Final Exam

The final exam is available here.

Note that the wording on problem 2 is changed slightly. One must assume that f is homogeneous in part a, and that I is a homogeneous ideal in part b) [see problem 1 for a definition of homogeneous].

Office hours on 05/05/2010

My office hours for Wednesday May 5 will be from 1 PM to 2 PM in Malott 112.

Saúl

Updated example file

The updated Macaulay2 example file from class is available here.

Homework #6 correction

I had the numbers wrong in the additional problem that was assigned. The correct problem is now listed in the homework (I changed the 'b' vector from [-323,1035,654] to [-323,389,8]).

Homework #6

Sorry for the delay in assigning the homework. Posting the problem set completely slipped my mind until Sunday evening. The due date for this homework will be May 6th to account for the delayed assignment time.

6.1: 4
6.2: 4
6.3: 5,7,10
7.1: 5,8

1. Find an optimal integer solution to the following integer programming problem using the cost function c(s_1,s_2,s_3,s_4,s_5) = 10s_1+100s_2+100s_3+10s_4+s_5, where
2s_1+5s_2-3s_3+s_4-2s_5=-323
s_1+7s_2+2s_3+3s_4+s_5=389
4s_1-2s_2-s_3-5s_4+3s_5=8

HW 5, what to hand in

I have received questions about what needs to be hand in for the computer problems. Here is what I think you need FOR ALL THREE PROBLEMS: The input (hypothesis, conclusions, edges, etc), the output, and a few sentences explaining what the output means and why it gives you (or not) what you want. It would be nice if you mark these clearly, so it is easier for me to find. Also, it is perfectly find to hand in printouts of what the input/output (no need to write them down with your own handwriting).

Good luck with your HW.

Saúl

Homework #5 correction

There was a slight problem with question #2. Here is the corrected version:

2. a) Using a computer algebra system, show that the graph with edges
{(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,7),(3,4),(3,6),(3,9),(5,6),(6,8),(6,9),(7,8),(8,9)}

is 3-colorable. Use the Grobner basis you find to give an explicit 3-coloring. How many colorings are there?
b) Show that if you add edge (1,3) to the graph, it is still 3-colorable, and that the coloring is now unique up to permutation of the colors.