mat280-hw3

# mat280-hw3 - MAT 280 — UC Davis Winter 2011 Longest...

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Unformatted text preview: MAT 280 — UC Davis, Winter 2011 Longest increasing subsequences and combinatorial probability Homework Set 3 Homework due: Wednesday 3/16/11 1. Prove the summation identity n X k =0 (- 1) k k + α n k = n ! α ( α + 1) ... ( α + n ) ( n ≥ 0) , which was used in the proof of the Bessel function identity (2.40) in the lecture notes. Hint: First prove, and then use, the following “binomial inversion principle”: if ( a n ) n ≥ and ( b n ) n ≥ are sequences such that a n = ∑ n k =0 (- 1) k ( n k ) b k for all n , then the symmetric equation b n = ∑ n k =0 (- 1) k ( n k ) a k also holds for all n . 2. Prove Hadamard’s inequality from elementary linear algebra: if A is an n × n matrix whose column vectors are v 1 ,...,v n ∈ R n , then | det( A ) | ≤ n Y j =1 || v j || 2 . What is the geometric meaning of this inequality? 3. Let Ω be a countable set. Let M : Ω × Ω → R be a kernel which is symmetric (i.e., M ( x,y ) = M ( y,x ) for all x,y ∈ Ω) and positive-definite (i.e., if we thinkΩ) and positive-definite (i....
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mat280-hw3 - MAT 280 — UC Davis Winter 2011 Longest...

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