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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

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.

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.

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,. ,

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.