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# hw1 - 5 Suppose a ∈ R k b ∈ R k and x ∈ R k Find all...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 1 D UE : Thurs. Sept. 16, 2010. Late papers will be accepted until 1:00 PM Friday . 1. Let x 0 = 1 and define x k : = 3 x k - 1 + 4, k = 1 , 2 ,... . Show that x k < 4 and that the x k are increasing. 2. Show that 1 + 1 2! + 1 3! + ··· + 1 k ! < 3. 3. Let A : = ( a i j ) be a n × n matrix of complex numbers with | a i j | ≤ M and let a ( k ) i j be the elements of A k , k = 1 , 2 ,... . Find an estimate for | a ( k ) i j | in terms of k , n , and M . 4. a) Let z , w , v C and define d ( z , w )) : = | z - w | 1 + | z - w | . Show that d ( z , v ) d ( z , w )+ d ( w , v ) [triangle inequality] . b) Let S be an arbitrary set with p , q , r S . Say there is a function g : S × S R that satisfies the triangle inequality g ( p , r ) g ( p , q )+ g ( q , r ) . Define d ( p , q )) : = g ( p , q ) 1 + g ( p , q ) . Show that this function d ( p , q ) also satisfies the triangle inequality.
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Unformatted text preview: 5. Suppose a ∈ R k , b ∈ R k , and x ∈ R k . Find all c ∈ R k and r > 0 (depending on a and b) such that | x-a | = 2 | x-b | is satisﬁed if and only if | x-c | = r . As an alternate, you may prefer the following generalization . For real λ > 0, λ 6 = 1, consider the points x ∈ R k that satisfy | x-a | = λ | x-b | . Show that these points lie on a sphere. Part of this is to ﬁnd the center and radius of this sphere in terms of a , b and λ . What if λ = 1? [Last revised: September 18, 2010] 1...
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