Monday, December 01, 2003
Use the greedy algorithm to form a sequence of integer triples (1,2,3), (4,6,8), (7,10,13), (11,15,19), (12,17,22), (14,20,26), (9,16,23), (21,29,37), ... such that 1. the n-th triple is an arithmetic progression with common difference n; and 2. no integer is allowed to appear in more than one triple. Question: does the number 5 ever appear? (All numbers <5000 other than 5 do appear!)
(If you do the same thing for pairs, you get (1,2), (3,5), (4,7), (6,10), (8,13), (9, 15), (11, 18), (12, 20), ...; the n-th pair is ([n phi], [n phi^2]), where phi is the golden ratio.)
posted by Lionel Levine at 9:41 PM
Saturday, September 06, 2003
Let n be an integer and let p(x) be a monic polynomial of degree d. Show that the (d+1)-by-(d+1) Toeplitz determinant det(p(n+i-j))_{i,j=1}^{d+1} is independent of both n and p. (!)
posted by Lionel Levine at 9:55 PM
Sunday, August 03, 2003
Show that the sum of the signs of all permutations in S_n which have no fixed points is (n-1)*(-1)^(n-1). (Hint: consider det(J-Id) where J is the all ones matrix)
posted by Lionel Levine at 9:28 PM
Monday, March 03, 2003
Show that p(n), the number of partitions of n, is equal to the sum of mu(lambda_i), where mu is the Mobius function and the sum runs over all parts of all partitions of n+1.
posted by Lionel Levine at 12:46 PM
Wednesday, February 26, 2003
The determinant of (gcd(m,n)), m,n running from 1 to N, is equal to phi(N)phi(N-1)...phi(1) (Stanley EC1, p. 161). Is there a similar formula for det(lcm(m,n))?
posted by Lionel Levine at 8:39 PM
Wednesday, February 19, 2003
Let K and L be number fields, and M their tensor product over Q. Under what conditions is M a field?
posted by Lionel Levine at 9:59 PM
Sunday, February 09, 2003
Let's say that a graph G has the "unique path property" if there's a unique shortest path connecting any two vertices of G. Show that G has the unique path property if and only if every even cycle of G has at least two internal edges.
posted by Lionel Levine at 5:58 PM
Pretty sure these are true but I'll make them PODASIPS just in case! Prove or disprove:
1. S^n cannot be covered by fewer than n+2 open hemispheres.
2. If S^n = union of H_alpha is any covering of the sphere by open hemispheres, there is a subcover S^n = union of H_alpha_i consisting of n+2 of them.
posted by Lionel Levine at 5:49 PM
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