Polynomial phase modulated (PPM) signals have been shown to provide improved error rate performance with respect to conventional modulation formats under additive white Gaussian noise and fading channels in single-input single-output (SISO) communication systems. In this dissertation, systems with two and four transmit antennas using PPM signals were presented. In both cases we employed full-rate space-time block codes in order to take advantage of the multipath channel. For two transmit antennas, we used the orthogonal space-time block code (OSTBC) proposed by Alamouti and performed symbol-wise decoding by estimating the phase coefficients of the PPM signal using three different methods: maximum-likelihood (ML), sub-optimal ML (S-ML) and the high-order ambiguity function (HAF). In the case of four transmit antennas, we used the full-rate quasi-OSTBC (QOSTBC) proposed by Jafarkhani. However, in order to ensure the best error rate performance, PPM signals were selected such as to maximize the QOSTBC’s minimum coding gain distance (CGD). Since this method does not always provide a unique solution, an additional criterion known as maximum channel interference coefficient (CIC) was proposed. Through Monte Carlo simulations it was shown that by using QOSTBCs along with the properly selected PPM constellations based on the CGD and CIC criteria, full diversity in flat fading channels and thus, low BER at high signal-to-noise ratios (SNR) can be ensured. Lastly, the performance of symbol-wise decoding for QOSTBCs was evaluated. In this case a quasi zero-forcing method was used to decouple the received signal and it was shown that although this technique reduces the decoding complexity of the system, there is a penalty to be paid in terms of error rate performance at high SNRs.

The problem of polynomial representability of functions is central to many branches of mathematics. If the underlying set is a finite field, every function can be represented as a polynomial. In this thesis we consider polynomial representability over a special class of finite rings, namely, Zpn, where p is a prime and n is a positive integer. This problem has been studied in literature and the two notable results were given by Carlitz(1965) and Kempner(1921).While the Kempner’s method enumerates the set of distinct polynomial functions, Carlitz provides a necessary and sufficient condition for a function to be polynomial using Taylor series. Further, these results are existential in nature. The aim of this thesis is to provide an algorithmic characterization, given a prime p and a positive integer n, to determine whether a given function over Zpn is polynomially representable or not. Note that one can give an exhaustive search algorithm using the previous results. Our characterization involves describing the set of polynomial functions over Zpn with a ‘suitable’ generating set. We make use of this result to give an non-exhaustive algorithm to determine whether a given function over Zpn is polynomial representable.nβ

This thesis presents a computational study to examine lateral impact behavior of pressurized pipelines and to determine influence of internal pressure on the impact behaviors of pipelines. More than 300 numerical simulations were carried out on mild steel pipe models with different internal pressure levels and were struck at the mid-span and at the one quarter span positions. The computational results for the first time systematically revealed the effects of internal pressure, impact position, and outside diameter on the lateral impact behavior of the pipeline models. It inspects effects of important parameters such as the outside diameter and internal pressure. Quartic polynomial functions are applied to formulate the maximum crushing force (F), permanent displacement (W), and absorbed energy (E) of the pressurized pipelines during the impact problem. Response surfaces are plotted based on the generated quartic polynomial functions and the quality (accuracy) of those functions are verified through several techniques.

I present a thorough examination of the unconditional deterministic polynomial-time algorithm for determining whether an input number is prime or composite proposed by Agrawal, Kayal and Saxena in their paper [1]. All proofs cited have been reworked with full details for the sake of completeness and readability.

<p> A definition of polynomial differentiability of an arc in the real affine plane at a point is given. The differentiable points are classified with respect to the intersection and support properties of certain families of osculating polynomials. For a given point of an arc, these properties are used to define a certain n-tuple of integers, the characteristic of that point. It is shown that the polynomial order of polynomially differentiable interior point of an arc is at least as great as the sum of the digits of its characteristic.</p> / Thesis / Doctor of Philosophy (PhD)

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Mechanical systems always operate under some degree of uncertainty, which can be due to the inherent properties of the system parameters, to random inputs or external excitations, to poorly known parameters in the interface between different systems, or to inadequate knowledge of the dynamic process. Also, mechanical systems are large and highly nonlinear, while the magnitude of uncertainties may be very large. This dissertation addresses the critical need for understanding of the stochastic nature of mechanical system, especially vehicle and terramechanics systems, and need for developing efficient computational tools to model mechanical systems in the presence of parametric and external uncertainty. This dissertation investigates the influence of parametric and external uncertainties on vehicle dynamics and terramechanics. The uncertainties studied include parametric uncertainties, stochastic external excitations, and random variables between vehicle-terrain and vehicle-soil/snow interface. The methodology developed has been illustrated on a stochastic vehicle-terrain interaction model, a stochastic vehicle-soil interaction model, two stochastic tire-snow interaction models, and two stochastic tire-force relations. The uncertainties are quantified and propagated through vehicle and terramechanics systems using a polynomial chaos approach. Algorithms which can predict the geometry of the contact patch and the interfacial forces and torques on the vehicle-soil interfaces are developed. All stochastic models and algorithms are simulated for various scenarios and maneuvers. Numerical results are analyzed from the computational effort point of view, or from the angle of vehicle dynamics and terramechanics, and provide a deeper understanding of the evolution of stochastic vehicle and terramechanics systems. They can also be used in guiding vehicle design and development. This dissertation represents a pioneer study on stochastic vehicle dynamics and terramechanics. Moreover, the methodology developed is not limited to such systems. Any mechanical system with uncertainties can be treated using the polynomial chaos approach presented, considering their specific characteristics. / Ph. D.

A hierarchical approach is typically followed in design of consumer products. First, a manufacturer sets performance targets for the whole system according to customer surveys and benchmarking of competitors' products. Then, designers cascade these targets to the subsystems or the components using a very simplified model of the overall system. Then, they try to design the components so that they meet these targets. It is important to have efficient tools that check if a set of performance targets for a component corresponds to a feasible design and determine the dimensions and mass of this design. This dissertation presents a methodology for developing two tools that link performance targets for a design to design variables that specify the geometry of the design. The first tool (called translator A) predicts the stiffness and mass of an automotive joint, whose geometry is specified, almost instantaneously. The second tool (called translator B) finds the most efficient, feasible design whose performance characteristics are close to given performance targets. The development of the two translators involves the following steps. First, an automotive joint is parameterized. A set of physical parameters are identified that can completely describe the geometry of the joint. These parameters should be easily understood by designers. Then, a parametric model is created using a CAD program, such as Pro/Engineer or I-Deas. The parametric model can account for different types of construction, and includes relations for styling, packaging, and manufacturing constraints. A database is created for each joint using the results from finite element analysis of hundreds or thousands of joint designs. The elements of the database serve as examples for developing Translator A. Response surface polynomials and neural networks are used to develop translator A. Stepwise regression is used in this study to rank the design variables in terms of importance and to obtain the best regression model. Translator B uses optimization to find the most efficient design. It analyzes a large number of designs efficiently using Translator A. The modified feasible direction method and sequential linear programming are used in developing translator B. The objective of translator B is to minimize the mass of the joint and the difference of the stiffness from a given target while satisfying styling, manufacturing and packaging constraints. The methodologies for Translators A and B are applied to the B-pillar to rocker and A-pillar to roof rail joints. Translator B is demonstrated by redesigning two joints of actual cars. Translator B is validated by checking the performance and mass of the optimum designs using finite element analysis. This study also compares neural networks and response surface polynomials. It shows that they are almost equally accurate when they are used in both analysis and design of joints. / Ph. D.

The main limitations in performing uncertainty analysis of CFD models using conventional methods are associated with cost and effort. For these reasons, there is a need for the development and implementation of efficient stochastic CFD tools for performing uncertainty analysis. One of the main contributions of this research is the development and implementation of Intrusive and Non-Intrusive methods using polynomial chaos for uncertainty representation and propagation. In addition, a methodology was developed to address and quantify turbulence model uncertainty. In this methodology, a complex perturbation is applied to the incoming turbulence and closure coefficients of a turbulence model to obtain the sensitivity derivatives, which are used in concert with the polynomial chaos method for uncertainty propagation of the turbulence model outputs. / Ph. D.

In this thesis we investigate different domination-related graph polynomials, like the connected domination polynomial, the independent domination polynomial, and the total domination polynomial. We prove some basic properties of these polynomials and obtain formulas for the calculation in special graph classes. Furthermore, we also prove results about the calculation of the different graph polynomials in product graphs and different representations of the graph polynomials. One focus of this thesis lays on the generalization of domination-related polynomials. In this context the trivariate domination polynomial is defined and some results about the bipartition polynomial, which is also a generalization of the domination polynomial, is presented. These two polynomials have many useful properties and interesting connections to other graph polynomials. Furthermore, some more general domination-related polynomials are defined in this thesis, which shows some possible directions for further research. / In dieser Dissertation werden verschiedene, zum Dominationspolynom verwandte, Graphenpolynome, wie das zusammenhängende Dominationspolynom, das unabhängige Dominationspolynom und das totale Dominationspolynom, untersucht. Es werden grundlegende Eigenschaften erforscht und Sätze für die Berechnung dieser Polynome in speziellen Graphenklassen bewiesen. Weiterhin werden Ergebnisse für die Berechnung in Produktgraphen und verschiedene Repräsentationen für diese Graphenpolynome gezeigt. Ein Fokus der Dissertation liegt auf der Verallgemeinerung der verschiedenen Dominationspolynome. In diesem Zusammenhang wird das trivariate Dominationspolynom definiert. Außerdem werden Ergebnisse für das Bipartitionspolynom bewiesen. Diese beiden Polynome haben viele interessante Eigenschaften und Beziehungen zu anderen Graphenpolynomen. Darüber hinaus werden weitere multivariate Graphenpolynome definiert, die eine mögliche Richtung für weitere Forschung auf diesem Gebiet aufzeigen.