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Since the mid 1980's many statisticians have studied methods for combining parametric and nonparametric esimates to improve the quality of fits in a regression problem. Notably in 1987, Einsporn and Birch proposed the Model Robust Regression estimate (MRR1) in which estimates of the parametric function, ƒ, and the nonparametric function, 𝑔, were combined in a straightforward fashion via the use of a mixing parameter, λ. This technique was studied extensively at small samples and was shown to be quite effective at modeling various unusual functions. In 1995, Mays and Birch developed the MRR2 estimate as an alternative to MRR1. This model involved first forming the parametric fit to the data, and then adding in an estimate of 𝑔 according to the lack of fit demonstrated by the error terms. Using small samples, they illustrated the superiority of MRR2 to MRR1 in most situations. In this dissertation we have developed asymptotic convergence rates for both MRR1 and MRR2 in OLS and GLS (maximum likelihood) settings. In many of these settings, it is demonstrated that the user of MRR1 or MRR2 achieves the best convergence rates available regardless of whether or not the model is properly specified. This is the "Golden Result of Model Robust Regression". It turns out that the selection of the mixing parameter is paramount in determining whether or not this result is attained. / Ph. D.

The problem of rough surface scattering and propagation over rough terrain in ducting environments has been receiving considerable attention in the literature. One popular method of modeling this problem is the parabolic wave equation (PWE) method. In this method, the Helmholtz wave equation is replaced by a PWE under the assumption of predominant forward propagation and scattering. The resulting PWE subjected to the appropriate boundary condition(s) is then solved, given an initial field distribution, using marching techniques such as the split-step Fourier algorithm. As is obvious from the assumption on which it is based, the accuracy of the PWE approximation deteriorates in situations involving appreciable scattering away from the near-forward direction, i.e. when the terrain under consideration is considerably rough. The backscattered field is neglected in all PWE-based models. An alternative and more rigorous method for modeling the problem under consideration is the boundary integral equation (BIE) method, which is formulated in two steps. The first step involves setting up an integral equation (the magnetic field integral equation, MFIE, or the electric field integral equation EFIE) governing currents induced on the rough surface by the incident field and solving for these currents numerically. The resulting currents are then used in the appropriate radiation integrals to calculate the field scattered by the surface everywhere in space. The BIE method accounts for all orders of multiple scattering on the rough surface and predicts the scattered field in all directions in space (including the backscattering direction) in an exact manner. In homogeneous media, the implementation of the BIE approach is straightforward since the kernel (Green's function or its normal derivative) which appears in the integral equation and the radiation integrals is well known. This is not the case, however, in inhomogeneous media (ducting environments) where the Green's function is not readily known. Due to this fact, there has been no attempt, up to our knowledge, at using the BIE (except under the parabolic approximation) to model the problem under consideration prior to the work presented in this thesis. In this thesis, a closed-form approximation of the Green's function for a two-dimensional ducting environment formed by the presence of a linear-square refractivity profile is derived using the asymptotic methods of stationary phase and steepest descents. This Green's function is then modified to more closely model the one associated with a physical ducting medium, in which the refractivity profile decreases up to a certain height, beyond which it becomes constant. This modified Green's function is then used in the BIE approach to study low grazing angle (LGA) propagation over rough surfaces in the aforementioned ducting environment. The numerical method used to solve the MFIE governing the surface currents is MOMI, which is a very robust and efficient method that does not require matrix storage or inversion. The proposed method is meant as a benchmark for people studying forward propagation over rough surfaces using the parabolic wave equation (PWE). Rough surface scattering results obtained via the PWE/split-step approach are compared to those obtained via the BIE/MOMI approach in ducting environments. These comparisons clearly show the shortcomings of the PWE/split-step approach. / Ph. D.

The open bin packing problem (OBPP) is a new variant of the well-known bin packing problem. In the OBPP, items are packed into bins so that the total content before the last item in each bin is strictly less than the bin capacity. The objective is to minimize the number of bins used. The applications of the OBPP can be found in the subway station systems in Hong Kong and Taipei and the scheduling in manufacturing industries. We show that the OBPP is NP-hard and propose two heuristic algorithms instead of solving the problem to optimality. We propose two offline algorithms in which the information of the items is known in advance. First, we consider the First Fit Decreasing (FFD) which is a good approximation algorithm for the bin packing problem. We prove that its asymptotic worst-case performance ratio is no more than 3/2. We observe that its performance for the OBPP is worse than that of the BPP. Consequently, we modify it by adding the algorithm that the set of largest items is the set of last items in each bin. Then, we propose the Modified First Fit Decreasing (MFFD) as an alternative and prove that its asymptotic worst-case performance ratio is no more than 91/80. We conduct empirical tests to show their average-case performance. The results show that in general, the FFD and MFFD algorithms use no more than 33% and 1% of the number of bins than that of optimal packing, respectively. In addition, the MFFD is asymptotically optimal when the sizes of items are (0,1) uniformly distributed. / Ph. D.

Recently, many new types of wireless networks have emerged for both civil and military applications, such as cognitive radio networks, MIMO networks. There is a strong interest in exploring the optimal performance of these new emerging networks, e.g., maximizing the network throughput, minimizing network energy consumption. Exploring the optimal performance objectives of these new types of wireless networks is both important and intellectual challenging. On one hand, it is important for a network researcher to understand the performance limits of these new wireless networks. Such performance limits are important not only for theoretical understanding, but also in that they can be used as benchmarks for the design of distributed algorithms and protocols. On the other hand, due to some unique characteristics associated with these networks, existing analytic techniques may not be applied directly to obtain the optimal performance. As a result, new theoretical results, along with new mathematical tools, need to be developed. The goal of this dissertation is to make a fundamental advance on network performance optimization via exploring a series of optimization problems. Based on the scale of the underlying wireless network, the works in this dissertation are divided into two parts. In the first part, we study the asymptotic capacity scaling laws of different types of wireless networks. By "asymptotic", we mean that the number of nodes in the network goes to infinity. Such asymptotic capacity scaling laws offer fundamental understandings on the trend of maximum user throughput behavior when the network size increases. In the second part of this dissertation, we study several optimization problems of finite-sized wireless networks. Under a given network size, we accurately characterize some performance limits (e.g., throughput, energy consumption) of wireless networks and provide solutions on how to achieve the optimal objectives. The main contributions of this dissertation can be summarized as follows, where the first three problems are on asymptotic capacity scaling laws and the last three problems are optimization problems of finite-sized wireless networks. <b>1. Capacity Scaling Laws of Cognitive Radio Ad Hoc Networks.</b> We first study the capacity scaling laws for cognitive radio ad hoc networks (CRNs), i.e., how each individual node's maximum throughput scales as the number of nodes in the network increases. This effort is critical to the fundamental understanding of the scalability of such network. However, due to the heterogeneity in available frequency bands at each node, the asymptotic capacity is much more difficult to develop than prior efforts for other types of wireless networks. To overcome this difficulty, we introduce two auxiliary networks ζ and α to analyze the capacity upper and lower bounds. We derive the capacity results under both the protocol model and the physical model. Further, we show that the seminal results developed by Gupta and Kumar for the simple single-channel single-radio (SC-SR) networks are special cases under the results for CRNs. <b>2. Asymptotic Capacity of Multi-hop MIMO Ad Hoc Networks.</b> Multi-input multi-output (MIMO) is a key technology to increase the capacity of wireless networks. Although there has been extensive work on MIMO at the physical and link layers, there has been limited work on MIMO at the network layer (i.e., multi-hop MIMO ad hoc network), particularly results on capacity scaling laws. In this work, we investigate capacity scaling laws for MIMO ad hoc networks. Our goal is to find the achievable throughput of each node as the number of nodes in the network increases. We employ a MIMO network model that captures spatial multiplexing (SM) and interference cancellation (IC). We show that for a MIMO network with n randomly located nodes, each equipped with γ antennas and a rate of W on each data stream, the achievable throughput of each node is Θ(γW/√<span style="text-decoration:overline;"> n ln n</span></span>). <b>3. Toward Simple Criteria for Establishing Capacity Scaling Laws.</b> Capacity scaling laws offer fundamental understanding on the trend of user throughput behavior when the network size increases. Since the seminal work of Gupta and Kumar, there have been tremendous efforts developing capacity scaling laws for ad hoc networks with various advanced physical layer technologies. These efforts led to different custom-designed approaches, most of which were intellectually challenging and lacked universal properties that can be extended to address scaling laws of ad hoc networks with a different physical layer technology. In this work, we present a set of simple yet powerful general criteria that one can apply to quickly determine the capacity scaling laws for various physical layer technologies under the protocol model. We prove the correctness of our proposed criteria and validate them through a number of case studies, such as ad hoc networks with directional antenna, MIMO, cognitive radio, multi-channel and multi-radio, and multiple packet reception. These simple criteria will serve as powerful tools to networking researchers to obtain throughput scaling laws of ad hoc networks under different physical layer technologies, particularly those to appear in the future. <b>4. Exploiting SIC forMulti-hopWireless Networks.</b> There is a growing interest on exploiting interference (rather than avoiding it) to increase network throughput. In particular, the so-called successive interference cancellation (SIC) scheme appears very promising, due to its ability to enable concurrent receptions from multiple transmitters and interference rejection. However, due to some stringent constraints and limit, SIC alone is inadequate to handle all concurrent interference. We advocate a joint interference exploitation and avoidance approach, which combines the best of interference exploitation and interference avoidance, while avoiding each's pitfalls. We discuss the new challenges of such a new approach in a multi-hop wireless network and propose a formal optimization framework, with cross-layer formulation of physical, link, and network layers. This framework offers a rather complete design space for SIC to squeeze the most out of interference. The goal of this effort is to lay a mathematical foundation for modeling and analysis of a joint interference exploitation and avoidance scheme in a multi-hop wireless network. Through modeling and analysis, we develop a tractable model that is suitable for studying a broad class of network throughput optimization problems. To demonstrate the practical utility of our model, we conduct a case study. Our numerical results affirm the validity of our model and give insights on how SIC can optimally interact with an interference avoidance scheme. <b>5. Throughput Optimization with Network-wide Energy Constraint.</b> Conserving network wide energy consumption is becoming an increasingly important concern for network operators. In this work, we study network-wide energy conservation problem which we hope will offer insights to both network operators and users. Specifically, we study how to maximize network throughput under a network-wide energy constraint for a general multi-hop wireless network. We formulate this problem as a mixed-integer nonlinear program (MINLP). We propose a novel piece-wise linear approximation to transform the nonlinear constraints into linear constraints. We prove that the solution developed under this approach is near optimal with guaranteed performance bound. <b>6. Bicriteria Optimization in Multi-hop Wireless Networks.</b> Network throughput and energy consumption are two important performance metrics for a multi-hop wireless network. Current state-of-the-art is limited to either maximizing throughput under some energy constraint or minimizing energy consumption while satisfying some throughput requirement. However, the important problem of how to optimize both objectives simultaneously remains open. In this work, we take a multicriteria optimization approach to offer a systematic study on the relationship between the two performance objectives. We show that the solution to the multicriteria optimization problem characterizes the envelope of the entire throughput energy region, i.e., the so-called optimal throughput-energy curve. We prove some important properties of the optimal throughput-energy curve. For case study, we consider both linear and nonlinear throughput functions. For the linear case, we characterize the optimal throughput-energy curve precisely through parametric analysis, while for the nonlinear case, we use a piece-wise linear approximation to approximate the optimal throughput-energy curve with arbitrary accuracy. Our results offer important insights on exploiting the trade-off between the two performance metrics. / Ph. D.

Approximate formulae are derived for obtaining confidence intervals for the probabilities of a multinomial distribution. The approach used is to consider the Chi-square goodness of fit statistic as a function of the population parameters and to invert this function to obtain a set of simultaneous confidence intervals for the parameters The confidence coefficient for the set of simultaneous confidence intervals obtained by this procedure is conservative, i.e., the true probability that every interval covers its corresponding parameter will in general be greater than the coefficient obtained by this method. As the sample size increases the intervals will converge on the population parameters and will estimate them exactly in the limit. / Master of Science

Taylor vortex flow is one of the basic problems of nonlinear hydrodynamic stability. In contrast with the wide region of wavenumber predicted by the linear theory, experiments show that Taylor vortex flow only appears in a small region containing the critical wavenumber ß_er This phenomenon is called wave selection. In this work, several high-order perturbation methods and a numerical method are established. Both evolution and steady state of the How caused by single or several disturbances are studied. The existence of multiple steady states for disturbances with small wavenumber is discovered and proved. The stable and unstable steady state solutions and some associated phenomena such as jump phenomenon and hysteresis phenomenon are found. and explained. In the small region, the wavenumbers and initial amplitudes of disturbances determine the wavenumber of the flow. But outside this region, only the wavenumbers of the disturbances have effect on the wave selection. These results indicate that unstable solutions play a key role in wave selection. The side-band stability curve produced by the high-order perturbation methods is accurate at low Taylor numbers but incorrect at relatively high Taylor numbers. The relation of the unstable solutions and side-band stability is discussed. Besides, the overshoot and the oscillation phenomena during evolution are studied in detail. Connections between this work and experiments are discussed. / Ph. D.

An asymptotically-exact methodology is presented for obtaining the cross-sectional stiffness matrix of a pre-twisted, moderately-thick beam having rectangular cross sections and made of transversely isotropic material. The beam is modelled with-out assumptions from 3-D elasticity. The strain energy of the beam is computed making use of the constitutive law and the kinematical relations derived with the inclusion of geometrical nonlinearities and initial twist. Large displacements and rotations are allowed, but small strain is assumed. The Variational Asymptotic Method (VAM) is used to minimize the energy functional, thereby reducing the cross section to a point on the reference line with appropriate properties, yielding a 1-D constitutive law. In this method as applied herein, the 2-D cross-sectional analysis is performed asymptotically by taking advantage of a material small parameter and two geometric small parameters. 3-D strain components are derived using kinematics and arranged as orders of the small parameters. Warping functions are obtained by the minimisation of strain energy subject to certain set of constraints that renders the 1-D strain measures well-defined. Closed-form expressions are derived for the 3-D non-linear warping and stress fields. The model is capable of predicting interlaminar and transverse shear stresses accurately up to first order.