November 2013 - mathfun@mailman.xmission.com

[math-fun] An interesting challenge
by Victor Miller 10 Nov '13

10 Nov '13

I just saw the following problem on the POW (problem of the week) list run by Stan Wagon. This looks like an interesting computing challenge: Problem 1171 Four Unary Functions Call n, a positive integer, "good" if one can start with n and use the operations factorial, square root, floor and ceiling to obtain n, using at least one of the first two operations. For example, 7 is good because Ceiling[Sqrt[7!]]! after extracting 7 square roots gives 6.3, and so a final ceiling gives 7 (see http://mathforum.org/wagon/fall13/p1171.jpg) Show that n is good for all n <= K, where K is as large as you can get. Note: Here factorial can operate only on integers. So while the example above, using the gamma function, works using Sqrt[7!]!instead of Ceiling[Sqrt[7!]]!, that is not legal. The assertion that all positive integers are good is a conjecture of Don Knuth and Richard Hess. Note that the actual representations are not sought, just the proof that the representation exists. It has been conjectured by D. Knuth and R. Hess that all positive integers are good. Source: Crux Math, M506, 38:8 Oct 2012, 310-313

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Re: [math-fun] What is the tennis-ball curve?
by Bill Gosper 09 Nov '13

09 Nov '13

On 2013-11-08 18:38, Fred Lunnon wrote: > For arbitrary sphere radius r , the curve has four nodal singularities > where x = 0 or y = 0 : these are real and obvious where the curve intersects > itself when r > 1 , but safely complex and out of harm's way when r < 1 . > Rather unexpectedly, there are four more real singularities at the > inflections where z = 0 and x = y : these correspond to a pair of lines > on the Enneper surface which superficially appear innocuous. > > There is a theorem that any TB curve must have (at least) four inflections, > see > http://www.qedcat.com/archive/165.html > --- it intrigues me that these points turn out to be so special here. > > Despite getting down and dirty with Magma's algebraic geometry feature, > I haven't yet managed to decide whether there are further singularities. > [Don't even think about trying to understand what schemes are --- just scope > the examples, then hack them!] > > Having gone this far, I couldn't resist putting the r ~ 0.2307718797455473 > curve onto a sphere. OK, I know it looks boring --- that's the whole point! > https://www.dropbox.com/s/ypsqc07tisw2gf2/tennis_ball.jpg > > Fred Lunnon > That's a pretty curve, but it accommodates a large inscribed circle, which requires large stretching to "map" onto the sphere. Minimizing stretching might motivate those racetrack shaped pieces we've seen. Recall years ago when the subject was the baseball stitch, we came up with *one piece* covers of arbitrarily low stretch, based on sphericons and fruit peels. gosper.org/onepiececover.png , gosper.org/tennis.gif It seems to me someone should manufacture tennis balls like this, just for the novelty. --rwg > > On 11/6/13, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: >> The Enneper-sphere tennis-ball curve is smooth, being the intersection >> of two algebraic surfaces. However it lacks a rational parameterisation, >> since (according to Maple) the plane curve |[x, y, z]|^2 = r^2 , qua >> function of parameters u,v , has genus = 8 rather than 0 . >> >> In practice such considerations are irrelevant, since a template needs >> to be computed only once, and to working tolerance. >> >> To "unroll" a strip of the corresponding spherical region onto a plane >> template requires a decision to be made about the appropriate projection. >> There doesn't appear to be a canonical answer to the latter question: >> it depends upon how much the cover material can be expected to stretch >> across its central line of symmetry, as opposed to wrinkling up along >> its boundary. >> >> Fred Lunnon >> >> >> On 11/6/13, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: >>> Customary typo correction --- should have read >>> >>> For the nice tennis-ball curve with parallel osculating planes >>> at its tips, the exact sphere radius is the root of >>> (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 >>> --- rather less than 1/4 . At the tips [x,y,z] ~ >>> [0, +0.1849108100, -0.1380711874] , >>> [0, -0.1849108100, -0.1380711874] , >>> [+0.1849108100, 0, +0.1380711874] , >>> [-0.1849108100, 0, +0.1380711874] ; >>> so the cuboid boxing the curve is somewhat flattened. >>> >>> >>> >>> On 11/6/13, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: >>>> Update on Enneper-sphere intersections: >>>> >>>> For the nice tennis-ball curve with parallel osculating planes >>>> at its tips, the exact sphere radius is the root of >>>> (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 >>>> --- rather less than 1/4 . At the tips >>>> [x,y,z] ~ [0, +/- 0.1849108100, +/- 0.1380711874] , >>>> so the cuboid boxing the curve is somewhat flattened. >>>> >>>> For the extreme waisted curve with tacnodes at its tips, >>>> dividing the sphere into four teardrop regions, radius 1 turns >>>> out to be exact. >>>> >>>> Note that the usual parameterisation for Enneper's surface scales >>>> all coordinates involved up by a factor 3 . >>>> >>>> Magma script and Maple graphic are available on request. >>>> >>>> Fred Lunnon >>>> >>>> >>>> On 11/4/13, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: >>>>> I'd always casually assumed that the tennis-ball / baseball curve >>>>> was probably some simple space quartic that everybody but me knew >>>>> about. >>>>> But since it seems this is not the case, I'll put in my two-penn'orth. >>>>> >>>>> The classic minimal Enneper's surface intersects concentric spheres >>>>> in a family of such curves. With the parameterisation in the Wikipedia >>>>> page, a sphere of small radius meets it in an approximate circle; >>>>> a sphere of radius (approx?) 1/4 meets it in a typical tennis-ball >>>>> curve, >>>>> with expected symmetry and arcs parallel at the extremities; >>>>> a sphere of radius (approx?) 1 meets it in a curve with touching arcs; >>>>> for larger radius the curve has 4 self-intersections. >>>>> >>>>> See >>>>> http://www.indiana.edu/~minimal/maze/enneper.html >>>>> http://en.wikipedia.org/wiki/Enneper_surface >>>>> >>>>> The degree of Enneper's surface equals 9, so presumably these curves >>>>> have degree 18. >>>>> >>>>> Fred Lunnon >>>>> >>>>> >>>>> On 11/3/13, Henry Baker <hbaker1(a)pipeline.com> wrote: >>>>>> Is a tennis ball seam the same shape as a baseball seam? >>>>>> >>>>>> http://math.arizona.edu/~rbt/baseball.PDF >>>>>> >>>>>> "Designing a Baseball Cover" >>>>>> >>>>>> Richard B. Thompson >>>>>> >>>>>> College Mathematics Journal, Jan. 1998. >>>>>> >>>>>> At 09:09 AM 11/3/2013, rkg wrote: >>>>>>>Dear funsters, >>>>>>> A tennis-ball appears to be made from 2 congruent >>>>>>>pieces of material, seamed together in a curve. Are possible >>>>>>>equations to the curve known? I'd like a smooth algebraic >>>>>>>equation, probably of degree 4, and preferably with a maximum >>>>>>>number of rational points on it. A first approximation might >>>>>>>be to take a sphere of radius root(3) and centre at (0,0,0) >>>>>>>and take the 8 great circle arcs (1,1,1) to (-1,1,1) to >>>>>>>(-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), >>>>>>>(1,1,-1) and back to (1,1,1). However, this isn't smooth >>>>>>>at the 8 corners of the cube, and I think that it doesn't >>>>>>>even partition the sphere into two congruent pieces. >>>>>>> >>>>>>> Is this well-known to those who well know it? What >>>>>>>do the tennis-ball manufacturers do? R. >>>>>> >>>>>> >>>>>> _______________________________________________ >>>>>> math-fun mailing list >>>>>> math-fun(a)mailman.xmission.com >>>>>> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>>>>> >>>>> >>>> >>> >> >

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Re: [math-fun] bizarre algebraic trivium
by Axel Vogt 08 Nov '13

08 Nov '13

> Message: 7 > Date: Sat, 2 Nov 2013 16:41:51 -0700 > From: Bill Gosper <billgosper(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] bizarre algebraic trivium I am using Maple. Denote the polynomial in that Root by p, the given root by x0. Using Sturm sequences one sees, that p has exactly 2 real roots, both are positive and 1 < x0 is confirmed by the system (the other one is smaller than 1). Now substitute x = 1/X^8 (ok, a bit strange) and assuming 0 < X that becomes 0 = -X^(31/4)*sqrt(sqrt(X+1)+sqrt(X))+1 Maple finds those roots, 2 of them are Reals, one is -1 and the other one is X0:= RootOf(_Z^30-_Z^29+_Z^28-_Z^27+_Z^26-_Z^25+_Z^24-_Z^23+_Z^22-_Z^21+_Z^20 -_Z^19+_Z^18-_Z^17+_Z^16+_Z^15-_Z^14+_Z^13-_Z^12+_Z^11-_Z^10 +_Z^9-_Z^8+_Z^7-_Z^6+_Z^5-_Z^4+_Z^3-_Z^2+_Z-1, index = 1); Feeding ln(x)/ln(x^2-2) with x = (1/X0)^8 is simplified to -8/15 by Maple.

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[math-fun] Elektrische Heimat
by Hans Havermann 07 Nov '13

07 Nov '13

I've been wowed this week by Jens Dauterstedt's recent artistic 3D renderings. If you like that sort of thing: http://www.elektrische-heimat.de/Kategorien/3D-Genera/441/

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[math-fun] bizarre algebraic trivium
by Bill Gosper 07 Nov '13

07 Nov '13

NeilB ran an interesting experiment on the OEIS. One of his lesser results: In[642] := Solve[x == Sqrt[1 + Sqrt[1 + Sqrt[Sqrt[Sqrt[x]]]]]] Out[642] = {{x -> Root[1 + #1 + #1^2 + #1^3 + #1^4 + #1^5 + #1^6 + #1^7 + #1^8 + #1^9 + #1^10 + #1^11 + #1^12 + #1^13 + #1^14 - 255 #1^15 - 255 #1^16 + 769 #1^17 + 769 #1^18 - 1023 #1^19 - 1023 #1^20 + 769 #1^21 + 769 #1^22 - 351 #1^23 - 351 #1^24 + 97 #1^25 + 97 #1^26 - 15 #1^27 - 15 #1^28 + #1^29 + #1^30 &, 2]}} In[659]:= Log[x^2 - 2, x] == -8/15 /. %642[[1]] Out[659]= Log[Root[1 + #1 + #1^2 + #1^3 + #1^4 + #1^5 + #1^6 + #1^7 + #1^8 + \ #1^9 + #1^10 + #1^11 + #1^12 + #1^13 + #1^14 - 255 #1^15 - 255 #1^16 + 769 #1^17 + 769 #1^18 - 1023 #1^19 - 1023 #1^20 + 769 #1^21 + 769 #1^22 - 351 #1^23 - 351 #1^24 + 97 #1^25 + 97 #1^26 - 15 #1^27 - 15 #1^28 + #1^29 + #1^30 &, 2]]/ Log[-2 + Root[ 1 + #1 + #1^2 + #1^3 + #1^4 + #1^5 + #1^6 + #1^7 + #1^8 + #1^9 + \ #1^10 + #1^11 + #1^12 + #1^13 + #1^14 - 255 #1^15 - 255 #1^16 + 769 #1^17 + 769 #1^18 - 1023 #1^19 - 1023 #1^20 + 769 #1^21 + 769 #1^22 - 351 #1^23 - 351 #1^24 + 97 #1^25 + 97 #1^26 - 15 #1^27 - 15 #1^28 + #1^29 + #1^30 &, 2]^2] == -(8/15) In[660]:= N[%, 1111] Out[660]= True Whereas this is straightforward to very, I don't see how a simplifier could do the LHS except by numerically guessing the -8/15 . Could it somehow Factor those two "logands" into the -8th and 15th powers of a certain root of -1 + 2 x^16 + x^31? --rwg

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Re: [math-fun] French Hens: open problem on space-filling curves / IFS
by Bill Gosper 05 Nov '13

05 Nov '13

We should mention Bob Floyd's times 4 solution, http://merganser.math.gvsu.edu/calculus/summation/cubes.html from Knuth, Vol 1. Also, I suspect the late Robert Ammann could have cooked this problem with a magic frac-tile. --rwg Retraction: Vijayaraghavan's name is not on the nested radical constant, except on some crank pages that were apparently confused by Google-glimpsing [...] No closed-form expression is known for this constant. "It was discovered by T. Vijayaraghavan [...] in http://oeis.org/search?q=A072449 . SPOILER WARNING I got zero responses to the √(x + √(4 x + √(16 x + √(64 x + √(256 x + √...))))) evaluation puzzle, leaving me perplexed as to whether it was too easy or too hard, (and whether or not it's old). The proof should be easy, once you see the answer: 1 + √x. On 2013-11-03 07:06, Michael Kleber wrote: > Since Joerg mentioned Ventrella's book _Brain-Filling Curves: A Fractal > Bestiary_, let me take this opportunity to re-mention an open problem I'm > fond of. (At least, I think it's still open.) > > Recall that 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+...+n)^2. That means that > one 1x1 square, two 2x2 squares, ..., n nxn squares together all have the > same total area as one square of side length 1+2+...+n. Robert Wainright's > "Partridge problem" asks whether you can realize this numerical identity as > a dissection of a big square into the requisite set of smaller squares. It > turns out to be possible for all n >= 8; you can see some pictures of > solutions on pages by Funsters at > http://www2.stetson.edu/~efriedma/mathmagic/0802.html or > http://www.mathpuzzle.com/partridge.html. (The "all n >=" part is very not > obvious even once you have an n=8 solution.) > > The name "partridge problem" comes from "four calling birds, three french > hens, two turtledoves, and a partridge in a square tree", of course. > > This identity can manifest itself as a tiling problem not just for squares, > but for an arbitrary planar figure P. If kP means a copy of P scaled > linearly by a factor of k, then the question is whether you can use one P, > two 2P's, ..., n nP's to cover a (1+2+...+n)P. If so then we say P has > "partridge number" n (or maybe only for the minimal such n, depending on > who you ask). For example, the square has partridge number 8, the > equilateral triangle has partridge number 7, the 30-60-90 right triangle > has a partridge number of 4, and so on. The 30-60-90 find is by Patrick > Hamlyn, and is a truly beautiful puzzle; I've had a physical copy sitting > on my desk for years. > > Anyway, the open problem part of all this! > > Patrick Hamlyn's 30-60-90 is the polygon with the smallest known partridge > number. As far as I know, it is open whether there are polygons OR > fractals with partridge numbers 2 or 3. > > The fact that 2 is open, even for polygons, really bugs me. I really want > to know whether there is a polygon P such that a P and two 2P's fit > together to make a 3P. > > (For a while I thought I had a proof of impossibility, and I was all set to > write a paper called "On the Necessity of French Hens", and that my proof > fell apart is a great disappointment, though I have to admit more for the > title of the non-paper than for its theorem.) > > Anyway, the fractal version of the partridge-2 or partridge-3 problem is > very much along the lines of Ventrella's book. In particular, I think that > if a solution exists then then must be a space-filling curve that realizes > it. > > --Michael > > > > On Sun, Nov 3, 2013 at 9:19 AM, Joerg Arndt <arndt(a)jjj.de> wrote: > >> A pointer to a neat picture book featuring >> very many "space filling" curves: >> >> Jeffrey J. Ventrella: >> Brain-Filling Curves: A Fractal Bestiary, >> LuLu.com, (2012). >> >> The book can be downloaded for free at >> >> http://archive.org/download/BrainfillingCurves-AFractalBestiary/BrainFillin… >> Warning: it is a hefty 235 Megabytes! >> >> Two pertinent URLs are >> http://www.fractalcurves.com/HorrorVacui.html (book as html) >> http://www.brainfillingcurves.com/ (needs scripting activated) >> >> On page 107 (top) there is a curve corresponding >> to the L-system (and turns of 120 degrees) >> Start: F >> Rules: >> F --> F-F+F+FF-FF >> + --> + >> - --> - >> >> Here is a rendering >> http://jjj.de/3frac/R7-x1-curve.pdf >> >> What Ventrella does not show is how neatly 3 of >> these combine into either >> http://jjj.de/3frac/R7-x1-closed-trihook.pdf >> or >> http://jjj.de/3frac/R7-x1-closed-gosper-island.pdf >> >> The better known splittings of Gosper's island are >> http://jjj.de/3frac/gosper-split7.pdf >> and possibly (3 renderings) >> http://jjj.de/3frac/R7-2-gosper.pdf >> http://jjj.de/3frac/R7-2-sty1-gosper.pdf >> http://jjj.de/3frac/R7-alt-hex-gosper.pdf >> Here is a bubbly one: >> http://jjj.de/3frac/R7-bubble-gosper.pdf >> >> (cf. http://www.fractalcurves.com/Root7.html ) >> >> >> For the truly bored there are more images under >> http://jjj.de/3frac/ >> >> Best, jj >> >>

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[math-fun] Brain-Filling Curves and Gosper splittings
by Joerg Arndt 04 Nov '13

04 Nov '13

A pointer to a neat picture book featuring very many "space filling" curves: Jeffrey J. Ventrella: Brain-Filling Curves: A Fractal Bestiary, LuLu.com, (2012). The book can be downloaded for free at http://archive.org/download/BrainfillingCurves-AFractalBestiary/BrainFillin… Warning: it is a hefty 235 Megabytes! Two pertinent URLs are http://www.fractalcurves.com/HorrorVacui.html (book as html) http://www.brainfillingcurves.com/ (needs scripting activated) On page 107 (top) there is a curve corresponding to the L-system (and turns of 120 degrees) Start: F Rules: F --> F-F+F+FF-FF + --> + - --> - Here is a rendering http://jjj.de/3frac/R7-x1-curve.pdf What Ventrella does not show is how neatly 3 of these combine into either http://jjj.de/3frac/R7-x1-closed-trihook.pdf or http://jjj.de/3frac/R7-x1-closed-gosper-island.pdf The better known splittings of Gosper's island are http://jjj.de/3frac/gosper-split7.pdf and possibly (3 renderings) http://jjj.de/3frac/R7-2-gosper.pdf http://jjj.de/3frac/R7-2-sty1-gosper.pdf http://jjj.de/3frac/R7-alt-hex-gosper.pdf Here is a bubbly one: http://jjj.de/3frac/R7-bubble-gosper.pdf (cf. http://www.fractalcurves.com/Root7.html ) For the truly bored there are more images under http://jjj.de/3frac/ Best, jj

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[math-fun] Nested radical puzzle
by Bill Gosper 04 Nov '13

04 Nov '13

I can't get NeilB to okay or deny this posting, but neither can I resist: While investigating the nagging problem of evaluating the nested radical constant In[62]:= Fold[Sqrt[#1 + #2] &, 0, Reverse[Range[7]]] Out[62]= Sqrt[1 + Sqrt[2 + Sqrt[3 + Sqrt[4 + Sqrt[5 + Sqrt[6 + Sqrt[7 + ...]]]]]]] sometimes given the name of our old pal (Videryutupronownstice) Vijayaraghavan, NeilB searched OEIS and found, among other things, this goodie: Puzzle: Evaluate In[68]:= Fold[Sqrt[#1 + #2] &, 0, Reverse[x*4^Range[0, 5]]] Out[68]= Sqrt[x + Sqrt[4 x + Sqrt[16 x + Sqrt[64 x + Sqrt[256 x + Sqrt[...]]]]]] Is this well known? If not, it might make a nice math contest problem. --rwg Neil's search also turned up more of those "bizarre algebraic trivia". Correction to earlier request: Subj:[math-fun] can someONE with access to The Symmetries of Things [...] Chit-chat: Someone thought http://gosper.org/IMG_0146.JPG was looking downhill. It's actually steeply up (13%? Steeper than an Aerobie glideslope, anyway). Around the next bend it's steep enough to have run the giant slinky a few steps. The grassy bank on the right is too steep for my hiking boots in the dry season. You can see the tree on top seems to be leaning back. The weed-whacking braceros must use some sort of cleats. --rwg

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[math-fun] Where's *my* nobel prize in economics?
by Warren D Smith 03 Nov '13

03 Nov '13

>I was just listening (time-shifted, obviously) to NPR Planet Money's discussion of the recent Nobel Prize in Economics. >Aside from the fact that Shiller & Fama won the same prize at the same time for diametrically opposing views (kind of like giving 2 different people prizes for P=NP and P/=NP, simultaneously), the discussion raised some interesting issues about the rationality/irrationality of stock market prices. -- I can perform the way more impressive feat of actually contradicting MYSELF, solo!

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[math-fun] Economics Nobel Prize & metastability
by Henry Baker 03 Nov '13

03 Nov '13

I was just listening (time-shifted, obviously) to NPR Planet Money's discussion of the recent Nobel Prize in Economics. Aside from the fact that Shiller & Fama won the same prize at the same time for diametrically opposing views (kind of like giving 2 different people prizes for P=NP and P/=NP, simultaneously), the discussion raised some interesting issues about the rationality/irrationality of stock market prices. http://www.npr.org/blogs/thetwo-way/2013/10/14/233934871/americans-win-econ… "Americans Win Economics Nobel For Interpreting Stock Prices" --- Basically, Shiller says that there can be "bubbles" (irrational prices), while Fama says that there can't be, because if there were irrational prices, then some rational entity would step in and make huge profits from this mispricing. I can understand why the Nobel Prize committee gave out both prizes, because I believe that both Shiller and Fama are correct. However, neither the Committee, nor NPR, could explain why this should be possible. I think that it takes an engineer to understand this problem. Consider _waves_. When waves travel through a _linear_ medium, _every wavelength is independent_ (the principle of Superposition), and in particular, the _power_ in one wavelength cannot be converted into power in another wavelength in a perfectly linear system. Exchanging power among wavelengths requires _nonlinearity_. Hence, a water wave can build up (due to wavelength/frequency dispersion even in a linear medium) and subsequently "crash" due to the nonlinearities in water wave equations. But the fundamental property of _waves_ requires _momentum_, without which there would be no waves. This is because the fundamental wave equation is a second order differential equation, and requires the combination of an elastic medium together with a momentum-carrying mass. This wave may quickly die out due to any _first-order_ term in the differential equation, but it is the _second-order_ term that defines the basic property of waves. We now return to the stock market and prices. We can model this system as a system of difference equations which have inevitable _time delays_. With modern high frequency trading (HFT), these time delays can be microseconds, but these time delays are still present. _Any difference equation with time delays may be subject to oscillations_, because any action is predicated on data that is old, and therefore subject to phase shift. When the phase shift reaches 180 or 360 degrees (depending upon the sign) for a given frequency, the system can oscillate and/or potentially diverge (complex & real parts of exp(z), respectively). We now look more carefully at how the stock market works. The stock exchange has an _order book_, which is a current list of bids and asks; each bid gives a price and a size of transaction; each ask gives a price and a size of transaction. When the bid price of one order exceeds the ask price of another order, a _transaction_ takes place, which has the side effect of erasing min(bid_size,ask_size) shares from the order book. The stock exchange then waits for another order (bid or ask), which either gets added to the book, or causes a transaction to happen. The _quoted_ prices on the stock exchange are the highest bid and the lowest ask prices, _regardless of size_. In particular, the quoted prices could depend upon orders worth only $100 on a company whose total market capitalization could be tens of billions of dollars. This is the very definition of "pricing on the margin". What does it take to _change the quoted price_, i.e., "move the market" ? Suppose I wanted to change the quoted price of the ACME company from its current price of $100 to $200. What would I have to do? If I wanted to change the price quickly, I would have to _buy all of the "ask" shares in the order book whose currently asked price was less than $200_, and I would have to issue a bid order whose size was large enough to actually accomplish this. Depending upon how "thinly traded" ACME shares were, this amount could be anywhere between a few hundred dollars to a few hundred million dollars. Basically, any sharp "uptick" would require a buy order large enough to hit an "air pocket" -- a "hole" in the order book price list where there are extremely small sizes of stock available for sale. Causing such a sharp uptick in stock prices would be a stupid thing to do, unless I were afraid that an order even larger than my order would be coming a few microseconds/milliseconds right after me. Thus, I would a) have to believe that the stock would be more than doubling soon, and b) that I had only a split-second jump on the market. If I believed that the stock would be doubling, but that I was the only one in possession of this knowledge, I would buy more judiciously & quietly, hoping that my purchases would drive up the market very slowly -- if at all. In this way, I could purchase my shares at an overall lower aggregate price, prior to the market as a whole becoming aware of the knowledge that I already had. But after my first (small) purchase, I'm now engaged in a _competitive_ market, wherein every transaction is monitored and reacted to by every other player. I.e., we all now go through a series of steps in time where prices are posted, and we all react to these changing prices according to our varying strategies. It is in these _reactions to prices_ that _time delays_ are encountered, which can now produce _oscillations_ due to _momentum_. One player may see a price uptick as an opportunity to sell, while another player may see a price uptick as a signal that someone thinks the stock is undervalued, and uses this presumption as an opportunity to buy. In this case, the seller is _counter-cyclical_ (a "dampening" force), while the buyer is _cyclical_ (an "amplifying" force). A non-cyclical player would have an a priori theory of the _intrinsic_ value of the stock, and would buy/sell accordingly. A cyclical (i.e., "momentum") player would not necessarily have any notion of _intrinsic_ value for the stock, but would attempt to guess whether the stock was headed up or down based upon previous price/volume behavior. We now ask an objective question: what is the ratio of cyclical to non-cyclical players in the market? The answer can be objectively determined: the overwhelming volume of players (measured by dollar volume) in the stock market are _cyclical_ ("momentum" players), and have no notion of intrinsic value (at least over a relatively large range of prices). Whether this fact supports Shiller or Fama requires a bit more analysis. The vast majority of time, the cyclical players beat the non-cyclical players, partially _because_ they know that the vast majority of players are cyclical. When a water wave is headed up, it carries all of the water molecules with it, regardless of how smart they are, how good their computer simulations are, or how well they can see how far above mean sea level they are. Indeed, from the perspective of any water molecule in the wave, no matter how steep the slope is, it appears to getting monotonically steeper, so _until the moment the wave crashes_, there is still as much money to be made (analogizing the water wave with a price bubble) at the last instant before the crash, as at any earlier point on the rising wave. --- >From a purely "rational" point of view, price movements should _never_ be smooth. In a world with "perfect" information and "rational" players, every new bit of information should change everyone's perception of price instantly, and in a discontinuous way. If I maintain a "discounted cash flow" model for the ACME company, and I hear about some fact that will affect the company seriously in 5 years, I should still "discount" this fact backwards to the present and instantly and discontinuously change my notion of intrinsic value for ACME. That price movements of ACME appear to be smooth at all is due to 2 factors: a) many, many different players have different intrinsic valuations of ACME, and so transactions (and hence price movements) will only occur for those whose intrinsic valuations are close to the current market price; and b) many "market-makers", who have no notion of intrinsic valuation, but are (usually) the quintessential damping mechanism, which attempts to "smooth" price variations about some longer-term "moving average". Market-makers would ordinarily be wiped out, except that they are _fully hedged_ (no net exposure to the market price), and bid/ask _spreads_, which enable them to make small amounts of money even in a completely flat market, so long as there is some volume. The real question is why there are any players whose intrinsic valuations are above the market ? The answer is that these players simply don't have access to enough money to actually move the market. Having the _information_ to move the market isn't the same thing as being _able_ to move the market. These intrinsic players are exactly in the position of the very smart water insect (Maxwell's flea?) who can see the wave is about to break, but isn't in any position to do anything about it (e.g., extract energy). --- Metastability. Metastability in an electronic circuit, or more general "decision-making system", is the name for the indecision that can arise from the time it takes the system to _amplify_ a tiny bit of information into a signal large enough to "move the needle" -- i.e., to affect the larger-scale world. Metastability typically occurs in _arbiters_ and _synchronizers_, where the bit of information is "which signal, A or B, came first?" must be amplified into a large-scale change of some perceptible state. The time to detect and amplify this bit of information may be related in a non-linear way to the time difference between the arrival of the signals, so it could take an unbounded amount of time to amplify this unboundedly small difference to a perceptible level. http://en.wikipedia.org/wiki/Metastability_in_electronics Metastability in electronics --- _The exact same kind of metastability happens in the stock market._ Very small bits of information may take a very long time to become amplified to the point of perceptible changes in stock market prices. This is because the few players who have access to this information simply don't have enough access to capital to _make a big enough splash_, and therefore can't move the market directly. The market is eventually moved -- if at all -- by those _counter-cyclical_ (i.e., "momentum") players who notice these almost imperceptible price signals of the small-but-knowledgeable players and _amplify_ them into larger-scale movements. Thus, the very momentum that enables information transmission in stock market prices also makes the market vulnerable to bubbles and crashes. So Shiller & Fama are both right.

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