Let S2k(N) be the space of cusp forms of weight 2k and level N. Atkin-Lehner theory shows that S2k(N) can be decomposed into the oldspace and its orthogonal complement newspace. Again, from Atkin-Lehner theory, it follows that there exists a basis of newspace whose elements are simultaneous eigenforms of all the Hecke operators. Such eigenforms when normalized are called primitive forms. In 1932, Petersson introduced a harmonic weighted sum of the Fourier coefficients of an orthogonal basis B2k(N) for S2k(N), denoted by _2k;N . Petersson connected _2k;N to Kloosterman sums and Bessel functions, which is now known as the Petersson trace formula. The Petersson trace formula shows that _2k;N is independent of the choice of orthogonal basis. It is known that the oldspace decomposes into the images of newspaces of different levels under the scaling operator Bd where d is a proper divisor of N. It is of interest to derive a Petersson-type trace formula for primitive forms. In 2001, H. Iwaniec, W. Luo and P. Sarnak obtained an expression of Petersson-type trace formula for primitive forms in terms of _2k;N , when the level N is squarefree. Their method is to construct a special orthogonal basis for S2k(N). Using their approach, D. Rouymi has extended similar results to the case of prime power level in 2011. In this thesis, the case of arbitrary levels is investigated. Analogously, a special orthogonal basis is constructed and a Petersson-type trace formula for primitive forms in terms of _2k;N is found. The result established in this thesis recovers the results of H. Iwaniec, W. Luo and P. Sarnak, and D. Rouymi respectively for the cases of squarefree and prime power levels. / published_or_final_version / Mathematics / Master / Master of Philosophy

The large sieve method has been used extensively, beginning with Bombieri in 1965, to provide bounds on the number of zeros of Dirichlet L-functions near the line σ = 1. Using the Kuznetsov trace formula and the work of Deshouillers and Iwaniec on Kloosterman sums, it is possible to derive large sieve inequalities for the Fourier coefficients of Maass cusp forms, which may then similarly be used to study the corresponding Hecke-Maass L-functions. Following an approach developed by Gallagher for Dirichlet L-functions, this thesis shows how the large sieve method may be used to prove a zero density estimate, averaged over the Laplace eigenvalues, for Maass cusp forms of weight zero for the congruence subgroup Γ₀(q) for any positive integer q.

The motivation of this thesis is to look for a signature of Lorentz violation, hopefully observable, in the gravitational waves emitted by cosmic strings. Aspects of cosmic strings are reviewed, in particular how focused bursts of gravitational radiation are emitted when a cusp forms on the string. The same phenomenon is then studied in an effective field theory with Lorentz violation called Einstein-aether theory. This is a simple theory with a dynamic preferred frame, but it retains rotational and diffeomorphism invariance. The linearized version of the theory produces five wave modes. We study the usual transverse traceless modes which now have a wave speed that can be lower or greater than the speed of light. This altered speed produces distinctive features in the waves. They depend on two free parameters: roughly the wave speed and the acceleration of the string cusp. The profile of the wave is analyzed in detail for different values of the parameters and explained by close comparison with the string motion.

The motivation of this thesis is to look for a signature of Lorentz violation, hopefully observable, in the gravitational waves emitted by cosmic strings. Aspects of cosmic strings are reviewed, in particular how focused bursts of gravitational radiation are emitted when a cusp forms on the string. The same phenomenon is then studied in an effective field theory with Lorentz violation called Einstein-aether theory. This is a simple theory with a dynamic preferred frame, but it retains rotational and diffeomorphism invariance. The linearized version of the theory produces five wave modes. We study the usual transverse traceless modes which now have a wave speed that can be lower or greater than the speed of light. This altered speed produces distinctive features in the waves. They depend on two free parameters: roughly the wave speed and the acceleration of the string cusp. The profile of the wave is analyzed in detail for different values of the parameters and explained by close comparison with the string motion.

The motivation of this thesis is to look for a signature of Lorentz violation, hopefully observable, in the gravitational waves emitted by cosmic strings. Aspects of cosmic strings are reviewed, in particular how focused bursts of gravitational radiation are emitted when a cusp forms on the string. The same phenomenon is then studied in an effective field theory with Lorentz violation called Einstein-aether theory. This is a simple theory with a dynamic preferred frame, but it retains rotational and diffeomorphism invariance. The linearized version of the theory produces five wave modes. We study the usual transverse traceless modes which now have a wave speed that can be lower or greater than the speed of light. This altered speed produces distinctive features in the waves. They depend on two free parameters: roughly the wave speed and the acceleration of the string cusp. The profile of the wave is analyzed in detail for different values of the parameters and explained by close comparison with the string motion. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The subject of this work is a theoretical analysis of the Pearcey function. In optics, thin lens theory supposes that all rays focus at a unique point where the field converges. For a real lens, the focal point is replaced by a cusp, which is the end point of a caustic curve dividing the bright field region from the dark. My particular interest is the pattern of nodal points within the cusp. By investigating the stationary points for the cusp catastrophe, asymptotic approximations are found for the Pearcey function. This in turn leads to the development of finding the positions of nodal points inside, and outside a caustic. Also values for $|P|$ on a small circle surrounding a node are examined and show reasonable accuracy of order $10^{-8}$. / Thesis / Master of Science (MSc) / Identifying the nodes of the Pearcey function.

The purpose of the present study was to examine the perceptual processes associated with gender differences in leadership emergence recognition. Prior research has indicated that females are less likely to be identified as an emerging leader, even when they display identical leadership behaviors as that of their male counterparts. Unlike most of the previous research performed in this area which has obtained only static snapshots of leadership recognition, the present study used a nonlinear dynamic modeling technique, called cusp catastrophe theory. It was predicted that a nonlinear model would account for more variance than a linear model. Furthermore, it was also predicted that participants would be more resistant to recognizing a female as an emerging leader, as compared to a male. This effect was expected to be greater for male participants than female participants. Participants included 19 organizational members, who watched videos of either a male or female emerging as the leader of a four-person group. Participants recorded their perceptions of leadership through a dynamic measure. In accordance with cusp catastrophe theory, results were analyzed using the program GEMCAT II (General Multivariate Methodology for Estimating Catastrophe Models). Contrary to expectations, none of the predictions were supported. It is suggested that this was primarily due to methodological issues, rather than the relevance of cusp catastrophe modeling for leadership perceptions. Recommendations for future work in this area are provided. / Master of Science

The thesis describes the study of the rare charged kaon decay K00 e4 on the NA62 experiment at CERN. This decay is interesting from the chiral perturbation theory point of view. In this work the data from 2007 recorded on NA62 ex- perimental setup were analyzed. The measurement of branching ratio for this process normalized to K00 3π decay has been performed. Also the form factor measurement has been done and the cusp singularity has been observed. 1

Title: Stochastic Catastrophe Model Cusp Author: Jan Voříšek Department: Department of Probability and Mathematical Statistics Supervisor: Prof. Ing. Miloslav Vošvrda, CSc., Czech Academy of Sciences, Institute of Information Theory and Automation Abstract: The goal of this thesis is to analyze the stochastic cusp model. This task is divided into two main topics. The first of them concentrates on the stationary density of the cusp model and statistical testing of its bimodality, where power and size of the proposed tests are simulated and compared with the dip test of unimodality. The second main topic deals with the transition density of the stochastic cusp model. Comparison of approximate maximum likelihood approach with traditional finite difference and numerical simulations indicates its advantage in terms of speed of estimation. An approximate Fisher information matrix of general stochastic process is derived. An application of the cusp model to the exchange rate with time-varying parameters is estimated, the extension of the cusp model into stochastic bimodality model is proposed, and the measure of probability of intrinsic crash of the cusp model is suggested. Keywords: stochastic cusp model, bimodality testing, transition density ap- proximation

This thesis provides improvements to computational fluid dynamics accuracy and ef- ficiency through two main methods: a new boundary condition verification procedure and preconditioning techniques. First, a new verification approach that addresses boundary conditions was developed. In order to apply the verification approach to a large range of arbitrary boundary condi- tions, it was necessary to develop unifying mathematical formulation. A framework was developed that allows for the application of Dirichlet, Neumann, and extrapolation bound- ary condition, or in some cases the equations of motion directly. Verification of boundary condition techniques was performed using exact solutions from canonical fluid dynamic test cases. Second, to reduce computation time and improve accuracy, preconditioning algorithms were applied via artificial dissipation schemes. A new convective upwind and split pressure (CUSP) scheme was devised and was shown to be more effective than traditional precon- ditioning schemes in certain scenarios. The new scheme was compared with traditional schemes for unsteady flows for which both convective and acoustic effects dominated. Both boundary conditions and preconditioning algorithms were implemented in the context of a "strand grid" solver. While not the focus of this thesis, strand grids provide automatic viscous quality meshing and are suitable for moving mesh overset problems.