Theorem 3 Clause Samples

Theorem 3. Let δ′, 1, and 2 > 0 be constants. Then for all sufficiently large E : {0, 1}n × {0, 1}d → {0, 1}r, exists with d ≤ Δ1n and r ≥ (δ′ — Δ2)n, such that for all random variables T ∈R T with T ⊆ {0, 1}n and with H∞(T ) > δ′n H(E(T, V )|V ) ≥ r — 2—n1/2−o(1) .

Theorem 3. Suppose ▇▇▇▇▇ and ▇▇▇ perform the standard SIDH key-agreement protocol as described §3.

Theorem 3. (▇▇▇▇). Suppose E/Q is an elliptic curve with a Q-rational torsion point P of odd prime order A, and suppose P is not contained in the kernel of reduction modulo A. Suppose SE = ∅. Suppose that D is a negative square-free integer coprime to ANE and satisfies

Theorem 3. The ( n + 1)-Provable Broadcast algorithm in Algorithms 2 and 3 satisfies Integrity, Validity, Provability, and Termination. Moreover, the protocol has linear communication complexity with an O(1)-sized proof.

Theorem 3. 2.1 The following bounds apply to |Zn(·)|: |Zn(t)| ≤ (i) There is an absolute constant c > 0 such that for any t ≥ n1/4(log n)2, cn t3 (ii) For any 0 < ε < 1/5 there exists Cε > 0 such that for t = Cε(n log n)1/5, we have |Zn(t)| ≤ nt(3/5+ε).

Theorem 3. Algorithm OVERABUNDANTWORDS solves problem ALLOVER- ABUNDANTWORDSCOMPUTATION in time and space O(n), and this is time-optimal. OverabundantWords(x,ρ) 1 T(x) BuildSuffixTree(x) 2 for each node v T(x) do 3 D(v) word-depth of v

Theorem 3. If there exists a positive constant c such that NT −1 ∆t||pn+1||2 ≤ c and ∆t is sufficiently small, then there exist two positive constants c1, c2 dependent on the space discretization and independent of ∆t such that the HOYq schemes for q = 0, 1, 2 applied to the ▇▇▇▇▇▇ problem satisfy ||2 + ▇ ▇▇ −▇ ▇ ▇=▇▇ ∆t||EU,q||2 ≤ c ∆t2q+3, NT −1 Σ n=n0 ∆t||EP,q||2 ≤ c ∆t2q+2. (3.33)

Theorem 3. If the operator 𝑁 is potential on 𝐷(𝑁 ) relative to bilinear form (2.3), then the corresponding ▇▇▇▇▇▇▇▇-▇▇▇▇▇▇▇▇▇▇▇▇ action is given by 𝐹𝑁 [𝑢] = ∫︁𝑡1 [︂ 1 (𝑀4′′(𝑡) − 𝑀2(𝑡)) (𝑢′(𝑡))2 + 𝑀4(𝑡)(𝑢′′(𝑡))2 + 𝐵𝑀 (𝑡, 𝑢(𝑡))]︂

Theorem 3. Suppose a dividend will be paid during the life of an option. Let D denotes its present value. Then we have for European option S − D − Ee−r(T −t) ≤ c ≤ S (3.5) − S + D + Ee−r(T −t) ≤ p ≤ Ee−r(T −t). (3.6) c + Ee−r(T −t) = p + S − D. (3.7) S − D − E ≤ C − P ≤ S − Ee−r(T −t) (3.8) S − E ≤ C − P ≤ S − Ee−r(T −t) (3.9) if the put is exercised before the dividend being paid. Proof. 1. Proof of c S Ee−r(T −t) D. We consider two portfolios: I = c + D + Ee−r(T −t), J = S. Then at time T , I(T ) = max{ST − E, 0} + D + E = max{ST , E} + Der(T −t) J(T ) = ST + Der(T −t). Hence I(T ) ≥ J(T ). This yields I(t) ≥ J(t) for all t ≤ T . This proves c ≥ S − D − Ee−r(T −t). In other word, c is reduced by an amount D. 32 CHAPTER 3. BLACK-SCHOLES ANALYSIS