\filename{samplepaper.tex}
\edited{last by stolfi on Fri Mar  6 21:36:04 1987}

%  Sample paper to exorcise two-column TEX macros

\input /udir/stolfi/tex/geomacros

\pagetitlestuff{
  \title{A General Algorithm for the Solution of a Classical Problem\cp
    of Computer Science in {\:F O({\:C log n})} Best-Case\cp
    Asymptotic Time}
  \titlerule
  \author{B. C. Dull}
  \institution{Stanford University}
  \titlerule
  \vskip 20pt
 }

\columntitlestuff{
  \abstract{In this paper we make some general
    remarks concerning an algorithm for solving a  problem related
    to an important application of a well-known theoretical concept.\lp
    We also show that some seemingly unrelated mathematical problems
    have no appreciable relation to the topic of this paper.}

  \keywords{Garbage, gibberish, trash, irrelevant, inane, nonsense.}
   }

\support{The work of B. C. Dull, who is on leave from Sing Sing University,
N.J., was unwillingly supported by hundreds of old ladies, suburban residents,
and poorly guarded bank agencies scattered throughout the country.}

\botinsert{\vbox to 1.2 truein{}} % For ACM copyright notice, etc

% reference macros

\def\Boo{1}
\def\Gna{2}
\def\Foo{3}
\def\Baz{4}
\def\Sig{5}

\section{1. Basic concepts} % note: 1 argument only!

This section is better omitted on first reading. Readers not acquainted with
the language of relativistic catastrophe theory should read section 5 first,
then read sections 7, 2, 3, and 1.5, then again section 2, and finally the
paper's title.

\subsection{1.1. Piffles}

\definition{1.1}{A {\it piffle} is a sub-rational normalized quasi-convergent
recursively bi-infinite partial boffle that does not satisfy the strict
polynomial $Q$-property except on a semi-denumerably transfinite multiset of
non-monostrofic algebraically differentiable singularities of the first order.}

The following conjecture is due to Boojum [\Boo]:

\conjecture{1.1}{There exists at least a finite number of piffles.}

The purpose of this paper is to give a new proof of the following theorem,
which seems to be closely related to conjecture 1.1:

\theorem{1.2 {\rm (Garibaldi, 1873)}}{Let $n$ be an integer between $0$
(exclusive) and $1$ (inclusive). Let $m=n^2$, $p=1/n$, and $r=2-p$.  Let $X$ be
the set of all solutions $x$ of the equation $16x + 9m = (1+2p)^2$. Then:

\begindent
\item{} $\abs{ X}\leq \sqrt{r}$;
\item{} $\abs{ X}\geq \sqrt{r}$.}

\beginproof
{By M\um{\rm u}nchausen's theorem we have
$$\eqalignno{
  \null&X\inte\biglp\set{0,1}\uni\rset{x\relv x^2=x}\bigrp\cr
  \null\simil\null&X\inte\set{1,0}\cr
  \null\iso\null  &X\star\and X'\or X\prr\xor X\deg\ldotss,&(1)\cr}$$
where $X'$ is the Obvian kernel of $X$, that is, the set of all occurences of
the integer $7$ in the Fr\acute{\rm e}ch\acute{\rm e}t-Lobachevski sequence
$$\la X\sp X^2\sp X^{44}\sp X^7\sp X^{22/5}\\cdots\\ra,$$
and $X\prr$ is its contra-revolutional pseudo-inverse.

From the above equation we easily derive (in the Sh\hatac{\rm o}z\hatac{\rm
o}d-Volterra symbolism) the following probabilistic identities:
$$\eqalign{\null&(\forall u\in {\cal R})(\exists v\in {\cal Z}^2)\,
      {X\prr}^{\bar \infty}\subset {X\deg}^{\empty}\cr\vsk4
  \null\rmiff\null&\quad \Pr X\implies \Pr \hat X\cr\vsk6
  \null\rmif\null&\quad \lnot\liminf {\cal Z}{\dollar\choose\atsign}
      \implies X\impliedby X\star\cr\vsk6
  \null\rmand\null&\quad X \neqv \inter_{i=0}^{m^2} X_i \doubleimpl
  Q^{Q^Q}.\cr}$$

This is clearly a contradiction, so the theorem follows.}
\endproof

It is interesting to note that expression (1) has a limit (under
quasi-stationary conditions) that oscillates antipodally in a vanishing
neighborhood of Rademacher's characteristic quaternion,
$$\varphi = \log_M\lim_{x\to\infty^2}\lg\ln\sin\csc\max\del,$$
where
$$\del=\rset{x\relv x=\min_{y\in X}\sup\inf\gcd (A, \dist(x,y))}$$
and
$$A=\det\left[\matrix{x&127&\cdots&M_0\cr
\lambda_\Omega(ay)&\Delta&\cdots&\sqrt{\pi}\cr
\vdots&\vdots&&\vdots\cr
\doubleint f(\xi)\,d\infty&\hbox{\bi Z}&\cdots&418\cr}\right]$$
is the Eisenstein-Frankenstein screw-symmetric scalar tensor.

\section{2. Applications}

\subsection{2.1. Puffle diagonalization}

Equation (1) leads directly to an algorithm for the diagonalization of puffles
\foot\dag{A puffle is a tri-dimensional Hermite tableau over the field of
Fourier-integrable definitely transcendental piffle-valued polynomials.}, which
runs in $O(\infty)$ time for an input of size $n$. The version below uses
Paracelsius's alpha-beta deterministic backtracking method to compute the dual
isomorphism semigroup of the discrete manifold associated with the set $X$:

\algorithm{1}{Diagonalization of puffles.}

\algbody{
  \comm{The input is a puffle $P$ with non-zero antisymmetric permanent.
    The output is the diagonal form of $P$, encoded as a string of at
    most $n\choose{\lfloor{{\sqrt n}\over n}\rfloor}$ zeros.}
  \step0{Set $Q\gets \empty$, $\mu\gets\infty$, $Z\gets P$.}
  \step1{\sna{Main loop} While $Q$ is
      not diagonal of order $\mu^3$, repeat} % DON'T FORGET THIS CLOSE BRACE!
    \begblock
      \step2{If $\mu>0$, then} % DON'T FORGET THIS CLOSE BRACE!
      \begblock
        \step3{Set $\mu\gets\mu+1$.}
        \step4{Set $\mu\gets\mu-1$.}
      \endblock
      \unstep{Else}
      \begblock
        \step5{Set $\mu\gets\mu+0$, and replace $Q$
           by $\mathop{\hbox{\fx Scramble}}(Q)$, where
           $\hbox{\fx Scramble}$ is Gnat's amphisbaenic
           truncation procedure [\Gna].}
      \endblock
      \step6{Set $\mu\gets\mu^2/\mu$, $Q\gets Q\uni Q$.}
  \endblock
} % end of \algbody

\subsection{2.2. Heterocyclic racemization}

An important practical application of algorithm 1.1. is in computing the
polysorbency coefficients for the heterocyclic racemization of autotrophic and
endochronic oligosaccharides in the Kratz-Arrhenius reaction for the
homogeneous organometalloid-mediated inversion of aliphatic aril-arenes [\Foo,
\Baz]. The structural formula of a typical aril-dixenonium complex is given by
figure [{\bi 1}]. The coordination orbital resonances that stabilize the
vicinal dixenonium proto-cation, and all weakly-adsorbed hidrazide ligands at
meta- and ortho- hydrophilic sites have been omitted for clarity.

\numfig3cm{[{\bi 1}]}

The molecular structure was determined by grazing-in\-cid\-ence $X$-ray
transmutation of an amorphous crystallite of {\chem
$Pt_3^{3-}(Xe_2)^+[Rh_3(C_6O_6)_2]_9$} heated by the neutrino beam of a
Cusinart high-performance laser-driven free-electron synchro-cyclotron, in a
thermally stabilized atmosphere of pressurized vacuum.

The program below is an implementation of algorithm 1.1 in Rascal 5.7.81a, a
partially portable subset of the extended UCDC dialect of Pascal-68-IV. It
exploits the language extensions incorporated in the September 1981 Rascal
5.7.81a compiler (sub-optimizing version) for the BIGMAC-205 ternary array
processor under development by the W. C. Fields Laboratory, Little Dead Rock,
MO.).

{\prog
BEGIN
  var
    PP: Puffle,
    CC: ARRAY [1..n] OF QUATERNION;
  read (PP);
  CC$\up$2 \pgets \txdollar\txpound PP / \bull;
  write (CC[*])
END.
\endprog}

The only statement of the program above that requires some explanation is
$\prg{write (CC[*])}$\foot\dag{The command "$\prg{CC$\up$2 \pgets\
\txdollar\txpound PP / \bull}$" turns on a Paper-tape Punch for
right-to-left alphanumeric floating-point transput
($\prg{\pgets\txdollar\txpound PP}$), and establishes a Common Carrier
Communications Circuit $\prg{CC$\up$2}$) to the South Milwaukee Black Hole Data
Bank, which in the Rascal 5.7.81a environment is denoted by the symbol
``\bull''.}. This statement creates a two-sided stack of re-entrant
read-on-write-only semi-collectible unordered local references to a
pseudo-paged free-allocation virtually sequential zone, which is downloaded to
the main secondary coroutine space and mapped by the subsegment relocatable
descriptor reference table.

A straightforward reverse transposition of the unreachable linkage paths from
this table is then conditionally performed on the unmodified entries, prior to
their automatic output (in breadth-first top-down order) into the backup
front-end left-to-right pre-processor. The resulting overlayable interface is
then swapped in-place with the remote semaphore queue from the low-level I/O
macro-expander, and any duplicate page faults are combined into an interrupt
register from which they are eventually cleared by the background reformatting
protocol. See figure 12 below:

\fig 3cm{}{Figure 12. The virtual processor table and its associated coroutine
chains.}

\section{3. Conclusions}

We have shown that the problem of puffle diagonalization is hard with
non-negative probability. It is not clear whether any restricted subproblem
would be asymptotically simple enough to warrant further efforts.  We were able
to illustrate our main result with a couple of examples, some of which we
believe may be valid and/or pertinent. This result has many applications in
several branches of pure and applied computer science, notably the theory of
computation, compilers, VLSI design, personal workstations, finite element
analysis, performance evaluation, machine language debugging, font generation,
voice processing and encoding, database theory, cryptography, distributed
network architecture, weather forecasting, garbage collection algorithms,
computer tomography, symbolic integration, statistics, missile guidance
systems, robotics, computer education, and software engineering.

The theory of woffles (unnormalized piffles with contravariant retractions)
seems to be at least as rich and powerful as that of standard piffles, but its
development requires more sophisticated machinery than the one used in this
paper. It seems likely that the decidability of Green's conjecture on the
transcendentality of the zero-order Hermitian disintegrable continuum may be
non-trivial in woffle theory; in fact, we have almost completed a partial proof
of this assertion (that holds for some recursively denumerable subset of the
cases). This and other related developments will be reported
elsewhere\foot\ddag{We have been informed that Z. Glorfindel from the
University of Northern Greenland has independently found an infinite family of
counterexamples to theorem 1.2. As was pointed out by an anonymous referee,
this may explain the apparent contradictions in theorems 3.5, 3.7, 2.1, and
possibly also the anomalous behavior of algorithm 1 when applied to non-zero
inputs.}.

\section{5. References}

\ref [\Boo] {\refauth{Boojum, C. D.}
  \reftit{Salting jubjubs in glue.}
  \refpub{J. of Beavers and Butchers V. 1 N. 357 (1999).}
  \refrem{Charming; reminds me of smiles and soap.}}

\ref [\Gna] {\refauth{Gnat, Z. Z.}
  \reftit{Ueber die
  Unter\-vorschrieben\-vertragen\-rechnung\-gebildungs\-theorie von
  Differentialen und Integralen P\um{\it u}ffeln.}
  \refpub{Bull. Latvian Acad. Soc. Sci., V. 103 N. 15 (Sept. 1983), pp.
  935--1336.}
  \refrem{One of the best expositions of Puffle Theory for the layman.
  Sections 1 thru 65 and 67 thru 139 are particulary straightforward, and
  require only an elementary knowledge of the classical Green-Rabinovich formal
  calculus of non-linear integro-diffrential functors in bifurcated Riemann
  space.\rp
Sections 66 and 140 are devoted to the analysis of infinitesimally perturbed
puffles by means of Gaussian expansions of the underlying semialgebra of
discrete polycyclic eigenvalues. These sections are of somewhat specialized
interest and can be omitted on first reading.}}

\ref [\Sig] {\refauth{Sam-Wang I. Vorgott.}
  \reftit{A note on Piffles.}
  \refpub{Sigact Bulletin, who knows which issue.}
  \refrem{THE classic.}}

\vfill\eject

\section{Appendix A: Font tests}

\showfontbox

{\smallsize \showfontbox}

{\titlefont \showfontbox}

{\cmr \showfontbox}

\par\vfill\eject\end

