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Let A and B be bounded linear operators on a Hilbert space satisfying A ≥ B ≥ 0. The well-known Furuta inequality is given as follows: Let r ≥ 0 and p > 0; then A r 2 A min{1,p} A r 2 ≥ (A r 2 B p A r 2) min{1,p}+r p+r. In order to give a self-contained proof of it, Furuta (1989) proved that if 1 ≥ r ≥ 0, p > p 0 > 0 and 2p 0 + r ≥ p > p 0 , then (A r 2 B p… (More)

Consider a random k-conjunctive normal form F k (n, rn) with n variables and rn clauses. We prove that if the probability that the formula F k (n, rn) is satisfiable tends to 0 as n→∞, then r ⩾ 2.83, 8.09, 18.91, 40.81, and 84.87, for k = 3, 4, 5, 6, and 7, respectively. SAT 问题是第一个被证明的 NP 完全问题, 是理论计算机科学研究的核心问题之一。 自从上世纪九十年代初发现 NP 完全问题存在相变现象以来, 随机 k-SAT… (More)

Consider a random instance I of k-SAT with n variables and m clauses. Suppose that θ, c > 0 are any fixed real numbers. Let k = k(n) ≥ 1 2 + θ log 2 n. We prove that lim n→∞ P r(I is satifiable) = 1 m ≤ 1 − c √ n 2 k n ln 2 0 m ≥ 1 + c √ n 2 k n ln 2.

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