| A1. V is a real vector space of dimension 10. U1 ⊆ U2 are subspaces such that dim U1 = 3, dim U2 = 6. Let E be the vector space of all linear maps T: V → V which satisfy T(U1) ⊆ U1 and T(U2) ⊆ U2. Find dim E. | |
| A2. Show that for n=3, for any permutation π1 ≠ 1 of {1,2,3, ... , n}, there is a permutation π2 such that any permutation of {1,2,3, ... , n} can be obtained from π1 and π2 using only compositions. Show that this is also true for n = 5, but not for n = 4. | |
| A3. Define f: R → R by f(x) = 2x(1-x). Define fn(x) as f(f(...f(x))). Find limn→∞ ∫01 fn(x) dx. Find ∫01 fn(x) dx for each n. | |
| A4. f: R → R is twice differentiable and f(0) = 2, f '(0) = -2, f(1) = 1. Show that for some ξ ∈ (0,1) we have f(ξ) f '(ξ) + f "(ξ) = 0. | |
| A5. p(x) is a real polynomial of degree n with all zeros real. Show that (n-1) p'(x)2 ≥ n p(x)p"(x) for all x. When do we have equality? | |
| A6. f: [0,1] → R is continuous and satisfies x f(y) + y f(x) ≤ 1 for all x, y. Show that ∫01 f(x) dx ≤ π/4. Find a function which gives equality. | |
| B1. V is a real vector space and f, fi: V → R are linear for i = 1, 2, ... , k. Also f is zero at all points for which all of fi are zero. Show that f is a linear combination of fi. | |
| B2. S is the set of all real cubic polynomials f satisfying |f(±1)| ≤ 1, |f(±½)| ≤ 1. Find supS max|x|≤1 |f "(x)| and find all members of S which give equality. | |
| B3. Given 0 < c < 1, define f(x) = x/c for x ∈ [0,c], (1-x)/(1-c) for x ∈ [c,1]. Let fn(x) = f(f(...f(x))). Show that for each n there is fn has a non-zero finite number of fixed points. | |
| B4. Let An = {1, 2, .. , n}. How many functions f: An → An satisfy f(k) ≤ f(k+1) and f(k) = f(f(k+1)) for k < n? | |
| B5. S is a family of balls in Rn (n > 1) such that the intersection of any two contains at most one point. Show that the set of points belonging to at least two members of S is countable. | |
| B6. f: (0,1) → [0, ∞) is zero except at a countable set of points a1, a2, a3, ... . Let bn = f(an). Show that if ∑ bn converges, then f is differentiable at at least one point. Show that for any sequence divergent sequence bn of non-negative reals, we can find a sequence an such that the function f is nowhere differentiable. |
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