Proof of Theorem 3 Sample Clauses

Proof of Theorem 3. 2.5.1 Our proof relies on the backward induction. It is trivial that equation (3.43)- (3.46) hold for t = T as in chapter 3.1.4. Assuming equation (3.39) and equa- tions (3.43)-(3.46) hold for t ≥ k + ∆t, we now examine the case for t = k. Let u = (uk, u∗k+∆t, ..., u∗T ), from definition (3.37), (3.38) and the Tower Property, the utility function can be written as J(k, Xk, ψk; u) = E [Xu] — γ V ar [Xx] k,Xk,ψk T = Ek,X ,ψ hE 2 k,Xk,ψk T uk Xu∗ i k k k+∆t,Xk+∆t

Proof of Theorem 3. 2.5.2 From equation (3.2), we can determine the expected wealth and the second moment at time t + ∆t based on time t as Et,X ,ψ [Xu ] = Σ (1 + µt,i∆t)ψt,i ut,i (3.69) t t t+∆t Σ = αt,2ψt,2ut,2 + rt,1Xt — αt,Li ψt,Li (3.70) Li The variance of wealth at time t + ∆t in terms of those on time t is V art,X ,ψ [Xx ] = Σ σt,iσt,jρij ∆tψt,iut,iψt,jut,j (3.71) t t t+∆t i,j Σ t,2 i i =σ2 ∆t(ψt,2ut,2)2 — 2 Σ Dt,2L ψt,L ψt,2ut,2 Li + Dt,LiLj ψt,Liψt,Lj (3.72) Xx,Xx Thus we have

Proof of Theorem 3. 2.1 Roughly speaking, condition (2) says that the harmonic measures in Ω and Ω∗ are comparable in the sense that their ratio is bounded above and below. Condition

Proof of Theorem 3. 1.1 The total number of subsets of [n] having fewer than n1/4(log n)2 elements is 2o(n1/3). Therefore we can focus on B3-sets of size n1/4(log n)2 ≤ t < n1/3. In particular, by Theorem 3.2.1(i), |Zn| ≤ 2 o(n1/3) n1/3 Σ + t=n1/4(log n)2 cn t t3

Proof of Theorem 3. 2.1 The proof of Theorem 3.2.1 uses the following strategy. Suppose that a B3-set S ⊂ [n] of cardinality s is given and one would like to extend it to a larger B3-set. We will show that if S satisfies a boundedness condition (see Definition 3.3.9 below), then the number of such extensions is fairly small. Moreover, we also prove that almost all B3-sets of cardinality s are sufficiently bounded in the sense of Definition 3.3.9. Consequently, in order to provide an upper bound for the number Zn(t) of B3-sets of size t in [n], for some t > s, we

Proof of Theorem 3. 1.5. For a given value of g, the family given by Proposition 3.2.5 and Lemma 3.4.5 comprises a positive proportion of all hyperelliptic curves with a rational Weierstrass point, since the latter is defined by finitely many congruence conditions. By Corollary 3.2.3, at least 25% of the curves in this family have rank r ≤

Proof of Theorem 3. 2 With Lemma 3.7 at our disposal, we obtain a lower bound for the size of the set {1 ≤ n < x : G(n) ∈ P2}. We wish to apply Lemma 3.4 and Lemma 3.7 to equation (3.5) to obtain a lower bound for W (A, z). We may do this for each term in (3.5) but the short sum x1−s≤p<x λ/2 log x p X .0 − log p Σ S(A , z). However, in this case, we make the estimate Σ x S(Ap, z) p log(x/p), log x yielding the bound O . sx . For notational convenience, set log z α := 1 + γ0 and γ := . log x By partial summation, we obtain . .α Σ A γ Σ∫ 1 ∫ u u − t γ .α − u − t Σ dt du W ( , z) > V (z)x f + f γ γ 1 t t t u ∫ 1 . γ .α − u Σ .α − u ΣΣ du − γ (1 − 2u) u F + uF ∫ u γ u 1 − (1 − u) F .α − u Σ du Σ − sΣ 1 γ u =: V (z)x(W − s), where we have let λ tend to 2, which is permitted by continuity. Since ΓG ƒ= 0, we have that V (z) = log−1 x by Xxxxxxx’ Theorem and we wish to show that W > 0. We observe that W decreases monotonically as α increases from 1, so we wish to find γ < 1 such that W|α=1 > 0. However, we will not immediately substitute α = 1 into the above formula. Instead, we will choose γ = α and ∫ Σ take the limit as α tends to 1 from the right. Using that if 3 ≤ s ≤ 5, and sF (s) = 2eC .1 + s−1 du log(u − 1) u sf (s) = 2eC .log(s − 1) + ∫ s−1 ∫ t−1 log(u − 1) du dt Σ . . if 4 ≤ s ≤ 6, we obtain 3 2 u t αeC W = log 5 Σ

Proof of Theorem 3. 2 With Lemma 3.7 at our disposal, we obtain a lower bound for the size of the set {1 ≤ n < x : G(n) ∈ P2}. −

Proof of Theorem 3