زائر

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الهوامش

الفصل الأول: الفعالية اللامعقولة

(1)
In 2012 the accountancy company Deloitte carried out a survey: Measuring the Economic Benefits of Mathematical Science Research in the UK. At that time, 2–8 million people were employed in mathematical science occupations: pure and applied mathematics, statistics, and computer science. The mathematical sciences contributed £208 billion (gross value added) to the UK economy in that year – just under £250 billion in 2020 money, around $300 billion. Those 2–8 million people made up 10% of the British workforce, and contributed 16% of the economy. The largest sectors were banking, industrial research and development, computer services, aerospace, pharmaceuticals, architecture,and construction. The report’s examples include smartphones, weather forecasting, healthcare, movie special effects, improving athletic performance, national security, managing epidemics, Internet data security, and making manufacturing processes more efficient.
(3)
The formula is

where    is the value of the random variable,    is the mean, and    is the standard deviation.
(4)
Vito Volterra was a mathematician and physicist. In 1926 his daughter was courting Umberto D’Ancona, a marine biologist, and later they married. D’Ancona had discovered that during the First World War, the proportion of predatory fish (sharks, rays, swordfish) that fishermen were catching increased, even though they were doing less fishing overall. Volterra wrote down a simple calculus-based model for how the populations of predators and prey change over time, which showed that the system goes round and round in a cycle of predator explosions and prey crashes. Crucially, on average the number of predators increases, proportionately, more than the number of prey.
(5)
No doubt Newton used physical intuition as well, and historians tell us that he probably pinched the idea from Robert Hooke, but there’s no point in being a one-trick pony.

الفصل الثاني: كيف يختار السياسيون ناخبيهم؟

(2)
Time wasn’t the only issue. At the Constitutional Convention of 1787, which led to the Electoral College system, though not by that name, James Wilson, James Madison, and others felt that a popular vote would be best. However, there were practical problems about who would be allowed to vote, with big differences of opinion between Northern and Southern states.
(3)
In 1927 E. P. Cox used the same quantity in palaeontology to assess how round sand grains are, which helps distinguish windblown sand from waterborne sand, providing evidence for environmental conditions in prehistoric times. See E. P. Cox. ‘A method of assigning numerical and percentage values to the degree of roundness of sand grains,’ Journal of Paleontology 1 (1927) 179–183. In 1966 Joseph Schwartzberg proposed using the ratio of the perimeter of a district to the circumference of the circle of the same area. This is the reciprocal of the square root of the Polsby-Popper score, so it ranks districts in the same way, though with different numbers. See J. E. Schwartzberg, ‘Reapportionment, gerrymanders, and the notion of “compactness”,’ Minnesota Law Review 50 (1966) 443–452.
(4)
By enclosing a hill, a curved surface, she crammed even more area into her circle.
(5)
V. Blåsjö, ‘The isoperimetric problem,’ American Mathematical Monthly 112 (2005) 526–566.
(6)
For a circle of radius   ,
(7)
N. Stephanopoulos and E. McGhee, ‘Partisan gerrymandering and the efficiency gap,’ University of Chicago Law Review 82 (2015) 831–900.
(8)
M. Bernstein and M. Duchin, ‘A formula goes to court: Partisan gerrymandering and the efficiency gap,’ Notices of the American Mathematical Society 64 (2017) 1020–1024.
(9)
J. T. Barton, ‘Improving the efficiency gap,’ Math Horizons 26.1 (2018) 18–21.
(10)
In the early 1960s John Selfridge and John Horton Conway independently found an envy-free method of cake division for three players:
  • (1)
    Alice cuts the cake into three pieces that she considers of equal value.
  • (2)
    Bob either passes, if he thinks two or more pieces are tied for largest, or trims what he considers to be the largest piece to create such a tie. Trimmings are called ‘leftovers’ and set aside.
  • (3)
    Charlie, Bob, and Alice, in that order, choose a piece that they think is largest or tied largest. If Bob didn’t pass in step 2 he must choose the trimmed piece, unless Charlie chose it first.
  • (4)
    If Bob passed at step 2 there are no leftovers and we’re done. If not, either Bob or Charlie took the trimmed piece. Call this person the ‘non-cutter’ and the other the ‘cutter’. The cutter divides the leftovers into three pieces that he considers equal.
  • (5)
    Players choose one of these pieces in the order non-cutter, Alice, cutter. No player has any reason to envy what another player receives: if they do, they got their tactics wrong and should have chosen differently. For a proof, see: en.wikipedia.org/wiki/Selfridge-Conway_procedure.
(11)
S. J. Brams and A. D. Taylor, The Win-Win Solution: Guaranteeing Fair Shares to Everybody, Norton, New York (1999).
(12)
Z. Landau, O. Reid, and I. Yershov, ‘A fair division solution to the problem of redistricting,’ Social Choice and Welfare 32 (2009) 479–492.
(13)
B. Alexeev and D. G. Mixon, ‘An impossibility theorem for gerrymandering,’ American Mathematical Monthly 125 (2018) 878–884.

الفصل الثالث: دع الحمامة تقود الحافلة!

(1)
B. Gibson, M. Wilkinson, and D. Kelly, ‘Let the pigeon drive the bus: pigeons can plan future routes in a room,’ Animal Cognition (2012) 379–391.
(2)
My favourite example is a politician who made a huge fuss about money being wasted on what he called ‘lie theory’–pronouncing ‘lie’ as in ‘untruth’, which is what he thought it was about. Not so. Sophus Lie (pronounced ‘lee’) was a Norwegian mathematician, whose work on continuous groups of symmetries (Lie groups) and associated algebras (guess what) is fundamental to large parts of mathematics and even more so to physics. The politician’s misconception was quickly pointed out … and he carried on exactly as before.
(3)
For technical reasons my remark about jigsaws doesn’t solve the prize problem. If it did, I’d have got there first.
(4)
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco (1979).
(5)
G. Peano, ‘Sur une courbe qui remplit toute une aire plane,’ Mathematische Annalen 36 (1890) 157–160.
(6)
Some care needs to be taken because some real numbers don’t have unique representations as decimals–for instance 0.500000… = 0.499999…. But that’s easy to sort out.
(7)
E. Netto, ‘Beitrag zur Mannigfaltigkeitslehre,’ Journal für die Reine und Angewandte Mathematik 86 (1879) 263–268.
(8)
H. Sagan, ‘Some reflections on the emergence of space-filling curves: the way it could have happened and should have happened, but did not happen,’ Journal of the Franklin Institute 328 (1991) 419–430. For an explanation, see: A. Jaffer, ‘Peano space-filling curves,’ http://people.csail.mit.edu/jaffer/Geometry/PSFC
(9)
J. Lawder, ‘The application of space-filling curves to the storage and retrieval of multi-dimensional data,’ PhD Thesis, Birkbeck College, London (1999).
(10)
J. Bartholdi, ‘Some combinatorial applications of spacefilling curves,’ www2.isye.gatech.edu/~jjb/research/mow/mow.html
(11)
H. Hahn, ‘Über die allgemeinste ebene Punktmenge, die stetiges Bild einer Strecke ist,’ Jahresbericht der Deutschen Mathematiker-Vereinigung, 23 (1914) 318–322. H. Hahn, ‘Mengentheoretische Charakterisierung der stetigen Kurven,’ Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien 123 (1914) 2433–2489. S. Mazurkiewicz, ‘O aritmetzacji kontinuów’, Comptes Rendus de la Société Scientifique de Varsovie 6 (1913) 305–311 and 941–945.
(12)
Published in 1998: S. Arora, M. Sudan, R. Motwani, C. Lund, and M. Szegedy, ‘Proof verification and the hardness of approximation problems,’ Journal of the Association for Computing Machinery 45 (1998) 501–555.
(13)
L. Babai, ‘Transparent proofs and limits to approximation,’ in: First European Congress of Mathematics. Progress in Mathematics 3 (eds. A. Joseph, F. Mignot, F. Murat, B. Prum, and R. Rentschler) 31–91, Birkhauser, Basel (1994).
(14)
C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus, ‘Intriguing properties of neural networks,’ arXiv:1312.6199 (2013).
(15)
A. Shamir, I. Safran, E. Ronen, and O. Dunkelman, ‘A simple explanation for the existence of adversarial examples with small Hamming distance,’ arXiv:1901.10861v1 [cs.LG] (2019).

الفصل الرابع: مسألة كونيجسبرج وزرع الكُلى

(1)
Not to be confused with the graph of a function, which is a curve relating a variable x to the value     of the function. Like the parabola for   .
(2)
Thanks to Robin Wilson for gently pointing this out when I got it wrong in one of my books.
(3)
Provided you know which region to start from, it’s enough just to list the bridge symbols, in the order they’re crossed. Consecutive bridges determine a common region, to which they both connect.
(4)
This is fairly easy to prove using Euler’s characterisation of open tours. The main idea is to break a hypothetical closed tour by cutting out one bridge. Now you have an open tour, and the bridge concerned originally joined the two ends.
(5)
The rest of this chapter is based on: D. Manlove, ‘Algorithms for kidney donation,’ London Mathematical Society Newsletter 475 (March 2018) 19–24.

الفصل الخامس: حلِّق آمنًا في الفضاء الإلكتروني

(1)
The exact date when Fermat stated his Last Theorem isn’t certain, but it’s often taken to be 1637.
(2)
The same can be said of much ‘applied’ mathematics too. However, there’s a difference: the attitude of the mathematician. Pure mathematics is driven by the internal logic of the subject: not merely monkey curiosity, but a feeling for structure and a sense of where our understanding has significant gaps. Applied mathematics is mainly driven by problems arising in the ‘real world’, but it’s more willing to tolerate unjustified shortcuts and approximations in search of an answer, and the answer may or may not have practical implications. As this chapter illustrates, however, a topic that seems completely useless at some moment in history can suddenly become vital to practical issues when culture or technology changes. Moreover, mathematics is an interconnected whole; even the pure/applied distinction is an artificial one. A theorem that seems useless in its own right may inspire, or even imply, results of great utility.
(3)
The answer is:
p = 12,277,385,900,723,407,383,112,254,544,721,901,362,713,421, 995,519
q = 97,117,113,276,287,886,345,399,101,127,363,740,261,423,928, 273,451
I found these two primes by trial and error, and multiplied them together, using a symbolic algebra system on a computer. This took a few minutes, mostly me changing digits at random until I stumbled across a prime. Then I told the computer to find the factors of the product, and it ran for ages with no result.
(4)
If    is a prime power     then    For a product of prime powers, multiply these expressions together for all the different prime powers in the prime factorisation of      For instance, to find     write     Then
(5)
For more detail about the issues involved, see Ian Stewart, Do Dice Play God?, Profile, London (2019), Chapters 15 and 16.
(6)
L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, ‘Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,’ Nature 414 (2001) 883–887.
(7)
F. Arute and others, ‘Quantum supremacy using a programmable superconducting processor,’ Nature 574 (2019) 505–510.
(8)
J. Proos and C. Zalka, ‘Shor’s discrete logarithm quantum algorithm for elliptic curves,’ Quantum Information and Computation 3 (2003).
(9)
M. Roetteler, M. Naehrig, K. Svore, and K. Lauter, ‘Quantum resource estimates for computing elliptic curve discrete logarithms,’ in: ASIACRYPT 2017: Advances in Cryptology, Springer, New York (2017), 214–270.

الفصل السادس: مستوى الأعداد

(1)
For instance, -25 has a square root 5i, because

In fact, it has a second square root, -5i, for similar reasons.
(2)
Algebraists regularise the situation by saying that the square root of zero is zero, with multiplicity two. That is, the same value occurs twice, in a meaningful but technical sense. An expression like     has two factors,     times     which respectively give two solutions     and     to the equation     Similarly, the expression     has two factors,     times     They just happen to be the same.
(3)
For real     the function     obeys the differential equation     with initial condition     If we define the exponential function for complex     so that the same equation holds, which is sensible, and set     then     Since multiplying by     rotates complex numbers through a right angle, the tangent to     as     varies is at right angles to     so the point     describes a circle of radius 1 centred at the origin. It rotates round this circle at a constant speed of one radian per unit of time, so at time     its position is at angle     radians. By trigonometry, this point is  
(4)
More precisely, there has to be an ‘inner product’, which determines distances and angles.

الفصل السابع: أبي، هل يمكنك ضرب الثلاثيات؟

(1)
The fastest supercomputer in 1988 was the Cray Y-MP, costing $20 million (over $50 million in today’s money). It would struggle to run a Windows operating system.
(2)
K. Shoemake, ‘Animating rotation with quaternion curves,’ Computer Graphics 19 (1985) 245–254.
(3)
L. Euler, ‘Decouverte d’un nouveau principe de mecanique’ (1752), Opera Omnia, Series Secunda 5, Orel Fusili Turici, Lausanne (1957), 81–108.
(4)
The half-angle property is important in quantum mechanics, where one formulation of quantum spin is based on quaternions. If the wave function of a particle of the kind known as a fermion is rotated through 360°, its spin reverses. (This is distinct from rotating the particle itself.) The wave function must rotate through 720° to return the spin to its original value. The unit quaternions form a ‘double cover’ of the rotations.
(5)
C. Brandt, C. von Tycowicz, and K. Hildebrandt, ‘Geometric flows of curves in shape space for processing motion of deformable objects,’ Computer Graphics Forum 35 (2016) 295–305.

الفصل الثامن: الزُّنبُركات

(1)
T. Takagi and M. Sugeno, ‘Fuzzy identification of systems and its application to modeling and control,’ IEEE Transactions on Systems, Man, and Cybernetics 15 (1985) 116–132.

الفصل العاشر: ابتسم، من فضلك!

(1)
This is JFIF encoding, used for the web. Exif coding, for cameras, also includes ‘metadata’ describing the camera settings, such as date, time, and exposure.
(2)
A. Jain and S. Pankanti, ‘Automated fingerprint identification and imaging systems,’ in: Advances in Fingerprint Technology (eds. C. Lee and R. E. Gaensslen), CRC Press, (2001) 275–326.

الفصل الحادي عشر: هل اقتربنا من الوصول إلى هناك؟

(1)
N. Ashby, ‘Relativity in the Global Positioning System,’ Living Reviews in Relativity 6 (2003) 1; doi: 10.12942/lrr-2003-1.
(2)
More precisely,       where the sum is over all configurations of spin variables.

الفصل الثاني عشر: إيزينج وذوبان ثلوج القطب الشمالي

(1)
Setting   , where   is Boltzmann’s constant, the formula is:
(2)
The formula is:

where     is the strength of the external field and     is the strength of the interactions between spins. In the absence of an external field     so       so the whole fraction is  
(3)
Y.-P. Ma, I. Sudakov, C. Strong, and K. M. Golden, ‘Ising model for melt ponds on Arctic sea ice,’ New Journal of Physics 21 (2019) 063029.

الفصل الثالث عشر: اتصل بعالم الطوبولوجيا!

(1)
S. Tanaka, ‘Topological analysis of point singularities in stimulus preference maps of the primary visual cortex,’ Proceedings of the Royal Society of London B 261 (1995) 81–88.
(2)
‘Lobster telescope has an eye for X-rays,’ https://www.sciencedailycom/releases/2006/04/060404194138.htm
(3)
Technically, the curve is the image, under a map from a disc to the sphere, of the boundary of the disc. The curve can cross itself and the disc can get scrunched up.
(4)
J. J. Berwald, M. Gidea, and M. Vejdemo-Johansson, ‘Automatic recognition and tagging of topologically different regimes in dynamical systems,’ Discontinuity, Nonlinearity, and Complexity (2014) 413–426.
(5)
F. A. Khasawneh and E. Munch, ‘Chatter detection in turning using persistent homology,’ Mechanical Systems and Signal Processing 70 (2016) 527–541.
(6)
C. J. Tralie and J. A. Perea, ‘(Quasi) periodicity quantification in video data, using topology,’ SIAM Journal on Imaging Science 11 (2018) 1049–1077.
(7)
S. Emrani, T. Gentimis, and H. Krim, ‘Persistent homology of delay embeddings and its application to wheeze detection,’ IEEE Signal Processing Letters 21 (2014) 459–463.
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