http://hdl.handle.net/10397/10 Search the Channel search http://repository.lib.polyu.edu.hk/jspui/simple-search http://hdl.handle.net/10397/5674 Title: The study of a class of analytical solutions for six fluid dynamical systems<br/><br/>Authors: Yuen, Man Wai<br/><br/>Abstract: The systems of gas dynamics (Euler, Euler-Poisson, Navier-Stokes, Navier-Stokes-Poisson equations) and of shallow water (Camassa-Holm and Degasperis-Procesi equations) are important basic models in fluid mechanics and astrophysics. Constructing analytical or exact solutions for the partial differential equations is a vital part in nonlinear sciences. Indeed, scientists and mathematicians are eager to seek analytical solutions for better understanding of the evolution of these kinds of systems. In this PhD thesis, I consider the construction of analytical solutions for the above systems. As these systems share similar mathematical structures in some aspects, I will exhibit some common features among them, including certain blowup and stability phenomena. In detail, I attempt to employ the well-known separation method to its fullest extent and introduce a novel pertubational method to seek analytical solutions with free boundaries. The main idea is to reduce the nonlinear partial differential systems into several ordinary or functional differential equations, or to simpler partial differential equations under some suitable assumptions on the functional structures of the solutions. After proving the existence of solutions of the corresponding simpler differential equations, the analytical solutions for the original nonlinear systems are constructed. One of the applications of such analytical solutions is to test numerical methods designed for these systems. Another application is to provide samples of concrete solutions so as to affirm or support theoretical hypotheses or conjectures about these complicated systems.; A substantial percentage of the results presented in this thesis have appeared in print. In total, sixteen published papers (not counting preprints; see the lists in the next three pages) are the direct outcome of work done during my PhD study. The fact that these results are well-received by referees and editors attests to the great interest of others in these analytical solutions. The most significant contributions of this thesis are as follows: - I am the first to reduce the compressible density-dependent Navier-Stokes equations in RN to new 1 + N differential functional equations, which lead to solutions with elliptical symmetry and drift phenomena. - I am the first to obtain self-similar solutions in explicit form for the 2-component shallow water systems. - We construct the first rotational solutions in explicit form for the 2-dimensional Euler- Poisson equations and demonstrate the principle that rotation can prevent blowup. The thesis is organized as follows: - A brief introduction of the above six models is provided. - The separation method is applied to construct solutions with free boundaries for the systems of gas dynamics and shallow water. - In addition, some solutions with rotation are constructed for the 2-dimensional Euler-Poisson and 3-dimensional Euler equations. - Based on the separation method, a novel pertubational method is used to obtain more general classes of analytical solutions for the 1-dimensional Euler and Camassa-Holm equations. - Finally a summary is provided to conclude the works done and other related works in the PhD studies, together with some future research insights for the further development of this thesis is included.<br/><br/>Description: 7, 140 leaves : ill. ; 30 cm.; PolyU Library Call No.: [THS] LG51 .H577P AMA 2012 Yuen http://hdl.handle.net/10397/5673 Title: Coordinated inventory-transportation supply chain models<br/><br/>Authors: Yeung, Hon Keung<br/><br/>Abstract: Supply chain, which is a flow of materials, information and funds between different parties, is one of the important issues in today's business and industrial sectors. In a supply chain, a vendor is required to produce items to satisfy the needs of buyers. If vendor and buyers operate independently to minimize their own costs, it may not be optimal to the system as a whole. Most of the literature has found that a supply chain can achieve better system cost performance through coordination of vendor and buyers, hence effective coordination plays an important role in the successful operation of supply chains. Chan and Kingsman (2005, 2007) proposed a synchronized cycles model for the coordination of a single-vendor multi-buyer supply chain in which vendor and buyers synchronize the production and ordering cycles so as to minimize the total system cost. The synchronized cycles model performs better than independent optimization as well as common order cycle model developed by Banerjee and Burton (1994) in terms of total system cost. Furthermore, the synchronized cycles model addresses some of the shortcomings of previous coordination models. For example, the model considers vendor as a manufacturer producing an item to supply multiple heterogeneous buyers and tackles the discrete vendor inventory depletion into the model. This issue was rarely addressed in the literature (see Sarmah et. al.(2006)). In the synchronized cycles model, the process of finding the optimal solution involves the determination of production cycle NT of vendor, ordering cycle kiT and ordering time ti of buyers where ki are integer factors of N. Due to the complexity of the model, it is very difficult to find the optimal solution analytically. Chan and Kingsman (2007) proposed a heuristic algorithm to find a "near-optimal" solution. The algorithm has been found to be competitive when compared with genetic algorithm. However, it is believed that there are still rooms for improvement in the algorithm in terms of the "optimal" solution and computational time.; Transportation is also a key component in a supply chain. Most of the literature of single-vendor multi-buyer coordination usually assumed that transportation cost is a constant (i.e $/order) for simplicity. Truck capacity and truck transportation cost were not considered. Different transportation modes such as less-than-truckload (LTL) and full-truckload (FTL) have been studied, but are limited to single-vendor single-buyer supply chain. It is rarely mentioned that transportation mode with truck capacity and truck cost are applied to a coordinated single-vendor multi-buyer supply chain. Finally, environmental problem is a key issue nowadays as people become more concerned about environmental performance. However, existing supply chain models which put stress on financial performance did not pay much attention to the environment. For instance, more frequent deliveries can reduce average inventory level in a supply chain but cause more air pollution during transportation. Also, holding too many stocks consume more materials and resources. Hence, raw materials wastage and energy wastage should be taken into consideration in supply chain models. It is worth addressing and incorporating these environmental measures into a coordinated supply chain system.<br/><br/>Description: xxiv, 210 leaves : ill. ; 30 cm.; PolyU Library Call No.: [THS] LG51 .H577M AMA 2012 Yeung http://hdl.handle.net/10397/5672 Title: Exact penalty function methods and their applications in search engine advertising problems<br/><br/>Authors: Ma, Cheng<br/><br/>Abstract: The penalty function method is one of the most fundamental and useful tools in the modern optimization and has developed into a major research field since 1950s. The study of penalty functions has proliferated in many interesting areas within mathematical optimization society. Nowadays, researchers in optimization fields still pursue unremittingly new breakthroughs in theoretical and algorithmic aspects of penalty function methods. However, it should be mentioned that the currently existing exact penalty functions have a disadvantage that the evaluation of the merit function either needs Jacobian (e.g., augumented Lagrangian penalty functions) or is no longer smooth (l1 or l∞ penalty functions). "It would be a major theoretic breakthrough in nonlinear programming if a simple continuously differentiable function could be exhibited with the property that any unconstrained minimum is a solution of the constrained problem."Evans, Gould and Tolle [21]. So far, to some extent, the breakthrough of the above quotation has been achieved. Recently, Huyer and Neumair in [38] proposed a new exact and smooth penalty function through adding an auxiliary variable ε to deal with equality constrained minimization problem. The proposed new penalty function enjoys several good properties: (1) good smoothness and exactness properties; (2) bounded below under reasonable conditions; (3) combination of regularization with penalty techniques, which are not possessed by the classical simple and exact penalty functions. Moreover, the new penalty function only involves the information of objective function and constraints function rather than the one of gradient and Hessian matrix. Nevertheless, the new exact penalty function is different from the definition of traditional penalty function, namely, the values of penalty terms are zero on the feasible set and positive outside the feasible set. In spite of significant differences between the new penalty function and the classical simple and exact penalty functions, naturally, a question arises: What's on earth the relationship between them? In this thesis, motivated by Huyer and Neumair's work [38], we extend the norm function term of the exact penalty function in [38] to a class of convex functions with a unified framework for some barrier-types and exterior-type penalty functions. We characterize necessary and sufficient conditions for the exact penalty property. Interestingly, we also explore the equivalence between this class of penalty functions and the traditional simple exact penalty functions in the sense of exactness property. These results clarify that this class of penalty functions not only have exactness property as the classical simple penalty function, but also possess the smoothness property, which is not shared by the latter. Furthermore, since the class of penalty functions are bounded below, a revised penalty function method is established. In addition, we verify that, under certain conditions, the proposed algorithm terminates at the optimal solution of the primal problem after finitely many iterations; while in the absence of these conditions, a perturbation theorem for this algorithm can be derived. As a corollary, the global convergence property is presented-namely, every accumulation point of the sequence generated by the algorithm is an optimal solution of the primal problem. The numerical outputs verify the correctness of our developed theory as desired.; We propose a new exact and smooth penalty function for semi-infinite programming problems. The main feature of our penalty function is that we only need to add one variable ε to handle infinitely many constraints. The merit function is considered as a function of x and ε simultaneously which has good smoothness and exactness properties, without involving gradient and Jacobian matrices. We derive another useful property that the minimizer (x*,ε*) of the penalty problem satisfies ε* = 0 if and only if x* solves semi-infinite programming problem. This property demonstrates that the introduced new variable ε can be viewed as an indicator variable of a local (global) minimizer of semi-infinite programming problem. Alternatively, under some mild conditions, the local exactness proof is shown. The numerical results demonstrate that it is an effective and promising approach for solving constrained semi-infinite programming problems. Similarly, we also apply a new exact penalty function to tackle the min-max programming problem and establish necessary and sufficient conditions for the exactness property. In addition, we characterize the second-order sufficient conditions for the local exactness property. We model and explore the search-based advertising auction as a large scale integer programming problem with more realistic situations, e.g., multiple slots, advertisers with choice behavior and the popular generalized second price mechanism etc.. And then, we apply the new penalty function to this proposed integer programming. In addition, we give numerical simulations to address managerial insights on both operational and theoretical aspects and compare the numerical performances with currently existing algorithms for search engine advertising problems.<br/><br/>Description: xv, 129 p. : ill. ; 30 cm.; PolyU Library Call No.: [THS] LG51 .H577P AMA 2012 Ma http://hdl.handle.net/10397/5671 Title: Semiparametric regression analysis of longitudinal data with informative observation times<br/><br/>Authors: Deng, Shirong<br/><br/>Abstract: Longitudinal data often occur in a long-term study where each individual is measured repeatedly at distinct time points rather than continuous times and also the observation times and censoring times may vary from subject to subject. Many researchers have considered the analysis of such longitudinal data under the assumption that observation process is independent of response process completely or conditional on covariates, which may not be true in practice. This thesis investigates semiparametric analysis of longitudinal data when the response process is correlated with the observation times. We develop a new class of semiparametric mean models for longitudinal data which allows for the interaction between the observation history and covariates, leaving patterns of the observation process to be arbitrary. Although panel count data is a special case of longitudinal data, it has particular features which can not be described by general longitudinal models. Thus, to analyze the panel count data, we propose a new class of flexible semiparametric regression models by incorporating the interaction between the observation history and some covariates to the mean model of the recurrent event process, without any formation restriction on the informative observation process. For inference on the regression parameters and the unknown baseline functions involved in both longtidunial data and panel count data models, spline-based least square estimation approachs are proposed, respectively, and asymptotic properties including the consistency, rate of convergence and asymptotic normality of the proposed estimators are established for both models. Simulation studies demonstrate that the proposed inference procedures perform well for both models. The analyses of a bladder tumor data are presented to illustrate the proposed methods. Furthermore, it would be desirable to develop estimation procedures for panel count data with informative observation times, and also with time-dependent covariates and informative censoring times. Thus we extend the joint frailty models proposed by Zhao and Tong (2011) to panel count data with the time-dependent covariates and informative observation and censoring times. A novel estimating equation approach that does not depend on the distribution of frailty variables and the link function is proposed for estimation of parameters, and the asymptotic properties of the proposed estimators are established. The performance of proposed inference procedure are demonstrated by some simulation studies and illustrated by the analysis of a bladder tumor data.<br/><br/>Description: x, 159 p. : ill. ; 30 cm.; PolyU Library Call No.: [THS] LG51 .H577P AMA 2012 Deng