Abstract in English / In this dissertation we apply linear evolution equations to the Newtonian derivative, Caputo time fractional derivative and $-time fractional derivative. It is notable that the most utilized fractional order derivatives for modelling true life challenges are Riemann- Liouville and Caputo fractional derivatives, however these fractional derivatives have the same weakness of not satisfying the chain rule, which is one of the most important elements of the match asymptotic method [2, 3, 16]. Furthermore the classical bounded perturbation theorem associated with Riemann-Liouville and Caputo fractional derivatives has con rmed not to be in general truthful for these models, particularly for solution operators of evolution systems of a derivative with fractional parameter ' that is less than one (0 < ' < 1) [29]. To solve this problem, we introduce the derivative with new parameter, which is de ned as a local derivative but has a fractional order called $-derivative and apply this derivative to linear evolution equation and to support what we have done in the theory, we utilize application to population dynamics and we provide the numerical simulations for particular cases. / Mathematical Sciences / M.Sc. (Applied Mathematics)

Contrary to popular belief, we can’t all be winners. Suppose 6 people compete in a chess tournament in which all pairs of players compete directly and no ties are allowed; i.e., 6 people compete in a ‘round robin tournament’. Each player is assigned a ‘score’, namely the number of games they won, and the ‘score sequence’ of the tournament is a list of the players’ scores. Determining whether a given potential score sequence actually is a score sequence proves to be difficult. For instance, (0, 0, 3, 3, 3, 6) is not feasible because two players cannot both have score 0. Neither is the sequence (1, 1, 1, 4, 4, 4) because the sum of the scores is 16, but only 15 games are played among 6 players. This so called ‘tournament score sequence problem’ (TSSP) was solved in 1953 by the mathematical sociologist H. G. Landau. His work inspired the investigation of round robin tournaments as directed graphs. We study a modification in which the TSSP is cast as a system of inequalities whose solutions form a polytope η-dimensional space. This relaxation allows us to investigate the possibility of fractional scores. If, in a ‘round-robin’-ish tournament, Players A and B play each other 3 times, and Player A wins 2 of the 3 games, we can record this interaction as a 2/3 score for Player A and a 1/3 score for Player B. This generalization greatly impacts the nature of possible score sequences. We will also entertain an interpretation of these fractional scores as probabilities predicting the outcome of a true round robin tournament. The intersection of digraph theory, polyhedral combinatorics, and linear programming is a relatively new branch of graph theory. These results pioneer research in this field.

For decades, the importance of information technology (IT) for organizations has continuously increased. To ensure effective IT management and operations, to enable the business, and to stay competitive, many large organizations appoint a Chief Information Officer (CIO) as their highest-ranked IT executive (Preston et al., 2008). While many small to medium-sized enterprises (SMEs) face similar challenges concerning IT, they often lack the resources to employ a full-time IT executive (Bhagwat & Sharma, 2007; Cragg et al., 2013). However, Kamariotou and Kitsios (2022) find that strategic planning and alignment with IT are important determinants for increasing an SME’s business value. An increasing number of SMEs started to employ CIOs part-time to avoid the risk of not staying competitive and losing business value (Moise, 2021; Pratt, 2022). Such so-called “Fractional CIOs” are often highly experienced and skilled individuals working for multiple SMEs simultaneously (Kratzer et al., 2022). By contracting a Fractional CIO, SMEs benefit in various ways. First, they get an experienced IT leader at a fraction of the cost. Second, they remain flexible concerning billable hours and the time horizon of the engagement. Third, they are receiving an outside-in perspective and might profit from the Fractional CIO’s experience gained from simultaneous engagements. Even though the Fractional CIO role benefits SMEs, adoption and awareness are still low. Further, substantial research on the novel role is missing. This dissertation aims to establish a new stream in CIO research that explores the role of the Fractional CIO. Our comprehensive literature review of the CIO research field provides the basis for this research (Kratzer et al., 2023b). As we explore this novel phenomenon, we are also interested in how to best report research findings. We, therefore, aim to conceptualize techniques to convey novel phenomena in information systems (IS) research in an illustrative way. Hence, we posed the following overarching research questions: RQ1: How can novel phenomena be communicated in an illustrative yet rigorous way in research papers? RQ2: What is the state-of-the-art in the CIO research field, and what are further avenues for research? RQ3: What concepts provide SMEs with the experienced IT executive management they need? RQ4: What makes Fractional CIO engagements successful? RQ5: How does the role of the Fractional CIO take shape in practice in the German market? These overarching research questions are answered through six consecutive publications. 1) 'Literary Sketches in Information Systems Research: Conceptualization and Guidance for Using Vignettes as a Narrative Form' This paper is conceptual and has a methodological focus on the use of vignettes. The paper contributes to the usage of vignettes as a narrative form in academic research, providing a taxonomy that structures vignettes and identifies three archetypes. By conceptualizing vignettes through a taxonomy and archetypes, we create transparency for the usage of vignettes, encourage the use of vignettes, and provide recommendations how to use them more rigorously. Long term, this might contribute to a better exchange between academia and practice through better comprehensibility of academic papers. 2) 'Four Decades of Chief Information Officer Research: A Literature Review and Research Agenda Based on Main Path Analysis' In this paper, we conducted a bibliographic literature review using main path analysis, which helped to objectively identify existing topics based on their importance. We also illustrated the knowledge flow in CIO research by identifying major and emerging research streams and analyzing their evolution over time. Additionally, we aggregated central papers in CIO research and developed a research agenda to provide guidance for future research. Overall, this paper helps to advance the understanding of the CIO research field and provides insights for researchers and practitioners. 3) 'The Fractional CIO in SMEs: Conceptualization and Research Agenda' With this paper, we were the first to conceptualize the role of the Fractional CIO and to develop a research agenda for this novel research field. The results from this paper contribute to research and practice in several ways. First, we are the first authors to conceptualize the role of the Fractional CIO, derive a definition, compare it to existing CIO role research, and derive four engagement types. Second, we develop a research agenda to guide future research in the new Fractional CIO research field. Third, we contribute to practice by promoting the role’s awareness among potential Fractional CIOs and organizations. Fourth, we show that Fractional CIO services are versatile and can benefit organizations of different sizes and maturity levels, and there are diverse possibilities for evolution. 4) 'What Makes Fractional CIO Engagements in SMEs Successful? – A Research Framework' In this paper, we developed a research framework for Fractional CIO engagement success. It makes contributions to research and practice. First, we propose a research framework for Fractional CIO engagement success and, therefore, develop a common ground for future research. Second, we provide practical advice for Fractional CIOs and SMEs regarding factors for engagement success. Third, we raise awareness about the Fractional CIO role that may benefit many SMEs around the world. 5) 'Factors for Fractional CIO Engagement Success' In this paper, we used Q-methodology to empirically evaluate the relative importance of factors influencing Fractional CIO engagement success and to identify different viewpoints on engagement success. Our paper makes several contributions to research and practice. First, our paper is the first to identify and evaluate the importance of factors for Fractional CIO engagement success. Therefore, it serves as a basis for further research in the new Fractional CIO research field. Second, we find that upper echelon theory (Hambrick & Mason, 1984) can be applied to executives independent of their contractual relationship in cases of comparable responsibilities. Third, by thoroughly applying Q-methodology, we showcase an exploratory tool for identifying opinions and preferences of participants and clustering them accordingly. This approach provides guidance for other IS scholars to apply it. Fourth, we develop preliminary archetypes of Fractional CIOs and, based on that, provide tentative advice for Fractional CIOs and organizations that hire them. These preliminary archetypes enabled us to identify further theoretical explanations of this phenomenon. In addition to transaction cost theory, agency theory, and upper echelon perspective, we found that stewardship theory and dynamic capabilities explain additional characteristics of Full-ownership CIOs and Change Agents. 6) 'Mehr als Vollzeit: Fractional CIOs in KMUs' This paper summarizes the results of our previous studies regarding the Fractional CIO role in German language and conducts semi-structured interviews with three Fractional CIOs from Germany to complement it with a perspective on the German market. Overall, we find that the Fractional CIO role in the German market is performed similarly to the international market. The interviewed German Fractional CIOs/CTOs could not clearly identify any hurdles that might explain the low adoption of the role. However, all three German Fractional CIOs agree that German SMEs would strongly benefit from Fractional CIOs. (References to be found in the full text)

[EN] In this thesis, we consider two problems: we first explore the application of visibility graphs for describing the orbits of a discrete dynamical system that is governed by a fractional version of the logistic equation. We also study how to use this type of graphs to study response time series from the perspective of psychology. The preliminaries and introduction of these visibility graphs are presented in Chapter 1, where we revisit some basic facts from network science related to them. In the first part of this thesis, we analyze a phenomenon of mathematical nature. Wu and Baleanu introduced a fractional discrete dynamical system inspired by the fractional difference logistic equation. In order to study the trajectories of this model under this perspective of network science, in Chapter 2, we first review the most used fractional derivatives (Riemann-Liouville, Caputo, and Gründwald-Letnikov). Later, we show how to consider discrete fractional derivatives. Within our work, we present an alternative way of deducing the governing equation with respect to the one shown by Wu and Baleanu. We revisit the Wu-Baleanu equation in Chapter 3, focused on the visibility graphs of trajectories generated under different values of the scaling factor and the fractional exponent. We also study the existing connections between these parameters and the fitting with the degree distribution of the corresponding visibility graphs. When chaos is present, we link them with the exponent obtained when fitting the degree distribution to a power-law of the form x^(¿¿). With this approach, we provide an integrated vision of the dynamics of a family of fractional discrete dynamical systems that cannot be obtained from single Feigenbaum diagrams computed for each scaling factor and fractional exponent. We also connect the power-law exponent of the degree distribution fitting with the Shannon entropy of the visibility graphs degree distribution. In the second part, we analyze the response times of students to a binary decision task from the perspective of network science. We analyze the properties of the natural visibility graphs associated with their reaction time series. We observe that the degree distribution of these graphs usually fits a power-law distribution p(x) = x^(¿¿). We study the range in which parameter ¿ occurs and the changes of this exponent with respect to the age and gender of the students. Besides, we also study the links between the parameter ¿ and the ex-Gaussian distribution parameters that best fits each subject's response times. Finally, we outline some conclusions and perspectives of future research in both parts in Chapter 6. / [ES] En esta tesis, hemos considerado dos problemas: primero exploramos la aplicación de los grafos de visibilidad para describir las órbitas de un sistema dinámico discreto que está gobernado por una versión fraccionaria de la ecuación logística. Además, también estudiamos cómo usar este tipo de grafos para estudiar series temporales de tiempos de respuesta desde una perspectiva psicológica. Los preliminares, así como una introducción a estos grafos de visibilidad, se presentan en el Capítulo 1, donde revisitamos algunos hechos básicos de la ciencia de redes relacionados con dichos grafos. En la primera parte de esta tesis, analizamos un fenómeno de naturaleza matemática. Wu y Baleanu introdujeron un sistema dinámico discreto fraccionario inspirado en la ecuación logística con derivadas fraccionarias. Con el propósito de estudiar las trayectorias de este modelo desde la perspectiva de la ciencia de redes, en el Capítulo 2, primero revisamos las derivadas fraccionarias más utilizadas (Riemann-Liouville, Caputo y Gründwald-Letnikov). Posteriormente, mostramos cómo considerar derivadas fraccionarias discretas. En nuestro trabajo, presentamos una forma alternativa de deducir la ecuación gobernante con respecto a la presentada por Wu y Baleanu. Revisitamos la ecuación de Wu-Baleanu en el Capítulo 3, centrado en los grafos de visibilidad de trayectorias generadas a partir de distintos valores del factor de escala y del exponente fraccionario. También estudiamos la existencia de conexiones entre estos parámetros y el ajuste de la distribución de los grados de los correspondientes grafos de visibilidad. Cuando el caos está presente, los enlazamos con el exponente obtenido al ajustar la distribución de los grados a una ley de potencias de la forma x^(¿¿). A través de este enfoque, proporcionamos una visión integrada de la dinámica de una familia de sistemas dinámicos discretos fraccionarios que no se pueden obtener a partir de diagramas de Feigenbaum individuales calculados para cada factor de escala y exponente fraccionario. Además, relacionamos el exponente de la ley de potencias del ajuste de la distribución de grados con la entropía de Shannon de la distribución de grados de los grafos de visibilidad. En la segunda parte, analizamos el tiempo de respuesta de un grupo de estudiantes que realizaron una tarea de decisión binaria desde la perspectiva de la ciencia de redes. Estudiamos las propiedades de los grafos de visibilidad natural asociados con sus correspondientes series de tiempos de respuesta. Observamos que la distribución de los grados de estos grafos normalmente sigue una distribución ley de potencias p(x) = x^(¿¿). Analizamos el rango en el cual el parámetro ¿ se mueve y los cambios de este exponente con respecto a la edad y el sexo de los estudiantes. Por otro lado, también estudiamos la relación entre el parámetro ¿ y los parámetros de la distribución ex-Gaussiana que mejor se ajusta al tiempo de respuesta de cada sujeto. Finalmente, destacamos algunas conclusiones y perspectivas de investigación futura en ambas líneas de trabajo en el Capítulo 6. / [CAT] En aquesta tesi, hem considerat dos problemes: primer explorem l'aplicació dels grafs de visibilitat per a descriure les òrbites d'un sistema dinàmic discret que està governat per una versió fraccionària de l'equació logística. A més a més, també estudiem com emprar aquest tipus de grafs per a analitzar sèries temporals de temps de resposta des d'una perspectiva psicològica. Els preliminars, així com una introducció a aquests grafs de visibilitat, es presenten al Capítol 1, on revisitem alguns fets bàsics de la ciència de xarxes relacionats amb ells. En la primera part d'aquesta tesi, analitzem un fenomen de naturalesa matemàtica. Wu i Baleanu van introduir un sistema dinàmic discret fraccionari inspirat en l'equació logística amb derivades fraccionàries. Amb el fi d'estudiar les trajectòries d'aquest model des d'una perspectiva de la ciència de xarxes, en el Capítol 2, primer revisem les derivades fraccionàries més utilitzades (Riemann-Liouville, Caputo i Gründwald-Letnikov). Posteriorment, mostrem com considerar derivades fraccionàries discretes. Al nostre treball, presentem una forma alternativa de deduir l'equació governant respecte a la presentada per Wu i Baleanu. Revisitem l'equació de Wu-Baleanu al Capítol 3, focalitzat en els grafs de visibilitat de trajectòries generades a partir de valors diferents del factor d'escala i de l'exponent fraccionari. També estudiem l'existència de connexions entre aquests paràmetres i l'ajust de la distribució dels graus dels corresponents grafs de visibilitat. Quan el caos hi és, els enllacem amb l'exponent que hem obtés en ajustar la distribució dels graus a una llei de potències de la forma x^(¿¿). Des d'aquesta perspectiva, proporcionem una visió integrada de la dinàmica d'una família de sistemes dinàmics discrets fraccionaris que no es poden obtenir a partir de diagrames de Feigenbaum individuals calculats per a cada factor d'escala i exponent fraccionari. A més a més, relacionem l'exponent de la llei de potències de l'ajust de la distribució de graus amb l'entropia de Shannon de la distribució de graus dels grafs de visibilitat. A la segona part, analitzem el temps de resposta d'un grup d'estudiants que realitzaren una tasca de decisió binària des del punt de vista de la ciència de xarxes. Estudiem les propietats dels grafs de visibilitat natural associats amb les seues corresponents sèries temporals de temps de resposta. Observem que la distribució dels graus d'aquests grafs normalment segueix una distribució llei de potències p(x) = x^(¿¿). Analitzem el rang en què el paràmetre ¿ es mou i els canvis d'aquest exponent respecte a l'edat i el sexe dels estudiants. D'altra banda, també estudiem la relació entre el paràmetre ¿ i els paràmetres de la distribució ex-Gaussiana que millor fita el temps de resposta de cada subjecte. Finalment, destaquem algunes conclusions i perspectives d'investigació futura en ambdues línies de treball en el Capítol 6. / Mira Iglesias, A. (2021). Applications of Visibility Graphs for the representation of Time Series [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/176012

Thesis (PhD)--University of Stellenbosch, 2002. / ENGLISH ABSTRACT: This study describes the design, construction, complete industrialisation and operation of a dual bed vacuum swing adsorption (VSA) demonstration plant, which operates at atmospheric and sub-atmospheric conditions. All design objectives as set out initially were met. The plant removes contaminants such as methanol and water from neutral wine spirit. Neutral wine spirit is a key component of various local and international spirituous products which include liqueurs, gin, vodka, fortified wines and brandy. Neutral wine spirit can chemically be described as the azeotropic mixture of ethanol and water, which occurs at an ethanol content of 96.4 vol. %. Methanol is naturally present in all products from the vine. Fermentation and distillation concentrate methanol even more, and due to physical and chemical characteristics, its separation consumes as much as 45 % of total production costs. Neutral wine spirit is produced by the proven technology of continuous atmospheric distillation. Continuous improvement of the distillation process is limited due to the physical constraints of an old facility, but also due to previous design philosophies and approaches. The VSA plant consists of two adsorbers, packed to a total height of 1.71 m and a diameter of 0.4 m. Adsorption took place at 100 °C and regeneration at the same bed temperature with purified nitrogen gas at 170 °C and a vacuum of 17 kPa (abs). Experiments were divided into Group I and Group II experiments. Group I investigated the ability to separate methanol and water from the azeotrope and to which efficiency it occurred. It consisted of 120 adsorption cycles of 5 minutes each and 60 samples were drawn for analyses. Breakthrough was not allowed to occur. The azeotropic feed was consistently dehydrated to a water content < 0.05 wt %, while methanol was reduced to < 4 mg/100mLAA. The type of 3A molecular sieve (MS 564 CS) was specifically selected to ensure analytic as well as organoleptic compliance with the product specification. Molecular sieve 4A was removed due to organoleptic problems with the product. Group" experiments were performed in the format of a sensitivity analysis. The effects of various process parameters on the methanol breakthrough curves were individually assessed. Eighteen experiments were performed over a period of 8 days, with 86 samples drawn. The duration of an adsorption cycle was 30 minutes, allowing methanol breakthrough to occur. Water was preferentially adsorbed. Negative methanol bed loadings during high water loadings confirmed that water was able to displace methanol molecules. In the presence of water, molecular sieve 3A was capable of adsorbing 0.6 mg methanol/100mLAA, while in the absence of water with synthetically dosed methanol, molecular sieve 3A achieved a maximum loading of 12.3 mg methanol/100mLAA. The latter corresponded with a maximum methanol feed content of 1118 mg/100mLAA. In general, quicker breakthrough occurred at higher flow rates and feed concentrations. Continuous breakthrough caused bed contamination and a 24-hour thermal regeneration was performed following experiment 12. The feed flow rate was increased from the theoretical 50 f/hr to 70 f/hr without any additional capital layout. Selected process conditions were found to be effective in continuously separating methanol from ethanol. Depending on the strategy of integration, profitability studies shows a Return on Investment of between 110.1% - 220.8% for the adsorption project. Adsorption is superior to distillation in the separation of methanol. Due to the level of innovation involved, it is recommended that the contents of this study remain confidential and patent protection is to be extended. This dissertation speaks to both the wine making as well as the chemical engineering fraternity. It seeks to provide credibility to both parties, by clarifying the unknown issues fundamental to the respective disciplines. / AFRIKAANSE OPSOMMING: Hierdie studie definieer die ontwerp, vervaardiging en volledige industrialisasie van 'n dubbelbed vakuum adsorpsie demonstrasie aanleg (VSA) wat by atmosferiese en sub-atmosferiese kondisies bedryf word. Alle ontwerpsdoelwitte is bereik. Die aanleg verwyder selektief metanol en water vanuit neutrale wynspiritus. Neutrale wynspiritus is 'n sleutelkomponent van verskeie spiritualieë in die plaaslike en internasionale wyn en spiritus bedryf. Hierdie produkte sluit in likeurs, jenewer, vodka, gefortifiseerde wyne en brandewyn. Chemies, kan neutrale wynspiritus beskryf word as die azeotropiese mengsel van etanol en water teen 96.4 vol. %. Metanol het 'n natuurlike teenwoordigheid in alle produkte vanaf die wynstok. Gisting en distillasie konsentreer metanol tot 'n hoër mate en weens fisiese en chemiese eienskappe word metanol teen hoë koste vanaf die etanol stroom geskei. Die metanol verwyderingskomponent beloop soveel as 45 % van die produksiekoste van die totale proses. Neutrale wynspiritus word deur die gevestigde tegnologie van kontinue atmosferiese distillasie geproduseer. Kontinue verbetering van die proses word beperk deur die fisiese ouderdom en toestand van die fasiliteite, maar is ongelukkig ook die resultaat van vorige ontwerpsfilosofieë en benaderings. Die adsorbeerders is gepak tot 'n hoogte van 1.71 m met 'n deursnit van 0.4 m. Adsorpsie het by 100°C plaasgevind en regenerasie by dieselfde bedtemperatuur met stikstofgas by 170°C en 'n vakuum van 17 kPa (abs). Eksperimentele werk is in Groep I en Groep II eksperimente verdeel. Groep I het die effektiewe prosesvermoë om metanol en water vanuit die azeotroop te verwyder ondersoek. Dit het bestaan uit 120 adsorpsie siklusse van 5 minute elk. Sestig monsters is getrek vir analise. Deurbreek van metanol was nie toe gelaat om plaas te vind nie. Die azeotropiese toevoer is konsekwent tot 'n water inhoud < 0.05 massa % gedehidrateer is. 'n Metanol inhoud < 4 mg/100 mLAA is bereik. Die tipe 3A molekulêre sif (MS 564 CS) was spesifiek vir die toepassing geselekteer om sodoende 'n analities sowel as organolepties aanvaarbare produk te lewer. Molekulêre sif 4A was verwyder weens die vorming van produk wangeure. Groep II eksperimente is in die vorm van 'n sensitiwiteits analise uitgevoer. Die effek van verskeie veranderlikes is individueel op die metanol deurbreekkurwe getoets. Agtien eksperimente is oor 'n tydperk van 8 dae gedoen, met 'n totaal van 86 monsters wat getrek is. 'n Adsorpsie siklus het 30 minute geduur en het deurbreek van metanol toegelaat. Water is by voorkeur geadsorbeer. Negatiewe metanol bed ladings tydens hoë waterteenwoordigheid toon dat water wel metanolmolekule op 'n adsorpsie-posisie kan verplaas. In die teenwoordigheid van water is 'n bedlading van 0.6 mg metanol/100mLAA verkry, met 'n maksimum van 12.3 mg metanol/100mLAA in die afwesigheid van water. Laasgenoemde verteenwoordig 'n toevoer met 'n metanol inhoud van 1118 mg/100mLAA. In die algemeen is gevind dat 'n toename in toevoer vloeitempo en konsentrasie die tyd vir deurbreek verkort. Kontinue deurbreek het kontaminasie van die bed teweeg gebring en 'n termiese regenerasie is vir 24 ure na eksperiment 12 gedoen. Die teoretiese ontwerps vloeitempo was 50 .elhr, maar resultate het getoon dat die aanleg tot 70 .t'/hr kan verwerk sonder addisionele koste. Die geselekteerde proseskondisies was effektief in die versekering van die kontinue skeiding van metanol en etanol. Die lewensvatbaarheidstudie toon, afhangende van die strategie van integrasie, 'n Opbrengs op Belegging van tussen 110.1% - 220.8%. Adsorpsie het 'n beter skeidingsvermoë as konvensionele distillasie vir die verwydering van metanol vanuit etanol. Weens die vlak van innovasie betrokke, word dit voorgestel dat die inhoud van hierdie studie vertroulik gehou word en dat patent beskerming verleng sal word. Hierdie verhandeling spreek tot beide die wynmaak sowel as chemiese ingenieurs dissiplines. Daar is gepoog om geloofwaardigheid vir beide partye te skep deur die onbekende aspekte van albei dissiplines aan te spreek.

Fractional crystallization, magma mixing, assimilation of continental crust, and how those processes modify volatile budgets, control the evolution of magma. As a consequence, the understanding of these processes, their magnitudes, and timescales is critical for interpreting ancient magma systems, their eruptions, and the potential future volcanic activity. In this dissertation I present the results of three projects. The first explores how magmatic processes affect magma reservoirs and eruption dynamics. The second explores where in the storage system and how often these processes occur. The third explores how volatile budgets are modified by processes such as crystallization, mixing, and assimilation. Volcán Popocatépetl (central México) erupted ~14100 14C yr BP producing the Tutti Frutti Plinian Eruption (TFPE). The eruption tapped two different silicic magmas that mixed just prior and during the eruption. The influx of mass and volatiles generated during the mixing of both magmas overpressured the reservoir, which was weakened at the top. The weakened reservoir relaxed while magma was tapped and collapsed to form a caldera at the surface. Although it is known that fractional crystallization, mixing, and assimilation can greatly modify magmas, the frequency and intensity of these events is not known. I investigated the magmatic processes responsible for the evolution of magmas erupted during five Plinian events of Popocatépetl volcano. Results show that during the last 23 ky magma was stored in two different zones, and was variably modified by replenishments of mafic magma. Interestingly, little evidence for large mafic inputs triggering explosive eruptions was found. Each of these processes alters the abundances of volatiles and introduces different types of volatiles to the system. Hence, the volatile budget of magma can have a rich and complex history. To investigate how volatile budgets evolve in active magma systems, I analyzed the abundances of volatiles (H2O, CO2, F, Cl, and S) in numerous glass inclusions trapped in phenocrysts. Results show that the magmas that produced the last five Plinian eruptions at Popocatépetl volcano evolved by crystallization and magma mixing, assimilation of the local carbonate basement is not chemically appreciable. Mixing with mafic magmas added substantial CO2 and S to the system, dewatered the magma, yet produced little change in the F contents of the magmas. / text

The dissertation considers an indirect approach for the estimation of the cointegrating parameters, in the sense that the estimators are jointly constructed along with estimating other nuisance parameters. This approach was proposed by Robinson (2008) where a bivariate local Whittle estimator was developed to jointly estimate a cointegrating parameter along with the memory parameters and the phase parameters (discussed in chapter 2). The main contributions of this dissertation is to establish, similar to Robinson (2008), a joint estimation of the memory, cointegrating and phase parameters in stationary and nonstationary fractionally cointegrated models in a multivariate framework. In order to accomplish such task, a general shape of the spectral density matrix, first noted in Davidson and Hashimzade (2008), is utilised to cover multivariate jointly dependent stationary long memory time series allowing more than one cointegrating relation (discussed in chapter 3). Consequently, the notion of the extended discrete Fourier transform is adopted based on the work of Phillips (1999) to allow for the multivariate estimation to cover the non stationary region (explained in chapter 4). Overall, the estimation methods adopted in this dissertation follows the semiparametric approach, in that the spectral density is only specified in a neighbourhood of zero frequency. The dissertation is organised in four self-contained chapters that are connected to each other, in additional to this introductory chapter: • Chapter 1 discusses the univariate long memory time series analysis covering different definitions, models and estimation methods. Consequently, parametric and semiparametric estimation methods were applied to a univariate series of the daily Egyptian stock returns to examine the presence of long memory properties. The results show strong and significant evidence of long memory in the Egyptian stock market which refutes the hypothesis of market efficiency. • Chapter 2 expands the analysis in the first chapter using a bivariate framework first introduced by Robinson (2008) for long memory time series in stationary system. The bivariate model presents four unknown parameters, including two memory parameters, a phase parameter and a cointegration parameter, which are jointly estimated. The estimation analysis is applied to a bivariate framework includes the US and Canada inflation rates where a linear combination between the US and Canada inflation rates that has a long memory less than the two individual series has been detected. • Chapter 3 introduces a semiparametric local Whittle (LW) estimator for a general multivariate stationary fractional cointegration using a general shape of the spectral density matrix first introduced by Davidson and Hashimzade (2008). The proposed estimator is used to jointly estimate the memory parameters along with the cointegrating and phase parameters. The consistency and asymptotic normality of the proposed estimator is proved. In addition, a Monte Carlo study is conducted to examine the performance of the new proposed estimator for different sample sizes. The multivariate local whittle estimation analysis is applied to three different relevant examples to examine the presence of fractional cointegration relationships. • In the first three chapters, the estimation procedures focused on the stationary case where the memory parameter is between zero and half. On the other hand, the analysis in chapter 4, which is a natural progress to that in chapter 3, adjusts the estimation procedures in order to cover the non-stationary values of the memory parameters. Chapter 4 expands the analysis in chapter 3 using the extended discrete Fourier transform and periodogram to extend the local Whittle estimation to non stationary multivariate systems. As a result, the new extended local Whittle (XLW) estimator can be applied throughout the stationary and non stationary zones. The XLW estimator is identical to the LW estimator in the stationary region, introduced in chapter 3. Application to a trivariate series of US money aggregates is employed.

Background: Studies of assessing dietary fluoride intake in children have employed different dietary methods mainly “2-day duplicate” and “3-day food diary” methods. However, none of these methods have been validated or standardised. Main aims: The main aims of the current study were to develop a better understanding of strengths and weaknesses of dietary assessment methods “2-day duplicate plate” and “3-day food diary” by comparing dietary fluoride intake estimated by each method and evaluate the validity of the two methods for estimating dietary fluoride intake in young children. Methods: Sixty one healthy 4-6 year old children living in fluoridated area of the north-east of England since birth were recruited via 10 primary schools. Dietary information was collected using “2-day duplicate plate” and “3-day food diary” methods. Two 24-h urine samples and two samples of post brushing expectorate (a mixture of saliva, toothpaste and water used to rinse after brushing) from each child. Completeness of 24-h urine samples was checked using urinary excretion of creatinine and urinary flow rate. Validity of the two dietary assessment methods was checked by measuring urinary excretion of nitrogen and potassium as independent validity checks. Total daily fluoride intake from diet and toothpaste ingestion and urinary fluoride excretion was determined for each child. Results: All participated children completed all aspects of the study. According to the validity criteria, dietary data of 58 (95%) children, when collected by the 3-day food diary, were considered valid. However, when the dietary data were collected by the 2-day duplicate plate method, the data for 46 (75%) children were viewed as valid. Mean total dietary fluoride intake was 0.533 mg/d by the 3-day food diary method and 0.583 mg/d by the 2-day duplicate plate method. No statistically significant difference in total dietary fluoride intake was observed between the two methods. The mean difference in estimated dietary fluoride intake by the two dietary assessment methods was -0.050 mg/d with 95% limits of agreement of -0.501 to + 0.401 mg/d. Conclusion: Either the 3-day food diary or the 2-day duplicate plate method can be used when investigating mean total daily dietary fluoride intake of a population. However the methods cannot be used interchangeably at the individual level.

The dynamic approach to fractional Brownian motion (FBM) establishes a link between non-Poisson renewal process with abrupt jumps resetting to zero the system's memory and correlated dynamic processes, whose individual trajectories keep a non-vanishing memory of their past time evolution. It is well known that the recrossing times of the origin by an ordinary 1D diffusion trajectory generates a distribution of time distances between two consecutive origin recrossing times with an inverse power law with index m=1.5. However, with theoretical and numerical arguments, it is proved that this is the special case of a more general condition, insofar as the recrossing times produced by the dynamic FBM generates process with m=2-H. Later, the model of ballistic deposition is studied, which is as a simple way to establish cooperation among the columns of a growing surface, to show that cooperation generates memory properties and, at same time, non-Poisson renewal events. Finally, the connection between trajectory and density memory is discussed, showing that the trajectory memory does not necessarily yields density memory, and density memory might be compatible with the existence of abrupt jumps resetting to zero the system's memory.

This thesis is comprised of three main parts: The Hermite-Hadamard inequality on discrete time scales, the fractional Hermite-Hadamard inequality, and Karush-Kuhn- Tucker conditions on higher dimensional discrete domains. In the first part of the thesis, Chapters 2 & 3, we define a convex function on a special time scale T where all the time points are not uniformly distributed on a time line. With the use of the substitution rules of integration we prove the Hermite-Hadamard inequality for convex functions defined on T. In the fourth chapter, we introduce fractional order Hermite-Hadamard inequality and characterize convexity in terms of this inequality. In the fifth chapter, we discuss convexity on n{dimensional discrete time scales T = T1 × T2 × ... × Tn where Ti ⊂ R , i = 1; 2,…,n are discrete time scales which are not necessarily periodic. We introduce the discrete analogues of the fundamental concepts of real convex optimization such as convexity of a function, subgradients, and the Karush-Kuhn-Tucker conditions. We close this thesis by two remarks for the future direction of the research in this area.