Article 343 of eunet.jokes:
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Path: puukko!santra!tut!enea!mcvax!cernvax!ethz!heiser
From: heiser@ethz.UUCP (Gernot Heiser)
Newsgroups: rec.humor,sci.math,eunet.jokes
Subject: Re: Math Jokes
Message-ID: <464@ethz.UUCP>
Date: 4 Jun 88 12:08:44 GMT
References: <3440@pasteur.Berkeley.Edu> <2932@phoenix.Princeton.EDU> <1155@bentley.UUCP> <1156@bentley.UUCP> <546@osupyr.mast.ohio-state.edu> <583@picuxa.UUCP>
Reply-To: heiser@iis.UUCP (Gernot Heiser)
Organization: ETH Zuerich, Switzerland
Lines: 184
The following is from a book whose title I don't recall. The book is in German
but the article is actually a translation from the original by H. Petard which
appared in the American Monthly 54, 466 (1938). Unfortunately our library is
lacking some years of this journal around WW 2, so I had to re-translate the
stuff into English. (That will make you people share the experience of reading
German translations of books on Einstein which also usually re-translate
Einstein's words :-) ).
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. Lets 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. Lets 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 diintegrate. 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
--
Gernot Heiser Phone: +41 1/256 23 48
Integrated Systems Laboratory CSNET/ARPA: heiser%ifi.ethz.ch@relay.cs.net
ETH Zuerich EARN/BITNET: GRIDFILE@CZHETH5A
CH-8092 Zuerich, Switzerland EUNET/UUCP: {uunet,...}!mcvax!ethz!iis!heiser
Article 426 of eunet.jokes:
Path: puukko!santra!tut!enea!mcvax!steven
From: steven@cwi.nl (Steven Pemberton)
Newsgroups: eunet.jokes
Subject: Re: Catching a Lion with computer science
Message-ID: <388@piring.cwi.nl>
Date: 6 Jul 88 11:24:18 GMT
References: <2024@sics.se>
Reply-To: steven@cwi.nl (mcvax!steven.uucp)
Distribution: eunet
Organization: CWI, Amsterdam
Lines: 66
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.
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 ensures 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.)
Steven Pemberton, CWI, Amsterdam; steven@cwi.nl
Article 426 of eunet.jokes:
Path: puukko!santra!tut!enea!mcvax!steven
From: steven@cwi.nl (Steven Pemberton)
Newsgroups: eunet.jokes
Subject: Re: Catching a Lion with computer science
Message-ID: <388@piring.cwi.nl>
Date: 6 Jul 88 11:24:18 GMT
References: <2024@sics.se>
Reply-To: steven@cwi.nl (mcvax!steven.uucp)
Distribution: eunet
Organization: CWI, Amsterdam
Lines: 66
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.
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 ensures 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.)
Steven Pemberton, CWI, Amsterdam; steven@cwi.nl