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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 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.

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.

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

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

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

15 Nov 2011: n}, let the non-negative numbers A(n)k and the“nodes” x(n)k [a, b] be given such that for each polynomial P , the error. ... such that for every f C([0, 1]), there exists a unique series.

21 Feb 2011: Show that if there exists a vector e such that e,n(s),b(s) are linearly dependent forall s I, then α is a plane curve. ... Show that the curve joining A and Bwith length and such that together with AB bounds the largest possible area is an arc 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

6 May 2011: c) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... Prove the following:. (a) G is homogeneous: given any x, y G there is a homeomorphism φ : G G such thatφ(x) = y.