University of Macau / Faculty of Science and Technology / Department of Mathematics

A Study of Problem-Solving Strategies and Errors in Inequalities for Junior High School Students The aim of this study is to investigate students in learning in inequalities with one unknown, as well as to collect corresponding strategies and errors in problem solving. The subjects of this study were nine-grade students from junior high school. Six classes were selected from three schools with total of 204 students. This investigator used a paper-and-pencil test in first round data collection. In the second round, some students were interviewed, to further understand students¡¦ way of thinking and reasons in errors produced in problem-solving procedures. Hopefully, results can be used as reference for junior high school math teacher to plan future teaching and to prepare teaching materials. The results of the study are three: students solved linear inequalities by using 12 different strategies; students¡¦ errors can be divided into 11 types; and, the reasons for errors are mainly understanding and transforming information from problems and the determination on solutions. The students also found it difficult to understand negative fractions and negative decimals relationships (no matter in word problems or in calculation problems). In this study, those who fail to solve problems involving inequalities with one unknown are those who cannot translate algebraic expressions or keywords. They produced errors 5 typical cases: determining objectives, integrating mathematics knowledge, using a problem solving method, calculating process, and, determining solution.

published_or_final_version / Mathematics / Master / Master of Philosophy

Establishing consensus is a key probleminmulti-agent systems (MASs). This thesis proposes a novel methodology based on convex optimization in the form of linear matrix inequalities (LMIs) for establishing consensus in linear and nonlinear MAS in the presence of model uncertainties, i.e., robust consensus. Firstly, this thesis investigates robust consensus for uncertain MAS with linear dynamics. Specifically, it is supposed that the system is described by a weighted adjacency matrix whose entries are generic polynomial functions of an uncertain vector constrained in a set described by generic polynomial inequalities. For continuous-time dynamics, necessary and sufficient conditions are proposed to ensure the robust first-order consensus and the robust second-order consensus, in both cases of positive and non-positive weighted adjacency matrices. For discrete-time dynamics, necessary and sufficient conditions are provided for robust consensus based on the existence of a Lyapunov function polynomially dependent on the uncertainty. In particular, an upper bound on the degree required for achieving necessity is provided. Furthermore, a necessary and sufficient condition is provided for robust consensus with single integrator and nonnegative weighted adjacency matrices based on the zeros of a polynomial. Lastly, it is shown how these conditions can be investigated through convex optimization by exploiting LMIs. Secondly, local and global consensus are considered in MAS with intrinsic nonlinear dynamics with respect to bounded solutions, like equilibrium points, periodic orbits, and chaotic orbits. For local consensus, a method is proposed based on the transformation of the original system into an uncertain polytopic system and on the use of homogeneous polynomial Lyapunov functions (HPLFs). For global consensus, another method is proposed based on the search for a suitable polynomial Lyapunov function (PLF). In addition, robust local consensus in MAS is considered with time-varying parametric uncertainties constrained in a polytope. Also, by using HPLFs, a new criteria is proposed where the original system is suitably approximated by an uncertain polytopic system. Tractable conditions are hence provided in terms of LMIs. Then, the polytopic consensus margin problem is proposed and investigated via generalized eigenvalue problems (GEVPs). Lastly, this thesis investigates robust consensus problem of polynomial nonlinear system affected by time-varying uncertainties on topology, i.e., structured uncertain parameters constrained in a bounded-rate polytope. Via partial contraction analysis, novel conditions, both for robust exponential consensus and for robust asymptotical consensus, are proposed by using parameter-dependent contraction matrices. In addition, for polynomial nonlinear system, this paper introduces a new class of contraction matrix, i.e., homogeneous parameter-dependent polynomial contraction matrix (HPD-PCM), by which tractable conditions of LMIs are provided via affine space parametrizations. Furthermore, the variant rate margin for robust asymptotical consensus is proposed and investigated via handling generalized eigenvalue problems (GEVPs). For each section, a set of representative numerical examples are presented to demonstrate the effectiveness of the proposed results. / published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

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This thesis concerns the solution of variational inequalities (VIs) with analytic center cutting plane methods (ACCPMs). A convex feasibility problem reformulation of the variational inequality is used; this reformulation applies to VIs defined with pseudo-monotone, single-valued mappings or with maximal monotone, multi-valued mappings. / Two cutting plane methods are presented: the first is based on linear cuts while the second uses quadratic cuts. The first method, ACCPM-VI (linear cuts), requires mapping evaluations but no Jacobian evaluations; in fact, no differentiability assumption is needed. The cuts are placed at approximate analytic centers that are tracked with infeasible primal-dual Newton steps. Linear equality constraints may be present in the definition of the VI's set of reference, and are treated explicitly. The set of reference is assumed to be polyhedral, or is convex and iteratively approximated by polyhedra. Alongside of the sequence of analytic centers, another sequence of points is generated, based on convex combinations of the analytic centers. This latter sequence is observed to converge to a solution much faster than the former sequence. / The second method, ACCPM-VI (quadratic cuts), has cuts based on both mapping evaluations and Jacobian evaluations. The use of such a richer information set allows cuts that guide more accurately the sequence of analytic centers towards a solution. Mappings are assumed to be strongly monotone. However, Jacobian approximations, relying only on mapping evaluations, are observed to work very well in practice, so that differentiability of the mappings may not be required. There are two versions of the ACCPM-VI (quadratic cuts), that differ in the way a new analytic center is reached after the introduction of a cut. One version uses a curvilinear search followed by dual Newton centering steps. The search entails a full eigenvector-eigenvalue decomposition of a dense matrix of the order of the number of variables. The other version uses two line searches, primal-dual Newton steps, but no eigenvector-eigenvalue decomposition. / The algorithms described in this thesis were implemented in the M ATLAB environment. Numerical tests were performed on a variety of problems, some new and some traditional applications of variational inequalities.