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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
141

Viridiofungins and xeniolide F: target oriented synthesis using different rearrangement reactions of a common substrate class / Xeniolid F und Viridiofungine: Unterschiedliche Umlagerungsreaktionen führen ausgehend von einer gemeinsamen Substratklasse zu sehr verschiedenen Bausteinen für die Naturstoffsynthese

Pollex, Annett 10 October 2006 (has links) (PDF)
The present dissertation covers the total synthesis of viridiofungin triesters and studies toward the total synthesis of xeniolide F. In both cases, sigmatropic rearrangements of α-allyloxy substituted α,β-unsaturated esters are employed: for the viridiofungin ester synthesis a [2,3]-Wittig rearrangement and for the xeniolide F synthesis a catalytic asymmetric Claisen rearrangement CAC. For both rearrangement reactions the historical development, main characteristic and important variations are discussed. The viridiofungin triester synthesis represents a convergent and highly flexible route toward these natural products. The [2,3]-Wittig rearrangement allowed the diastereoselective synthesis of the polar head group with two adjacent stereogenic centers. The E-configured double bond was formed by a Julia-Kocienski olefination. During the studies toward the total synthesis of xeniolide F a new, diastereoselective strategy for the generation of allyl vinyl ethers with E-configured vinyl ether double bond was established employing rhodium catalyzed OH-insertion and an E-selective Horner-Wadsworth-Emmons olefination. Under the conditions of the catalytic asymmetric Claisen rearrangement (CAC) this highly substituted allyl vinyl ether rearranged diastero- and enantioselectively to the corresponding a-keto ester. This example clearly illustrates the high potential of the CAC as synthetic tool for natural product synthesis. / In der vorliegenden Arbeit wird die Totalsynthese von Tirestern der Viridiofungine A, A2 und A4 sowie die Synthese eines Schlüsselintermediates für die Totalsynthese von Xeniolid F dargestellt. In beiden Fällen wird ausgehend von einem α-allyloxysubstituierten α,β-ungesättigten Ester eine Umlagerungsreaktion als Schlüsselschritt eingesetzt: im Falle der Viridiofunginester eine diastereoselektive [2,3]-Wittig-Umlagerung, bei den Arbeiten zur Totalsynthese von Xeniolid F eine diastero- und enantioselektive, katalytische Claisenumlagerung. Für beide Umlagerungsreaktionen werden ausführlich die theoretischen Hintergründe sowie die historische Entwicklung und wichtige Varianten besprochen. Mit der Viridiofungintriestersynthese wird eine konvergente und bezüglich der lipophilen Seitenkette sehr flexible Syntheseroute vorgestellt. Die [2,3]-Wittig-Umlagerung konnte dabei erfolgreich für die diastereoselektive Synthese der hochsubstituierten, polaren Kopfgruppe der Viridiofunginester mit zwei benachbarten stereogenen Zentren (davon eines quartär) eingesetzt werden. Zur Bildung der E-konfigurierten Doppelbindung wurde die Julia-Kocienskie-Olefinierung ausgenutzt. Bei den Arbeiten zur Totalsynthese von Xeniolid F wurde eine neuartige Strategie zur diastereoselektiven Synthese eines Allylvinylethers mit E-konfigurierter Vinyletherdoppelbindung eingesetzt. Die Horner-Wadsworth-Emmons-Olefinierung (HWE-Olefinierung) generierte dabei E-selektiv die Vinyletherdoppelbindung. Das für die HWE-Olefinierung benötigte Phosphonat wurde durch rhodiumkatalysierte OH-Insertion aus einem Allylalkohol und einem Diazaphosphonoacetat hergestellt. Der hochsubstituierte Allylvinylether wurde unter den Bedingungen der katalytisch asymmetrischen Claisenumlagerung umgesetzt und führte mit exzellenter Diastereo- und Enantioselektivität zum entsprechenden α-Ketoester. Anhand dieses Beispiels konnte das Potential der katalytisch asymmetrischen Claisenumlagerung zum Aufbau von hochfunktionalisierten Bausteinen für die Naturstoffsynthese verdeutlicht werden.
142

Stability Rates for Linear Ill-Posed Problems with Convolution and Multiplication Operators

Hofmann, B., Fleischer, G. 30 October 1998 (has links) (PDF)
In this paper we deal with the `strength' of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according_to_M. Z. Nashed [15] the ill-posedness of type I if A is not compact, but we have R(A) 6= R(A) for the range R(A) of A; and the ill-posedness of type II for compact operators A: From our considerations it seems to follow that the problems with noncompact operators A are not in general `less' ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed in [10] are derived from the decay rate of the nonincreasing sequence of singular values of A. Since singular values do not exist for noncompact operators A; we introduce stability rates in order to have a common measure for the compact and noncompact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in the noncompact case. Moreover using increasing rearrangements of the multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for the multiplication operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared for compact operators and multiplication operators.
143

Η χρωμοσωματική εξέλιξη του ποντικού Mus musculus domesticus στο Robertsonian σύστημα της Δυτικής Πελοποννήσου

Μήτσαινας, Γεώργιος Π. 04 December 2008 (has links)
Ο καρυότυπος του οικιακού ποντικού Mus musculus domesticus είναι τυπικά ακροκεντρικός εντούτοις χαρακτηρίζεται από τη συχνή εμφάνιση Rb συντήξεων σε φυσικούς πληθυσμούς. Εξαιτίας αυτών μειώνεται ο 2n από τον τυπικό 2n=40 έως και σε 2n=22 και προκύπτουν στη φύση Rb φυλές, οι οποίες όταν έχουν κοινή προέλευση δημιουργούν Rb συστήματα. Στόχος της διδακτορικής διατριβής ήταν ο λεπτομερής καθορισμός της περιοχής εξάπλωσής του Rb συστήματος της Δ. Πελοποννήσου, η Rb σύσταση και οι σχέσεις των Rb φυλών του και η σχέση του με τον ακροκεντρικό πληθυσμό που το περιβάλλει. Επίσης, να προσδιοριστεί η φυλογενετική πορεία που ακολουθήθηκε στο Rb σύστημα και η πιθανή ειδογένεση που δρομολογείται σε αυτό. Γι’ αυτόν το λόγο, έγινε κυτταρολογική μελέτη σε 232 άτομα του ποντικού από 40 τοποθεσίες της Ελλάδας με τη χρήση των τεχνικών G – και C – ζώνωσης. Βρέθηκε ότι το Rb σύστημα της Δ. Πελοποννήσου αποτελείται κυρίως από 3 Rb φυλές με 2n=24, 28 και 30 και έχει ως κέντρο εξέλιξης την περιοχή της Πάτρας, από όπου εκτείνεται για τουλάχιστον 55 km προς τα Β.-ΒΑ. και για πάνω από 70 km προς τα Ν. Οι παραπάνω Rb φυλές συνδέονται φυλογενετικά μέσω 5 κοινών Rb συντήξεων και είναι πιθανό στην εξέλιξη του Rb συστήματος να έχουν συμμετάσχει και Αμοιβαίες Ανταλλαγές Ολόκληρων Βραχιόνων (WART). Ενδέχεται η μετάβαση από το Rb σύστημα στον ακροκεντρικό πληθυσμό να γίνεται απότομα, ενώ ειδογενετικές διαδικασίες θα ήταν δυνατό να πραγματοποιηθούν μεταξύ των Rb φυλών με 2n=24 και 2n=28, που χαρακτηρίζονται από μερική ομολογία βραχιόνων. / The karyotype of the house mouse Mus musculus domesticus is typically acrocentric, yet it is characterized by the frequent appearance of Rb fusions in natural populations. Due to them, 2n is reduced from the typical 2n=40 even down to 2n=22 and Rb races are formed in the nature, which when linked through common decent, form Rb systems. The goal of this doctorate thesis was the detailed definition of the distribution area of the Rb system of W. Peloponnese, of the Rb constitution and relationship among its Rb races and of its relationship with the surrounding acrocentric population. Also, to clarify the phylogenetic process that was followed in this Rb system and the possible speciation that is under way. Thus, a karyological study was implemented on 232 individuals of the house mouse from 40 Greek localities, using the G – and C – banding staining techniques. It was found that the Rb system of W. Peloponnese consists mainly of 3 Rb races with 2n=24, 28 and 30 and its centre of evolution must lie in the wider Patras area, from where it extends for at least 55 km to the N-NE and for more than 70 km to the S. The above Rb races are phylogenetically related through 5 Rb fusions they have in common and it is possible that Whole Arm Reciprocal Translocations (WART) have contributed to the evolution of this Rb system. The transition from the Rb system to the surrounding typical acrocentric population may occur abruptly, whereas speciation processes could take place between the Rb races with 2n=24 and 2n=28, which are charaterized by monobrachial homology.
144

Target identification and validation studies in chemical biology & Synthesis of medium-sized ring containing compounds via oxidative fragmentation

Liu, Gu January 2010 (has links)
Part I of this thesis describes the development of bioactive small molecules of relevance to the study of the apicomlexan parasite Toxoplasma gondii into useful chemical tools. The research includes the target identification and validation studies, using both chemical and biological methods. Chapter 1 provides an overview of chemical genetics with a particular emphasis on methods for the identification of the protein targets of bioactive small molecules. The concept of biochemical protein target identification techniques was introduced with a detailed discussion of interesting applications from the literature. Chapter 2 focuses on the development of a tetrahydro-β-carboline based lead molecule into a chemical tool through target identification studies. The structure activity relationship (SAR) data associated with this core structure, the design of a chemical inducer of dimerisation (CID) and the synthesis of this CID are discussed in detail. Chapter 3 described work done to identify the potential protein target(s) of Conoidin A. Experiments to assess whether Conoidin A can inhibit a proposed target in vitro are also included. Further optimisation of this structural class to develop more potent inhibitors is discussed in the second part of this chapter. Part II of this thesis describes the development of methods for the synthesis of medium-sized ring containing compounds using oxidative fragmentation and rearrangement strategies. Chapter 5 provides an overview of the existing oxidative fragmentation methodology, with an emphasis on the use of oxidative fragmentation reactions for the synthesis of medium-sized ring systems (8-11 ring atoms). Chapter 6 focuses on using the established oxidative fragmentation method in the oxizino carbazolone system to investigate the diasteroselectivity of this reaction. Possible mechanisms for this transformation are investigated and discussed using both chemical and computational methods. An interesting rearrangement reaction has also been observed during this study. Chapter 7 focuses on developing an asymmetric oxidative fragmentation method, for use in the diazabenz[e]aceathrylenes system. Asymmetric oxidative fragmentation reactions using [Ru(pybox)(pydic)] catalysts are discussed. Attempts to optimise the enantiomeric excesses of the reaction by varying reaction conditions and substituents in the substrate are also included.
145

Relações federativas de poder: uma análise histórico-comparativa do Brasil

Souza, Celso Florêncio de 29 September 2017 (has links)
Submitted by CELSO FLORÊNCIO DE SOUZA (celso.florencio@jacarei.sp.gov.br) on 2017-09-29T11:19:10Z No. of bitstreams: 1 CELSO FLORÊNCIO DE SOUZA - Completo.pdf: 2485152 bytes, checksum: fc3620179ebb52d47c4c099d412a5f50 (MD5) / Approved for entry into archive by ÁUREA CORRÊA DA FONSECA CORRÊA DA FONSECA (aurea.fonseca@fgv.br) on 2017-10-03T18:50:10Z (GMT) No. of bitstreams: 1 CELSO FLORÊNCIO DE SOUZA - Completo.pdf: 2485152 bytes, checksum: fc3620179ebb52d47c4c099d412a5f50 (MD5) / Made available in DSpace on 2017-10-16T12:27:03Z (GMT). No. of bitstreams: 1 CELSO FLORÊNCIO DE SOUZA - Completo.pdf: 2485152 bytes, checksum: fc3620179ebb52d47c4c099d412a5f50 (MD5) Previous issue date: 2017-09-29 / This thesis follows the three-article model, having as its background power relations within the federations. The first article talks to the literature on federalism from the perspective of vertical power relations, with a view to understanding the relations between various levels of power. Although there is already a large body of literature on the theme, the theoretical gap of the first article consists of power rearrangements (centralization and decentralization), considering all players and not just this relation between central government and subnational governments. The second article is a reflection on horizontal power relations at the state level, with the following guiding question: “What are the causal conditions that lead some states to have increased political power before their peers?” To do this, two empirical cases of great representativeness were selected: Brazil and the United States of America (USA), analyzed since their independence having the path dependence literature as a basis, considering only causal conditions at the macro level. The third article is also inserted in the horizontal power relations, nevertheless, its analysis field consists in the Brazilian municipalities. The latter have great contemporary relevance, due to their key role in the execution of public policies. / Esta tese segue o modelo de três artigos, tendo como plano de fundo as relações de poder dentro das federações. O primeiro artigo dialoga com a literatura acerca do federalismo pela ótica das relações verticais de poder, com vistas a compreender as relações entre diferentes níveis de poder. Embora já exista ampla literatura relativa ao tema, a lacuna teórica do primeiro artigo consiste nos rearranjos de poder (centralização e descentralização), considerando todos os atores e não apenas essa relação entre governo central e governos subnacionais. O segundo artigo constitui uma reflexão sobre as relações horizontais de poder no plano estadual, com a seguinte questão norteadora: “Quais são as condições causais que levam alguns estados a ter um aumento de poder político perante seus pares?”. Para tanto, foram selecionados dois casos empíricos de grande representatividade: Brasil e Estados Unidos da América (EUA), analisados desde sua independência com base na literatura de rota dependência, considerando apenas condições causais no nível macro. O terceiro artigo também está inserido nas relações horizontais de poder, no entanto, seu campo de análise consiste nos municípios brasileiros. Estes apresentam grande relevância contemporânea, devido ao seu papel fundamental na execução de políticas públicas.
146

Transfert de chiralité dans les réarrangements en cascade d'ènediynes / Chirality transfer in cascade rearrangements of enediynes

Campolo, Damien 13 December 2013 (has links)
La synthèse asymétrique d’aza-hétérocycles (tétrahydro-isoquinoléines et naphtodiazépines) a été réalisée grâce à la mise en œuvre d’un processus faisant intervenir des réactions radicalaires et polaires en cascade à partir des ènediynes portant un centre stéréogène. Ce processus implique successivement : la formation d’un ényne-allène (via une migration-1,3 de proton, une réaction d’un alcyne terminal avec un carbénoïde de cuivre, ou encore une réaction d’homologation de Crabbé)/ la cyclisation de Saito-Myers/ le transfert-1,5 d’un atome d’hydrogène/ la recombinaison du biradical résultant. Les deux dernières étapes élémentaires de ce réarrangement étaient idéalement adaptées à l’application d’une stratégie basée sur le phénomène de mémoire de chiralité. Des études mécanistiques basées sur des expériences de marquage isotopique et des calculs théoriques ont permis de mieux comprendre les paramètres qui contrôlent la régio- et la stéréosélectivité de la réaction. L’ambition de contrôler par cette voie, via une double mémoire de chiralité, deux centres stéréogènes nous a conduits à étudier le transfert de la chiralité axiale d’un motif allénique judicieusement substitué. Cette étude a permis de découvrir une cycloisomérisation originale catalysée par le cuivre (I) conduisant à des fulvènes chiraux via un double transfert de chiralité (centrique-axial-centrique). / The asymmetric synthesis of azaheterocycles (tetrahydorisoquinolines and naphthodiazepines) was successfully achieved via the polar/radical cross-over rearrangement of enediynes bearing a stereogenic center. This process involves successively : enyne-allene formation (via 1,3-proton shift, reaction of a terminal alkyne group with carbenoids or Crabbé homologation)/Saito-Myers cyclization/1,5-hydrogen atom transfer/biradical recombination. It was ideally suited to apply a strategy based on the memory of chirality phenomenon. Mechanistic studies based on isotopic labelling and theoretical calculations enabled to go deeper into the understanding of the parameters controlling the regio- and the stereoselectivity of the reaction. The ambition to control two stereogenic centers via double memory of chirality, led us to investigate the transfer of the axial chirality of a designed allenic moiety. This study led to the discovery of an original copper (I)-mediated cycloisomerization leading to chiral fulvenes and proceeding via central-to-axial-to-central double chirality transfer.
147

Détection à grande échelle des réarrangements génomiques et élucidation de leurs mécanismes

Tremblay-Belzile, Samuel 04 1900 (has links)
No description available.
148

Citogenética em peixes de cabeceiras de dois riachos pertencentes à bacia do Alto rio Paraná, região de Toledo (PR) / Cytogenetic in fishes from two headwaters streams belonging to the Upper Paraná river basin, Toledo (PR)

Yano, Cássia Fernanda 25 February 2011 (has links)
Made available in DSpace on 2017-07-10T18:13:25Z (GMT). No. of bitstreams: 1 Cassia Fernanda Yano.pdf: 1692770 bytes, checksum: 2f6626f3bb741520d86130373963a00f (MD5) Previous issue date: 2011-02-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Cytogenetics studies were carried out in two genera of fishes from Pindorama and Lopei streams, Upper Paraná River basin, with cytossistematic and ecologic-evolutionary approaches. In Heptapterus mustelinus, the first cytogenetics characterization in the genus was carried out, verifying 54 chromosomes (26m+18sm+4st+6a), which is a diploid number not observed among the Heptapteridae species till now studied. Differently from the others members of this family, multiple Ag-NORs and centromeric heterochromatin in almost all the chromosome of the complement were observed, being these heterochromatin CMA3+/DAPI- in nine chromosome pairs, beyond that coincident with Ag-NORs. These data show a different way in the karyotypic evolution of H. mustelinus in relation to the others Heptapteridae genus. In the genus Astyanax, an interpopulation comparative analysis was carried out in Astyanax aff. paranae, Astyanax fasciatus and Astyanax altiparanae from Pindorama and Lopei streams, in way to observe the chromosomal differentiation and the evolutionary trends in the three species. It was verified 48 and 50 chromosome in Astyanax aff. paranae and A. fasciatus, respectively, with interpopulation differences in the karyotypic formulae. The diploid number obtained to A. altiparanae was 50 chromosomes, with the same karyotypic formulae in both populations. The three species presented multiple Ag-NORs, with interpopulation differences observed in Astyanax aff. paranae and A. fasciatus. In relation to the heterochromatin distribution pattern, interpopulation differences were verified. The occurrence of Ag-NORs/CMA3+ was observed in the three species, besides DAPI+ bands in A. altiparanae. The results showed that Astyanax aff. paranae and A. fasciatus presented more conspicuous interpopulation differences than A. altiparanae populations, probably due bionomic characteristic that avoid the presence of Astyanax aff. paranae and A. fasciatus to upper segments in the streams, leading to the fixation of chromosomal rearrangements in these populations. However, the absence of interpopulation karyotypic differences in A. altiparanae suggests gene flow between the two populations. By the way, the present study showed that cytogenetics is an important tool in understanding the chromosomal mechanism involved in the evolutionary process in fishes group, and that the association to ecological studies is fundamental to the comprehension of dispersion process of some group of fishes. / Estudos citogenéticos foram realizados em dois gêneros de peixes nos riachos Pindorama e Lopei, pertencentes à bacia do Alto rio Paraná, com diferentes abordagens: citossistemática e ecológico-evolutiva. Em Heptapterus mustelinus foi realizada a primeira caracterização citogenética no gênero, tendo sido verificado 54 cromossomos (26m+18sm+4st+6a), número diplóide não relatado entre os heptapterídeos até o momento estudados. Diferentemente da maioria das espécies da família, foram observadas Ag-RONs múltiplas e heterocromatina presente na região centromérica da maioria dos cromossomos do complemento, sendo estas heterocromatinas CMA3+/DAPI- em nove pares cromossômicos, além daquelas coincidentes com as Ag-RONs. Estes dados revelam um caminho diferente na evolução cariotípica de H. mustelinus em relação aos demais gêneros de Heptapteridae. No gênero Astyanax foi realizada uma análise comparativa entre populações de Astyanax aff. paranae, Astyanax fasciatus e Astyanax altiparanae coletadas nos riachos Pindorama e Lopei, verificando-se as tendências da evolução e diferenciação cromossômica nas três espécies. Nas populações de Astyanax aff. paranae e A. fasciatus foram verificados 48 e 50 cromossomos, respectivamente, mas com diferenças interpopulacionais nas fórmulas cariotípicas em ambas as espécies. O número diplóide para A. altiparanae foi de 50 cromossomos com mesma fórmula cariotípica em ambas as populações. As três espécies apresentaram Ag-RONs múltiplas, com diferenças interpopulacionais observadas em Astyanax aff. paranae e A. fasciatus. Em relação ao padrão de distribuição de heterocromatina, diferenças interpopulacionais foram verificadas. A ocorrência de Ag-RONs/CMA3+ foi observada nas três espécies, além de bandas DAPI+ em A. altiparanae. Os dados demonstraram que Astyanax aff. paranae e A. fasciatus apresentaram diferenças cariotípicas interpopulacionais mais conspícuas do que as populações de A. altiparanae, provavelmente por características bionômicas que restringem Astyanax aff. paranae e A. fasciatus a trechos superiores de riachos, com a fixação de rearranjos cromossômicos nas populações. Entretanto, a ausência de diferenças cariotípicas interpopulacionais em A. altiparanae sugerem fluxo entre as duas populações. De forma geral ficou evidente que a citogenética é uma importante ferramenta para o entendimento dos mecanismos cromossômicos envolvidos no processo evolutivo dentro dos grupos de peixes, e que a associação a estudos de ecologia é fundamental para compreender os processos envolvidos na dispersão de algumas espécies.
149

Microscopic origin of the rheological and surface properties of embryonic cell aggregates / Origine microscopique des propriétés rhéologiques et de surface des agrégats de cellules embryonnaires

Stirbat, Tomita Vasilica 28 September 2012 (has links)
Ces travaux de thèse portent sur l'étude expérimentale des propriétés physiques et de la biomécanique des agrégats cellulaires embryonnaires. Le but de cette thèse était d'une part de mieux comprendre l'origine biologique de la viscosité et de la tension de surface tissulaire, d'autre part d'étudier quantitativement en détail l'élasticité cellulaire par des nouvelles mesures de rhéologie en cisaillement. Un premier chapitre concerne les mesures de tension de surface tissulaire par la méthode de compression et de viscosité tissulaire par analyse de la cinétique de fusion de deux agrégats en faisant varier comme paramètre principal la contractilité cellulaire que certains suspectent comme étant la principale origine biologique de ces paramètres. Nous utilisons le formalisme du DITH (Haris, 1976: Differential Interfacial Tension Hypothesis) pour interpréter les données. Le deuxième chapitre concerne les mesures rhéologiques en cisaillement à l'aide d'un rhéomètre commercial plan-plan sur plusieurs centaines ou milliers d'agrégats cisaillés ensembles. Nous montrons que les cellules deviennent moins rigides pour une déformation minimale d'environ 4%, mais sur l'échelle de l'heure les cellules sont capables de se rigidifier à nouveau. Ces expériences sont analysées à l'aide d'un modèle de ressorts qui cassent sous contrainte puis se ressoudent à contrainte nulle / This thesis focuses on the experimental study of physical properties and biomechanics of embryonic cell aggregates. The aim of this thesis was on one hand to better understand the biological origin of tissue viscosity and tissue surface tension, and on the other hand to study quantitatively in detail cell elasticity by means of new rheological measurements in shear. A first chapter deals with measurements of tissue surface tension by tissue compression method and tissue viscosity by analysis of the fusion kinetics of two aggregates. We vary key parameters such as cell contractility that some people suspect to be the main biological origin of these parameters. We use the formalism of DITH (Haris, 1976: Differential Interfacial Tension Hypothesis) to interpret the data. The second chapter deals with rheological measurements in shear using a commercial plate-plate rheometer over several hundred of aggregates. We showed that cells become softer after a minimal deformation of 4% is reached, and can harden again on the timescale of hour. These experiences are analyzed using a model of springs that break under stress and then reattach at zero strain
150

Stability Rates for Linear Ill-Posed Problems with Convolution and Multiplication Operators

Hofmann, B., Fleischer, G. 30 October 1998 (has links)
In this paper we deal with the `strength' of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according_to_M. Z. Nashed [15] the ill-posedness of type I if A is not compact, but we have R(A) 6= R(A) for the range R(A) of A; and the ill-posedness of type II for compact operators A: From our considerations it seems to follow that the problems with noncompact operators A are not in general `less' ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed in [10] are derived from the decay rate of the nonincreasing sequence of singular values of A. Since singular values do not exist for noncompact operators A; we introduce stability rates in order to have a common measure for the compact and noncompact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in the noncompact case. Moreover using increasing rearrangements of the multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for the multiplication operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared for compact operators and multiplication operators.

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