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Bibliography: p. 145-148 This thesis deals with the squat of a moving ship, that is, the downward displacement and angle of trim caused by its forward motion. The aim is to be able to predict accurately the squat of any ship at any given speed and water depth. (introduction)

In the present work, we develop and illustrate powerful, but straightforward, formal methods for deriving asymptotic expansions from differential equations. In the second chapter, the ‘inverse Frobenius method' for deriving Stokes expansions is exemplified. The main body of this thesis, however, consists of the development of the new Green-Liouville-Melin transform method, and its detailed application to modified Bessel functions, parabolic cylinder functions, Whittaker functions, Poiseuille functions, confluent hypergeometric functions, and also to periodic Mathieu functions and oblate spheroidal wave functions, all with at least one parameter large+. The wide scope of the method is evinced by the fact that treatment of the essentially eigenvalue problem posed by the two last-named cases requires no additional techniques. This method, as will be explained in detail in chapter 3, yields Green-type, transitional and uniform expansions. The transitional expansions found in this way are usually of a simpler form than those derived by alternative processes (e.g. perturbation theory). To state an example, the asymptotic expansions for the periodic Mathieu functions ce(z,h) and se(z,h) valid near |z| = 1/2π that have been obtained in earlier work contain the complicated parabolic cylinder functions (c.f. Meixner 1948, Sips 1949, Dingle and Müller 1962). By contrast, our methods yield expansions of comparable applicability, but involving only elementary functions. To demonstrate their usefulness, we have fed these expansions into a digital computer and obtained extensive tables for ce(z,h) and se(z,h) in the range 50°≤ z ≤90° . Extracts from these tables and comparisons with correct results are given in §8.71. Following the chapters on the introduction and applications of the Mellin transform technique, there is some preliminary work on a new method for determining the general term in Green-type expansions. The method is illustrated by detailed calculations for modified Bessel and parabolic cylinder functions. In the final chapter, we present certain suggestions for further work.

Multi-lable learning (ML), multi-instance multi-label learning (MIML), large network learning and random under-sampling system are four active research topics in machine learning which have been studied intensively recently. So far, there are still a lot of open problems to be figured out in these topics which attract worldwide attention of researchers. This thesis mainly focuses on several novel methods designed for these research tasks respectively. Then main difference between ML learning and traditional classification task is that in ML learning, one object can be characterized by several different labels (or classes). One important observation is that the labels received by similar objects in ML data are usually highly correlated with each other. In order to exploring this correlation of labels between objects which might be a key issue in ML learning, we consider to require the resulting label indicator to be low rank. In the proposed model, nuclear norm which is a famous convex relaxation of intractable matrix rank is introduced to label indicator in order to exploiting the underlying correlation in label domain. Motivated by the idea of spectral clustering, we also incorporate information from feature domain by constructing a graph among objects based on their features. Then with partial label information available, we integrate them together into a convex low rank based model designed for ML learning. The proposed model can be solved efficiently by using alternating direction method of multiplier (ADMM). We test the performance on several benchmark ML data sets and make comparisons with the state-of-art algorithms. The classification results demonstrate the efficiency and effectiveness of the proposed low rank based methods. One step further, we consider MIML learning problem which is usually more complicated than ML learning: besides the possibility of having multiple labels, each object can be described by multiple instances simultaneously which may significantly increase the size of data. To handle the MIML learning problem we first propose and develop a novel sparsity-based MIML learning algorithm. Our idea is to formulate and construct a transductive objective function for label indicator to be learned by using the method of random walk with restart that exploits the relationships among instances and labels of objects, and computes the affinities among the objects. Then sparsity can be introduced in the labels indicator of the objective function such that relevant and irrelevant objects with respect to a given class can be distinguished. The resulting sparsity-based MIML model can be given as a constrained convex optimization problem, and it can be solved very efficiently by using the augmented Lagrangian method (ALM). Experimental results on benchmark data have shown that the proposed sparse-MIML algorithm is computationally efficient, and effective in label prediction for MIML data. We demonstrate that the performance of the proposed method is better than the other testing MIML learning algorithms. Moreover, one big concern of an MIML learning algorithm is computational efficiency, especially when figuring out classification problem for large data sets. Most of the existing methods for solving MIML problems in literature may take a long computational time and have a huge storage cost for large MIML data sets. In this thesis, our main aim is to propose and develop an efficient Markov Chain based learning algorithm for MIML problems. Our idea is to perform labels classification among objects and features identification iteratively through two Markov chains constructed by using objects and features respectively. The classification of objects can be obtained by using labels propagation via training data in the iterative method. Because it is not necessary to compute and store a huge affinity matrix among objects/instances, both the storage and computational time can be reduced significantly. For instance, when we handle MIML image data set of 10000 objects and 250000 instances, the proposed algorithm takes about 71 seconds. Also experimental results on some benchmark data sets are reported to illustrate the effectiveness of the proposed method in one-error, ranking loss, coverage and average precision, and show that it is competitive with the other methods. In addition, we consider the module identification from large biological networks. Nowadays, the interactions among different genes, proteins and other small molecules are becoming more and more significant and have been studied intensively. One general way that helps people understand these interactions is to analyze networks constructed from genes/proteins. In particular, module structure as a common property of most biological networks has drawn much attention of researchers from different fields. However, biological networks might be corrupted by noise in the data which often lead to the miss-identification of module structure. Besides, some edges in network might be removed (or some nodes might be miss-connected) when improper parameters are selected which may also affect the module identified significantly. In conclusion, the module identification results are sensitive to noise as well as parameter selection of network. In this thesis, we consider employing multiple networks for consistent module detection in order to reduce the effect of noise and parameter settings. Instead of studying different networks separately, our idea is to combine multiple networks together by building them into tensor structure data. Then given any node as prior label information, tensor-based Markov chains are constructed iteratively for identification of the modules shared by the multiple networks. In addition, the proposed tensor-based Markov chain algorithm is capable of simultaneously evaluating the contribution from each network. It would be useful to measure the consistency of modules in the multiple networks. In the experiments, we test our method on two groups of gene co-expression networks from human beings. We also validate biological meaning of modules identified by the proposed method. Finally, we introduce random under-sampling techniques with application to X-ray computed tomography (CT). Under-sampling techniques are realized to be powerful tools of reducing the scale of problem especially for large data analysis. However, information loss seems to be un-avoidable which inspires different under-sampling strategies for preserving more useful information. Here we focus on under-sampling for the real-world CT reconstruction problem. The main motivation is to reduce the total radiation dose delivered to patient which has arisen significant clinical concern for CT imaging. We compare two popular regular CT under-sampling strategies with ray random under-sampling. The results support the conclusion that random under-sampling always outperforms regular ones especially for the high down-sampling ratio cases. Moreover, based on the random ray under-sampling strategy, we propose a novel scatter removal method which further improves performance of ray random under-sampling in CT reconstruction.

M.Ing. / A steady-state model was developed to predict the performance of the highpressure side' of vapour-compression air-conditioning systems. The model consists of two segments; the compressor model and condenser model. The compressor model consists of a single empirical equation, for reciprocating compressors, operating with R-22, and having a cooling capacity from 2.6 to 3.5 kW. An important advantage of the approach is that the compressor performance indexes are based on operating conditions. The condenser model displays an exact method to determine physical dimensions of heat exchangers, and a simple, accurate manner to calculate the heat transfer variables. The correctness of the condensing temperature is obtained by an iterative procedure, using terms from both the compressor and condenser studies. The feasibility of the proposed model is demonstrated via a comparison with experimental data and a simulation study. Results indicate a good correlation between the mathematical model and its counterparts.

University of Macau / Faculty of Science and Technology / Department of Computer and Information Science

The design and optimisation of inventory replenishment systems has already been exhaustively studied by the operational research community. Many classical mathematical methods and simulation techniques have been developed and introduced in the literature. However, what can be observed is the fact that in a real case scenario the lead-time, deterioration of goods and demand for product are likely to be time-varying and uncertain, which traditionally have not necessarily been reflected in the model formulations. Therefore, in response to the dynamical nature of inventory systems, the potential of algorithms based on control theory to reduce the undesirable influences of system uncertainties on inventory level stability, have been investigated /proposed. Consequently, the mapping of the inventory problem into the control theory domain, for cost-benefit inventory trade-off achievement has been realised. Although, the application of control theory in inventory optimisation appears to be beneficial, there are certain reasons why the approach has gained yet little attention among the operational research community. One reason is that it cannot be adopted easily by researchers who are unfamiliar with control theory and another is due to a communication gap which exists between the control theory and operational research communities. Prompted by these observations, the thesis presents a novel, systematic mathematical approach for finding the optimal order quantities. The proposed approach has been mathematically demonstrated to be equivalent in study-sate to model-based predictive control, which is one of the more well-established productive control techniques with industrial application today. The mathematically reduced approach attempts to bridge the identified gap to fulfil the lacking dual perceptions of both communities. It enables the straightforward benefits afforded by predictive control without the necessity to become familiarised with principles of control theory. The method is shown to be applicable for both perishable and non-perishable inventory. Although the novel technique was inspired by MPC and noticing the MPC patterns in the mathematical description, the resulting proposal is no longer MPC. It is in fact a minimum variance approach, or dear beat controller, with an incorporated Smith predictor. Therefore using the adjective ‘predictive’ in the title of the thesis refers to both, the inspiration of MPC and the predictive nature of the minimum variance controller to accommodate lead time, being incorporated within an inherent Smith predictor. The developed approach is considered to be transferable to other applications, where similar model formulations may be applicable.

Lie group analysis is arguably the most systematic vehicle for finding exact solutions of differential equations. Using this approach one has at one's disposal a variety of algorithms that make the solution process of many differential equations algorithmic. Vital properties of a given differential equation can often be inferred from the symmetries admitted by the equation. However, Lie group analysis has not enjoyed wide-spread application to systems of first-order ordinary differential equations. This is because such systems typically admit an infinite number of Lie point symmetries, and there is no systematic way to find even a single nontrivial one-dimensional Lie symmetry algebra. In the few applications available, the approach has been to circumvent the problem by transforming a given system of first-order ordinary differential equations into one in which at least one of the equations is of order two or greater. It is therefore fair to say that the full power of Lie group analysis has not been sufficiently harnessed in the solution of systems of first-order ordinary differential equations. In this dissertation we review some applications of Lie group analysis to systems of first order ordinary differential equations. We shed light on the integration procedure for first-order systems of ordinary differential equations admitting a solvable Lie algebra. We do this via instructive examples drawn from mathematical epidemiology models. In particular we revisit the work of Nucci and Torrisi [54] and improve the exposition of the Lie-symmetry-inspired solution of a mathematical model which describes a HIV transmission. To aid implementation of the integration strategy for systems of ordinary differential equations, we have developed ad-hoc routines for finding particular types of admitted symmetries and checking if a given symmetry is indeed admitted by a system of ordinary differential equations.

This dissertation is an extension of Leland and Pyle's (1977) signaling model. It introduces the length of the retention period to which the entrepreneur commits in the prospectus as a signal of firm value, in addition to the retention level. The analysis uses concepts of game theory to examine a two-period model where an entrepreneur seeks to issue shares on the market and invest in a productive project that generates outcomes which are publicly announced at the end of the next two periods. The entrepreneur can retain some of her firm's shares and trade them later on the secondary market, after information has been released about the outcomes. The length of the retention period is found to be a signaling mechanism that complements ownership retention. Depending on the information structure of the firm, a longer retention period may reduce or increase the retention level necessary for separation. The model also shows that there are realistic situations in which entrepreneurs prefer to retain a portion of their firm's shares for longer than the minimum retention period imposed by regulations, and others in which she prefers the shortest period possible. The optimal combination of under-diversification and commitment is shown to depend on the information structure and the probability distribution of outcomes of the firm. The empirical implications of the model are tested on the set of firms that made an initial public offering in 1981. Although the results of the tests are generally consistent with the predictions of the model, they are not strong enough to reject the null hypotheses. / Arts, Faculty of / Vancouver School of Economics / Graduate

This thesis deals with Attack and Defence Models involving four different pay-off functions In each model, by means of standard min-max and convexity arguments, the optimal attack strategy, optimal defence strategy, and value are calculated. / Science, Faculty of / Mathematics, Department of / Graduate