HOMEWORK: (click here for comments / solutions to the HW)
Please spend at least 2 hours a night reading the material/looking at the proofs/making sure you understand the details. Below is a tentative reading list and homework assignments. It is subject to changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.
Week 11: November 14 to 18
Video: Mon: https://youtu.be/U9Hg7Up5gzM Wed: Fri:
Read: Chapter 16, http://www.maa.org/sites/default/files/pdf/upload_library/22/Hasse/00029890.di011943.01p0581t.pdf
Homework: Work on presentation
Week 10:
Nov 7 to 11, 2019:
No class Friday - use that to decide what you want to present and reach out to
me Read:
Chapters 15 and 16 Video:
Homework: Due November 14: \#1. In the plane we have \(\overrightarrow{v}^{(0)}
= (3,-2)\), \(\overrightarrow{v}^{(1)}= (1,5)\) and \(\overrightarrow{v}^{(2)}
= (-7,1)\). Let \(\overrightarrow{x}\) have cartesian coordinates
\((c_1,c_2)\) and barycentric coordinates \((x_0,x_1,x_2)\). Write the
cartesian coordinates in terms of the barycentric coordinates, and vice
versa. \#2: Consider the map from the unit circle (all points \((x,y): x^2 +
y^2 \le 1\) to itself given by \(f(x,y) = ((y-1/2)^2/8, (x-1/2)^2/8)\). Does
this map have any fixed points? Why or why not. If yes find or approximate
it. \#3: Consider \(x_n = \cos(n)\) (measured in radians). The Bolzano-Weierstrass
Theorem asserts it has a subsequence which converges to a point in \([-1,
1]\). Explicitly find such a subsequence. Is it easier, harder or the same
to prove the analogous statement for \(y_n = \sin(n)\)? Do so. Hint: you may
use properties of \(\pi\), such as its decimal or continued fraction
expansion. Also, if you have a homework exemption could be a good time to
use it....
Week 9: Oct 31 to Nov 4, 2020:
Read: Chapters 15 and 16
Videos: Mon: https://youtu.be/Dz2UC47Dxjw Wed: Fri: No class, watch video:
Homework: Due Monday November 7: Problem \#1: Exercise 8.8.18. Frequently in problems we desire two distinct tuples, say points \((a_1, \dots, a_k) \neq (\alpha_1, \dots, \alpha_k)\). Find a way to incorporate such a condition within the confines of integer linear programming. Problem \#2: Medical Residencies: Imagine there are \(P\) people who have just graduated from medical school and \(H\) hospitals. We are trying to match medical students with hospitals. Each student ranks the hospitals and each hospital ranks the students. Formulate this assignment problem as a linear programming problem; you may need to make some assumptions to finish the modeling. There are a lot of ways to do this; what do you want to maximize? Does a feasible solution always exist, and if so when? Does the existence of a feasible solution depend on the function you want to optimize? Problem \#3: Exercise 9.4.5. Modify the decomposition problem so that we write S as a sum of non-negative summands, but now we want to maximize the product of the squares of the summands; what is the answer? \#4: Exercise 9.4.13. Generalize the knapsack problem so that in addition to needing the total weight to be below a critical threshold, there is also a volume constraint. Set this up as a linear programming problem. \#5: Write down linear constraints for the event \(A\) or \(B\) or \(C\) must happen. \#6: Consider an \(n \times n \times n\) chesscube. Write down a linear programming problem to figure out how many hyperpawns can safely be placed given that \(n\) hyperqueens are placed in the chesscube. Note the hyperqueens can attack diagonally, horizontally, vertically, and forward-backly.
Week 8: Oct 24 to 28, 20
22Videos: Mon: Wed: Fri:
To be done by the end of classes: Choose a person in the class to write a letter of recommendation for, and ask them. Everyone must write one letter. Confirm with me when you agree to write a letter for someone. You are not writing the letter now, just finding people. If no one asks you to write a letter I will assign someone to you.
Homework: Due Monday November 7: Problem \#1: Exercise 8.8.18. Frequently in problems we desire two distinct tuples, say points \((a_1, \dots, a_k) \neq (\alpha_1, \dots, \alpha_k)\). Find a way to incorporate such a condition within the confines of integer linear programming. Problem \#2: Medical Residencies: Imagine there are \(P\) people who have just graduated from medical school and \(H\) hospitals. We are trying to match medical students with hospitals. Each student ranks the hospitals and each hospital ranks the students. Formulate this assignment problem as a linear programming problem; you may need to make some assumptions to finish the modeling. There are a lot of ways to do this; what do you want to maximize? Does a feasible solution always exist, and if so when? Does the existence of a feasible solution depend on the function you want to optimize? Problem \#3: Exercise 9.4.5. Modify the decomposition problem so that we write S as a sum of non-negative summands, but now we want to maximize the product of the squares of the summands; what is the answer? \#4: Exercise 9.4.13. Generalize the knapsack problem so that in addition to needing the total weight to be below a critical threshold, there is also a volume constraint. Set this up as a linear programming problem. \#5: Write down linear constraints for the event \(A\) or \(B\) or \(C\) must happen. \#6: Consider an \(n \times n \times n\) chesscube. Write down a linear programming problem to figure out how many hyperpawns can safely be placed given that \(n\) hyperqueens are placed in the chesscube. Note the hyperqueens can attack diagonally, horizontally, vertically, and forward-backly.
Week 7: Oct 17 to Oct 21, 2022: No class Friday (take home exam due Monday Oct 31)
Read Chapters 13 and 14
Homework: Prepare for take home exam
Week 6: Oct 10 to Oct 14, 2022:
Read Chapters 10, 11 and 12
Video: Mon: Wed: Fri: Mountain Day
Homework: Due Monday Oct 17: \#1: Formulate Sudoku as a linear programming problem (you can do either \(4\times 4\) or \(9\times 9\) Sudoku). \#2: Consider the \(3 \times 3\) constraint matrix \(A\) where the first row is 1, 2, 3, the second row is 4, 5, 6 and the third row 7, 8, 9 (thus it's the numbers 1 through \(3^2\)). Let the vector b equal \((1, 1, 1)^T\). Find all basic feasible solutions to \(A x = b\) with \(x \ge 0\). \#3: Let's revisit the chess problem from class. Consider an \(n \times n\) chess board. We want to put down \(n\) queens and maximize the number of pawns that can be safely placed on the board. Set this up as a linear programming problem. \#4: Do Exercise 6.6.30. WORTH ZERO POINTS: ONLY DO IF YOU ARE THINKING OF DOING A PROJECT! Hand in a short write-up saying who is in your group and what you will be studying / doing. Give a brief outline of what you think you'll need to learn, what data you think you'll need to gather, what you've done so far .... Describe why you feel your group has the necessary skill sets to complete the task, or if not what your plan is to remedy that. As some of the projects are still coming it, it may not be possible to do this as of now.
Videos: Mon:https://youtu.be/xNHDsEz290Q Wed: https://youtu.be/YPUHL-ayNz8 Fri: see for fun Special Second Lecture: Lookup Tables, Trigonometry, Taylor Series: https://youtu.be/EsQGxTwTNrQ
Homework problems: Due Monday, September 19: #0: Write a program to generate Pascal's triangle modulo 2. How far can you go? Can you use the symmetries to compute it quickly? You do not need to hand this in. From the textbook: #1: Exercise 1.7.4 (there are many trig tables online: see for example http://www.sosmath.com/tables/trigtable/trigtable.html), and read BUT DO NOT DO Exercise 1.7.5. #2: Exercise 1.7.7. #3: Exercise 1.7.18. #4: Exercise 1.7.34. #5: Exercise 1.7.26. #6: Exercise 1.7.36 (unassigned).
Welcome slides for first class. First Lecture: https://youtu.be/2omU3nDcboA (2022) https://youtu.be/G2ufnra8TLI (2019)
THE PROBLEMS BELOW ARE FROM THE 2019 ITERATION
Week 13: November 25
Videos: Mon: https://youtu.be/YOs8QYU3Wq8
Week 12: November 18-22
Videos: Mon: http://youtu.be/_L72XJXxGCc Wed: https://youtu.be/CBhzj3pH5_8 and first 28.5 minutes of https://youtu.be/I-CkX-dLysw Fri: https://youtu.be/qIVEjIr64jY
Work on projects or study material for exam
Read Chapters 17 and 18
Video: Mon: https://youtu.be/IOcI2c3QQbw Wed: https://youtu.be/HX6JOfIW9bw Fri: No class, work on project
Homework: Should have a draft of your letter of recommendation, hand in or email outline of project to me, keep working on project if doing one.
Week 10: Nov 4 to 8, 2019:
Read: Chapters 15 and 16
Video: Mon: https://youtu.be/q54Nz86rhCg Wed: https://youtu.be/5TmLQpts_jg Fri: https://youtu.be/dFeQkE5p8a8
Homework: Due November 11: \#1. In the plane we have \(\overrightarrow{v}^{(0)} = (3,-2)\), \(\overrightarrow{v}^{(1)}= (1,5)\) and \(\overrightarrow{v}^{(2)} = (-7,1)\). Let \(\overrightarrow{x}\) have cartesian coordinates \((c_1,c_2)\) and barycentric coordinates \((x_0,x_1,x_2)\). Write the cartesian coordinates in terms of the barycentric coordinates, and vice versa. \#2: Consider the map from the unit circle (all points \((x,y): x^2 + y^2 \le 1\) to itself given by \(f(x,y) = ((y-1/2)^2/8, (x-1/2)^2/8)\). Does this map have any fixed points? Why or why not. If yes find or approximate it. \#3: Consider \(x_n = \cos(n)\) (measured in radians). The Bolzano-Weierstrass Theorem asserts it has a subsequence which converges to a point in \([-1, 1]\). Explicitly find such a subsequence. Is it easier, harder or the same to prove the analogous statement for \(y_n = \sin(n)\)? Do so. Hint: you may use properties of \(\pi\), such as its decimal or continued fraction expansion. Also, if you have a homework exemption could be a good time to use it....
Week 9: Oct 28 to Nov 1, 2019:
Read: Chapters 15 and 16
Videos: Mon: https://youtu.be/c5GEXuUepC8 Wed: https://youtu.be/SBlQbQBnOLM Fri: https://youtu.be/m9ZjLUhnklM
Homework: Due Monday November 4: Problem \#1: Exercise 8.8.18. Frequently in problems we desire two distinct tuples, say points \((a_1, \dots, a_k) \neq (\alpha_1, \dots, \alpha_k)\). Find a way to incorporate such a condition within the confines of integer linear programming. Problem \#2: Medical Residencies: Imagine there are \(P\) people who have just graduated from medical school and \(H\) hospitals. We are trying to match medical students with hospitals. Each student ranks the hospitals and each hospital ranks the students. Formulate this assignment problem as a linear programming problem; you may need to make some assumptions to finish the modeling. There are a lot of ways to do this; what do you want to maximize? Does a feasible solution always exist, and if so when? Does the existence of a feasible solution depend on the function you want to optimize? Problem \#3: Exercise 9.4.5. Modify the decomposition problem so that we write S as a sum of non-negative summands, but now we want to maximize the product of the squares of the summands; what is the answer? \#4: Exercise 9.4.13. Generalize the knapsack problem so that in addition to needing the total weight to be below a critical threshold, there is also a volume constraint. Set this up as a linear programming problem. \#5: Write down linear constraints for the event \(A\) or \(B\) or \(C\) must happen. \#6: Consider an \(n \times n \times n\) chesscube. Write down a linear programming problem to figure out how many hyperpawns can safely be placed given that \(n\) hyperqueens are placed in the chesscube. Note the hyperqueens can attack diagonally, horizontally, vertically, and forward-backly.
Week 8: Oct 21 to 25, 2019: No class Friday due to Midterm
Keep working on your projects.
Videos: Mon: https://youtu.be/fZAQf0iH8I8 Wed: https://youtu.be/t6eBC43jx4U
To be done by the end of classes: Choose a person in the class to write a letter of recommendation for, and ask them. Everyone must write one letter. Confirm with me when you agree to write a letter for someone. You are not writing the letter now, just finding people. If no one asks you to write a letter I will assign someone to you.
Homework: Due Monday November 4: Problem \#1: Exercise 8.8.18. Frequently in problems we desire two distinct tuples, say points \((a_1, \dots, a_k) \neq (\alpha_1, \dots, \alpha_k)\). Find a way to incorporate such a condition within the confines of integer linear programming. Problem \#2: Medical Residencies: Imagine there are \(P\) people who have just graduated from medical school and \(H\) hospitals. We are trying to match medical students with hospitals. Each student ranks the hospitals and each hospital ranks the students. Formulate this assignment problem as a linear programming problem; you may need to make some assumptions to finish the modeling. There are a lot of ways to do this; what do you want to maximize? Does a feasible solution always exist, and if so when? Does the existence of a feasible solution depend on the function you want to optimize? Problem \#3: Exercise 9.4.5. Modify the decomposition problem so that we write S as a sum of non-negative summands, but now we want to maximize the product of the squares of the summands; what is the answer? \#4: Exercise 9.4.13. Generalize the knapsack problem so that in addition to needing the total weight to be below a critical threshold, there is also a volume constraint. Set this up as a linear programming problem. \#5: Write down linear constraints for the event \(A\) or \(B\) or \(C\) must happen. \#6: Consider an \(n \times n \times n\) chesscube. Write down a linear programming problem to figure out how many hyperpawns can safely be placed given that \(n\) hyperqueens are placed in the chesscube. Note the hyperqueens can attack diagonally, horizontally, vertically, and forward-backly.
Week 7: Oct 14 to Oct 18, 2019: No class Monday (Reading Period)
Read Chapters 13 and 14
Videos: Wed: Linearization I: https://youtu.be/L8oU7-0y7kQ Fri: https://youtu.be/rZZkfEHZWXU
Homework: Due Monday Oct 21: \#1: Formulate Sudoku as a linear programming problem (you can do either \(4\times 4\) or \(9\times 9\) Sudoku). \#2: Consider the \(3 \times 3\) constraint matrix \(A\) where the first row is 1, 2, 3, the second row is 4, 5, 6 and the third row 7, 8, 9 (thus it's the numbers 1 through \(3^2\)). Let the vector b equal \((1, 1, 1)^T\). Find all basic feasible solutions to \(A x = b\) with \(x \ge 0\). \#3: Let's revisit the chess problem from class. Consider an \(n \times n\) chess board. We want to put down \(n\) queens and maximize the number of pawns that can be safely placed on the board. Set this up as a linear programming problem. \#4: Do Exercise 6.6.30. \#5 WORTH ZERO POINTS: ONLY DO IF YOU ARE THINKING OF DOING A PROJECT! Hand in a short write-up saying who is in your group and what you will be studying / doing. Give a brief outline of what you think you'll need to learn, what data you think you'll need to gather, what you've done so far .... Describe why you feel your group has the necessary skill sets to complete the task, or if not what your plan is to remedy that. As some of the projects are still coming it, it may not be possible to do this as of now.
Week 6: Oct 7 to Oct 11, 2019:
Read Chapters 10, 11 and 12
Video: Mon: https://youtu.be/sYQ_xLmTGM8 Wed: NO CLASS Fri: Mountain Day
Homework: Due Monday Oct 14: Date changed to Monday Oct 21: \#1: Formulate Sudoku as a linear programming problem (you can do either \(4\times 4\) or \(9\times 9\) Sudoku). \#2: Consider the \(3 \times 3\) constraint matrix \(A\) where the first row is 1, 2, 3, the second row is 4, 5, 6 and the third row 7, 8, 9 (thus it's the numbers 1 through \(3^2\)). Let the vector b equal \((1, 1, 1)^T\). Find all basic feasible solutions to \(A x = b\) with \(x \ge 0\). \#3: Let's revisit the chess problem from class. Consider an \(n \times n\) chess board. We want to put down \(n\) queens and maximize the number of pawns that can be safely placed on the board. Set this up as a linear programming problem. \#4: Do Exercise 6.6.30. WORTH ZERO POINTS: ONLY DO IF YOU ARE THINKING OF DOING A PROJECT! Hand in a short write-up saying who is in your group and what you will be studying / doing. Give a brief outline of what you think you'll need to learn, what data you think you'll need to gather, what you've done so far .... Describe why you feel your group has the necessary skill sets to complete the task, or if not what your plan is to remedy that. As some of the projects are still coming it, it may not be possible to do this as of now.
Homework problems: Due Monday, September 16: #0: Write a program to generate Pascal's triangle modulo 2. How far can you go? Can you use the symmetries to compute it quickly? You do not need to hand this in. From the textbook: #1: Exercise 1.7.4 (there are many trig tables online: see for example http://www.sosmath.com/tables/trigtable/trigtable.html), and read BUT DO NOT DO Exercise 1.7.5. #2: Exercise 1.7.7. #3: Exercise 1.7.18. #4: Exercise 1.7.34. #5: Exercise 1.7.26. #6: Exercise 1.7.36 (unassigned).
Welcome slides for first class. First Lecture: https://youtu.be/G2ufnra8TLI
THE PROBLEMS BELOW ARE FROM THE 2016 ITERATION
Week 10: Nov 7 to 11, 2016:
Read: Chapters 15 and 16
Homework: Due November 14: \#1. In the plane we have \(\overrightarrow{v}^{(0)} = (3,-2)\), \(\overrightarrow{v}^{(1)}= (1,5)\) and \(\overrightarrow{v}^{(2)} = (-7,1)\). Let \(\overrightarrow{x}\) have cartesian coordinates \((c_1,c_2)\) and barycentric coordinates \((x_0,x_1,x_2)\). Write the cartesian coordinates in terms of the barycentric coordinates, and vice versa. \#2: Consider the map from the unit circle (all points \((x,y): x^2 + y^2 \le 1\) to itself given by \(f(x,y) = ((y-1/2)^2/8, (x-1/2)^2/8)\). Does this map have any fixed points? Why or why not. If yes find or approximate it. \#3: Consider \(x_n = \cos(n)\) (measured in radians). The Bolzano-Weierstrass Theorem asserts it has a subsequence which converges to a point in \([-1, 1]\). Explicitly find such a subsequence. Is it easier, harder or the same to prove the analogous statement for \(y_n = \sin(n)\)? Do so. Hint: you may use properties of \(\pi\), such as its decimal or continued fraction expansion. Also, if you have a homework exemption could be a good time to use it....
Week 9: Oct 31 to Nov 4, 2016:
Read: Chapters 15 and 16
Week 8: Oct 24 to 28, 2016:
Keep working on your projects.
Make sure you have chosen a chapter of the book, confirmed it with me, and start working on that.
Choose a person in the class to write a letter of recommendation for, and ask them. Everyone must write at least one letter, if you wish to do more you may. Confirm with me when you agree to write a letter for someone. You are not writing the letter now, just finding people. If no one asks you to write a letter I will assign someone to you.
Homework: Due Monday October 31: Problem \#1: Exercise 8.8.18. Frequently in problems we desire two distinct tuples, say points \((a_1, \dots, a_k) \neq (\alpha_1, \dots, \alpha_k)\). Find a way to incorporate such a condition within the confines of integer linear programming. Problem \#2: Medical Residencies: Imagine there are \(P\) people who have just graduated from medical school and \(H\) hospitals. We are trying to match medical students with hospitals. Each student ranks the hospitals and each hospital ranks the students. Formulate this assignment problem as a linear programming problem; you may need to make some assumptions to finish the modeling. There are a lot of ways to do this; what do you want to maximize? Does a feasible solution always exist, and if so when? Does the existence of a feasible solution depend on the function you want to optimize? Problem \#3: Exercise 9.4.5. Modify the decomposition problem so that we write S as a sum of non-negative summands, but now we want to maximize the product of the squares of the summands; what is the answer? \#4: Exercise 9.4.13. Generalize the knapsack problem so that in addition to needing the total weight to be below a critical threshold, there is also a volume constraint. Set this up as a linear programming problem. \#5: Write down linear constraints for the event \(A\) or \(B\) or \(C\) must happen. \#6: Consider an \(n \times n \times n\) chesscube. Write down a linear programming problem to figure out how many hyperpawns can safely be placed given that \(n\) hyperqueens are placed in the chesscube. Note the hyperqueens can attack diagonally, horizontally, vertically, and forward-backly.
Week 7: Oct 17 to Oct 24, 2016:
Read Chapters 13 and 14
Homework: Due Monday Oct 24: No written assignment. Continue making progress on your group project, and start working on your book chapter.
Week 6: Oct 10 to Oct 14, 2016:
Read Chapters 10, 11 and 12
Homework: Due Monday Oct 17: \#1: Formulate Sudoku as a linear programming problem (you can do either \(4\times 4\) or \(9\times 9\) Sudoku). \#2: Consider the \(3 \times 3\) constraint matrix \(A\) where the first row is 1, 2, 3, the second row is 4, 5, 6 and the third row 7, 8, 9 (thus it's the numbers 1 through \(3^2\)). Let the vector b equal \((1, 1, 1)^T\). Find all basic feasible solutions to \(A x = b\) with \(x \ge 0\). \#3: Let's revisit the chess problem from class. Consider an \(n \times n\) chess board. We want to put down \(n\) queens and maximize the number of pawns that can be safely placed on the board. Set this up as a linear programming problem. \#4: Do Exercise 6.6.30. \#5: Hand in a short write-up saying who is in your group and what you will be studying / doing. Give a brief outline of what you think you'll need to learn, what data you think you'll need to gather, what you've done so far .... Describe why you feel your group has the necessary skill sets to complete the task, or if not what your plan is to remedy that.