<p dir="ltr">This thesis presents applications of one-point quadrature domains to encryption and decryption as well as a method for estimating average temperature. In addition, it investigates methods for finding explicit formulas for certain functions and introduces results regarding quadrature domains for harmonic functions and for double quadrature domains. We use properties of quadrature domains to encrypt and decrypt locations in two dimensions. Results by Bell, Gustafsson, and Sylvan are used to encrypt a planar location as a point in a quadrature domain. A decryption method using properties of quadrature domains is then presented to uncover the location. We further demonstrate how to use data from the decryption algorithm to find an explicit formula for the Schwarz function for a one-point area quadrature domain. Given a double quadrature domain, we show that the fixed points within the area and arc length quadrature identities must be the same, but that the orders at each point may differ between these identities. In the realm of harmonic functions, we demonstrate how to uncover a one-point quadrature identity for harmonic functions from the quadrature identity for a simply-connected one-point quadrature domain for holomorphic functions. We use this result to state theorems for the density of one-point quadrature domains for harmonic functions in the realm of smooth domains with $C^{\infty}$-smooth boundary. These density theorems then lead us to discuss applications of quadrature domains for harmonic functions to estimating average temperature. We end by illustrating examples of the encryption process and discussing the building blocks for future work.</p>

In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring. We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, we restrict our attention to those domains U that are connected. We extend a result of I. Amemiya to see that the Lie ring structure is determined by the conformal structure of U. In a similar vein to our ring considerations, we see that, again for certain domains U of usual interest, the Lie ring A(U) has a unique Polish topology for which it is a Polish Lie ring. Again, the Lie ring A(U) imposes topological restrictions on C. That is, C must have its usual topology when sitting inside any Polish Lie ring isomorphic to A(U). In the last chapter, we introduce a new ideal of subsets of Polish spaces consisting of what we call residually null sets. From this ideal, we introduce an algebra consisting of what we call R-sets which is consistently a strict extension of the algebra of Baire property sets. We show that the algebra of R-sets is closed under the Alexandrov-Suslin operation and generalize Pettis' Theorem. From this, we provide new automatic continuity results and give a generalization of a result of D. Montgomery which shows that minimal assumptions on the continuity of group operations of an abstract group G with a Polish topology imply that G is actually a Polish group. We also see that many results pertaining to the algebra of Baire property sets generalize to the context of R-sets.

Wir untersuchen die analytische Struktur von Feynman Integralen als mengenwertige holomorphe Funktionen mit topologischen Methoden, spezifisch mit Techniken für singuläre Integrale. Der Hauptfokus liegt auf dem Ein-Schleifen-Fall. Zunächst geben wir einen gründlichen Überblick über die Theorie der singulären Integrale und füllen einige Lücken in der Literatur. Anschließend untersuchen wir die Topologie von endlichen Vereinigungen und Schnitten von bestimmten nicht-degenerierten affinen komplexes Quadriken, welche die relevante Geometrie von Ein-Schleifen Feynman Integralen darstellen. Wir etablieren einige grundsätzliche topologische Eigenschaften und führen eine Kompaktifizierung von Bündeln solcher Räume und eine Whitney Stratifizierung dieser ein. Des Weiteren berechnen wir die Homologiegruppen der Fasern durch eine Dekomposition in die auftretenden Schnitte komplexer Sphären. Das Einführen einer CW-Dekomposition einer spezifischen Faser führt zu einer kombinatorischen Studie, welche es uns erlaubt explizite Generatoren in Sinne dieser CW-Strukture zu berechnen. Unter Verwendung dieser Generatoren berechnen wir die relevanten Schnittindizes, welche im Ramifizierungsproblem auftreten. Durch Anwendung dieser Resultate auf Ein-Schleifen Feynman Integrale finden wir die klassischen Landau Gleichungen wieder und erhalten einen vollständigen Beweis von Cutkoskys Theorem. Des Weiteren untersuchen wir, wie viel dieses Mechanismus sich auf den Mehr-Schleifen Fall überträgt. Insbesondere betrachten wir zwei Beispiele von Mehr-Schleifen Integralen und erhalten Resultate die über den aktuellen Stand der Literatur hinaus gehen. / We investigate the analytic structure of Feynman integrals as multivalued holomorphic functions with topological methods, specifically with techniques for singular integrals. The main focus lies on the one-loop case. First, we conduct a thorough review of the theory of singular integrals, filling some gaps in the literature. Then, we investigate the topology of finite unions and intersections of certain non-degenerate affine complex quadrics which constitute the relevant geometry of one-loop Feynman integrals. We establish some basic topological properties and introduce a compactification of bundles of such spaces and a Whitney stratification thereof. Furthermore, we compute the homology groups of the fibers via a decomposition into the direct sum of all occurring intersections of complex spheres. Introducing a CW-decomposition of a specific fiber leads to a combinatorial study, allowing us to obtain explicit generators in terms of this CW-structure. Using these generators, we compute the relative intersection indices that occur in the ramification problem. Applying these results to one-loop Feynman integrals, we retrieve the classical Landau equations and obtain a full proof of Cutkosky's Theorem. Furthermore, we investigate how much of this machinery applies to the multi-loop case. In particular, we consider two examples of multi-loop integrals and obtain results beyond the current state of the literature.

Dans ce mémoire, nous allons nous concentrer sur le sujet de l’universalité en analyse complexe. Tout d'abord, nous allons énumérer de nombreux résultats découverts dans ce domaine, tout en soulignant que, dans la plupart des cas, les preuves d'existence des éléments universels sont implicites et non pas constructives. Nous examinerons en détail une preuve spécifique de l'existence des séries universelles de Taylor qui se voulait constructive et nous déterminerons si tel est le cas ou non. Pour atteindre cet objectif, nous introduirons un nouvel élément universel que nous appellerons les séries numériques universelles. Ce sont des séries complexes telles que leurs sommes partielles sont denses dans le plan complexe. Nous donnerons une preuve constructive de l'existence de ces éléments et, afin de déterminer pleinement si la preuve susmentionnée de l'existence des séries universelles de Taylor est constructive, nous allons la comparer avec notre preuve de l'existence des séries numériques universelles. Enfin, nous examinerons les propriétés topologiques et algébriques des séries numériques universelles, en montrant sous quelles conditions elles sont topologiquement génériques et algébriquement génériques dans l'ensemble de toutes séries formelles à termes complexes. / This master's thesis will be centered around the subject of universality in complex analysis. First, we will provide a summary of many of the results that have been discovered in the field of universality. We will show that, in most cases, the proofs of existence of the universal elements are not constructive, but, rather, implicit. We will perform an in-depth analysis of a specific proof of the existence of Universal Taylor series which was intended to be constructive and we will determine whether or not this goal was achieved. To do this, we will introduce a new universal element, which we will call Universal numerical series. These are complex numerical series such that the partial sums of the series are dense in the complex plane. We will give a constructive proof of the existence of these elements and, in order to fully determine whether the aforementioned proof of the existence of the Universal Taylor series is constructive, we will compare it to our proof of the existence of the Universal numerical series. Finally, we will examine the topological and algebraic properties of the Universal numerical series, showing under which conditions they are topologically generic and algebraically generic in the set of all complex numerical series.

The modified mapping-collocation (MMC) method is applied to analyze a circumferential crack in an isotropic curved beam. Based on the method a MATLAB code was developed to obtain the stress field. Incorporating the stress correlation technique, the opening and sliding fracture mode stress intensity factors (SIF)s of the crack for the beam under pure bending moment load case are calculated. The MMC method aims to solve two-dimensional problems of linear elastic fracture mechanics (LEFM) by combining the power of analytic tools of complex analysis (Muskhelishvili formulation, conformal mapping, and extension arguments) with simplicity of applying the boundary collocation method as a numerical solution approach. Qualitatively, a good agreement between the computed stress contours and the fringe shapes obtained from the photoelastic experiment on a plexiglass specimen is observed. Quantitatively, the results are compared with that of ANSYS finite element analysis software. The effect of crack size, crack position and beam thickness variation on SIF values and mode mixity is investigated.

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.

Dans ce mémoire nous étudions les hypersurfaces Levi-plates analytiques dans les surfaces algébriques complexes. Il s'agit des hypersurfaces réelles qui admettent un feuilletage par des courbes holomorphes, appelé le feuilletage de Cauchy Riemann (CR). Dans un premier temps nous montrons que si ce dernier admet une dynamique chaotique (i.e. s'il n'admet pas de mesure transverse invariante) alors les composantes connexes de l'extérieur de l'hypersurface sont des modifications de domaines de Stein. Ceci permet d'étendre le feuilletage CR en un feuilletage algébrique singulier sur la surface complexe ambiante. Nous appliquons ce résultat pour montrer, par l'absurde, qu'une hypersurface Levi-plate analytique qui admet une structure affine transverse dans une surface algébrique complexe possède une mesure transverse invariante. Ceci nous amène à conjecturer que les hypersurfaces Levi-plates dans les surfaces algébriques complexes qui sont difféomorphes à un fibré hyperbolique en tores sur le cercle sont des fibrations par courbes algébriques. / In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. These are real hypersurfaces that admit a foliation by holomorphic curves, called Cauchy Riemann foliation (CR). First, we show that if this foliation admits chaotic dynamics (i.e. if it doesn't admit an invariant transverse measure), then the connected components of the complement of the hypersurface are Stein. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse invariant measure. This leads us to conjecture that Levi-flat hypersurfaces in complex algebraic surfaces that are diffeomorphic to a hyperbolic tori bundle over the circle are fibrations by algebraic curves.

La fonction de Möbius est définie par$$\mu(n)= \begin{cases} 1 & \textit{si $n=1$},\\ (-1)^k& \textit{si n est le produit de k nombres premiers distincts,}\\ 0 & \textit{si n contient un facteur carré. } \end{cases}$$Nous avons démontré que pour $x \ge \exp( 10^9) $ et $h=x^{1-\frac{1}{16000}}$, il existe dans chaque intervalle $[x-h,x]$ des entiers $n_1$ avec $\mu(n_1)=1$ et des entiers $n_2$ avec $\mu(n_2)=-1$.\\Ce résultat est une conséquence d'un résultat plus général.\\Pour $x \ge \exp(4\times 10^6)$, $\frac{1}{\sqrt{\log x}} \le \theta \le \frac{1}{2000}$, $h=x^{1-\theta}$ et $Q=(x/h)^{\frac{1}{20}}$, nous avons \\$$\sum_{q \leq Q} \log(Q/q)\sum_{\chi mod q}^*\left| \sum_{x.-h\le n \le x} \mu(n) \chi(n) \right| \leq 10^{20} h \theta \log(x) \exp( \frac{-1}{300 \theta}); $$la somme $\sum^*$ portant sur les caractères primitifs sauf l'éventuel caractère exceptionnel.\\Et en particulier pour $x \ge \exp( 10^9)$,$$ \left | \sum_{x.-x^{1-\frac{1}{16000}}\le n \le x} \mu(n) \right | \le \frac{1}{100} x^{1-\frac{1}{16000}}.\\$$ / The Möbius function is defined by$$\mu(n)= \begin{cases} 1 & \textit{if $n=1$},\\ (-1)^k& \textit{if n is a product of k distinct prime numbers,}\\ 0 & \textit{if n contains a square factor. } \end{cases}$$We demonstrate that for $x \ge \exp( 10^9) $ and $h=x^{1-\frac{1}{16000}}$, it exists in each interval $[x-h,x]$ integers $n_1$ with $\mu(n_1)=1$ and integers $n_2$ with $\mu(n_2)=-1$.\\This result is a consequence of a more general result. \\For $x \ge \exp(4\times 10^6)$, $\frac{1}{\sqrt{\log x}} \le \theta \le \frac{1}{2000}$, $h=x^{1-\theta}$ et $Q=(x/h)^{\frac{1}{20}}$, we have \\ $$\sum_{q \leq Q} \log(Q/q)\sum_{\chi mod q}^*\left| \sum_{x-h \le n \le x} \mu(n) \chi(n) \right| \leq 10^{20} h \theta \log(x) \exp( \frac{-1}{300 \theta}); $$the sum $\sum^*$ relating to primitive characters except for possible exceptional character.\\And in particular for $x \ge \exp( 10^9)$,$$\left | \sum_{x-.x^{1-\frac{1}{16000}}\le n \le x} \mu(n) \right | \le \frac{1}{100} x^{1-\frac{1}{16000}}.$$

Les travaux de cette thèse relèvent du domaine de la théorie des opérateurs, et se situent à l'interface de l'analyse complexe, de la théorie des semi-groupes et de la théorie de l'universalité. Le premier résultat principal de cette thèse relève de l'étude des opérateurs de composition sur des espaces de fonctions holomorphes : nous déterminons le spectre d'un opérateur de composition par un symbole de Koenigs sur l'espace des fonctions holomorphes sur le disque unité, et en déduisons des informations sur la forme générale du spectre des opérateurs de composition par un symbole de Koenigs sur des espaces de Banach de fonctions holomorphes. L'outil principal que nous développons pour notre étude est une description des projections spectrales associées à ces opérateurs. Le second résultat principal de cette thèse relève de la théorie de l'universalité : nous étendons aux semi-groupes d'opérateurs la notion d'opérateur universel, et établissons l'existence d'un semi-groupe universel pour les semi-groupes quasi-contractifs en exhibant un semi-groupe sur un espace de fonctions holomorphes. Nous élargissons ensuite ce résultats aux semi-groupes d'opérateurs concaves / The works in this thesis address topics from operator theory and involves ideas and notions arising from complex analysis, the theory of operator semigroups and the theory of universality. The first main result of this thesis relates to the study of composition operators on spaces of holomorphic functions: we compute the spectrum of an operator of composition by a Koenigs's symbol acting on the space of holomorphic functions on the open unit disk, and derive from it the general description of the spectrum of composition operators on Banach spaces of holomorphic functions. The key tool we develop in this study is a description of spectral projections associated with such operators.The second main result of this thesis relates to the thoery of universality: we extend to operator semigroups the notion of universality. Then, we prove the existence of a universal semigroup for quasi-contractive operators semigroups. We then show a similar result for concave semigroups

Целью диссертационного исследования является разработка концептуальной платформы повышения конкурентоспособности производственного предприятия на основе совместного применения принципов Бережливого производства и стратегии концентрической диверсификации с использованием методологии проектного управления. Объектом исследования является рыночная деятельность промышленного предприятия ОАО «Северский гранитный карьер». Предметом исследования является система организационно-экономических отношений, возникающих в процессе перехода промышленного предприятия к стратегии концентрической диверсификации. Научная новизна диссертационного исследования заключается в разработке теоретико-методического обеспечения повышения конкурентоспособности промышленного предприятия, направленного на успешную и эффективную концентрическую диверсификацию деятельности хозяйствующего субъекта на основе интеграции концепции Бережливого производства с использованием методологии проектного управления. / The purpose of the dissertation research is the development of a conceptual platform for increasing the competitiveness of a production enterprise on the basis of the joint application of Lean Manufacturing Principles and the strategy of concentric diversification using the methodology of project management. The object of research is the market activity of an industrial enterprise. The subject of the study is the system of organizational and economic relations that arise during the transition of an industrial enterprise to the strategy of concentric diversification. The scientific novelty of the dissertation research is to develop a theoretical and methodological support for increasing the competitiveness of an industrial enterprise aimed at successful and effective concentric diversification of the business entity's activities based on the integration of the Lean Manufacturing concept using the project management methodology.