Archive for the ‘mathematics’ Category
Negative probabilities
Was planning to write, but just dumping open tabs for now.
Negative Probabilities by Dan Piponi (sigfpe)’s blog A Neighborhood of Infinity: as usual, the clearest exposition.
Abramsky and Brandenburger, An Operational Interpretation of Negative Probabilities and No-Signalling Models. Gives an interpretation. I read this (section 2) and thought I understood, but when I tried to explain in my own words on a new example, faltered. Should read this again.
Gábor J. Székely, Half of a Coin: Negative Probabilities. Simple example, and an interpretation as “difference”.
Mark Burgin, Interpretations of Negative Probabilities. [Haven’t read]
Post by John Baez. Less than his usual standard. :-(
Colliding balls approximate pi
Found via G+, a new physical experiment that approximates , like Buffon’s needle problem: The Pi Machine.
Roughly, the amazing discovery of Gregory Galperin is this: When a ball of mass collides with one of ball , propelling it towards a wall, the number of collisions (assuming standard physics idealisms) is , so by taking , we can get the first digits of . Note that this number of collisions is an entirely determinstic quantity; there’s no probability is involved!
Here’s a video demonstrating the fact for (the blue ball is the heavier one):
The NYT post says how this discovery came about:
Dr. Galperin’s approach was also geometric but very different (using an unfolding geodesic), building on prior related insights. Dr. Galperin, who studied under well-known Russian mathematician Andrei Kolmogorov, had recently written (with Yakov Sinai) extensively on ball collisions, realized just before a talk in 1995 that a plot of the ball positions of a pair of colliding balls could be used to determine pi. (When he mentioned this insight in the talk, no one in the audience believed him.) This finding was ultimately published as “Playing Pool With Pi” in a 2003 issue of Regular and Chaotic Dynamics.
The paper, Playing Pool With π (The number π from a billiard point of view) is very readable. The post has, despite a “solution” section, essentially no explanation, but the two comments by Dave in the comments section explain it clearly. And a reader sent in a cleaned-up version of that too: here, by Benjamin Wearn who teaches physics at Fieldston School.
Now someone needs to make a simulation / animation graphing the two balls in phase space of momentum. :-)
I’d done something a while ago, to illustrate The Orbit of the Moon around the Sun is Convex!, here. Probably need to re-learn all that JavaScript stuff, to make one for this. Leaving this post here as a placeholder.
Or maybe someone has done it already?
Prefatory apprehension
Robert Recorde’s 1557 book is noted for being the first to introduce the equals sign =, and is titled:
The Whetstone of Witte: whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers.
Its title page (see http://www.maa.org/publications/periodicals/convergence/mathematical-treasures-robert-recordes-whetstone-of-witte, see also the full book at https://archive.org/stream/TheWhetstoneOfWitte#page/n0/mode/2up) contains this verse:
Original spelling
Though many ſtones doe beare greate price, The grounde of artes did brede this ſtone: |
Modern spelling
Though many stones do bear great price, The ground of arts did breed this stone; |
Apparently the full title contains a pun (see http://www.pballew.net/arithm17.html): “the cossike practise” in the title refers to algebra, as the Latin cosa apparently meaning “a thing” was used to stand for an unknown, abbreviated to cos — but the Latin word cos itself means a grindstone.
The author again reminds readers not to blame his book, at the end of his preface:
To the curiouſe ſcanner.
If you ought finde, as ſome men maie, But if you mende not that you blame, |
Authors are either anxious about how their book is received, or make sure to be pointedly uncaring.
Sir Arthur Conan Doyle, in a mostly forgettable volume of poetry (Songs of the Road, 1911), begins:
If it were not for the hillocks
You’d think little of the hills;
The rivers would seem tiny
If it were not for the rills.
If you never saw the brushwood
You would under-rate the trees;
And so you see the purpose
Of such little rhymes as these.
Kālidāsa of course begins his Raghuvaṃśa with a grand disclaimer:
kva sūryaprabhavo vaṃśaḥ kva cālpaviṣayā matiḥ /
titīrṣur dustaram mohād uḍupenāsmi sāgaram // Ragh_1.2 //mandaḥ kaviyaśaḥ prārthī gamiṣyāmy upahāsyatām /
prāṃśulabhye phale lobhād udbāhur iva vāmanaḥ // Ragh_1.3 //atha vā kṛtavāgdvāre vaṃśe ‘smin pūrvasūribhiḥ /
maṇau vajrasamutkīrṇe sūtrasyevāsti me gatiḥ // Ragh_1.4 //
But the most nonchalant I’ve seen, thanks to Dr. Ganesh, is this gīti by Śrīkṛṣṇa Brahmatantra Yatīndra of the Parakāla Maṭha, Mysore:
nindatu vā nandatu vā
mandamanīṣā niśamya kṛtim etām
harṣaṃ vā marṣaṃ vā
sarṣapamātram api naiva vindema
Screw you guys. :-)
Big O() notation: a couple of sources
This post contains, just for future reference, a couple of primary sources relevant to the (“Big O”) notation:
- Some introductory words from Asymptotic Methods in Analysis by de Bruijn
- An letter from Donald Knuth on an approach to teaching calculus using this notation.
Visualizing product of permutations
A simple pedagogical trick that may come in handy: represent a permutation using arrows (curved lines) from to for each . Then, the product of two permutations can be represented by just putting the two corresponding figures (sets of arrows) one below the other, and following the arrows.
The figure is from an article called Symmetries by Alain Connes, found via the Wikipedia article on Morley’s trisector theorem (something entirely unrelated to permutations, but the article covers both of them and more).
I’m thinking how one might write a program to actually draw these: if we decide that the “height” of the figure is some , then each arrow needs to go from some to (using here the usual screen convention of coordinate increasing from left to right, and coordinate increasing from top to bottom). Further, each curve needs to have vertical slope at its two endpoints, so that successive curves can line up smoothly. The constraint on starting point, ending point, and directions at the endpoints defines almost a quadratic Bezier curve, except that here the two directions are parallel. So it’s somewhere between a quadratic and the (usual) cubic Bezier curve, which is given by the start point, end point, and derivatives at the start and end point. (Here we only care about the direction of the derivative; we can pick some arbitrary magnitude to fix the curve: the larger we pick, the more smooth it will look at the ends, at the cost of smoothness in the interior.)
Even knowing the curve, how do we generate an image?
The idea of logarithms, and the first appearance of e
[Incomplete post: about 10% written, about 90% left.]
The notion of the number , the exponential function , and logarithms are often conceptual stumbling blocks even to someone who has an otherwise solid understanding of middle-school mathematics.
Just what is the number ? How was it first calculated / where did it first turn up? Premature exposure to its numerical value
only serves to deepen the mysteriousness and to make it seem arbitrary.
Here a historical perspective helps: as is often the case, here too, the first appearance is simpler and more well-motivated than the accounts in dry textbooks. This is from this account by Matthew P. Wiener (originally posted on USENET somewhere, as quoted by MJD). I’m just going to quote it directly for now, and edit it later:
Napier, who invented logarithms, more or less worked out a table of logarithms to base , as follows:
0 1 2 3 4 5 6 7 8 9 10 ... 1 2 4 8 16 32 64 128 256 512 1024 ...The arithmetic progression in the first row is matched by a geometric progression in the second row. If, by any luck, you happen to wish to multiply 16 by 32, that just happen to be in the bottom row, you can look up their “logs” in the first row and add 4+5 to get 9 and then conclude 16·32=512.
For most practical purposes, this is useless. Napier realized that what one needs to multiply in general is for a base—the intermediate values will be much more extensive. For example, with base 1.01, we get:
0 1.00 1 1.01 2 1.02 3 1.03 4 1.04 5 1.05 6 1.06 7 1.07 8 1.08 9 1.09 10 1.10 11 1.12 12 1.13 13 1.14 14 1.15 15 1.16 16 1.17 17 1.18 18 1.20 19 1.21 20 1.22 21 1.23 22 1.24 23 1.26 24 1.27 25 1.28 26 1.30 27 1.31 28 1.32 29 1.33 30 1.35 31 1.36 32 1.37 33 1.39 34 1.40 35 1.42 [...] 50 1.64 51 1.66 52 1.68 53 1.69 54 1.71 55 1.73 [...] 94 2.55 95 2.57 96 2.60 97 2.63 98 2.65 99 2.68 100 2.70 101 2.73 102 2.76 103 2.79 104 2.81 105 2.84 [...]So if you need to multiply 1.27 by 1.33, say, just look up their logs, in this case, 24 and 29, add them, and get 53, so 1.27·1.33=1.69. For two/three digit arithmetic, the table only needs entries up to 9.99.
Note that is almost there, as the antilogarithm of 100. The natural logarithm of a number can be read off from the above table, as just [approximately] the corresponding exponent.
What Napier actually did was work with base .9999999. He spent 20 years computing powers of .9999999 by hand, producing a grand version of the above. That’s it. No deep understanding of anything, no calculus, and pops up anyway—in Napier’s case, was the 10 millionth entry. (To be pedantic, Napier did not actually use decimal points, that being a new fangled notion at the time.)
Later, in his historic meeting with Briggs, two changes were made. A switch to a base was made, so that logarithms would scale in the same direction as the numbers, and the spacing on the logarithm sides was chosen so that . These two changes were, in effect, just division by .
In other words, made its first appearance rather implicitly.
(I had earlier read a book on Napier and come to the same information though a lot less clearly, here.)
I had started writing a series of posts leading up to an understanding of the exponential function (here, here, here), but it seems to have got abandoned. Consider this one a contribution to that series.
The functional equation f(x+y) = f(x)f(y)
Suppose satisfies . What can we say about ?
Putting gives
which can happen if either or . Note that the function which is identically zero satisfies the functional equation. If is not this function, i.e., if for at least one value of , then plugging that value of (say ) into the equation gives . Also, for any , the equation forces as well. Further, so for all .
Next, putting gives , and by induction . Putting in place of in this gives which means (note we’re using here). And again, . So , which completely defines the function at rational points.
[As , it can be written as for some constant , which gives for rational .]
To extend this function to irrational numbers, we need some further assumptions on , such as continuity. It turns out that being continuous at any point is enough (and implies the function is everywhere): note that . Even being Lebesgue-integrable/measurable will do.
Else, there are discontinuous functions satisfying the functional equation. (Basically, we can define the value of the function separately on each “independent” part. That is, define the equivalence class where and are related if for rationals and , pick a representative for each class using the axiom of choice (this is something like picking a basis for , which corresponds to the equivalence class defined by the relation ), define the value of the function independently for each representative, and this fixes the value of on . See this article for more details.)
To step back a bit: what the functional equation says is that is a homorphism from , the additive group of real numbers, to , the multiplicative monoid of real numbers. If is not the trivial identically-zero function, then (as we saw above) is in fact a homomorphism from , the additive group of real numbers, to , the multiplicative group of positive real numbers. What we proved is that the exponential functions are precisely all such functions that are nice (nice here meaning either measurable or continuous at least one point). (Note that this set includes the trivial homomorphism corresponding to : the function identically everywhere. If is not this trivial map, then it is in fact an isomorphism.)
Edit [2013-10-11]: See also Overview of basic facts about Cauchy functional equation.