268 lines
9.4 KiB
Plaintext
268 lines
9.4 KiB
Plaintext
A Contribution to the Mathematical Theory of Big Game Hunting
|
|
=============================================================
|
|
|
|
Problem: To Catch a Lion in the Sahara Desert.
|
|
|
|
1. Mathematical Methods
|
|
|
|
1.1 The Hilbert (axiomatic) method
|
|
|
|
We place a locked cage onto a given point in the desert. After that
|
|
we introduce the following logical system:
|
|
Axiom 1: The set of lions in the Sahara is not empty.
|
|
Axiom 2: If there exists a lion in the Sahara, then there exists a
|
|
lion in the cage.
|
|
Procedure: If P is a theorem, and if the following is holds:
|
|
"P implies Q", then Q is a theorem.
|
|
Theorem 1: There exists a lion in the cage.
|
|
|
|
1.2 The geometrical inversion method
|
|
|
|
We place a spherical cage in the desert, enter it and lock it from
|
|
inside. We then performe an inversion with respect to the cage. Then
|
|
the lion is inside the cage, and we are outside.
|
|
|
|
1.3 The projective geometry method
|
|
|
|
Without loss of generality, we can view the desert as a plane surface.
|
|
We project the surface onto a line and afterwards the line onto an
|
|
interiour point of the cage. Thereby the lion is mapped onto that same
|
|
point.
|
|
|
|
1.4 The Bolzano-Weierstrass method
|
|
|
|
Divide the desert by a line running from north to south. The lion is
|
|
then either in the eastern or in the western part. Let's assume it is
|
|
in the eastern part. Divide this part by a line running from east to
|
|
west. The lion is either in the northern or in the southern part.
|
|
Let's assume it is in the northern part. We can continue this process
|
|
arbitrarily and thereby constructing with each step an increasingly
|
|
narrow fence around the selected area. The diameter of the chosen
|
|
partitions converges to zero so that the lion is caged into a fence of
|
|
arbitrarily small diameter.
|
|
|
|
1.5 The set theoretical method
|
|
|
|
We observe that the desert is a separable space. It therefore
|
|
contains an enumerable dense set of points which constitutes a
|
|
sequence with the lion as its limit. We silently approach the lion in
|
|
this sequence, carrying the proper equipment with us.
|
|
|
|
1.6 The Peano method
|
|
|
|
In the usual way construct a curve containing every point in the
|
|
desert. It has been proven [1] that such a curve can be traversed in
|
|
arbitrarily short time. Now we traverse the curve, carrying a spear,
|
|
in a time less than what it takes the lion to move a distance equal to
|
|
its own length.
|
|
|
|
1.7 A topological method
|
|
|
|
We observe that the lion possesses the topological gender of a torus.
|
|
We embed the desert in a four dimensional space. Then it is possible
|
|
to apply a deformation [2] of such a kind that the lion when returning
|
|
to the three dimensional space is all tied up in itself. It is then
|
|
completely helpless.
|
|
|
|
1.8 The Cauchy method
|
|
|
|
We examine a lion-valued function f(z). Be \zeta the cage. Consider
|
|
the integral
|
|
|
|
1 [ f(z)
|
|
------- I --------- dz
|
|
2 \pi i ] z - \zeta
|
|
|
|
C
|
|
|
|
where C represents the boundary of the desert. Its value is f(zeta),
|
|
i.e. there is a lion in the cage [3].
|
|
|
|
1.9 The Wiener-Tauber method
|
|
|
|
We obtain a tame lion, L_0, from the class L(-\infinity,\infinity),
|
|
whose fourier transform vanishes nowhere. We put this lion somewhere
|
|
in the desert. L_0 then converges toward our cage. According to the
|
|
general Wiener-Tauner theorem [4] every other lion L will converge
|
|
toward the same cage. (Alternatively we can approximate L arbitrarily
|
|
close by translating L_0 through the desert [5].)
|
|
|
|
2 Theoretical Physics Methods
|
|
|
|
2.1 The Dirac method
|
|
|
|
We assert that wild lions can ipso facto not be observed in the Sahara
|
|
desert. Therefore, if there are any lions at all in the desert, they
|
|
are tame. We leave catching a tame lion as an execise to the reader.
|
|
|
|
2.2 The Schroedinger method
|
|
|
|
At every instant there is a non-zero probability of the lion being in
|
|
the cage. Sit and wait.
|
|
|
|
2.3 The nuclear physics method
|
|
|
|
Insert a tame lion into the cage and apply a Majorana exchange
|
|
operator [6] on it and a wild lion.
|
|
|
|
As a variant let us assume that we would like to catch (for argument's
|
|
sake) a male lion. We insert a tame female lion into the cage and
|
|
apply the Heisenberg exchange operator [7], exchanging spins.
|
|
|
|
2.4 A relativistic method
|
|
|
|
All over the desert we distribute lion bait containing large amounts
|
|
of the companion star of Sirius. After enough of the bait has been
|
|
eaten we send a beam of light through the desert. This will curl
|
|
around the lion so it gets all confused and can be approached without
|
|
danger.
|
|
|
|
3 Experimental Physics Methods
|
|
|
|
3.1 The thermodynamics method
|
|
|
|
We construct a semi-permeable membrane which lets everything but lions
|
|
pass through. This we drag across the desert.
|
|
|
|
3.2 The atomic fission method
|
|
|
|
We irradiate the desert with slow neutrons. The lion becomes
|
|
radioactive and starts to disintegrate. Once the disintegration
|
|
process is progressed far enough the lion will be unable to resist.
|
|
|
|
3.3 The magneto-optical method
|
|
|
|
We plant a large, lense shaped field with cat mint (nepeta cataria)
|
|
such that its axis is parallel to the direction of the horizontal
|
|
component of the earth's magnetic field. We put the cage in one of the
|
|
field's foci. Throughout the desert we distribute large amounts of
|
|
magnetized spinach (spinacia oleracea) which has, as everybody knows,
|
|
a high iron content. The spinach is eaten by vegetarian desert
|
|
inhabitants which in turn are eaten by the lions. Afterwards the
|
|
lions are oriented parallel to the earth's magnetic field and the
|
|
resulting lion beam is focussed on the cage by the cat mint lense.
|
|
|
|
[1] After Hilbert, cf. E. W. Hobson, "The Theory of Functions of a Real
|
|
Variable and the Theory of Fourier's Series" (1927), vol. 1, pp 456-457
|
|
[2] H. Seifert and W. Threlfall, "Lehrbuch der Topologie" (1934), pp 2-3
|
|
[3] According to the Picard theorem (W. F. Osgood, Lehrbuch der
|
|
Funktionentheorie, vol 1 (1928), p 178) it is possible to catch every lion
|
|
except for at most one.
|
|
[4] N. Wiener, "The Fourier Integral and Certain of itsl Applications" (1933),
|
|
pp 73-74
|
|
[5] N. Wiener, ibid, p 89
|
|
[6] cf e.g. H. A. Bethe and R. F. Bacher, "Reviews of Modern Physics", 8
|
|
(1936), pp 82-229, esp. pp 106-107
|
|
[7] ibid
|
|
--
|
|
|
|
4 Contributions from Computer Science.
|
|
|
|
4.1 The search method
|
|
|
|
We assume that the lion is most likely to be found in the direction to
|
|
the north of the point where we are standing. Therefore the REAL
|
|
problem we have is that of speed, since we are only using a PC to
|
|
solve the problem.
|
|
|
|
4.2 The parallel search method.
|
|
|
|
By using parallelism we will be able to search in the direction to the
|
|
north much faster than earlier.
|
|
|
|
4.3 The Monte-Carlo method.
|
|
|
|
We pick a random number indexing the space we search. By excluding
|
|
neighboring points in the search, we can drastically reduce the number
|
|
of points we need to consider. The lion will according to probability
|
|
appear sooner or later.
|
|
|
|
4.4 The practical approach.
|
|
|
|
We see a rabbit very close to us. Since it is already dead, it is
|
|
particularly easy to catch. We therefore catch it and call it a lion.
|
|
|
|
4.5 The common language approach.
|
|
|
|
If only everyone used ADA/Common Lisp/Prolog, this problem would be
|
|
trivial to solve.
|
|
|
|
4.6 The standard approach.
|
|
|
|
We know what a Lion is from ISO 4711/X.123. Since CCITT have specified
|
|
a Lion to be a particular option of a cat we will have to wait for a
|
|
harmonized standard to appear. $20,000,000 have been funded for
|
|
initial investigastions into this standard development.
|
|
|
|
4.7 Linear search.
|
|
|
|
Stand in the top left hand corner of the Sahara Desert. Take one step
|
|
east. Repeat until you have found the lion, or you reach the right
|
|
hand edge. If you reach the right hand edge, take one step
|
|
southwards, and proceed towards the left hand edge. When you finally
|
|
reach the lion, put it the cage. If the lion should happen to eat you
|
|
before you manage to get it in the cage, press the reset button, and
|
|
try again.
|
|
|
|
4.8 The Dijkstra approach:
|
|
|
|
The way the problem reached me was: catch a wild lion in the Sahara
|
|
Desert. Another way of stating the problem is:
|
|
|
|
Axiom 1: Sahara elem deserts
|
|
Axiom 2: Lion elem Sahara
|
|
Axiom 3: NOT(Lion elem cage)
|
|
|
|
We observe the following invariant:
|
|
|
|
P1: C(L) v not(C(L))
|
|
|
|
where C(L) means: the value of "L" is in the cage.
|
|
|
|
Establishing C initially is trivially accomplished with the statement
|
|
|
|
;cage := {}
|
|
|
|
Note 0:
|
|
This is easily implemented by opening the door to the cage and shaking
|
|
out any lions that happen to be there initially.
|
|
(End of note 0.)
|
|
|
|
The obvious program structure is then:
|
|
|
|
;cage:={}
|
|
;do NOT (C(L)) ->
|
|
;"approach lion under invariance of P1"
|
|
;if P(L) ->
|
|
;"insert lion in cage"
|
|
[] not P(L) ->
|
|
;skip
|
|
;fi
|
|
;od
|
|
|
|
where P(L) means: the value of L is within arm's reach.
|
|
|
|
Note 1:
|
|
Axiom 2 esnures that the loop terminates.
|
|
(End of note 1.)
|
|
|
|
Exercise 0:
|
|
Refine the step "Approach lion under invariance of P1".
|
|
(End of exercise 0.)
|
|
|
|
Note 2:
|
|
The program is robust in the sense that it will lead to
|
|
abortion if the value of L is "lioness".
|
|
(End of note 2.)
|
|
|
|
Remark 0: This may be a new sense of the word "robust" for you.
|
|
(End of remark 0.)
|
|
|
|
Note 3:
|
|
|
|
From observation we can see that the above program leads to the
|
|
desired goal. It goes without saying that we therefore do not have to
|
|
run it.
|
|
(End of note 3.)
|
|
(End of approach.)
|