Theorem 5. 2. Assume that δ > 0 or δ = 0, γ − r > σ2 and 0 ≤ k ≤ h( q ). Then the value of stock loan with automatic termination clause, cap and margin is given by ( 5.6) and (5.7). Moreover, if L > b then the stopping time τb ∧ τL is the optimal exercise time. If L ≤ b then τL is the optimal exercise time.
Theorem 5. .2.1. If Mm is a von Mangoldt plane, then
Theorem 5. 3.1. If Mm is a von Mangoldt plane with a point where Gm < 0 and such that lim inf m(r) > 0, then
Theorem 5. 12. The ring C∞ is faithfully flat over its subring O. X Corollary 5.13. On a complex analytic manifold X , the sheaf A0 of rings of O smooth functions is faithfully flat over the sheaf X of rings of holomorphic func- tions. The authors are grateful to Xxxxxxx Xxxxxxxx and Xxxxx Xxxxxxxxxx for valu- able comments on the subject of this paper.
Theorem 5. Using a coin toss arbiter, the contract has weak game-theoretic security for γ = 1 and λ = x.
Theorem 5. For any USKE scheme uske = (UEnc, UDec) and hash function H, let the encryption scheme uske′ = (UEnc′, UDec′) be defined as follows: 1. It samples ks, kd randomly for its initial key k0′ = (ks, kd). 2. Then, the first time it uses UEnc′ or UDec′, it computes k0 dprf(ks, kd), and uses k0 as the first input to UEnc, UDec, with the corresponding message or ciphertext. 3. Then, for the next use of UEnc′ (resp. UDec′), it uses the key k1 (resp. k1′ ) output by XXxx (resp. UDec) in (2) above as input to UEnc (resp. UDec) again; and proceeds like this for all subsequent computations. l−12 · If uske is (t, εcpa∗)-CPA* secure and dprf is modelled as a random oracle H, then uske′ is (t, εGSD)-adaptive GSD secure, where εGSD = 2N 2 εcpa∗ + mN , with N the number of nodes, m the number of oracle queries to H, l the length of dPRF keys. The theorem can be proved almost identically to that of [3, Theorem 3]. We provide a sketch below for exposition. Proof (Sketch). We prove GSD security by a sequence of hybrids inerpolating between the real game GSD0 where the challenge query is answered with real key ky(= (ky, ky)) and the random game GSD1 where it is answered with an 0 s d K independent uniformly random key in 2 (where we assume the dPRF key space is the same as the USKE key space). – Define G0 := GSD0, the real GSD game. – Let k′ ∈ K2 and v be the challenge node. For 1 ≤ i ≤ indeg(v) we define the hybrid game Gi as follows: The game is similar to Gi−1 except that the i-th query of the form (encrypt, u, v) is answered by UEnc(ku, k′(= (ks′ , k′ )).
Theorem 5. .1.1. For all k ≥ 2, the number of k-regular partitions of n ∈ N is given by 2π pk(n) = k ΣD|k D<k 2 ∞ Σ m gcd(k,m)=D s<ΣJ(k,D) pj(s)A(m, k − 1, n, s, D)L(m, k − 1, n, s, D).
Theorem 5. There exists an oblivious polarization method for parameters α and β if and only if there exists a one-way secret-key agreement protocol secure on (α, β). Moreover, there exists an efficient oblivious polarization method if and only if there exists a protocol with efficient encoding (i.e., Xxxxx is efficient). We prove Theorem 5 in both directions separately, and start by showing that a polarization method implies the existence of a one-way secret-key agreement protocol:
Theorem 5. For even N and even m it is true that: H Q (N, Nρ , m, , m ,) − ρ = 1 − Q (N, N (1 − ρ ), m, , m ,) − h(N, N (1 − ρ ), m, m ) − ρ H
