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AfterMath issue 5 is out

It’s now the end of Semester 2 of 2008, all examinations are over and done with, and issue 5 of AfterMath is out for your reading pleasure. You can grab the latest issue here, hot off the digital press.

This issue contains articles covering topics such as:

  • the Hairy Ball Theorem
  • a differential equation variant of Gronwall’s inequality
  • a survey on Maxima
  • using PARI/GP to study number theory and cryptography

Mark Ioppolo’s article, “The Hairy Ball Theorem,” contains a proof of a version of Henri Poincare’s conjecture popularly known as the Hairy Ball Theorem. Poincare was able to provide a proof [5] of the version he conjectured, and Brouwer [1] provides a proof of a generalization of Poincare’s version. The proofs in [1, 5] rely upon rather advanced techniques, whereas Milnor’s proof [4] uses a clever argument that relies on less sophisticated techniques. Ioppolo’s article aims to recount Milnor’s clever arguments.

The article “Differential Inequalities” by Wilson Ong considers Gronwall’s inequality, a result originally published in [3]. Gronwall’s inequality is usually stated as follows [2]: Let x, \Psi and \chi be real continuous functions defined in \,[a, b], with \chi (t) \geq 0 for t \in [a, b]. Suppose that on \,[a, b] we have the inequality

x(t) \leq \Psi(t) + \displaystyle{\int_a^t} \chi(s) x(s) \, ds


x(t) \leq \Psi(t) + \displaystyle{\int_a^t} \chi(s) \Psi(s) \exp \left[ \displaystyle{\int_s^t} \chi(u) \, du \right] \, ds

This inequality is clearly an inequality involving integrals. Ong’s exposition covers a variant of this, where the Gronwall type inequalities involve first and second order differential equations.

In issue 4 of AfterMath, there’s an article on using the computer algebra system (CAS) Maxima to study basic linear algebra. In this issue (issue 5), Alasdair McAndrew’s article “A Morsel of Maxima” presents an overview of this venerable CAS. McAndrew first provides a sketch of Maxima’s history, followed by an outline of Maxima’s state as of March 2007. The article then considers various interfaces to Maxima, including command line based and graphical user interfaces. The remainder of the article surveys features of Maxima that are useful in studying undergraduate mathematics: arithmetic, polynomial and rational functions, calculus, linear algebra, basic Maxima programming, solving equations, differential equations, plotting, and help and documentation.

And there’s the article “Number Theory and Cryptography using PARI/GP” written by yours truly. It introduces various PARI/GP commands that are useful for studying elementary number theory and undergraduate cryptography, in particular the RSA public key cryptosystem. But be warned that the exposition on RSA and cryptography is for educational purposes only, and readers who require more detailed discussions should consult specialized texts on the subject.

Issue 5 also contains a section on mathematics jokes, as well as a section containing some basic mathematical problems.


  1. L. E. J. Brouwer. Uber Abbildung von Mannigfaltigkeiten. Math. Ann., pages 97-115, 1912.
  2. S. S. Dragomir. Some Gronwall Type Inequalities and Applications. Nova Science Publishers, Hauppauge, New York, USA, 2003.
  3. T. H. Gronwall. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math., 20(2):293-296, 1919.
  4. J. Milnor. Analytic proofs of the “Hairy Ball Theorem” and the Brouwer Fixed Point Theorem. The American Mathematical Monthly, 85(7):521-524, 1978.
  5. H. Poincare. Sur les courbes definies par les equations differentielles. J. Math. Pures. Appl., 4(1):167-244, 1885.
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