February 2010 - mathfun@mailman.xmission.com

[math-fun] Symbol for rotation?
by Dan Asimov 11 Feb '10

11 Feb '10

I wonder if there is a standard symbol in math or physics or another field for one rotation. (Of course this is 2pi radians, but for certain purposes it would be more convenient to refer solely to the number of rotations, not radians.) If there is no standard symbol for this, I propose a circle with a dot in the center. Comments? --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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Re: [math-fun] Symbol for rotation?
by Dan Asimov 11 Feb '10

11 Feb '10

Bob Palais wrote an article in the Math Intelliencer (Summer 2001) called "Pi is wrong!", in which he argued that 2pi is a more natural constant and suggested a symbol for it (a pi modified to have a third vertical stroke). < http://www.math.utah.edu/%7Epalais/pi.pdf > But that's not quite what I'm looking for; I want to bypass radians entirely. --Dan Marc wrote: << << Someone whose name I forget (do you recall Neil?) came by the OEISF booth in SF and in passing mentioned that he was having fun rewriting various things using "the correct value of pi", namely 2pi. Alas I don't recall the symbology, but I liked his adoption of "turn" for the unit of full rotation (previously I've resorted to "cycle", from pre-metricked-up audio)... [I wrote:] << I wonder if there is a standard symbol in math or physics or another field for one rotation. (Of course this is 2pi radians, but for certain purposes it would be more convenient to refer solely to the number of rotations, not radians.) If there is no standard symbol for this, I propose a circle with a dot in the center. Comments? >> >> ________________________________________________________________________________________ "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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Re: [math-fun] Symbol for rotation?
by Marc LeBrun 11 Feb '10

11 Feb '10

Someone whose name I forget (do you recall Neil?) came by the OEISF booth in SF and in passing mentioned that he was having fun rewriting various things using "the correct value of pi", namely 2pi. Alas I don't recall the symbology, but I liked his adoption of "turn" for the unit of full rotation (previously I've resorted to "cycle", from pre-metricked-up audio)... ======== At 11:52 AM 2/11/2010, Dan Asimov wrote: >I wonder if there is a standard symbol in >math or physics or another field for one rotation. > >(Of course this is 2pi radians, but for certain >purposes it would be more convenient to refer >solely to the number of rotations, not radians.) > >If there is no standard symbol for this, I propose >a circle with a dot in the center. Comments? > >--Dan > >________________________________________________________________________________________ >"Outside of a dog, a book is man's best friend. Inside of a dog, >it's too dark to read." --Groucho Marx > >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] An alternating symmetric polynomial ?
by Eugene Salamin 09 Feb '10

09 Feb '10

Puzzle: Does there exist a nonzero polynomial P(x,y,z) that is symmetric in its arguments and is alternating in the sense that P(x,x,z) = 0 identically ? -- Gene

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[math-fun] OEIS tweaks: triangular socks
by Bill Gosper 08 Feb '10

08 Feb '10

A029549 formula: (cosh((4*n+2)*log(1+sqrt(2)))-3)/16 . = binomial(A001652+1,2) = 2*binomial(A053141+1,2). (Correcting(?) M. Harris formula.) Mathematica: Intersection[#,2*#]&@Table[Binomial[n,2],{n,999999}] Macsyma: (makelist(binom(n,2),n,1,999999),intersection(%%,2*%%)) -------------------------------- A001652 formula: (sinh((2*n+1)*log(1+sqrt(2)))-1)/2 = (sqrt(1+8*A029549)-1)/2 binomial(A001652+1,2) = 2*binomial(A053141+1,2) = A029549 . See A053141. ------------------- A053141 formula: sqrt(2)*cosh((2*n+1)*log(1+sqrt(2))/4 - 1/2 = (sqrt(1+4*A029549)-1)/2 . Jason Holt observes: A pair drawn from a drawer with A053141(n)+1 red socks and A001652(n) - A053141(n) blue socks will likely as not be matching reds: A053141+1 A053141 1 --------- ------- = -, n>0. A001652+1 A001652 2 --rwg CHINA TREES CHINESE RAT CATHERINE'S CHESTERIAN HERITANCES

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Re: [math-fun] cybernetics
by Henry Baker 05 Feb '10

05 Feb '10

Essentially all "insurance" is tail-stuffing. When you purchase insurance, you pay a modest premium which covers your losses (typically up to some maximum limit), which insurance eliminates your losses during the period of coverage (at least until you reach the maximum limit). In some cases where the probability distribution function is well-understood and the tail can be integrated -- e.g., a "normal" distribution -- then one could conceivably remove the need for a maximum limit, since the probability of achieving an event of many, many sigmas away from the mean would be less than 1/the number of elementary particles in the universe. Since most of us believe in thermodynamics, we know that while it is theoretically _possible_ for all of the air molecules in the room where we reside to group into one small corner, leaving us to suffocate, the likelihood of this happening exceeds a large number of lifetimes of the universe. So truly normal curves are pretty safe bets. A party which "writes" insurance integrates the most likely portion of the distribution curve, includes a profit margin, and that becomes the basis for the premium; the insurance writer also includes the integration limits as its maximum limits in the insurance policy. Insurance writers often write insurance whose limits are too large for them to handle. In these cases, they will "reinsure" the portions of the tail that they are still on the hook for with another party. If reinsurers do their homework, they make sure that the risks they are being asked to insure are uncorrelated. However, this is really, really hard to discover. An analogy is in order. A company wishing to have redundant fiber optic communications between New York and Boston will go to a lot of trouble to purchase fiber from two different companies. However, they may have no way of knowing that both companies route their fiber through the same physical bridge in Connecticut. A gasoline truck hits the bridge & burns it down & wipes out both fibers. (This is a true story.) As reinsurance gets passed through more & more levels, it becomes impossible to know or find out exactly what correlations there are between policies. A number of reinsurers were wiped out in the 1980's in this way. AIG became a big writer of insurance on debt defaults -- so-called "credit default swaps" (none dare call it "insurance", since that would put these derivatives under the control of a different regulator who actually did understand insurance, and would have required much higher reserves) -- and due to AIG's size, there wasn't anyone else big enough to purchase reinsurance from. But because of the number, size and connectedness of AIG's customers, AIG became TBTF (too big to fail), and de facto put a number of govts on the hook as the reinsurers of last resort. We know in hindsight that the premiums AIG received were probably far too low to cover even their foreseeable risks, much less the really fat tail risks. And those fat tail risks should have required far higher reserves to cover them. For example, some special purpose companies (by definition "limited liability") were capitalized with something like $1-2 million, and proceeded to write insurance on billions of dollars of debt. The folks at Goldman Sachs, et al, try to achieve an essentially "neutral" position every evening, so if the overnight markets either go up or down sharply, they won't take losses from movements in either direction. GS et al utilize huge computers & lots of clever people to try to minimize the correlation of all of their bets, and buy & sell derivatives to achieve these ends. If their models are good & their software is accurate, they minimize risks to themselves and pass that risk onto their counterparties, who also try to do the same thing for themselves. Since each party is trying to insure itself against losses, it builds up a web of interconnected counterparties who are all tied together, like mountain climbers with rope. If any one climber loses his footing, the rope to the others will easily hold his weight & he is saved. However, if a large enough subset of climbers all fall at the same time, their sheer bulk will take down the remainder. The only solution is to have the rope tied to a much more massive object -- e.g., the mountain itself, or more likely Uncle Sam. But Uncle Sam didn't realize he was on the hook for these potential losses, and didn't charge any insurance premium at all. At 05:30 PM 2/4/2010, Dan Asimov wrote: >Having worked for 4 years as a rocket scientist >in high finance myself (which apparently qualified >me for my next job -- at NASA), I don't think that's >how it works. > >The fact is, no one knows how to estimate those risks, >even close. As most people know, probability applies >to the real world only in special cases. The status >of financial markets is constantly changing over time, >in different statutory and regulatory and economic >environments, most of which have never been seen before. >Smart quants will make their best effort to model risks >accurately, but no one really knows how to do it. > >The blame, IMHO, belongs to the people directly above them, >who give them their marching orders and who are or should >be well aware of the impossibilty of quantifying risk, >or even quantifying the error in the best estimates >of risk. > >--Dan > ><< >The math wizards of Wall Street are employed at enormous salaries to do "tail-stuffing", wherein all minor risks (i.e., those that might reduce their bonuses) are pushed down into the probability distribution "tail", where the probability remains low, but the expected value is significantly increased. Net result: nice (more-or-less) smooth privatized profits, with state-subsidized catastrophic losses. "I be gone, you be gone, we all be gone". >>>

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Re: [math-fun] cybernetics
by Dan Asimov 05 Feb '10

05 Feb '10

Having worked for 4 years as a rocket scientist in high finance myself (which apparently qualified me for my next job -- at NASA), I don't think that's how it works. The fact is, no one knows how to estimate those risks, even close. As most people know, probability applies to the real world only in special cases. The status of financial markets is constantly changing over time, in different statutory and regulatory and economic environments, most of which have never been seen before. Smart quants will make their best effort to model risks accurately, but no one really knows how to do it. The blame, IMHO, belongs to the people directly above them, who give them their marching orders and who are or should be well aware of the impossibilty of quantifying risk, or even quantifying the error in the best estimates of risk. --Dan << The math wizards of Wall Street are employed at enormous salaries to do "tail-stuffing", wherein all minor risks (i.e., those that might reduce their bonuses) are pushed down into the probability distribution "tail", where the probability remains low, but the expected value is significantly increased. Net result: nice (more-or-less) smooth privatized profits, with state-subsidized catastrophic losses. "I be gone, you be gone, we all be gone". >> ________________________________________________________________________________________ "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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Re: [math-fun] cybernetics
by Steve Witham 05 Feb '10

05 Feb '10

>From: Henry Baker <hbaker1(a)pipeline.com> > >I think that this entire area You mean the entire "complex systems" area rather than cybernetics, right? >has been subsumed by fractals, basins of attraction, etc. About cybernetics, this page http://en.wikipedia.org/wiki/Control_theory has lots of subtopics and a history overview from before Maxwell thru after Weiner. I.e. it seems to be both thriving and hard core. By the way I didn't mean to say that human-design or analysis of hierarchical control systems was a dead end, only that it didn't lead to to the grand unifying paradigm for biological, psychological, social and engineered systems that some hoped it would. >Most of the folks in economics aren't aware of how chaotic complex >systems can become, and they naively continue thinking that these >systems can be "controlled"/"regulated". If we would only >"regulate" the economy, we could somehow avoid the chaos. I think chaos ideas are a clear fit in classical economics. I'm agnostic about what most economists know. >From: Dan Asimov <dasimov(a)earthlink.net> > >The control theory I'm aware of is quite specific: It addresses >the problem of how to steer most efficiently from one point >to another in some phase space -- assumed locally Euclidean -- >when the allowable steering options are subject to infinitesimal >constraints. (Example: How do you best roll a sphere on a plane >without slippage so its net motion is some prescribed >rotation plus translation? Here "best" is defined by minimizing >some cost function defined on all allowable itineraries.) More generally: maintain some relationship between goal inputs and the current state given your sensors and actuators and other limitations and prejudices up to the designer. As I said, the wikipedia page gives an idea of the variety, from thermostats to inverse kinematics to AI. >Don't know much economics. Are there standard economics models >as "multilayered feedback systems" ? It's important to distinguish what economies do from what we can model about them. Good economists know in a common-sense way that economies are multi-level (for instance, a corporation has employees, managers and stockholders), but I'm not aware of any useful models that have that same character themselves. I guess economics overlaps science and kookery. >Despite having done research in differential dynamical systems, >I never really understood what the field of "Complex Systems" >or for that matter "Chaos Theory" were (and extend my >skepticism to those fields as well). At least notions like chaotic and "strongly mixing" have precise definitions with scientific (will the solar system eventually eject a planet?) and engineering ("divide and concur" algorithms) applications. ><< >The funny thing is, hierarchical feedback systems are making a >comeback with people like Jeff Hawkins, only it turns out every >little piece of knowledge is its own feedback loop, > >> > >How so? Just in the technicalities of Hawkins' (synthesizing others') ideas. Everything's a minicolumn that has learned to do something in the process of two-way interactions up and down the cortical hierarchy. >I like the idea of hierarchical feedback systems (though maybe >they can always be relabeled so as to be non-hierarchical?). Right, the distinction between a system in which hierarchy tends to build up, and one that somebody designed or analyzed in a layered picture. >In any case they're probably useful in a field I'd love to >see more progress in -- causal inference. Like/not like Jeff Hawkins (_On Intelligence_ and Numenta)? --Steve

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Re: [math-fun] cybernetics
by Henry Baker 04 Feb '10

04 Feb '10

Steve: I've espoused the same economic regulation = forest fire management myself before, but I haven't heard others use it. Do you have a reference to such a blog? Redundancy in complex systems is a good way to keep small amounts of damage from becoming catastrophic. Margin/buffer space is a good way to keep minor infractions from becoming catastrophic. Freedom of individual action (computer science terminology "optimistic concurrency") keeps a complex system from seizing up due to gridlock. The math wizards of Wall Street are employed at enormous salaries to do "tail-stuffing", wherein all minor risks (i.e., those that might reduce their bonuses) are pushed down into the probability distribution "tail", where the probability remains low, but the expected value is significantly increased. Net result: nice (more-or-less) smooth privatized profits, with state-subsidized catastrophic losses. "I be gone, you be gone, we all be gone". Unlike the pilots of airliners who (currently, at least) take the same risk of life & limb as the passengers, the wizards of Wall Street are playing video games with the economy in a manner eerily similar to Predator drone pilots. Neither wizard nor drone pilot has any "skin in the game". At 02:29 PM 2/4/2010, Steve Witham wrote: >>I think that this entire area has been subsumed by fractals, basins of attraction, etc. >> >>Most of the folks in economics > >For instance, today I read a blog post comparing regulation to wildfire >management: there used to be a policy of preventing wildfires, but the >result was bigger and bigger accumulations of kindling until you had >a gigantic wildfire; current philosophy says let the little fires burn >and limit damage at the edges. Similarly, downturns are a natural way >of weeding out deadwood businesses, contracts, employment arrangements, >products, etc. (again, standard micro view), and propping up the losers >in small downturns probably pushes us toward large ones. > >Like forests, economies have evolved fallbacks; although the world >economic situation evolves much more quickly, I think there's a kind of >meta-regulation going on. I.e. pricing, trade, investment, etc. are >short-to-medium-term activities that produce some mixture of stability >and chaos, but there's a evolving-toward-the-edge-of-chaos thing going >on that effectively adjusts long-term strategies. My guesswork.

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Re: [math-fun] cybernetics
by Steve Witham 04 Feb '10

04 Feb '10

>From: Dan Asimov <dasimov(a)earthlink.net> > >I know, or think I know, Norbert Wiener coined the word cybernetics >and wrote a book about it. [...] Me too. >But [...] the dictionary definition of cybernetics: > ><< >: the science of communication and control theory that is concerned >especially with the comparative study of automatic control systems >(as the nervous system and brain and mechanical-electrical >communication systems) > >> The hard results were and still are in control theory. I know Wiener's no slouch, I think what happened was people hoped to get traction by looking at feedback loops, loops controlling the parameters of loops, etc., in hierarchies... but didn't get anywhere. So it is cybernetics itself (not just this conference) that has kooks. For instance people who want to manage economies with feedback systems, without really getting that economies already are multilayered feedback systems attempting to "route around damage" as the saying goes. It's similar to how Complex Systems had some interesting results and then kooks. The funny thing is, hierarchical feedback systems are making a comeback with people like Jeff Hawkins, only it turns out every little piece of knowledge is its own feedback loop, so (assuming they work) you don't get general high-level results, just general infrastructure designs and lots of specific stuff for a system to learn. --Steve

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