Timeline for Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
Current License: CC BY-SA 3.0
30 events
| when toggle format | what | by | license | comment | |
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| Sep 2 at 12:52 | answer | added | Doradus | timeline score: 1 | |
| Aug 29, 2016 at 17:59 | comment | added | Douglas Zare | @JoeSilverman: Computing the volume of a ball as an integral of volumes of balls of one lower dimension does not seem much different from the disk method for computing volumes of revolution. It's not trivial to go from the volume formulas to the statement that most of the ball/sphere is near the equator; the easy statement is that most of the ball is near the surface. Usually one does wait until multi-variable calculus to compute things like the centroid of a hemisphere. (The first moment is easy after you compute the volume.) | |
| Aug 29, 2016 at 17:18 | answer | added | Noah Stein | timeline score: 5 | |
| Aug 29, 2016 at 15:04 | comment | added | Joe Silverman | @DouglasZare Not wanting to belabor the point, but although the integrals are certainly 1-variable, the interpretation as a distribution of points on a high dimensional sphere seems well beyond what is normally covered in single variable calculus. Indeed, even defining the volume of a hypersurface in $\mathbb R^n$ is generally done in the context of multi-variable calculus. | |
| Aug 29, 2016 at 11:19 | comment | added | Yemon Choi | @RW If you like, we could continue this discussion in chat or over email or, if anyone still uses it, on tea.mathoverflow.net However, as someone who has needed character theory of finite groups that is in Isaacs's book, apparently standard knowledge to various algebraists on MathOverflow, but certainly not part of a functional analyst's toolkit, I feel quite strongly that one person's "obviously not research level" is another person's "exactly what I need to get me past a roadblock in my own research". | |
| Aug 29, 2016 at 10:05 | comment | added | Douglas Zare | @JoeSilverman: No, I really meant calculus of one variable. Integrals like $\int_0^{\pi/2} \cos^{d+1} \theta ~d\theta$ are covered in calculus of one variable. | |
| Aug 29, 2016 at 9:05 | answer | added | Mikhail Katz | timeline score: 5 | |
| Aug 28, 2016 at 23:03 | comment | added | R W | There is an obvious contradiction here as I hope we are not yet in the brave new world where anything which is "not part of the usual general UG mathematical education" is considered as "research level". Anyway, this is not an appropriate place for this kind of discussion. | |
| Aug 28, 2016 at 21:35 | comment | added | Yemon Choi | Moreover, to those who seem to have voted or are voting to close because the question is "not research level", doesn't this fit under one of MO's original remits: "questions that an expert from another area could answer easily, which are not part of the usual general UG mathematical education"? | |
| Aug 28, 2016 at 21:34 | comment | added | Yemon Choi | @ChristianRemling While one can understand the question asked here as answered by the same circle of ideas as "concentration of measure", I think it is excessive to regard this as a duplicate of those earlier questions | |
| Aug 28, 2016 at 20:42 | answer | added | Robert Mastragostino | timeline score: 17 | |
| Aug 28, 2016 at 18:50 | review | Close votes | |||
| Aug 29, 2016 at 9:08 | |||||
| Aug 28, 2016 at 18:32 | comment | added | Christian Remling | Duplicate of mathoverflow.net/questions/248151/… (asked by the OP) and of mathoverflow.net/questions/210291/… | |
| Aug 28, 2016 at 17:45 | answer | added | user97804 | timeline score: 10 | |
| Aug 28, 2016 at 17:24 | history | reopened |
Joseph O'Rourke Stanley Yao Xiao KConrad paul garrett Yemon Choi |
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| Aug 28, 2016 at 17:22 | comment | added | Amir Sagiv | I don't think answers in this depth and intuition would have been available in Math.Stackexchange. Maybe it should be re-opened? | |
| Aug 28, 2016 at 17:07 | comment | added | Anthony Quas | Here's a 3 word answer: the Central Limit Theorem | |
| Aug 28, 2016 at 16:59 | review | Reopen votes | |||
| Aug 28, 2016 at 17:26 | |||||
| Aug 28, 2016 at 15:50 | history | closed |
Douglas Zare Wolfgang Michael Albanese Alexey Ustinov R W |
Not suitable for this site | |
| Aug 28, 2016 at 15:47 | answer | added | KConrad | timeline score: 18 | |
| Aug 28, 2016 at 15:47 | comment | added | Joe Silverman | @DouglasZare I assume you mean "multi-variable", not "single variable". But I must respectfully disagree with you that this is too elementary for MO. i never saw it in either undergraduate or graduate classes and only ran across it when working with lattices in cryptography. (My guess is the that OP found this statement in such an article, but you are right that he/she should have given the reference.) | |
| Aug 28, 2016 at 15:32 | comment | added | Douglas Zare | How did you see this, numerical experiments or did you see someone claim this? If it was numerical experiments, you should state how you were generating the random vectors and how you measured their orthogonality. If you heard this claim, you should cite the source and state the result as clearly as you can. Either way, I think this is too well-known and elementary to be considered research-level since this is sometimes mentioned ("most of a sphere is near the equator") in a single variable calculus class when one computes the volumes of higher dimensional spheres by integrating. | |
| Aug 28, 2016 at 14:32 | history | edited | Joe Silverman | CC BY-SA 3.0 |
Made title more descriptive, added "lattice" tag
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| Aug 28, 2016 at 13:33 | answer | added | Joe Silverman | timeline score: 29 | |
| Aug 28, 2016 at 13:06 | review | Close votes | |||
| Aug 28, 2016 at 15:51 | |||||
| S Aug 28, 2016 at 12:24 | history | suggested | Amir Sagiv | CC BY-SA 3.0 |
changed tags + title
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| Aug 28, 2016 at 12:14 | comment | added | Amir Sagiv | I think that there should be some constraints over the way the vectors are chosen, e.g. being uniformly chosen from the unit ball. Then you also need to specify what do you mean by "approximately orthogonal" - almost certainly orthogonal? Mean of the inner product converges to zero? | |
| Aug 28, 2016 at 12:11 | review | Suggested edits | |||
| S Aug 28, 2016 at 12:24 | |||||
| Aug 28, 2016 at 12:02 | review | First posts | |||
| Aug 28, 2016 at 12:14 | |||||
| Aug 28, 2016 at 12:01 | history | asked | YONGSEEN KIM | CC BY-SA 3.0 |