A NEW UPPER BOUND FOR RANDOM (2 + p)-SAT BY FLIPPING TWO VARIABLES

The random (2 + p)-SAT model has been proposed [18] to study the possible relation between the “order” of phase transitions and computational complexity. It was also claimed that there exists pc > 0, such that for p < pc the random (2 + p)-SAT instance behaves like 2-SAT. Later, Achlioptas et al. [3] obtained the first rigorous results that 0.4 ≤ pc ≤ 0.695, the methods they use are the first moment method and the simple Unit-Clause algorithm. In this paper, we try to optimize the local maximality condition of the truth assignments when implementing the first moment method. We prove that the phase transition point of clauses-to-variables ratio r (dependent on p) can be improved. Moreover, we show that the upper bound of pc can be reduced to 0.6846. This fact implies that, for a constant λ < 1, a random (2 + p)-SAT formula with λn 2-clauses and 2.17n 3-clauses is almost surely unsatisfiable.

On Local Maximality for the Coefficient a6

Recently a number of authors have studied the application of Grunsky’s coefficient inequalities to the study of the Bieberbach conjecture for the class of normalized regular univalent functions f(z) in the unit circle |z|< 1Charzynski and Schiffer [2] applied this result to give an elementary proof of the inequality | a4 | Ȧ 4. One of the present authors [8] proved that if a2 is real non-negative then A natural first step in the study of the inequality for a coefficient is to prove local maximality for a2 near to 2.