This thesis focuses on convex sets and convex cones defined using hyperbolic polynomials. We first review some of the theory of convex sets in $\R^d$ in general. We then review some classical algebraic theorems concerning polynomials in a single variable, as well as presenting a few more modern results about them. We then discuss the theory of hyperbolic polynomials in several variables and their associated hyperbolicity cones. We survey various ways to build and decompose hyperbolic cones and we prove that every nontrivial hyperbolic cone is the intersection of its derivative cones. We conclude with a brief discussion of the set of extreme rays of a hyperbolic cone.

The pattern reduction (PR) algorithm we proposed previously, which works by eliminating patterns that are unlikely to change their membership during the convergence process, is obviously one of the most efficient methods for reducing the computation time of clustering algorithms. However, it is limited to problems with solutions that can be binary or integer encoded, such as combinatorial optimization problems. As such, this study is aimed at developing a new pattern reduction algorithm, called pattern reduction over continuous space, to get rid of this limitation. Like the PR, the proposed algorithm consists of two operators: detection and compression. Unlike the PR, the detection operator is divided into two steps. The first step is aimed at finding out subsolutions that can be considered as the candidate subsolutions for compression. The second step is performed to ensure that the candidate subsolutions have reached the final state so that any further computation is eventually a waste and thus can be compressed. To evaluate the performance of the proposed algorithm, we apply it to metaheuristics for clustering.

Linearized equations of motion are important in engineering applications, especially with respect to stability analysis and control design. Traditionally, the full, nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. Here, this development is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems, where a hybrid system is described with both discrete and continuous generalized coordinates. The results presented require only velocity level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest. This study shows that the entire nonlinear equations of motion do not have to be generated in order to construct the linearized equations of motion. Several examples are presented to illustrate application of these results to both continuous and hybrid system problems.

Infinite dimensional systems such as flexible airplane wings and Vertical Axis Wind Turbine (VAWT) blades may require control to improve performance. Traditional control techniques use position and velocity information feedback, but velocity information for infinite dimensional systems is not easily attained. This research investigates the use of reduced-sensing control for these applications. Reduced-sensing control uses feedback of position measurements and an associated filter state to stabilize the system dynamics. A filter state is a nonphysical entity that appends an additional ordinary differential equation to the system dynamics. Asymptotic stability of a system using this control approach is confirmed through a sequence of existing mathematical tools. These tools include equilibrium point solutions, Lyapunov functions for stability and control, and Mukherjee and Chen's Asymptotic Stability Theorem. This thesis research investigates the stability of a beam representing an airplane wing or a VAWT blade controlled using feedback of position and filter state terms only. Both of these infinite dimensional systems exhibit asymptotic stability with the proposed reduced-sensing control design. Additionally, the analytical stability response of the VAWT is verified through numerical simulation.

Conventionally, the free distance is taken as the principle criterion for computing the error of convolutional and linear block codes. In other words, a larger free distance implies a better correction ability for the error correction codes. Distance spectrum is also an important factor for Maximum likelihood decoding. In this thesis, distance spectrum for different convolutional codes and CPM systems are investigated by us. Experiments results has demonstrate that a better correction ability of a shorter free distance does exist in some cases if they have a better distance spectrum. We also improve the fast algorithm for computing the distance spectrum developed by M. Cedervall and R. Johannesson. Their success is based upon the strategy of a traveling along the coding tree to find the distance spectrum. However, they need a new traveling for ever new distance computation. In contrast, we compute all the distance spectrum just in one travel by taking the advantage of the storing nodes of previous distance computation.

Combined coding with modulation is an important topic. It is verified in this thesis that a combined decoder and demodulation Viterbi receiver has a better error probability than a cascade of two separate Viterbi decoder and demodulator. Conventionally, the free distance is taken as the principle criterion for computing the error probability for coding or modulation. In many cases, distance spectrum needs to be provided for analyze the Maximum likelihood decoding. However, it is difficult for computing the distance spectrum for a combined coding with modulation because of a nonlinear structure inside. In this thesis, we first build an augmented trellis for the combined coding with modulation. Applying the concept of difference by exclusive OR and regular subtraction to the augmented trellis, we build an improved virtual trellis. As a consequence the distance spectrum for our problem can be computed because of the linear structure of the virtual trellis. The distance spectrum for different convolutional codes and CPM systems are investigated by us. Experiments results have demonstrate that a better distance spectrum implies a better error ability.

Let X. Y be compact Hausdorff spaces, and E, F be Banach spaces. A linear map T¡GC (X¡AE)¡÷C (Y¡AF) is separating if ¡üTf(y)¡ü¡üTg(y)¡ü¡×0 whenever ¡üf(x)¡ü¡üg(x)¡ü¡×0, for every x belonging to X, y belonging to Y. Gau, Jeang and Wong prove that a biseparating linear bijection T is a weighted composition oprator Tf¡×hf¡³£p where h is a function from Y into the set of inveritable linear operators from E onto F and £p is a homeomorphism from Y onto X. In this thesis, we extend this result to the case that continuous functions are defined to a locally compact Hausdorff space, which is either £m-compact or first countable. Moreover, we give a short proof of a recent result of Mrcun. Finally, we give an alternative approach to an Araujo's result concerning biseparating maps of smooth functions appeared in Adv. Math.

Combined coding with modulation is an important topic. Conventionally, the free distance is taken as the principle criterion for computing the error probability for coding or modulation. In many cases, distance spectrum needs to be provided for analyze the Maximum likelihood decoding. However, it is difficult for computing the distance spectrum for a combined coding with modulation because of a nonlinear structure inside. In this thesis, we study the order of distance spectrum members to find some limited number of members to present the whole distance spectrum. In our previous work, we have built an augmented trellis for the combined coding with modulation. Applying the concept of difference by exclusive OR and regular subtraction to the augmented trellis, we build an improved virtual trellis. In this thesis, we expend the concept of subtraction to a pair relation. Thus, this augmented trellis is first composed of paired states and transition lines. Then, we use a partition principle to group the states and lines. Finally, the complex trellis is reduced to a reasonable structure. We therefore can apply distance spectrum computing algorithm to find the distance spectrum. The distance spectrum for different convolution codes and CPM systems are investigated by us. Experiments results have demonstrate this distance spectrum is more accurate than before.

Signal model and observation model are commonly used to describe a dynamic system model in system identification or estimation such as Kalman filtering. The signal model is usually described by a linear dynamical equation driven by generating noise. The observation model is composed of a linear transformed signal and an additive white Gaussian noise. In this thesis, we set the generating noise to be a white binary sequence. This discrete generating noise makes the generating signal to be discrete. In contrast, the conventional generating signal is continuous. Discrete signal is simpler than the continuous signal. However, there still are too many states for this discrete signal. Therefore, defining the states and reducing the number of states are important in our work. In this thesis, we apply the tree structure to define the states. The number of states is reduced by focusing on the most probable working states. Afterwards, we apply two methods to recover the white sequence using the observation data. One is the Viterbi method; the other is Extended Kalman filter. Both methods are based upon the concept of signal states. Finally, we compare the error rates with the signal generated by continues phase modulation method.

The advances of communication and computer techniques have enabled the development of low-cost, low-power, multifunctional sensor nodes that are small in size and capable of communicating in short distances. A sensor network is composed of a large number of sensor nodes that are densely deployed either inside the phenomenon to be observed or very close to it. Sensor networks open up new opportunities to observe and interact with the physical world around us. Despite the recent advances in sensor network applications and technology, sensor networks still suffer from the major problems of limited energy. It is because most sensor nodes use battery as their energy srouce and are inconvenient and sometimes difficult to be replaced when the battery run out. Understanding the events, measures, and tasks required by certain applications has the potential to provide efficient communication techniques for the sensor network. Our focus in this work is on the efficient processing of continuous queries, by which query results have to be generated according to the sampling rate specified by the user for an extended period of time. In this thesis, we will deal with two types of continuous queries. The first type of queries requires data from all sensor nodes; while the other is only interested in the data returned by some selected nodes. To answer these queries, data have to be sent to the base station at some designated rate, which may consume much energy. Previous works have developed two methods to reduce the energy consumption. They both base on the error range which the user can tolerate to determine whether current sensing data should be transmitted. While the first uses simple cache method, the second uses complex multi-dimensional model. However, the proposed methods required the user to specify the error range, which may not be easy to specify. In addition, the sensed data reported by the sensors were assumed to be accurate, which is by no means true in the real world. This thesis is based on Kalman filter to correct and predict sensing data. As a result, the sampling frequency of each sensor is dynamically adjusted, referred to as resampling which systematically determine the data sensing/transferring rate of sensors. We evaluate our proposed methods using empirical data collected from a real sensor network.