November 2008 - mathfun@mailman.xmission.com

Re: [math-fun] Doin' the Hilbert walk
by Dan Asimov 13 Nov '08

13 Nov '08

It may be worth mentioning that for any given dimension = d the d-dimensional Hilbert polygonal curves [0,1] -> [0,1]^d approach a limit as the number of vertices approaches infinity. Which is, of course, a continuous surjection H_d : [0,1] -> [0,1]^d. Although this is old hat to some of us, when it was first discovered (around 1900) it stunned the world of math, forcing people to figure out what the concept of dimension really meant. By making adjustments to the Hilbert polygons, one can easily arrange that the limit function is one-to-one (but of course no longer onto). So the limit function is a homeomorphism of [0,1] onto its image. I suspect the Hausdorff dimension of such a one-one curve can be any real number in [1,d). (Can you confirm this, rwg? Below are some free anagrams for your trouble.) --Dan nawab / bwana bluegill / gullible aquiline / quiniela _____________________________________________________________________ "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] What use is algebra, anyway?
by Fred lunnon 13 Nov '08

13 Nov '08

As I waited at the checkout of our local wholefood shop, a lady who I know to be both literate and equipped with a technical background --- besides a number of other attributes which, though perhaps not immediately relevant to the discussion, hardly serve to encourage coherent impromptu cogitation on the part of an elderly academic --- conveyed to me an enquiry posed by her teenage daughter, along the lines of: "You're a mathematician" (always an ominous opener, I have observed) "--- what's the use of algebra?" I think I did eventually manage to marshall a reasonably convincing reply. But I wondered what would other people here have told her? Fred Lunnon

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Re: [math-fun] What use is algebra, anyway?
by Dan Asimov 13 Nov '08

13 Nov '08

Fred wrote: << ... "You're a mathematician" (always an ominous opener, I have observed) "--- what's the use of algebra?" I think I did eventually manage to marshall a reasonably convincing reply. But I wondered what would other people here have told her? >> For any student who has any chance of going on into a profession that makes use of algebra (not just science but, e.g., carpentry and finance), taking algebra makes sense. But I'm not at all sure it makes sense for it to be a required course for *all* students. IF it does make sense to require for all students, I'd like to see it incorporated gently into the math curriculum from the earliest grades: If we know what a certain number is, we use it. If we don't, we use x. And if there's room for only one algebra-level course in a student's schedule, I'd much rather it were statistics and logical thinking -- aka how to read a newspaper skeptically -- which has a much more concrete payoff in daily life than algebra. --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] What use is algebra, anyway?
by Henry Baker 12 Nov '08

12 Nov '08

FYI -- some thoughts from the NY Times: http://www.nytimes.com/2004/12/12/weekinreview/12mcne.html?scp=12&sq=algebr… December 12, 2004 "In the 1986 movie, 'Peggy Sue Got Married,' Kathleen Turner, an unhappily married wife and mother, magically returns to relive her senior year as the most popular girl at Buchanan High." "She leaves a math test blank, and when her teacher (described in the screenplay as 'an officious little creep') demands an explanation, answers: 'Mr. Snelgrove, I happen to know that in the future, I will never have the slightest use for algebra. And I speak from experience.' " http://query.nytimes.com/mem/archive-free/pdf?res=9C06E1DC1E3FE63BBC4E51DFB… May 26, 1878 "Horse Feeding by Algebra" http://query.nytimes.com/mem/archive-free/pdf?res=9A07E5DB163EE033A2575AC2A… April 29, 1904 "A correspondent of our neighbor The Sun has just been 'appalled,' he says, by the discovery that algebra is taught to children in the public schools of this city. This dismal fact, hitherto carefully concealed from him, he explosively divulges to a world which he obviously expects to share his own horror, for he asserts as a self-evident truth that the teaching of algebra is 'a real cruelty to the child--besides being a waste of time.' " http://query.nytimes.com/mem/archive-free/pdf?res=9402E2DC1330E132A25754C1A… January 17, 1901 "TOO MUCH MATHEMATICS" "Isaac Levene Demonstrated by Algebra that He Did Not Owe for a Meal" "Isaac Levene of 82 West Ninetieth Street ate 90 cents' worth of food in a Sixth Avenue restaurant yesterday afternoon. When the check was presented to him he made a mathematical problem of it, and after figuring it out by algebra, arrived at the conclusion that he owed nothing. He tried to demonstrate his process to the waiter, but the waiter did not understand algebra, and called the proprietor of the restaurant, with whom Levene went over his figures, but even the proprietor could not get the thing right, and called in a policeman. Levene labored with the policeman for a brief time, but the latter knew less about algebra than either of the others, and took Levene to the Jefferson Market Police Court." " 'It's a simple algebraic problem, Judge,' said Levene to Magistrate Flammer, when asked why he did not pay the 90 cents. If I ate an amount equal to 2 X plus 3 Y, and afterward it was said it was equivalent to 4 XZY minus 3 XZ, then by equation I would be found to own nothing.' " "The Magistrate concluded that there was something defective in Levene's reasoning, and sent him to Bellevue to be examined." At 08:17 PM 11/11/2008, Fred lunnon wrote: >As I waited at the checkout of our local wholefood shop, a lady who >I know to be both literate and equipped with a technical background >--- besides a number of other attributes which, though perhaps not >immediately relevant to the discussion, hardly serve to encourage >coherent impromptu cogitation on the part of an elderly academic >--- conveyed to me an enquiry posed by her teenage daughter, >along the lines of: > > "You're a mathematician" (always an ominous opener, I have observed) > "--- what's the use of algebra?" > >I think I did eventually manage to marshall a reasonably convincing >reply. But I wondered what would other people here have told her? > >Fred Lunnon

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[math-fun] Tail coefficients, Bell numbers, and the Wilson Jr primality test
by Fred lunnon 12 Nov '08

12 Nov '08

I've mentioned this generalisation of Bell numbers before in math-fun, so I'll try to avoid taking up too much bandwidth repeating myself. There is a fuller discussion of G_k(n), including a short table and Maple procedures for computing them, in Maple Advisor at http://mapleadvisor.com/cgi-bin/moin.cgi/Bell%20Numbers . They seem to have made their original appearance (with a different origin, for positive k only) in a paper by Gill Williamson (1976), where they were applied to "ranking" or indexing set partitions by natural numbers. I encountered them in a completely different setting: the efficient computation of Bell numbers B(n) modulo p, for small prime p and large natural n. Case k = 0 is simply the usual Bell numbers, and other small values of k also yield well-known sequences: G_{-2}(n) = 1,-1,2,-3,7,-10,31,-21,204,307,2811,... [OEIS A126617]; (*) G_{-1}(n) = 1,0,1,1,4,11,41,162,715,3425,... [OEIS A000296]; (*) G_0(n) = 1,1,2,5,15,52,203,877,... Bell numbers B(n) [OEIS A000110]; G_1(n) = 1,2,5,15,52,203,877,... shifted Bell numbers B(n+1); G_2(n) = 1,3,10,37,151,674,3263,... [OEIS A005493]; G_3(n) = 1,4,17,77,372,1915,10481,60814,... [OEIS A005494]; G_4(n) = 1,5,26,141,799,4736,29371,... [OEIS A045379]; the 2-D sequence is found at [OEIS ], although for k >= 0 only, and so omitting the special cases marked (*) above. A simple O(n^2) recursive rule for generating these numbers is G_k(n+1) = G_{k+1}(n) + k G_k(n) with G_k(0) = 1; or for somewhat larger n, the Dobinski summation is available: G_m (n) = (1/e) \sum_{i >= 0} (i+m)^n / i!; and there are various other nice umbral relations involving them. In this capacity, a generalisation of Touchard's recurrence has corollary: for all k and prime p, G_k(p) = k+2 (mod p); leading immediately to a strengthening of the (somewhat computationally impractical) converse criterion suggested a year or so ago by David Wilson. That possibility was dashed by the counter-example n = 21361 = 41 x 521, for which B(n) = 2 (mod n) --- case k = 0 of the above. However, taking k = 1 we find for this n, G_1(n) = B(n+1) = 3 (mod n) is actually false; so (tongue firmly in cheek) we proudly unveil the new improved Wilson-Lunnon primality-test (conjectural) --- Is there any composite number n for which simultaneously B(n) = 2 (mod n) and B(n+1) = 3 mod n ? And if so, is there any n for which G_k(n) = k+2 (mod n) for all 0 <= k < n ? [Note that, for given n, G_k(n) mod n qua function of k has period n.] A second excuse for my telling you all this concerns a striking property of the "new" sequences for k < 0: for small n their signs alternate with n, but from some n >= 2 m + 1 say, the terms apparently become all positive. For 0 < -k <= 40, these critical values are at m = 0,0,4,8,13,18,24,30,36,42, 49,56,63,70,77,85,92,100,108,115, 123,131,140,148,156,164,173,181,190,198, 207,216,225,234,243,252,261,270,279,288, 297,... Notice that m = m(-k) appears remarkably close to the ceiling of 2|k| log |k| + 3; and what I should like to know is --- are these things true; and if so, why? Fred Lunnon [12/11/08]

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Re: [math-fun] Financial Crisis = Mathematical Malpractise ?
by Dan Asimov 09 Nov '08

09 Nov '08

Henry wrote: << [Scholes's LTCM disaster first, now his Platinum Grove failure. Fool us once, shame on you. Fool us twice, shame on us. So much for mathematical economic genius. Perhaps the Nobel Prize in Economics should be given posthumously, to avoid additional embarrassment. Paul Krugman is next at bat...] ... >> I don't think there is any shame in not being able to foretell the future, and especially in the financial world. I don't consider this to be Scholes's failure. The only mistake here imo is the expectation that a Nobel economist have superhuman powers. --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] 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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[math-fun] Computer Mathematics Systems (was Computer Algebra Systems)
by N. J. A. Sloane 06 Nov '08

06 Nov '08

I still have the 1st edition of "Numerical Recipes" by Press et al., another classic book, and very much to my taste since the programs are written in Fortran, still my favorite language for numerical analysis. Plus I have all the programs on my computer, still good after 22 years (the programs, that is - the computer is a linux x86-64 now). But I see from the web site for the book that it is in its 3rd edition, and the language is now C++ ! Does anyone know if the book has been changed much, and if anything is gained by switching to C++ (which is not a language I'm familiar with)? Neil

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[math-fun] Computer Algebra Systems
by Fred lunnon 06 Nov '08

06 Nov '08

Folks, I'll apologise straight away for raising a topic which does not really belong here. And my first question is: are there any general CAS discussion groups around any more out there? I currently run Maple and Magma on an Apple G4 Powerbook laptop, which I'm thinking shortly of superseding with maybe a Pro desktop box. What I'm looking for is some guidance about whether this is a good idea from the point of view of maintaining a CAS engine. Bill Gosper previously observed that there were serious issues with all extant systems. Bear with me while I outline a selection of those I labour under at the moment. I use Maple version 9. I daren't ever update Quicktime (Apple video viewer) because newer versions disable the Maple loader (don't ask!). The online Maple help doesn't --- although it did work on a PC I also once used, and it does work on version 10. Version 9 is notoriously glitchy, but the infernal licensing software on version 10 (and 11 presumably) has banjaxed me just before a presentation so often that I flatly refuse ever to have anything to do with it again. Second question: would Maple version 12 on a current Mac be any improvement in these and other respects? I also use Magma. On a modern PC, Magma is seriously fast; on an Apple under OS X 3.9, it is horribly crippled. One reason for this appears to be that the version of gcc (C-compiler) as customised by Apple is unable to compile the GMP multi-precision package used by Magma ... Oh, and by the way, it's a waste of time spending an evening trying to install any gcc updates, because an assembler environment enquiry required is missimg; and the link GNU give for supplying same is broken --- or was last time I tried, and I'm not in any hurry to *$£@?!% well try again ... Third question: have Apple yet managed to install a C compiler which actually compiles GMP? I'd be grateful to hear from any others who have wrestled with such questions in the recent past! Fred Lunnon

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[math-fun] DLMF -- online successor to Abramowitz & Stegun book
by rcs＠xmission.com 05 Nov '08

05 Nov '08

NIST recently released a 5-chapter preview of the DLMF online "tables" project (an online successor to the Abramowitz & Stegun book of tables). The chapters available include the Gamma Function and Number Theory. http://dlmf.nist.gov I was looking it over, and came away strangely dissatisfied. I was trying to put my finger on why, but I can't really explain it. The website doesn't have the same thrill as flipping through the book. The typesetting is reasonable. I can't fault dropping out the numeric tables, and replacing them with pointers to the original paper book, along with augmented references. They've added a section on software for computing the functions, with pointers to ACM algorithms[$$]. I might wish for a more thorough introduction to the subject of the Gamma Function, but the original A&S book certainly didn't do more than scratch the surface. And the introductory material could well get in the way of looking up some fine detail. What's wrong with this picture? Rich

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