cobos Diaz." Jena : Thüringer Universitäts- und Landesbibliothek Jena, 2011. http://d-nb.info/1017972079/34.
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Pelander, Anders. "A Study of Smooth Functions and Differential Equations on Fractals." Doctoral thesis, Uppsala : Department of Mathematics, Uppsala university [distributör], 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7590.
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Kombrink, Sabrina [Verfasser], Marc [Akademischer Betreuer] Keßeböhmer, and Manfred [Akademischer Betreuer] Denker. "Fractal Curvature Measures and Minkowski Content for Limit Sets of Conformal Function Systems / Sabrina Kombrink. Gutachter: Marc Keßeböhmer ; Manfred Denker. Betreuer: Marc Keßeböhmer." Bremen : Staats- und Universitätsbibliothek Bremen, 2011. http://d-nb.info/1071898752/34.
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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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Carnovale, Marc. "Arithmetic Structures in Small Subsets of Euclidean Space." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555657038785892.
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Maman, Delphine. "Généricité et prévalence des propriétés multifractales de traces de fonctions." Phd thesis, Université Paris-Est, 2013. http://tel.archives-ouvertes.fr/tel-00974540.
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L'analyse multifractale est l'étude des propriétés locales des ensembles de mesures ou de fonctions. Son importance est apparue dans le cadre de la turbulence pleinement développée. Dans ce cadre, l'expérimentateur n'a pas accès à la vitesse en tout point d'un fluide mais il peut mesurer sa valeur en un point en fonction du temps. On ne mesure donc pas directement la fonction vitesse du fluide, mais sa trace. Cette thèse sera essentiellement consacrée à l'étude du comportement local de traces de fonctions d'espaces de Besov : nous déterminerons la dimension de Hausdorff des ensembles de points ayant un exposant de Hölder donné (spectre multifractal). Afin de caractériser facilement l'exposant de Hölder et l'appartenance à un espace de Besov, on utilisera la décomposition de fonctions sur les bases d'ondelettes.Nous n'obtiendrons pas la valeur du spectre de la trace de toute fonction d'un espace de Besov mais sa valeur pour un ensemble générique de fonctions. On fera alors appel à deux notions de généricité différentes : la prévalence et la généricité au sens de Baire. Ces notions ne coïncident pas toujours, mais, ici on obtiendra les mêmes résultats. Dans la dernière partie, afin de déterminer la forme que peut prend un spectre multifractal, on construira une fonction qui est son propre spectre
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Gaudric, Julien. "Morphométrie des anévrismes de l’aorte thoracique : de l’anatomie scanographique à la modélisation numérique." Electronic Thesis or Diss., Sorbonne université, 2019. http://www.theses.fr/2019SORUS574.
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Rationnel : L’étude de l’anatomie de la crosse aortique a évolué pour optimiser le traitement des anévrismes par endoprothèse.Objectifs : I : caractériser les modifications morphométriques de la crosse induites par un anévrisme. II : évaluer la faisabilité des dispositifs endovasculaires actuels pour traiter les anévrismes de la crosse. III : créer un outil automatisé de mesure pour évaluer les angulations induites par ces déformations. IV : valider un modèle numérique de simulation 0D de la mécanique vasculaire en le confrontant à des données in vivo.Résultats : I : nous avons montré sur une étude scannographique que les anévrismes thoraciques s’accompagnaient d’un étirement bi directionnel de la paroi et d’une rotation de la crosse antérieure ou postérieure selon la localisation de l’anévrisme. II : parmi 56 malades opérés chirurgicalement d’anévrismes de la crosse, l’étude rétrospective de leur scanner montrait qu’aucun n’avait les critères anatomiques pour permettre une endoprothèse branchée. III : nous avons créé un logiciel de calcul automatisé des points de courbure maximale de l’aorte à partir d’une analyse continue du rayon de courbure de la ligne centrale. Sa pertinence a été validée par sa concordance avec la détermination visuelle des points. IV : nous avons confronté la déformation de l’onde de pression artérielle après clampage et déclampage de l’aorte chez 11 patients avec une bonne corrélation et accord entre un modèle numérique et des enregistrements continus intravasculaires. Conclusion : Les progrès dans l’analyse de la conformation de l’aorte et de la mécanique vasculaire sont nécessaires à l’adaptation de nouveaux substituts endovasculairesRational: Research on the anatomy of the aortic arch has been fueled by the need of a comprehensive analysis of this structure in the setting of endovascular repair. Aneurysmal disease causes distortions in areas where the implantation of stent grafts undergo major stress. Objectives: I: To characterize the morphometric modifications of the aortic arch induced by a thoracic aneurysm. II: To evaluate the feasibility of current endovascular devices in treating aortic arch aneurysms. III: To create an automated measurement tool for assessing the angulations induced by these deformations. IV: To validate a 0D numerical simulation model of vascular mechanics by comparing its predictions with in vivo data.Results: I: In a study of 78 CT scan, thoracic aneurysms were associated with bi-directional wall stretching and anterior or posterior rotation according to the aneurysm’s location. II: A retrospective study of the CT scans of 56 patients who underwent aortic arch aneurysm surgical repair showed that none of these patients had the anatomical criteria for a stent graft implantation. III: An automated software for calculating the aortic angulations from a continuous analysis of the curvature radius of the central line was developed and validated against the visual assessment of points. IV: Changes in the morphology of blood pressure waves after aortic clamping and unclamping were studied in 11 patients with a good correlation and agreement between the numerical model and continuous intravascular measurements. Conclusion: Advances in the analysis of aortic geometry and the simulation of vascular mechanics are necessary for the adaptation of new endovascular devices
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Körber, Martin Julius. "Phase-Space Localization of Chaotic Resonance States due to Partial Transport Barriers." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-218817.
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Classical partial transport barriers govern both classical and quantum dynamics of generic Hamiltonian systems. Chaotic eigenstates of quantum systems are known to localize on either side of a partial barrier if the flux connecting the two sides is not resolved by means of Heisenberg's uncertainty. Surprisingly, in open systems, in which orbits can escape, chaotic resonance states exhibit such a localization even if the flux across the partial barrier is quantum mechanically resolved. We explain this using the concept of conditionally invariant measures by introducing a new quantum mechanically relevant class of such fractal measures. We numerically find quantum-to-classical correspondence for localization transitions depending on the openness of the system and on the decay rate of resonance states. Moreover, we show that the number of long-lived chaotic resonance states that localize on one particular side of the partial barrier is described by an individual fractal Weyl law. For a generic phase space, this implies a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of phase space.
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Saxcé, Nicolas de. "Sous-groupes boréliens des groupes de Lie." Thesis, Paris 11, 2012. http://www.theses.fr/2012PA112179.
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Dans cette thèse, on étudie les sous-groupes boréliens des groupes de Lie et leur dimension de Hausdorff. Si G est un groupe de Lie nilpotent connexe, on construit dans G des sous-groupes de dimension de Hausdorff arbitraire, tandis que si G est semisimple compact, on démontre que la dimension de Hausdorff d'un sous-groupe borélien strict de G ne peut pas être arbitrairement proche de celle de GGiven a Lie group G, we investigate the possible Hausdorff dimensions for a measurable subgroup of G. If G is a connected nilpotent Lie group, we construct measurable subgroups of G having arbitrary Hausdorff dimension, whereas if G is compact semisimple, we show that a proper measurable subgroup of G cannot have Hausdorff dimension arbitrarily close to the dimension of G
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Yang, Xiaochuan. "Etude dimensionnelle de la régularité de processus de diffusion à sauts." Thesis, Paris Est, 2016. http://www.theses.fr/2016PESC1073/document.
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Dans cette thèse, on étudie diverses propriétés dimensionnelles de la régularité de processus de difusions à sauts, solution d’une classe d’équations différentielles stochastiques à sauts. En particulier, on décrit la fluctuation de la régularité höldérienne de ces processus et celle de la dimension locale pour la mesure d’occupation qui leur est associée en calculant leur spectre multifractal. La dimension de Hausdorff de l’image et du graphe de ces processus ont aussi étudiées.Dans le dernier chapitre, on applique une nouvelle notion de dimension de grande échelle pour décrire l’asymptote à l’infini du temps de séjour d’un mouvement brownien en dimension 1 sous des frontières glissantesIn this dissertation, we study various dimension properties of the regularity of jump di usion processes, solution of a class of stochastic di erential equations with jumps. In particular, we de- scribe the uctuation of the Hölder regularity of these processes and that of the local dimensions of the associated occupation measure by computing their multifractal spepctra. e Hausdor dimension of the range and the graph of these processes are also calculated.In the last chapter, we use a new notion of “large scale” dimension in order to describe the asymptotics of the sojourn set of a Brownian motion under moving boundaries
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Le, Thanh Hoang Nhat. "Sur la dimension de Minkowski des quasicercles." Phd thesis, Université d'Orléans, 2012. http://tel.archives-ouvertes.fr/tel-00762750.
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"Spectral analysis on fractal measures and tiles." 2012. http://library.cuhk.edu.hk/record=b5549619.
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在這篇論文中，我們將會首先討論什麼概率測度μ 上的L²空間會存在指數型正交基(exponential orthonormal basis) ，而μ 則稱為一個譜測度若指數型正交基存在。這個問題源於1974年的Fuglede猜想和Jorgensen與Pedersen對分形譜測度存在性的研究。我們有興趣理解怎麼樣的測度會是譜測度，而對於沒有指數型正交基的測度，我們則探討它們會否存在更廣義且在Fourier分析中常用的指數型基，如Riesz基或Fourier框架(Fourier frame) 。我們知道一個測度可以唯一分解成離散、奇異和絶對連續三部份。我們首先証明譜測度肯定是純型(pure type) 。若測度是絶對連續，我們對有Fourier框架的測度的密度給出一個完全的刻直到。這個結果對研究Gabor框架的問題都有幫助。對於離散且只有有限個非零質量原子的測度，我們証明它們全都都有Riesz基。在最困難作出一般討論的奇異測度中，我們透過譜測度與離散測度的卷積找出了有Riesz基但沒有指數型正交基的奇異測度。我們進而探討自彷函數送代系統(affine IFSs) ，我們証明到如果一個自彷函數送代系統是測度分離且有Fourier框架，那麼它在每一個函數的概率權都是一樣的。我們亦証明了Laba-Wang猜想在絶對連續的自相似測度上是正確的。這些結果都表示了一個有Fourier框架的測度都應該在其支撐上有一定的均勻性。
在論文的第二部分我們會探討自彷tile其Fourier變換的零點集。在自彷tile的研究中，其中一個基本問題就是刻劃其數字集(digit set)使得那自彷函數送代系統的不變集能以平移密鋪空間。透過Kenyon條件，我們可將這個問題轉化成理解Fourier變換的零點集。男一方面，指數型正交基的存在性亦需要我們探討Fourier變換的零點集，而自彷tile 的Fourier變換是可以明確寫出來的。這使自彷tile成為一個很好去研究tilings和譜測度相互關係的好例子。
我們利用了分圓多項式(cyclotomic polynomials)對一維自彷tile的零點集進行了一個詳細的研究。從這裡我們把tile的數字集寫成某些分圓多項式的乘積。這個乘積亦可以一個樹上的切集(blocking)表示出來。這個表示亦把tile數字集的乘積形式(product-forms) 一般化成高階乘積形式。我們証明了在任何維數的tile數字集都是整數tile(即它們能平移密鋪整數集Z) 。這個結果讓我們能使用Coven和Meyerowitz所提出的整數tile分解方法，來使tile數字集完整刻劃成高階模乘積形式(high order modulo product-forms) 當數字集的數目為p[superscript α]q而p，q則是質數。由於我們對零點集亦完全清楚，這對在自彷tile上尋找完備的指數型正交基提供了一個新的方向。
In this thesis, we will first consider when a probability measure μ admits an exponential orthonormal basis on its L² space (μ is called spectral measures).This problem originates from the conjecture of Fuglede in 1974, and the discovery of Jorgensen and Pedersen that some fractal measures also admit exponential orthonormal bases, but some do not. It generates a lot of interest in understand- ing what kind of measures are spectral measures. For those measures failing to have exponential orthonormal bases, it is interesting to know whether such mea- sures still have Riesz bases and Fourier frames, which are generalized concepts of orthonormal bases with wide range of uses in Fourier analysis.
It is well-known that a measure has a unique decomposition as the discrete, singular and absolutely continuous parts. We first show that spectral measures must be of pure type. If the measure is absolutely continuous, we completely classify the class of densities of the measures with Fourier frames. This result has new applications to topics in applied harmonic analysis, like the Gabor analysis. For the discrete measures with finite number of atoms, we show that they all have Riesz bases. For the case of singular measure, which is the most difficult one, we show that there exist measures with Riesz bases but not orthonormal bases by considering convolution between spectral measures and discrete measures. We then investigate affine iterated function systems (IFSs), we show that if an IFS has measure disjoint condition and admits a Fourier frame, then the probability weights are all equal. Moreover, we also show that the Łaba-Wang conjecture is true if the self-similar measure is absolutely continuous. These results indicate that measures with Fourier frames must have certain kind of uniformity on the support.
In the second part of the thesis we study the zero sets of Fourier transform of self-affine tiles. One of the fundamental problems in self-affine tiles is to classify the digit sets so that the attractors form tiles. This problem can be turned to study the zeros of the Fourier transform via the Kenyon criterion. On the other hand, existence of exponential orthonormal bases requires us to know the zero sets of the Fourier transform. Self-affine tiles are translational tiles arising from IFSs with its Fourier transform written explicitly. It therefore serves as an ideal place to investigate the relation of tilings and spectral measures.
We carry out a detail study in the zero sets of the one-dimensional tiles using cyclotomic polynomials. From this we characterize the tile digit sets through some product of cyclotomic polynomials represented in terms of a blocking in a tree, which is a generalization of the product-form to higher order. We show that tile digit sets in any dimension are integer tiles. This result allows us to use the decomposition method of integer tiles by Coven and Meyerowitz to provide the explicit classification of the tile digit sets in terms of the higher order modulo product-forms when number of the digits is p[superscript α]q, p, q are primes. Since the zero sets are completely known, this provides us a new approach to study the existence of complete orthogonal exponentials in the self-affine tiles on R¹.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Lai, Chun Kit.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2012.
Includes bibliographical references (leaves 128-135).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstract also in Chinese.
Chapter 1 --- Introduction --- p.9
Chapter 1.1 --- Background and Motivations --- p.9
Chapter 1.2 --- Results on spectral measures --- p.13
Chapter 1.3 --- Results on self-affine tiles --- p.16
Chapter 2 --- Fourier Frames: Absolutely Continuous Measures --- p.21
Chapter 2.1 --- Beurling densities --- p.22
Chapter 2.2 --- Law of pure type --- p.28
Chapter 2.3 --- Absolutely continuous F-spectral measures --- p.31
Chapter 2.3.1 --- Proof by Beurling densities --- p.32
Chapter 2.3.2 --- Proof by translational absolute continuity --- p.35
Chapter 2.4 --- Applications to applied harmonic analysis --- p.40
Chapter 2.5 --- Remarks and open questions --- p.42
Chapter 3 --- Fourier Frames: Discrete and Singular Measures --- p.45
Chapter 3.1 --- Discrete measures --- p.46
Chapter 3.2 --- Convolutions with discrete measures --- p.50
Chapter 3.3 --- Self-affine measures --- p.56
Chapter 3.4 --- Iterated function systems on R¹ --- p.65
Chapter 3.5 --- Concluding remarks --- p.70
Chapter 4 --- Spectral structure of tile digit sets --- p.74
Chapter 4.1 --- Preliminaries --- p.76
Chapter 4.2 --- Modulo product-forms --- p.81
Chapter 4.3 --- Higher order product-forms --- p.86
Chapter 4.4 --- Φ-tree, blocking and kernel polynomials --- p.90
Chapter 5 --- Classifications of tile digit sets --- p.101
Chapter 5.1 --- Tile digit sets --- p.101
Chapter 5.2 --- The p[superscript α]q[superscript β] integer tiles --- p.105
Chapter 5.3 --- Tile digit sets for b = p[superscript α]q --- p.112
Chapter 5.4 --- Self-similar measures: Absolute continuity --- p.122
Chapter 5.5 --- Remarks --- p.126
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"On fractal curvature measures." 2013. http://library.cuhk.edu.hk/record=b5884465.
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Du, Yangge.Thesis (M.Phil.)--Chinese University of Hong Kong, 2013.
Includes bibliographical references (leaves 88-91).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstracts also in Chinese.
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"Topological structure and Lipschitz equivalence of fractal sets." 2012. http://library.cuhk.edu.hk/record=b5549659.
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在該論文中，我們探討了自相似集和自仿集兩類基本分形集的拓撲結構。我們主要研究了他們的連通性、全不連通性以及李普希茲等價性。我們首先研究了一類由正方塊迭代生成的自相似集的拓撲，我們稱這種自相似集為分形方塊。通過研究它的 torus-like 結構， 我們用連通分支把分形方塊的拓撲結構分成三種情形，同時我們還給出了一系列簡單有效的判別方法。這對於進一步研究其李普希茲等價類非常有用。
另外一個方面，基於之前對由相鄰共線性數字集生成的自仿集的連通性的研究工作，我們嘗試研究非相鄰共線性數字集，我們處理里一類特殊的由行列式絶對值為 3 的擴張矩陣 A 生成的二維自仿集。通過逐項檢驗 A 的每個特徵多項式，我們得到了該自仿集的連通性的一個完整的刻畫。
最近，關於自相似集的李普希茲等價的研究，引起了很多的關注。在這篇論文中，我們藉助符號空間上自帶的“擴張樹“的結構和它的雙曲邊界，拓寬了該問題的研究框架。通過邊的重排技巧，我們構造了兩個擴張樹之間的擬等距同構以便於證明他們雙曲邊界之間的李普希茲等價。最終，我們解決了更加一般的自相似集，甚至自仿集之間的李普希茲等價問題。
In the thesis, we explore the topological structure of the self-similar sets and self-a±ne sets, two basic classes of fractals. We study their connectedness, total disconnectedness and Lipschitz equivalence.
We first initiate a new study on the topology of a class of self-similar sets generated by nested squares, which we call fractal squares. By studying the torus-like structure, we obtain some useful and simple criteria to classify the fractal squares into three types through connected components. That is very useful for the Lipschitz classification.
In another direction, motivated by the previous work on the self-affine sets associated with consecutive collinear digit sets, we make a first attempt to study the non-consecutive collinear digit sets. We deal with the special case that the planar self-a±ne sets are generated by expanding matrices A with j det(A)j = 3. By checking the characteristic polynomial of A case by case, we obtain a complete characterization for the self-a±ne sets to be connected or disconnected.
Recently there is a lot of interest to study the Lipschitz equivalence of self- similar sets. In this thesis, we provide a broader framework of the study through the concept of augmented (rooted) tree and its hyperbolic boundary. Making use of a technique of rearrangement of the edges, we construct a near-isometry between the two trees to show their boundaries are Lipschitz equivalent. Finally we establish the Lipschitz equivalence on more general self-similar sets and even self-affine sets.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Luo, Jun.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2012.
Includes bibliographical references (leaves 107-111).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstract also in Chinese.
Chapter 1 --- Introduction --- p.3
Chapter 2 --- Topological structure of fractal squares --- p.9
Chapter 2.1 --- Introduction --- p.9
Chapter 2.2 --- Classification of F by connected components --- p.11
Chapter 2.3 --- F containing line segments --- p.17
Chapter 2.4 --- H[superscript c] and its components --- p.19
Chapter 2.5 --- Algorithm and Examples --- p.26
Chapter 2.6 --- Remarks and open questions --- p.32
Chapter 3 --- Connectedness of self-a±ne sets --- p.33
Chapter 3.1 --- Introduction --- p.33
Chapter 3.2 --- Preliminaries --- p.37
Chapter 3.3 --- Integer collinear digit sets --- p.41
Chapter 3.4 --- General collinear digit sets --- p.48
Chapter 3.5 --- Other results on --- p.49
Chapter 3.6 --- Remarks and open questions --- p.52
Chapter 4 --- Lipschitz equivalence and total disconnectedness --- p.53
Chapter 4.1 --- Introduction --- p.53
Chapter 4.2 --- Augmented trees --- p.57
Chapter 4.3 --- Statements of main theorems --- p.63
Chapter 4.4 --- Proofs of main theorems --- p.72
Chapter 4.5 --- Examples --- p.80
Chapter 4.6 --- Remarks and open questions --- p.91
Chapter 5 --- More on Lipschitz equivalence --- p.94
Chapter 5.1 --- Dust-like self-similar sets --- p.94
Chapter 5.2 --- Lipschitz classification of fractal squares --- p.99
Bibliography --- p.107
49
Koudela, Libor. "O pojetí křivky." Doctoral thesis, 2012. http://www.nusl.cz/ntk/nusl-312075.
Full textAbstract:
The notion of a curve played important role in the history of mathematical thought. This dissertation is focused on the conception of a curve in analysis, point set theory and topology. The rectification of curves and the notion of arc length are considered in connection with the history of analysis from antiquity to the beginning of the 20th century. "Measurement of curves" is also discussed from the measure-theoretic viewpoint and various definitions of linear measure and fractional dimension are described. Historically, there are two main approaches to understanding curves. Jordan defined a curve as a continuous image of a closed interval. However, his definition appeared to be too wide, since it was met by objects such as the Peano curve. In the point set theory, a curve is considered to be a one-dimensional continuum. The development of the dimension theory and the continuum theory, starting with the pioneering work of Bolzano, was motivated by the search for rigorous topological definition of a curve, a surface etc. Among "pathological" curves, that were often introduced as counterexamples in the development of modern analysis, we can find early examples of fractals. The fractal theory motivated further study of mathematical properties of these curves in the late 20th century, such as self-similarity and...
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Senthil, Raani K. S. "Lp-Asymptotics of Fourier Transform Of Fractal Measures." Thesis, 2015. http://etd.iisc.ernet.in/2005/3673.
Full textAbstract:
One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in Rn. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let f in Cc∞(dσ), where dσ is the surface measure on the sphere Sn-1 Rn.Then the modulus of the Fourier transform of fdσ is bounded above by (1+|x|)(n-1)/2. Also fdσ in Lp(Rn) for all p > 2n/(n-1) . This result can be extended to compactly supported measure on (n-1)-dimensional manifolds with appropriate assumptions on the curvature. Similar results are known for measures supported in lower dimensional manifolds in Rn under appropriate curvature conditions. However, the picture for fractal measures is far from complete. This thesis is a contribution to the study of asymptotic properties of the Fourier transform of measures supported in sets of fractal dimension 0 < α < n for p ≤ 2n/α.
In 2004, Agranovsky and Narayanan proved that if μ is a measure supported in a
C1-manifold of dimension d < n, then the Fourier transform of μ is not in Lp(Rn) for 1 ≤ p ≤ 2n/d. We prove that the Fourier transform of a measure μ supported in a set E of fractal dimension α does not belong to Lp(Rn) for p≤ 2n/α. As an application we obtain Wiener-Tauberian type theorems on Rn and M(2). We also study Lp-asymptotics of the Fourier transform of fractal measures μ under appropriate conditions and give quantitative versions of the above statement by obtaining lower and upper bounds for the following
limsup L∞ L-k∫|x|≤L|(fdµ)^(x)|pdx