Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 230-235). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 141-155). Available also in a digital version from Dissertation Abstracts.

Suppose $f,g_1,\cdots,g_p$ are holomorphic functions over $\Omega\subset\cxC^n$. Then there raises a natural question: when can we find holomorphic functions $h_1,\cdots,h_p$ such that $f=\sum g_jh_j$? The celebrated Skoda theorem solves this question and gives a $L^2$ sufficient condition. In general, we can consider the vector bundle case, i.e. to determine the sufficient condition of solving $f_i(x)=\sum g_{ij}(x)h_j(x)$ with parameter $x\in\Omega$. Since the problem is related to solving linear equations, the answer naturally connects to the Cramer's rule. In the first part we will give a proof of division theorem by projectivization technique and study the generalized fundamental inequalities. In the second part we will apply the skills and the results of the division theorems to show some applications. / Mathematics

A comprehensive and rigorous description of instantaneous balance of electromagnetic power defined as the derivative of energy with respect to time is offered by the Poynting's theorem. Such theorem is expressed as the sum of a series of volume integrals representing the volume densities of densities of different components of electromagnetic power and the power flow through the general surface surrounding the entire domain in which the Poynting's vector expresses the instantaneous power leaving the domain (the positive normal is the outward normal to the enclosing surface). The original feature of the present approach is the introduction in the electromagnetic power balance and conservation of the electromechanical energy conversion by the use of the flux derivatives of the fields [D with vector arrow] and [B with vector arrow]. For the moving points (rotors) involved in electromechanical energy conversion, the surface of integration is driven together with them and [permittivity] and [permeatility] remain substantially constant--(a point in movement maintains its properties as [formula]). Then the balance of energy (and power) can be written at each infinitesimal time interval for the electromagnetic energy in which case the elementary mechanical work is produced by mechanical forces of electromagnetic origin. The thermal energy accounts for the Joule (and hysteresis) losses in the system. A treatment of the flow of electromagnetic energy is given for a complete of illustrative relationship in time and frequency domain. / text

One key element of underwater transducer design is the acoustic baffle. Acoustic baffles isolate a structure, such as a submarine hull, from noise and vibration produced by the active elements of the transducer and vice versa. Baffle materials must meet many conflicting requirements such as the need to be lightweight while providing high acoustic isolation. Currently Syntactic Acoustic Damping Material (SADM) is widely used as the primary acoustic baffle material. However, SADM baffles have many undesirable characteristics such as high density, poor machinability, high lead content and depth dependent acoustical behavior. The study of baffle materials is an under-represented area of sonar design. Most sonar transducer research focuses on the electrically active materials and their response to a variety of conditions. Relatively fewer studies have been devoted to understanding the effects of the supporting and baffle materials. This work considers the effects of the entire hydrophone system on the response while developing a method for aiding in proper system material selection. This was accomplished by first developing a model for a transducer's response in a variety of conditions. The response was validated with numerical finite-element models and experiments. Next, a generic model was developed that allows any number of layers with any material to be analyzed. This generic model is applied in concert with a material optimization method to aid in the selection of materials that will improve the transducer's response. The tools are finally applied to a simple real world problem to illustrate its strengths and weaknesses. / text

The Riemann integral has many flaws, some that becomes visible in the fundamental theorem of calculus. The main point of this essay is to introduce the gauge integral, and prove a much more suitable version of that theorem. / Riemannintegralen har många brister. Vissa utav dessa ser man i integralkalkylens huvudsats. Huvudmålet med denna uppsats är att introducera gauge integralen och visa en mer lämplig version av huvudsatsen.

In this dissertation we consider several limiting criteria forn-dimensional diffusion processes defined as solutions of stochasticdifferential equations. Our main interest is in criteria for polynomialand exponential rates of convergence to the steady state distributionin the total variation norm. Resulting criteria should place assumptionsonly on the coefficients of the elliptic differentialoperator governing the diffusion.Coupling of Harris chains is one of the main methods employed in thisdissertation.

In this thesis, we study time-dependent empirical processes, which extend the classical empirical processes to have a time parameter; for example the empirical process for a sequence of independent stochastic processes {Yi : i ∈ N}: (1) ν_n(t, y) = n^(−1/2 )Sigma[1_(Y i(t)¬<=y) – P(Yi(t) <= y)] from i=1 to n, t ∈ E, y ∈ R. In the case of independent identically distributed samples (that is {Yi(t) : i ∈ N} are iid), Kuelbs et al. (2013) proved a Central Limit Theorem for ν_n(t, y) for a large class of stochastic processes. In Chapter 3, we give a sufficient condition for the weak convergence of the weighted empirical process for iid samples from a uniform process: (2) α_n(t, y) := n^(−1/2 )Sigma[w(y)(1_(X (t)<=y) – y)] from i=1 to n, t ∈ E, y ∈ [0, 1] where {X (t), X1(t), X2(t), • • • } are independent and identically distributed uniform processes (for each t ∈ E, X (t) is uniform on (0, 1)) and w(x) is a “weight” function satisfying some regularity properties. Then we give an example when X (t) := Ft(Bt) : t ∈ E = [1, 2], where Bt is a Brownian motion and Ft is the distribution function of Bt. In Chapter 4, we investigate the weak convergence of the empirical processes for non-iid samples. We consider the weak convergence of the empirical process: (3) β_n(t, y) := n^(−1/2 )Sigma[(1_(Y (t)<=y) – Fi(t,y))] from i=1 to n, t ∈ E ⊂ R, y ∈ R where {Yi(t) : i ∈ N} are independent processes and Fi(t, y) is the distribution function of Yi(t). We also prove that the covariance function of the empirical process for non-iid samples indexed by a uniformly bounded class of functions necessarily uniformly converges to the covariance function of the limiting Gaussian process for a CLT.

We consider various infinite series and examine their arithmetic nature. Series of interest are of the form $$\sum_{n =0}^{\infty} \frac{f(n)A(n)}{B(n)}, \;\;\;\; \sum_{n \in \mathbb{Z}}\frac{f(n)A(n)}{B(n)}, \;\;\;\; \sum_{n=0}^{\infty} \frac{z^n A(n)}{B(n)}$$ where $f$ is algebraic valued periodic function, $A(x), B(x) \in \overline{\mathbb{Q}}[x]$ and $z$ is an algebraic number with $|z| \leq 1$. We also examine multivariable extensions $$\sum_{n_1, \ldots, n_k = 0}^{\infty} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}$$ and $$\sum_{n_1, \ldots, n_k \in \mathbb{Z}} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}.$$ These series are all very natural things to write down and we would like to understand them better. We calculate closed forms using various techniques. For example, we use relations between Hurwitz zeta functions, digamma functions, polygamma functions, Fourier analysis, discrete Fourier transforms, among other objects and techniques. Once closed forms are found, we make use of some of the well-known transcendental number theory including the theorem of Baker regarding linear forms in logarithms of algebraic numbers to determine their arithmetic nature. In one particular setting, we extend the work of Bundschuh \cite{bundschuh} by proving the following series are all transcendental for positive $c \in \mathbb{Q} \setminus \mathbb{Z}$ and $k$ a positive integer: $$\sum_{n \in \mathbb{Z}} \frac{1}{(n^2 + c)^k}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^4 - c^4)^{2k}}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^6 - c^6)^{2k}}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^3 \pm c^3)^{2k}}$$ $$\sum_{n \in \mathbb{Z}} \frac{1}{n^3 \pm c^3}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^3 -1}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^4 -1}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^6 -1}.$$ Bundschuh conjectured that the last three series are transcendental, but we offer the first unconditional proofs of transcendence. We also show some conditional results under the assumption of some well-known conjectures. In particular, for $A_i(x), B_i(x) \in \overline{\mathbb{Q}}[x]$ with each $B_i(x)$ has only simple rational roots, if Schanuel's conjecture is true, the series (avoiding roots of the denominator) $$\mathop{{\sum}}_{n_1, \ldots, n_k =0}^{\infty} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}$$ is either an effectively computable algebraic number or transcendental. We also show that Schanuel's conjecture implies that the series $$\sum_{n \in \mathbb{Z}} \frac{A(n)}{B(n)}$$ is either zero or transcendental, when $B(x)$ has non-integral roots. We develop a general theory, analyzing various infinite series throughout. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-04-23 10:45:19.735

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