Relaxation oscillations are highly non-linear oscillations, which appear to feature many important biological phenomena such as heartbeat, neuronal activity, and population cycles of predator-prey type. They are characterized by repeated switching of slow and fast motions and occur naturally in singularly perturbed ordinary differential equations, which exhibit dynamics on different time scales. Traditionally, slow-fast systems and the related oscillatory phenomena -- such as relaxation oscillations -- have been studied by the method of the matched asymptotic expansions, techniques from non-standard analysis, and recently a more qualitative approach known as geometric singular perturbation theory. It turns out that relaxation oscillations can be found in a more general setting; in particular, in slow-fast systems, which are not written in the standard form. Systems in which separation into slow and fast variables is not given a priori, arise frequently in applications. Many of these systems include additionally various parameters of different orders of magnitude and complicated (non-polynomial) non-linearities. This poses several mathematical challenges, since the application of singular perturbation arguments is not at all straightforward. For that reason most of such systems have been studied only numerically guided by phase-space analysis arguments or analyzed in a rather non-rigorous way. It turns out that the main idea of singular perturbation approach can also be applied in such non-standard cases. This thesis is concerned with the application of concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method to the study of relaxation oscillations in slow-fast systems beyond the standard form. A detailed geometric analysis of oscillatory mechanisms in three mathematical models describing biochemical processes is presented. In all the three cases the aim is to detect the presence of an isolated periodic movement represented by a limit cycle. By using geometric arguments from the perspective of dynamical systems theory and geometric desingularization based on the blow-up method analytic proofs of the existence of limit cycles in the models are provided. This work shows -- in the context of non-trivial applications -- that the geometric approach, in particular the blow-up method, is valuable for the understanding of the dynamics of systems with no explicit splitting into slow and fast variables, and for systems depending singularly on several parameters.

在這篇論文中，我們討論具有初始值條件的二階微分方程 □□□□□ 和系統微分方程 □□□□□等問題。 我們利用能量方法（即能量是常數的特性）來探討上述方程解之特性。例如：生成時間、爆破和爆破速率以及局部解的漸進行為。 / In this paper we shall consider the initial value problem for second order differential equation of the form □□□□□ and the system □□□□□ . We shall discuss the blow-up properties, such as the life-span, the blow-up rate and blow-up constants, and the asymptotic behavior of the global solution by using the energy method.

本研究中討論了非線性微分方程式之解的正則性。在這之中發現了一些有趣的現象，得到了方程式解可以做任意次的微分，並且得到對該解任意次微分後其值趨近到無限大時之爆破速率、爆破常數及當其值遞減至零時的爆破速率、爆破常數。 / In this paper we work with the regularity of solutions for the non-linear ordinary differential equation u''-u^p=0 for some well-defined functions u^p. We have found some interesting phenomena, u belongs to C^q for any q in positive integer, blow-up constant, blow-up rate, null point and decay rate of u^(n) are obtained in this work, through that we get the characterization for these equations in this case.

On étudie l'existence de blow-up et l'ensemble des points de blow-up pour une équation de type chaleur dégénérée ou non avec un terme source uniforme fonction nonlinéaire de la température instantanée en un point fixé du domaine. L'étude est conduite par les méthodes d'analyse classique (fonctions de Green, développements en fonctions propres, principe du maximum) ou fonctionnelle (semi-groupes d'opérateurs linéaires). / We study existence of blow-up and blow-up sets for a (degenerate or not) heat-like equation with a uniform source term non linear function of the instantaneous temperature at a given point of the domain. The techniques are relevant from either classical analysis (Green functions, eigenfunction expansions, maximum principle) or functional analysis (semi-groups of linear operators).

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-17T15:40:30Z No. of bitstreams: 1 arquivototal.pdf: 1244185 bytes, checksum: d60955ede563305b1f641dd53d7154a7 (MD5) / Made available in DSpace on 2017-08-17T15:40:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1244185 bytes, checksum: d60955ede563305b1f641dd53d7154a7 (MD5) Previous issue date: 2017-03-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In thisworkwestudytheasymptoticbehaviorofthesolutionsofpartialdi erential equations slowlynon-dissipative,wewillgiveabriefhistoricaloverviewonthetheme, wewillpresentanintroductiontothetheoryofsemigroupsofboundedlinearoperators in Banachspaces,fractionalpowersofsectoroperatorsandresultsontheexistenceand uniqueness ofsolutionsofabstractCauchysemilinearproblemsoftheparabolictype. / Neste trabalhoestudamosocomportamentoassint oticodassolu c~oesdeequa c~oes diferenciais parciaislentamenten~aodissipativas,daremosumbreveapanhadohist orico sobre otema,apresentaremosumaintrodu c~ao ateoriadesemigruposdeoperadores lineares limitadosemespa cosdeBanach,pot^enciasfracion ariasdeoperadoressetoriais e resultadossobreexist^enciaeunicidadedesolu c~oesdeproblemasabstratosdeCauchy semilineares dotipoparab olico.

<div>Forensic entomology is a subdiscipline of entomology that involves the use of insect behavior and developmental data to aid in criminal investigations. Genetic data has become increasingly important to the field as there has been a push for DNA-based species identification methods of forensically relevant insects. Genetic data can also elucidate population structure and relatedness of these insects, and such knowledge can contribute to the development of more specific datasets for insects in different regions. The first study presented here investigated the phylogenetics of sister species <i>Lucilia cuprina</i> and <i>Lucilia sericata</i> to identify possible subspecies divisions and issues with DNA-based identifications in the United States. The initial aim of this study was to identify genetic differences between specimens of <i>L. cuprina</i> that preferred live versus carrion flesh. Flies collected from Indiana, USA and South Africa were sequenced and analyzed. Upon sequencing of the genes <i>COI, Period,</i> and <i>28s,</i> our results indicated that <i>L. cuprina</i> from Indiana possess a unique combination of nuclear and mitochondrial haplotypes that suggest a unique lineage, possibly indicating modern hybridization with <i>L. sericata. </i>The inability of both nuclear and mitochondrial genes to distinguish between <i>L. cuprina</i> and <i>L. sericata</i> raises questions about the capabilities of DNA-based species identifications within this genus. Additionally, the inability of these genes to distinguish between specimens that preferred live versus carrion flesh highlights a need for continued research of these behavioral differences. The second study presented here investigated the population structure and relatedness of house flies in the American southwest in relation to a civil lawsuit where neighbors of a poultry farm alleged that flies were emanating from the farm to their homes. <i>Musca domestica</i> (house fly) specimens were collected from the chicken farm and from locations in varying directions and distances from the farm. Amplified fragment length polymorphism (AFLP) analysis was performed and the data were used in a number of analyses. Population reallocation simulations generally indicated that samples from different locations were not genetically different enough from other locations to allocate to their true origin population over others. Kinship analysis showed differences in samples collected in a later season that indicate a genetic bottleneck over time. Population structure analysis indicated the presence of two intermixing genetic populations in the dataset. AMOVA revealed that the majority of genetic variation laid within, rather than among, populations. A Mantel test revealed no significant correlation between genetic and geographic distances. These results indicate that the <i>M. domestica</i> population in this region of southwestern America is large and intermixing, with no clear genetic distinctions between specimens collected at the poultry farm versus the surrounding locations. In regard to the civil lawsuit, it was not possible to conclude that the flies did not emanate from the poultry farm. In a broader perspective, these data can be utilized to develop pest management strategies in this region. Overall, the data from both studies presented here will be useful to forensic investigations, development of more specific and detailed data and identification techniques, and pest control measures.</div>

Relaxation oscillations are highly non-linear oscillations, which appear to feature many important biological phenomena such as heartbeat, neuronal activity, and population cycles of predator-prey type. They are characterized by repeated switching of slow and fast motions and occur naturally in singularly perturbed ordinary differential equations, which exhibit dynamics on different time scales. Traditionally, slow-fast systems and the related oscillatory phenomena -- such as relaxation oscillations -- have been studied by the method of the matched asymptotic expansions, techniques from non-standard analysis, and recently a more qualitative approach known as geometric singular perturbation theory. It turns out that relaxation oscillations can be found in a more general setting; in particular, in slow-fast systems, which are not written in the standard form. Systems in which separation into slow and fast variables is not given a priori, arise frequently in applications. Many of these systems include additionally various parameters of different orders of magnitude and complicated (non-polynomial) non-linearities. This poses several mathematical challenges, since the application of singular perturbation arguments is not at all straightforward. For that reason most of such systems have been studied only numerically guided by phase-space analysis arguments or analyzed in a rather non-rigorous way. It turns out that the main idea of singular perturbation approach can also be applied in such non-standard cases. This thesis is concerned with the application of concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method to the study of relaxation oscillations in slow-fast systems beyond the standard form. A detailed geometric analysis of oscillatory mechanisms in three mathematical models describing biochemical processes is presented. In all the three cases the aim is to detect the presence of an isolated periodic movement represented by a limit cycle. By using geometric arguments from the perspective of dynamical systems theory and geometric desingularization based on the blow-up method analytic proofs of the existence of limit cycles in the models are provided. This work shows -- in the context of non-trivial applications -- that the geometric approach, in particular the blow-up method, is valuable for the understanding of the dynamics of systems with no explicit splitting into slow and fast variables, and for systems depending singularly on several parameters.

Thesis (PhD) -- University of Stellenbosch, 2000. / ENGLISH ABSTRACT: The secular songs of John Blow (1649-1708): An Edition The aim of this thesis is to assemble the 109 secular songs of John Blow in one anthology, to transcribe them into modem notation and in doing so to make them accessible for modem use and further research. A significant feature of this collection is a group of 13 songs which have not been printed previously and which are available only in manuscript form in special collections in Great Britain. Other songs published during Blow's lifetime are likewise found in special collections which are not accessible to the public. Many of these songs are hard to decipher because of ageing. In some cases the paper is so thin that the notes show through from the back to the detriment of readibility. Where the manuscripts as well as contemporary publications exist, significant comparisons could be made, e.g. with In vain, brisk God of Love (vol. 2: 147) where the MUMS 118 manuscript could be compared with the published version in Choice Ayres and Songs printed by Godbid and Playford (jnr) in 1683. An important 'discovery' was finding that an autograph manuscript, Ah me, undone! (Lhl Add. 31457), does not comprise an individual song as listed by Watkins Shaw (1980), but an excerpt from the song Happy the Man who languishing (1700). This made it possible to compare an original manuscript by Blow with a publication of the same song by Playford. The 20th century has seen renewed interest in Blow's work: Frederick George Edwards (1902) and William Cummings (1908), in particular, started this revival in interest. Harold Watkins Shaw took the lead from 1936 and Leland Clarke (1947) was responsible for the next phase. Since 1975, Bruce Wood has been the main researcher of Blow's anthems. Anthony Rooley, director of the Consort of Musicke in London and Peter Holman, director of the Parley of Instrument, have contributed greatly to the recent (1987-1999) revival of interest in Blow's music with their performances and recordings making use of original instruments. This thesis, as well as my Master's thesis (Grobler 1993), forms part of the most recent stage of research into Blow's works. Volume 1: In the first chapter of the thesis the secular song of the English Restoration (1660- 1714) is presented in perspective. Blow's stylistic characteristics as they manifest themselves in his secular songs are discussed. The criticism that this style evoked from music critics through the years, especially Charles Burney (1726-1814), is put into historical perspective. Stylistic characteristics of the song, the influence of French and Italian vocal music, as well as the strong influence of Charles II's preferences on . court composers' music, are highlighted. The function of the song in Restoration society is discussed. In the second chapter Blow's contribution to the different song types are discussed in detail: the solo songs, songs for two voices and dialogues, songs for more than two voices and songs for incidental theatre music. The editorial process followed in transcribing the songs is explained. This is based on the methods suggested in Caldwell (1985) and described in the Musica Britannica. A discussion of the performing practice of the song contributes towards understanding the Restoration song. The textual commentary deals with aspects, such as notation and provides more information about the manuscripts and publications which form the basis of this investigation. A systematic index of sources and songs is provided. Volume 2: In this volume the 109 songs are presented chronologically in modem edited form. The songs reflect the original manuscript or publication as clearly as possible; old English spelling has been retained but archaic English letter forms have been modernised. Clef signs, time signatures, and key signatures, as well as accidentals, have been used according to modem practice. The figured bass is given as featured in the sources and is not realised or expanded. / AFRIKAANSE OPSOMMING: Die sekulêre liedere van John Blow (1649-1708): 'n edisie Die hoofdoel van die proefskrif is om John Blow se 109 sekulêre liedere toeganklik te maak vir uitvoerings- en navorsingsdoeleindes deur hulle vir die eerste keer as een versameling bymekaar te bring, getranskribeer in moderne musieknotasie. Van besondere belang is die insluiting van dertien van Blow se liedere wat nog nie voorheen gepubliseer is nie en slegs in manuskripvorm in spesiale versamelings in Engeland beskikbaar is, en liedere wat wel tydens Blow se lewe gepubliseer is, maar eweneens nie toeganklik is vir die algemene publiek nie. Die leesbaarheid van sommige liedere word bemoeilik as gevolg van die ouderdom en toestand van die papier. In 'n paar gevalle is die papier so dun dat die agterste notebeeld deurslaan en die leesbaarheid erg belemmer. Waar 'n manuskrip en kontemporêre weergawe van 'n lied bestaan, kon betekenisvolle vergelykings getref word. 'n Belangrike 'ontdekking' wat gemaak is, is dat die outograaf-manuskrip vanAh me, Undone (Lbl Add. 31457) nie 'n afsonderlike lied is soos wat Watkins Shaw (1980) dit gelys het nie, maar 'n gedeelte is van die lied Happy the man, who languishing (1700). Dit maak dit moontlik om in dié een geval Blow se oorspronklike komposisie te vergelyk met Playford se gepubliseerde weergawe daarvan. Dit is eers gedurende die twintigste eeu dat belangstelling in Blow se musiekgeleidelik begin posvat het. Frederick George Edwards (1902) en veral William Cummings (1909) het 'n belangrike rol in die Blow-herlewing gespeel. Harold Watkins Shaw neem vanaf 1936 die leiding met Leland Clarke (1949), 'n Amerikaner, wat die die volgende fase verteenwoordig. Bruce Wood staan vanaf 1975 aan die voorpunt met sy navorsing oor Blow se Anthems. Anthony Rooley, direkteur van die Consort of Musicke in Londen asook Peter Holman, direkteur van die Parley of Instruments het vanaf 1987 met hulle uitvoerings en opnames 'n groot rol gespeel om Blow se musiek weer aan die publiek bekend te stel. Hierdie tesis, asook my Magister-verhandeling (Grobler 1993) vorm ook deel van hierdie nuwe belangstelling in Blowen Engelse Restourasie-musiek. Volume 1: In die eerste hoofstuk word die sekulêre lied van die Engelse Restourasie-era (1660- 1714) in perspektief geplaas. Saam met 'n bespreking oor bydraes van John Blow, Henry Purcell en hulle tydgenote tot die genre, word die ontwikkeling van die lied en uitvoeringspraktyke daarvan krities ondersoek. Terselfdertyd word insae verkry in die' tydgenootlike liedpublikasies in Engeland gedurende die laat 17de en vroeg 18de eeu. Daarna volg 'n bespreking van Blow se styleienskappe soos dit veral in die sekulêre lied manifesteer. Kritiek wat Blow se werkswyse deur die eeue van musiek-kritici soos bv. Charles Burney (1726-1814) uitgelok het, word in historiese perspektief geplaas. Blow se bydrae tot die verskillende liedtipes word breedvoerig bespreek: die sololied, die twee-stemmige lied en dialogue, die lied vir meer as twee stemme en die lied as bykomstige teatermusiek. Die styleienskappe van hierdie liedere, die invloed van die Franse en Italiaanse vokale komposisies en die sterk invloed wat Charles II se voorkeure op die hofkomponiste se musiek gehad het, word uitgewys. Die funksie van die lied in die destydse samelewing kom ook onder die loep. Die redaksionele werkswyse wat gevolg is met die transkribering van die liedere is gebaseer op moderne benaderings tot dié veld soos Caldwell (1985) dit voorstaan en toegepas word in die Musica Britannica (1953, 1993, 1996). 'n Bespreking oor die uitvoeringspraktyk van die lied wil 'n bydrae lewer tot die begrip van die term 'Song' soos dit in die Restourasie-tydperk verstaan is. Die tekstuele kommentaar lig aspekte soos notasie in die liedere uit en gee toeligting oor manuskripte en publikasies wat die ondersoek ten grondslag lê. Ten slotte word al 109liedere, asook liedmusiek, en -publikasies in 'n sistematiese indeks saamgevat. Volume 2: In dié volume verskyn die 109 sekulêre liedere chronologies in moderne geredigeerde vorm. Die liedere word sover moontlik weergegee soos dit in die oorspronklike manuskripte en publikasies voorkom: die ouer Engelse spelling word behou maar argaïese lettervorme is gemoderniseer. Sleuteltekens, tydmaattekens, toonsoorttekens en skuiftekens is in ooreenstemming met huidige praktyke. Die besyferde bas IS weergegee soos dit in die bronne voorkom, en is nie aangevul of gerealiseer nie.

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The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence. In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will be measured via a direct approach which involves the derivation of representation formulae for solutions to the problem. For a real parameter lambda, several classes of initial data are considered. These include the class of smooth functions with either zero or nonzero mean, a family of piecewise constant functions, and a large class of initial data with a bounded derivative that is, at least, continuous almost everywhere and satisfies Holder-type estimates near particular locations in the domain. Amongst other results, our analysis will indicate that for appropriate values of the parameter, the curvature of the data in a neighborhood of these locations is responsible for an eventual breakdown of solutions, or their persistence for all time. Additionally, we will establish a nontrivial connection between the qualitative properties of L-infinity blow-up in ux, and its Lp regularity. Finally, for smooth and non-smooth initial data, a special emphasis is made on the study of regularity of stagnation point-form solutions to the two and three dimensional incompressible Euler equations subject to periodic or Dirichlet boundary conditions.