most recent 30 from http://mathoverflow.net 2013-05-19T03:30:11Z http://mathoverflow.net/feeds/user/16793 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126415/using-fourier-transform-to-speed-up-calculation-of-forces-following-an-inverse-sq Matthias Goergens 2013-04-03T17:02:48Z 2013-04-03T17:36:20Z

<p>Suppose I have $n$ electric point charges in, say, two dimensions. Is there any algorithm (and I have a hunch that it might be related to the Fourier transform) to compute the net forces that act on each point charge in less than $O(n^2)$, preferably something like $O(n \log n)$? Thanks!</p> <p>An approximation might be good enough for my use case.</p>

http://mathoverflow.net/questions/97857/partial-backups Matthias Goergens 2012-05-24T18:11:26Z 2012-05-24T18:11:26Z

<p>Suppose you have some storage medium of a given size M, and can make some kind of backup on another medium of size B with M > B. You can choose the scheme to determine the contents of the backup.</p> <p>After you made that partial backup, an adversary (or a random process) will make a number of changes to your original medium. Given the changed medium and your partial backup, your task is to restore the original state of your medium. How many changes could you undo? What is the theoretical maximum? And how successful are the schemes you can come up with?</p> <p>I have toyed with this question for a while. Obviously, in general you can not hope to undo more than B changes. Viewed more mathematical, I am looking for a systematic code that works with huge block sizes.</p>

http://mathoverflow.net/questions/71502/circumference-of-convex-shapes Matthias Goergens 2011-07-28T15:46:46Z 2011-08-23T16:17:03Z

<p>Here is a puzzle I found in "Mitteilungen der DMV" (roughly "Letters of the German Society of Mathematicians") issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch rabbits), and he claims that less than 5% of his subjects could solve it in under 1 hour. He tested it on students of mathematics, professors of mathematics, computer science and engineering.</p> <p>See if you have more luck. The problem is deceptively simple:</p> <p>Suppose you have a triangle ABC and a point D inside the triangle. Prove: The perimeter of ABC is smaller than the circumference of ABD.</p> <p>I am currently working on a generalization: Given two convex shapes s and S, where S totally encloses s. Proof that the perimeter of s is no bigger than the perimeter of S.</p> <p>(Or alternatively, for a shapes with straight edges: Proof that the perimeter of the convex hull of a set of points increases monotonically (but not strictly monotonically) when adding points to the set.)</p> <p>Please try to find an elementary proof for the special case of the triangle.</p> <p>Edit: Thanks for all the nice answers. By now I found a really elementary proof on my own that just uses the triangle inequality twice.</p>

http://mathoverflow.net/questions/97857/partial-backups Matthias Goergens 2012-05-29T17:40:43Z 2012-05-29T17:40:43Z

Jykri, as for the adversary model, one scenario I had in mind was: a company wants to offer an online back up / restore service to customers. With a complete backup you'd have to wait for a full initial upload before the backup becomes useful. But with an (extensible) partial backup scheme, you can revert some changes to the initial data right away. (To keep the scheme simple at this point, we disallow changes halfway into the upload.)

http://mathoverflow.net/questions/97857/partial-backups Matthias Goergens 2012-05-29T17:32:51Z 2012-05-29T17:32:51Z

Douglas, that's the result I want to achieve eventually. But I don't see what kind of systematic code would do that. Also in the case of very small B's I don't see how that would work. Correcting a single bit error/change in M requires more than a single bit in B, because you are essentially encoding the position of the flipped bit, which needs log M space.

http://mathoverflow.net/questions/71502/circumference-of-convex-shapes Matthias Goergens 2011-07-28T17:48:40Z 2011-07-28T17:48:40Z

Thanks for the answers. That was a quite fast. The proof I found for the triangle case looked at the ellipses around A and B, and uses their convexity. Noam's reasoning reminds me very much of my studies of convex optimization. I'll have to look up the book that Deane' mentioned.

http://mathoverflow.net/questions/71502/circumference-of-convex-shapes/71504#71504 Matthias Goergens 2011-07-28T17:41:15Z 2011-07-28T17:41:15Z

Mark, Thanks. I am interested in the general result, but also in a very elementary proof for the case of triangles.

http://mathoverflow.net/questions/71502/circumference-of-convex-shapes/71505#71505 Matthias Goergens 2011-07-28T17:39:16Z 2011-07-28T17:39:16Z

Noam, yes, the intuitive argument is what motivated my generalisation.

http://mathoverflow.net/questions/71502/circumference-of-convex-shapes Matthias Goergens 2011-07-28T17:38:36Z 2011-07-28T17:38:36Z

Richard, I've heard of the boxes in boxes problem, and it suggests a different generalization than the one I mentioned. I wonder whether the boxes in boxes results holds for all dimensions.

http://mathoverflow.net/questions/71502/circumference-of-convex-shapes Matthias Goergens 2011-07-28T17:36:51Z 2011-07-28T17:36:51Z

Yes, I mean perimeter. Sorry, I read that problem in German, and normally don't do much geometry in English. I fixed it above. Thanks!