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July 17

graph embedding dimension

Any graph with n vertices can be embedded in Eⁿ⁻¹ so that its edges are a subset of the Delaunay triangulation; this is trivial. So there must be, for each graph, a minimum k for which such an embedding exists in E^k. Is there a name for this k? —Tamfang (talk) 06:49, 17 July 2013 (UTC)[reply]

My motive here, by the way, is to extend (something akin to) the notion of a planar graph. —Tamfang (talk) 05:08, 20 July 2013 (UTC)[reply]

Matrices with one negative entry per row and column

Let A be an n x n matrix with exactly n negative entries, which are distributed so that there is exactly one negative entry in each row and in each column (all other entries are 0 or positive). In general A need not be invertible - for example

A={\begin{pmatrix}2&-1\\-2&1\end{pmatrix}}

is not invertible. Are there additional conditions that can be imposed on the row sums of A that will ensure that A is always invertible ? For example, if we add the condition that all row sums are negative ? Or all row sums are 0 or negative but not all are 0 ? Gandalf61 (talk) 10:25, 17 July 2013 (UTC)[reply]

It reminds me of the problem of placing the maximum number of rooks of a chessboard so that no two attack one other. — 79.113.233.129 (talk) 23:05, 17 July 2013 (UTC)[reply]

It would appear that assuming the row sum is negative can puts some sign and size constraints on what can appear in the kernel, but I didn't notice anything really that worth reporting; although, I'm sure that with some consideration, these wouldn't be hard to improve on. It's completely elementary, but if you're assuming that there is a single strictly negative permutation of entries, then you can work with a matrix with -1's down the diagonal and everything else positive; it only changes the magnitude of the row sums.Phoenixia1177 (talk) 06:14, 20 July 2013 (UTC)[reply]

Meant to add this: it seems like adding a condition to rule out all negative entried vectors from being in the kernel would immensely increase the odds of it being invertible.Phoenixia1177 (talk) 06:38, 20 July 2013 (UTC)[reply]

Is 27182818284590452354 special?

There is a writer seen on the Internet who is occupied with calculus, information theory and game theory and who consistently uses as pseudonym a particular number 27182818284590452354. That looks like a large integer that could be shown as a 65-bit binary sequence, or with fewer digits in another number base such as octal or hex, but does it have any special significance? I am not equipped to factorise it but I can see that it is not a prime number. Theoretically every finite sequence of digits can be found among Transcendental numbers but is this sequence already known anywhere? Should we look for a coded message? The leading pair 27... is too large for an alphabet substitution cryptogram and I don't see ASCII codes. If the number has no significance then it's hard to see a rational motive for the person who is using it as a rather aggressive Internet entity, so can we inspect the numerical signature for a clue to the signer's agenda? DreadRed (talk) 17:03, 17 July 2013 (UTC)[reply]

See e (mathematical constant). -- Toshio Yamaguchi 17:09, 17 July 2013 (UTC)[reply]

I don't see anything remarkable about the prime factorization. This applet allows you to factor it yourself (within a fraction of a second). -- Toshio Yamaguchi 17:30, 17 July 2013 (UTC)[reply]

It's the first 20 digits of e. Or 2 x 13 x 439 x 3967 x 600336 035933.--Gilderien Chat|List of good deeds 21:43, 17 July 2013 (UTC)[reply]