Cette thèse est dédiée à l’étude de la violation CP dans les processus ūd→tb̄ e⁻ μ⁻ et W⁻→ l-1 l-2 (q′¯ q)⁺. Il y a des expériences qui cherchent à identifier ces processus au grand collisionneur de hadrons, ou LHC (Large Hadron Collider). Une conséquence de l’observation de ces désintégrations impliquait une brisure de la conservation du nombre leptonique, car ∆L = 2. Si le neutrino est de type Majorana, il est sa propre antiparticule et on peut observer la double désintégration bêta sans émission de neutrino (0νββ). Ce processus est caractérisé par ∆L = 2. Si nous observons des processus similaires à 0νββ, ∆L = 2, au LHC, il y a aussi la possibilité de chercher la violation CP. La mesure de ce type de violation CP nous donnerait des informations sur de la Nouvelle Physique (NP) qui ne pourraient pas être facilement obtenues autrement. Il existe trois types de violation CP : directe, indirecte et cinématique (produit triple). Ce travail relève essentiellement mes études sur deux processus de violation CP, l’un direct et l’autre produit triple. Elles sont présentées dans deux articles qui sont inclus dans cette thèse. Le premier article est consacré au produit triple dans le processus : ūd→tb̄ e⁻ μ⁻ . Le modèle que nous avons considéré n’implique pas de neutrino de Majorana. À l’aide de MadGraph, nous avons étudié la violation CP pour les deux processus mentionnés pour deux versions du LHC, haute luminosité HL-LHC (14 TeV) et haute énergie HE-LHC (27 TeV),et pour un futur collisionneur FCC-hh (100 TeV). Nous trouvons que le produit triple n’est pas mesurable au HL-LHC, peut être mesurable au HE-LHC, et est certainement mesurable au FCC-hh. Dans le deuxième article, nous avons examiné comment générer la violation CP avec des neutrinos droitiers. Le deuxième article porte sur l’étude de la violation CP directe dans la désintégration W⁻→ l-1 l-2 (q′¯ q)⁺ en impliquant deux neutrinos N1 et N 2. Nous avons trouvé que la valeur de l’asymétrie CP peut être mesurée avec une précision de 3σ et que 0.1% ≤ ACP ≤ 10%. / This thesis is dedicated to the study of CP violation in these processes ūd→tb̄ e⁻ μ⁻ et W⁻→ l-1 l-2 (q′¯ q)⁺. There are experiments which search to identify these processes at the "Large Hadron Collider" (LHC). The outcome of the observation of these decays would imply violation of lepton number conservation, ∆L = 2. If the neutrino is a Majorana particle, it is its own antiparticle and we can observe processes that violate the conservation of the lepton number, ∆L = 2. The neutrinoless double beta decay (0νββ), is a process which is characterized by ∆L = 2, occurs when the neutrinos are Majorana particles. If we observe 0νββ-like processes at the LHC, there is also the possibility of looking for CP violation. Measuring this type of CP violation would give us information about New Physics that could not be easily obtained using other methods. There are three types of CP violation, namely: direct, indirect and kinematic (triple product). This research presents studies concerning direct and triple product CP violation. The first paper is devoted to triple products in the process: ūd→tb̄ e⁻ μ⁻. In addition, the model considered does not involve a Majorana neutrino. Using MadGraph, we have numerically studied and evaluated the CP violation for the two mentioned processes for three types of accelerators: high luminosity HL-LHC (14 TeV), high energy HE-LHC (27 TeV) and for a future FCC-hh collider (100 TeV). We found that the triple product is not measurable at the HL-LHC, may be measurable at HE-LHC, and is certainly measurable at FCC-hh. In the second article, we investigated to see how to generate CP violation with right-handed neutrinos. The second article is on the study of direct CP violation the decay W⁻→ l-1 l-2 (q′¯ q)⁺ by involving two neutrinos N1and N2. It is also observed that the value of CP asymmetry can be measured with an accuracy of 3σ and as 0.1% ≤ ACP ≤ 10%.

This thesis consists of seven chapters. Chapter 1 is an introduction to the infinite products. Here we provide a proof for representing sine function as an infinite product. This chapter also describes the notation used throughout the thesis as well as the method used to prove the identities. Each of the other chapters may be read independently, however some chapters assume familiarity with the Jacobi triple product identity. Chapter 2 is about the Jacobi triple product identity as well as several implications of this identity. In Chapter 3 the quintuple product identity and some of its special cases are derived. Even though there are many known proofs of this identity since 1916 when it was first discovered, the proof presented in this chapter is new. Some beautiful formulas in number theory are derived at the end of this chapter. The simplest two dimensional example of the Macdonald identity, A2, is investigated in full detail in Chapter 4. Ian Macdonald first outlined the proof for this identity in 1972 but omitted many of the details hence making his work hard to follow. In Chapters 5 and 6 we somewhat deviate from the method which uses the two specializations to evaluate the constant term and prove Winquist's identity and Macdonald's identity for G2. Some of the work involved in proving G2 identity is new. Finally in Chapter 7 we discuss the work presented with some concluding remarks as well as underlining the possibilities for the future research. Throughout the thesis we point to the relevant papers in this area which might provide different strategies for proving above identities.

Abstract This dissertation consists of four articles on irrationality measures. In the first paper we derive explicit irrationality measures by using the simple continued fraction expansions in a completely new way. In the second and third articles we use Padé approximations to construct irrationality measures. In the second paper we obtain an explicit irrationality measure for the values of q-exponential series, for which the earlier corresponding results are not as explicit. Furthermore, we construct a restricted irrationality measure for the values of q-exponential series, which is an improvement on the earlier results in the restricted case. In the third article we derive the best possible asymptotic restricted irrationality exponent for the values of Jacobi's triple product. In the last paper we consider Cantor series. We generalize the earlier results by deriving Sondow's irrationality measure for some Cantor series. / Tiivistelmä Tämä väitöskirja koostuu neljästä artikkelista, jotka kaikki käsittelevät irrationaalisuusmittoja. Ensimmäisessä artikkelissa irrationaalisuusmittoja johdetaan uudella tavalla irrationaalilukujen yksinkertaisista ketjumurtolukuesityksistä. Toisessa ja kolmannessa artikkelissa irrationaalisuusmitat konstruoidaan Padé-approksimaatioiden avulla. Toisessa artikkelissa saadaan eksplisiittinen irrationaalisuusmitta q-eksponenttisarjan arvoille, joiden vastaavat aikaisemmat irrationaalisuusmitat eivät ole näin eksplisiittisiä. Lisäksi samassa artikkelissa konstruoidaan q-eksponenttisarjan arvoille rajoitettu eksplisiittinen irrationaalisuusmitta, mikä parantaa aikaisempia tuloksia rajoitetussa tapauksessa. Kolmannessa artikkelissa johdetaan paras mahdollinen asymptoottinen irrationaalisuuseksponentti Jacobin kolmitulon arvoille. Viimeisessä artikkelissa käsitellään Cantorin sarjoja. Siinä yleistetään aikaisempia tuloksia johtamalla Sondowin irrationaalisuusmitta tietylle joukolle Cantorin sarjoja.

This dissertation presents results from a thorough study of the tip-gap turbulent flow structure in a low-speed linear compressor cascade wind tunnel at Virginia Tech that includes a moving belt system to simulate the relative motion between the tip and the casing. The endwall pressure measurements and the surface oil flow visualizations were made on a stationary endwall to obtain the flow features and to determine the measurement profiles of interest. A custom-made miniature 3-orthogonal-velocity-component fiber-optic laser-Doppler velocimetry (LDV) system was used to measure all three components of velocity within a 50 mm spherical measurement volume within the gap between the endwall and the blade tip, mainly for the stationary wall with 1.65% and 3.30% tip gaps as well as some initial experiments for the moving wall. Since all of the vorticity in a flow originates from the surfaces under the action of strong pressure gradient, it was very important to measure the nearest-wall flow on the endwall and around the blade tip. The surface skin friction velocity was measured by using viscous sublayer velocity profiles, which verified the presence of an intense lateral shear layer that was observed from surface oil flow visualizations. All second- and third-order turbulence quantities were measured to provide detailed data for any parallel CFD efforts. The most complete data sets were acquired for 1.65% and 3.30% tip gap/chord ratios in a low-speed linear compressor cascade. This study found that tip gap flows are complex pressure-driven, unsteady three-dimensional turbulent flows. The crossflow velocity normal to the blade chord is nearly uniform in the mid tip-gap and changes substantially from the pressure to suction side. The crossflow velocity relies on the local tip pressure loading that is different from the mid-span pressure loading because of tip leakage vortex influence. The tip gap flow is highly skewed three-dimensional flow throughout the full gap. Normalized circulation within the tip gap is independent of the gap size. The tip gap flow interacts with the primary flow, separates from the endwall, and rolls up on the suction side to form the tip leakage vortex. The tip leakage vortex is unsteady from the observation of the TKE transport vector and oil flow visualizations. The reattachment of tip separation vortex on the pressure side strongly depends on the blade thickness-to-gap height ratio after the origin of tip leakage vortex but is weakly related to it before the origin of tip leakage vortex for a moderate tip gap. Other than the nearest endwall and blade tip regions, the TKE does not vary much in tip gap. The tip leakage vortex produces high turbulence intensities. The tip gap flow correlations of streamwise and wall normal velocity fluctuations decrease significantly from the leading edge to the trailing edge of the blade due to flow skewing. The tip gap flow is a strongly anisotropic turbulent flow. Rapid distortion ideas can not apply to it. A turbulence model based on stress transport equations and experimental data is necessary to reflect the tip gap flow physics. For the moving endwall, relative motion skews the inner region flow and is decorrelated with the outer layer flow. Hence, the TKE and correlations of streamwise and wall normal velocity fluctuations decrease. / Ph. D.

This dissertation reviews the theory of fast matrix multiplication from a multilinear-algebraic point of view, as well as recent fast matrix multiplication algorithms based on discrete Fourier transforms over nite groups. To this end, the algebraic approach is described in terms of group algebras over groups satisfying the triple product Property, and the construction of such groups via uniquely solvable puzzles. The higher order singular value decomposition is an important decomposition of tensors that retains some of the properties of the singular value decomposition of matrices. However, we have proven a novel negative result which demonstrates that the higher order singular value decomposition yields a matrix multiplication algorithm that is no better than the standard algorithm. / Mathematical Sciences / M. Sc. (Applied Mathematics)

This dissertation reviews the theory of fast matrix multiplication from a multilinear-algebraic point of view, as well as recent fast matrix multiplication algorithms based on discrete Fourier transforms over nite groups. To this end, the algebraic approach is described in terms of group algebras over groups satisfying the triple product Property, and the construction of such groups via uniquely solvable puzzles. The higher order singular value decomposition is an important decomposition of tensors that retains some of the properties of the singular value decomposition of matrices. However, we have proven a novel negative result which demonstrates that the higher order singular value decomposition yields a matrix multiplication algorithm that is no better than the standard algorithm. / Mathematical Sciences / M. Sc. (Applied Mathematics)

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal. / Cette thèse est constituée de quatre articles qui ont tous été publiés. Le sujet principal est l’étude d’asymétries de brisure CP/T via les produits triples dans les désintégrations des hadrons contenant un quark b (mésons B ou hypérons Ab) et ce, dans le modèle standard de la physique des particules et au-delà. / This thesis is comprised of four articles, ail of which have been published. The main subject is the study, in and beyond the standard model of particle physics, of CP/T-violation via triple products in decays of hadrons containing a b quark (B mesons or Ab hyperons).