Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 61).

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In chapter I known asymptotic forms and expansions of the hypergeometric function obtained by Erdélyi [5], Hapaev [10,11], Knottnerus [15L Sommerfeld [25] and Watson [28] are discussed. Also the asymptotic expansions of the hypergeometric function occurring in gas-flow theory will be discussed. These expansions were obtained by Cherry [1,2], Lighthill [17] and Seifert [2J]. Moreover, using a paper by Thorne [28] asymptotic expansions of ₂F₁(p+1, -p; 1-m; (1-t)/2), -1 < t < 1, and ₂P₁( (p+m+2)/2, (p+m+1)/2; p+ 3/2-, t⁻² ), t > 1, are obtained as p-»» and m = -(p+ 1/2)a, where a is fixed and 0 < a < 1. The : expansions are in terms of Airy functions of the first kind. The hypergeometric equation is normalized in chapter II. It readily yields the two turning points t₁, i = 1,2. If we consider,the case the a=b is a large real parameter of the hypergeometric function ₂F₁(a,b; c; t), then the turning points coalesce with the regular singularities t = 0 and t = ∞ of the hypergeometric equation as | a | →∞. In chapter III new asymptotic forms are found for this particular case; that is, for ₂F₁ (a, a; c;t) , 0 < T₁ ≤ t < 1, and ₂F₁ (a,a+1-c; 1; t⁻¹), 1 < t ≤ T₂ < ∞ , as –a → ∞ . The asymptotic form is in terms of modified Bessel functions of order 1/2. Asymptotic expansions can be obtained in a similar manner. Furthermore, a new asymptotic form is derived for ₂F₁ (c-a, c-a; c; t), 0 < T₁ ≤ t < 1, as –a → ∞, this result then leads to a sharper estimate on the modulus of n-th order derivatives of holomorphic functions as n becomes large. / Science, Faculty of / Mathematics, Department of / Graduate

Approved for public release; distribution is unlimited / This thesis investigates the viability of a new method for numerically computing the input impedance and the currents on simple antenna structures. This technique considers the antenna between two ground planes and uses multiregion cylindrical harmonic expansions with tangential field continuity to obtain the surface currents and input impedance. The computed results are compared to the results obtained from the Numerical Electromagnetics Code for various physical parameters to assess computational accuracy. / http://archive.org/details/computationofmon00hurl / Lieutenant, United States Navy

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

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The notion of asymptotic structure of an infinite dimensional Banach space was introduced by Maurey, Milman and Tomczak-Jaegermann. The asymptotic structure consists of those finite dimensional spaces which can be found everywhere `at infinity'. These are defined as the spaces for which there is a winning strategy in a certain vector game. The above authors introduced the class of asymptotic $\ell_p$ spaces, which are the spaces having simplest possible asymptotic structure. Key examples of such spaces are Tsirelson's space and James' space. We prove some new properties of general asymptotic $\ell_p$ spaces and also compare the notion of asymptotic $\ell_2$ with other notions of asymptotic Hilbert space behaviour such as weak Hilbert and asymptotically Hilbertian. We study some properties of smooth functions defined on subsets of asymptotic $\ell_\infty$ spaces. Using these results we show that that an asymptotic $\ell_\infty$ space which has a suitably smooth norm is isomorphically polyhedral, and therefore admits an equivalent analytic norm. We give a sufficient condition for a generalized Orlicz space to be a stabilized asymptotic $\ell_\infty$ space, and hence obtain some new examples of asymptotic $\ell_\infty$ spaces. We also show that every generalized Orlicz space which is stabilized asymptotic $\ell_\infty$ is isomorphically polyhedral. In 1991 Gowers and Maurey constructed the first example of a space which did not contain an unconditional basic sequence. In fact their example had a stronger property, namely that it was hereditarily indecomposable. The space they constructed was `$\ell_1$-like' in the sense that for any $n$ successive vectors $x_1 < \ldots < x_n$, $\frac{1}{f(n)} \sum_{i=1}^n \| x_i \| \leq \| \sum_{i=1}^n x_i \| \leq \sum_{i=1}^n \| x_i \|,$ where $ f(n) = \log_2 (n+1) $. We present an adaptation of this construction to obtain, for each $ p \in (1, \infty)$, an hereditarily indecomposable Banach space, which is `$\ell_p$-like' in the sense described above. We give some sufficient conditions on the set of types, $\mathscr{T}(X)$, for a Banach space $X$ to contain almost isometric copies of $\ell_p$ (for some $p \in [1, \infty)$) or of $c_0$. These conditions involve compactness of certain subsets of $\mathscr{T}(X)$ in the strong topology. The proof of these results relies heavily on spreading model techniques. We give two examples of classes of spaces which satisfy these conditions. The first class of examples were introduced by Kalton, and have a structural property known as Property (M). The second class of examples are certain generalized Tsirelson spaces. We introduce the class of stopping time Banach spaces which generalize a space introduced by Rosenthal and first studied by Bang and Odell. We look at subspaces of these spaces which are generated by sequences of independent random variables and we show that they are isomorphic to (generalized) Orlicz spaces. We deduce also that every Orlicz space, $h_\phi$, embeds isomorphically in the stopping time Banach space of Rosenthal. We show also, by using a suitable independence condition, that stopping time Banach spaces also contain subspaces isomorphic to mixtures of Orlicz spaces.

Thesis (Ph. D.)--Carleton University, 1992. / Also available in electronic format on the Internet.

This thesis aims to resolve some open questions about sonic boom, and particularly secondary sonic boom, which arises from long-range propagation in a non-uniform atmosphere. We begin with an introduction to sonic boom modelling and outline the current state of research. We then proceed to review standard results of gas dynamics and we prove a new theorem, similar to Kelvin's circulation theorem, but valid in the presence of shocks. We then present the definitions used in sonic boom theory, in the framework of linear acoustics for stationary and for moving non-uniform media. We present the wavefront patterns and ray patterns for a series of analytical examples for propagation from steadily moving supersonic point sources in stratified media. These examples elucidate many aspects of the long-range propagation of sound and in particular of secondary sonic boom. The formation of `fold caustics' of boomrays is a key feature. The focusing of linear waves and weak shock waves is compared. Next, in order to address the consistent approximation of sonic boom amplitudes, we consider steady motion of supersonic thin aerofoils and slender axisymmetric bodies in a uniform medium, and we use the method of matched asymptotic expansions (MAE) to give a consistent derivation of Whitham's model for nonlinear effects in primary boom analysis. Since for secondary boom, as for primary, the inclusion of nonlinearities is essential for a correct estimation of the amplitudes, we then study the paradigm problem of a thin aerofoil moving steadily in a weakly stratified medium with a horizontal wind. We again use MAE to calculate approximations of the Euler equations; this results in an inhomogeneous kinematic wave equation. Returning to the linear acoustics framework, for a point source that accelerates and decelerates through the sound speed in a uniform medium we calculate the wavefield in the `time-domain'. Certain other motions of interest are also illustrated. In the accelerating and in the manoeuvring motions fold caustics that are essentially the same as those from steady motions in stratified atmospheres again arise. We also manage to pinpoint a scenario where a `cusp caustic' of boomrays forms instead. For the accelerating motions the asymptotic analysis of the wavefield reveals the formation of singularities which are incompatible with linear theory; this suggests the re-introduction of nonlinear effects. However, it is a formidable task to solve such a nonlinear problem in two or three dimensions, so we solve a related one-dimensional problem instead. Its solution possesses an unexpectedly rich structure that changes as the strength of nonlinearity varies. In all cases however we find that the singularities of the linear problem are regularised by the nonlinearity.