An independent set in a graph G is a set of pairwise nonadjacent vertices, and the maximum size, alpha(G), of an independent set in G is called the independence number. Given a graph G and weight matrix A of G with entries from some field F, the maximum dimension of an A-isotropic subspace, known as the Witt index of A, is an upper bound on alpha(G). Since any weight matrix can be used, it is natural to seek the minimum upper bound on the independence number of G that can be achieved by a weight matrix. This minimum, iota_F^*(G), is called the minimum Witt index of G over F, and the resulting bound, alpha(G)<= iota_F^*(G), is called the isotropic bound. When F is finite, the possible values of iota_F^*(G) are determined and the graphs that attain the isotropic bound are characterized. The characterization is given in terms of graph classes CC(n,t,c) and CK(n,t,k) constructed from certain spanning subgraphs called C(n,t,c)-graphs and K(n,t,k)-graphs. Here t is the term rank of the adjacency matrix of G. When F=R, the isotropic bound is known as the Cvetkovi\'c bound. It is shown that it is sufficient to consider a finite number of weight matrices A when determining iota_R^*(G) and that, in many cases, two weight values suffice. For example, if the vertex set of G can be covered by alpha(G) cliques, then G attains the Cvetkovi\'c bound with a weight matrix with two weight values. Inequalities on alpha and iota_F^* resulting from graph operations such as sums, products, vertex deletion, and vertex identification are examined and, in some cases, conditions that imply equality are proved. The equalities imply that the problem of determining whether or not alpha(G)=iota_F^*(G) can be reduced to that of determining iota_F^*(H) for certain crucial graphs H found from G. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-09-04 15:38:47.57

In 1974, H. Ruderman posed the following question: If $(2^m-2^n)|(3^m-3^n)$, then does it follow that $(2^m-2^n)|(x^m-x^n)$ for every integer $x$? This problem is still open. However, in 2011, M. R. Murty and V. K. Murty showed that there are only finitely many $(m,n)$ for which the hypothesis holds. In this thesis, we examine two generalizations of this problem. The first is replacing 2 and 3 with arbitrary integers $a$ and $b$. The second is to replace 2 and 3 with arbitrary algebraic integers from an imaginary quadratic field. In both of these cases we have shown that there are only finitely many $(m,n)$ for which the hypothesis holds. To get the second result we also generalized a result by Bugeaud, Corvaja and Zannier from the integers to imaginary quadratic fields. In the last half of the thesis we use the abc conjecture and some related conjectures to study some exponential Diophantine equations. We study the Pillai conjecture and the Erd\"{o}s-Woods conjecture and show that they are implied by the abc conjecture and that when we use an effective version, very clean bounds for the conjectures are implied. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-04-13 12:04:14.252

The estimation of power spectral densities is a critical step in many speech enhancement algorithms. The demand for multi-channel speech enhancement systems is high with applications in teleconferencing, cellular phones, and hearing aids. The first objective of the thesis is to develop a general multi-channel framework to solve for the diffuse noise power spectral densities whenever the spatial correlation or coherence matrix is pre-estimated and the number of speakers is less than the number of microphones. The second objective is to develop closed-form analytical solutions. The performance of the developed algorithms is evaluated with pre-existing algorithms using prescribed performance measures.

The human genome contains tens of thousands of gene loci which code for an even greater number of protein and RNA products. The highly complex temporal and spatial expression of these genes makes possible all the biological processes of life. Altered gene expression by mutation or deregulation is fundamental for the development of many human diseases. The ultimate aim of this thesis was to identify gene expression changes relevant to cancer. The advent of genome-wide expression profiling techniques, such as microarrays, has provided powerful new tools to identify such changes and researchers are now faced with an explosion of gene expression data. Processing, comparing and integrating these data present major challenges. I approached these challenges by developing and assessing novel methods for cross-platform analysis of expression data, scalable subspace clustering, and curation of experimental gene regulation data from the published literature. I found that combining results from different expression platforms increases reliability of coexpression predictions. However, I also observed that global correlation between platforms was generally low, and few gene pairs reached reasonable thresholds for high-confidence coexpression. Therefore, I developed a novel subspace clustering algorithm, able to identify coexpressed genes in experimental subsets of very large gene expression datasets. Biological assessment against several metrics indicates that this algorithm performs well. I also developed a novel meta-analysis method to identify consistently reported genes from differential expression studies when raw data are unavailable. This method was applied to thyroid cancer, producing a ranked list of significantly over-represented genes. Tissue microarray analysis of some of these candidates and others identified a number of promising biomarkers for diagnostic and prognostic classification of thyroid cancer. Finally, I present ORegAnno (www.oreganno.org), a resource for the community-driven curation of experimentally verified regulatory sequences. This resource has proven a great success with ~30,000 sequences entered from over 900 publications by ~50 contributing users. These data, methods and resources contribute to our overall understanding of gene regulation, gene expression, and the changes that occur in cancer. Such an understanding should help identify new cancer mechanisms, potential treatment targets, and have significant diagnostic and prognostic implications.

The development and application of environmental models to help understand natural systems, and support decision making, is commonplace. A difficulty encountered in the development of such models is determining which physical and chemical processes to simulate, and on what temporal and spatial scale(s). Modern computing capabilities enable the incorporation of more processes, at increasingly refined scales, than at any time previously. However, the simulation of a large number of fine scale processes has undesirable consequences: first, the execution time of many environmental models has not declined despite advances in processor speed and solution techniques; and second, such complex models incorporate a large number of parameters, for which values must be assigned. Compounding these problems is the recognition that since the inverse problem in groundwater modeling is non-unique the calibration of a single parameter set does not assure the reliability of model predictions. Practicing modelers are, then, faced with complex models that incorporate a large number of parameters whose values are uncertain, and that make predictions that are prone to an unspecified amount of error. In recognition of this, there has been considerable research into methods for evaluating the potential for error in model predictions arising from errors in the values assigned to model parameters. Unfortunately, some common methods employed in the estimation of model parameters, and the evaluation of the potential error associated with model parameters and predictions, suffer from limitations in their application that stem from an emphasis on obtaining an over-determined, parsimonious, inverse problem. That is, common methods of model analysis exhibit artifacts from the propagation of subjective a-priori parameter parsimony throughout the calibration and predictive error analyses. This thesis describes theoretical and practical developments that enable the estimation of a large number of parameters, and the evaluation of the potential for error in predictions made by highly parameterized models. Since the focus of this research is on the use of models in support of decision making, the new methods are demonstrated by application to synthetic applications, where the performance of the method can be evaluated under controlled conditions; and to real-world applications, where the performance of the method can be evaluated in terms of trade-offs in computational effort versus calibration results and the ability to rigorously yet expediently investigate predictive error. The applications suggest that the new techniques are applicable to a range of environmental modeling disciplines. Mathematical innovations described in this thesis focus on combining complementary regularized inversion (calibration) techniques with novel methods for analyzing model predictive error. Several of the innovations are founded on explicit recognition of the existence of the calibration solution and null spaces – that is, that with the available observations there are some (combinations of) parameters that can be estimated; and there are some (combinations of) parameters that cannot. The existence of a non-trivial calibration null space is at the heart of the non-uniqueness problem in model calibration: this research expands upon this concept by recognizing that there are combinations of parameters that lie within the calibration null space yet possess non-trivial projections onto the predictive solution space, and these combinations of parameters are at the heart of predictive error analysis. The most significant contribution of this research is the attempt to develop a framework for model analysis that promotes computational efficiency in both the calibration and the subsequent analysis of the potential for error in model predictions. Fundamental to this framework is the use of a large number of parameters, the use of Tikhonov regularization, and the use of subspace techniques. Use of a large number of parameters enables parameter detail to be represented in the model at a scale approaching true variability; the use of Tikhonov constraints enables the modeler to incorporate preferred conditions on parameter values and/or their variation throughout the calibration and the predictive analysis; and, the use of subspace techniques enables model calibration and predictive analysis to be undertaken expediently, even when undertaken using a large number of parameters. This research focuses on the inability of the calibration process to accurately identify parameter values: it is assumed that the models in question accurately represent the relevant processes at the relevant scales so that parameter and predictive error depend only on parameter detail not represented in the model and/or accurately inferred through the calibration process. Contributions to parameter and predictive error arising from incorrect model identification are outside the scope of this research.

This thesis investigates the capabilities of the Collapse Prediction Relay (CPR-D) and also investigates the use of system identification for detection of oscillatory instability. Both the CPR-D and system identification are based on system measurements and do not require modelling of the power system. Measurement based stability monitors can help to avoid instability and blackouts, in cases where the available system model can not predict instability. The CPR-D uses frequency patterns in voltage oscillation to detect system instability. The relay is based on non-linear dynamics Theory. If a collapse is predicted, measures could be taken to prevent a blackout. The relay was tested using the output of simulators and was later installed in a substation. The data from laboratory tests and site installations is analysed enabling a detailed evaluation of the CPR-D.Oscillatory instability can be detected by monitoring the damping ratio of oscillations in the power system. Poor damping indicates a smaller stability margin. Subspace identification is used to estimate damping ratios. The method is tested under different conditions and using several power system models. The results show that using several measurements gives more accurate estimates and requires shorter data windows. A selection method for measurements is proposed in the thesis.

This thesis consists of four research papers. In the first paper we study whether certain properties of sets (functions) are separably determined. In our results we use the "method of elementary submodels". In the second paper we generalize some results concerning Valdivia compacta (equivalently spaces with a commutative retractional skeleton) to the context of spaces with a retractional skeleton (not necessarily commutative). The third paper further studies the structure of spaces with a projectional (resp. retractional) skeleton. Under certain conditions we prove the existence of a "simultaneous projectional skeleton" and we use this result to prove other statements concerning the structure of spaces with a projectional (resp. retractional) skeleton. In the last paper we study the method of elementary submodels in a greater detail and we compare it with the "method of rich families". 1

This thesis proposes novel methods for identification and control of multivariable systems including multiple delays and describes their application to control of general anesthesia administration. First, an identification method for multivariable systems whose input and output paths have different time delays is presented. Second, a state predictor for multivariable systems whose input and output paths have different time delays is proposed. Third, the state predictor is used for constructing a state-predictive servo control system for controlled processes whose output paths have different time delays. A robust stability analysis method of the state-predictive servo control system is also examined. Furthermore, based on results of these theoretical studies, control systems for use in general anesthesia administration are developed. First, an identification method for multivariable systems whose input and output paths have different time delays is proposed. This method comprises two steps. In the first step, the delay lengths are estimated from the impulse response matrix identified from input and output (I/O) sequences using a subspace identification algorithm. In the second step, I/O sequences of a delay-free part are constructed from the original sequences and the delay estimates, and the system matrices of the delay-free part are identified. The proposed method is numerically stable and efficient. Moreover, it requires no complex optimization to obtain the delay estimates, nor does it require an assumption about the structure of the system matrices. Second, a state predictor is proposed for multivariable systems whose input and output paths have different time delays. The predictor consists of a full-order observer and a prediction mechanism. The former estimates a vector consisting of past states from the output. The latter predicts the current state from the estimated vector. The prediction error converges to zero at an arbitrary rate, which can be determined using pole assignment method, etc. In the proposed predictor, the interval length of the finite interval integration fed to the observer is shorter than that of an existing delay-compensating observer. Consequently, the proposed predictor is more numerically accurate than the delay-compensating observer. Using the proposed state predictor, a design method of a state-predictive servo controller is described for multivariable systems whose output paths have different time delays. Furthermore, a sufficient stability condition of the state-predictive servo control system against parameter mismatches is derived. Using a characteristic equation of the perturbed closed-loop system, a stability margin can be given on a plane whose axes correspond to the magnitudes of the mismatches on system matrices and on delay lengths. In the remainder of this thesis, development of anesthesia control systems is described to illustrate an application of the theoretical results described above. First, a hypnosis control system is presented. This system administers an intravenous hypnotic drug to regulate an electroencephalogram-derived index reflecting the patient’s hypnosis. The system comprises three functions: i) a model predictive controller that can take into account effects of time delay adequately, ii) an estimation function of individual parameters, and iii) a risk-control function for preventing undesirable states such as drug over-infusion or intra-operative arousal. Results of 79 clinical trials show that the system can reduce the total amount of drug infusion and maintain hypnosis more accurately than an anesthesiologist’s manual adjustment. Second, a simultaneous control system of hypnosis and muscle relaxation is described. For development of this system, a multivariable model of hypnosis and muscle relaxation is identified using the method proposed in this thesis. Then a state-predictive servo control system is designed for controlling hypnosis and muscle relaxation. Finally, the control system’s performance is evaluated through simulation. The resultant simultaneous control system satisfies the performance specifications of settling time, disturbance rejection ability, and a robust stability range. Although this system is not fully developed, the procedure of constructing this control system demonstrates the effectiveness of the proposed methods: the identification method for systems whose input and output paths have different time delays and the design and stability analysis methods of the state-predictive servo control system. / Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第13820号 / 工博第2924号 / 新制||工||1432(附属図書館) / 26036 / UT51-2008-C736 / 京都大学大学院工学研究科電気工学専攻 / (主査)教授 小林 哲生, 教授 萩原 朋道, 准教授 古谷 栄光 / 学位規則第4条第1項該当

This research is related to the control of energy consumption and efficiency in building Heating Ventilation and Air Conditioning (HVAC) systems and is primarily concerned with controlling the function of heating. The main goal of this thesis is to develop a control system that can achieve the following two main control functions: a) detection of unexpected indoor conditions that may result in unnecessary power consumption and b) energy efficiency control regarding optimal balancing of two parameters: the required energy consumption for heating, versus thermal comfort of the occupants. Methods of both orientations were developed in a multi-zone space composed of nine zones where each zone is equipped with a wireless node consisting of temperature and occupancy sensors while all the scattered nodes together form a wireless sensor network (WSN). The main methods of both control functions utilize the potential of the deterministic subspace identification (SID) predictive model which provides the predicted temperature of the zones. In the main method for detecting unexpected situations that can directly affect the thermal condition of the indoor space and cause energy consumption (abnormal situations), the predictive temperature from the SID model is compared with the real temperature and thus possible temperature deviations that indicate unexpected situations are detected. The method successfully detects two situations: the high infiltration gain due to unexpected cold air intake from the external surroundings through potential unforeseen openings (windows, exterior doors, opened ceilings etc) as well as the high heat gain due to onset of fire. With the support of the statistical algorithm for abrupt change detection, Cumulative Sum (CUSUM), the detection of temperature deviations is accomplished with accuracy in a very short time. The CUSUM algorithm is first evaluated at an initial approach to detect power diversions due to the above situations caused by the aforementioned exogenous factors. The predicted temperature of the zone from the SID model utilized appropriately also by the main method of the second control function for energy efficiency control. The time needed for the temperature of a zone to reach the thermal comfort zone threshold from a low initial value is measured by the predicted temperature evolution, and this measurement bases the logic of a control criterion for applying proactive heating to the unoccupied zones or not. Additional key points for the control criterion of the method is the occupation time of the zones as well as the remaining time of the occupants in the occupied zones. Two scenarios are examined: the first scenario with two adjacent zones where the one is occupied and the other is not, and the second scenario with a multi-zone space where the occupants are moving through the zones in a cascade mode. Gama and Pareto probability distributions modeled the occupation times of the two-zone scenario while exponential distribution modeled the cascade scenario as the least favorable case. The mobility of the occupants modeled with a semi-Markov process and the method provides satisfactory and reasonable results. At an initial approach the proactive heating of the zones is evaluated with specific algorithms that handle appropriately the occupation time into the zones.

The existence and uniqueness of Lagrangian invariant subspaces of Hamiltonian matrices is studied. Necessary and sufficient conditions are given in terms of the Jordan structure and certain sign characteristics that give uniqueness of these subspaces even in the presence of purely imaginary eigenvalues. These results are applied to obtain in special cases existence and uniqueness results for Hermitian solutions of continuous time algebraic Riccati equations.