We have determined the melting slopes as a function of pressure for MgO up to a pressure of 135 GPa, and for LiF up to a pressure of 100 GPa using the Lindemann law. Values of melting temperature have also been calculated from the melting slopes using Euler’s finite difference calculus method. It is found that the melting slope decreases continuously with the increase in pressure giving a nonlinear pressure dependence of the melting temperature. Values of bulk modulus and the Grüneisen parameter appearing in the Lindemann law of melting have been determined using the Stacey reciprocal K-primed equation of state and the Shanker reciprocal gamma relationship. The results for melting temperatures of MgO and LiF at different pressures are compared with the available experimental data. Values of melting temperatures at different pressures determined from the Al’tshuler relationship for the volume dependence of the Grüneisen parameter have also been included in the comparison presented.
We initiate a complexity theoretic study of the language based graph reachability problem (L-Reach) : Fix a language L. Given a graph whose edges are labelled with alphabet symbols of the language L and two special vertices s and t, test if there is path P from s to t in the graph such that the concatenation of the symbols seen from s to t in the path P forms a string in the language L. We study variants of this problem with different graph classes and different language classes and obtain complexity theoretic characterizations for all of them. Our main results are the following:• Restricting the language using formal language theory we show that the complexity of L-Reach increases with the power of the formal language class. We show that there is a regular language for which the L-Reach is NL-complete even for undirected graphs. In the case of linear languages, the complexity of L-Reach does not go beyond the complexity of L itself. Further, there is a deterministic context-free language L for which L-DagReach is LogCFL-complete.• We use L-Reach as a lens to study structural complexity. In this direction we show that there is a language A in TC 0 for which A-DagReach is NPcomplete. Using this we show that P vs NP question is equivalent to P vs DagReach −1 (P) 1 question. This leads to the intriguing possibility that by proving DagReach −1 (P) is contained in some subclass of P, we can prove an upward translation of separation of complexity classes. Note that we do not know a way to upward translate the separation of complexity classes.
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