May 2010 - mathfun@mailman.xmission.com

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

10 May '10

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. Tanh is also arbitrary but "in the family" at least. I wondered whether the function is lumpy, "unstable", or particularly tied to the base e. I decided to compare it to the base-two version of the same function: sinh2 = ( 2^x + 2^-x ) / 2 = sinh( x ln 2 ) tahn2 = ( 2^x - 2^-x ) / ( 2^x + 2^-x ) = tanh( x ln 2 ) f2(x) = (2/(2-ln(2))) sinh2( f2( x - tanh2( x/2 ) ) ) The functions are impossible to visualize with linear or log scales, but their inverse functions make nice scales to graph each other by! So I plot fe^-1(y) vs. f2^-1(y). The theory is that, if I scale the inverse functions so that they match where y = MAX_FLOAT = 2^1024, then fe() and f2() have two different recursion periods, so if they are glitching or specialized to their bases, their glitches shouldn't coincide, and they will rat on each other. The short answer is that, over the range of floating point, the two functions look pretty smooth to each other. I play with the initial slopes and some different visualizations here: http://www.tiac.net/~sw/2010/03/Superbola/superbolavi.pdf & here's the code: http://www.tiac.net/~sw/2010/03/Superbola/superbolavi.py Unlike sinh and sinh2, these two functions aren't just scaled versions of each other. They wobble compared to each other through about a cycle over the range I was able to plot. I suspect that's the beat frequency between the two scaled recursion spacings. To see whether the wobbling increases or decreases over a longer period, I'd need a way to calculate with and convert between towers- of-powers notations-- is that something that's been worked out before? Is it even possible? > From: Fred lunnon <fred.lunnon(a)gmail.com> > > I didn't want to get to involved in detail at this stage, since I'm not sure > how relevant it might be. But the essence is in the rider, that "it's > desirable (in some sense) to minimise the growth of the derivatives with n". and > some of the features here remind me of my investigation in 2004 into > blending pairs of functions for graphics purposes. I'm curious about your technique, but fe() and f2() are sort of self-blending. I don't see any rationalization for them, but they get their smoothness nearly naturally. --Steve p.s. "blended" in coffee shops and restaurants means "processed in a blender," which irks me.

1 0

[math-fun] Mma exact triadic dragon
by Bill Gosper 09 May '10

09 May '10

w = 1/2 + I/(2 Sqrt[3]); Trid[t_, a_: 1, b_: 0] := Trid[t, x_: 0, y_: 0] = (Trid[t, s1_: 1, s0_: 0] = (b - s0)/(s1 - a); Module[{u = (t + 1)*3/2, n}, n = Floor[u]; u = 2*(u - n) - 1; ComplexExpand[ Switch[n, 0, w*(Trid[u, Expand[a*w], Expand[b + a*w - a]] + 1) - 1, 1, Trid[u, Expand[a/Sqrt[3]/I], b]/Sqrt[3]/I, 2, w*(Trid[u, Expand[a*w], Expand[b - a*w + a]] - 1) + 1, 3, 1]]]) Exactly maps rational -1 <= t <= 1 onto the tridaic dragon with the same endpoints, which is an complete definition since a continuous function is determined by its values on a dense set. Try, e.g., ListLinePlot[{Re[#], Im[#]} & /@ Table[Trid[t], {t, 1/81/3 - 1, 1, 2/81/3}], Axes -> None, AspectRatio -> Automatic] for self-avoidance, and ListLinePlot[{Re[#], Im[#]} & /@ Table[Trid[t], {t, -1, 1, 1/3^n/2}], Axes -> None] n>2 for interesting texture. Puzzles: What is the bounding box of the dragon's range? Give inverse images for a point on a hrizontal edge and a vertical edge. In how many points does this range "curve" intersect this box? Unsolved, but probably easy: What is Trid[theta], the radix 3 stutter-free Morse number formed by pairwise grouping the bits of the Thue number 0.110100110010110... and mapping 11->2, 10->1, 01->1, and 00->0 (ceiling[n/2] = digit sum)? Bonus: Self-avoiding Z-zerpent: ListLinePlot[{Re[#], Im[#]} & /@ Table[Trid[t], {t, -2/3, 0, 2/81/3}], Axes -> None, AspectRatio -> Automatic] --rwg >Omigod, I've been using Vaseline five years past expiration! To make matters worse, the jar fails to list the percentage of inert ingredients! Answers: Trid[13/40] = (-1 - i 3 rt 3)/8, Trid[39/40] = (9 - i rt3)/8, e.g. # intersections = oo! (Subpuzzle: countable?)

1 0

Re: [math-fun] Financial trading in a relativistic world
by Dan Asimov 09 May '10

09 May '10

Thanks for posting that, Henry -- it is pretty exciting. Though I bet it's even more so for those who unlike me understand what's going on. (I worked in high finance for a bank '87-'91, but that was before the repeal of much of Glass-Steagall.) I can't help wondering if there's a known correlation between a) attempted terrorist incidents and b) stock market crashes shortly afterwards. --Dan Henry wrote: << Here's an audio blow-by-blow from the S&P trading pit, approx. 11 minutes; pretty exciting: . . . >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

1 0

Re: [math-fun] Financial trading in a relativistic world
by Axel Vogt 09 May '10

09 May '10

I think it is deeper an you find some discussion between practitioner at http://www.nuclearphynance.com/Show%20Post.aspx?PostIDKey=142075 (though it is not much about automated trading and hedging, more about that very event and rules related)

2 1

[math-fun] Financial trading in a relativistic world
by Henry Baker 09 May '10

09 May '10

It appears that 100msec (fiber optic speed approx 1/2 way around the world) was too slow to keep up with the financial markets last Thursday. http://fridayinvegas.blogspot.com/2010/05/market-speed-bumps.html?showComme… Financial trading is based on a Newtonian model where pricing is the same everywhere at the same time. Fiber optic speed is about 2/3 c, so one could presumably get a slight edge by using radio waves. If one were on the other side of the Earth, one could presumably get an edge if one could somehow utilize neutrinos to communicate. But sooner or later, this Newtonian pricing model has to be dispensed with. I've seen zero discussion of this issue, as of yet.

1 0

Re: [math-fun] Innumeracy, regulation & education
by Henry Baker 08 May '10

08 May '10

When someone says "negative feedback loop" instead of "vicious cycle", they just admitted that they didn't really understand the concept of feedback loops. Since these are the folks writing the rules, I'm not about to give them a pass on this mistake. If you're searching for _mines_, I'll give you more credit for finding more mines. However, how do you search for "things likely to exacerbate a chaotic system"? How do you even know the system is chaotic in the first place? How do you find the regions that are chaotic -- before stepping into them? And these folks think that they are knowledgeable and clever enough to design a "systemic regulator"? Re antibiotics & crime: it is well-known/well-understood that there are significant political forces arrayed against eliminating all crime; you wouldn't need policemen, prosecutors, prison guards, etc. We fought & won the war against certain diseases -- primarily through vaccinations -- only to have a certain percentage of the population now refuse to vaccinate their children; these diseases are now coming back. MRSA exists because people refuse to finish their antibiotic regimens, allowing a small % of the bugs to mutate & become resistant. The drug companies have now concluded that _curing_ people of a disease isn't particularly profitable, so they now focus on "managing" the disease, so the patient has to take their medicine for the rest of their lives; at least these companies now have an incentive to keep the patients alive as long as possible! Re stabilizing control through feedback: there is usually a tradeoff between speed of control and instability. If you want your system to be able to react quickly, it has to be closer to the point of instability. If you want to avoid all instability (perhaps impossible), you end up with a very sluggish system. Example: high performance fighter aircraft, which require the ability to quickly change course, are inherently unstable without active (computer) control. There is a point for every system where you are willing to put up with some risk of instability in order for the system to be able to respond more quickly to a changing environment. An analogy can be made to converging sequences. My heuristic is that faster convergence may entail more violent initial swings (Gibbs phenomenon?). Smooth, monotonic convergence may be possible, but is more likely to be slower convergence. (Its been a long time since undergrad days -- is there a theorem about this?) I should have expanded my last point about these regulatory regimes requiring more design tools & testing. If Boeing or Toyota released an airplane or an automobile that hadn't been thoroughly tested & it crashed, there would be a huge demand for heads to roll. Yet politicians & bureaucrats routinely inflict untested theories on the public all the time, leaving immense damage in their wake. They don't even have the courtesy of testing these theories on a small % of the population before inflicting them on everyone simultaneously. It's as if Microsoft (or MacAfee) forced every user to upgrade his systems every night _with the latest untested batch of changes_. At 01:50 PM 5/8/2010, Dan Asimov wrote: >>> It seems hardly worthy of mention if someone was trying to say "vicious cycle" but instead said "negative feedback loop". << > >>> This sounds like saying that because our minesweepers are flawed and can't even detect some mines at all, we shouldn't even try to find them. << > >>> There's certainly some truth to this and closely related ideas. For example, recent studies have shown that often, the more that law enforcement tries to clamp down on violent, powerful drug organizations, the more virulent and powerful they tend to become. (A bit like taking a course of antibiotics for an infection, but stopping before it's fully wiped out. Which is not to say we should engage in a battle to the finish with such drug organizations . . ..) And certainly there are computer-automated trading routines whose behavior in many situations is not known and could be (and could have already been) disastrous. I said *some* truth, because there is much regulation that seems successful, like your most basic law enforcement, laws that make many utilities *public* utilities so we don't end up getting gouged for a glass of water, laws that prevent 51% or the population from tyrannizing the other 49%, etc.

1 0

[math-fun] Innumeracy, regulation & education
by Henry Baker 08 May '10

08 May '10

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. I'm not at all confident that these folks have a clue, in which case any cure they propose is highly likely to be worse than the disease. Given that these people are often from the best universities (Ivy League, etc.), I think their innumeracy demonstrates that our university educational system has failed society miserably, at least when it comes to some relatively fundamental mathematical concepts. Example 1. A pronouncement that the stock market crash of last Thursday was caused by a "negative feedback loop". Of course, it was far more likely to have been caused by a "positive feedback loop"; negative feedback loops are typically self-stabilizing, while positive feedback loops are the ones that explode exponentially. Example 2. After all the incessant talk of "chaos" and "fractals" over the past 30+ years, these basic ideas still don't seem to have percolated into brains of the intelligentsia. When a system is chaotic, there may be no amount of "regulation" that can fix it. If a butterfly wing flap can cause a hurricane to hit New Orleans instead of Tallahassee, what do you expect a regulator to do, kill all the butterflies with DDT? Example 3. We've known since GÃ¶del that every sufficiently powerful logical system can have undecidable consequences. In the mean time, we've lowered the bar for "sufficiently powerful" to a very low level by showing that extremely simple systems qualify -- this is particularly so if we consider "practically undecidable" instead of just "undecidable". Furthermore, the source of this undecidability is (in effect) the ability of the system to model & reason about itself. This provides a powerful heuristic: the first place to look for instability in a system is where a system is attempting to regulate itself -- e.g., Congress trying to make rules that apply to Congress itself. How is it that someone can graduate from a top university with a major in economics or law today and not be aware of these three concepts? How could someone who didn't understand these concepts _ever_ be expected to design regulations that might make things better? I'm not ranting about regulation, per se. I fully support the concept of umpires who enforce rules, but these rules have to be very carefully designed. Such rule design is more important than the design of a modern jetliner, and deserves computer-aided design tools that are at least as sophisticated as those used by Boeing in designing jetliners.

2 1

Re: [math-fun] Innumeracy, regulation & education
by Dan Asimov 08 May '10

08 May '10

Henry wrote: << 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. I'm not at all confident that these folks have a clue, in which case any cure they propose is highly likely to be worse than the disease. Given that these people are often from the best universities (Ivy League, etc.), I think their innumeracy demonstrates that our university educational system has failed society miserably, at least when it comes to some relatively fundamental mathematical concepts. Example 1. A pronouncement that the stock market crash of last Thursday was caused by a "negative feedback loop". Of course, it was far more likely to have been caused by a "positive feedback loop"; negative feedback loops are typically self-stabilizing, while positive feedback loops are the ones that explode exponentially. >> It seems hardly worthy of mention if someone was trying to say "vicious cycle" but instead said "negative feedback loop". << Example 2. After all the incessant talk of "chaos" and "fractals" over the past 30+ years, these basic ideas still don't seem to have percolated into brains of the intelligentsia. When a system is chaotic, there may be no amount of "regulation" that can fix it. If a butterfly wing flap can cause a hurricane to hit New Orleans instead of Tallahassee, what do you expect a regulator to do, kill all the butterflies with DDT? >> This sounds like saying that because our minesweepers are flawed and can't even detect some mines at all, we shouldn't even try to find them. << Example 3. We've known since Gödel that every sufficiently powerful logical system can have undecidable consequences. . . . This provides a powerful heuristic: the first place to look for instability in a system is where a system is attempting to regulate itself -- e.g., Congress trying to make rules that apply to Congress itself. >> There's certainly some truth to this and closely related ideas. For example, recent studies have shown that often, the more that law enforcement tries to clamp down on violent, powerful drug organizations, the more virulent and powerful they tend to become. (A bit like taking a course of antibiotics for an infection, but stopping before it's fully wiped out. Which is not to say we should engage in a battle to the finish with such drug organizations . . ..) And certainly there are computer-automated trading routines whose behavior in many situations is not known and could be (and could have already been) disastrous. I said *some* truth, because there is much regulation that seems successful, like your most basic law enforcement, laws that make many utilities *public* utilities so we don't end up getting gouged for a glass of water, laws that prevent 51% or the population from tyrannizing the other 49%, etc. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

1 0

[math-fun] Anagram from the May 2010 "The Enigma" (puzzlers.org)
by Thane Plambeck 07 May '10

07 May '10

FOR MASTER MATHLETES [*6'1 4 7] JANGLER, Nashville TN -- Thane Plambeck tplambeck(a)gmail.com http://thaneplambeck.typepad.com/

2 1

Re: [math-fun] Anagram from the May 2010 "The Enigma" (puzzlers.org)
by Dan Asimov 07 May '10

07 May '10

Cute. Thane wrote: << FOR MASTER MATHLETES [*6'1 4 7] JANGLER, Nashville TN >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

1 0