Other Research Areas in Mathematics

  "When the Trivial Is Nontrivial" by William Capecchi and me, submitted to the Pi Mu Epsilon Journal in May 2011 for final approval.  We investigate abelian groups whose only multiplication making them into rings is the trivial multiplication.  Click here for a pre-print.

     "Idempotents a la Mod," accepted as a Student Research Project for the College Mathematics Journal, to appear in 2012.  This article leads students through the investigation of how many idempotents there are in the integers (mod n).  An idempotent is an element x such that x*x = x.  Click here for a pre-print.

     "Sublimital Analysis," Mathematics Magazine, vol. 81 # 5 (December 2008), 369 - 373.  It shows the possible sets of limits of subsequences of a sequence of real numbers match the closed sets.  This can be generalized to separable metric spaces.  Click here for a pre-print.

    “Deconstructing bases: fair, fitting, and fast bases,” Mathematics Magazine, vol. 76 #5, (December 2003), 380-385.  This article generalizes the idea of a base using elementary infinite series.  A variable base is a sequence {bn} of positive integers such that bn divides bn+1.  For example, in ordinary base 10 each bn is 10^n.  Allowing the ratio of increase rn+1 = bn+1/bn to vary provides many more interesting possibilities.  Then 0.a1a2a3...{bn} is just the sum of the series an/bn from n = 1 to infinity.  We need to limit the an by 0 < an < rn - 1.  For example, if bn = n!, all rational numbers have finite representations and e = 2.0111111..., a more memorable value than the base 10 value of 2.71828182846...   JSTOR Link:

  "Rhombic Penrose Tilings Can Be 3-Colored", is a joint paper with Stan Wagon, in the American Mathematical Monthly 107 (March 2000) 251 - 253.   We show that any tiling of the plane made of parallelograms meeting edge-to-edge can always be 3-colored.  In particular, Penrose tilings made of rhombi can be 3-colored. JSTOR link:

   "Coloring space tilings of generalized parallelograms," appeared in Geombinatorics 10 (2000) #2, 75-76.  This short article gives a conjecture generalizing the 3-coloring result of the Monthly article above.  However, the conjecture that any analogue tiling in n-dimensions needs at most n+1 colors was disproven for three dimensions. 

    "The possibility of impossible pyramids," Mathematics Magazine 73 (June 2000) #3, 185 - 193 discusses questions concerning the existence of pyramids having given edge lengths.  It first considers the Euclidean case, which has "impossible pyramids."  For example, suppose we have six sticks, five of which are of length 8 and the last of length 14.  Any three of these can make a triangle, but there is no way to build a pyramid.  The article then considers pyramids in the taxicab metric and, more generally, in the p-metrics, for 1 < p and the infinity metric.  JSTOR Link:

    I wrote two expository papers that were published in the Saint John's Faculty Journal, Symposium.  Click here for the text of the historical essay, "Changing modes of thought: Non-Euclidean geometry and the liberal arts," published in 1989.  Click here for the text of the second, "Beauty Bare," which seeks to explain the nature of mathematical research to a general audience through examples from my own research.  It was published in 1993.  (Unfortunately, the pdf files have the text sideways.)

    I published “A fractal is,” which qualifies as a poem, in The Mathematical Intelligencer, vol. 20 #2 (Spring, 1998), 22.  Click here for a copy.

Mathematical Biology

     I am a novice in this area, but I taught a course on it in the fall of 2000.  My interests are right now centered around population dynamics.  Student projects during and after the class include wolf meta-populations, positive assortative mating, the dynamics of the trp operator (which regulates the production of tryptophan), counting formulas for measuring the efficiency of wasp nest construction, combining logistic growth with a Leslie matrix model, and modeling mosquito populations.
     Jennifer Klein and I published "Taking the sting out of wasp nests: a dialogue on modeling in mathematical biology" in College Mathematics Journal, vol. 34 # 3 (May 2003), 207 - 215.
    JSTOR Link:

    Several students since that class have done summer research modeling competing populations of mosquitoes, the malarial parasite they host and other populations.

Dissertation Topic

    My dissertation investigated finitely additive probability measures.  I haven't worked in this area for over 25 years.  My only publication in this area is "Ultrafilter Limits and Finitely Additive Probability," published in Proceedings of the American Mathematical Society, Vol. 84, No. 4 (Apr., 1982), pp. 560-562.  JSTOR Link: