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24 Nov 2023: Let a,b K such that the polynomials f = Xm a, g = Xm b are irreducible. ... Use asuitable Lagrange resolvent to find a Q(ζ3) such that K = Q(ζ3, 3.

14 Oct 2014: 8. Let A1, A2, A3,. be sets such that A1 A2. ... c there is an x with a < x < b such that f(x) = c.

5 Nov 2019: A = {x M : (xn) in A such that xn x} ,. Define the interior of A (in M) to be the set. ... Deduce that if A and B are disjoint closed subsets of M, then there exist disjoint opensubsets U and V of M such that A U and B V.

3 Mar 2020: Show that f is integrable. What is. 10f(x)dx? 4. Give an example of a continuous function f : [0,) [0,) such that0f(x)dx. ... Show. that for every > 0 and every closed interval I [a,b] of positive length, there isa closed interval J I such that J has

24 Nov 2006: Show that, for such a map,. dimH f (M ) 61α. ... Prove that the limit set for such a group is totallydisconnected.

10 Nov 2006: Does it follow that every open set must be. closed? Give an example of such a metric space. ... Show that thereexists a norm on C(I) such that the map sending f to y0.

12 Nov 2020: 5. Let A Rn be such that every continuous function f : A R is bounded. ... V such that A Uand B V.

28 Nov 2006: ii) Suppose that f : R2 R is such that D1f = f/x exists and is bounded near (a,b), and. ... is a function l : U M such that exp(l(A)) = A for all A U.

12 Feb 2021: 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

23 Feb 2009: Show that for any b C, there exists a sequence of pointszn 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.