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- MATH THEORY OF BIG GAME HUNTING
- 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.)
- --
- brought to you by Anon
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