We investigate reaction-diffusion models for populations whose members undergo two specific processes: dispersal and settling. Systems of this type occur throughout biological science, in contexts ranging from ecology to cell biology.Here we consider three distinct applications, namely: / • animal translocation, / • the invasion of a domain by precursor and differentiated cells, and / • the development of tissue-engineered cartilage. / Mathematical modelling of these systems provides an understanding of the population-level patterns that emerge from the behaviour of individuals. / A multi-species reaction-diffusion model is developed and analysed for each of the three applications. We present numerical results, which are illuminated through analytical results derived for simplified or limiting cases. For these special cases, results are obtained using analytical techniques including perturbation analysis, travelling wave analysis and phase plane methods. These analytic results provide a more complete understanding of system behaviour than numerical results alone. Emphasis is placed on connecting modelling results with experimental observations. / The first application considered is animal translocations. Translocations are widely used to reintroduce threatened species to areas where they have disappeared. A variety of different dispersal and settling mechanisms are considered, and results compared. The model is applied to a case study of a double translocation of the Maud Island frog, Leiopelma pakeka. Results suggest that settling occurs at a constant rate, with repulsion playing a significantrole in dispersal. This research demonstrates that mathematical modelling of translocations is useful in suggesting design and monitoring strategies for future translocations, and as an aid in understanding observed behaviour. / The second application we investigate is the invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric (intestinal) nervous system, but is presented in general terms and is of broader applicability. Regions of the parameter space are characterised according to existence, shape and speed of travelling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, these results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells. / As a third application, we propose a model for the development of cartilage around a single chondrocyte. The limited ability of cartilage to repair when damaged has led to the investigation of tissue engineering as a method for reconstructing cartilage. As in healthy cartilage, the model predicts a balance between synthesis, transport, binding and decay of matrix components. Our observations could explain differences observed experimentally between various scaffold media. Modelling results are also used to predict the minimum chondrocyte seeding density required to produce functional cartilage. / In summary, we develop reaction-diffusion models for dispersing and settling populations for three biological applications. Numerical and analytical results provide an understanding of population-level behaviour. This thesis demonstrates that mathematical modelling of biological systems can further understanding of biological systems and help to answer questions posed by experimental research.

This dissertation computationally and analytically investigates ways to model solar cells when the lateral motion of charge carriers and light are relevant. We focus on back-contact perovskite solar cells, and assessing the experimental technique of scanning photocurrent microscopy as a means to investigate them. Solar cells are three-dimensional objects frequently modelled as being one-dimensional. However, for more complex designs of solar cell or if the cell is only point-illuminated, one-dimensional modelling is insufficient. In the first study, some conditions for reducing the complexity of two-dimensional drift-diffusion simulations are investigated for a back-contact perovskite cell. Analytic expressions for the relationship in both the low extraction velocity and high extraction velocity regimes are demonstrated, and the conditions where these approximations break down are investigated. These findings are then applied a point-excited film with an extended electrode, a problem encountered during scanning photocurrent microscopy. We show the current recorded in this case should decay exponentially with the distance between excitation and electrode, with a decay constant that can be related to device parameters. The characteristic equilibration time for the system to reach this current is demonstrated to increase linearly with distance. Between this gradient and the exponent, information about the diffusion and recombination mechanics can be extracted from a variety of systems. Photon recycling is the process in whereby photogenerated carriers recombine to generate light that is absorbed again within the solar cell. In the second section, we apply the findings of the first section to show that experimental results published elsewhere are best explained by photon recycling in methylammonium lead iodide perovskite back-contact solar cells. However we do not have an established theoretical model for long-ranged lateral optical transport in these solar cells. Three models are developed: a bimolecular model for unscattered, coherent transport; a photon diffusion model for frequently scattered, noncoherent light; and a monomolecular, assisted-diffusion model. The modal nature of coherent optical transport is considered and modifications to previous one-dimensional theories are made. The nature of the photon diffusion model is discussed, as are theoretical shortcomings. All three models are then solved numerically and compared to experimental results. The low-scattering photon diffusion models correspond well to the experiment. The third investigation involves the performance of different architectures of back-contact perovskite cells. These cells potentially offer increased current due to less shadowing by front electrodes. We compare them to each other and to traditional vertical structures. It is found that, in terms of internal transport, the back-contact solar cells give less efficient performance than the vertical design. The best of the back-contact cells investigated is a flat interdigitated design. The increase in efficiency from optical factors would have to exceed 10% for the overall efficiency of back-contact cells to be higher than vertical devices. We also develop a model of photon recycling appropriate for short-ranged, bulk 2D transport and demonstrate that in perovskites, it produces little change in power conversion efficiency (and small changes in short-circuit voltage) when compared with the standard drift-diffusion equations with the second-order recombination constant is adjusted.

<p> The estimation of physiological parameters that characterize electrical signal propagation in the heart is an important component of the inverse problem in electrocardiography. Recent studies show that some patterns in cardiac electrical signals (e.g. spiral waves) are associated with the re-entrance phenomenon seen in cardiac arrhythmia. Therefore, further research in this field will lead to improved detection and diagnosis of cardiac diseases and conditions. </p> <p> Electrical activity in the heart is initiated at the SA node and an electrical impulse propagates to the atria causing their mechanical contraction. Subsequent contraction of the ventricles (systole) followed by relaxation (diastole) completes the heart cycle. Evidence of electrical activity in cardiac cells is shown by a potential difference across the cell membrane that changes when ·ionic currents flow through the membrane's channels. This electrical activation of the heart can be modeled using a diffusion model in which the physiological parameters (e.g., conductivity) govern the resulting spatiatemporal process. </p> <p> In this thesis we derive an inverse model for the electrical activation of the heart using the Fitzhugh-N agumo diffusion equations which account for the dynamics of spiral waves in excitable media such as, in our case, cardiac cells. The electric potential is expressed through activator and inhibitor variables and we simulate the measurements of the electromagnetic field are on the torso surface. A signal processing model is derived where the physiological parameters are deterministic or stochastic, and the resulting physiological measurements are a function of space, time, and the parameters. </p> <p> We estimate these unknown parameters using an optimization algorithm that minimizes the cost function of the model. For our estimation we use Least Squares and we derive the Maximum Likelihood Estimator. We measure the performance using mean square error, and we compute the Cramer-Rao Lower Bound, which shows the minimum variance attainable. </p> <p> In our simulations we use a finite element mesh of a human torso to describe a realistic geometry to generate the potentials on the surface. Our results indicate that estimating the physiological parameters of a diffusion equation from the measurements taken outside the torso are feasible. This further suggests that ECG/MCG signals can be used to provide detailed information about the physiological properties of the electrical impulse generated in the heart and aid in diagnosis of various pathological conditions including arrhythmia. </p> / Thesis / Master of Applied Science (MASc)

Financial innovation is an important research topic modern economics. Financial innovation is an ongoing process where new financial products, services and procedures are created and it concerns important financial factors such as the regulatory restrictions, the relationship between financial innovation and the functionality of financial markets, the inefficiency of markets promoted by globalization and unexpected changes of economic status and financial intermediary. The previous literature that deals with financial innovation is relatively constrained compared to the significance of the issue, which is a surprise considering the relative abundance of such research on other sectors of finance and economics. The consequences of financial innovations concern the functional framework of capital markets, the microeconomic and the macroeconomic functional frameworks. This thesis studies the influence of diffusion of financial innovation to market participants’ frictions and their values, through a theoretical, mathematical and empirical framework. We derive a novel measure of the influence of financial innovation to the market participants based on their correlation friction patterns. The main objective is to highlight a number of aspects and dimensions of this field. In particularly, we aim to present: i) the theoretical framework on the role of financial innovation at the financial structure (the fundamental generating root causes and the effects on the function of financial markets, etc), and ii) the parameterization of the influence of financial innovation to market participants through a mathematical and econometric framework based on the participants’ minimum need for change, the diffusion rate and time parameter. We undertake an extensive empirical analysis about the influence of introduction and diffusion of a financial innovation to market participants. The findings lead us to the conclusion that the parametric function, which is followed in order to show the influence of diffusion of financial innovation, has a statistically significant impact on returns and volatility of financial and economic indices. / Οι χρηματοοικονομικές καινοτομίες αποτελούν σήμερα ένα κρίσιμο πεδίο έρευνας στο οικονομικό γίγνεσθαι. Η χρηματοοικονομική καινοτομία είναι μια τρέχουσα διαδικασία ανάδειξης νέων χρηματοοικονομικών προϊόντων/υπηρεσιών και διαδικασιών και αφορά βασικούς τομείς του χρηματοοικονομικού συστήματος όπως τους κανονιστικούς περιορισμούς που αντιμετωπίζει μια αγορά, τη λειτουργία των χρηματοοικονομικών αγορών και τη διαχείριση κινδύνου, τις απροσδόκητες μεταβολές μεταβλητών και την χρηματοοικονομική διαμεσολάβηση. Η προηγούμενη βιβλιογραφία και εμπειρική έρευνα των τελευταίων ετών που αναφέρεται στη χρηματοοικονομική καινοτομία είναι σχετικά μικρή σε σχέση με τη σημαντικότητα του ζητήματος, κάτι που αποτελεί έκπληξη λαμβάνοντας υπόψη την σχετική αφθονία παρόμοιων μελετών για άλλους τομείς της χρηματοοικονομικής. Οι συνέπειες των χρηματοοικονομικών καινοτομιών είναι σημαντικές και αφορούν το λειτουργικό πλαίσιο των αγορών, το μικροοικονομικό πλαίσιο λειτουργίας των επιχειρήσεων και το μακροοικονομικό πλαίσιο λειτουργίας των επιχειρήσεων και του κράτους. Αντικείμενο της διδακτορικής διατριβής είναι η επίδραση της διάχυσης μιας χρηματοοικονομικής καινοτομίας στη διαμόρφωση των χρηματοοικονομικών τριβών της αγοράς. Ο κύριος στόχος είναι να αναδείξουμε μια σειράς πτυχών και διαστάσεων αυτού του πεδίου και, κυρίως: i) το θεωρητικό πλαίσιο του ρόλου των χρηματοοικονομικών καινοτομιών στο χρηματοοικονομικό περιβάλλον (τα θεμελιώδη γενεσιουργά αίτια και τις βασικές επιπτώσεις στη λειτουργία των χρηματοοικονομικών αγορών, κτλ) και, ii) την παραμετροποίηση της επίδρασης της χρηματοοικονομικής καινοτομίας στους συμμετέχοντες της αγοράς μέσα από ένα μαθηματικό και οικονομετρικό πλαίσιο βασισμένο στο ελάχιστο κατώτερο όριο ανάγκης για αλλαγή, στο ποσοστό διάχυσης, στις χρηματοοικονομικές τριβές μεταξύ των συμμετεχόντων της αγοράς και, του χρόνου. Διεξάγοντας μια εκτεταμένη εμπειρική ανάλυση για την επίδραση της εισαγωγής και διάχυσης μιας χρηματοοικονομικής καινοτομίας στους συμμετέχοντες μιας αγοράς, τα ευρήματα της παρούσης διατριβής οδηγούν στο συμπέρασμα ότι η παραμετρική απεικόνιση που ακολουθείται για να δείξει την επίδραση της διάχυσης της χρηματοοικονομικής καινοτομίας έχει μια στατιστικά σημαντική επίπτωση στις αποδόσεις και τη μεταβλητότητα χρηματοοικονομικών και οικονομικών δεικτών.

Στόχος της εργασίας είναι η μελέτη και η παρουσίαση των μοντέλων διάχυσης καθώς και η εφαρμογή τους στα ευρυζωνικά δίκτυα, με απώτερο σκοπό τη μελέτη της αξιοπιστίας τους ως προς τις εκτιμήσεις τους αλλά και την πρόβλεψη του ποσοστού διείσδυσης της ευρυζωνικότητας στην Ελλάδα για τα επόμενα χρόνια. / -

Η παρούσα διπλωματική εργασία αναφέρεται στην πρόβλεψη της διείσδυσης της ευρυζωνικότητας στις αστικές περιοχές της Ελλάδας. Η εν λόγω πρόβλεψη γίνεται με την χρήση μερικών μοντέλων διάχυσης. Βασικός στόχος είναι να προβλέψουμε όσο το δυνατόν καλύτερα πως θα συμπεριφερθεί η διείσδυση των ευρυζωνικών υπηρεσιών στο εγγύς μέλλον στις αστικές περιοχές της Ελλάδας. Για την μελέτη αυτή χρησιμοποιήθηκαν στοιχεία από έρευνες που καθορίζουν το ποσοστό που κατέχουν οι αστικές περιοχές σε σχέση με τον Ελλαδικό χώρο αλλά και το ποσοστό των αστικών περιοχών που έχουν πρόσβαση σε ευρυζωνικές υπηρεσίες σε σχέση με όλες τις αστικές περιοχές. Αστική περιοχή θεωρήθηκε η περιοχή στην οποία υπάρχουν τουλάχιστον 500 νοικοκυριά ανά τετραγωνικό χιλιόμετρο. Επίσης σε κάθε νοικοκυριό θεωρείτε ότι ζουν κατά μέσο όρο 2.5 άνθρωποι. Επίσης γίνεται μια εκτενής αναφορά στα δίκτυα, την αρχιτεκτονική των δικτύων καθώς και στις ευρυζωνικές υπηρεσίες και στις ευρυζωνικές τεχνολογίες πρόσβασης που υπάρχουν στις μέρες μας. Στο δεύτερο κεφάλαιο γίνεται αναφορά σε όλα τα μοντέλα διάχυσης και λεπτομερής περιγραφή καθενός από αυτά και των παραμέτρων τους. Ακολούθως γίνεται αναφορά στην ευρυζωνικότητα στην παγκόσμια κοινότητα και στην εξέλιξη που είχε και έχει στην Ελλάδα. Αναφέρονται μελέτες που αφορούν την διείσδυση της σε κάθε περιοχή καθώς και σε όλη την Ευρώπη. Επίσης γίνεται αναφορά σε μελέτες που κατηγοριοποιούν τις ευρυζωνικές γραμμές ανά ταχύτητα πρόσβασης καθώς και με βάση τις εταιρίες που προσφέρουν τις πιο πάνω υπηρεσίες. Στο τέλος γίνεται μια προσπάθεια πρόβλεψης της διείσδυσης της ευρυζωνικότητας με την βοήθεια τριών βασικών μοντέλων διάχυσης και έγινε ομαλοποίηση των εν λόγω γραφημάτων και υπολογισμών των παραμέτρων του κάθε μοντέλου με την βοήθεια αριθμητικών μεθόδων. / This dissertation concerns the provision of broadband penetration in urban areas of Greece. This prediction is use of some diffusion models. The main objective is to predict the best way possible how to behave in the penetration of broadband services in the near future in urban areas of Greece. For this study used data from surveys that determine the percentage occupied by urban areas in relation to Greece and the percentage of urban areas have access to broadband services in relation to all urban areas. Urban area was the area in which there are at least 500 households per square kilometer. Also in the household think they live on average 2.5 people. Also there is an extensive reference to networks, network architecture and broadband services and broadband access technologies that exist today. The second chapter refers to all models of diffusion and detailed description of each of them and their parameters. Then refer to broadband in the world community and the development that had been in Greece. Mentioned studies of penetration in each region and across Europe. Also refer to studies that track the speed of broadband lines per access and based on the companies offering the above services. In the end, we try predicting the penetration of broadband with the help of three basic models of diffusion and became normalization of these graphs and calculations of the parameters of each model with the help of numerical methods.

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This thesis presents the modeling of credit risk by using structural approach. Three fundamental questions of credit risk literature are analyzed throughout the research: modeling single firm credit risk, modeling portfolio credit risk and credit risk pricing. First we analyze these questions under the assumptions that firm value follows a geometric Brownian motion and the interest rates are constant. We discuss the weaknesses of the geometric brownian motion assumption in explaining empirical properties of real data. Then we propose a new extended model in which asset value, volatility and interest rates follow affine jump diffusion processes. In our extended model volatility is stochastic, asset value and volatility has correlated jumps and interest rates are stochastic and have jumps. Finally, we analyze the modeling of single firm credit risk and credit risk pricing by using our extended model and show how our model can be used as a solution for the problems we encounter with simple models.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Os Modelos da Onda CinemÃtica e da Onda Difusiva foram aplicados em um rio natural, para estudar a propagaÃÃo de uma onda de cheia neste corpo hÃdrico. Esses modelos sÃo derivaÃÃes do Modelo da Onda DinÃmica, a partir de simplificaÃÃes nas EquaÃÃes de Saint Venant, onde alguns termos sÃo desprezados. No processo de soluÃÃo das equaÃÃes diferenciais, pertinentes aos modelos, foi usado o MÃtodo das DiferenÃas Finitas, sendo que o esquema de aproximaÃÃo explicita foi aplicado para a onda cinemÃtica, enquanto que o esquema de aproximaÃÃo implÃcita foi aplicado para a onda difusiva. Para esta pesquisa, um programa computacional, em linguagem FORTRAN, foi desenvolvido e permitiu que viÃrias simulaÃÃes fossem realizadas, para diferentes cenÃrios encontrados nos rios naturais. Estudos para verificar a sensibilidade dos modelos, com respeito aos parÃmetros hidrÃulicos da bacia, foram realizados. TambÃm foi verificada a influÃncia da linearizaÃÃo das equaÃÃes diferenciais, que compÃem os modelos, nÃs cÃlculos das variÃveis de controle. Os resultados mostraram que o modelo da onda cinemÃtica à mais sensÃvel ao coeficiente de rugosidade das paredes do canal, enquanto que o modelo da onda difusiva à mais sensÃvel para parÃmetros da declividade de fundo do canal, onde este parÃmetro atua diretamente no processo de amortecimento da onda em propagaÃÃo. Os resultados mostraram ainda que, para os cenÃrios usados nas simulaÃÃes, o processo de linearizaÃÃo das equaÃÃes diferenciais nÃo afeta, consideravelmente, a soluÃÃo dos modelos.