\input TR8TwoColFormat.tex \input GeoMacros \begfile{SamplePaper.tex of October 12, 1983 3:19 am --- Stolfi} % Sample paper to exorcise two-column TEX macros \inputtty % type "i \def\printfilestamp{F}" to supress filestamp on page 1 \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: \begitems \item $\leftv X\rightv\leq \sqrt{r}$; \item $\leftv X\rightv\geq \sqrt{r}$.} \proof{By M\um{\rm u}nchausen's theorem we have $$\eqalignno{ \nullX\inte\biglp\set{0,1}\uni\rset{x\relv x^2=x}\bigrp\cr \null\simil\nullX\inte\set{1,0}\cr \null\iso\null X\star\and X\pr\or X\prr\xor X\deg\ldotss,(1)\cr}$$ where $X\pr$ 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. \pp From the above equation we easily derive (in the Sh\hatac{\rm o}z\hatac{\rm o}d-Volterra symbolism) the following probabilistic identities: $$\eqalign{\null(\forall u\in \Rscr)(\exists v\in \Zscr^2)\, {X\prr}^{\bar \infty}\subset {X\deg}^{\empty}\cr\vsk4 \null\rmiff\null\quad \Pr X\implies \Pr \hat X\cr\vsk6 \null\rmif\null\quad \lnot\liminf \Zscr{\dollar\choose\atsign} \implies X\impliedby X\star\cr\vsk6 \null\rmand\null\quad X \neqv \inter{i=0}^{m^2} Xi \doubleimpl Q^{Q^Q}.\cr}$$ \pp This is clearly a contradiction, so the theorem follows.} 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, $$\varphi = \logM\lim{x\to\infty^2}\lg\ln\sin\csc\max\del,$$ where $$\del=\rset{x\relv x=\min{y\in X}\sup\inf\gcd (A, \dist(x,y))}$$ and $$A=\det\left[\matrix4{x127\cdotsM0\cr \lambda\Omega(ay)\Delta\cdots\sqrt{\pi}\cr \vdots\vdots\vdots\cr \doubleint f(\xi)\,d\infty\hbox{\bi Z}\cdots418\cr}\right]$$ 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 $Pt3^{3-}(Xe2)^+[Rh3(C6O6)2]9$} 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 / $\bullet$; 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 / $\bullet$}$" turns on a Paper-tape Punch for right-to-left alphanumeric floating-point transput ($\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 ``$\bullet$''.}. 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: \capfig3cm{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 {\small \showfontbox} {\titlefont \showfontbox} {\cmr \showfontbox} \par\vfill\eject\end