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18 Jan 2021: Proof. Let f : J0(N) A be the quotient map. There exists a mapg : A J0(N) such that fg = [n] for some n > 0. ... and for a given f, it’s possible that some Vf,p are of the form (trivial) (cyclotomic) while others aren’t.

3 Mar 2021: Prove that for anyn 2 there is some pair (A,B) and a constellation on (A,B) made of precisely n circles. ... 2). (c) Deduce that there is some δ > 0 such that, in any hyperbolic triangle, the union of the δ-neighbourhoods of twoof the sides completely

16 Mar 2021: Z. Rn. eix eiy2. ||2sd 6 Cn, |x y|2. b) Show that if s = n2 k for some k 2 Z>0, 0 < < 1, then. ... d) () A simple crystal consists of identical atoms placed at the lattice points of some.

15 Oct 2021: some function which maps an interval (a,b) with a < b to thereal line R. ... Thenf(x) = C for some C R. Proof. Pick c,d with a < c < d < b.

15 Oct 2021: supx(a,b). f ′i(x) Mfor some M independent of i. Show that F is uniformly equicontinuous. ... supx[a,b]. |fi(x)| K. for some K independent of i. Then F has a subsequence which converges uniformly.

15 Oct 2021: 1.1 Convolutions 3. ii) Since zj 0, there exists some ρ > 0 such that zj Bρ(0) for all j. ... 1.1 Convolutions 5. Lemma 4. Suppose f L1loc.(Rn) and g Ck0 (Rn) for some k 0.

15 Oct 2021: interval t (s,s), and that moreover the solution depends continuously on s.Later we’ll state more precisely some sufficient conditions on a,b,c such that we can ... and that for any y > 1supR|h′| a shock has formed for some x.

15 Oct 2021: With some reasonable furtherconditions (see the Appendix) we can in fact pick out τ uniquely. ... Lemma 1.12. Suppose f L1loc.(Rn) and g Ck0 (Rn) for some k 0.

15 Oct 2021: Supposethat the xi satisfy ||xi|| < r for all i and some r > 0. ... Similarly, (dj)j=0 is monotone decreasingand bounded below so dj d for some d.

15 Oct 2021: v) We need some topology on D′() such that the above operations are continuous. ... By Lemma 2.9 there exists some compactK and N N, C > 0 such that:.