Wikipedia:Reference desk/Archives/Mathematics/2016 May 20

Mathematics desk
< May 19	<< Apr | May | Jun >>	May 21 >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

May 20

Power tower

Let $x=(5/10)^{(7/10)^{(9/10)^{(11/10)^{...}}}}$. Does $x$ converge to a finite value? 24.255.17.182 (talk) 05:49, 20 May 2016 (UTC)

0. The first 3 terms equal 0.646176... and then the powers exceed 1 so every step gets smaller. -- SGBailey (talk) 07:17, 20 May 2016 (UTC)

Well, but how do you know they get smaller without bound (above 0)? They could still approach a nonzero limit. They don't, I think, if I've understood the rule for the successive exponents, but "every step gets smaller" is not a sufficient argument. --Trovatore (talk) 07:30, 20 May 2016 (UTC)

Oh, wait, I was associating the exponents backwards. Oops. That makes it harder. Not sure of the answer; will think about it more. --Trovatore (talk) 07:32, 20 May 2016 (UTC)

I think SGBaily is right that, after the first three steps, the part past 9/10 increases (without bound? I think so). In that case, the part starting with 9/10 goes to 0. So in the limit, the expression approaches $(5/10)^{(7/10)^{0}}=(5/10)^{1}$. Still some details to check.... --Trovatore (talk) 07:37, 20 May 2016 (UTC)

You are correct. SGBailey forgot that exponentiation is right-associative.

If you have a power tower $a^{a^{a^{a\ldots }}}$, it goes to infinity for every $a>e^{1/e}=1.44467...$. So in the OP's power tower, the part that starts with 15/10 is greater than the power tower of all 15/10, so it is infinite. This means the parts starting with 13/10, and with 11/10, are infinite.

Then that part starting with 9/10 goes to 0, 7/10 goes to 1, and the whole tower becomes $5/10=1/2$. -- Meni Rosenfeld (talk) 08:33, 20 May 2016 (UTC)

Zero: 11/10>1, 21/10>2, .... This and the fact that the sequence is monotonically decreasing after the first three terms means that the sequence is less than $a^{n}$ with $a=(5/10)^{(7/10)^{(9/10)}}<1$ and n = 1,2,3,..., which converges to zero.--Wikimedes (talk) 11:45, 21 May 2016 (UTC)

No: even from your own LaTeX code you should be able to tell you're doing the association wrong. (As has already been discussed in the posts preceding your own.) --JBL (talk) 15:24, 21 May 2016 (UTC)

Well, you're correct that I was going from left to right. I took the sequence as written to mean x_1 = .5, x_2 = .5^.7 ~=.615..., x_3 = .615...^.9 = .646..., etc. Three editors are saying that my interpretation is nonstandard, so unless the OP chimes in to say that my interpretation is actually what was meant, I'll leave it to those who are familiar with the standard interpretation. (Or is my conclusion wrong even from my interpretation?)--Wikimedes (talk) 20:50, 21 May 2016 (UTC)

Your argument is correct for exponents nested the other way. (The reason the convention is the way it is is as follows: (a^b)^c = a^(bc) and so there is no need for a tower when nested that way; repeated exponentiation on the outside is "just" repeated multiplication in the exponent.) --JBL (talk) 21:33, 21 May 2016 (UTC)

Knot theory

Can knot theory help people untie complex knots in the real world? I know knot theoretic knots are closed but do they have applications to real knots? 2001:630:12:2428:7139:A0F9:1688:C7C2 (talk) 15:30, 20 May 2016 (UTC)

Somewhat. Here's an interesting paper [1] using knot theory to understand real-world knots arising from random agitation. Turns out prime knots are highly over represented compared to compound knots! My WP:OR is that understanding basic mathematics of basic knots helps me deploy knots better in the real world. Knot sum and satellite knot (and analogous operations) can be used as a sort of syntax that can be useful doing something like securing a load or tying up a bear bag. Real-world knots are often examples of tangle_(knot theory), and things analogous to Reidemeister moves help in untying real knots. Basically, untying real knots is a practiced skill. Like any practiced skill, knowing some theory can help, but the extent to which it helps is debatable and a somewhat a matter of opinion. If you know a little knot theory, then I suggest you start learning more about real knots and then you can decide for yourself :)

Now, you didn't ask this, but there are some cool applications of knot theory to real-world scientific problems where things are knot-like. DNA is a prime example. Knot theory can be used to understand structural/informational properties of DNA supercoiling, how circular DNA moves in gel electrophoresis, and lots of other things. If you want more along those lines let me know and I'll be happy to provide refs. SemanticMantis (talk) 16:04, 20 May 2016 (UTC)

Equal distribution (infinitely many sides on a die)

(moved from science desk) If I roll infinite sided die with faces labelled from zero to one are the odds of it coming up higher than, say, 0.8 equal to it coming up lower than 0.8? I mean there as many numbers above 0.8 as there are below so it seems like it they should be. 88.164.119.137 (talk) 14:52, 20 May 2016 (UTC)

probably move to math...do you mean the die will have every real number between 0 and 1 (an infinite sum, I do believe)??..I'm curious how this will be answered..it's possible the question will be considered nonsensical..68.48.241.158 (talk) 15:16, 20 May 2016 (UTC)

The article almost surely almost surely provides some background to the problems when dealing with the intersection between infinite sets and finite sets. --Jayron32 15:19, 20 May 2016 (UTC)

It depends a bit on how you conceptualize this die. But the reasonable and naive thing to do is to identify it with the uniform distribution on [0,1]. That is analogous to the fair die in the case of finitely many sides. In that case rolling the die is the same as throwing a dart (whose tip is a point) at the unit interval. The probability of the dart hitting any given number is zero, and the probability of it hitting an interval is the size of that interval. So, for your example, the probability of getting a roll in (0.8,1] is 0.2, while P(x in [0,0.8))=0.8. The number of points in the interval is not as important as the size or measure of the interval. This is the distinction between cardinality of a set versus the measure of a set. So the measure of the set is the size of the event_(probability), and we use that to compute probabilities of given events. For intervals, the measure of the interval is just its length (l((a,b))=|b-a|), but for more complicated sets/events, you'd use tools form measure theory to compute the probabilities. SemanticMantis (talk) 15:43, 20 May 2016 (UTC)

op here. Thank-you for that clear and well sourced response. Does the same reasoning hold for a countably infinite sided die (faces labelled with, say, all the positive integers)? 92.90.17.111 (talk) 15:59, 20 May 2016 (UTC)

yes the cardinality couldn't matter...if infinite the chance of landing on a particular number would still have to be zero...and the discussion as far as measure would have to be same too...(it's possible there are fine points involved here that haven't been mentioned)..68.48.241.158 (talk) 16:10, 20 May 2016 (UTC)

or maybe I'm wrong as far as the measure as there's no defined interval like between 0 and 1...?? But I'm not sure this question is technically sensical..68.48.241.158 (talk) 16:17, 20 May 2016 (UTC)

I think the move from [0,1] to $\mathbb {N}$ is importantly different - the former accepts a uniform distribution while the latter does not. Never underestimate the weird things that can happen when switching back and forth between sets of countable and uncountable cardinality ;) SemanticMantis (talk) 16:39, 20 May 2016 (UTC)

You're welcome! For the follow up - No, I don't think so. What's the uniform distribution on the natural numbers? We can define it on an interval, but not on the whole line, and not on the positive integers. To put a probability distribution on a set we have to have an integrable function on it. For a discrete uniform distribution, we just set the probability at 1/n for n many choices. But there's no obvious way to extend that to all $\mathbb {N}$, because the infinite sum of a finite value will always be infinite. See here [2] [3] for some discussion of why we can't have uniform distributions on the whole line. It comes down to the axioms of probability. Improper priors are a way to look at things that are almost like probability distributions, but are not. Here [4] is a paper that examines things that are in some sense "almost uniform" distributions on the natural numbers. I haven't read that but it does look promising. Another thing to do would just be to have the die not be fair. Then you could put e.g. a Geometric_distribution on it. If you want more on this sub-question, I do recommend making a new post on the math desk, and including a link to this question if you do so. SemanticMantis (talk) 16:39, 20 May 2016 (UTC)

Thank-you for another excellent and interesting response. If I knew how to link to this question, I would certainly do as you suggest for my followup and repose the same question on the math desk. 2A01:E34:EF5E:4640:CDC3:3824:8EB5:8B7C (talk) 17:22, 20 May 2016 (UTC)

You know, now I'm curious what anyone else might come up with for the countably many sides case, i.e. analogs of a uniform distribution on N. So I'm moving it over presently. SemanticMantis (talk) 17:49, 20 May 2016 (UTC)

further discussion

Anyone have any other ideas for how a "fair" $\mathbb {N}$-sided die might behave or be modeled? Thanks, SemanticMantis (talk) 17:49, 20 May 2016 (UTC)

See natural density. --JBL (talk) 18:18, 20 May 2016 (UTC)

So that is roughly "fair", but fails to be countably additive, right? I think it would be finitely additive though, and if so, then it fits in well with the Schirokauer and Kadane piece I linked above, though I'm not sure if the natural density would satisfy the various uniformity and shift invariance properties they discuss. The idea that all finite sets have density zero and arithmetic progressions ax+b have density 1/a matches pretty well with what we'd intuitively want though, and this is a conceptually easier notion than the things they discuss. Thanks! SemanticMantis (talk) 19:39, 20 May 2016 (UTC)

Yes, that's right: finite sets have density 0, but of course countably infinite sets can have positive density, so no countable additivity. In the Schirokauer--Kadane piece, I think you want to see Example 2.5 to find the (different) name by which they are calling natural density. --JBL (talk) 22:25, 20 May 2016 (UTC)

Ah yes, I was skimming too quickly, and missed Ex. 2.5 - thanks again. SemanticMantis (talk) 14:14, 23 May 2016 (UTC)

My comprehension falters already on reading "infinite sided die with faces labelled" where I don't distinguish between "sides" and "faces" and can imagine only a spherical ball with no place on which to write a label. (A so-called spherical dice that has internal trickery that loads it to settle in a finite number of positions seems ruled out.) Unless the die is constrained to roll in only one direction, its settling state after rolling needs to be defined in two, not one, dimensions. If we are interested in the Euclidean norm of the complex-number settling state of a sphere of unit circumference, it lies in the range 0 to 1.4142... and the value 0.8 is somewhat displaced from the center of the range at 56.57...%. AllBestFaith (talk) 22:42, 20 May 2016 (UTC)

I'm not a mathematician but...what!? Is this for real???68.48.241.158 (talk) 01:07, 21 May 2016 (UTC)

No, there's nothing of value. --JBL (talk) 01:29, 21 May 2016 (UTC)

So long story short then: a fair die labelled with the real numbers between zero and one behaves pretty much as you'd expect (more chance of a roll between 0 and 0.8 than between 0.8 and 1) whereas a fair die labelled with the naturals doesn't make sense?2A01:E34:EF5E:4640:CDC3:3824:8EB5:8B7C (talk) 07:24, 21 May 2016 (UTC)

I think that's basically right, at least superficially..a problem in conceptualizing this is that neither die could actually exist...so it seems we're talking about an impossible object but then going on to try to describe how it behaves, which seems a tad nonsensical at least to me..68.48.241.158 (talk) 12:30, 21 May 2016 (UTC)

I dont think you don't need to get hung up on a particular physical interpretation; the first is just "select a number randomly from the (uncountable infinite) reals between zero and one" the second "choose a number randomly from the (countably infinite) naturals". Seems like an equal distribution is possible for the first, but not for the second. 2A01:E34:EF5E:4640:CDC3:3824:8EB5:8B7C (talk) 16:34, 21 May 2016 (UTC)

Up to scale, there is only one invariant measure, the Haar measure, on $\mathbb {Z}$. Since $\mathbb {Z}$ is not compact, this is not a probability measure. In contrast, identifying $[0,1]$ with the circle group (which me may clearly do almost surely), there is a unique invariant probability measure. The integers can, however, be embedded into a compact topological group like the p-adic integers, which each has a unique invariant probability measure. But in those cases, individual integers have probability zero. For example, the 2-adic integers consist of all infinite sequences of coin tosses, and the probability measure is the law of sequence of Bernoulli trials. An individual integer is defined by its binary expansion as a null-terminating sequence of bits. The probability of an individual such null-terminated sequence occurring as a random sequence of bits is zero. However, there is a great deal of other kinds of important probability associated with this space, such as random walks. Sławomir Biały (talk) 13:24, 23 May 2016 (UTC)