In this paper we investigate the topological structure of the Graev free topoiogical group over ;... more In this paper we investigate the topological structure of the Graev free topoiogical group over ;he rationals. We show that this free group fails to be a k-space and fails to carry the weak topology generalied by its subspaces of words of length less than or equal to n. As tools in this investigation we establish some properties of net convergence in free groups and also some propertics of certain canonical maps which are closely related to the topological structure of free groups. j A,M.S. (M(X) Subj. Class.: 22A99,20E05, 55D50,55620.
International Journal of Mathematical Education in Science and Technology, 2002
ABSTRACT Given three points in the plane, interest is in the locus of all points for which the su... more ABSTRACT Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y, and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given points, for which the constant is a minimum. This point is determined for isosceles triangles, the general case is discussed, although all details are not revealed. Student problems, such as supplying the missing details mentioned and further generalizations, are suggested.
International Journal of Mathematical Education in Science and Technology, 2002
ABSTRACT Harmonic oscillator equations of the form ÿ + 2y = h(t) where is a real constant and h(t... more ABSTRACT Harmonic oscillator equations of the form ÿ + 2y = h(t) where is a real constant and h(t) is a continuous, piecewise smooth, periodic ‘forcing’ function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving ‘term-by-term’ by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated ‘term-by-term’. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research.
International Journal of Mathematical Education in Science and Technology, Jul 1, 2003
ABSTRACT Non-linear second-order differential equations whose solutions are the elliptic function... more ABSTRACT Non-linear second-order differential equations whose solutions are the elliptic functions sn(t, k), cn(t, k) and dn(t, k) are investigated. Using Mathematica, high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions.
International Journal of Mathematical Education in Science and Technology, Jun 16, 2009
ABSTRACT We report on a study of the forced van der Pol equation by solving numerically the diffe... more ABSTRACT We report on a study of the forced van der Pol equation by solving numerically the differential equation for a variety of values of the parameters ϵ,F and ω. In doing so, many striking and interesting trajectories can be discovered and phenomena such as frequency entrainment, almost periodic solutions, space filling trajectories and seemingly chaotic behaviour are explored. These examples naturally give rise to computer laboratory problems suitable for student research and small group projects.
International Journal of Mathematical Education, Nov 10, 2010
ABSTRACT Harmonic oscillator equations of the form ÿ + 2y = h(t) where is a real constant and h(t... more ABSTRACT Harmonic oscillator equations of the form ÿ + 2y = h(t) where is a real constant and h(t) is a continuous, piecewise smooth, periodic ‘forcing’ function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving ‘term-by-term’ by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated ‘term-by-term’. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research.
Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the... more Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied:i. if h: UA → UB and g: VA → VB
In this paper we investigate the topological structure of the Graev free topoiogical group over ;... more In this paper we investigate the topological structure of the Graev free topoiogical group over ;he rationals. We show that this free group fails to be a k-space and fails to carry the weak topology generalied by its subspaces of words of length less than or equal to n. As tools in this investigation we establish some properties of net convergence in free groups and also some propertics of certain canonical maps which are closely related to the topological structure of free groups. j A,M.S. (M(X) Subj. Class.: 22A99,20E05, 55D50,55620.
International Journal of Mathematical Education in Science and Technology, 2002
ABSTRACT Given three points in the plane, interest is in the locus of all points for which the su... more ABSTRACT Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y, and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given points, for which the constant is a minimum. This point is determined for isosceles triangles, the general case is discussed, although all details are not revealed. Student problems, such as supplying the missing details mentioned and further generalizations, are suggested.
International Journal of Mathematical Education in Science and Technology, 2002
ABSTRACT Harmonic oscillator equations of the form ÿ + 2y = h(t) where is a real constant and h(t... more ABSTRACT Harmonic oscillator equations of the form ÿ + 2y = h(t) where is a real constant and h(t) is a continuous, piecewise smooth, periodic ‘forcing’ function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving ‘term-by-term’ by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated ‘term-by-term’. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research.
International Journal of Mathematical Education in Science and Technology, Jul 1, 2003
ABSTRACT Non-linear second-order differential equations whose solutions are the elliptic function... more ABSTRACT Non-linear second-order differential equations whose solutions are the elliptic functions sn(t, k), cn(t, k) and dn(t, k) are investigated. Using Mathematica, high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions.
International Journal of Mathematical Education in Science and Technology, Jun 16, 2009
ABSTRACT We report on a study of the forced van der Pol equation by solving numerically the diffe... more ABSTRACT We report on a study of the forced van der Pol equation by solving numerically the differential equation for a variety of values of the parameters ϵ,F and ω. In doing so, many striking and interesting trajectories can be discovered and phenomena such as frequency entrainment, almost periodic solutions, space filling trajectories and seemingly chaotic behaviour are explored. These examples naturally give rise to computer laboratory problems suitable for student research and small group projects.
International Journal of Mathematical Education, Nov 10, 2010
ABSTRACT Harmonic oscillator equations of the form ÿ + 2y = h(t) where is a real constant and h(t... more ABSTRACT Harmonic oscillator equations of the form ÿ + 2y = h(t) where is a real constant and h(t) is a continuous, piecewise smooth, periodic ‘forcing’ function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving ‘term-by-term’ by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated ‘term-by-term’. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research.
Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the... more Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied:i. if h: UA → UB and g: VA → VB
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