Enhancing Mathematical Modeling Activities in Classroom Instruction

The ability of students in mathematical modeling was enhanced through activities that involved systems of linear equations with two variables. Students involved were in form four, at the final year of the ordinary secondary school level where they were expected to have mastered the knowledge on systems of linear equations with two variables. Students’ knowledge on ill-conditioned linear systems was explored as well as their knowledge on practical problems in linear equations. Then after, mathematics subject teachers guided students to identify practical problems in linear equations of two variables. Students were put into groups to think of problems in real life and come up with solutions. The solutions were related to the real situations in the environment and each group had to make a presentation in the class. Problems in transportation, manufacturing, production and diet were identified by students and the results presented for discussion. It came out clearly that students acquired knowledge on solving real life problems at the end of the activities. Before these activities, students had theoretical knowledge on solving problems with two unknowns without relating these to real life problems. While knowledge on independent and inconsistent systems was known to students, enthusiasm was noted among students at the end of the activities when they got involved in real aspects of solutions obtained. It was concluded that with more time availed in the school curricula, students can acquire useful knowledge on mathematical modeling to achieve problem-thinking skills that involve real life situations.

A general iterative solver for unbalanced inconsistent transportation problems

The transportation problem, as a particular case of a linear programme, has probably the highest relative frequency with which appears in applications. At least in its classical formulation, it involves demands and supplies. When, for practical reasons, the total demand cannot satisfy the total supply, the problem becomes unbalanced and inconsistent, and must be reformulated as e.g. finding a least squares solution of an inconsistent system of linear inequalities. A general iterative solver for this class of problems has been proposed by S. P. Han in his 1980 original paper. The drawback of Han’s algorithm consists in the fact that it uses in each iteration the computation of the Moore-Penrose pseudoinverse numerical solution of a subsystem of the initial one, which for bigger dimensions can cause serious computational troubles. In order to overcome these difficulties we propose in this paper a general projection-based minimal norm solution approximant to be used within Han-type algorithms for approximating least squares solutions of inconsistent systems of linear inequalities. Numerical experiments and comparisons on some inconsistent transport model problems are presented.

Modified Han algorithm for inconsistent linear inequalities

In this paper we present a modified version of S. P. Han iterative method for solving inconsistent systems of linear inequalities. Our method uses an iterative Kaczmarz-type solver to approximate the minimal norm least squares solution of the problems involved in each iteration of Han’s algorithm. We prove some convergence properties for the sequence of approximations generated in this way and present numerical experiments and comparisons with Han’s and other direct solver based methods for inconsistent linear inequalities.

Investigating the links of internal and external reliability with the system conditionality in Gauss-Markov models with uncorrelated observations

The relationship between internal response-based reliability and conditionality is investigated for Gauss-Markov (GM) models with uncorrelated observations. The models with design matrices of full rank and of incomplete rank are taken into consideration. The formulas based on the Singular Value Decomposition (SVD) of the design matrix are derived which clearly indicate that the investigated concepts are independent of each other. The methods are presented of constructing for a given design matrix the matrices equivalent with respect to internal response-based reliability as well as the matrices equivalent with respect to conditionality. To analyze conditionality of GM models, in general being inconsistent systems, a substitute for condition number commonly used in numerical linear algebra is developed, called a pseudo-condition^number. Also on the basis of the SVD a formula for external reliability is proposed, being the 2-norm of a vector of parameter distortions induced by minimal detectable error in a particular observation. For systems with equal nonzero singular values of the design matrix, the formula can be expressed in terms of the index of internal response-based reliability and the pseudo-condition^number. With these measures appearing in explicit form, the formula shows, although only for the above specific systems, the character of the impact of internal response-based reliability and conditionality of the model upon its external reliability. Proofs for complementary properties concerning the pseudo-condition^number and the 2-norm of parameter distortions in systems with minimal constraints are given in the Appendices. Numerical examples are provided to illustrate the theory.

Analysis of Partition Coefficients of Ternary Liquid-Liquid Equilibrium Systems and Finding Consistency Using Uniquac Model

Analysis of Partition Coefficients of Ternary Liquid-Liquid Equilibrium Systems and Finding Consistency Using Uniquac Model The objective of this study is to investigate the change in partition coefficient with a change in the concentration of the solute in a liquid system consisting of two relatively immiscible solvents. To investigate the changes in the partition coefficients, the data of the partition coefficients at infinite dilution and the ternary Liquid-Liquid Equilibrium (LLE) data at finite concentrations of the solute should be consistent. In this study, 29 ternary systems that are found in literature and for which the partition coefficients at infinite dilution and the ternary LLE data cannot be predicted accurately by the universal quasi-chemical (UNIQUAC) model are identified. On the basis of this model, some consistent and inconsistent ternary systems are introduced. Three inconsistent systems, namely hexane-butanol-water, CCl4 (carbon tetrachloride)-PA (propanoic acid)-water, and hexane-PA-water, are chosen for detailed analysis in this study. The UNIQUAC activity coefficient model is used to represent these data over a range of concentrations. The results show large errors, exhibiting the inability of this model to correlate the data. Furthermore, some ternary systems in which cross behavior of solutes between two phases observed are identified.