While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to zeta function regularization and explore more fully the relationship between operators in physics and classical zeta functions of mathematics. In so doing, we highlight intriguing connections to number theory that arise.

The thesis focuses on service area analysis. The most fitted marketing mix for a particular market area and a marketing plan setting for XY Company are aims of the thesis. The company runs education field and service business. The marketing plan and mix should get the company to leading position in the market, using modern marketing tools. In a theoretical part of the thesis the basic issues are specified, the marketing mix and marketing planning are explained in general, including a description of an each marketing mix element and necessary analyses for marketing planning. In a practical part of the thesis the company and its background are defined and the final marketing plan is concluded.

This thesis aims to provide an introduction to the field of potential theory at an undergraduate level, by studying an important mathematical problem in the field, namely the Dirichlet problem. By examining the historical development of different methods for solving the problem in increasingly general contexts, and the mathematical concepts which were established to support these methods, the aim is to provide an overview of various basic techniques in the field of potential theory, as well as a summary of the fundamental results concerning the Dirichlet problem in Euclidean space. / Målet med detta arbete är att vara en introduktion på kandidatnivå till ämnesfältet potentialteori, genom att studera ett viktigt matematiskt problem inom potentialteori, Dirichletproblemet. Genom att undersöka den historiska utvecklingen av olika lösningsmetoder för problemet i mer och mer allmänna sammanhang, i kombination med de matematiska koncepten som utvecklades för att användas i dessa lösningsmetoder, ges en översikt av olika grundläggande tekniker inom potentialteori, samt en sammanfattaning av de olika matematiska resultaten som har att göra med Dirichletproblemet i det Euklidiska rummet.

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Les plasmons de surface sont des modes non-radiatifs qui vivent à l'interface d'un diélectrique et d'un métal. Ils peuvent confiner la lumière à des échelles sub-longueur d'onde. Néanmoins, leur propagation reste limitée par les pertes inhérentes au métal qui entraînent une absorption rapide du mode. L'objet de cette thèse est l'étude du couplage des plasmons de surface dans des structures planes métallo-diélectriques. L'obtention des propriétés des différents modes nécessite le prolongement des solutions dans le plan complexe définissant la constante de propagation. La méthode mise en œuvre consiste à déterminer les pôles de la matrice de diffusion en utilisant les intégrales de Cauchy. Une première solution pour résoudre le problème de propagation des plasmons de surface consiste à coupler ces modes entre eux. Dans un milieu symétrique, lorsque l'épaisseur d'un film métallique devient suffisamment faible, le couplage entre les modes plasmons existants sur chaque surface devient possible. L'un des deux modes couplés ainsi créé, dit plasmon longue portée, a une longueur de propagation supérieure à celle du plasmon de surface usuel tandis que l'autre, dit plasmon à courte portée, se propage moins. Nous présentons une configuration permettant l'excitation du mode longue portée sans le mode courte portée grâce à une couche métallique déposée sur un substrat infiniment conducteur. Cette excitation peut alors être effectuée dans l'air et permettre des applications comme la détection et la caractérisation de molécules. Ensuite, nous présentons le couplage entre deux guides d'ondes diélectriques, et plus particulièrement la théorie des modes couplés, étendue au cas de la polarisation transverse magnétique. Nous considérons aussi le cas de la PT symétrie. La dernière partie de ce mémoire présente la démonstration du régime de couplage fort entre un plasmon de surface et un mode guidé. Nous mettons alors en évidence une augmentation de la longueur de propagation pour le mode hybride plasmon dont le confinement reste celui d'un mode de surface. Un gain linéaire est ensuite ajouté dans les différentes couches de la structure pour en étudier l'effet. L'ajout de gain dans la couche intermédiaire entre les deux modes couplés a pour conséquence l'exaltation de la longueur de propagation des modes et plus particulièrement du mode hybride plasmon qui peut alors se propager au-delà du millimètre. / Surface plasmon polaritons are non radiative modes which exist at the interface between a dielectric and a metal. They can confine light at sub-wavelength scales. However, their propagation is restricted by the intrinsic losses of the metal which imply a rapid absorption of the mode. The aim of this thesis is the study of the coupling of surface plasmons in metallo-dielectric planar structures. Obtaining the properties of the modes implies the extension of the solutions to the complex plane of propagation constants. The method used consists in determining the poles of the scattering matrix by means of Cauchy's integrals. The first solution to solve the problem of propagation of the surface plasmons consists in coupling these modes to one another. In a symmetric medium, when the thickness of the metallic film becomes thin enough, the coupling between the plasmon modes which exist on each side becomes possible. One of the coupled modes which is created, the so-called long range surface plasmon, has a bigger propagation length than the usual plasmon whereas the other coupled mode, named short range surface plasmon, has a smaller propagation length. We present a configuration which allows the excitation of the long range surface plasmon without the short range mode with a metallic layer deposited on a perfect electric conductor substrate. This excitation can be done in air and allows applications, such as the detection and the characterisation of molecules. Then, we present the coupling between dielectric waveguides, and, in particular, the coupled-mode theory in the case of the transverse magnetic polarisation. We consider also the case of PT symmetric structure. The last part of this work presents the demonstration of the strong coupling regime between a surface plasmon and a guided mode. We demonstrate an increase of the propagation length of the hybrid surface plasmon, which still has the confinement of a surface mode. A linear gain is added in the different layers of the structure. When the gain is added in the layer between both coupled modes allows an enhancement of the propagation lengths of the modes, and more precisely of the hybrid surface plasmon mode, which can propagate at the millimeter scale.

In this thesis, the relationship between several high-latitude phenomena and the ionospheric conductance in both hemispheres is studied theoretically and experimentally. </p>Theoretically, the high-latitude electrodynamics is studied by considering currents in the magnetosphere-ionosphere system resulting from the ionospheric sheet current redistribution between the conjugate ionospheres. It is shown that strong flow between the conjugate ionospheres, the interhemispheric currents (IHC), can be set up if the conductance distribution is asymmetric in the conjugate ionospheric regions. Such conditions are typical for solstices owing to the differences in the solar illumination. Analytical and numerical modeling shows that IHCs can appear in the regions of strong conductance gradient, more specifically around the solar terminator line, and that the intensity of the IHCs can be comparable to the intensity of the well known Region 1/Region 2 currents. The effect of IHC excitation on observable magnetic perturbations on the ground is investigated. It is shown that in the vicinity of the solar terminator line, the pattern of magnetic perturbation can be such that an apparent equivalent current vortex can be detected. In addition, strong conductance gradients are shown to affect significantly the quality of the ionospheric plasma flow estimates from the ground-based magnetometer data. </p>Experimentally, the effect of the nightside ionospheric conductance on occurrence of substorms, global storm sudden commencement and radar auroras is investigated. To characterize substorm occurrence, new parameters, the derivatives of the classical AE and AO indices, are introduced. It is shown that the seasonal and diurnal variations of these parameters are controlled by the total nightside ionospheric conductance in the conjugate regions. The substorm onsets preferentially occur at low levels of the total conductance, which is consistent with the idea of the substorm triggering through the magnetosphere-ionosphere feedback instability. It is hypothesized that the total conductance affects the global storm onsets as well. To check this idea, the 33-year sudden storm commencement (SSC) data are considered. The semiannual, annual, semidiurnal, and diurnal variations in the SSC occurrence rate are found to be significant and these components exhibit a strong relationship with the total conductance of the high-latitude ionospheres. Finally, the SuperDARN midnight echo occurrence is shown to correlate, for some radars, with the total conductance minima and presumably with electric field maxima, which is consistent with general expectation that the F-region irregularities occur preferentially during times of enhanced electric fields. The gradients of the high-latitude conductance can also lead to significant errors in the plasma convection estimates from the ground-based magnetometers, and to investigate this effect a statistical assessment of the difference between the true plasma convection (SuperDARN) and the magnetometer-inferred equivalent convection direction is performed. The largest differences are found for the transition region between the dark and sunlit ionospheres and in the midnight sector where strong conductance gradients are expected due to particle precipitation. Consideration of regular conductance gradients due to solar illumination improves the agreement between the radar and magnetometer data. Finally, an attempt is made to demonstrate the effects of conductance upon the properties of traveling convection vortices (TCVs). Joint SuperDARN and magnetometer data reveal that there is resemblance between the magnetometer and radar inferred TCV images on a scale of thousands of kilometers. However, on a smaller scale of hundreds of kilometers, significant differences are observed.

I komplex analys finns det ett antal olika konvergenser varav vi tittar närmare på några här. Bland annat hur likformig konvergens medför punktvis konvergens men att det omvända ej gäller. Vi tittar också på vad de har för samband med lokal likformig konvergens och normal konvergens dvs. likformig konvergens på kompakta delmängder. Slutligen kommer vi att se på vad som gäller för familjer och kommer då in på lokalt begränsad, ekvikontinuitet, Arzela/Ascoli, Montels och Runges satser. Vi kommer här även se exempel på hur stort fel det egentligen kan bli för punktvisa konvergenta följder. De får normalt inte en gränsfunktion som är analytisk men vi ser både i Exempel 3.19 och Korollarium 3.23 att dessa ger resultat som är analytiska nästan överallt. / This report will describe four different types of convergence. The types described are pointwise, local uniformly, uniformly and normal convergence. The different convergences are explored in a way of how they relate to each other. Finally this report will also investigate how this applies to normal families and the theories of Arzela/Ascoli, Montel and Runge. We will here see examples of how wrong it really can go for pointwise convergent sequences. They do usually not have a limit that is analytic but from both Example 3.19 and Corollary 3.23 we will see that they give functions that in fact are analytic almost everywhere.

In this thesis, the relationship between several high-latitude phenomena and the ionospheric conductance in both hemispheres is studied theoretically and experimentally. </p>Theoretically, the high-latitude electrodynamics is studied by considering currents in the magnetosphere-ionosphere system resulting from the ionospheric sheet current redistribution between the conjugate ionospheres. It is shown that strong flow between the conjugate ionospheres, the interhemispheric currents (IHC), can be set up if the conductance distribution is asymmetric in the conjugate ionospheric regions. Such conditions are typical for solstices owing to the differences in the solar illumination. Analytical and numerical modeling shows that IHCs can appear in the regions of strong conductance gradient, more specifically around the solar terminator line, and that the intensity of the IHCs can be comparable to the intensity of the well known Region 1/Region 2 currents. The effect of IHC excitation on observable magnetic perturbations on the ground is investigated. It is shown that in the vicinity of the solar terminator line, the pattern of magnetic perturbation can be such that an apparent equivalent current vortex can be detected. In addition, strong conductance gradients are shown to affect significantly the quality of the ionospheric plasma flow estimates from the ground-based magnetometer data. </p>Experimentally, the effect of the nightside ionospheric conductance on occurrence of substorms, global storm sudden commencement and radar auroras is investigated. To characterize substorm occurrence, new parameters, the derivatives of the classical AE and AO indices, are introduced. It is shown that the seasonal and diurnal variations of these parameters are controlled by the total nightside ionospheric conductance in the conjugate regions. The substorm onsets preferentially occur at low levels of the total conductance, which is consistent with the idea of the substorm triggering through the magnetosphere-ionosphere feedback instability. It is hypothesized that the total conductance affects the global storm onsets as well. To check this idea, the 33-year sudden storm commencement (SSC) data are considered. The semiannual, annual, semidiurnal, and diurnal variations in the SSC occurrence rate are found to be significant and these components exhibit a strong relationship with the total conductance of the high-latitude ionospheres. Finally, the SuperDARN midnight echo occurrence is shown to correlate, for some radars, with the total conductance minima and presumably with electric field maxima, which is consistent with general expectation that the F-region irregularities occur preferentially during times of enhanced electric fields. The gradients of the high-latitude conductance can also lead to significant errors in the plasma convection estimates from the ground-based magnetometers, and to investigate this effect a statistical assessment of the difference between the true plasma convection (SuperDARN) and the magnetometer-inferred equivalent convection direction is performed. The largest differences are found for the transition region between the dark and sunlit ionospheres and in the midnight sector where strong conductance gradients are expected due to particle precipitation. Consideration of regular conductance gradients due to solar illumination improves the agreement between the radar and magnetometer data. Finally, an attempt is made to demonstrate the effects of conductance upon the properties of traveling convection vortices (TCVs). Joint SuperDARN and magnetometer data reveal that there is resemblance between the magnetometer and radar inferred TCV images on a scale of thousands of kilometers. However, on a smaller scale of hundreds of kilometers, significant differences are observed.

Η παρούσα διπλωματική εργασία αποτελείται από δύο μέρη. Στο πρώτο μέρος θα μελετήσουμε βασικές έννοιες και θεωρήματα από την θεωρία δυναμικού. Έννοιες όπως το δυναμικό, τα polar σύνολα, η συνάρτηση Green ενός συνόλου και η χωρητικότητα ενός συνόλου είναι αναγκαίες ώστε να οδηγηθούμε στο περίφημο θεώρημα των Bernstein-Walsh, το οποίο δίνει την ταχύτητα της πολυωνυμικής προσέγγισης αναλυτικών συναρτήσεων σε συμπαγή σύνολα με συνεκτικό συμπλήρωμα. Στο δεύτερο μέρος, μελετάμε ένα αποτέλεσμα των Γ. Κωστάκη και Ν. Τσιρίβα, για μία έννοια σχετική με τις καθολικές σειρές Taylor, τις διπλά καθολικές σειρές Taylor. Συγκεκριμένα, για δοσμένη γνησίως αύξουσα ακολουθία φυσικών αριθμών (λn), μια ολόμορφη συνάρτηση f στον ανοιχτό μοναδιαίο δίσκο λέγεται διπλά καθολική σειρά Taylor, ως προς τις ακολουθίες (n),(λn), αν για κάθε συμπαγές σύνολο Κ, υποσύνολο του μιγαδικού επιπέδου, ξένο με τον δίσκο και με συνεκτικό συμπλήρωμα και για κάθε ζεύγος συναρτήσεων (g1,g2) συνεχών στο Κ, ολόμορφων στο εσωτερικό του Κ, υπάρχει υπακολουθία των φυσικών αριθμών (μn), τέτοια ώστε (S_{μn}(f,0),S_{λ_{μn}}(f,0)) προσεγγίζουν ομοιόμορφα τις (g_{1},g_{2}) (όπου S_{n}(f,0) το n-οστό μερικό άθροισμα του αναπτύγματος Taylor της f με κέντρο το 0). Το κεντρικό λοιπόν αποτέλεσμα είναι ότι για δοσμένη ακολουθία (λn), η οικογένεια των διπλά καθολικών σειρών Taylor, ως προς τις ακολουθίες (n),(λn), είναι Gδ και πυκνή στο σύνολο των ολόμορφων συναρτήσεων στον ανοιχτό μοναδιαίο δίσκο (ειδικότερα είναι μη-κενή) αν και μόνο αν το ανώτερο όριο limsup_{n}(λn/n) είναι άπειρο. Εργαλείο-κλειδί για το παραπάνω αποτέλεσμα είναι το Θεώρημα Bernstein-Walsh. / --

L’objectif de cette thèse est d’étudier l’évolution de la fissure sous les effets de plasticité et du champ de contrainte non-uniforme à l’aide du modèle de zone cohésive. Dans un premier temps, l’évolution de la fissure au sein du matériau élastoplastique est explorée dans le cadre de l’approche variationnelle. Les solutions sont explicitées dans le cas d’une barre 1D sous traction simple grâce aux conditions de stabilité locale d’énergie au premier ordre et au second ordre. Cette étude nous permet de mettre en lumière l’effet de la plasticité sur le comportementadoucissant du matériau dès que la fissure cohésive apparaît. En effet, la réponse globale de la barre sous déplacement imposé est stable seulement si la longueur de la barre est inférieure à une longueur critique. Cette dernière est démontrée indépendante du module d’écrouissage plastique mais dépend du module Young et de la dérivée seconde de la densité d’énergie de fissure. Les formulations énergétiques peuvent être généralisées pour la structure 3D. Dans ce cas, les critères de plasticité et de cohésif deviennent les courbes dans le plan des contraintes de Mohr. La comparaison des courbes nous permet d’étudier la nucléation de fissure cohésive au sein du domaine plastifié. Dans un deuxième temps, les effets de la non-uniformité du champ de contrainte sur la nucléation de la fissure au sein de la structure élastique sont mis en évidence. On construit la solution analytique en utilisant la technique à deux échelles et l’analyse complexe. L’évolution de la fissure purement cohésive et partiellement non-cohésive est contrôlée par le gradient du champ de contrainte lié à une longueur caractéristique. L’utilisation des différentes lois cohésives dans le problème est explorée. La sensibilité de la solution à la taille du défaut préexistant est également étudiée. Finalement, des résultats analytiques sont validés par les simulations numériques et le modèle de zone cohésive en mode mixte est implémenté dans Code_Aster. / The aims of this work is to study the cracks evolution under plasticity and nonuniform stress field effects by using cohesive zone model. Firstly, basing on the variational approach, the crack evolution in the elastoplastic material is investigated. The solutions for 1D beam under simple tension is expressed explicitly through the first and the second orders stability conditions of energy. This study shows us the plasticity effects on the material softening behavior as soon as crack appears. In fact, the global solution of the beam under described displacement is stable only if the beam length is lower than a characteristic length. This length is independent of plasticity hardening module but depends on Young modulus and on the second derivative of crack energy density. The energy formulations can be generalized for 3D structure. In this case, the plasticity and cohesive criteria become two curves in Mohr’s stresses plane. The comparison between theses curves allows us to consider the crack nucleation in the plastified domain. Secondly, the non-uniform stress field effects on the crack nucleation in the elastic material is highlighted. The analytical solution is established by using two-scales techniqueand complex analysis. The evolution of fully cohesive crack and partially non-cohesive crack is controlled by the stress gradient, which is related to a characteristic length. Different cohesive laws are used in our study. The sensitivity of solution to preexisting imperfection size is also explored. Finally, analytical results are validated by numerical simulations and the cohesive zone model in mixed mode is implemented in Code_Aster.