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27 Jan 2020: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists 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 Jan 2020: iii) Let w = a1b2a2b1a F. Using the method of Stallings folding, orotherwise, produce an explicit homomorphism f : F Q to a finite groupQ such that f(b) / f(〈w〉). ... b) For a commutative ring R with unity, define. T(R) =. {(a b0 d. ) SL2(R). }. (i)

5 Jun 2020: Ân such that Ln(xn1 ) log Qn(B(xn1,D)) for allxn1 An. ... Conversely, for any sequence of “admissible” distributions {Qn}, there is a sequenceof codes {Cn,Ln} operating at distortion level D, such that, for (almost) any realization ofthe source,

9 Nov 2020: Now suppose b G(L) is such that νb X(T) is dominant, hence defined over Zp. ... Itsuffices to prove that also descends to R(a,b). The coordinates ti induce a Frobenius-equivariant map RG R(a,b), such that the compositeR(a,b) RG

31 Jul 2020: where a,b,c,d,e Z. Provided that f is squarefree, such an equation defines a genus one curveover Q. ... By the proofof Lemma 12, the effect of such a transformation on (n,d,e) pZ3p is measure preserving.

4 Mar 2020: i) There exists an open set U B such that µ{0,1}N (UB) < ε2. ... so a B, a contradiction.Therefore, there exists m > 1 such that g 6 HAm.

23 Nov 2020: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.

5 Jun 2020: Yπ(n)π(n) , (6). whereM 1 is the largest integer such thatM 2 divides N ,and theYi are now binary. ... Actually Chebyshev came pretty close to proving the PNT.For example, using Theorem 1 in a slightly refined form, hewas able to find explicit

3 Nov 2020: 2. Given (fi)iI an arbitrary collection of continuous functions [0, 1] R such thatsupiI |fi(x)| < at each x [0, 1], show that there is an interval [a,b] [0, ... 1] witha < b such that supx[a,b] supiI |fi(x)| <. 3. Let X be a closed subspace of 1. Assume

11 Nov 2020: We now let µ X(T) and let b G(L) such that [b] B(G,µ). ... Reversing the above process we obtain Z′ Σtop(Xw′(b′))) such that StabU′(J′) = StabZ′(J′).