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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

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

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.

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

21 Nov 2011: Show that there is nodecodable coding c such that all codewords have length 2 or less. ... However,the power required is such that the ship’s force shields must be switched off for a minuteduring each test.

24 Nov 2011: Such a basis {σ(y)} is said to be a normal basis for F/K.). ... Use (ii) to show that there exists b K such thatdet(σiσjQ1(b)) ̸= 0.

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.

16 Aug 2011: Then there existsan injection φ : I(A) I(C) such that |φ(B)| = |B| for all B I(A). ... such that |φ(B)| = |B| for allB I(A).

29 Jul 2011: We find a subgroup H of order prime to l such that Γ = Γ0H. ... any such factor would act trivially on V =. Wi, contradicting thatG0(Fl)/Z0 acts faithfully.

21 Dec 2011: A GL5(K) and µ K such that [A,µ Diag(Im,πI5m)B]Φis an integral model of the same or smaller level. ... Let a < b < cbe chosen such that {i,j,a,b,c} = {1,. ,

16 Oct 2011: Definition 5.2. A Heisenberg group is a Galois invariant subgroup H SLn(K)such that. ... Then thereare projective representations χV : E[5] PGL(V ) and χW : E[5] PGL(W)such that.

29 Jul 2011: qni1ai} for i = 1,. ,r are disjoint. Choose a matrixΦ(σ,a) such that. ... such that ẽ1,. , ẽm lie in s̃ and ẽm1,. , ẽn lie in ψ̃.

9 May 2011: variables a,b,c such that v g = gv, respectively v g = gT v, for all g G. ... variables a,b,c,d such that vg = χ(g)rgv, respectivelyv g = χ(g)rgT v, for all g G.

12 Jul 2011: Say ρ is a direct sum ρU ρW. If no such exists, we sayρ is indecomposable. ... And have n1. nr such r-tuples, eachgiving a 1-dimensional representation. Examples.