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This thesis covers the topic of the computation of Galois groups over the rationals. Beginning with the classic algorithm by R. Stauduhar, we then review the theory necessary to explain the modular algorithm by K. Yokoyama. More precisely, we discuss the notion of the universal splitting ring of a polynomial. For a separable polynomial, we then study idempotents in the universal splitting ring. The modular algorithm involves computations in the ring of p-adic integers. Examples are given for polynomials of degree 3 and 4.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22982号 / 理博第4659号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 熊谷 隆, 教授 長谷川 真人, 小澤 登高 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Spectral properties of integral operators on spaces of analytic functions on the unit disk of the complex plane have been studied since 1918. In this dissertation we determine spectral pictures and resolvent estimates for Ces`aro-like operators on the weighted Bergman spaces and show in particular that some of these operators are subdecomposable. Moreover, in a special case, we show that some of these operators are subnormal, some are normaloid, and some are subscalar. We also determine the spectrum and essential spectrum as well as resolvent estimates for a class of integral operators acting on Banach spaces of analytic functions on the unit disk, including the classical Hardy and weighted Bergman spaces, analytic Besov spaces as well as certain Dirichlet spaces and generalized Bloch spaces. Our results unify and extend recent work by Aleman and Persson, [4], Ballamoole, Miller and Miller, [6], and Albrecht and Miller, [3]. In [3], another class of integral operators were investigated in the setting of the analytic Besov spaces and the little Bloch space where the spectra, essential spectra together with one sided analytic resolvents in the Fredholm regions of these operators were obtained along with an explicit strongly decomposable operator extending one of these operator and simultaneously lifting the other. In this disseration, we extend this spectral analysis to nonseparable generalized Bloch spaces using a modification of a construction due to Aleman and Persson, [4].

For decades noise-induced hearing loss has been a concern of the Department of Defense (DoD). My research investigates noise generation and dispersion in supersonic jets and focuses on the fluid-dynamic regime typical of high-performance turbojet and turbofan engines. The goal of my research is to understand how dispersion and propagation of wavepackets can be modified by noise reduction strategies based on secondary injections of fluid with a different temperature from the main jet. The research is organized into three studies that focus on instability, large eddy simulations, and resolvent modes. The first study is a computational investigation of the role of thermal non-uniformity on the development of instability modes in the shear-layer of a supersonic $M= 1.5$, $Re=850,000$ jet. Cold fluid is injected at the axis of a heated jet to introduce radial non-uniformity and control the spatial development of the shear layer. The mean flow is analyzed with an efficient 2D and 3D Reynolds-averaged Navier-Stokes (RANS) approach using the SU2 code platform for 3 different cases -baseline, centered, and offset injection. Different turbulence models are tested and compared with the experiments. The coherent perturbation is analyzed using linear parallel and parabolized stability equations (PSEs). The second study investigates novel formulations of large eddy simulation models using an arbitrary high order discontinuous Galerkin scheme. The LES analysis focuses on both numerical issues (such as convergence against the polynomial order of the mesh), modeling issues (such as the choice of subgrid model), and underlying physics (such as vortex stretching and noise generation). Wall models are used to capture the viscous sublayer at the nozzle. The Ffowcs Williams-Hawkings (FW-H) method is used for far-field noise predictions for all cases. Three-dimensionality is studied to investigate how injection in the shear layer acts to create a rotational inviscid core and affects the mixing of the cold fluid and noise dispersion. The third study extends the (first) instability study by considering (global) resolvent modes. Such optimally forced modes of the turbulent mean flow field will identify the turbulent coherent structures (wavepackets) for different turbulence models at $M=1.5$. The LES simulations performed in the second study will be used to extract the mean flow and the dynamic modes for comparison. My research plan is to perform the resolvent analysis of the axisymmetric mean flow fields for the thermally activated case (i.e., the centered injection) and compare it to the baseline jet case. Different turbulence models will be investigated to determine the correct alignment of dynamic and resolvent modes. Finally, I will consider the three-dimensional, non-axisymmetric mean flow created by offset injection described in the second study, which requires evaluating the convolution products of resolvent modes and base flow. Such three-dimensional resolvent compressible modes have never been identified in the context of supersonic jets. / Doctor of Philosophy / For decades noise-induced hearing loss has been a concern of the Department of Defense (DoD). Research in this area is critical to US national security and valued by both the aircraft industry and government. The noise generated during take-off and landing is hazardous to the crew personnel who work around this vicinity. A reduction of noise can significantly decrease medical expenditure and allow the aircraft industry to meet the stringent community noise requirements. Among the various techniques of noise reduction analyzed over the years, thermal non-uniformity stands out for its simple implementation and cost-effectiveness, especially in after-burner turbojets. Thermal non-uniformity with a cold secondary stream introduces low-velocity fluid in a supersonic jet by locally increasing the density while matching the mass flow rate. Changes to the velocity profile are localized; different regions of the jet emit sound at different frequencies and radiation angles, thus the link between injection location and noise control is not well understood. Using different computational tools this research investigates the link connecting thermal non-uniformity, turbulent production, and sound generation. Injection at different radial locations affects the two mechanisms of sound radiation in different ways. The first mechanism, the Kelvin Helmholtz instability, can be studied as an eigenvalue problem that represents the spatial growth of normal modes. De-coherence of these modal fluctuations can be obtained by injecting secondary fluid directly into the shear layer. This injection mode is called offset injection. The present research shows that the thickening of the shear layer due to low-velocity fluid delays the formation of Kelvin-Helmholtz modes in the offset case. Thus, the outskirts of the jet produce pressure fluctuations with a lower spectral energy density. The second mechanism, the Orr instability, can be analyzed as non-modal growth of acoustic perturbation forced by the breakdown of the core of the jet. LES and stability analysis shows that centered injection is highly effective in reducing the Orr radiation. Resolvent modes explain that the rationale is the delay and reduction of a secondary resonant peak between spatial eddies and forcing caused by changes in the mean profile responsive to secondary injection. Our analysis also explains why the offset injection is more effective at a low polar angle, while centered injection reduces acoustic radiation towards high polar angles. Parametric studies of different injection strategies, i.e., location and number of injection ports are performed to demonstrate the best strategy for noise level reductions.

Neste trabalho estudaremos a existência de soluções fracas, semi-clássicas e clássicas, conceitos introduzidos no texto para uma classe de sistemas integro-diferenciais do tipo neutro com retardamento não limitado modelados na forma d/dt D(t, xt) = AD(t, xt) + ∫t0 B(t - s)D(s, xs)ds + g(t, xt), t ∈ (0, a), x0 = φ ∈ B, d/dt (x(t) + F(t, xt)) = Ax(t) + ∫t0 B(t - s)x(s)ds + G(t, xt), t ∈ (0, a), x0 = φ ∈ B, onde A é um operador linear fechado densamente definido em um espaço de Banach X, cada B(t) : D(B(t)) ⊂ X → X, t ≥ 0 é um operador linear fechado, a história xt : (-∞, 0] → X, xt(θ) = x(t + θ), pertence a um espaço de fase abstrato B definido axiomaticamente e D, F, g, G : [0, a] × B → X são funções apropriadas. Para obter alguns de nossos resultados, estudamos a existência e propriedades qualitativas de uma família resolvente de operadores lineares limitados (R(t))t≥0, para o sistema integro-diferencial d/dt (x(t) + ∫t0 N(t - s)x(s)ds) = Ax(t) + ∫t0 B(t - s)x(s) ds, t ∈ (0, a), x(0) = x0, onde (N(t)) t≥0 é uma família de operadores lineares limitados em X. Mencionamos que este tipo de sistemas aparece no estudo da condução de calor em materiais com memória amortecida. / In this work we study the existence of mild, semi-classical and classical solution, concepts introduced be later for a class of abstract neutral functional integrodifferential systems with unbounded delay in the form d/dt D(t, xt) = AD(t, xt) + ∫t0 B(t - s)D(s, xs)ds + g(t, xt), t ∈ (0, a), x0 = φ ∈ B, d/dt (x(t) + F(t, xt)) = Ax(t) + ∫t0 B(t - s)x(s)ds + G(t, xt), t ∈ (0, a), x0 = φ ∈ B, where A : D(A) ⊂ X → X is a closed linear densely defined operator in a Banach space X, each B(t) : D(B(t)) ⊂ X → X, is a closed linear operator, the history xt : (-∞, 0] → X, xt(θ) = x(t + θ), belongs to some abstract phase space B defined axiomatically and D, F, g :[0, a] × B → X are appropriate functions. To establish some of our results, we studied the existence and qualitative properties of a resolvent of bounded linear operators (R(t))t≥0, for a system in the form d/dt (x(t) + ∫t0 N(t - s)x(s)ds) = Ax(t) + ∫t0 B(t - s)x(s) ds, t ∈ (0, a), x(0) = x0, where (N(t)) t≥0 is a family of bounded linear operators on X. We mention that this class of system arise in the study of heat conduction in material with fading memory.

Nous proposons deux méthodes pour calculer le spectre des ondes de cisaillement dans les cristaux phononiques (CP) 1D et 2D. Commençant notre étude par les CP 1D, nous développons la méthode des impédances scalaires pour la couche sur le substrat 1D.Le focus principal de ce travail est sur les CP 2D : en particulier, on considère la couche sur le substrat 2D, la plaque à conditions libres 2D et la couche entre les deux substrats 2D. Comme la matrice propagateur M à travers la cellule unitaire obtenue via l’expansion des ondes planes dans une coordonnée peut avoir des composants très grandes, notre approche consiste à la substituer par sa résolvante R= (zI−M)−1qui est numériquement stable (où z est un nombre complexe hors des pecM). Deux autres outils centraux définis par la résolvante, le projecteur spectral P de t propagateur Md pour les ondes évanescentes, entrent en jeu pour le cas des CP avec un substrat. La méthode de la résolvante, fournissant les équations de dispersion et du champ d’ondes en termes de R,P de tMd, a de multiples avantages. Elle est d’une bonne précision grâce à la solution exacte dans une coordonnée, efficace grâce à la réduction du problème à une seule cellule unitaire, même pour un substrat semi-infini, et polyvalente, puisque applicable pour les structures uniformes ou périodiques à 1D ou 2D. De plus, la méthode peut être généralisée aux CP à 3D et aux ondes vectorielles.Dans les exemples numériques, nous calculons les bandes d’arrêt de basse fréquence et les comparons avec les profils de symétrie axiale et les profils perturbées. / We propose two methods for calculating the shear waves spectra in 1D and 2D phononiccrystal (PC) wave guides. Starting this study with 1D PC, we consider the 1D-periodic coated substrate. Here we develop scalar impedance method providing efficient means for analysis and calculation of dispersion spectrum. The main focus of our work in on the 2D PC’s: the 2D PC layer on a substrate, the freePC plate and the PC plate sandwiched between two substrates.Since the propagator Mover a unit cell approximated by Fourier harmonics in one coordinate can have very large components, we introduce its resolvent R= (zI−M)−1(z is a complex number outside of specM) as a numerically stable substitute. Another two key tools given in terms of there solvent, a spectral projector Pd and propagator Md for the decreasing modes, come intoplay in the case of a wave guide with a substrate. The resolvent method providing simple dispersion and wave field equations in termsof R,Pd and Md has several advantages. It is of a good precision due to the exact solution in one direction, computationally cheap due to the reduction of the problem to one unitcell even in a semi-infinite substrate, and versatile since it is applicable to uniform, 1D- or 2D-periodic structures. More over, it is extendible to P/SVwaves and 3D PC.In numerical examples, we model low-frequency band gaps and compare them for the mirror-symmetric and perturbed profiles.

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This thesis concerns the stability of plane Couette flow and pipe Poiseuille flow in three space dimensions. The mathematical model for both flows is the incompressible Navier--Stokes equations. Both analytical and numerical techniques are used. We present new results for the resolvent corresponding to both flows. For plane Couette flow, analytical bounds on the resolvent have previously been derived in parts of the unstable half-plane. In the remaining part, only bounds based on numerical computations in an infinite parameter domain are available. Due to the need for truncation of this infinite parameter domain, these results are mathematically insufficient. We obtain a new analytical bound on the resolvent at s=0 in all but a compact subset of the parameter domain. This is done by deriving approximate solutions of the Orr--Sommerfeld equation and bounding the errors made by the approximations. In the remaining compact set, we use standard numerical techniques to obtain a bound. Hence, this part of the proof is not rigorous in the mathematical sense. In the thesis, we present a way of making also the numerical part of the proof rigorous. By using analytical techniques, we reduce the remaining compact subset of the parameter domain to a finite set of parameter values. In this set, we need to compute bounds on the solution of a boundary value problem. By using a validated numerical method, such bounds can be obtained. In the thesis, we investigate a validated numerical method for enclosing the solutions of boundary value problems. For pipe Poiseuille flow, only numerical bounds on the resolvent have previously been derived. We present analytical bounds in parts of the unstable half-plane. Also, we derive a bound on the resolvent for certain perturbations. Especially, the bound is valid for the perturbation which numerical computations indicate to be the perturbation which exhibits largest transient growth. The bound is valid in the entire unstable half-plane. We also investigate the stability of pipe Poiseuille flow by direct numerical simulations. Especially, we consider a disturbance which experiments have shown is efficient in triggering turbulence. The disturbance is in the form of blowing and suction in two small holes. Our results show the formation of hairpin vortices shortly after the disturbance. Initially, the hairpins form a localized packet of hairpins as they are advected downstream. After approximately $10$ pipe diameters from the disturbance origin, the flow becomes severely disordered. Our results show good agreement with the experimental results. In order to perform direct numerical simulations of disturbances which are highly localized in space, parallel computers must be used. Also, direct numerical simulations require the use of numerical methods of high order of accuracy. Many such methods have a global data dependency, making parallelization difficult. In this thesis, we also present the process of parallelizing a code for direct numerical simulations of pipe Poiseuille flow for a distributed memory computer. / QC 20100825

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