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19 Feb 2024: 8. Find two totally ordered sets such that neither is isomorphic to a subset of the other.Can you find three such sets? ... Is there a countable ordinal α such thatα is the αth fixed point?

11 Jan 2024: 6. 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

29 Jan 2024: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n N withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

8 Feb 2024: i) f(z)/z 0 as |z|.(ii) There exists b C and > 0 such that for every z C, |f(z) b| >. (iii) f = u iv and |u(z)| > ... Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n.

8 Feb 2024: Regarding T as a map from C into C, show that there existunique complex numbers A,B such that for every z C, T(z) = Az Bz. ... Can such a map be the restriction tothe strip of a Möbius map?

15 Jan 2024: 6. 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

24 Jan 2024: 3. Let f,g : R R be such that f(x) as x a and g(y) k as y. ... Construct a function f : R R that takes every value on every interval – in otherwords for every a < b and every c there is an x with a < x < b such

14 Feb 2024: 8. Find two totally ordered sets such that neither is isomorphic to a subset of the other.Can you find three such sets? ... Is there a countable ordinal α such thatα is the αth fixed point?

7 Mar 2024: b) Find a condition on 𝑠 such that 𝛿𝑥 𝐻𝑠(R𝑛). c) Show that 𝐻𝑡 (R𝑛) is continuously embedded in 𝐻𝑠(R𝑛) for 𝑠 < 𝑡. ... Δ2𝑢 𝑢 = 𝑓. b) Show further that there exists 𝐶 > 0 such that ‖𝑢‖𝐻 𝑠4

24 Jan 2024: Does 10f exist? 3. Give an example of a continuous function f : [0,) [0,), such that0f. ... Must f vanishidentically?[We say g vanishes near a and b if there exists > 0 such that g(x) = 0 for x 6(a ,b ).]. 1. 2 ANALYSIS I EXAMPLES 4.