Things Important to the Creation of a Webpage

Some New Features of Maple 6

Experiments with Two-by-Two Matrices

• Show that trace of AB = trace of BA
• Show that (AB - BA)(AB - BA) is scalar
• Find all matrices commuting with .
• Show that A^-1BA can never be diagonal.
• Show that there is no real invertible matrix S with S^-1S triangular.

Spreadsheet in Maple

• Use the spreadsheet in Maple to experiment with the 3x + 1 problem.
• How is the number pattern related to the sucessive derivatives of tan(x) formed?
• How is the number pattern related to the sucessive derivatives of cot(x) formed?
• Express tan(nx) as a rational function of tan(x) for n = 1,2,3,4,5,6.
• Study the number pattern associated with the indefinite integral of the function xⁿe^x.
• List the Chebyshev polynomials T_n(x) for n = 0,1,2,3,4,5,6.

Factoring

• Find an exponent n such that some coefficients appearing in the factorization of xⁿ - 1 are different from 0, -1 and 1.

Reference: 吳振奎，數學中的美--數學美學初探，天津教育，(1996), p. 143.
• Factor (((x² - a)² - a)² - a)² - a - x.

Reference: Ramanujan's Notebooks IV, p. 42.