Size reduction and partial decoupling of systems of equations

A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely to be factorizable or integrable. A variation of this method is applicable to non-linear systems. Modifications to improve efficiency are given and examples are shown. This procedure can be used in connection with the computation of the radical of a differential ideal (differential Groebner basis)

Reduced kinetic mechanisms for modelling LPP combustión in gas turbines

Reduced kinetic mechanisms for modelling LPP combustión in gas turbine

A comparative study of students' understanding of trigonometry in the United Kingdom and the Turkish Republic

Research focus: This is a comparative study of English and Turkish 16-18 year old students' performance in trigonometry: finding unknown lengths or angles from diagrams, simplification of expressions and solving word problems. It is also concerned with the culture of learning because students' learning histories are shaped by curricula and national cultures of education. Methodology: This is a comparative study with exploratory and descriptive enquiry purposes. It employs an interpretitivist paradigm with a naturalistic mode of enquiry. A wide variety of instruments were used to address two sets of research questions with two different samples: students and teachers. The primary focus of this study is students' performance. Data collected from teachers is used to gain insight into how students learn. Four written tests were used to collect data from the student sample. The foci of these tests were: algebra, simplification of trigonometric expressions, finding unknown quantities in right-angled triangles and solving word problems. Interviews and concurrent verbal protocols were conducted with a subset of the student sample to explore reasoning behind the answers in the tests. Two questionnaires with follow up interviews and classroom observations were employed to collect data from the teacher sample. Main Findings: Turkish students' performance in the algebra and simplification of trigonometric expressions tests was considerably better than English students' performance: 71 % of Turkish students' answers in the algebra test were correct, compared with 44% in the case of English students; 33% of Turkish students' answers in the trigonometry test were correct, compared with 18% in the case of English students. Turkish and English students' performance in the right-angled triangles test were similar, 66% of Turkish and 68% of English answers were correct. English students' performance in the trigonometry word problem test was considerably better than Turkish students' performance: 63% of English students' answers were correct, compared with 46% of Turkish students' answers. Despite these differences, the interviews and verbal protocols revealed a uniformity of approach, from both countries' students, to simplifying trigonometric expressions and answering trigonometry word problems. Document analysis and classroom observations revealed significant differences in the trigonometry curricula and the privileging of techniques, e.g. calculator methods in England and surd forms in Turkey. Discussion: The Discussion section focuses on three issues: the nature of trigonometry in the two countries, a model of students' manner of simplifying trigonometric expressions and students' methods of solving trigonometry word problems. With regard to the first focus, an analysis of similarities and differences in curricula, teaching approaches and the 'tools' students use suggests that trigonometry in England and Turkey are substantively distinct areas of mathematics. With regard to the second and third foci, models are developed which are, despite the radical differences in the 'trigonometry' in the two countries, independent of the nationality of the students

A Kinetic Model for Toluene Oxidation Comprising Benzylperoxy Benzoate Ester as Reactive Intermediate in the Formation of Benzaldehyde

During the oxidation of toluene under semibatch conditions, the formation of benzyl alcohol is initially equal to the rate of formation of benzaldehyde. As the overall conversion increases the benzyl alcohol concentration at first decreases much faster than benzaldehyde, but this decrease slows down causing the benzyl alcohol concentration to reduce to zero only very slowly. To account for this phenomenon a new reaction pathway has been proposed where the formation of benzaldehyde out of benzylhydroperoxide is catalysed by benzoic acid. Incorporation of this new reaction in a model improves the description of benzyl alcohol concentration prophiles while maintaining good predictions for benzaldehyde and benzoic acid