Testwiki:Reference desk/Archives/Mathematics/2013 May 15

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May 15

Dijkstra's algorithm

Does the tree produced by Dijkstra's algorithm contain the shortest path between any two vertices or only between the starting vertex and another vertex? --107.207.240.197 (talk) 00:10, 15 May 2013 (UTC)

From a single source to all other vertices, but see the Floyd–Warshall algorithm. 2602:306:33A0:D150:226:8FF:FEE2:4E6A (talk) 01:14, 15 May 2013 (UTC)

In a triangle the shortest path between two vertices is the line between them. And all three shortest paths together don't form a tree. Dmcq (talk) 13:08, 15 May 2013 (UTC)

Decomposing a simply-connected space

Say I can produce a simply-connected space Y by gluing together finitely many copies of a space X in some way. Am I able to conclude that X is simply-connected? This seems far too good to be true -- I imagine I am just lacking the imagination to come up with a counter-example. Does anyone have a good one? (...or a proof of the contrary!) Thanks, Icthyos (talk) 17:50, 15 May 2013 (UTC)

Found a counter-example: take X to be the wedge of a 2-sphere with a 1-sphere. Glue two of them together by sticking the 1-sphere of one copy around a great circle on the 2-sphere of the other copy, and vice versa. Icthyos (talk) 19:04, 15 May 2013 (UTC)

You can get the fundamental group of a nice union of subspaces by the Seifert-van Kampen Theorem, as the free product of the groups of the subspaces, amalgamated by the fundamental group of their intersection. For your example, you get the trivial group as a free product of two free groups on one letter, amalgamated by the group of a figure 8, their intersection, which is a free group on two letters. Another similar example is gluing two solid tori to get a 3-sphere, as described here, for example. John Z (talk) 05:37, 16 May 2013 (UTC)

Expected value convergence question

Suppose $X_1,X_2,X_3,\dots$ are identical independent variables that take take value 0 with probability .5 and take value 1 with probability .5. Consider the following:$\underset{n\to \mathrm{\infty }}{\lim }\frac{n/2-\sum _{i=1}^n{X}_i}{n^{\delta }}$. For $\delta =1$, the law of large numbers tells us this limit almost surely exists (and equals 0). For $\delta =0$, the limit almost surely does not exist. What's the cut-off value for $\delta$ where the limit ceases to almost surely exist?--80.109.106.49 (talk) 19:16, 15 May 2013 (UTC)

It seems to me that the Law of the iterated logarithm answers this question - the answer is δ > 1/2. 96.46.198.58 (talk) 21:06, 15 May 2013 (UTC)

That will do it. Thanks.--80.109.106.49 (talk) 09:34, 16 May 2013 (UTC)