Search

13 Nov 2011: Showthat there is a vector x such that x, α(x), α2(x),. , ... 9. (i) Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linearfunctional θ V such that θ(x) 6=

17 Nov 2011: NB: maybexn < 0. ). 14. Let S PN be such that ifA, B S then A B or B A. ... Can S be uncountable?Is there an uncountable familyT PN such thatA B is finite for all distinctA, B T?

13 Nov 2011: Explain geometrically why this is thecase. 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that f/x exists and is

20 Feb 2011: Using partial fractions evaluate the integral. Hence deduce that f (a) = f (b). ... 5. Let f be an entire function such that for some a C and r > 0, f takes no values in the discD(a, r).

24 Feb 2011: 9. Let R be an integral domain. The highest common factor (hcf) of non-zero elements a and b in R is anelement d in R such that d divides both ... ii) Show that can be identified with the set Fq {} in such a way that GL2(Fq) acts on as thegroup of

18 Oct 2011: On the other hand, show that if fn does not converge uniformly to f, then we can find aconvergent sequence xm x in [a,b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges

5 Nov 2011: Recall that the existence of such a mini-mizer was proved in lecture.). ... Thus, for any fixed δ (0, 1/2), In,δ/In 0 asn. (b) Choose numbers a1, b1 such that 0 < a1 < a < b < b1 < 1, and set, for x R

12 Oct 2011: show that for every β L we can find a, b, c, d K such that β = a bαc dα. ... 1.16. Let K, L be subfields of a field M such that M/K is finite.

13 Oct 2011: 1. Calculate d = (a, b) and find integers x and y such that d = ax by when. ... 13. Let A be the group (Z/65520Z). Determine the least positive integer n such thatgn = 1 for all g in A.

10 Jan 2011: In this and the following questions, you arenot being asked to find such an isomorphism explicitly.). ... Here O(n) denotesthe subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form.(b) Explicitly construct such a