| A1. Calculate det A, where A = (aij) is the (n+1) x (n+1) matrix with aij = a + |i-j|d. | |
| A2. Find ∫-ππ sin nx dx/( (2x + 1) sin x). | |
| A3. V is a finite dimensional vector space. If A: V → V is linear and V2 = 1, show that there is a basis of V consisting of eigenvectors of A. Find the maximum number of linear maps A: V → V which satisfy A2 = 1 and such that each pair commute. | |
| A4. The sequence a1, a2, a3, ... is defined by a1 = 1, an = (a1an-1 + a2an-2 + ... + an-1a1)/n. Show that 2/3 ≤ lim sup |an|1/n < 1/√2. | |
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A5. (1) a, b are real numbers such that b ≤ 0 and 1 + ax + bx2 ≥ 0 for all x ∈ [0,1]. Show that limn→∞ n ∫01 (1 + ax + bx2)n dx = -1/a if a < 0 and ∞ if a ≥ 0.
(2) f: [0,1] → [0,∞) has a continuous second derivative and f "(x) ≤ 0 for all x ∈ [0,1]. L = limn→∞ n ∫01 f(x)n dx exists and 0 < L < ∞. Show that f ' has constant sign and minx∈[0,1] |f '(x)| = 1/L. |
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| A6. For E ⊆ R2, define C(E) = inf ∑ diam(Ei), where inf is taken over all finite families of subsets Ei of R2 such that E ⊆ ∪ Ei. f: E → L is a contraction onto L if f is surjective and |f(x) - f(y)| ≤ |x - y| for all x, y ∈ E. Define K(E) = sup |L|, where the sup is taken over all closed line segments L such that there is a contraction of E onto L. Show that if L is a closed line segment then C(L) = |L|. Show that C(E) ≥ K(E). There is a compact E for which C(E) > K(E). | |
| B1. f: [0,1] → [0,1] is continuous. Define xn+1 = f(xn). Show that the sequence xn converges iff lim(xn+1 - xn) = 0. | |
| B2. θ is a positive real and k a positive integer. Show that if cosh kθ and cosh(k+1)θ are rational, then so is cosh θ. | |
B3. Let G be the subgroup of GL2(R) (the group of all 2 x 2 invertible matrices) generated by:
A = 2 0 and B = 1 1
0 1 0 1
H consists of those matrices in G which have diagonal elements both 1. Show that H is an abelian subgroup of G, but is not finitely generated.
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| B4. B is a bounded closed convex subset of R2 which is symmetric wrt the origin. Its boundary is the curve Γ. B has the property that the ellipse of maximal area contained in B is the disk D with center the origin, radius 1 and boundary the unit circle C. Show that any arc of C with length ≥ π/2 must contain a point of Γ. | |
| B5. Show that limx→∞ ∑1∞ nx/(n2 + x)2 = 1/2. Show that there is a constant c > 0 such that |∑1∞ nx/(n2 + x)2 - 1/2| ≤ c/x for all x ≥ 1. | |
| B6. a1, a2, a3, ... is such that all an > 0 and ∑1∞ an < ∞. Show that ∑1∞ (a1a2...an)1/n < e ∑1∞ an. Show that for any ε > 0 we can find such a sequence with ∑1∞ (a1a2...an)1/n > (e - ε) ∑1∞ an |
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