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10 Jan 2018: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

23 Feb 2018: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

9 Oct 2018: Show that f isbounded. 11. Show that there is an infinite set S of positive integers such that the sum of any two distinct elements. ... b) a collection of 100 horizontal lines and 100 vertical lines such that the 10,000 points of intersection are.

19 Jan 2018: Show that I Thas a square root, i.e., there exists S B(X) such that S2 = I T. ... 14+. Let X be a separable infinite-dimensional Banach space. Suppose thatthere is a constant C such that every finite-dimensional subspace E of X satisfiesd(E,n2 ) C, where

4 Feb 2018: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

29 Oct 2018: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f

8 Oct 2018: 1. Let E be a Lebesgue measurable subset of the real line with positive Lebesgue measure m(E).Show that for every > 0 there exists an open interval (a,b) such ... A Dirac mass at x X is themeasure δx on B such that δx(A) = 1 or 0 according as x A or x /

12 Oct 2018: x). On the other hand, show that if fn does not convergeuniformly to f then we can find a convergent sequence xn x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)

9 Oct 2018: B. Prove that (A4B)4C = A4(B4C). 7. Let A1, A2, A3,. be sets such that A1 A2. ... c there is an x with a < x < b such that f(x) = c.

26 Nov 2018: 13. Let S PN be such that if A,B S then A B or B A. ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A,B T?

23 Feb 2018: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

10 Oct 2018: Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

5 Oct 2018: rationals. and let β = α4. Findrationals a,b,c such that β = a bα cα2. ... Show that there is no intermediateextension K M L.(ii) Let α be such that |K(α) : K| is odd.

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. ... Letx1,x2 Rn. By the Picard–Lindelöf theorem, we know that there is (0, 1] and differentiable functionsf1,f2 : [0,] B such that dfjdt = F(t,fj

31 Jan 2018: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

5 Feb 2018: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

25 Oct 2018: Find all such solutions. 2. Let a, b, c, d N. ... 3. Find integers x, y and z such that 56x 63y 72z = 1.

23 Nov 2018: Does it follow that thereexists c (a,b) such that f(b) f(a) = f′(c)(ba)? ... continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I).

25 Sep 2018: b) We recall that a norm N in Rn is a function from Rn to R0 such that. - ... Show that forany given function f : [n] R, the subset of distributions such that Ef(X) [a,b],where X p, is a convex set.

12 Oct 2018: α L,show that there exists an integer r 0 such that αpr F. ... Prove thatthere do not exist elements λ1,. ,λm of L (not all zero) such that.

5 Nov 2018: Use ψ to build, foreach interval I = [a,b] and > 0 a C-function ψI, on R such that. ... Fn. Finally let F = σ(C). Show that there exists a unique probability measure µ on (,F) such thatµ(B) = µ1.

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 ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

18 Jan 2018: to V4 V3; in this and the followingquestions, you are not 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

20 Nov 2018: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 13. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

10 Oct 2018: Ch. > A. g. f. Dk. > B. with g epic, there exists a (necessarily unique) t: D A such that ft = k and tg = h. ... in [C, Set]), there exists i I such that the restriction of αto Gi is still epimorphic.

22 Mar 2018: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y.

19 Jan 2018: 49. Let T B(X,Y ) be a non-compact operator. Prove that there exist a boundedsequence (xn) in X and an ε > 0 such that ‖Txm Txn‖ > ε for all ... invertible operator T : X X such that Txn = yn for all n N.

26 Nov 2018: 2) Convert the following CFG to CNF:. S aSbb | T, T bTaa | S |. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF:. ... b) Show that the language L := {anbncn | n 1} is not a CFL.

23 Oct 2018: b) By replacing A in the construction of B with a suitably-chosen set, construct a setC N such that neither C nor NC are r.e. ... Use this to show there exists some m N such that Wm has exactly m elements.

17 Aug 2018: Proof. As the action of Γ on X is cocompact, let x0 X and r > 0 such that X = Γ B (x0,r).Define S = {γ Γ, γB (x0, 3r) B ... for x X, there exists r > 0 such that B (x,r) is CAT(0)).

17 Aug 2018: If. 5. Geometry of Coxeter groups Alexis Marchand. γ Γ is such that γK K 6= , then γB(1,r) B(1,r) 6= , so d(1,γ) < 2r. ... for x X, there exists r > 0 such that B (x,r) is CAT(0)).

4 Apr 2018: Definition. Given (a,b,c,d) C4 such that ad bc 6= 0 we can define a functionf : C C as follows:. ... Moregenerally, b nZ is the set of integers x such that x b is a multiple of n.19.

16 Nov 2018: 14 SIMON WADSLEY. Definition. Given (a,b,c,d) C4 such that ad bc 6= 0 we can define a functionf : C C as follows:. ... Moregenerally, b nZ is the set of integers x such that x b is a multiple of n.24.

28 Nov 2018: identity). (G3) For each a G there exists a b G such that a b = e. ... ii) If a G then e a = a. (iii) If a,b,b′ G are such that a b = e and a b′ = e, then b = b′.

4 Apr 2018: sense. Throughout the paper, we employ the well-known fact that the right adjoint of a strongmonoidal functor between monoidal categories has a unique monoidal structure such that theunit and counit ... Definition 2.4. (1) A bimonoid in a braided V is an

16 Jul 2018: such. that G is residually finite) has solvable word problem. The conjugacy and membership. ... the explicit definition above suffices for most concrete categories such as Sets, Groups,.