The aim of this short note is mainly pedagogical. It summarizes some knowledge about Boolean satisfiability (SAT) and the P=NP? problem in an elementary mathematical language. A convenient scheme to visualize and manipulate CNF formulae is introduced. Also some results like the formulae for the number of unsatisfied clauses and the number of solutions might be unknown. Introduction. Specificall...

Berg and Ulfberg and Amano and Maruoka have used CNF-DNF-approximators to prove exponential lower bounds for the monotone network complexity of the clique function and of Andreev's function. We show that these approximators can be used to prove the same lower bound for their non-monotone network complexity. This implies P not equal NP.

We develop polynomial time Monte-Carlo algorithms which produce good approximate solutions to enumeration problems for which it is known that the computation of the exact solution is very hard. We start by developing a Monte-Carlo approximation algorithm for the DNF counting problem, which is the problem of counting the number of satisfying truth assignments to a formula in disjunctive normal f...

In this work we study Randers spacetimes of Berwald type and analyze Pfeifer Wohlfarth's vacuum field equation Finsler gravity for class. We show that in case the is equivalent to vanishing Ricci tensor, analogously Einstein gravity. This implies considered Rutz's coincide scenario. also construct all exact solutions Berwald-Randers gravity, which turn out be composed a CCNV (covariantly consta...

A $p$-group $G$ is called a $mathcal{CAC}$-$p$-group if $C_G(H)/H$ is ‎cyclic for every non-cyclic abelian subgroup $H$ in $G$ with $Hnleq‎ ‎Z(G)$‎. ‎In this paper‎, ‎we give a complete classification of‎ ‎finite $mathcal{CAC}$-$p$-groups‎.