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9 Oct 2018: 6. Show that R(s,t) 6(st2s1. )for all s, t > 2. ... Suppose that (i) |A| = 3 for each A A; and (ii) |AB| = 1for all distinct A, B A.

4 Feb 2018: Show that the sumof the residues of g at all its poles equals zero. ... Let p(z) = z5 z. Find all z such that |z| = 1 and Im p(z) = 0.

19 Jan 2018: 4. Show that T B(H) is normal if and only if ‖Tx‖ = ‖Tx‖ for all x H. ... 8. Let S and T be compact normal operators on a complex Hilbert space withdim ES(λ) = dim ET (λ) for all λ.

20 Feb 2018: φ(x) =. [A(x) B(x)C(x) D(x). ]with A(y) = Ir, B(y) = C(y) = D(y) = 0. ... Hence show that A(x) is invertible for all x insome open neighbourhood U of y.

4 Feb 2018: every z C, |f(z) b| > ε.(iii) f = u iv and |u(z)| > |v(z)| for all z C. ... 9. (i) Let w C, and let γ, δ : [0, 1] C be closed curves such that for all t [0, 1],|γ(t) δ(t)| < |γ(t) w|.

25 Sep 2018: b) The level sets of a real-valued function on a set C are defined, for all t R, as{x C : f(x) t}. ... ϕ(b) = inf{f(x) : g(x) b, x X}.Assuming ϕ(b) is finite for all b, show that the function ϕ is convex.

26 Nov 2018: 8. Let. n=1xn be convergent. If xn > 0 for all n, show that. ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A,B T?

18 Jan 2018: b) Find all 1-dimensional representations of G.(c) Let ψ : Fp C be a non-trivial 1-dimensional representation of the cyclic group. ... d) Prove that the collection of representations constructed in (b) and (c) gives a com-plete list of all irreducible

22 Oct 2018: b) Show that S1 S1 is homeomorphic to a graph.(c) Draw all the covering spaces of S1 of degree 2 or 3. ... d) Draw all the covering spaces of S1 S1 of degree 2 or 3.2.

24 Oct 2018: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... f(A) = trA for all A Matn,n(F).(b) Now let V be a non-zero finite dimensional real vector space.

13 Nov 2018: Supposethat there is a constant K such that ‖F(t,x) F(t,y)‖ K‖xy‖ for all t [0, 1] and all x,y B. ... assumethat there exist constants K and α (0, 1) such that ‖F(t,x)F(t,y)‖ K‖xy‖α for all t [0, 1] and allx,y B.

25 Oct 2018: Find all such solutions. 2. Let a, b, c, d N. ... 6. Solve (i.e., find all solutions to) these congruences:-(i) 77x 11 (mod 40), (ii) 12y 30 (mod 54),.

5 Feb 2018: Show that the sumof the residues of g at all its poles equals zero. ... Let p(z) = z5 z. Find all z such that |z| = 1 and Im p(z) = 0.

20 Nov 2018: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all nn real matrices. ... sn forms a basis for Pn.(iii) For all 1 k n, sk spans the orthogonal complement of Pk1 in Pk.(iv) sk is an

27 Feb 2018: b) Find the fundamental unit in Q(. 10). Determine all the integer solutions ofthe equations x2 10y2 = m for m = 1, 6 and 7. ... 6. Find all integer solutions of the equations y2 = x3 13 and y2 = x5 10.

8 Oct 2018: 2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b R}. ... One says that f is Riemann integrable if all Riemann sums, for varying P, converge to the samelimit as τ(P) 0

5 Feb 2018: every z C, |f(z) b| > ε.(iii) f = u iv and |u(z)| > |v(z)| for all z C. ... 9. (i) Let w C, and let γ, δ : [0, 1] C be closed curves such that for all t [0, 1],|γ(t) δ(t)| < |γ(t) w|.

19 Jan 2018: 8. Suppose X and Y are metric spaces. A mapf : X Y is an isometric embedding ifdX(x1,x2) = dY (f(x1),f(x2)) for all x1,x2 X. ... 10. Let f : Rn Rn be a C1 map. Suppose that there is some constant µ < 1 such that‖Df|x I‖op < µ for all x Rn.

22 Mar 2018: Show that the collection of all such subsets of X form a base for a topology on X. ... Let. Y := {f : A {0, 1} : f(α) = 0 for all but countably many α A} X.

10 Oct 2018: a (b c) = (a b) (a c). for all a, b, c L). ... ts=1 nsas where all the as are in Ai and the ns are integers.