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

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Unformatted text preview: 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 = so that the axiom x, x = 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 Luenbergers 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 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.pointing outwards....
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2-hw116 - MATHEMATICS 116, FALL 2007 CONVEXITY AND...

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