I said in class, yet forgot to post on the blog, that homework #5 is due on April 22nd. By the way, I think I will assign one more homework on the 22nd that will be due May 4th. The final will also be assigned on May 4th, and will be due on the day the final is scheduled to be for our class, which is Thursday May 13th.
Also, here is the Macaulay2 file that illustrates the automated geometric proving and colorability of graphs applications from class.
I have also placed the book An Introduction to Grobner Bases by Adams and Loustaunau on reserve in the math library. The material on colorability of graphs, as well as the applications to linear programming are covered in chapter 2 of that book.
Wednesday, April 14, 2010
Friday, April 9, 2010
Homework #5
Problems from the text:
4.5: 3,8,11,12
4.6: 9
6.4: 3,8
Additional problems:
1. Using the Grobner basis technique from class (and of course, a computer algebra system), prove that Euler's nine point circle theorem is true. You may use either the fraction field method, or the saturation method to do this problem, but since the conclusions do not follow strictly from the hypotheses, you must use one or the other. A nice picture and discussion of the theorem is available here.
The statement is the following. Let ABC be a triangle. Then the midpoints of the sides, the feet of the altitudes, and the midpoints of the line segments joining the orthocenter (which is the intersection of the altitudes) and the vertices of the triangle, all lie on a single circle. The circle theorem of Appolonius is Euler's theorem in the case of a right triangle.
When I worked this problem out, I had 3 'u' variables, and 20 'x' variables, and therefore 20 'hypotheses'. There should be six 'conclusions' that you have to check. One way to proceed would be to consider the circle that goes through the midpoints of the sides, and show that the other 6 points are on this circle.
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),(3,6),(3,9),(5,6),(6,8),(7,8),(8,9)} is 3-colorable. Use the Grobner basis you find to give an explicit 3-coloring.
b) Show that if you add a new edge {(1,3)} to the graph, it is still 3-colorable, and that the coloring is now unique up to permutation of the colors.
3) Use the method of linear programming we learned in class to solve the following system of equations in the nonnegative integers:
3s_1+2s_2+s_3+2s_4=10
4s_1+3s_2+s_3=11
2s_1+4s_2+2s_3+s_4=10
4.5: 3,8,11,12
4.6: 9
6.4: 3,8
Additional problems:
1. Using the Grobner basis technique from class (and of course, a computer algebra system), prove that Euler's nine point circle theorem is true. You may use either the fraction field method, or the saturation method to do this problem, but since the conclusions do not follow strictly from the hypotheses, you must use one or the other. A nice picture and discussion of the theorem is available here.
The statement is the following. Let ABC be a triangle. Then the midpoints of the sides, the feet of the altitudes, and the midpoints of the line segments joining the orthocenter (which is the intersection of the altitudes) and the vertices of the triangle, all lie on a single circle. The circle theorem of Appolonius is Euler's theorem in the case of a right triangle.
When I worked this problem out, I had 3 'u' variables, and 20 'x' variables, and therefore 20 'hypotheses'. There should be six 'conclusions' that you have to check. One way to proceed would be to consider the circle that goes through the midpoints of the sides, and show that the other 6 points are on this circle.
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),(3,6),(3,9),(5,6),(6,8),(7,8),(8,9)} is 3-colorable. Use the Grobner basis you find to give an explicit 3-coloring.
b) Show that if you add a new edge {(1,3)} to the graph, it is still 3-colorable, and that the coloring is now unique up to permutation of the colors.
3) Use the method of linear programming we learned in class to solve the following system of equations in the nonnegative integers:
3s_1+2s_2+s_3+2s_4=10
4s_1+3s_2+s_3=11
2s_1+4s_2+2s_3+s_4=10
Friday, April 2, 2010
Tuesday, March 30, 2010
Friday, March 12, 2010
Monday, March 8, 2010
Tuesday, March 2, 2010
Macaulay2 script
The Macaulay2 script solving the Lagrange multiplier problem can be found here.
One can perform this computation in a Sage Notebook screen with macaulay2 enabled as I mentioned in an earlier blog post, or you can try installing Macaulay2 and emacs on your personal machine by following the installation instructions on the Macaulay2 Home Page.
If you have any problems setting it up, then I can try and help you via email.
One can perform this computation in a Sage Notebook screen with macaulay2 enabled as I mentioned in an earlier blog post, or you can try installing Macaulay2 and emacs on your personal machine by following the installation instructions on the Macaulay2 Home Page.
If you have any problems setting it up, then I can try and help you via email.
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