MATH527_HW5 - Serge Ballif MATH 527 Homework 5 October 5...

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Serge Ballif MATH 527 Homework 5 October 5, 2007 Problem 1. Show that if A is a nonsingular 3 by 3 matrix having nonnegative entries, then A has a positive eigenvalue. Consider the closed positive octant P of the sphere S 2 . Since A has nonnegative entries, matrix multiplication gives us a continuous map f A : P P with rule of assignment f A ( p ) = Ap k Ap k . The space P is homeomorphic to the closed unit disk in the plane B 2 = { ( x, y ) | x 2 + y 2 1 } . Hence, by the Brouwer fixed point theorem, f A must have a fixed point p (i.e. f A ( p ) = p = 1 · p ). Therefore, the number k Ap k is an eigenvalue for the eigenvector p . Problem 2. Let p : E B be a continuous and surjective map. Suppose U is an open set of B that is evenly covered by p . Show that if U is connected, then the partition p - 1 ( U ) into slices is unique. Let { A α } and { B β } be partitions of p - 1 ( U ) into slices. Then each set A α and each set B β is connected, since they are homeomorphic to U . Let a 0 A ∈ { A α } be an element such that p ( a 0 ) = u U . Then there exists a set B ∈ { B β } that contains the point a 0 . Claim 1. A = B . Proof. Suppose A 6 = B . Then without loss of generality we can assume that there is an element a A that is not in B . Hence, each a in A - B is in some set B a ∈ { B β } . Then the set A S a A - B B a together with the set A B form a separation of A . This contradicts the fact that A is connected. Thus,
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