October 2008 - mathfun@mailman.xmission.com

[math-fun] Financial Crisis = Mathematical Malpractise ?
by Henry Baker 09 Nov '08

09 Nov '08

FYI -- According to Nassim Nicholas Taleb, the author of "The Black Swan: The Impact of the Highly Improbable", most of the blame for the current financial crisis can be laid at the feet of the "quants": the mathematicians and physicists who have been drawn to Wall Street over the past several decades. The very short version of his argument: Blinded by the elegance of Gaussian distributions, which have nice properties and some closed-form solutions, the quants ignored the actual long-tailed/fractal/hyperbolic nature of the events they were trying to model. Through ignorance or incompetence, they didn't realize that while parameter estimation for Gaussian processes is relatively easy & more-or-less stable, parameter estimation for long-tailed distributions is inherently ill-conditioned: no matter how many samples you look at, the very next sample can completely overwhelm the net impact of _all_ of the previous samples, rendering meaningless every previous calculation. "Hundred year" events seem to happen in economics with distressingly higher frequency than expected: the Latin American financial crisis in the early 1980's wiped out sum total of the entire banking industry's profits _ever_ made up until that time; the 1987 stock market crash in which Taleb made his "FU" money; the 1998 crisis which wiped out the Nobel-prize-winning firm of Long Term Capital Management and nearly caused a crisis eerily similar to today's; and, of course, the current sub-prime CDO mess. An almost-humorous example of the difference between Gaussian ("Mediocristan", see below) & non-Gaussian ("Extremistan", see below) models is his analysis of the financial performance of a world-renowned Las Vegas casino. While the returns from its _gambling_ activities were boringly predictable (due to their Gaussian behavior), the worst event in the casino's history was the loss of its major draw show due to a tiger partially eating one of the performers. The casino had insurance against the tiger eating the audience, but not against the tiger eating the performer. What's amazing to me (HB) is how such "Gaussian" thinking continues to flourish & be the mainstream curriculum at economics and business schools after Mandelbrot and all the subsequent Mandelbrot-wannabees made such a strong case for fractal patterns in economic models. Yet Black-Scholes continues to be utilized today in option pricing (e.g., for employee stock options) and "modern portfolio theory" (MPT) continues preaching that 10-15 investments will provide you with the same diversification as 100-500 investments. "For every problem, there is a solution that is simple, elegant and wrong." H.L. Mencken. ------------------------- http://www.guardian.co.uk/books/2008/sep/28/businessandfinance.philosophy Profile: Nassim Nicholas Taleb The new sage of Wall Street The trader turned author has emerged as the guru of the global financial meltdown. Not only is he riding high in the bestseller lists, his theory of black swan events has become the most seductive guide to our uncertain times * Edward Helmore * The Observer, * Sunday September 28 2008 On Friday afternoon, Nassim Nicholas Taleb could be found on the veranda of a hotel bar in Louisville, Kentucky, knocking back bourbon in the warm afternoon sunlight. No wonder the Lebanese-born trader turned author feels like relaxing: his book, The Black Swan: The Impact of the Highly Improbable, has become a huge success. A book of economics and philosophy, it's found a vast audience, speaks to its time and has become something of a key text to help understand the crisis in market capitalism. Not only does the book sit high in the US bestseller lists, but as a mark of its impact, the term 'black swan' has joined 'tipping point' and 'long tail' by having a life of its own. The financial meltdown is now routinely referred to as a 'black swan event'. As influential financial website bloomberg.com noted earlier this month: 'One hears folks from New York to Ulaanbaatar buzzing about black swans.' Taleb's central thesis is that a small number of unexpected events - the black swans - explains much of import that goes on in the world. We need to understand just how much we will never understand is the line. 'The world we live in,' he likes to say, 'is vastly different from the world we think we live in.' The title refers to the medieval belief that all swans were white, hence black swan was a metaphor for something that could not exist, a metaphor that shifted into a perceived impossibility that came to pass when black swans were discovered in the 17th century. Taleb does not think of himself as an ideological or even philosophical writer, though his book contains elements from both realms of thinking; he prefers the role of mischievous intellectual whose observations stretch effortlessly from reason to superstition. In the midst of the economic upheaval, Taleb's acerbic attitude to bankers and economists has won him a new following and provided a comfort to many puzzling over the apparent unpredictability of recent events. We are, he believes, suckers. 'The tools we have to understand what's happening on Wall Street were developed over the last couple of centuries,' he told the audience at Kentucky's Idea Festival last week. 'We need new tools. We will have to finance the losses because of a huge misunderstanding.' That misunderstanding, he explains in his book, is partly based on our belief that bankers and financial analysts are somehow blessed with superior knowledge. While 'peasants know they can't predict the future', Wall Street bankers believe they can. 'Banks hire dull people and train them to be even more dull. If they look conservative, it's only because their loans go bust on rare, very rare occasions.' But, Taleb believes, bankers are not conservative at all. They are just 'phenomenally skilled at self-deception by burying the possibility of a large, devastating loss under the rug'. In his estimation of the scale of the disaster: 'The banking system, betting against black swans, has lost more than $1 trillion - more than was ever made in the history of banking.' When it comes to finance, collective wisdom has shown itself to be close to astrology - based on nothing. But according to Taleb, unpredictable events - 9/11, the dotcom bubble, the current financial implosion - are much more common than we think. In Taleb's coinages, most people live in 'Mediocristan,' a fake model of reality where no rare events occur, and not in 'Extremistan', the complex real world where unpredictable and devastating events can dictate the outcome. One of Taleb's favourite allegorical tales is the story of the turkey and the butcher. As previously described by Bertrand Russell, a turkey may get used to the idea of being fed but when, the day before Christmas, it is slaughtered, it will incur 'a revision of belief'. No one, he believes, is more guilty of living like turkeys in the false security of Mediocristan than economists and financial risk management analysts who rely on computer models that don't account for rare devastating events. Taleb's personal black swan event came when Lebanon was engulfed by civil war in 1975, and he spent several war years in the basement of the family home studying. The son of a wealthy, highly educated Greek Orthodox family, he received degrees at Wharton, Pennsylvania, and the University of Paris before becoming a Wall Street trader. Then came 19 October 1987 - Black Monday. The event reinforced his belief in chance events. It was a sizable black swan, he says, that 'had vastly more influence on my thought than any other event in history'. Shorting the market made him nearly $40m. A brush with throat cancer preceded the publication of his first book, Fooled by Randomness, that became a surprise hit in 2001. It established Taleb as a new thinker. The author even found his views adopted by strategic planners at the Pentagon. In February 2002, then Defence Secretary Donald Rumsfeld offered this hybrid of Taleb thought: 'There are known knowns. There are things we know that we know. There are known unknowns. That is to say, there are things that we now know we don't know. But there are also unknown unknowns. There are things we do not know we don't know.' Taleb is full of inconsistencies - happily so. 'My problem is what my mother kept telling me I'm too messianic in my views,' he concedes. He claims not to read newspapers, yet his website (www.fooledbyrandomness.com) is crammed with links to his press cuttings (he says he get news from people at parties). He's paid up to $60,000 for speaking to businessmen at conferences, yet he claims to distrust anyone who wears a tie (following the example of the Silicon Valley types he admires more, he sticks to black polo necks). And of course there is the ultimate contradiction: where he believes no one can predict black swan events, he's eager to cite instances where he alone has done so. 'One simply cannot predict history beyond a short time horizon,' he offered his audience in Louisville last week. 'It might be useful to be able to predict war. But tension does not necessarily lead to war, but often to peace and to denouement.' To establish his credentials as sage of our current predicament, Taleb frequently refers to an August 2003 article in the New York Times in which he correctly predicted the quasi-governmental US insurance giant Fannie Mae had underestimated the risk of a rise in interest rates that would destroy the value of their portfolios. 'The fact that they have not blown up in the past doesn't mean that they're not going to blow up in the future,' he said. 'The math is bogus.' (The company was taken over by the US government last month.) The footnote on page 225 of The Black Swan reads: 'When I look at their risks, [Fannie Mae] seems to be sitting on a barrel of dynamite, vulnerable to the slightest hiccup. But not to worry - their large staff of scientists deem these events "unlikely".' Even globalisation, which was supposed to make the economic machine more resilient, may be having the opposite effect, he thinks. 'Things are way too efficient, so the smallest mistake blows up. Tomorrow, if there's a problem in Bangalore, we're toast for a long time, you see? 'Economics is a tragedy for me. Because look at how the whole world now is designed according to some ideas that have not proved adequate. The whole financial system. We don't understand economic policy, do you realise that? When Alan Greenspan lowered interest rates thinking it would help the economy, all it did was push banks to take risks-hidden risks.' Taleb is not the only author competing to provide the intellectual weight to the current confusion. This autumn, Michael Lewis, author of Liar's Poker and Moneyball, seeks to describe the collision of market forces, human idiosyncrasies and greed in Panic: The Story of Modern Financial Insanity. His thinking: risky loans and easy money almost always lead to disaster - whether it's the crash of '87, the Asian currency crisis of 1999 or the sub-prime mortgage fallout. And one might feel that with hindsight it's not hard to predict that a banking industry that drops lending standards, securitises bad loans without transparency and sells them on to massively leveraged institutions and hedge funds will end in trouble. Perhaps Taleb's black swan theory promises to be more enduring because it does not, he argues, depend on hindsight. In any case, he has his own in-built defence mechanism. 'My major hobby,' he says, 'is teasing people who take themselves and the quality of their knowledge too seriously and those who don't have the courage to sometimes say, "I don't know..."' The Taleb Lowdown Born: In 1960 into an influential Greek Orthodox Levantine family from Amioun, Lebanon. Son of oncologist Dr Najib Taleb and Minerva Ghosn, daughter and granddaughter of former Deputy Prime Ministers of Lebanon. Family influence reduced during the Lebanese civil war. Education: An MBA from the University of Pennsylvania, PhD from Paris University. Best of times: The current market crash vindicated Taleb's economic predictions, making him, according to the Times, 'the hottest thinker in the world' with a $4m advance on his next book and worldwide speaking engagements ($60,000 a pop). Worst of times: The frosty reception for The Black Swan. The American Statistician dedicated an entire issue to criticising Taleb's black swan theory. What he says 'I am interested in how to live in a world we don't understand very well - in other words, while most human thought [particularly since the Enlightenment] has focused us on how to turn knowledge into decisions, I am interested in how to turn lack of information, lack of understanding, and lack of "knowledge" into decisions - how not to be a "turkey".' What they say: 'Beneath Mr Taleb's blustery rhetoric lives a surprisingly humble soul who has chosen to follow a demanding and somewhat lonely path.' The Wall Street Journal.

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Re: [math-fun] Distance ratio question
by Dan Asimov 01 Nov '08

01 Nov '08

David, Thanks for your info re the two circle-packing problems. This is tricky stuff; it seemed like there was a simple argument showed they are equivalent. As for your problem with getting your posts posted in math-fun, the person to contact about that is Rich at <rschroe(a)sandia.gov>, who is the list's founder and manager. Good luck. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Pfaffians
by Eugene Salamin 29 Oct '08

29 Oct '08

Out of curiosity, I checked the permanent of an antisymmetric matrix. The permanent has the same expansion into n! terms as the determinant, except that the coefficient is always +1 rather than the sign of the permutation. For an odd order antisymmetric matrix, the permanent vanishes, for the same reason as the determinant. For an even order antisymmetric matrix, I factored the order 4, 6, and 8 permanents using Maple. They do not factor. Gene

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Re: [math-fun] Life on the Edge
by Andrew Trevorrow 28 Oct '08

28 Oct '08

Attempting to break the world record for a late reply... Early last year, Franklin T. Adams-Watters described an interesting cellular automaton defined on the edges of a square grid and asked: > Can anyone find other interesting patterns? It would be especially > interesting if someone can find an "airplane gun", which generates > airplanes at regular intervals. Any other patterns that move, > and any other periodic patterns - especially with a period greater > than two - would also be of interest. Franklin, you'll be glad to hear that Golly 2.0b includes two new rules, one called LifeOnTheEdge and another called LifeOnTheSlope. See life-on-the-edge.rle and life-on-the-slope.rle, both in Patterns/Other-Rules. Andrew

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[math-fun] Golly 2.0b
by Andrew Trevorrow 28 Oct '08

28 Oct '08

Golly is an open source, cross-platform application for exploring Conway's Game of Life and other cellular automata. A beta version of Golly 2.0 is now available for testing: http://www.trevorrow.com/golly/golly-2.0b-win.zip (Windows 2000+) http://www.trevorrow.com/golly/golly-2.0b-mac.zip (Mac OS 10.4+) http://www.trevorrow.com/golly/golly-2.0b-gtk.tar.gz (Linux/GTK+ i386) There are a number of significant changes in this release: - Golly now supports multiple algorithms, and universes with more than 2 cell states. Tom Rokicki has extended Gosper's hashlife algorithm to allow up to 256 states, and that means Golly can handle many new types of CA. - The new JvN algorithm is a hashlife-based implementation of John von Neumann's 29-state CA. Inside Patterns/JvN you can find sphinx.mc, a self-replicating machine created by William R. Buckley, along with other smaller (faster) replicators created by Renato Nobili and Tim Hutton using 32-state variants of JvN's CA. - The new Generations algorithm supports an entire family of rules familiar to MCell users (Brian's Brain, Star Wars, Transers, etc). - The new RuleTable and RuleTree algorithms allow you to add new rules by creating .table or .tree files in the Rules folder. A number of example rules are supplied, including Langtons-Loops, Evoloop, and WireWorld. See Help > File Formats for more details about these files. (Much thanks to Tim Hutton who wrote the RuleTable algorithm and created all the .table rules.) - Golly gives you full control of how colors are assigned to cell states. Color schemes can be algorithm-specific, rule-specific or layer-specific. You also have the option of displaying icons (small bitmaps) instead of colored squares, but only at scales 1:8 or 1:16. - The Set Rule dialog can be expanded to show algorithm-specific help that includes clickable rules and clickable patterns. See Help > Changes for the complete list of changes. Please report any problems or suggestions to me (andrew(a)trevorrow.com) or Tom (rokicki(a)gmail.com) Or even better, subscribe to the golly-test mailing list (http://lists.sourceforge.net/lists/listinfo/golly-test) and post your feedback there. Have fun! Andrew (on behalf of the Golly Gang)

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Re: [math-fun] Pfaffians
by Eugene Salamin 27 Oct '08

27 Oct '08

I rather like the discussion of alternate bilinear forms and Pfaffians in Nathan Jacobson's "Basic Algebra", vol. 1. A bilinear form B over a vector space V is alternate if B(x,x) = 0 for all x in V. This implies B(x,y) = -B(y,x), but the reverse implication is true only for characteristic not 2. A bilinear form is degenerate if for some x, B(x,y) = 0 for all y in V. In a basis (e[1], ..., e[n]), a bilinear form is represented by a matrix B[i,j] = B(ei, ej), and if the form is alternate, its matrix is antisymmetric. A degenerate form yields a singular matrix. Since we want to show that the determinant is a square, we are interested only in nondegenerate alternate forms on a 2n-dimensional space V. The first task is to show that there exists a basis in which the matrix consists of n blocks along the diagonal, where each 2 x 2 block is [[0, 1], [-1, 0]], other matrix elements being zero. Since B is nondegenerate, there are vectors u1, v1 such that B(u1, v1) /= 0, and by suitable scaling we may assume B(u1, v1) = 1 = -B(v1, u1). Since B(x, ax) = 0, u1 and v1 must be independent. This gives us the first block. Suppose we have already found linearly independent vectors (u1, v1, ..., uk, vk) such that B(ui, vi) = 1 = -B(vi, ui), and B(x, y) = 0 for all other choices. Let Vk be the 2k-dimensional subspace spanned by these vectors, and let Vk[perp] be the subspace of all x such that B(x, y) = 0 for all y in V. Then the nondegeneracy of B implies that V is the direct sum of Vk and Vk[perp]. Jacobson shows this by a calculation. Since the restriction of B to Vk[perp] is nondegenerate, there exist vectors u[k+1], v[k+1] in Vk[perp] that extend the list of linearly independent vectors and have the required properties. So we have a basis in which B is represented by a matrix S consisting of n blocks, each [[0, 1], [-1, 0]]. We have det S = 1. If x = sum(x[i] e[i]), y = sum(y[j] e[j]) are vectors, B(x, y) = sum(x[i] B(e[i], e[j]) y[j]) = sum(x[i] B[i,j] y[j]) = x^t B y. (^t denotes transpose) Under a change of basis given by a matrix P, the matrix of B transforms as B' = P^t B P. So our antisymmetric matrix B can be expressed as B = P^t S P, and then det B = (det P^t) (det S) (det P) = (det P)^2. Thus det B is a square, but this is as an element of the field. However, the matrix whose determinant we wish to show is a square is one in which B[i,j] = x[i,j]. Let R be the ring Z[x[i,j]] of polynomials with integer coefficients in the (2n)(2n-1)/2 indeterminates x[i,j], i < j. Let F = Q(x[i,j]) be the fraction field of R. We want to show that det B is a square in R, but so far we only know that det B is a square in F, i.e. there are polynomials f and g in R such that det B = (f / g)^2. It is a theorem that if D is an integral domain with unique factorization, then so is the ring D[x1, ..., xn] of polynomials in n indeterminates over D. (Jacobson proves this in the same book.) Since Z is a UFD, so is R. Then we may assume common factors to be cancelled in the fraction f / g. So we have (g^2) (det B) = (f^2). Now, det B is a polynomial in x[i,j]; it is an element of R. If g is not a unit (a divisor of 1), then it has a prime factor p, and this p must divide f. This contradicts the choice of f and g, so g is a unit (+1 or -1), and det B = f^2 is a square in R. The sign of f is fixed by f(S) = 1, and then Pf B = f(B), and the Pfaffian is a polynomial in the elements of B. Gene

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[math-fun] Fwd: Roland Silver has died
by rcs＠xmission.com 26 Oct '08

26 Oct '08

Foo. :-( ----- Forwarded message from whitfield.diffie(a)sun.com ----- Date: Sun, 26 Oct 2008 12:50:40 -0700 From: Whitfield Diffie <whitfield.diffie(a)sun.com> Reply-To: Whitfield Diffie <whitfield.diffie(a)sun.com> Subject: Roland Silver has died To: Richard Schroeppel <rcs(a)xmission.com> Cc: Hilarie Orman <ho(a)alum.mit.edu> Sunday 26 October 2008 at 12:46 We got a call from Susie McCarthy yesterday to let us know that Roland had died of throat cancer. He was in Boston and it was on Thursday, I think. Whit ----- End forwarded message -----

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Re: [math-fun] Pfaffians
by Dan Asimov 26 Oct '08

26 Oct '08

Gene asked: << The determinant of a matrix is a polynomial in its elements. For an antisymmetric matrix of odd order, the determinant is zero, while for an antisymmetric matrix of even order, this polynomial factors into the square of another polynomial, called the Pfaffian. Does anyone know of a simple, easy to follow, proof that the determinant factors as asserted? >> Here's a cite for what appears to be an elementary 1-page proof: Skew-Symmetric Determinants Author(s): S. Parameswaran Source: The American Mathematical Monthly, Vol. 61, No. 2 (Feb., 1954), p. 116 I believe there is a more informative but more advanced proof in "Characteristic Classes" by J. Milnor (Princeton orange series). --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Distance ratio question
by Dan Asimov 26 Oct '08

26 Oct '08

Kevin Buzzard posted this to sci.math.research, and I am taking the liberty of reposting it here: << The following came up in an undergraduate problem-solving group: given 6 distinct points in the plane we can consider the 6-choose-2 distances between pairs of distinct points. If M is the largest of these distances, and m is the smallest, then show that M/m>=sqrt(3). In fact sqrt(3) isn't optimal, but I don't know what the optimal number is. One might ask what the answer is with n>=3 points in the plane: a compactness argument shows that there will be some r(n)>=1 such that M/m is always at least r(n) and furthermore such that r(n) is attained by some configuration of n points. . . . . . . >> (I think I have an asymptotic formula for the optimal ratio as n -> oo, but so far not a proof. I'm curious if anyone else comes up with a conjectural asymptotic formula that is asymptotic to mine. Putative asymptotic formula is way, way below.) --Dan Guess: r(n) ~ sqrt(3n/2) as n -> oo. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Distance ratio question
by Dan Asimov 25 Oct '08

25 Oct '08

By the way, Kevin Buzzard & I just realized that this M/m problem is (trivially) equivalent to the "Penny Problem": -- Given n non-overlapping* unit disks in the plane, what is the smallest R = R(n) for which they can be arranged inside of a disk of radius R ? Neil, did you do any numerical work on this question? --Dan _____________________________________ * other than possible tangencies. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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