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%Theory that Obviates Mathematical Analysis
%a Plain TeX file of an article D. Zeilberger
 
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\bf
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{Mathematical Genitalysis: A Powerful New Combinatorial
Theory that Obviates Mathematical Analysis}
\rm
\bigskip
\centerline{ {\it Doron ZEILBERGER}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
%\break
{\eighttt zeilberg@math.temple.edu} \hfill \break
{\eighttt http://www.math.temple.edu/\Tilde zeilberg   .}
April 1, 1998.  Exclusive to the author's website.
} 
}
 
{\it  \qquad\qquad\qquad\qquad\qquad\qquad
  Dedicated to my beloved gurus, Paul Halmos and
Steven Krantz
 }
 
{\bf Abstract:} A Powerful new combinatorial theory,
superseding, and sometimes trivializing, mathematical analysis,
is introduced. It is illustrated by an exact determination
of Bloch's constant, a generalization of Kaczynski's
boundary functions from one 
to several complex variables, and a generalization of
Petryshyn's theorem about the variational solvability
of elliptic boundary value problems at resonance from
the quasi-linear case to the fully non-linear case.
 
 
{\bf Introduction} 
 
With the possible
exception of topology (formerly called
{\it analysis situs}),
mathematical analysis is the deepest and most technically
challenging area of mathematics. It is also wrought with
many logical and foundational problems. In this
{\it research report}, I will outline a new
{\it combinatorial} theory, that I have christened
{\it genitalysis}, that promises to simplify, and in
many cases {\it trivialize}, most, and potentially all,
of the current open problems in analysis.
 
Like many novel ideas, the central idea of genitalysis is
embarrassingly simple. It is `symbolic discretization',
which is roughly the `symbolic' analog of `numerical discretization',
that has been used in applied and computational mathematics
at least since Euler, but that has become a powerful and
mature theory only in the last thirty years, spurred, of course,
by the amazing {\it number crunching} capabilities of
modern electronic computers.
 
The idea behind numerical discretization is to approximate
a continuous problem by its discrete analog, on a
{\it grid} of a small spacing $h$. More efficient
approximations can be achieved by {\it multigrid} methods.
 
The $20^{th}$ century was the century of {\it numerical}
computations, and Fortran, and later Pascal, C, C++, and
Java ruled. The $21^{th}$ century, by contrast, is going
to be the century of {\it symbolic} computations, and
Maple, Mathematica, Macsyma, and their likes will conquer
the mathematical world. The idea behind
`symbolic discretization' is to solve {\it symbolically},
and hence {\it rigorously}, the discrete analog, and
at the very end let the mesh-size(s) $h$, tend to zero.
The issue of convergence, in all the examples that
we have encountered so far, is routine, because of
the explicit nature of the symbolic solution.
It would be very interesting to have an {\it a priori}
guarantee of convergence of our method.
 
Obviously, by the law of conservation of difficulty,
the `trivialization' comes at a price.
Our method requires extensive computer
resources, as well as efficient software engineering.
A very preliminary {\it beta version}, a Maple package
called {\tt DIBRA6}, will soon be available from
the author's website.
 
The exact forms of the symbolic expressions at specific
cases are extremely complicated (to a human), but also
irrelevant, since the passage to the limit $h \rightarrow 0$
is performed {\it silently} and completely {\it automatically}.
 
But enough of generalities. Let's go to business.
My choice of examples may seem idiosyncratic, and it is
certainly subjective: these are the examples that we
liked the best! We were also motivated by the fame of the
two complex analysts and the one non-linear mathematician
with whom these examples are associated.
 
 
{\bf The Exact Value of Bloch's Constant}
 
Andr\'e Bloch(1893-1948) is one of the most eminent complex 
analysts who has ever lived. In 1924, he showed that a covering
surface over the $w$-plane obtained from a mapping
$w=F(z)=z+ \dots$ that is one-to-one, conformal, and holomorphic
in $|z|<1$ always contains a {\it univalent} (schlicht) disk whose
radius $B$ is a positive number independent of $F$. The
supremum of such constants is called {\it Bloch's constant}, ${\cal B}$.
It was a major open problem to determine its exact value.
Previously only lower and upper bounds were known. Using
the present method, we determined that 
$$
{\cal B}= {{3^{1/4} \pi^{1/4} \Gamma(1/3)^{1/2}\Gamma(11/12)^{1/4}}
\over {2^{7/8} \Gamma(1/4)^{1/2} \Gamma(1/12)^{1/4} } }\quad.
$$
 
{\bf  Generalization of Kaczynski's Theorem to $C^{n}$}
 
Another celebrated complex-analyst is T.J. Kaczynski(1942-).
Bagemihl and Piranian gave an example of a harmonic function
having a boundary function defined on $|z|=1$ that is not
of Baire class $0$ or $1$, and they asked whether there exists
a {\it bounded} harmonic function
having a boundary function defined on $|z|=1$ that is not
of Baire class $0$ or $1$. Kaczynski[K] gave a brilliant
affirmative answer by means of an intricate and ingenious  construction.
Using his ideas, and symbolic discretization, 
and to Kaczynski's (possible) dismay, 100 hours of CPU time on a Cray 3,
we were able to extend
his construction to several complex variables.
 
 
{\bf Petryshyn's Theory of Generalized Topological Degree
Generalized from Semilinear to Nonlinear Equations}
 
Equally brilliant, but perhaps not as widely known, is
the non-linear Rutgers analysts W.V. Pertryshyn(1929-).
In a series of brilliant papers, that culminated in
the monograph [P], that appeared in the very prestigious
Tracts series of Cambridge University Press, Petryshyn
developed a theory of topological degree for semilinear
equations. Using our new method, and the package DIBRA6,
we were able to extend his results from {\it semilinear}
equations to (fully) {\it nonlinear} equations.
 
\eject
{\bf Conclusion}
 
We hope to be able, in the future, to present many more
examples. Meanwhile, let us note that the present
approach, being essentially {\it empirical}, lends
credence to the extremely influential and very important
{\it Calculus Reform Movement} and {\it NCTM standards}.
Of course, the proofs are completely rigorous, but
{\it a posteriori} trivial and uninteresting. The
crucial part is {\it finding} the proofs, and this
is done empirically. It is regrettable that some otherwise
very wise people cling so desperately and 
fanatically to the sinking ship of
{\it proof}.
 
{\bf References}
 
[K] T.J. Kaczynski, {\it Boundary functions and sets
of curvilinear convergence for continuous functions},
Trans. Amer. Math. Soc. {\bf 141}(1969), 107-125.
 
[P] W.V. Petryshyn,{\it ``Generalized Topological Degree
and Semilinear Equations''}, Cambridge Tract in Mathematics
{\bf 117}, Cambridge University Press, 1995.
 
 
 
 
 
 
\bye
 
 
 
 
