| A1. εn is a sequence of positive reals which tends to zero. Find limn→∞ (1/n) ∑1n ln(k/n + εn). | |
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A2. ∑ an converges. Do the following sums necessarily converge:
a1 + a2 + a4 + a3 + a8 + a7 + a6 + a5 + a16 + a15 + ... + a9 + a32 + ... a1 + a2 + a3 + a4 + a5 + a7 + a6 + a8 + a9 + a11 + a13 + a15 + a10 + a12 + a14 + a16 + a17 + a19 + ... (the pattern is that the terms from a2n+1 to a2n+1 are permuted to put the odd before the even). |
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| A3. A, B are real n x n matrices such that A2 + B2 = AB and BA - AB is invertible. Show that n must be a multiple of 3. | |
| A4. Show α ∈ (1,2) has a unique representation as an infinite product (1 + 1/n1)(1 + 1/n2) ... where ni+1 ≥ ni2. Show that α is rational iff its infinite product has ni+1 = ni2 for all sufficiently large i. | |
| A5. For x ∈ Rn, we denote its coordinates by (x1, x2, ... , xn). Let S be the set of points in Rn satisfying ∑ xi = 0. Let S' be the set of points in S whose coordinates are all integral. For x ∈ Rn, define |x|p = (∑ |xi|p)1/p, and |x|∞ = max |xi|. Show that if x ∈ S such that max xi - min xi ≤ 1, show that |x|p ≤ |x + y|p for all p ≥ 1 (including ∞) and for all y ∈ S'. Show that for 0 < p < 1, we can find x ∈ S with max xi - min xi ≤ 1 and y ∈ S' such that |x|p > |x + y|p. | |
| A6. F is a collection of finite sets of positive integers such that for A, B ∈ F, we have A ∩ B ≠ ∅. Is there a finite set Y such that for any A, B ∈ F we have A ∩ B ∩ Y ≠ ∅. Suppose all members of F have the same size? | |
| B1. f: R → R has continuous third derivative and satisfies f(x) ≥ 0 for all x, f(0) = f '(0) = 0, and f "(0) > 0. Let g(x) be the derivative of (√f(x))/f '(x) for x ≠ 0, and g(0) = 0. Show that g is bounded in some neighborhood of 0. Is this still true if f has a continuous second derivative, but not necessarily a third derivative? | |
B2. M is an invertible 2n x 2n matrix. We write:
M = A B and M-1 = E F
C D G H
where A, B, C, D, E, F, G, H are n x n arrays. Show that det M det H = det A.
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| B3. Show that ∑1∞ (-1)n-1 (sin ln n)/nα converges iff α > 0. | |
| B4. Let Mn be the set of all real n x n matrices. Let f: Mn → R be linear. Show that there is a unique C ∈ Mn such that f(A) = tr(AC) for all A ∈ Mn. If f also satisfies f(AB) = f(BA) for all A, B ∈ Mn, show that f(A) = λ tr(A) for some real λ. | |
| B5. X is any set and f is a bijection on X. Show that there are mappings g, g' : X → X such that f = gg', gg = 1, g'g' = 1. | |
| B6. Let f: [0,1] → R be continuous. We say f crosses the axis at x if f(x) = 0 but in any neighborhood of x there are points y, z with f(y) < 0 < f(z). Give an example of a function which crosses the axis infinitely often. Can f cross the axis uncountably often? |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
1 Dec 2003