Talk:Random walk - Knowledge (XXG)

2856:; I think the contents are OK for now but I'll probably edit it a bit for readability before moving it to mainspace) with which I would like to replace the current one. On the other hand the material is perhaps a bit technical for a "entry-level" page such as this one and it may be preferable to create a new page for this content, rewriting the material on the main page to be much more lightweight. I personally favour the latter option, in fact I would probably put a large part of the current content in separate pages (in particular the stuff on Pólya's walk feels too massive) and try to make this page more accessible (this seems like a rather long-term project so I cannot claim I'll be able to do a substantial part of this myself). Opinions and comments? 1819:, although there is a force pulling the two objects together, they never meet. In the same way although there is always a certain non-zero probability that the random walk will return to the origin the probability does not sum to one even after considering an infinite sum in the same way that an infinitely long integral of the force over time applied to an object at it's escape velocity will only, manage to cancel out the momentum, not reverse it (in order to meet). So just think of the third dimension as the "probabilistic" 85: 64: 184: 174: 153: 31: 2174:

but simple random walk for the same case on p213, simple binomial random walk on p393 and binomial random walk on p395, while Feller Vol 1 p363 uses generalized random walk for cases where steps may be of any integer size. In addition, Feller Vol 2 p190 uses general random walk for cases where the step size may have any (1-dimensional) distribution. In all cases the steps are independent across times.

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a graph. Will our drunkard reach his home? It turns out that under rather mild conditions, the answer is still yes. For example, if the lengths of all the blocks are between a and b (where a and b are any two finite positive numbers), then the drunkard will, almost surely, reach his home. Notice that we do not assume that the graph is planar, i.e. the city may contain tunnels and bridges.

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independent of each other there will be an infinite number of occurrences of both dimensions crossing the origin at the same time. This analogy can be extended to any number of dimensions where the third dimension will cross the plane, representing the original 2 dimensions, an infinite number of times.

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The language used here is fairly standard fare for a mathematics discussion. The "Imagine..." is setting up an analogy to the problem, and the question "Will the drunkard ever get back to his home from the bar?" then asked is not rhetorical, it is also analogy to the equivalent hypothesis, "A random

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Then I have another consistency problem, with the picture that shows "steps" of a brownian motion (it's in both articles). If a brownian motion has steps, then it's a random walk, it's not the scaling limit. If we are speaking of brownian motion as a scaling limit, then there can't be any discernable

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Assume now that our city is no longer a perfect square grid. When our drunkard reaches a certain junction he picks between the various available roads with equal probability. Thus, if the junction has seven exits the drunkard will go to each one with probability one seventh. This is a random walk on

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Does the "move distribution" for a generic random walk have to be independent of the current position? If so (or if not so) could this be specified somewhere in the introduction. If so then presumably the sample space has to follow certain symmetry conditions which depend on the "move distribution."

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Thanks for the reply. I think this formula is interesting because it says that when the mean is 0 you can expect stay reasonably close to the starting point (within sqrt(n)) no matter what the distribution. In the general case you can be all over the place so the corresponding formula would probably

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There does need to be some clarification of terminology etc., if the article is to be extended (or split) to deal with more general types of random walks. For example, Feller Vol 2 p 192, use ordinary random walk for the case where steps are of sizes -1,+1 only (with possibly unequal probabilities),

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What mathematicans are usually concerned with is if the random walk visits a given point infinitely many times, not just one time. In particular, a simple symmetric random walk on the d-dimensional lattice doesn't visit any point infinitely often for dimension greater than two. The section on higher

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Indeed, there is no perfect consistency in the terminology in textbooks. But that's not a big deal. The article currently is ordered by increasing generality. There is some point to be made that for this particular subject the more general setup of simple random walks on graphs is a better starting

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This is far too narrow since RW's can be correlated, biased and have any step-length distribution. However, there is some inconsistency in the literature about definitions (for example: is a random walk a kind of Markov process? or are some Markov processes a limited form of a random walk?) and I

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Nobody's claiming that the probability of returning to the origin is zero. The claim is that in 3 or more dimensions, the probability is less than one. The flaw in your "take each dimension separately" argument is that, although the path will almost certainly cross the x-y plane infinitely often,

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From this I would conclude that a random walk will always return to the point of origin regardless of the number of dimensions if an infinite amount of time is allowed. In other words there always exists the chance that a random walk will take the shortest possible path to it's origin at any point

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In a one dimensional random walk every point (including the origin) is crossed an infinite number of times in an infinite amount of time. If we add a second dimension the line representing the point in the original 1 dimension will be crossed an infinite number of times. Since both dimensions are

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You're probably right, except that the root-mean-square is *exactly* sqrt n. I wouldn't remove it, it is one of those intuitive statements that are useful to know. Note that I did not replace "average" by "root-mean-square", but rather supplemented the nontechnical statement with a technical one. —

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For a mere (uncorrelated) random walk, if the steps are constant and equal to 1 unit then for the distance from the starting point (net displacement): - the rms is equal to sqrt(n) in both 1 and 2 dimensions (the expected net squared displacement is equal to n) - the average distance asymptotes to

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OK, for that formula a zero mean is required for the step size, and I have put that in. (A zero mean is not the same as symmetry.) I guess some more general formulae could be put in to cover the more general case, but the obvious way to do it would not match in well with the rest of the article.

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Would someone please clarify the sentence leading up to the b/(a+b) formula. a and b are fixed positions. The current usage of "steps" could refer either to steps taken or a fixed number of steps from the origin. It would probably read better like this: "The expected number of steps until a one

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Sketch of inductive proof: Clear if n=0. Suppose true for some fixed n. In going from n to n+1, each summand x^2 is replaced by two summands (x-1)^2 + (x+1)^2 = 2x^2+ 2. The sum S of all 2^n terms therefore goes to 2S + 2*2^n. Since S = n*2^n by hypothesis, the new sum is n*2^(n+1) + 2^(n+1) =

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The statement is correct. The distance from the origin is the sum of a number of step. The variance of the distance is the sum of the variances of the step. The "root mean squared translation distance" is just the square root of the variance. A slightly odd terminology, which doesn't need the

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Surely the sqrt(2/pi) factor for the direct mean (as opposed to the rms) is only for the 1 dimensional walk (see the above referenced Mathworld 1-d walk article)? If so, this should definately be pointed out in this article. The proof that the rms is exactly sqrt(n) in 2d (and 3d) is easy to

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I created an account so I could submit my Random Walk animated gifs but now it tells me I must be Autoconfirmed. Can I just give these to someone to put on the page? I think they're perfect ... small and actually a Random Walk, unlike the Brownian motion simulated by Random Walk video.

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show...see the link in the Mathworld article to the 2d random walk. I have never seen a value for the 'direct' mean in the 2d walk...I assume it's probably next to impossible to calculate, hence why people only worry about the rms, but I assume it wouldn't be sqrt(2/pi).

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This section is extremely confusing and could really do with some rewriting. Or decent citations. I'm really interested in this subject and frustrated that all the sources I can find are hopelessly impenetrable; Someone should really fix this section up!

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In 1 dimension, starting from 0, for n steps there are 2^n possible paths p each with some ending point E(p). The sum over p of (E(p))^2 is exactly n*2^n (easy proof by induction). The mean value of (E(p))^2 is therefore n and the rms is therefore

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Well, I didn't really see the point here. A graph is a graphic representation of something and there is no "strict sense" in which it has to be one way or the other. The time axis is just as important as the space axis. Could you please clarify?

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especially the section about writing style in mathematics articles. While it is pretty standard for the mathematics community, it is inappropriate and unnecessary in an encyclopedia. Use of the "royal we" should be avoided whenever possible.

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I don't see anything about bounded random walks here. That seems a shame. A walk on a finite graph is finite, OK, but what about the continuous case? e.g. random walk in a space with one boundary (e.g. position can't go below zero).

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walk from point A will eventually reach point B". While it may seem "non-encyclopedic" to someone who does not frequently read lay-discussions of mathematics, this is probably the clearest way to illustrate this concept.

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But, in the more general case, you will be within a smallish distance (within sqrt(n)) of a known location (that depends on the mean step size) that drifts away from zero at a fixed rate, which is actually quite simple.

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If this is not what is intended here, perhaps the above-quoted statement could be modified to make clear just why such grids as the 3D one (and for that matter all the corresponding nD ones, n ≥ 3) do not qualify.

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The article could do with a mathematical definition somewhere, perhaps after the introduction? The descriptive definition is good, but it plunges into examples without a formal statement of what a random walk is.

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Well, the walk takes place in space, so it is only "upwards" or "downwards". It gets difficult for beginners to grasp this if they only see the planar representation. But I may be overly cautios and punctillious.

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is infinite it would take an infinite amount of kinetic energy to achieve an escape velocity (which is impossible, therefore the probability of returning is always one in less than or equal to two dimensions).

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I added a picture and sorted the sections with the more elementary on top. This makes the article a bit non-rigorous (with examples and pictures before the definition) but should make for a better read I hope.

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I can't provide a correct proof (that's what I'm looking for), but I can show you that your proof is invalid. It was a simple oversight: when you sum over the values, you square them first. If every value is

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for practically any definition of average. So the original formulation was quite correct. I made some "compromise" with Miguel, but thinking about it, maybe this sentence can be just removed? What do you say?

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Please, would you be so kind to give an example of a random walk under this heading, for example, the geometric distribution. It would especially be helpful if you could show how the probability distribution

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For steps distributed according to any distribution with a finite variance (not necessarily just a normal distribution), the root mean squared expected translation distance after n steps is sigma x sqrt(n)

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is finite it only takes a finite amount of kinetic energy to achieve an escape velocity. Whereas if the forces applied only in two dimensions the force of gravity would be proportional to 1/d, and since

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Thank you very much for your reply. I am not familiar with the term "root mean squared translation distance", but the formula seems to say it is sqrt(E(S_n^2)), where S_n is the position after n steps.

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What is wrong with this explanation, assuming that "In a one dimensional random walk every point (including the origin) is crossed an infinite number of times in an infinite amount of time." is true?

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sqrt(2n/pi) in 1 dimension but to sqrt(pi*n/4) in 2 dimensions (as consequences of the distribution of a khi law with 1 or 2 dof) - the expressions for correlated random walks are much more complex

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article. It is rather important to mention that each step is distributed according to a normal distribution, because that's what makes it "brownian motion" (the scaling limit of the random walk).

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thus there exists with equal probability that the random walker would take the shortest path to the origin from the farthest possible distance. How can their exist a chance that it would

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I don't agree that cities are infinite. Maybe the word can be changed to indicate that cities are vast, but not infinite. "Imagine now a drunkard walking around in the city. The city is

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Certainly all "blocks" have equal lengths. The statement "Notice that we do not assume that the graph is planar" seems to allow a grid like this 3D one. What properties of the graph

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dimensional random walk goes up to position b or down to position -a is ab. The probability that the random walk will go up to position b before going down to position a is ..."

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Can someone please explain how this is consistent with the fact that the symmetric random walk in 3D (i.e., on the 1-skeleton of the cubical tiling) is known to be transient?

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This article tells the reader to imagine things and asks rhetorical questions. This is not written from an encyclopedic POV, and these parts should be replaced or deleted.

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I revised/expanded the intro somewhat and totally changed the definition. Hopefully the intro covers the topic more broadly. As for the definition, it formerly said:

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I had the same difficulty understanding this so I had to read a book describing the proof. I ended up conceptualizing it as similar in nature to the issue of an

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is in my opinion terribly written, being vague and unstructured even though the contents are mostly fine. I have written a new version (you can see it at

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Could you clarify my mistake: if I step +1 with probability 1 then sigma is 0 and S_n=n so sqrt(E(S_n^2))=n, which does not seem to match the formula

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and completely ordered, and at every corner he chooses one of the four possible routes (including the one he came from) with equal probability."

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Should Langton's Ant be mentioned (and linked to) as another example of a random walker, or is that diverting too far from the main point?

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Of course I do. This is my attempt to revive discussion on this point. It makes no sense to create a new section to discuss a topic that

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could be derived in the framework of a random walk as well as the central moments and, if possible, the maximum likelihood estimator for

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The article says: "Will the drunkard ever get back to his home from the bar? It turns out that he will" Wouldn't "Will the drunkard

2505:"expected" where it was. But there is no need to assume symmetry, just the existence of the variance of the distribution of steps. 1585:. This means their exists a chance for a random walk to return to the origin once in any number of dimensions. Lets say k times: 1908: 1374:

Er... do you realize you replied to a comment more than 4 years old? I think it may be time to archive some things on this page. ~

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Can someone show a proof that the rms is exactly sqrt n? This is not a Poisson distribution, it's a random walk - see e.g.

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get back to his home from the bar? It turns out that he will" be more accurate? For me "ever" implies any probabilty : -->

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The direction from one point in the path to the next is chosen at random, and no direction is more probable than another.

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I tried to derive this property... Ouch ! Could somebody give me little hints of derivation ? Sincerely yours, Damien.

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It's only 2. that is the scaling limit for a random walk. 1. is just a particular type of random movement (follwing the

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there's no reason to suppose that any of those crossings will coincide with crossing the y-z plane and the z-x plane.

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I'm not sure that such a definition adds any clarity to the article. I wouldn't object if someone adds it, though. ~

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in time and space. For example after n steps the random walker could be at coordinate (0,0,n) with probability

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Rather than drunkard's walk, I remember the term "drunk sailor's walk" - or is this only used in German?

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on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join

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on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join

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Right; I've made this more precise by changing "Brownian motion" to "Wiener process" in that section.

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0) a chance that it could return to the origin in any finite number of steps from any conceivable (

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For example you couldn't have a Gaussian random walk on the interval because of the boundaries.

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it doesn't say cities are infinite, it says "imagine a city infinite". This is is mathematics. --

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I saw Firefox 13.0.1 cannot play the video, it previews, but when I click play, nothing happens.

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For example you couldn't have a Gaussian random walk on the interval because of the boundaries.

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0 would mean that the drunkard will get home - saying that he always gets home is redundant. --

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I just realized that the analogy works quite well in another regard. In three dimensions the

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http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter12.pdf

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1. The physical phenomenon that minute particles immersed in a fluid move about randomly; or

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Lets generalize to 'd' dimensions travelling 'n' units straight out and back to the origin

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This doesn't seem right. Should it no be "any symmetric distribution" (3rd moment = 0)?

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0 (which is trivially true) whereas "always" would more accurately describe "p = 1". --

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A random walk will always return to the origin regardless of the number of dimensions.

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So i've modified the rms statement. Please show that my proof is wrong and the sqrt

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As I understand it, with infinite time (as implied by "ever"), any probability : -->

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holds for forces. So the force at a distance 'd' is proportional to 1/d^2. Since

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never mentions a random walk returing to the origin an infinite number of times. --

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either, which is why I call into question the statement "A simple random walk on

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is a sequence of independent discrete random variables, the sequence of sums

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do not presume that the definition/notation I provide is canonical. Best,

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The random walk is a path constructed according to the following rules:

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Some clarification on the a and b formula in the one dimensional case

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2. The mathematical models used to describe those random movements."

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The distance from one point in the path to the next is a constant.

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dimension needs this distinction to be brought up and clarified.

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I just added a sentence at the end of the first paragraph of the

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of a random walker. I think such an analogy works quite well.

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You're right, the crossing coincidence is not certain, just

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OK now I understand the picture... Well, the caption in the

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I originally wrote that section. Thanks for fixing it up. ~

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How does this passage square with the transience in 3d ???

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http://mathworld.wolfram.com/RandomWalk1-Dimensional.html

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Even the Mathematica web site has a similar definition:

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http://qjmath.oxfordjournals.org/cgi/reprint/4/1/120.pdf

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point than the specific case of simple random walk on

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Knowledge (XXG) level-5 vital articles in Mathematics

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will cross every point an infinite number of times."

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The mean squared distance is 2426: 2382: 2338: 2205: 1953: 1896: 1789: 1737: 1648: 1577: 1488: 1072:) (ie: the square of the mean, plus the variance) 531:and i seem to have a simple proof that it's wrong: 502: 400: 380: 330: 1664: 1578:{\displaystyle p=\left({1 \over 2d}\right)^{2n}} 1195:Basically the reason is that this article says: 2792:I want to submit a couple of Random Walk .gif's 2469:Gaussian random walk - Is this really correct? 1489:{\displaystyle p=\left({1 \over 6}\right)^{n}} 812:A useful thing to note in the 1D case is that 2890:Knowledge (XXG) vital articles in Mathematics 1349:Knowledge (XXG):Manual of Style (mathematics) 548:steps can be anywhere in the range from 0 to 8: 2404: 2397: 2360: 2353: 2250:http://mathworld.wolfram.com/RandomWalk.html 1304:There doesn't seem to be any randomness in 147: 58: 2418: 2407: 2395: 2374: 2363: 2351: 2330: 2311: 2298: 2285: 2272: 2266: 2198: 2197: 2196: 2194: 1927: 1921: 1916: 1910: 1872: 1863: 1857: 1852: 1846: 1783: 1782: 1780: 1722: 1709: 1690: 1679: 1667: 1661: 1640: 1627: 1608: 1590: 1566: 1547: 1534: 1480: 1466: 1453: 1249:steps (i.e. the steps are infinitesimal). 493: 491: 393: 343: 310: 268: 2144:I found this article over the internet: 2854:User:Jean Raimbault/sandbox/Random walk 2692:Clarification of the general definition 2064:Clarification of the general definition 1937: 1880: 1185:clarification and consistency of figure 215:Knowledge (XXG):WikiProject Mathematics 149: 60: 19: 2885:Knowledge (XXG) level-5 vital articles 614:is right if you wish to reinstate the 552:inclusive. Taking the definition from 110:Knowledge (XXG):WikiProject Statistics 2905:C-Class vital articles in Mathematics 2427:{\displaystyle \{S\}_{n=1}^{\infty }} 2383:{\displaystyle \{x\}_{k=1}^{\infty }} 1262:article should be the same as in the 482:Actually, the average distance after 331:{\displaystyle P(N=n)=p(1-p)^{(n-1)}} 7: 1500:(p=0) return to the origin if their 1191:relation to brownian motion section. 932:. So the mean square distance, < 195:This article is within the scope of 90:This article is within the scope of 2915:High-importance Statistics articles 2256:However, here's a mathematical one: 1276:error/precision in average distance 841:So the number of moves to the left 49:It is of interest to the following 2930:High-priority mathematics articles 2419: 2375: 1948: 1922: 1858: 1674: 14: 597:must be strictly less than sqrt 218:Template:WikiProject Mathematics 182: 172: 151: 83: 62: 29: 20: 2920:WikiProject Statistics articles 1509:) distance from the origin. -- 579:. This is clearly wrong, since 235:This article has been rated as 130:This article has been rated as 113:Template:WikiProject Statistics 2895:C-Class level-5 vital articles 2206:{\displaystyle {\mathbb {Z} }} 2031:21:45, 28 September 2007 (UTC) 2000:14:58, 27 September 2007 (UTC) 1763:05:26, 10 September 2007 (UTC) 1671: 323: 311: 307: 294: 285: 273: 1: 2727:15:32, 19 February 2011 (UTC) 2168:09:17, 13 February 2008 (UTC) 1243:17:16, 12 February 2009 (UTC) 1179:15:31, 11 November 2005 (UTC) 209:and see a list of open tasks. 104:and see a list of open tasks. 2925:C-Class mathematics articles 2707:20:02, 16 October 2010 (UTC) 2140:The general case random walk 2135:19:57, 14 January 2008 (UTC) 2085:20:01, 16 October 2010 (UTC) 2057:09:15, 12 October 2011 (UTC) 1790:{\displaystyle \mathbb {Z} } 1328:13:40, 5 February 2007 (UTC) 1313:07:41, 21 January 2006 (UTC) 1298:17:09, 20 January 2006 (UTC) 1271:14:32, 27 October 2005 (UTC) 1254:13:45, 27 October 2005 (UTC) 1121:17:05, 5 February 2007 (UTC) 716:05:21, August 15, 2005 (UTC) 681:17:05, 5 February 2007 (UTC) 657:, giving an RMS distance of 472:15:48, 7 February 2009 (UTC) 455:17:46, 19 October 2008 (UTC) 258:Probabilistic interpretation 2910:C-Class Statistics articles 2866:16:46, 30 August 2016 (UTC) 2669:08:46, 31 August 2010 (UTC) 2651:17:39, 24 August 2010 (UTC) 2608:17:09, 24 August 2010 (UTC) 2572:16:59, 24 August 2010 (UTC) 2515:13:58, 24 August 2010 (UTC) 2496:11:14, 24 August 2010 (UTC) 2463:17:36, 16 August 2010 (UTC) 2243:16:17, 16 August 2010 (UTC) 1807:14:09, 9 October 2015 (UTC) 814:you must move on every step 787:comment added by ] (] • ]) 780:(n+1)*2^(n+1), as desired. 737:05:45, 5 October 2005 (UTC) 503:{\displaystyle {\sqrt {n}}} 381:{\displaystyle n=1,2,3,...} 2946: 2848:The current section about 2787:15:25, 17 April 2014 (UTC) 2107:There is a starting point. 1972:21:59, 25 March 2010 (UTC) 1833:21:15, 25 March 2010 (UTC) 1431:03:19, 18 April 2007 (UTC) 1404:15:57, 10 April 2011 (UTC) 1339:19:57, 12 March 2007 (UTC) 2772:23:50, 5 March 2011 (UTC) 2223:21:24, 15 June 2008 (UTC) 2184:16:41, 11 June 2008 (UTC) 1753:03:46, 29 June 2007 (UTC) 1386:06:26, 5 April 2011 (UTC) 1362:06:02, 5 April 2011 (UTC) 1154:10:14, 24 Jul 2004 (UTC) 1146:05:17, 24 Jul 2004 (UTC) 486:steps is of the order of 429:15:27, 17 June 2011 (UTC) 234: 167: 129: 78: 57: 2839:12:22, 6 July 2012 (UTC) 2818:00:59, 2 July 2011 (UTC) 2091:Redid intro and defition 1525:22:46, 3 June 2007 (UTC) 1514:03:46, 2 June 2007 (UTC) 622:13:38, 29 Apr 2005 (UTC) 540:The (absolute) distance 520:18:17, 16 Jul 2004 (UTC) 241:project's priority scale 586:should range from 0 to 198:WikiProject Mathematics 2880:C-Class vital articles 2844:Random walks on graphs 2428: 2384: 2340: 2207: 1955: 1898: 1791: 1750:ANONYMOUS COWARD0xC0DE 1739: 1650: 1579: 1511:ANONYMOUS COWARD0xC0DE 1490: 504: 402: 382: 332: 93:WikiProject Statistics 2850:random walk on graphs 2738:Random walk on graphs 2429: 2385: 2341: 2208: 1956: 1899: 1792: 1740: 1651: 1580: 1491: 922:binomial distribution 505: 403: 383: 333: 43:on Knowledge (XXG)'s 36:level-5 vital article 2394: 2350: 2265: 2193: 2045:Lamina-le-sédentaire 1909: 1845: 1779: 1660: 1589: 1533: 1452: 1347:Dolohov, please see 1319:Non-encyclopedic POV 1158:Cities are Infinite? 556:, the only way that 490: 392: 342: 267: 221:mathematics articles 2740:begins as follows: 2712:Bounded random walk 2635:not say very much 2535:Also, when you say 2423: 2379: 1926: 1862: 1288: 1235:David-Sarah Hopwood 116:Statistics articles 2486:comment added by 2424: 2403: 2380: 2359: 2336: 2203: 1951: 1938: 1912: 1894: 1881: 1848: 1839:inverse square law 1787: 1735: 1678: 1646: 1575: 1486: 704:, then the RMS is 500: 398: 378: 328: 190:Mathematics portal 45:content assessment 2825:Video not working 2821: 2804:comment added by 2641:comment added by 2606: 2461: 2443:Paraphrased from 2228:Formal Definition 2170: 2154:comment added by 2122: 2121: 2075:comment added by 2060: 2043:comment added by 2033: 2017:comment added by 2002: 1990:comment added by 1935: 1878: 1703: 1663: 1621: 1560: 1474: 1384: 1227:langevin equation 789: 498: 478:RMS Distance (1D) 436:Higher Dimensions 432: 415:comment added by 401:{\displaystyle p} 255: 254: 251: 250: 247: 246: 146: 145: 142: 141: 2937: 2820: 2798: 2653: 2600: 2498: 2455: 2433: 2431: 2430: 2425: 2422: 2417: 2389: 2387: 2386: 2381: 2378: 2373: 2345: 2343: 2342: 2337: 2335: 2334: 2316: 2315: 2303: 2302: 2290: 2289: 2277: 2276: 2212: 2210: 2209: 2204: 2202: 2201: 2149: 2098: 2087: 2059: 2037: 2012: 1985: 1960: 1958: 1957: 1952: 1936: 1928: 1925: 1920: 1903: 1901: 1900: 1895: 1879: 1877: 1876: 1864: 1861: 1856: 1796: 1794: 1793: 1788: 1786: 1744: 1742: 1741: 1736: 1728: 1727: 1726: 1721: 1717: 1716: 1708: 1704: 1702: 1691: 1677: 1655: 1653: 1652: 1647: 1645: 1644: 1639: 1635: 1634: 1626: 1622: 1620: 1609: 1584: 1582: 1581: 1576: 1574: 1573: 1565: 1561: 1559: 1548: 1495: 1493: 1492: 1487: 1485: 1484: 1479: 1475: 1467: 1378: 788: 781: 554:root-mean-square 509: 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2152:unsigned 2073:unsigned 2053:contribs 2041:unsigned 2027:contribs 2015:unsigned 1988:unsigned 1522:Filam3nt 1376:Amatulić 1310:GrafZahl 1164:infinite 1152:Pfortuny 967:- 4 < 959:= 4 < 785:unsigned 758:sqrt(n). 425:contribs 413:unsigned 2762:Thanks. 2235:Wjastle 1336:Dolohov 618:claim. 616:exactly 239:on the 134:on the 41:C-class 2542:Thanks 2127:Eliezg 1502:always 1118:Jheald 1087:= 1/2. 993:+ var( 978:+ < 956:): --> 920:has a 859:; and 710:sqrt n 708:, not 678:Jheald 645:, ,if 641:, not 601:. QED. 567:is if 544:after 518:Miguel 442:always 47:scale. 2831:Now3d 2474:: --> 1498:never 1396:Cliff 1392:still 1354:Cliff 1172:Taejo 1058:n p q 1043:n p q 1026:n p q 1010:: --> 992:: --> 982:: --> 977:: --> 966:: --> 944:: --> 936:: --> 930:n p q 714:Luqui 668:Sqrt( 465:Aseld 28:This 2862:talk 2835:talk 2810:talk 2783:talk 2768:talk 2764:Daqu 2723:talk 2719:mcld 2703:talk 2665:talk 2647:talk 2603:talk 2568:talk 2511:talk 2492:talk 2458:talk 2239:talk 2219:talk 2215:Oded 2180:talk 2160:talk 2131:talk 2081:talk 2049:talk 2023:talk 1996:talk 1968:talk 1829:talk 1803:talk 1656:and 1427:talk 1400:talk 1381:talk 1358:talk 1239:talk 1176:Talk 1056:= 4 1053:- 1) 1041:= 4 1028:- 4 1024:+ 4 1017:= 4 940:< 913:But 877:= 2 620:Boud 451:talk 421:talk 338:for 231:High 126:High 2756:are 1665:lim 1308:.-- 1229:). 1116:-- 1079:if 1060:+ 1049:(2 1045:+ 926:n p 651:= n 594:rms 573:= n 560:rms 408:. 2876:: 2864:) 2837:) 2816:) 2812:• 2785:) 2770:) 2748:" 2725:) 2717:-- 2705:) 2667:) 2649:) 2570:) 2513:) 2494:) 2420:∞ 2376:∞ 2321:⋯ 2241:) 2221:) 2182:) 2166:) 2162:• 2133:) 2118:” 2101:“ 2083:) 2055:) 2051:• 2029:) 2025:• 1998:) 1970:) 1949:∞ 1923:∞ 1914:∫ 1859:∞ 1850:∫ 1831:) 1805:) 1675:∞ 1672:→ 1429:) 1402:) 1360:) 1241:) 1174:| 1083:= 1075:= 1068:- 1035:+ 1011:+ 952:- 884:- 870:- 863:= 852:- 712:. 676:. 665:). 453:) 427:) 423:• 318:− 301:− 2860:( 2833:( 2808:( 2781:( 2766:( 2743:" 2721:( 2701:( 2663:( 2645:( 2605:) 2601:( 2566:( 2509:( 2490:( 2460:) 2456:( 2438:. 2415:1 2412:= 2409:n 2405:} 2401:S 2398:{ 2371:1 2368:= 2365:k 2361:} 2357:x 2354:{ 2332:n 2328:x 2324:+ 2318:+ 2313:3 2309:x 2305:+ 2300:2 2296:x 2292:+ 2287:1 2283:x 2279:= 2274:n 2270:S 2237:( 2217:( 2199:Z 2178:( 2158:( 2129:( 2079:( 2047:( 2021:( 1994:( 1966:( 1946:= 1943:x 1940:d 1933:x 1930:1 1918:1 1892:1 1889:= 1886:x 1883:d 1874:2 1870:x 1866:1 1854:1 1827:( 1801:( 1784:Z 1733:0 1730:= 1724:k 1719:) 1714:n 1711:2 1706:) 1700:d 1697:2 1693:1 1688:( 1683:( 1669:k 1642:k 1637:) 1632:n 1629:2 1624:) 1618:d 1615:2 1611:1 1606:( 1601:( 1596:= 1593:p 1571:n 1568:2 1563:) 1557:d 1554:2 1550:1 1545:( 1540:= 1537:p 1482:n 1477:) 1472:6 1469:1 1464:( 1459:= 1456:p 1425:( 1398:( 1383:) 1379:( 1356:( 1237:( 1085:q 1081:p 1077:n 1070:q 1066:p 1064:( 1062:n 1051:p 1047:n 1037:n 1033:p 1030:n 1022:p 1019:n 1013:n 1008:n 1005:R 1002:m 998:R 995:m 990:R 987:m 980:n 975:n 972:R 969:m 964:R 961:m 954:n 950:R 947:m 942:x 934:x 918:R 915:m 888:. 886:n 882:R 879:m 874:L 872:m 868:R 865:m 861:x 857:R 854:m 850:n 846:L 843:m 816:. 706:n 702:n 674:n 670:n 663:n 659:n 655:i 649:i 647:x 643:n 639:n 612:n 599:n 592:x 588:n 583:i 581:x 577:i 571:i 569:x 565:n 558:x 550:n 546:n 542:x 496:n 484:n 449:( 419:( 396:p 376:. 373:. 370:. 367:, 364:3 361:, 358:2 355:, 352:1 349:= 346:n 324:) 321:1 315:n 312:( 308:) 304:p 298:1 295:( 292:p 289:= 286:) 283:n 280:= 277:N 274:( 271:P 243:. 138:. 53::

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