May 2010 - mathfun@mailman.xmission.com

Re: [math-fun] Puzzle: In Julian's inverted gasket
by Bill Gosper 16 May '10

16 May '10

I assume you're talking about this: http://www.wm.uni-bayreuth.de/fileadmin/Sascha/Publikationen2/rare.pdf which as far as I know is still the state of the art: a heptagon with all integer distances. --Joshua Tnx! Interesting how close both sol'ns come to having 3 in a row. --rwg PS, the Triadic Dragon barely grazes its bounding box--in four uncountable Cantor sets! cooperationist cationotropies cephalometric petrochemical animal starch charlatanism ammonolytic commonality astonished dositheans on its heads osteologic tocologies saponified adipose fin low german angleworm

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[math-fun] Puzzle: In Julian's inverted gasket
by Bill Gosper 15 May '10

15 May '10

http://gosper.org/Inverted%20Sierpinski%20Triangle%20thick.png some triangles are biconvex and some biconcave, and some hard to tell. Are there any straight arcs? --rwg > Apropos lines and circles, Tatiana Shubin requests the status of the > restricted Anning's(?) > problem: How many points can lie in a plane with no three in a line nor four in a circle? Last she heard, it was up to 7 or 8.

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[math-fun] Exact satanic staircase
by Bill Gosper 14 May '10

14 May '10

Unboundedly declivitous near the Cantor set: http://gosper.org/satan.png SS[t_, a_: 1, b_: 0] := Module[{u = t*3, h}, If[1 <= u <= 2, 1 - t, SS[t, x_: 0, y_: 0] = (SS[t, s1_: 1, s0_: 0] = (b - s0)/(s1 - a); h = Floor[u/2]/3; SS[u - 6*h, a*2/3, b + a*h]*2/3 + h)]] Monitor[Plot[SS[x], {x, 0, 1}], Row[{ProgressIndicator[x, {0, 1}], x}]] (You can watch Plot skipping the smooth parts on the refinement passes.) --rwg Texture morphed Triadic dragon (sampled at beating frequency): http://gosper.org/morphdrag.png Self avoiding z-zerpents: http://gosper.org/sazz.png Julian redid http://gosper.org/Inverted%20Sierpinski%20Triangle%20thick.png

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[math-fun] Exact Devil's Staircase
by Bill Gosper 13 May '10

13 May '10

DS[t_, a_: 1, b_: 0] := Module[{u = t*3, h}, If[1 <= u <= 2, 1/2, DS[t, x_: 0, y_: 0] = (DS[t, s1_: 1, s0_: 0] = (b - s0)/(s1 - a); h = Floor[u/2]/2; DS[u - 4*h, a/2, b + a*h]/2 + h)]] E.g.; In[176]:= DS[3/13] Out[176]= 2/7 In[178]:= DS[.1] Out[178]= 3435973837/17179869184 In:= %-(1+2^-34)/5 Out= 0 My tutor Julian suggests 4*DS[x]-3*x as a continuous map of [0,1] onto [0,1] which preserves both endpoints and has slope -3 A.E. For good measure, it does this twice. (http://gosper.org/devil.png) It's tough to keep that guy's hands off my keyboard. But then why would I? (http://gosper.org/rosewindow.png) --rwg

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[math-fun] NIST's new version of Abramowitz & Stegun is up
by rcs＠xmission.com 13 May '10

13 May '10

NIST has had a multi-year project to produce a web-based successor to Abramowitz & Stegun. They've gone public with the result, at http://dlmf.nist.gov/ I guess my expectations were impossibly high. The DLMF seems inferior to the physical A&S book in many respects. The web version does take advantage of various web possibilities. It has some very pretty graphs. It seems to work well in my ancient Mozilla browser. Moving the mouse near a formula brings a popup that explains the notations. They have dissed numerical tables :-(, and have minimized numerical methods. As a formula compendium, they give the basics, but most often refer to other references. Many of these are online. (You may need your credit card, but it's faster than walking over to the library.) They cover the bare bones of interpolation. Numerical integration of ODEs stops after basic Runge-Kutta. I couldn't figure out from the Table of Contents which chapter had Dilogarithm, but it showed up immediately in the index. The Dilog section is still missing the two-variable functional equation, D's most important property. Plus ca change ... On the plus side, they have extensive pointers to mathematical software, and have taken care to mark Free and Pay categories. There are several new chapters, including multivariate theta functions, and functions with matrix arguments. I didn't see a button for "download the whole thing so I can use it offline". Browsability is so-so: they provide Next/Previous buttons so you can read through a chapter, and lots of crosslinks. I found too little information per page for my taste, but tastes vary. In terms of usefulness, I dunno: If I want to learn about the hyperbolic-zilch function, I might do better to use Wikipedia or Google. DLMF does provide an overall viewpoint and organization, a roadmap for newcomers. As a youngster, some of my favorite books were the CRC mathematical tables book, and similar compilations. I first encountered A&S in college, and it had a similar "draw", the attraction of a big book of mysterious but accessible symbols and associated knowledge. DLMF doesn't have that feel to it. There is a companion book from Cambridge, which I didn't investigate. It may be that the time for this kind of object has passed, that the knowledge is better organized on Wikipedia & the like. If we need compendia of formulas, there will be online versions of Gradshteyn & Ryzhik, the Bateman project, and Jahnke & Emde soon, if not already. Overall, a worthwhile attempt. Have a look. Rich

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Re: [math-fun] Financial trading in a relativistic world
by Henry Baker 11 May '10

11 May '10

You can't make this stuff up. Here may be an example of "self-application" in the real world. Nassim Taleb, who has written extensively on "Black Swans" & "fat-tailed distributions", may have himself inadvertently set off last Thursday's "flash crash" in the stock market !! Elsewhere in the article, it talks about high-frequency trading in places like Chicago, Europe, Austin, Kansas City, NJ, Santa Monica, NYC, etc. I can readily believe that the effective communications latencies among some of these locations are far north of 100 msecs, so it would be impossible for these various computers to agree on a price quickly enough to avoid instability. The Wall Street Journal BUSINESS MAY 10, 2010 Did a Big Bet Help Trigger 'Black Swan' Stock Swoon? By SCOTT PATTERSON And TOM LAURICELLA http://online.wsj.com/article/SB10001424052748704879704575236771699461084.h… Shortly after 2:15 p.m. Eastern time last Thursday, hedge fund Universa Investments LP placed a big bet in the Chicago options trading pits that stocks would continue their sharp declines. On any other day, this $7.5 million trade for 50,000 options contracts [big enough to pay off $4 billion in June if the S&P index hit 800] might have briefly hurt stock prices, though not caused much of a ripple. But coming on a day when all varieties of financial markets were deeply unsettled, the trade may have played a key role in the stock-market collapse just 20 minutes later. The trade by Universa, a hedge fund advised by Nassim Taleb, author of "Black Swan: The Impact of the Highly Improbable," led traders on the other side of the transactionÂincluding Barclays Capital, the brokerage arm of British bank Barclays PLCÂto do their own selling to offset some of the risk, according to traders in Chicago. Then, as the market fell, those declines are likely to have forced even more "hedging" sales, creating a tsunami of pressure that spread to nearly all parts of the market. ...

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Re: [math-fun] Innumeracy, regulation & education
by Steve Witham 11 May '10

11 May '10

Henry Baker <hbaker1(a)pipeline.com>: > I'm as upset as anyone by the financial crisis(es?) of the last > several years, but I'm even more upset by the lack of mathematical > sophistication of those in charge. Although economics, feedback and chaos are related to mathematics, I think mathematical sophistication is less important than understanding the basics of the underlying situations. Most of these things have been pretty well understood for centuries under different names. In fact, out-of-context math, or any out-of-context specialized knowledge, becomes a smokescreen in many cases. I also doubt worldwide computerized trading networks involve any principles or issues much different than trading by camel caravan involved. Relativity just means finite communication speed in this case, an old familiar problem. If people are setting up fragile systems, there's something more than mere speed going on. Ways of buffering shocks are also an old problem. One thing to look at if people aren't cautious enough is whether caution is being punished by regulation, or carelessness rewarded. There is a bit of math sophistication that asks, are the spikes and dips we see more than what statistics and theory say we should expect? Is there even any news here? --Steve

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Re: [math-fun] more less lumpy fun functions
by Steve Witham 11 May '10

11 May '10

Steve W. writes: ><< >This is the tower-of-exponentials I'm playing with now: > >(*) fe(x) = 2 sinh( fe( x - tanh( x/2 ) ) ) > >As x --> +oo, fe(x) --> e^f( x-1 ), should be --> e^fe( x-1 ), >as x --> -oo, fe(x) --> -e^-f( x+1 ) (or just -f(-x)), and should be --> -e^-fe( x+1 ) (or just -fe(-x)), and >as x --> 0, fe(x) --> 2 fe( x/2 ). > >To calculate despite the circular definition, I assume a straight line >with some slope when abs(x) < 2^-27, where sinh and tanh are linear >enough that the location of the seam is invisible to floating point. > >> From: Dan Asimov <dasimov(a)earthlink.net> >Is it clear that the functional equation I've labeled (*) has a unique >solution fe ? It's not unique. Lots of crazy solutions, but maybe this is unique: First pick an arbitrary initial slope, then take The limit of fe( x, init_slope ) = x * init_slope if |x| <= cutoff 2 sinh( fe( x - tanh( x/2 ) ) ) if |x| > cutoff as cutoff --> 0 (from above). I'm not even sure that that converges. The Python function is piecewise as above. There's a discontinuity (not just discontinuous derivatives) at the ends of the line segment, and at every echo of the ends, but the first one is on the order of one part in 2^-54, under the floating point noise, and the echoes of it are smaller than the echoes of that noise. --Steve

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Re: [math-fun] Terminology for exponentiation
by Steve Witham 10 May '10

10 May '10

>From: Dan Asimov <dasimov(a)earthlink.net> > >When we write a base x to a power c to get the number x^c >we say we "exponentiate x to the power c". Or just "raise" x to the c. >I seek an appropriate verb so the the power is the object, >reversing the emphasis: > >I.e., we use x *to do what* to c ??? Yarg, "base" ...something...on x? I've been thinking about, oh say, "towering" or "erecting" expressions like e^e^e^e^e^e^c (6 e^ 's). I haven't figured out whether the base even matters in the long run: is e^e^e^e^e^e^e^e "much different from" 2^2^2^2^2^2^2^2? --Steve

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Re: [math-fun] more less lumpy fun functions
by Dan Asimov 10 May '10

10 May '10

Steve W. writes: << This is the tower-of-exponentials I'm playing with now: (*) fe(x) = 2 sinh( fe( x - tanh( x/2 ) ) ) As x --> +oo, fe(x) --> e^f( x-1 ), as x --> -oo, fe(x) --> -e^-f( x+1 ) (or just -f(-x)), and as x --> 0, fe(x) --> 2 fe( x/2 ). To calculate despite the circular definition, I assume a straight line with some slope when abs(x) < 2^-27, where sinh and tanh are linear enough that the location of the seam is invisible to floating point. >> Is it clear that the functional equation I've labeled (*) has a unique solution fe ? --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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