User puzzly - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:21:42Z http://mathoverflow.net/feeds/user/22580 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108489/distribution-of-random-vectors Distribution of random vectors puzzly 2012-09-30T19:25:53Z 2012-09-30T19:25:53Z <p>Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).</p> <p>A vector $u\in S$ is chosen uniformly at random in the spherical disc with center $(0,0,1)$ and radius $\alpha$ on the sphere. After that a vector $v\in S$ is chosen uniformly at random on the spherical circle with center $u$ and radius $\beta$. </p> <p>Can we describe the distribution of $v$ on the sphere (e.g., by specifying its density function or otherwise)? Clearly, $v$ lies in the spherical disc around $(0,0,1)$ of radius $\alpha+\beta$, however its distribution doesn't seem to be uniform. Can we estimate somehow the most frequent angle that $v$ subtends with $(0,0,1)$? The answer should certainly depend on $\alpha$ and $\beta$.</p> <p>(By a "spherical disc" with center $c$ and radius $r$ we understand the set of points on the sphere whose distance from $c$ is at most $r$. Similarly, a "spherical circle" with center $c$ and radius $r$ is the set of points on the sphere whose distance from $c$ is exactly $r$.)</p> <p>This question arouse related to the distribution of the polarization vector in a randomly grained polycrystalline material (ceramics).</p> http://mathoverflow.net/questions/92883/small-4-chromatic-coin-graphs Small 4-chromatic coin graphs puzzly 2012-04-02T10:10:18Z 2012-04-04T14:29:44Z <p>A <strong><em>coin graph</em></strong> is a graph that can be represented by a set of disjoint, except possibly touching, <em>unit</em> disks in the plane (i.e. the disks are the vertices and the edges correspond to the pairs that touch each other). It's easy to show by induction that $\chi(G)\leq4$ for every coin graph $G$, as there's always a vertex of degree at most 3. </p> <p>My question is: what is the smallest order (i.e. the number of vertices) of a 4-chromatic coin graph? </p> <p>In this paper by Erdos <a href="http://www.renyi.hu/~p_erdos/1987-27.pdf" rel="nofollow">http://www.renyi.hu/~p_erdos/1987-27.pdf</a> there is a coin graph of order 19 that is 4-chromatic (see Figure 1) by I doubt it's the smallest one (it was constructed for a different purpose, having to do with the independence number). The question I asked was proposed for an IMO competition in 1979, see p. 138 question 73 in Djukic, Jankovic, Matic, Petrovic: the IMO Compendium (there is no solution there, however). </p> <p>Clearly, coin graphs are also <em>unit distance graphs</em>, for the definition see <a href="http://en.wikipedia.org/wiki/Unit_distance_graph" rel="nofollow">http://en.wikipedia.org/wiki/Unit_distance_graph</a>. The smallest 4-chromatic unit-distance graph is probably the Moser spindle <a href="http://en.wikipedia.org/wiki/Moser_spindle" rel="nofollow">http://en.wikipedia.org/wiki/Moser_spindle</a> that has 7 vertices. There is a similar notion of <strong><em>matchstick graphs</em></strong>: those are unit distance graphs drawn in the plane with non-crossing straight-line segments, see <a href="http://en.wikipedia.org/wiki/Matchstick_graph" rel="nofollow">http://en.wikipedia.org/wiki/Matchstick_graph</a> Note that the Moser spindle is NOT a matchstick graph, although it's planar and unit-distance. </p> <p>The second (related) question is: what is the smallest order of a 4-chromatic matchstick graph?</p> <p>I think the answer (to the second question) is 8. </p> http://mathoverflow.net/questions/92883/small-4-chromatic-coin-graphs/92931#92931 Comment by puzzly puzzly 2012-04-03T12:58:35Z 2012-04-03T12:58:35Z great, sounds convincing enough, except that i don't see (at least not without a bit of geometry) how |V(G)-V(C)|\geq 2 implies |V(C)|\geq 8. did u have in mind a trivial way to see that? http://mathoverflow.net/questions/92883/small-4-chromatic-coin-graphs Comment by puzzly puzzly 2012-04-02T12:22:53Z 2012-04-02T12:22:53Z if i understand correctly, your second construction is a matchstick graph, but not a coin graph, right?