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THE  CORUNDUMINIUM
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MMATH   ( RESEARCH )

  
 

 

  
       The items here are things I have had a great curiosity about but did not find in the literature.   They are here so I can point them out to or request feedback from interested colleagues.  Because many manuscripts are symbolically complex and I do not have math typing software attached to this website, I may just scan the pages and thumbnail them here for all to enjoy and maybe manipulate for printing in readable format.  Go ahead and print what you want - no copyright restrictions on these pages!  (Latest update:  December 26, 2008.)  
     
        e  contains a paper, "Transcendental Meditations", which treats the existence of the classical limit defining e and the more modern "area" definition.  It makes the connection in the simplest way I know of, one which my calculus students can grasp visually and intuitively.   
     
       Integrals of Products are Products of Integrals deals with the classical student error of integrating products by multiplying antiderivatives of its factors (a source of anguish to any calculus instructor)!  It turns out there is a curious existence theorem that can be made to sound like this, and in fact there are constructive methods that produce numerous examples whose verification is an interesting and challenging excursion through the "legitimate" methods of the subject.  This talk has been presented at regional meetings of the MAA and TexMATYC.   
     
       Pythagoreana is a collection of results related to Pythagorean triplets, including tilings of rectangles and even a square with pairwise non-similar Pythagorean triangles.  A difficult related problem is generation of triples of Pythagorean tangents where the sum of two equals the third.  Families of solutions are discussed, but because they lie on a surface that is not (Mordell theoretically) simple, it is unlikely any exhaustive parametrization of solutions will be found.  
        
       Bertrand refers to a proof of "Bertrand's Theorem", a brilliant result of celestial mechanics.  Simply put, the premise is that periodicity of bounded orbits for the two body problem implies Newton's inverse-square gravitation law.  The Nineteenth Century proof offered by Bertrand lacked rigor; and indeed the statement had to be tightened up a little; but his methods seem to be repairable, and repairable using only mathematics known at the time the result was first proposed (one point has been shown using Lebesgue theory, and that came later; but I think even that can be conformed).  (Final manuscript to appear when done, presumably in 2009.  
     
        Sarkovskii refers to a result in chaos theory, regarding what periods periodic points of endomorphisms of the real line or an interval can have.  It was astonishing that the implication is not chaotic at all - in fact, it linearly orders the positive integers and partitions the set of endomorphisms corresponding to "cuts" in the ordering.  The papers here will show how the problem of computing periods is related to lengths of cycles in directed graphs and, once this is understood, the constructions make the reason for the "Sarkovskii ordering" clear.   A secondary paper examines a particular "chaotic" quadratic polynomial whose periodic points can be explicitly computed.   
     
       Minimal Polynomials for Trig Functions is now published in primitive form.  Click on Minpolys to access it.  It includes a familiar derivation of the minimal polynomials for cosines of angles rationally commensurate with π.   From these, we have derived minimal polynomials for all the other basic functions, but the manuscript is not yet complete.  Check this page occasionally for developments.