CAAM 560
Homework Assignments
Fall 2012
Fall 2012 Reading Assignments
Read in the order listed in the assignment. Critique each of these three sections of our book, i.e., write some relatively brief comments about each chapter . You may use as a guide to what to write the following four questions:
1) Is the unit of value to the readers of the book, why or why not?
2) What did you like, if anything, about the unit?
3) What did you dislike, if anything, about the unit?
4) What changes, if any, would you like to see in the unit?
If you have a strong desire and good reason to not follow the suggested guide, then that is fine, as long as what you do makes sense.
As you read for the first time read for general understanding not to be able to reproduce the proofs. You will want to re-read later on, but that is standard in mathematics. Some of the proofs in either the chapters or the appendices are challenging. I have included them so you can gain an appreciation for the result and for mathematical proof techniques. I would like you to have a reading assignment a week, but we always fall short of that. Ideally, I would like you to read each chapter before we discuss it in class. The appendices make the book self-contained. If you have never seen the material in the appendices before then you will have to spend more time on the material. But it can, and has been, navigated by others with minimal background. I feel that you will learn more in this class than in most classes you have taken. I have failed if you do not leave feeling mathematics is elegant, beautiful, and useful. Much of the material is essentially new and is not readily available in other texts.
Reading Assignment 1.
Chapter 1, Chapter 2, and Appendix E.Due Monday, August 27, 2012
Reading Assignment 2.
Appendix A, Appendix B, and Chapter 3.
Reading Assignment 3.
Appendix C, Chapter 4, and Chapter 5.
Reading Assignment 4.
Read Chapters 6 and 7.
Reading Assignment 5.
Due Monday, November 5, 2012 (This assignment will be graded like one of the problem sets)
Read in this order Appendix D, Chapter 8, and Chapter 9. Give usual reading assignment comments . However, include the following specific comments
-Read carefully Section 8.8 An Abstract Analogy. Comment on this Section.
Do you think that it adds something to the book or does not really add anything?
-Read the two proofs of Theorem 9.4.1. Recall that the first proof is borrowed from Wilansky.
The second proof is mine. Contrast the two proofs. Is my proof the perfect proof of this theorem, why or why not?
-Carefully read and then comment on Davood’s proof of Theorem 9.5.1, the extended Farkas Theorem. Do you think that his proof is particularly clever? Recall that this theorem had been searched for by me for several years. It allows us to complete our multiplier theory.
Reading Assignment 6.
Due Monday, November 12, 2012 (This assignment will be graded like one of the problem sets)
Read Chapters 10, 11, 12, and 13. Chapters 10-13 can be found on the CAAM 560 Assignment page right below Fall 2011 Homework Assignments under the titles Chapter 10 Lecture , … , Chapter 13 Lecture.
Reading Assignment 7.
Due Monday, November 30, 2012 (The last day of class)
(This assignment will be graded like one of the problem sets)
- Read and comment on the three papers
1. William Karush’s master ‘s thesis.
2. A Characterization of Inner Product Spaces (by RAT).
3. The Isoperimetric Problem Revisited (by RAT).(I will use your comments in revision)
Fall 2012 Problem Set Assignments
Problem Set Assignment 1.
Chapter 1: Problems 3, 4, 6, 7, and 11. Due Wednesday, September 5, 2012
Problem Set Assignment 2.
Chapter 3: Problems 3,4,5,6, and 7. Due Wednesday September 12, 2012. Due Wednesday, September 12, 2012
Problem Set Assignment 3.
Chapter 1: Problems 18, 26, 31, and 32.
Problem Set Assignment 4.
Chapter 4: Problems 13, and 16. Work Problem 13 in two ways. The first is from the definition of convexity and the second is using differential characterizations.
Chapter 5: Problem 1
Chapter 6: Problem 4
Dragster Problem: A dragster runs a quarter mile track in a straight line attempting to accelerate hard and end up with as large a top speed at the end of the ¼ mile as possible. This top speed is measured from placing two clocks at the end of the strip (track). One clock is always placed at the end of the ¼ mile track and the other 66 feet inside the track (before the end) or 66 feet outside (after the finish line). Of course a simple difference formula is used to approximate the speed. It is generally felt that a dragster accelerates for the full ¼ mile or to the second clock when the second clock is past the end of the quarter mile. So you may assume this. Prove that if the second clock is placed inside the quarter mile, then the actual speed at the end of the quarter mile is greater than the estimated speed while if the second clock is placed outside the quarter mile the actual speed is less than the estimated speed. In the old days the second clock was outside, today it is required to be inside, why do you think that this is so?
Fall 2011 Homework Assignments
The problem sets are important. I want you to work together to discuss the solutions. However, you must write up the solutions to the problem set on your own. Moreover, I want them well-written and complete. You should be able to return to the write-up after some time and completely understand the write-up. In the beginning, we will schedule help sessions. Then, there will be a time when I feel that you have shown that you should be on your own. I am willing to help you as much as you need.
Lecture Notes
Lecture 0, 8/22/11
Lecture 1, 8/24/11
Lecture 2, 8/26/11
Lecture 3, 9/9/11
Lecture 4, 9/19/11
Lecture 5, 10/3/11
Lecture 6, The Remarkable Isoperimetic Problem and the Euler-Lagrange Equation Revisited, 10/12/11
The Euler-Lagrange Equation as a multiplier rule, 11/9/11
Chapter 9 Lecture
Chapter 10 Lecture
Chapter 11 Lecture
Chapter 12 Lecture
Chapter 13 Lecture
11/28/11 Lecture
Fall 2011 Reading Assignments
Reading Assingment 1
1. Appendix E, Chapter 1 and Chapter 3 Due Monday, August 29, 2011
Critique each of these three chapter of our book, i.e., write some relatively brief comments about each chapter that at least touches on the following four questions:
1) Is the unit of value to the readers of the book, why or why not?
2) What did you like, if anything, about the unit?
3) What did you dislike, if anything, about the unit?
4) What changes, if any, would you like to see in the unit?
Read for general understanding not to be able to reproduce the proofs. You will want to re-read later on, but that is standard in mathematics. Some of the proofs in either the chapters or the appendices are challenging. I have included them so you can gain an appreciation for the result and for mathematical proof techniques. I would like you to have a reading assignment a week, but we always fall short of that. Ideally, I would like you to read each chapter before we discuss it in class. The appendices make the book self-contained. If you have never seen the material in the appendices before then you will have to spend more time on the material. But it can, and has been, navigated by others with minimal background. I feel that you will learn more in this class than in most classes you have taken. I have failed if you do not leave feeling mathematics is elegant, beautiful, and useful. Much of the material is essentially new and is not readily available in other texts.
Reading Assingment 2
Reading Assignment 2: Chapter 4 and Chapter 5. Due Friday, September 2, 2011.
Reading Assingment 3
Reading Assignment 3: Appendix A,B,C and Chapter 2. Due Monday, September 19, 2011.
Reading Assingment 4
Reading Assignment 4: Appendix D and Chapter 6, 7, and 8. Please read Appendix D before you read Chapter 8 Due Monday, October 3, 2011.
Reading Assingment 5
Reading Assignment 5: Chapter 9 and 10 and make usual comments. Due Monday, November 7, 2011
Read carefully Section 8.8 An Abstract Analogy.
Comment specifically on this section. Does it belong in the book? Why or why not?
Read the proof of Theorem 9.4.1 in our book. Recall that I said that I borrowed it from Wilansky. Now contrast the Wilansky proof with the proof that I came up with, CAAM 560 proof. It is listed in this assignment section. Tell me why the quotient space was introduced? Also, is my proof the perfect proof, why do you think so, or do not think so?
Carefully read and then comment on Davood's proof of Theorem 9.5.1 , the Extended Farkas Theorem. Be aware that this result had been searched for (by me) for several years. It allows us to complete our multiplier theory.
Fall 2011 Problem Sets
Problem Set 1
CAAM 560 Problem Set 1: Due Wednesday, September 7, 2011
Three Ancient Greek Optimization Problems from Geometry.
1.Solve the isoperimetric problem for the rectangle, i.e.,
Of all rectangles with a fixed perimeter find the one of largest area..
2.Of all triangles with a given base and with a given altitude, hence of the same area, find the one with the smallest perimeter.
3. Consider a triangle ABC. Let D be a point on side AB, E a point on side BC, and F a point on side AC, Where should the point E be placed on side BC so that the area of parallelogram ADEF is maximal?
Chapter 1 Problems 6 and 7.
Chapter 3 Problems 3,4,5,7, and 8.
Problem Set 2
CAAM 560 Problem Set 2: Due Monday, September 26, 2011
Chapter 1: Problems 10, 18, 26, 31
Chapter 5: Problems 1
Dragster Problem: A dragster runs a quarter mile track in a straight line attempting to accelerate hard and end up with as large a top speed at the end of the ¼ mile as possible. This top speed is measured from placing two clocks at the end of the strip (track). One clock is always placed at the end of the ¼ mile track and the other 66 feet inside the track (before the end) or 66 feet outside (after the finish line). Of course a simple difference formula is used to approximate the speed. It is generally felt that a dragster accelerates for the full ¼ mile or to the second clock when the second clock is past the end of the quarter mile. So you may assume this. Prove that if the second clock is placed inside the quarter mile, then the actual speed at the end of the quarter mile is greater than the estimated speed while if the second clock is placed outside the quarter mile the actual speed is less than the estimated speed. In the old days the second clock was outside, today it is required to be inside, why do you think that this is so?
Problem Set 3
CAAM 560 Problem Set 3: Due Wednesday, October 12, 2011
Chapter 1: Problem 16, and 32
Chapter 4: Problem 16
Chapter 6: Problem 3 and 4
Chapter 7: Problem 4
Fall 2010 Reading Assignments
1. Chapters 1, 2 and 3
Critique each of these three chapter of our book, i.e., write some relatively brief comments about each chapter that at least touches on the following four questions:
1) Is the chapter of value to the readers of thebook, why or why not?
2) What did tou like, if anything, about the chapter?
3) What did you dislike, if anything, about the chapter?
4) What changes, if any, would you like to see in the chapter?
2. Appendices A, B, C, and E
This reading assignment will not officially be assigned until Problem Set 1 is due.
3. Read Appendix D, Chapter 4, Chapter 5, and Chapter 7.
Theorem and proof (10/15)
4. Read Chapters 7,8, and 9. Write the usual type of report on these three chapters.
Additional Comments
Read carefully the proofs of Proposition 8.3.1 and 8.3.2.
Comment specifically on these proofs.
Read carefully Section 8.8 An Abstract Analogy.
Comment specifically on this section. Does it belong in the book?
Read Theorem 0.1 and its proof listed in Reading Assignment 3. I say that the proof ( it is my proof) is perfect. Do you agree or disagree , why or why not?
Read the proof of Theorem 9.4.1 in our book. Recall that I said that I borrowed it from Wilansky. Now contrast the Wilansky proof with the proof that I came up with and is listed on our CAAM 560 webpage in the section entitled “Assignments”. Again is my proof the perfect proof, why do you think so, or do not think ?
Carefully read and then comment on Davood’s proof of Theorem 9.5.1 , the Extended Farkas Theorem.
Be aware that this result had been searched for (by me) for many years. It allows us to complete our multiplier theory.
2010 Problem Sets
1. Chapter 1 - #s 4,6,7 and Chapter 3 - #s 3,4,5,7
2. Chapter 1: # 4 Give a clean perfect write-up. There will not be any partial credit. # 11, # 26, and # 31.
Chapter 5: # 6
Dragster Problem: A dragster runs a quarter mile track in a straight line attempting to accelerate hard and end up with as large a top speed at the end of the ¼ mile as possible. This top speed is measured from placing two clocks at the end of the strip (track). One clock is always placed at the end of the ¼ mile track and the other 66 feet inside the track (before the end) or 66 feet outside (after the finish line). Of course a simple difference formula is used to approximate the speed. It is generally felt that a dragster accelerates for the full ¼ mile or to the second clock when the second clock is past the end of the quarter mile. So you may assume this. Prove that if the second clock is placed inside the quarter mile, then the actual speed at the end of the quarter mile is greater than the estimated speed while if the second clock is placed outside the quarter mile the actual speed is less than the estimated speed. In the old days the second clock was outside, today it is required to be inside, why do you think that this is so?
