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Since Hecke first proved his correspondence between Dirichlet series with functional equations and automorphic forms, there have been a great number of generalizations. Of particular interest is a generalization due to Bochner that gives a correspondence between Dirichlet series with any finite number of poles that satisfy the classical functional equation and automorphic integrals with (finite) log-polynomial sum period functions. In this dissertation, we extend Bochner's result to Dirichlet series with finitely many essential singularities. With some restrictions on the underlying group and the weight, we also prove a correspondence for Dirichlet series with infinitely many poles. For this second correspondence, we provide a technique to approximate automorphic integrals with infinite log-polynomial sum period functions by automorphic integrals with finite log-polynomial period functions. / Mathematics

<p dir="ltr">This thesis investigates the large deflection behavior of cantilever beams under various configurations and loading conditions. The primary objective is to uset an analytical model using elliptic integrals to solve the second-order non-linear differential equations that govern the deflection of these beams. The analytical model is implemented in Python and compared against Finite Element Analysis (FEA) results obtained from ANSYS, ensuring the accuracy and reliability of the model. The study examines multiple beam configurations, including straight and inclined beams, with both free and fixed tip slopes. Sensitivity analysis is conducted to assess the impact of key parameters, such as Young’s modulus, beam height, width, and length, on the deflection behavior. This analysis reveals critical insights into how variations in material properties and geometric dimensions affect beam performance. A detailed error analysis using Root Mean Square Error (RMSE) is performed to compare the analytical model's predictions with the FEA results. The error analysis highlights any discrepancies, demonstrating the robustness of the analytical approach. The results show that the analytical model, based on elliptic integrals, closely matches the FEA results across a range of configurations and loading scenarios. The insights gained from this study can be applied to optimize the design of cantilever beams in various engineering applications, including prosthetics, robotics, and structural components. Overall, this research provides a comprehensive understanding of the large deflection behavior of cantilever beams and offers a reliable analytical tool for engineers to predict beam performance under different conditions. The integration of Python-based numerical methods with classical elliptic integral solutions presents a useful approach that enhances the precision and applicability of beam deflection analysis.</p>

In this thesis we shall examine the role of measurability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bouuded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration

Since the time of Cauchy, integration theory has in the main been an attempt to regain the Eden of Newton. In that idyllic time [. . . ] derivatives and integrals were [. . . ] different aspects of the same thing. -Peter Bullen, as quoted in [24] The theory of integration has gone through many changes in the past centuries and, in particular, there has been a tension between the Riemann and the Lebesgue approach to integration. Riemann's definition is often the first integral to be introduced in undergraduate studies, while Lebesgue's integral is more powerful but also more complicated and its methods are often postponed until graduate or advanced undergraduate studies. The integral presented in this paper is due to the work of Ralph Henstock and Jaroslav Kurzweil. By a simple exchange of the criterion for integrability in Riemann's definition a powerful integral with many properties of the Lebesgue integral was found. Further, the generalized Riemann integral expands the class of integrable functions with respect to Lebesgue integrals, while there is a characterization of the Lebesgue integral in terms of absolute integrability. As this definition expands the class of functions beyond absolutely integrable functions, some theorems become more cumbersome to prove in contrast to elegant results in Lebesgue's theory and some important properties in composition are lost. Further, it is not as easily abstracted as the Lebesgue integral. Therefore, the generalized Riemann integral should be thought of as a complement to Lebesgue's definition and not as a replacement. / Ända sedan Cauchys tid har integrationsteori i huvudsak varit ett försök att åter finna Newtons Eden. Under den idylliska perioden [. . . ] var derivator och integraler [. . . ] olika sidor av samma mynt.-Peter Bullen, citerad i [24] Under de senaste århundradena har integrationsteori genomgått många förändringar och framförallt har det funnits en spänning mellan Riemanns och Lebesgues respektive angreppssätt till integration. Riemanns definition är ofta den första integral som möter en student pa grundutbildningen, medan Lebesgues integral är kraftfullare. Eftersom Lebesgues definition är mer komplicerad introduceras den först i forskarutbildnings- eller avancerade grundutbildningskurser. Integralen som framställs i det här examensarbetet utvecklades av Ralph Henstock och Jaroslav Kurzweil. Genom att på ett enkelt sätt ändra kriteriet for integrerbarhet i Riemanns definition finner vi en kraftfull integral med många av Lebesgueintegralens egenskaper. Vidare utvidgar den generaliserade Riemannintegralen klassen av integrerbara funktioner i jämförelse med Lebesgueintegralen, medan vi samtidigt erhåller en karaktärisering av Lebesgueintegralen i termer av absolutintegrerbarhet. Eftersom klassen av generaliserat Riemannintegrerbara funktioner är större än de absolutintegrerbara funktionerna blir vissa satser mer omständiga att bevisa i jämforelse med eleganta resultat i Lebesgues teori. Därtill förloras vissa viktiga egenskaper vid sammansättning av funktioner och även möjligheten till abstraktion försvåras. Integralen ska alltså ses som ett komplement till Lebesgues definition och inte en ersättning.

This thesis presents results in two apparently disparate mathematical fields which can both be examined -- and even united -- by means of pure analysis. Fractional calculus is the study of differentiation and integration to non-integer orders. Dating back to Leibniz, this idea was considered by many great mathematical figures, and in recent decades it has been used to model many real-world systems and processes, but a full development of the mathematical theory remains incomplete. Many techniques for partial differential equations (PDEs) can be extended to fractional PDEs too. Three chapters below cover my results in this area: establishing the elliptic regularity theorem, Malgrange-Ehrenpreis theorem, and unified transform method for fractional PDEs. Each one is analogous to a known result for classical PDEs, but the proof in the general fractional scenario requires new ideas and modifications. Fractional derivatives and integrals are not uniquely defined: there are many different formulae, each of which has its own advantages and disadvantages. The most commonly used is the classical Riemann-Liouville model, but others may be preferred in different situations, and now new fractional models are being proposed and developed each year. This creates many opportunities for new research, since each time a model is proposed, its mathematical fundamentals need to be examined and developed. Two chapters below investigate some of these new models. My results on the Atangana-Baleanu model proposed in 2016 have already had a noticeable impact on research in this area. Furthermore, this model and the results concerning it can be extended to more general fractional models which also have certain desirable properties of their own. Fractional calculus and zeta functions have rarely been united in research, but one chapter below covers a new formula expressing the Lerch zeta function as a fractional derivative of an elementary function. This result could have many ramifications in both fields, which are yet to be explored fully. Zeta functions are very important in analytic number theory: the Riemann zeta function relates to the distribution of the primes, and this field contains some of the most persistent open problems in mathematics. Since 2012, novel asymptotic techniques have been applied to derive new results on the growth of the Riemann zeta function. One chapter below modifies some of these techniques to prove asymptotics to all orders for the Hurwitz zeta function. Many new ideas are required, but the end result is more elegant than the original one for Riemann zeta, because some of the new methodologies enable different parts of the argument to be presented in a more unified way. Several related problems involve asymptotics arbitrarily near a stationary point. Ideally it should be possible to find uniform asymptotics which provide a smooth transition between the integration by parts and stationary phase methods. One chapter below solves this problem for a particular integral which arises in the analysis of zeta functions.

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.

MSc / Thesis (MSc (Physics))--University of Stellenbosch, 2009. / The propagation of electromagnetic radiation in homogeneous or periodically modulated media can be described by the coupled mode equations. The aim of this study was to derive analytical expressions modeling the solutions of the coupled-mode equations, as alternative to the generally used numerical and transfer-matrix methods. The path integral formalism was applied to the coupled-mode equations. This approach involved deriving a path integral from which a generating functional was obtained. From the generating functional a Green’s function, or propagator, describing the nature of mode propagation was extracted. Initially a Green’s function was derived for the propagation of modes having position independent coupling coefficients. This corresponds to modes propagating in a homogeneous medium or in a uniform grating formed by a periodic variation of the index of refraction along the direction of propagation. This was followed by the derivation of a Green’s function for the propagation of modes having position dependent coupling coefficients with the aid of perturbation theory. This models propagation through a nonuniform inhomogeneous medium, specifically a modulated grating. The propagator method was initially tested for the case of propagation in an arbitrary homogeneous medium. In doing so three separate cases were considered namely the copropagation of two modes in the forward and backward directions followed by the counter propagation of the two modes. These more trivial cases were used as examples to develop a rigorous mathematical formalism for this approach. The results were favourable in that the propagator’s results compared well with analytical and numerical solutions. The propagator method was then tested for mode propagation in a periodically perturbed waveguide. This corresponds to the relevant application of mode propagation in uniform gratings in optical fibres. Here two case were investigated. The first scenario was that of the copropagation of two modes in a long period transmission grating. The results achieved compared well with numerical results and analytical solutions. The second scenario was the counter propagation of two modes in a short period reflection grating, specifically a Bragg grating. The results compared well with numerical results and analytical solutions. In both cases it was shown that the propagator accurately predicts many of the spectral properties of these uniform gratings. Finally the propagator method was applied to a nonuniform grating, that is a grating for which the uniform periodicity is modulated - in this case by a raised-cosine function. The result of this modulation is position dependent coupling coefficients necessitating the use of the Green’s function derived using perturbation theory. The results, although physically sensible and qualitatively correct, did not compare well to the numerical solution or the well established transfer-matrix method on a quantitative level at wavelengths approaching the design wavelength of the grating. This can be explained by the breakdown of the assumptions of first order perturbation theory under these conditions.

This thesis is about double product integrals with pseudo rotational generator, and aims to exhibit them as unitary implementors of Bogolubov transformations. We further introduce these concepts in this abstract and describe their roles in the thesis's chapters. The notion of product integral, (simple product integral, not double) is not a new one, but is unfamiliar to many a mathematician. Product integrals were first investigated by Volterra in the nineteenth century. Though often regarded as merely a notation for solutions of differential equations, they provide a priori a multiplicative analogue of the additive integration theories of Riemann, Stieltjes and Lebesgue. See Slavik [2007] for a historical overview of the subject. Extensions of the theory of product integrals to multiplicative versions of Ito and especially quantum Ito calculus were first studied by Hudson, Ion and Parthasarathy in the 1980's, Hudson et al. [1982]. The first developments of double product integrals was a theory of an algebraic kind developed by Hudson and Pulmannova motivated by the study of the solution of the quantum Yang-Baxter equation by the construction of quantum groups, see Hudson and Pulmaanova [2005]. This was a purely algebraic theory based on formal power series in a formal parameter. However, there also exists a developing analytic theory of double product integral. This thesis contributes to this analytic theory. The first papers in that direction are Hudson [2005b] and Hudson and Jones [2012]. Other motivations include quantum extension of Girsanov's theorem and hence a quantum version of the Black-Scholes model in finance. They may also provide a general model for causal interactions in noisy environments in quantum physics. From a different direction "causal" double products, (see Hudson [2005b]), have become of interest in connection with quantum versions of the Levy area, and in particular quantum Levy area formula (Hudson [2011] and Chen and Hudson [2013]) for its characteristic function. There is a close association of causal double products with the double products of rectangular type (Hudson and Jones [2012] pp 3). For this reason it is of interest to study "forwardforward" rectangular double products. In the first chapter we give our notation which will be used in the following chapters and we introduce some simple double products and show heuristically that they are the solution of two different quantum stochastic differential equations. For each example the order in which the products are taken is shown to be unimportant; either calculation gives the same answer. This is in fact a consequence of the so called multiplicative Fubini Theorem Hudson and Pulmaanova [2005]. In Chapter two we formally introduce the notion of product integral as a solution of two particular quantum stochastic differential equations. In Chapter three we introduce the Fock representation of the canonical commutation relations, and discuss the Stone-von Neumann uniqueness theorem. We define the notion of Bogolubov transformation (often called a symplectic automorphism, see Parthasarathy [1992] for example), implementation of these transformations by an implementor (a unitary operator) and introduce Shale's theorem which will be relevant to the following chapters. For an alternative coverage of Shale's Theorem, symplectic automorphism and their implementors see Derezinski [2003]. In Chapter four we study double product integrals of the pseudo rotational type. This is in contrast to double product integrals of the rotational type that have been studied in (Hudson and Jones [2012] and Hudson [2005b]). The notation of the product integral is suggestive of a natural discretisation scheme where the infinitesimals are replaced by discrete increments i.e. discretised creation and annihilation operators of quantum mechanics. Because of a weak commutativity condition, between the discretised creation and annihilation operators corresponding on different subintervals of R, the order of the factors of the product are unimportant (Hudson [2005a]), and hence the discrete product is well defined; we call this result the discrete multiplicative Fubini Theorem. It is also the case that the order in which the products are taken in the continuous (non-discretised case) does not matter (Hudson [2005a], Hudson and Jones [2012]). The resulting discrete double product is shown to be the implementor (a unitary operator) of a Bogolubov transformation acting on discretised creation and annihilation operators (Bogolubov transformations are invertible real linear operators on a Hilbert space that preserve the imaginary part of the inner product, but here we may regard them equivalently as liner transformations acting directly on creation and annihilations operators but preserving adjointness and commutation relations). Unitary operators on the same Hilbert space are a subgroup of the group of Bogolubov transformations. Essentially Bogolubov transformations are used to construct new canonical pairs from old ones (In the literature Bogolubov transformations are often called symplectic automorphisms). The aforementioned Bogolubov transformation (acting on the discretised creation and annihilation operators) can be embedded into the space L2(R+) L2(R+) and limits can be taken resulting in a limiting Bogolubov transformation in the space L2(R+) L2(R+). It has also been shown that the resulting family of Bogolubov transformation has three important properties, namely bi-evolution, shift covariance and time-reversal covariance, see (Hudson [2007]) for a detailed description of these properties. Subsequently we show rigorously that this transformation really is a Bogolubov transformation. We remark that these transformations are Hilbert-Schmidt perturbations of the identity map and satisfy a criterion specified by Shale's theorem. By Shale's theorem we then know that each Bogolubov transformation is implemented in the Fock representation of the CCR. We also compute the constituent kernels of the integral operators making up the Hilbert-Schmidt operators involved in the Bogolubov transformations, and show that the order in which the approximating discrete products are taken has no bearing on the final Bogolubov transformation got by the limiting procedure, as would be expected from the multiplicative Fubini Theorem. In Chapter five we generalise the canonical form of the double product studied in Chapter four by the use of gauge transformations. We show that all the theory of Chapter four carries over to these generalised double product integrals. This is because there is unitary equivalence between the Bogolubov transformation got from the generalised canonical form of the double product and the corresponding original one. In Chapter six we make progress towards showing that a system of implementors of this family of Bogolubov transformations can be found which inherits properties of the original family such as being a bi-evolution and being covariant under shifts. We make use of Shales theorem (Parthasarathy [1992] and Derezinski [2003]). More specifically, Shale's theorem ensures that each Bogolubov transformation of our system is implemented by a unitary operator which is unique to with multiplicaiton by a scalar of modulus 1. We expect that there is a unique system of implementors, which is a bi-evolution, shift covariant, and time reversal covariant (i.e. which inherits the properties of the corresponding system of Bogolubov transformation). This is partly on-going research. We also expect the implementor of the Bogolubov transformation to be the original double product. In Evans [1988], Evan's showed that the the implementor of a Bogolubov transformation in the simple product case is indeed the simple product. If given more time it might be possible to adapt Evan's result to the double product case.

The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.