published_or_final_version / abstract / Electrical and Electronic Engineering / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy

No description available.

One of the differentiating features of PARAFAC decompositions is that, under certain conditions, unique solutions are possible. The search for uniqueness conditions for the PARAFAC Decomposition has a limited past, spanning only three decades. The complex structure of the problem and the need for tensor algebras or other similarly abstract characterizations provided a roadblock to the development of uniqueness conditions. Theoretically, the PARAFAC decomposition surpasses its bilinear counterparts in that it is possible to obtain solutions that do not suffer from the rotational problem. However, not all PARAFAC solutions will be constrained sufficiently so that the resulting decomposition is unique. The work of Kruskal, 1977, provides the most in depth investigation into the conditions for uniqueness, so much so that many have assumed, without formal proof, that his sufficient conditions were also necessary. Aided by the introduction of Khatri-Rao products to represent the PARAFAC decomposition, ten Berge and Sidiropoulos (2002) used the column spaces of Khatri-Rao products to provide the first evidence for countering the claim of necessity, identifying PARAFAC decompositions that were unique when Kruskals condition was not met. Moreover, ten Berge and Sidiropoulos conjectured that, with additional k-rank restrictions, a class of decompositions could be formed where Kruskals condition would be necessary and sufficient. Unfortunately, the column space argument of ten Berge and Sidiropoulos was limited in its application and failed to provide an explanation of why uniqueness occurred. On the other hand, the use of orthogonal complement spaces provided an alternative approach to evaluate uniqueness that would provide a much richer return than the use of column spaces for the investigation of uniqueness. The Orthogonal Complement Space Approach (OCSA), adopted here, would provide: (1) the answers to lingering questions about the occurrence of uniqueness, (2) evidence that necessity would require more than a restriction on k-rank, and (3) an approach that could be extended to cases beyond those investigated by ten Berge and Sidiropoulos.

<p>A method is developed for solving the two dimensional Helmholtz equation in a ductwith varying cross-section region bounded by a curved top and flat bottom, having oneregion inside. To compute the propagation of sound waves in a curved duct with a curvedinternal interface is difficult problem. One method is to transform the wave equation intoa solvable form and making the curved interface plane. To this end a local orthogonaltransformation is developed for the varying cross-sectional duct having one medium inside.This transformation is first used to make the curved top of the waveguide flat andto transform the Helmholtz equation into an initial value problem. Later on the local orthogonaltransformation is developed for a waveguide having two media inside with flattop, a flat bottom and a curved interface. This local orthogonal transformation is used toflatten the interface and also to transform the Helmholtz equation into a simple, solvableordinary differential equation. In this paper we present operator formulation for the partwith flat bottom and curved top including a curved interface. In the ordinary differentialequation with operators in coefficients, obtained after the transformation, all the operationsrelated to the transverse variable are treated as operators while the derivative withrespect to the range variable is kept.</p>

In this thesis, we study foldings of orthogonal polygons into orthogonal polyhedra. The particular problem examined here is whether a paper cutout of an orthogonal polygon with fold lines indicated folds up into a simple orthogonal polyhedron. The folds are orthogonal and the direction of the fold (upward or downward) is also given. We present a polynomial time algorithm to solve this problem. Next we consider the same problem with the exception that the direction of the folds are not given. We prove that this problem is NP-complete. Once it has been determined that a polygon does fold into a polyhedron, we consider some restrictions on the actual folding process, modelling the case when the polyhedron is constructed from a stiff material such as sheet metal. We show an example of a polygon that cannot be folded into a polyhedron if folds can only be executed one at a time. Removing this restriction, we show another polygon that cannot be folded into a polyhedron using rigid material.

The Point Spread Function (PSF) indirectly encodes the wavefront aberrations of an optical system and therefore is a metric of the system performance. Analysis of the PSF properties is useful in the case of diffractive optics where the wavefront emerging from the exit pupil is not necessarily continuous and consequently not well represented by traditional wavefront error descriptors such as Zernike polynomials. The discontinuities in the wavefront from diffractive optics occur in cases where step heights in the element are not multiples of the illumination wavelength. Examples include binary or N-step structures, multifocal elements where two or more foci are intentionally created or cases where other wavelengths besides the design wavelength are used. Here, a technique for expanding the electric field amplitude of the PSF into a series of orthogonal functions is explored. The expansion coefficients provide insight into the diffraction efficiency and aberration content of diffractive optical elements. Furthermore, this technique is more broadly applicable to elements with a finite number of diffractive zones, as well as decentered patterns.

Since an extratropical transition (ET) of a decaying tropical cyclone (TC) often results in a fast-moving, rapidly developing extratropical cyclone and amplification of synoptic-scale systems far downstream, proper forecasting of ET events is critical to forecast accuracy over large ocean regions. Past studies have linked forecast accuracy to the phasing of a decaying TC with favorable midlatitudes conditions. Because ET events are sensitive to the analyzed initial conditions, this phasing is examined using 11 member ensemble predictions available four times daily from the National Centers for Environmental Prediction, which were combined into a single 44 member ensemble based on a common forecast verification time. Recurring ET patterns within the 44 member ensemble were objectively identified using a combination of EOF and cluster analysis. Ensemble spread first appears near the point where the TC moves into the midlatitudes and then propagates downstream. Although ensemble spread in the forecast fields was large at extended forecast intervals, the ensemble spread, and the number of ET patterns identified in successive EPS predictions, decreased as the ET process became better defined. Within 48 hours of the ET event, the ensemble prediction system properly identified the ET pattern with a minimum ensemble spread. Similar to Klein et al. (2002), the shifts in the initial position of the TC and the subsequent dynamical coupling can explain differences between weak and strong ET reintensifications.

Microbial metabolism can be tailored to meet human specifications, but the degree to which these living systems can be repurposed is still unknown. Artificial biological control strategies are being developed with the goal of enabling the predictable implementation of novel biological functions (e.g., engineered metabolism). This dissertation project contributes genetic tools useful for modulating gene expression levels (extending promoters with UP elements) and isolating transcription and translation of engineered DNA from the endogenous cellular network (expression by orthogonal cellular machinery), which have been demonstrated in Escherichia coli for the production of lycopene, a 40-carbon tetraterpene carotenoid with antioxidant activity and a number of other desirable properties.