Theorem 3 Sample Clauses
Theorem 3. Let 0 < δ < 1, 1 < t1 < 1, 0 < t2 < 1, 1, 2 > 0, n1, n2 N and consider the setting of two independent, partially secret strings, say SI of length n1 and SII of length n2. Then, both strong and weak (n2, n2 ,2,n2 t2 , n2t2 s, 2—s/ ln 2, δ)
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. If the operator 𝑁 is potential on 𝐷(𝑁 ) relative to bilinear form (2.3), then the corresponding Xxxxxxxx-Xxxxxxxxxxxx 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. 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 Xxxxxx problem satisfy ||2 + x XX −0 X x=x0 ∆t||EU,q||2 ≤ c ∆t2q+3, NT −1 Σ n=n0 ∆t||EP,q||2 ≤ c ∆t2q+2. (3.33)
Theorem 3. (seq Event, ^, ()) is a trace algebra. Though simple, we note that the sequence-based trace model has been shown to be suffi- cient to characterise both untimed [40] and discrete time modelling languages [51]. →≥ A more complex model is that of piecewise continuous functions, for which we adopt and refine a model called timed traces (TT) [24]. A timed trace is a partial function of type R 0 Σ, for continuous state type Σ, which represents the system’s continuous evolution with respect to time. → In our model we also require that timed traces be piecewise continuous, to allow both continuous and discrete information. A timed trace is split into a finite sequence of continuous segments, as shown in Figure 4. Each segment accounts for a particular evolution of the state interspersed with discontinuous discrete events. This necessitates that we can describe limits and continuity, and consequently we require that Σ be a topological space, such as Rn , though it can also contain discrete topological information, like events. Continuous variables are projections such as x : Σ R. We give the formal model below. Definition 3.3 (Timed Traces). f : R≥0 → Σ ran(I ) ⊆ [0, t] | ∃ t • dom(f ) = [0, t) ∧ t > 0 ⇒ ∃ I : Roseq ∧ {0, t} ⊆ ran(I ) f (t) exists ∧ f cont-on [In, In+1) ∧ • ∀ n < #I − 1 • t→In−+1 ∀ ∈ • Roseq ¾ {x : seq R | ∀ n < #x − 1 • xn < xn+1} f cont-on [m, n) ¾ t [m, n) lim f (x) = f (t)
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. (Xxxx). 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. There exists a black-box quantum polynomial-time two-stage quantum algorithm such that for any adaptive Fiat-Shamir adversary , mak- ingqqueries to a uniformly random functionHwith appropriate domain and range, and for anyx ◦ ∈X: Pr x=x ◦ ∧v=accept: (x, v)→ ⟨S A,V⟩ (2q+ 1) 2 ≥ 1 Pr x=x ∧V H (x,π) : (x,π)→A H . Below, we apply the above general reduction to the respective standard defini- tions forsoundnessandproof of knowledge. Each property comes in the variants computationalandstatistical, for guarantees against computationally bounded or unbounded adversaries respectively, and one may consider the static or the adaptive case.