A technique has been developed for studying surface waves on liquids. The measuring device employed Fourier analyzes the surface wave being studied. This property of the technique has been verified by three independent tests. The method developed has been applied to the study of the Rayleigh-Taylor instability of fluid surfaces. The results of this study are in good agreement with theory. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

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This thesis consists of two unrelated parts. In the first part of this thesis, we give explicit expressions for the Fourier coefficients of Eisenstein series E∗(z, s, χ) twisted by modular symbols ⟨γ, f⟩ in the case where the level of f is prime and equal to the conductor of the Dirichlet character χ. We obtain these expressions by computing the spectral decomposition of an automorphic function closely related to E∗(z, s, χ). We then give applications of these expressions. In particular, we evaluate sums such as Σχ(γ)⟨γ, f⟩, where the sum is over γ ∈ Γ∞\Γ0(N) with c^2 + d^2 < X, with c and d being the lower-left and lower-right entries of γ respectively. This parallels past work of Goldfeld, Petridis, and Risager, and we observe that these sums exhibit different amounts of cancellation than what one might expect. In the second part of this thesis, given an elliptic curve E and a point P in E(R), we investigate the distribution of the points nP as n varies over the integers, giving bounds on the x and y coordinates of nP and determining the natural density of integers n for which nP lies in an arbitrary open subset of {R}^2. Our proofs rely on a connection to classical topics in the theory of Diophantine approximation.

Using C. Fefferman's embedding of a charge space in a measure space allows us to apply standard interpolation theorems to the establishment of norm inequalities for Besicovitch almost periodic functions. This yields a significant improvement to the results of A. Avantaggiati, G. Bruno and R. Iannacci. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

<p>This work is devoted to trigonometric series with monotone coefficients. The main problem is to study conditions under which a given series is the Fourier series of an integrable (or continuous) function.</p>

This work is devoted to trigonometric series with monotone coefficients. The main problem is to study conditions under which a given series is the Fourier series of an integrable (or continuous) function.

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