In Pursuit of the Unknown (64 page)

See Figure 61.

Fig 61
Combining choices in different ways. The information should be the same in either case.

The factor
in front of the final
H
is present because this second choice occurs only half the time, namely when
r
is chosen in the first stage. There is no such factor in front of the
H
just after the equals sign, because this refers to a choice that is always made – between
q
and
r
.

2
See Chapter 2
of: C.E. Shannon and W. Weaver.
The Mathematical Theory of Communication
, University of Illinois Press, Urbana 1964.

1
If the population
x
_t
is relatively small, so that is close to zero, then 1 −
x
_t
is close to 1. The next generation will therefore have a size close to
kx
_t
, which is
k
times as large as the current one. As the size of the population increases, the extra factor 1 −
x
_t
makes the actual growth rate smaller, and it drops to zero as the population approaches its theoretical maximum.

2
R.F. Costantino, R.A. Desharnais, J.M. Cushing, and B. Dennis. Chaotic dynamics in an insect population,
Science
275
(1997) 389–391.

3
J. Huisman and F.J. Weissing. Biodiversity of plankton by species oscillations and chaos,
Nature
402
(1999) 407–410.

4
E. Benincà, J. Huisman, R. Heerkloss, K.D. Jöhnk, P. Branco, E.H. Van Nes, M. Scheffer, and S.P. Ellner. Chaos in a long-term experiment with a plankton community,
Nature
451
(2008) 822–825.

1
The value of a call option is

where

The price of a corresponding put option is

Here
N
(
dj
) is the cumulative distribution function of the standard normal distribution for
j
= 1, 2, and
T
–
t
is the time to maturity.

2
Strictly, a Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel.

3
M. Poovey. Can numbers ensure honesty? Unrealistic expectations and the U.S. accounting scandal,
Notices of the American Mathematical Society
50
(2003) 27–35.

4
A.G. Haldane and R.M. May. Systemic risk in banking ecosystems,
Nature
469
(2011) 351–355.

The following figures are reproduced with the permission of the named copyright owners:

Figure 9
: Johan Hidding.

Figures 11
,
41
: Wikimedia Commons. Reproduced under the terms of the GNU Free Documentation License.

Figures 15
,
16
,
17
: Wang Sang Koon, Martin Lo, Shane Ross and Jerrold Marsden.

Figure 31
: Andrzej Stasiak.

Figure 43
: BMW Sauber Formula 1 Team.

Figure 53
: Willem Schaap.

Figure 59
: Jef Huisman and Franz Weissing,
Nature
402
(1999) 407-410.

The suffix ‘n' indicates a note. Italic page references refer to Figures.

A

abstract algebra
101
,
103
,
276
,
279–80

acceleration, defined
39–40

aerodynamics
86
,
168
,
170
,
172

air travel
167

aircraft design
172

(Taqi) al-Din Muhammad ibn Ma'ruf al-Shami al-Asadi
199

al-Katibi, Najm al-Din al-Qazwini
57

Alexander, James Waddell
101
,
103–4

algebra

abstract
101
,
103
,
276
,
279–80

Cardano and
75
,
77–8

notation of
78

relation to geometry
9

algorithms
157–60
,
264
,
320

Alhazen
57

‘alternate worlds'
258

American Psychological Association
125

analogue computers
172
,
270

analysis
51
,
82

see also
calculus

analysis of variance method
125

angular momentum
49

animal population dynamics
286–7
,
291–2
,
330n

Apollo missions
65–6
,
168

Arago, François
233

arbitrage pricing theory
304
,
306

Archimedes
3
,
57
,
169

Argand, Jean-Robert
81–2
,
87

Aristarchus of Samos
56

Aristotle
46–7

arithmetic, rules of
77

Arnold, Vladimir
289–90

Arp, Halton
241

arrow of time
208
,
211
,
213
,
318

Aryabhata
56

astronomy

chaotic dynamics in
290–1

geocentric universe
8

interpretations of the night sky
55–6

lasers in
263

tests of relativity
233–5

use of logarithms
29

atomic bomb
229

atomic theory.
see
molecules

‘average man'
116

B

Babylonian mathematics
6–7
,
80

Bachelier, Louis
302–4

banking system

acceptance of derivatives
297

modularity and
314

base, logarithms
27–8
,
30

beats
143
,
144

Belbruno, Edward
69
,
70