Conference rusure::math

437.0. "Big Game Hunting" by CORVUS::THALLER () Thu Jan 23 1986 10:47

	 A Contribution to the Mathematical Theory of Big Game Hunting

			H. Petard, Princeton, New Jersey

	This little known mathematical discipline has not, of recent years,
received in the literature the attention which, in our opinion, it deserves.
In the present paper we present some algorithms which, it is hoped, may be of
interest to other workers in the field.  Neglecting the more obviously trivia
methods, we shall confine our attention to those which involve significant
applications of ideas familiar to mathematicians and physicists. 
	The present time is particularly fitting for the preparation of an
account of the subject, since recent advances both in pure mathematics and in
theoretical physics have made available powerful tools whose very existence was
unsuspected by earlier investigators.  At the same time, some of the more
elegant classical methods acquire new significance in the light of modern
discoveries.  Like many other branches of knowledge to which mathematical
techniques have been applied in recent years, the Mathematical Theory of Big
Game Hunting has a singularly happy unifying effect on the most diverse
branches of the exact sciences. 
	For the sake of simplicity of statement, we shall confine our
attention to Lions (Felis Leo) whose habitat is the Sahara Desert.  The methods
which we shall enumerate will easily be seen to be applicable, with obvious
formal modification, to other carnivores and to other portions of the globe. 
The paper is divided into three parts, which draw their material respectively
from mathematics, theoretical physics, and experimental physics.
	The author desires to acknowledge his indebtness to the Trivia Club of
St. John's College, Cambridge, England; to the M.I.T. chapter of the Society
for Useless Research; to the F.o.I',' of Princeton University; and to numerous
individual contributors, know and unknown, conscious and unconscious. 

			1. Mathematical methods

	1. THE HILBERT, OR AXIOMIC, METHOD.  We placed a locked cage at a
given point of the desert.  We then introduce the following logical systems.
	AXIOM I. The class of lions in the Sahara Desert is non-void.
	AXIOM II. If there is a lion in the Sahara Desert, there is a lion
		  in the cage.
	RULE OF PROCEDURE. If p is a theorem and "p implies q" is a theorem,
			   then q is a theorem.
	THEOREM I. There is a lion in the cage.

	2. THE METHOD OF INVERSIVE GEOMETRY.  We place a *spherical* cage in
the desert, enter it, and lock it.  We perform an inversion with respect to the
cage.  The lion is then in the interior of the cage, and we are outside.

	3. THE METHOD OF PROJECTIVE GEOMETRY Without loss of generality, we
may regard the Sahara Desert as a plane.  Project the plane into a line, and
then project the line into an interior point of the cage.  The lion is projected
into the same point.

	4. THE BLZANO-WEIERSTRASS METHOD. Disect the desert by a line running
N-S.  The lion is either in the E portion or in the W portion; let us suppose
him to be in the W portion.  Bisect the portion by a line running E-W.  The lion
is either in the N portion or the S portion; let us suppose him to be in the N
portion.  We continue this process indefinitely, constructing a sufficiently
strong fence about the chosen portion at each step.  The diameter of the chosen
portions approaches zero, so that the lion is ultimately surrounded by a fence
of arbitrarily small perimeter.

	5. THE  "MENGENTHEORETISCH" METHOD.  We observe that the desert is a
separable space.  It therefore contains an innumerable dense set of points, from
which can be extracted a sequence having the lion as limit.  We then approach
the lion stealthily along this sequence, bearing with us suitable equipment.

	6. THE PEANG METHOD.  Construct, by standard methods, a continuous curve
passing through every point of the desert.  It has been remarked "that it is
possible to traverse such a curve in an arbitrarily short time.  Armed with a
spear, we traverse the curve in a time shorter than that in which a lion can
move his own length.

	7. A TOPOLOGICAL METHOD.  We observe that a lion has at least the
connectivity of the torus.  We transport the desert into four-space.  It is then
possible to carry out such a deformation that the lion can be returned to
three-space in a knotted condition.  He is then helpless.

	8. THE CAUCHY, OR FUNCTIONTHEORETICAL, METHOD.  We consider an analytic
lion-valued function f(z).  Let q be the cage.  Consider the integral

	
	       /
           1   |   f(z)
        ------ | ------- dz
        2(pi)i |  z - q
	       /C

where C is the boundary of the desert; its value is f(q), i.e. a lion in the
cage.

	9. THE WIENER TAUBERIAN METHOD.  We procure a tame lion, L0, of class
L(-oo, +oo), whose Fourier transform nowhere vanishes, and release it in the
desert.  L0 then converges to our cage.  By Wiener's General Tauberian
Theorem, any other lion, L (say), will then converge to the same cage.
Alternatively, we can approximate arbitrarily closely to L by transplanting
L0 about the desert.


			2. Methods from Theoretical Physics

	10. THE DIRAC METHOD.  We observe that wild lions are, ipso facto, not
observable in the Sahara Desert.  Consequently, if there are any lions in the
Sahara, the are tame.  The capture of a tame lion may be left as an exercise
for the reader.

	11. THE SCHRODINGER METHOD.  At any given moment there is a positive
probability that there is a lion in the cage.  Sit down and wait.

	12.  THE METHOD OF NUCLEAR PHYSICS.  Place a tame lion in the cage, and
apply a Majorana exchange operator between it and a wild lion.
   As a variant, let us suppose, to fix ideas, that we require a male lion.  We
place a tame lioness in the cage, and apply a Heisenberg exchange operator
which exchanges the spins.

	13. A RELATIVISTIC METHOD.  We distribute about the desert lion bait
containing large portion of the  Companion of Sirius.  When enough bait has been
taken, we project a beam of light across the desert.  This will bend right
around the lion, who will then become so dizzy that he can be approached
with impunity.

			3. Methods from Experimental Physics

	14. THE THERMODYNAMICAL METHOD.  We construct a semi-permeable membrane,
permeable to everything except lions, and sweep it across the desert.

	15. THE ATOM-SPLITTING METHOD.  We irradiate the desert with slow
neutrons.  The lion becomes radioactive, and a process of disintegration sets
in.  When the decay has proceeded sufficiently far, he will become incapable
of showing fight.

	16. THE MAGNETO-OPTICAL METHOD.  We plant a large lenticular bed of
catnip (Nepeta cataria), whose axis lies along the direction of the horizontal
component of the earth's magnetic field, and place a cage at one of its foci.
We distribute over the desert large quantities of magnetized spinach (Spinacia
oleracea), which , as is well known, has a high ferric content.  The spinach
is eaten by the herbivorous denizens of the desert, which are in turn eaten by
lions.  The lions are then oriented parallel to the earth's magnetic field,
and the resulting beam of lions is focused by the catnip upon the cage.