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\bf
\centerline{The Theory of Self-Avoiding Walks is Only \%66.81 Mathematical
(According to the Appel Test)}
\medskip
\it
\centerline{Shalosh B. EKHAD\footnote{$^1$}
{\baselineskip=9pt
\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122.
{\baselineskip=9pt
\eighttt ekhad@math.temple.edu ; 
\break http://www.math.temple.edu/\Tilde ekhad. }
May 11, 1997.
}}
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\rm
In a recent tour-de-force,
Andrew Appel[A] proposed the soon-to-be-famous
{\it Appel Factor} for determining the mathematical content
of an area. In a given collection of papers submitted to
a conference in that area, let $a_k$ be the number of
$k$-author papers where the authors' names are listed
alphabetically, and $b_k$ be the number of papers where the
order is not alphabetical. He defined
$$
L(p):=\prod_{k} \left ( (1-p) + {{p} \over {k!}} \right )^{a_k}
\left ( p -{{p} \over {k!}} \right )^{b_k} {{a_k+b_k} \choose {a_k}} \quad,
$$
and defined $p_{max}$ to be the value of $p$ ($0 \leq p \leq 1$)
maximizing $L(p)$. His rationale was that,
mathematicians and other theoreticians, being intrinsically
nicer and less competitive than their applied and experimental
counterparts, have the habit of listing the authors' names
alphabetically, while scientists have a pecking order.
Hence if there were a fraction of $p$ scientists  and
$1-p$ mathematicians, then $L(p)$ is the probability of
the observed distribution. By maximizing $L(p)$, we find the
$p$ that was most likely to produce the observed data.
 
Like most great discoveries, the discoverer did not realize
the full scope of his invention. Appel only applied his
test to conference proceedings (in computer science).
However, the bibliography of any interdisciplinary book provides
ideal grist for the Appel mill. One such area is
that of `Self-Avoiding-Walks', and the references 
in the modern classic [MS] supply authentic data.
By direct counting we found
$a_2=76$, $b_2=19$; $a_3=22$, $b_3=1$; $a_4=1$, $b_4=4$.
Hence $p_{max}=.3319075633$. \halmos
 
P.S. A small Maple package, {\tt appel},
implementing the Appel test, is available from the author's website.
 
\bigskip
 
{\bf REFERENCES}
 
[A] Andrew W. Appel, {\it Is POPL Mathematics or Science?},
ACM SIGPLAN Notices {\bf 27}(4), 87-89, April 1992.
{\tt http://www.cs.princeton.edu/\Tilde appel/papers/science.ps}.
 
[MS] N. Madras and G. Slade, {\it `` The Self Avoiding Walk''},
Birkhauser, 1993.
 
\bye
 
 
 
 
 
 
