Timeline for Expected number of identical vertex pairs with the same Euclidean distance on a randomly colored rectangular lattice
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 26, 2012 at 4:55 | vote | accept | Polyrhythm | ||
| Jul 18, 2012 at 11:09 | comment | added | Joseph O'Rourke | My (added) graph for several colors supports Gerhard's conclusion that "you will find the answer not very influenced by the number of colors used." | |
| Jul 17, 2012 at 22:17 | history | edited | Polyrhythm | CC BY-SA 3.0 |
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| Jul 17, 2012 at 22:11 | history | edited | Polyrhythm | CC BY-SA 3.0 |
Provided Gerhard Paseman's defintion for $P$; deleted 5 characters in body
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| Jul 17, 2012 at 21:15 | comment | added | Polyrhythm | @Gerhard Paseman, I think you've made a very good suggestion, however one of my interests here is to understand the number of colors, $k$, one needs to obtain a specific value of $P$. | |
| Jul 17, 2012 at 19:05 | comment | added | Gerhard Paseman | In fact, it is an interesting problem just to consider the multiplicity function m(d) suggested by Douglas Zare. When N/M is large, m(d) is often a simple function of m and n, and decreases something like 1/d^2, so you will find the answer not very influenced by the number of colors used. I recommend focusing on m(d) for M < N < 3M, say. Gerhard "Ask Me About Sysem Design" Paseman, 2012.07.17 | |
| Jul 17, 2012 at 18:58 | comment | added | Polyrhythm | @Gerhard Paseman, Given that the elements within the tuples are unordered, that you wouldn't count {1, color1, color2} and {1, color2, color1} as distinct, yes, exactly: $P+T =$ ${N*M}\choose{2}$ according to your definition of T. | |
| Jul 17, 2012 at 17:34 | comment | added | Gerhard Paseman | You might provide an explicit example for further clarity. Suppose I count the number of types T of (Distance, color1, color2) tuples that occur for a particular coloring. Then P + T = mn choose 2, by my interpretation of P. If so, that gives you a good start on bounds right there, especially for small colors. Gerhard "Did I Get That Right?" Paseman, 2012.07.17 | |
| Jul 17, 2012 at 17:09 | history | edited | Polyrhythm | CC BY-SA 3.0 |
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| Jul 17, 2012 at 16:49 | answer | added | Joseph O'Rourke | timeline score: 5 | |
| Jul 17, 2012 at 3:13 | comment | added | Polyrhythm | Honestly, I would be very surprised if someone posts an exact answer... though that's obviously very welcome. I'm interested in how well one can do on bounding the value of P. | |
| Jul 17, 2012 at 2:25 | history | edited | Polyrhythm | CC BY-SA 3.0 |
added 28 characters in body
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| Jul 17, 2012 at 2:20 | history | asked | Polyrhythm | CC BY-SA 3.0 |