July 2013 - mathfun@mailman.xmission.com

[math-fun] What are your favorite math apps for smart phones?
by Robert Baillie 09 Jul '13

09 Jul '13

I recently acquired an Android smart phone, and have been surpised at all the things it can do and the apps it can run. I installed "Primality Test" by the Mincemeat team. This is the only prime testing app I could find that works with numbers of arbitrary precision. I didn't find any factoring apps that go beyond 10^18, and those appear to do factoring only by trial division. Are there better ones out there? (I'm already working on one for my own amusement, but I'd like to find a good one). Just for fun, I installed a spectrum analyzer (Spectral Audio Analyzer by RadonSoft): it displays the frequencies in sounds that are picked up by the microphone. What are your favorite math apps? In particular, are there any calculators where you can type expressions (like "sqrt(-5) + sin(pi/7)^2") instead of just punching keys on a keypad? Bob Baillie

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[math-fun] privacy / Re: What are your favorite math apps for smart phones?
by Axel Vogt 08 Jul '13

08 Jul '13

> I recently acquired an Android smart phone ... <snipp> > Just for fun ... <snipp> ... > What are your favorite math apps? I will never understand why educated, adult persons give up privacy just for fun and 'gaming'

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[math-fun] Oz rules and grossones
by Fred lunnon 08 Jul '13

08 Jul '13

Whenever I have enquired about the rules for Australian Rules Football, the answer has come back that "there aren't any". http://theconversation.com/its-time-we-draft-aussie-rules-to-tackle-indigen… Can't help wondering if these fellows have taken the message to heart, and rather too literally: http://si.deis.unical.it/~yaro/numta2013/ProceedingsNUMTA2013.pdf WFL

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Re: [math-fun] a question of computation
by Henry Baker 05 Jul '13

05 Jul '13

The amount of technology involved in a modern 'smart phone' is truly prodigious. The GPS subsystem itself is remarkable, with gains involved of perhaps 150 dB (which enables even an _indoor_ cellphone to 'hear' the GPS satellites). Thar's also 'general relativity' in thar! Precise GPS positioning requires corrections based on general relativity. The signal processing involved in the sending & receiving the data is also prodigious. In addition to the amount of processing involved in the cellphone itself, you also have the amount of processing in the 'base station', which can be anything from a large box on a tall tower, to a little tiny thing that sits (I understand) on top of a phone booth (whatever that is!) Modern base station technology includes multiple radios, 'smart antennas' (which can do some version of 'beam-steering' -- essentially pointing an antenna at your cellphone, but doing so electronically instead of mechanically), etc. Much of this processing and the proliferation of base stations enable high speed communication _at very low radiated power levels_. My first cellphone was in my car in the 1980's and could do 3-4 watts of power. I think that most cellphones now use only perhaps 100 milliwatts, and far less than that when the base station is nearby. The processing of the cellphone camera is also pretty impressive. Those little boxes that show you where people's faces are require a huge amount of processing. Also, the 'video compression' algorithms for cellphone video require huge amounts of processing. I'll leave it to others on this list to comment on the amount of processing power required by various agencies to then monitor & store all of this data in the name of 'protecting' us. At 12:03 AM 7/5/2013, Simon Plouffe wrote: > So here is my question, in this example, how much computation is needed to >perform the call first and then the sending and reception of the mms >message containing the photo with the gps location : can this be evaluated >??

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[math-fun] a question of computation
by Simon Plouffe 05 Jul '13

05 Jul '13

Hello, I have a general question about computations, like many of the mathematicians around here we are constantly asked to answer. What is the use of all the computations you guys are doing anyway and what is the purpose of all this, etc. My favorite answer for this stupid question is : you know just take a smart phone, suppose you are somewhere and you receive a call and then you decide to send a photo of the place you are to that person , a 'mms' message. The message would contain the gps location + the image, etc. The amount of computation to do that simple thing is huge. This is where I loose most people because, to begin explaining all the mathematics behind this action is too much to explain in 2-3 phrases. So here is my question, in this example, how much computation is needed to perform the call first and then the sending and reception of the mms message containing the photo with the gps location : can this be evaluated ?? My guess is : something like 10^10, ? 10^12 ? Here the computation would include, the call process, the processing of the photo, the GPS thing, the reception, the location of the person you call etc. All of the operation combined. Any guess someone ? Best regards, Simon PLOUFFE

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[math-fun] binomial coefficient puzzle
by Fred lunnon 05 Jul '13

05 Jul '13

Prove sum_(j = 0)^(j = n) (2j)_C_j (2n-2j)_C_(n-j) = 4^n eg. with n = 4 , 1.20 + 2.6 + 6.2 + 20.1 = 64 Fred Lunnon

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Re: [math-fun] Exact N-body simulations?
by Bill Gosper 04 Jul '13

04 Jul '13

Henry Baker> All this talk of Gaussian integers reminded me of a problem that I'd dearly love to see solved. Consider N bodies, each with integer masses. Start them in initial positions in a 2D (for the moment, there's plenty of complexity in 2D) which are on integer coordinates. The attractive Newtonian gravitation force is proportional to m*m'/(dx^2+dy^2), so this force is rational. We need to compute new positions for each of the bodies in such a way that all the conservation laws are preserved exactly (if this is possible), yet the number of bits required doesn't grow too quickly (see below). We need to preserve the total energy, we need to preserve momentum & angular momentum, & we might like to preserve information (the system is reversible). Note that we don't necessarily require that delta-t be constant, although if it isn't, we may need to have another counter to represent the "clock" -- i.e., the global time. I don't know how to do this even in the 2-body case, but the Minsky circle hack may provide some guidance -- if we could get the two bodies to each draw a Minsky circle while revolving around one another, then we'd have a good start on the problem. In addition to solving the problem of finite arithmetic, we also need to figure out what happens in collisions. Taking a cue from Fredkin, we should probably have 100% elastic collisions in order to ensure reversibility. How does this square with Poincaré ? Well, I'm assuming that my N-body problem for finite N is Turing complete, so with enough bodies and enough "tape" (empty space), we should be able to compute any computable function. Minsky already provides one potential representation with his "counter machines". So we want a reasonably "efficient" Turing Machine (or Counter Machine) simulation so that we aren't wasting more than a fixed fraction of the bits required to represent the state (N pairs of integers or N gaussian integers). Has anyone attempted this sort of exact simulation before? ------- rwg> The variable speed of the Kepler solution is pretty unMinskylike, although there might be a transformation through which a nonphysical but errorless simulation can appear physical, analogous to the circularization methods in Minskys and Trinskys <http://www.blurb.com/books/2172660>. If http://www.freewebz.com/vitaliy/triApplet/triGrav3.html is any indication, the n=3 case in 2D is unstable with p ~ 1. (Unlike the Windows version, the Apple version seems capable of occasional gross integration errors. But both simulations seem always to blow up.) --rwg --------------- Probably just another false memory, but I thought I publicly fretted over these inevitable blowups. Privately, to the kids, Guys, are you too young to have seen this classic old three-body display hack? http://www.freewebz.com/vitaliy/triApplet/triGrav3.html It always starts with three equal masses, zero net momentum, and, I think, zero net angular momentum about the center, leaving only one degree of freedom in the random in[i]tial conditions. Although the numerical integration is remarkably stable (maybe through "cheating" by nulling out accumulated net momenta and maybe enforcing net energy conservation), it inevitably blows up, despite the existence of at least one known periodic solution. ("Stable" under perturbations O(10^-5).) It is intriguing to try figure out how it (apparently correctly) blows up, ejecting two tightly orbiting particles in one direction and one particle twice as fast in the opposite direction. (It's also interesting to guess which ejections are incomplete, with the particles eventually returning to the center.) The energy for the ejections comes from the tightness of the orbiting pair, which is somehow the result of all three of them nearly colliding, whose probability is enormously enhanced by the 2D constraint. Is the 3D p(eventual ejection) = 1 also? (With a much longer waiting time.) There are known periodic, nonplanar, equal mass three body solutions. Apparently, the relativistic *two* body problem is still incompletely solved! --Bill Duh, it's obvious why the triGrav3 simulations blow up so "unrealistically": Zero angular momentum! Realistic astronomic assemblages come with some "gyro-stabilization" that greatly rarefies the weird encounters that make triGrav3 so visually interesting and unstable. --rwg

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Re: [math-fun] computing non-rigorously defined quantities
by Henry Baker 02 Jul '13

02 Jul '13

Some wags would say that many (most?) large scale, long term computer simulations fall into this category, although you might have a hard time getting their proponents to agree with this assessment. (You know what I'm talking about!) On the other hand, Chris Anderson's now-famous rant "The End of Theory: The Data Deluge Makes the Scientific Method Obsolete" argues that data trumps theory: http://www.wired.com/science/discoveries/magazine/16-07/pb_theory "All models are wrong, but some are useful." So proclaimed statistician George Box 30 years ago, and he was right. Peter Norvig, Google's research director, offered an update to George Box's maxim: "All models are wrong, and increasingly you can succeed without them." At 08:41 AM 7/2/2013, James Propp wrote: >For some thoughts about the enterprise of computing quantities one hasn't rigorously defined, see >http://mathoverflow.net/questions/135536/procedure-based-as-opposed-to-defi… >(which I hope will elicit interesting responses over the next few days). > >Jim Propp

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[math-fun] When (-1)-choose-(-1) is 1/2
by James Propp 02 Jul '13

02 Jul '13

Following up on issues raised in the thread "Atrocious Mma 8&9, WolframAlpha bug", here are some thoughts on (-1)-choose-(-1): Under the Euler-measure-as-generalized-cardinality philosophy, (-1)-choose-(-1) could be interpreted as the "number" of ways, given a polyhedral set S of Euler measure -1, to choose a polyhedral subset S' of Euler measure -1. (Here and hereafter, "number" should be understood in a very loose way. E.g., the set of triples r,s,t with 0<r<s<t<1 is an open 3-cell and therefore has Euler measure (-1)^3 = -1, so we say that there are -1 such triples. For an introduction to this mode of thinking, mostly due to Schanuel, see http://arxiv.org/abs/math/0203289.) In 1 dimension (the easiest place to pursue these notions), the simplest kind of polyhedral set of Euler measure -1 is just an open interval. Let's take S = (0,1), for definiteness. Its polyhedral subsets S' of Euler measure -1 are unions of k points and k+1 open intervals (which can abut the k points), for some non-negative k. We stratify the collection of all such subsets by the number of "breakpoints". With 0 breakpoints, the only possibility is S' = (0,1) itself. With 1 breakpoint, we have subsets of the form (0,t) and (t,1) (with t strictly between 0 and 1). How many such values of t are there? -1 of them (in the Euler measure sense). So there are -2 subsets of (0,1) with Euler measure -1 that have 1 breakpoint. With 2 breakpoints, s and t (with s < t), the possibilities for S' are (s,t), (0,s] \union (t,1), (0,s) \union [t,1), (0,s) \union (s,t], and [s,t) \union (t,1). How many pairs s,t with 0<s<t<1 are there? +1 of them. So there are +5 subsets of (0,1) with Euler measure -1 that have 2 breakpoints. The sequence continues 1, -2, 5, -13, 35, -96, ..., and to count all the subsets S' we're interested in, we just have to add up these numbers. It turns out that the power series 1 - 2q + 5q^2 - 13q^3 + 35q^4 - 96q^5 + ... can be "evaluated" at q=1 by analytic continuation, because it is the expansion of 2/(1 + 3 q + Sqrt[1 + 2 q - 3 q^2]); substituting q=1, we get 1/2 as our answer. I suspect that in this theory (or rather the theory into which such flimflam will eventually evolve), the generalized Euler measure of the set of Euler-measure-k polyhedral subsets S' of an Euler-measure-n polyhedral set S depends not just on n and k but also on other properties of S. So it's wrongheaded to think of the above calculation as determining "the" right value of (-1)-choose-(-1); there will be other right values (such as 0) in other contexts. Jim Propp

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[math-fun] self-reproducing decimal numbers
by rcs＠xmission.com 02 Jul '13

02 Jul '13

This is from Erich Friedman, who is having trouble posting directly. --Rich ----- funsters, i'm looking for self-reproducing decimal numbers, numbers that can be formed using their decimal digits (both to the left and right of the decimal point) in order together with the 4 usual arithmetic operations, exponentiation, concatenation, and as many parentheses as you like. notice that decimal points are not allowed. some examples are: .28125 = (2 + 8 - 1) / 2^5 1562.5 = 1 x 5^6 / 2 / 5 i'll even accept repeating decimals like: .1(6) = 1 / 6 5208.(3) = 5^(-2 + 0 + 8) / 3 it's easy to find examples using lots of digits. i was curious how many examples there were using, say, 5 digits or fewer. erich friedman

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