Well-posedness of one-way wave equations and absorbing boundary conditions

A one-way wave equation is a partial differential which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. The construction of such equations can be reduced to the approximation of the square root of (1-s sup 2) on -1, 1 by a rational function r(s) = p sub m (s)/q sub n(s). Those rational functions r for which the corresponding one-way wave equation is well-posed are characterized both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if r(s) interpolates the square root of (1-s sup 2) at sufficiently many points in (-1,1), then well-posedness is assured. It follows that absorbing boundary conditions based on Pade approximation are well-posed if and only if (m, n) lies in one of two distinct diagonals in the Pade table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived from Chebyshev or least-squares approximation

Finite cyclicity of graphics through a nilpotent singularity of elliptic or saddle type

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

Splines with Free Knots, the Heat Equation, and the Gauß Transform

The Gauß (or Weierstraß) transform has applications in many fields of applied mathematics. One of its most important properties within approximation theory is the fact that it maps weak Chebychev spaces onto Chebychev spaces. The aim of this paper is twofold. First, after proving some elementary invariance properties of the Gauß transform, necessary and sufficient conditions for best approximation by (Gauß transformed) free knot spline spaces are given. Then, in Section 3, we develop a method for the numerical solution of an initial value problem for the heat equation. The present paper can be viewed as a continuation of two recent publications by Meinardus [5,6]

A study in higher education calculus and students' learning styles

This research is devoted to focussing on the influence of different learning style on the performance of undergraduate students in various parts of calculus. In carrying out the study, calculus materials were classified into four main categories (Z4,Z5,Z6,Cals) and, for the Iranian students, the results of their mathematical performance in the university entrance examination is labelled (En) to identify their grounding in high school mathematics at the beginning of the calculus course in higher education. Also, in the present study, students' performance (weakness) in the manipulation of mathematical notation and logical discussion is called (Z1) category and (Cal) indicates students' total achievement in calculus examination which is, in fact, the students' performance on the combination of the categories (Z4,Z5,Z6). These calculus categories are described in Chapter 5. However in short term, multi-conceptual and procedural tasks are classified as (Z4). The (Z5) category is defined as the translation processes between mathematical abstraction (analytic/symbolic) and (pictorial/visual) forms of calculus materials. Moreover, multi-skilled, transferable and procedural skills are labelled as (Z6) category. It should be noted that these categories are interrelated in a scheme to exhibit activities in calculus. 572 students participated in the experimental part of this study and were selected from two Iranian universities (Sabzvar University and Mashhad University) and Glasgow University in Scotland, U.K. During the period of the study, the samples of students were subjected to some psychological tests in order to assign their Field-dependent/Field-independent and Convergent/Divergent learning styles. It was found throughout the study that the most effective combination of learning styles which emerged from the interacting picture of all the psychological factors used in the research, were field-independent/convergent (F1+Con) in Iran, and field-independent/divergent (FI+Div) in Scotland in performing on the calculus. On the other hand, the combination of field-dependent and convergent styles (FD+Con) could lessen achievement in calculus by mathematics/physics students, and field-dependent and divergent styles (FD+Div) would lessen attainment in calculus by engineering students. In addition, when the mean scores in calculus categories were calculated for various groups of students with different learning styles, the convergent thinkers (Con) were found to be best in (Z6), while divergent thinkers (Div) exhibited higher performance in (Z5). These findings demonstrate that the Con/Div way of thinking is the most effective in influencing performance in different areas of calculus, the FI/FD factor takes the second position. All these findings have been combined to form a model which emerges at the end of this thesis. Moreover, in Chapters 3 and 4, a comparison is made between calculus in secondary (high school) and higher education in Iran and Scotland, focussing on content, teaching order, learning objectives and teaching methods

Nice Polynomials

We consider the problem of finding, constructing, and classifying nice polynomials. After a short history of previous results, we present a general property of nice polynomials which leads to an important modification of the concept of equivalence classes of nice polynomials. We give several important results on nice symmetric or antisymmetric polynomials with an odd number of roots, which dramatically increase the speed of a computer search for examples. We present complete solutions to the symmetric three root case, the general three root case, and the symmetric four root case. We also give the relations between the roots and critical points for the general four root case and the symmetric five, six, and seven root cases. Using the relations for the general three and four root cases, we state, without proof, the suggested pattern for the relations for the general N root case. We present several important examples antisymmetric polynomials with five distinct roots and the first known examples of nice polynomials with six distinct roots. To conclude our study, we present several open problems and new conjectures suggested by our results, examples, and computer searches

On the Galois group of the modular equation

This thesis looks at a method of generating infinitely many extensions of the rationals with Galois group PGL(_2)Z(_n)). Firstly, the Galois group of the modular equation over Q(j) is shown to be PGL(_2)(Z(_n) by considering the n-th division points on an elliptic curve. Then, using Hilbert's Irreducibility Theorem and work discussed by Lang, we show that there are infinitely many rational values of such that this Galois group does not reduce in size. Finally, an equation whose roots generate the same extension as the modular equation but which has much smaller coefficients is found, based on work by Cohn