The Viner—Wong Envelope Theorem
Eugene Silberberg
The envelope theorem. now the fundamental tool in modern duality analysis.
had its beginnings in Jacob Viner’s classic 1931 article on short» and long
14.102 Problem Set 3
Due Tuesday, October 18, in class
1. Lecture Notes Exercise 208: Find
numbers.
Rb
a
log(t)dt , where 0 < a < b are real
2. (Sundaram 4.4, page 110) Find and classify all critical
14.102 Problem Set 1 Solutions
1. Lecture Notes Exercise 12: Show that Q, the set of real rational numbers,
does not have the least upper-bound property.
Solution: To show this, we must show that ther
14.102 Problem Set 1
Due Thursday, September 22, in class
1. Lecture Notes Exercise 12: Show that Q, the set of real rational numbers,
does not have the least upper-bound property.
2. Is the set of re
14.102 Problem Set 2 Solutions
1. Lecture Notes Exercise 105: Given an m n matrix A, show that S(B)
S(A) and N (A0 ) N (B 0 ) whenever B = AX for some matrix X. What
is the geometric interpretation?
14.102 Problem Set 3
Due Tuesday, October 18, in class
1. Lecture Notes Exercise 208: Find
numbers.
Rb
a
log(t)dt , where 0 < a < b are real
Solution: This can be solved by integration by parts.
t, F
14.102 Problem Set 2
Due Thursday, October 6, in class
1. Lecture Notes Exercise 105: Given an m n matrix A, show that S(B)
S(A) and N (A0 ) N (B 0 ) whenever B = AX for some matrix X. What
is the ge
14.102 Midterm Exam Solutions
October 20, 2005
Real Analysis (15 points)
1. (2 points) Give an example of a bounded sequence with exactly three limit points.
Solution: One example is the sequence cfw_
Solutions to Review Questions for 14.102 Midterm
10/14/05
Note: For true/false questions you should either prove the statement
or provide a counterexample.
1
Real Analysis
1. Give an example of a rela
Review Questions for 14.102 Midterm
10/14/05
Note: For true/false questions you should either prove the statement
or provide a counterexample.
1
Real Analysis
1. Give an example of a relation R that i
14.102 Midterm Exam
October 20, 2005
Instructions: This is a closed book exam; you have 90 minutes to complete it.
Please answer all questions.
Suggestion: Read through all questions briefly before yo
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/22/2005
These notes are primarily based on those written by George Marios Angeletos
for the Harvard Math Camp in 1999 and 2000, and updated by St
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/13/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/8/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003/
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/27/2005
These notes are primarily based on those written by George Marios Angeletos
for the Harvard Math Camp in 1999 and 2000, and updated by St
14.102, Math for Economists
Fall 2005
Lecture Notes, 10/18/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 200
14.102, Math for Economists
Fall 2005
Lecture Notes, 10/4/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003
14.102, Math for Economists
Fall 2005
Lecture Notes, 10/13/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 200
14.102, Math for Economists
Fall 2005
Lecture Notes, 10/6/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/20/2005
These notes are primarily based on those written by George Marios Angeletos
for the Harvard Math Camp in 1999 and 2000, and updated by St
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/29/2005
These notes are primarily based on those written by George Marios Angeletos
for the Harvard Math Camp in 1999 and 2000, and updated by St