{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2-hw116

# 2-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

This preview shows pages 1–2. Sign up to view the full content.

MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #2 Last modified: October 3, 2007 Due Thursday, Oct. 4 at class 1. (a) Suppose that you modify the NSA budget system from the lecture outline by defining 0 S = θ so that the axiom x, 0 x = θ is satisfied. Does this make it into a vector space, or does it create a problem with one of the other axioms? (b) A conjecture: In a vector space X , the element z that satisfies z + z = z is unique. Either prove this from the axioms, or construct a counterexample. 2. Prove Luenberger’s axiom 1 from the other axioms. Note that axiom 3 is “one-sided.” Without axiom 1, this is still OK, but you must prove that θ is a left identity if you want to use it as one. 3. Luenberger, section 2.16, problem 4. The statement that you are to prove could have been used as the definition of “convex hull,” but Luenberger used a different definition on page 18. 4. Find the convex hull of the Koch snowflake, a famous fractal curve, and prove that your answer is correct. You construct the snowflake as follows. Start with an equilateral triangle of side A . In the middle third of each side, construct an equilateral triangle of side A 3 , pointing outwards. The snowflake now consists of 12 segments of length A 3 . In the mid- dle third of each segment, construct an equilateral triangle of side A 9 , pointing outwards.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}