# hw1 - ECE 5620 Spring 2011 Homework#1 Issued Jan 26 Due Feb...

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ECE 5620 Spring 2011 Homework #1: Issued Jan. 26, Due Feb. 4 at 5 p.m. Complete all four problems. 1. Prove that (1 - x ) y e - xy , for all x < 1 and y > 0 . 2. Convexity (a) Determine whether each of the following functions is convex, concave, neither, or both: x 2 , x + 5 , log( x ) , log(1 + x ) , x log( x ) . (b) Suppose f and g are concave functions, but not necessarily differen- tiable. For each of the following functions, either prove that the func- tion is concave or supply an example that shows that it is not. i. f + g ii. f · g iii. max( f,g ) iv. min( f,g ) (c) Show that if f is convex, but not necessarily differentiable, and g is afﬁne, then f ( g ( x )) is convex. (d) Show that if a function is both convex and concave, then it is afﬁne. Do not assume a priori that the function is differentiable. 3. The Probabilistic Method Let ~v 1 ,...,~v n be n given unit vectors in R n . Show that there exist n con- stants a 1 ,...,a n each with absolute value 1 such that ± ± ± ± ± ± ± ± ± ± n X i =1 a

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## This note was uploaded on 10/03/2011 for the course ECE 5620 at Cornell.

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hw1 - ECE 5620 Spring 2011 Homework#1 Issued Jan 26 Due Feb...

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