Satisfiability threshold for random regular NAE-SAT

We consider the random regular $k$-NAE-SAT problem with $n$ variables each appearing in exactly $d$ clauses. For all $k$ exceeding an absolute constant $k_0$, we establish explicitly the satisfiability threshold $d_*=d_*(k)$. We prove that for $d<d_*$ the problem is satisfiable with high probability while for $d>d_*$ the problem is unsatisfiable with high probability. If the threshold $d_*$ lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krzakala et al. (2007). Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs

The effect of import competition on firm productivity and innovation: does the distance to technology frontier matter?

How does foreign competition affect growth and innovation in China? Using our unique measures of proximity of Chinese firms and industries to the world technology frontier, we find that despite vast sectoral heterogeneity, Chinese manufacturing industries have undergone rapid technological upgrading over the period of 2000–06. The distance to the world production frontier of firms and industries plays an important role in shaping the nexus between the competition pressure from foreign imports and domestic firms' growth and innovation behaviour. Our results support the theoretical predictions of Aghion et al. (2005, The Quarterly Journal of Economics, pp. 701–728) that import competition stimulates the domestic firms' productivity growth and R&#38;D expenditure if firms and their industries are close to the world frontier, but discourages such incentives for laggard firms and industries. The two forces highlighted by the model operate for imports under the ordinary-trade regime, for collective and private firms, and for imports originated from high-income countries. Our findings are robust after controlling the influence of foreign investment, the reverse causality of regressors and the short-term business cycle fluctuations