| A1. A is an n x n real symmetric, invertible matrix with positive elements. Show that its inverse has at most n2 - 2n zero elements. B = (bij) is the n x n matrix with bij = 2 if i,j are both even, or i is even and i<j, or i is odd and i>j, and 1 otherwise. Show that B is symmetric and invertible and find the number of zeros in its inverse. | |
| A2. R is the reals. f: (a,b) → R has a continuous derivative. It tends to ∞ as x → a and -∞ as x → b. Also f '(x) + f(x)2 ≥ -1 for all x. Show that b - a ≥ π and give an example where there is equality. | |
| A3. S is a set of 2n-1 distinct irrationals. Show that we can find n elements x1, x2, ... , xn of S such that ∑aixi is irrational for all non-negative rational numbers ai such that ∑ai > 0. | |
| A4. R is the reals. F, G: Rn → Rn are linear maps such that for some k ≠ 0, FG - GF = kF (in other words, F(G(x)) - G(F(x)) = k F(x) for all x ∈ Rn). Let Fm denote the map with Fm(x) = F(F(...F(x))), iterated m times. Show that FmG - GFm = kmFm for all m and that Fm = 0 for some m. | |
| A5. f: [0,b] → R and g: R → R are continuous. g has period b. Show that limn→∞∫0b f(x) g(nx) dx = (1/b) ∫0b f(x) dx ∫0b g(x) dx. Find limn→∞∫0π sin x/(1 + 3cos2nx) dx. | |
| A6. f: [0,N] → R has continuous second derivative, and |f '(x)| < 1, f "(x) > 0 for all x. 0 ≤ m0 < m1 < ... < mk ≤ N are integers such that f(mi) are all integers. Put ai = mi - mi-1 and bi = f(mi) - f(mi-1). Prove that -1 < b1/a1 < b2/a2 < ... < bk/ak < 1. Show that for A > 1, there are at most N/A indices i such that ai > A. Show that there are at most 3N2/3 lattice points on the curve y = f(x). | |
| B1. f: [a,b] → R is continuous, f(a) = 0, and for some λ > 0 we have |f '(x)| ≤ λ |f(x)| for all x. Is it true that f is identically zero. | |
| B2. f: R2 → R is f(x,y) = (x2 - y2)e-x2-y2. Show that f attains its minimum and maximum. Find all points where f has both partial derivatives zero and determine at which of them f has a global or local minimum or maximum. | |
| B3. f: R → R has n+1 derivatives. Show that if a < b and ln(f(b) + f '(b) + ... + f(n)(b)) - ln(f(a) + f '(a) + ... + f(n)(a)) = b - a, then we can find c ∈ (a,b) such that f(n+1)(c) = f(c). | |
| B4. A is an n x n diagonal matrix, such that the distinct values on the diagonal are c1, c2, ... , ck, where ci appears di times and ∑ di = n. Show that the vector space of n x n matrices B which commute with A has dimension ∑ di2. | |
| B5. x1, x2, ... , xk are vectors in Rm with sum zero. Show that for some permutation xπ(i) of xi we have |∑xπ(i)| ≤ √(∑|xi|2), where |x| is the usual norm. | |
| B6. Find limn→∞ (1/n)ln2n ∑2n-2 1/(ln k ln(n-k) ) |
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