User akopyan - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:38:16Z http://mathoverflow.net/feeds/user/2158 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71479/1-lipschitz-length-preserving-isometry/71598#71598 Answer by akopyan for (1-Lipschitz) + (length-preserving) = isometry akopyan 2011-07-29T18:04:27Z 2011-07-29T19:51:15Z <p>If α is a convex shape and f is a 1-lipshitz map then perimeter of convex hull of f(α) ⩽ length of α. (Similar statement for higher dimension has been proven by Alexander)</p> http://mathoverflow.net/questions/50343/what-would-you-want-to-see-at-the-museum-of-mathematics/50405#50405 Answer by akopyan for What would you want to see at the Museum of Mathematics? akopyan 2010-12-26T07:53:05Z 2010-12-26T07:53:05Z <p><a href="http://en.wikipedia.org/wiki/Flexible_polyhedron" rel="nofollow">Flexible polyhedron</a>.</p> http://mathoverflow.net/questions/30511/ebook-readers-for-mathematics/30521#30521 Answer by akopyan for eBook readers for mathematics akopyan 2010-07-04T15:29:31Z 2010-07-04T15:29:31Z <p>iPad is good for pdf and djvu (and only for that). But you can't reed your favorite mathbooks on a beach, because its display is not e-ink.</p> http://mathoverflow.net/questions/30402/parabolic-envelope-of-fireworks/30492#30492 Answer by akopyan for Parabolic envelope of fireworks akopyan 2010-07-04T09:03:57Z 2010-07-04T09:57:09Z <p>It is easy to see that all these parabolas have the same directrix. Height of a directirix correspond to energy of the body. So you have the family of parabolas with the common point $P$ and the directrix $l$. It is easy to prove, (using just definition of parabola as a locus of points...) that all of the touched the parabola with the focus at $P$ and the directrix $l_1$, which parallel $l$ (actually $l$ is midline of $P$ and $l_1$).</p> <p>The same holds for sun-earth set. If Earth decides to fly in other direction (but with the same speed) its path will be always touch the fixed ellipse with foci in Sun and this position of Earth.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/21926#21926 Answer by akopyan for What are the most overloaded words in mathematics? akopyan 2010-04-20T03:49:48Z 2010-04-20T03:49:48Z <p>"Cauchy theorem"</p> http://mathoverflow.net/questions/21877/minimum-cover-of-partitions-of-a-set/21918#21918 Answer by akopyan for Minimum cover of partitions of a set akopyan 2010-04-20T01:31:25Z 2010-04-20T01:43:40Z <p>If $n=\frac{p^t-1}{p-1}$ and $k=p+1$ then we can consider the set $N$ as a finite projective space and our sets be lines in this space. </p> <p>I think that in other cases problem is hard.</p> http://mathoverflow.net/questions/20551/sources-for-bibtex-entries/20582#20582 Answer by akopyan for Sources for Bibtex entries akopyan 2010-04-07T02:02:29Z 2010-04-07T02:02:29Z <p>Another way is to use google with option "filetype:bib". Something like <a href="http://www.google.com/search?hl=en&amp;q=Kolmogorov+Arnold+filetype%3abib&amp;aq=f&amp;aqi=&amp;aql=&amp;oq=&amp;gs_rfai=" rel="nofollow">this</a>.</p> http://mathoverflow.net/questions/19046/open-source-mathematical-software/19052#19052 Answer by akopyan for Open source mathematical software. akopyan 2010-03-22T19:19:04Z 2010-03-22T19:19:04Z <p>I like <a href="http://edu.kde.org/kig/" rel="nofollow">kig</a>. Very useful software for simple geometric constructions. Also it can help if you want to make a figure for a paper. </p> http://mathoverflow.net/questions/8247/one-step-problems-in-geometry/18249#18249 Answer by akopyan for One-step problems in geometry akopyan 2010-03-15T06:49:03Z 2010-03-15T06:49:03Z <p>Every finite point set has Delauney triangulation (circumsphere of any simplex doesn't contain points from this set).</p> http://mathoverflow.net/questions/16681/is-ellipse-on-a-sphere-convex-proof/16710#16710 Answer by akopyan for Is ellipse on a sphere convex? (proof) akopyan 2010-02-28T17:54:53Z 2010-03-01T04:39:07Z <p>Yes it is. After central projection on the plane (Klein model for sphere) you obtain usual ellipse.</p> <p>Also you can show it using triangle inequality. All proofs from euclidean plane works. For example this one: Suppose $F_1$ and $F_2$ foci of the ellipse. Take any two points $A$ and $B$ inside and reflect $F_2$ with respect to the line $AB$. New point denote by $F_2'$. Take any point $X$ on the segment $AB$. Suppose ray $F_1X$ intersect the segment $F_2'A$ (the case $F_2'B$ is the same) in the point $Y$. We have, $$F_1X+F_2X=F_1X+F_2'X&lt; F_1X+XY+YF_2'=F_1Y+YF_2'&lt; F_1A+AY+YF_2'=F_1A+AF_2'$$</p> http://mathoverflow.net/questions/16691/geometrically-interpreting-the-answer-to-a-vector-calculus-question-involving-tan/16718#16718 Answer by akopyan for Geometrically interpreting the answer to a vector calculus question involving tangent line segments to ellipses. akopyan 2010-02-28T22:01:19Z 2010-02-28T22:10:11Z <p>There is a geometric way to show that $n$-gon circumscribed around an ellipse has minimal perimeter if it is inscribed in a confocal ellipse. From Poncelet porism (and generalization of optical property) it follows that we have continuous family of "minimal" polygons.</p> <p>If we know it, then it is easy to understand that the circumscribed rhomb (from your question) and the circumscribed rectangular (with perimeter $4(a+b)$) are minimal polygons. So, side of the rhomb equals $a+b$.</p> http://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics/15525#15525 Answer by akopyan for Your favorite surprising connections in Mathematics akopyan 2010-02-17T00:59:35Z 2010-02-17T00:59:35Z <p>The chromatic number of the <a href="http://en.wikipedia.org/wiki/Kneser%5Fgraph" rel="nofollow">Kneser graph</a> $KG_{n,k}$ is equal exactly $2n-k+2$. There are very simple proof based on Borsuk-Ulam theorem. </p> http://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics/15510#15510 Answer by akopyan for Your favorite surprising connections in Mathematics akopyan 2010-02-16T23:37:41Z 2010-02-16T23:37:41Z <p>There exist two binary trees with <a href="http://en.wikipedia.org/wiki/Tree%5Frotation" rel="nofollow">rotation distance</a> $2n-6$. The proof is unexpected and based on hyperbolic geometry (Sleator, Tarjan, Thurston (1988), "Rotation distance, triangulations, and hyperbolic geometry").</p> http://mathoverflow.net/questions/102935/a-weak-form-of-borsuks-conjecture Comment by akopyan akopyan 2012-07-30T08:06:57Z 2012-07-30T08:06:57Z Dear Prof. Kalai, how many faces in your example? http://mathoverflow.net/questions/71479/1-lipschitz-length-preserving-isometry/71598#71598 Comment by akopyan akopyan 2011-07-30T13:18:34Z 2011-07-30T13:18:34Z Here it is for high dimensional Euclidean case. <a href="http://www.cs.elte.hu/geometry/Workshop09/large_r5.pdf" rel="nofollow">cs.elte.hu/geometry/Workshop09/large_r5.pdf</a> http://mathoverflow.net/questions/71479/1-lipschitz-length-preserving-isometry/71481#71481 Comment by akopyan akopyan 2011-07-28T12:55:01Z 2011-07-28T12:55:01Z <i>Using that f is 1-Lipschitz and that [aa′] is a diameter, we get that β must be contained in the interior of α</i> Why? http://mathoverflow.net/questions/60968/is-there-a-generalized-feuerbach-point-for-an-irregular-non-euclidean-triangle Comment by akopyan akopyan 2011-04-12T06:31:20Z 2011-04-12T06:31:20Z I've found synthetic proof for it to the two dimensional non-Euclidean case. But article is in Russian. <a href="http://www.mccme.ru/free-books/matpros/mpd.pdf#page=155" rel="nofollow">mccme.ru/free-books/matpros/mpd.pdf#page=155</a> http://mathoverflow.net/questions/31354/theorems-in-euclidean-geometry-with-attractive-proofs-using-more-advanced-methods/31420#31420 Comment by akopyan akopyan 2010-07-11T17:19:41Z 2010-07-11T17:19:41Z This problem from IMO Shortlist. You can find solution for high school students in Prasolov book. But this problem is not so easy in hyperbolic geometry. http://mathoverflow.net/questions/31354/theorems-in-euclidean-geometry-with-attractive-proofs-using-more-advanced-methods Comment by akopyan akopyan 2010-07-11T06:03:52Z 2010-07-11T06:03:52Z Every theorem from classical geometry has short and elegant proof for &quot;high-school students&quot;. But one can find very difficult proof which use algebraic topology or category theory. http://mathoverflow.net/questions/30511/ebook-readers-for-mathematics/30521#30521 Comment by akopyan akopyan 2010-07-04T17:41:04Z 2010-07-04T17:41:04Z It is not compfortable to read from netbook. iPad is good as bookreader. http://mathoverflow.net/questions/8247/one-step-problems-in-geometry/18249#18249 Comment by akopyan akopyan 2010-03-16T02:42:31Z 2010-03-16T02:42:31Z I mean another proof. There is very nice proof through stereographic projection on sphere http://mathoverflow.net/questions/17006/linear-algebra-proofs-in-combinatorics/17029#17029 Comment by akopyan akopyan 2010-03-04T03:29:08Z 2010-03-04T03:29:08Z Alexey Kanel-Belov http://mathoverflow.net/questions/16691/geometrically-interpreting-the-answer-to-a-vector-calculus-question-involving-tan/16718#16718 Comment by akopyan akopyan 2010-03-01T19:11:29Z 2010-03-01T19:11:29Z 2 Agol, it is not. In this case you obtain two similar ellipses with different foci. Reason is, under a affine transformation length of segments doesn't preserve. http://mathoverflow.net/questions/16691/geometrically-interpreting-the-answer-to-a-vector-calculus-question-involving-tan/16718#16718 Comment by akopyan akopyan 2010-03-01T03:28:54Z 2010-03-01T03:28:54Z Yes, I'm coauthor of this book. Also, look for the famous book &quot;Geometry&quot; by Marcel Berger. In volume 2 you can find many interesting statements similar to the fact in the initial question. – akopyan 53 secs ago