def V(n):

return pi.n()^(n/2.)/gamma(n/2.+1)

def A(n):

return n*V(n)

G = plot(A, 0, 20, color = 'red', legend_label='surface of unit ball') + list_plot([A(n) for n in srange(20)], color = 'red')

G += plot(V, 0, 20, color = 'blue', legend_label='volume of unit ball') + list_plot([V(n) for n in srange(20)], color='blue')

G.show()

Let be an Hermitian matrix with eigenvalues and let the matrix be obtained from by removing the th row and the th column for some . Then the eigenvalues of satisfy the interlacing

.

Alasdair gave a little Matlab/Octave experiment to illiustrate this and we reproduce it here using SAGE (sticking to his notation for comparison).

sage: n = 4

sage: D = diagonal_matrix(range(1,n+1))

sage: D

[1 0 0 0]

[0 2 0 0]

[0 0 3 0]

[0 0 0 4]

sage: R = random_matrix(RR,n)

sage: R

[ 0.778692495501054 -0.135208587751857 0.257223918336617 0.774429743108338]

[ 0.423676053284628 0.417711760181128 0.241748509057067 0.961897138926902]

[ 0.761841048992683 0.728238125048413 -0.919597769483716 0.778538514077347]

[-0.459693778794198 -0.349046725059769 -0.969387932465504 -0.922931517520695]

sage: S = R - R.transpose()

sage: Q = (S-identity_matrix(n)).inverse()*(S+identity_matrix(n))

sage: M = Q*D*Q.transpose()

sage: M

[ 1.95237691577229 0.231093937356808 0.216951816905018 0.296266773415154]

[ 0.231093937356808 2.92188071228280 0.263114531184218 0.0747380763688735]

[ 0.216951816905018 0.263114531184218 2.90749330981640 -1.36941012136363]

[ 0.296266773415154 0.0747380763688735 -1.36941012136363 2.21824906212850]

sage: M.eigenvalues()

[3.99999999999999, 3.00000000000003, 1.99999999999998, 1.00000000000000]

sage: x = range(n)

sage: x.remove(randint(0,3)

sage: N = M.matrix_from_rows_and_columns(x,x)

sage: N

[ 1.95237691577229 0.216951816905018 0.296266773415154]

[0.216951816905018 2.90749330981640 -1.36941012136363]

[0.296266773415154 -1.36941012136363 2.21824906212850]

sage: N.eigenvalues()

[3.97504068833705, 2.09499000737375, 1.00808859200640]

Footnote: We had to take a little detour to delete some rows and columns from a matrix. It turns out, that this feature is already Ticket #11528 on the trac server of sage.

We considered patenting; we prepared a patent and it was nearly filed. Then I had an interaction with a big, multinational electronics company. I approached a guy at a conference and said, “We’ve got this patent coming up, would you be interested in sponsoring it over the years?” It’s quite expensive to keep a patent alive for 20 years. The guy told me, “We are looking at graphene, and it might have a future in the long term. If after ten years we find it’s really as good as it promises, we will put a hundred patent lawyers on it to write a hundred patents a day, and you will spend the rest of your life, and the gross domestic product of your little island, suing us.” That’s a direct quote.

Source: Andre Geim: in praise of graphene, interview with nature news, published online 7 October 2010, doi:10.1038/news.2010.525 (last accessed: 9 Oct 2010)

Based on surveys Barnes collected, the top five worries of parents are, in order:

1. Kidnapping
2. School snipers
3. Terrorists
4. Dangerous strangers
5. Drugs

But how do children really get hurt or killed?

1. Car accidents
2. Homicide (usually committed by a person who knows the child, not a stranger)
3. Abuse
4. Suicide
5. Drowning

FAO estimates that 1.02 billion people are undernourished worldwide in 2009.

FAO, the state of food insecurity in the world, 2009, p. 2.