Problems for Tuesday, October 13: January 2009, II-3
August 2008, I-1, I-2, II-1
And
these
leftover problems.
Problems for Thursday, October 15: August 2008, I-3, I-4, II-2 and II-3
Problems for Tuesday, October 20: August 2008, I-5, I-6, II-4 and II-5.
And
these
leftover problems (I added another hint).
Problems for Thursday, October 22: August 2008, II-6
January 2008, I-1, I-2, II-1
Problems for Tuesday, October 27: January 2008, I-3, I-4, II-2, II-3
Problems for Thursday, October 29: January 2008, I-5, I-6, II-4, II-5
Problems for Tuesday, November 3: January 2008, II-6
August 2007, I-1, I-2, II-1
Problems for Thursday, November 5: August 2007, I-3, I-4, II-2, II-4
Also: finish January 2008, II-6.
Problems for Tuesday, November 10: August 2007, I-5, I-6, finish
II-2(b), II-3, II-4
I regret that I do not see how to do August 2007, I-3, unless the given
semigroup S has the cancellation property. If it does, then, as in
class, for each
p,q in G we construct r in G with rq=p. In particular, we
have, for any x in G, an e_x in G with e_x x = x. For any
x,y, e_x xy = e_y xy = xy, so e_x=e_y is the identity of G. Call
it e. For each q in G, we have r in G with rq=e, so we have
inverses too.
I believe that the committee made a mistake in not including the
cancellation property in the hypothesis, and so will not ask you
to think about what happens without it.
Problems for Thursday, November 12: August 2007, II-5, II-6,
January 2007, I-1, II-1
Problems for Tuesday, November 17:
Finish August 2007, II-5. The hypothesis that f preserves the vertical
fibration means that f(x,y) = (u(x),v(x,y)).
Also: January 2007, I-2, I-3, II-2, II-3
Problems for Thursday, November 19:
January 2007, I-4, I-5, II-4, II-5
Problems for Tuesday, November 24:
January 2007, I-6, II-6
August 2006, I-1, II-1
Problems for Tuesday, December 1:
August 2006, I-4, I-5, II-2, II-3
Problems for Thursday, December 3:
August 2006, I-6, II-4, II-5
January 2006, I-1, II-1
Problems for Tuesday, December 8:
January 2006, I-2, I-3, II-2, II-3, II-5
Problems for Thursday, December 10:
August 2006, II-6
January 2006, I-4, I-5, I-6
Problems for Tuesday, December 15, 10 a.m. - 11:30 a.m., with Professor
Zinger:
August 2006, I-2, I-3
January 2006, II-4, II-6
Please leave your solutions under Professor Zinger's door (3-111) by
5 p.m. on Monday, December 14.
Good luck on comps!!!
