Theorem 4. Suppose that an [n, k, d] linear code is employed in Scheme 4.
Theorem 4. Let ts, ti such that 2ti + ts < n, ti ≤ ts < n . There exists an T2 = O(1) round deterministic protocol GCS∗ab that satisfies the following in the context of the graded consensus primitive (See Definition 2) in a ∆-synchronous network if the round distance between protocol initiation times of any two honest parties is at most 1. – Authenticated ts-validity. – Authenticated ts-termination. – Sabotaged ti-security. Furthermore, given ΠAW has at most O(λn2) bit complexity and constant round complexity, so too does GC∗Sab.
Theorem 4. Every conventional linear code of length n with minimum Ham- ming distance d can be converted to a code of length 2n with minimum 0-1 distance d by replacing every bit in the original codewords by a pair of bits, namely by replacing 0 by 01 and 1 by 10. We omit the proof since it is obvious. The code obtained from the linear code with the method of Theorem 4.3.5 is called a 0-1 code, and its codewords are called 0-1 codewords. Note that the number of 1’s is equal to that of 0’s in any 0-1 codeword. Xxxxx and Xxx can agree on a linear code. For any source state, a corresponding codeword is obtained according to the encoding rule of the linear code. Subsequently, the codeword can be changed into a 0-1 codeword. Then both Xxxxx and Xxx are able to construct the authenticator for the source state by taking together some bits from her or his initial string, where the positions of the bits are determined by the indices of 1-entries of the 0-1 codeword. More precisely, the authentication scheme can be described as follows. Let Xxxxx be the sender and Bob the receiver.
Theorem 4. Assume that δ > 0 and γ − r +δ ≥ 0 or δ = 0 and γ − r > σ2 . Define f by (4.11), b by Proposition 4.1 and Proposition 4.2. Then the initial value of stock loan with automatic termination clause is f (S 0).
Theorem 4. Let u0 ∈ Cc∞(R) and suppose that v is the unique smooth solution to (4.2.1). Then, there exists C1, C2 > 0 and µ1, µ2 > 0 such that, for every y ∈ R and τ > 0, . .
Theorem 4. Let F be the function field of a p-adic curve with p ƒ= 2 and D a division algebra over F with an involution of the first kind. Let L/F be a quadratic extension.
Theorem 4. If a ceded loss function f∗ ∈ Fπ satisfies f∗(x) and If∗(x) are both non-decreasing functions in x ≥ 0, and f∗ + uR = f∗(uI + uR + P0), (6) then f∗ is an optimal ceded loss function in Fπ, which maximizes the joint survival probability and satisfies (5). Proof: For any f ∈ Fπ, it holds that Jf = Pr{If (X) ≤ uI + Pf, f (X) ≤ uR + Pf}
Theorem 4. If n = 3t + δ, then there is an almost-surely terminating ABA protocol with expected running time O( t ).
Theorem 4. Let Nn+1 be a Riemannian manifold with sectional curvature bounded from below and let Mn be a complete, immersed, δ-stable, CMC-hypersurface with n = 3, 4 and δ < 7 , 19
Theorem 4. Let (Mn, g) be a two-sided Riemannian manifold. Suppose that q a smooth function on M. Then the following are equivalent:
