The Fractal Dimension of SAT Formulas

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental testing process. Recently, there have been some attempts to analyze the structure of these formulas in terms of complex networks, with the long-term aim of explaining the success of these SAT solving techniques, and possibly improving them. We study the fractal dimension of SAT formulas, and show that most industrial families of formulas are self-similar, with a small fractal dimension. We also show that this dimension is not affected by the addition of learnt clauses. We explore how the dimension of a formula, together with other graph properties can be used to characterize SAT instances. Finally, we give empirical evidence that these graph properties can be used in state-of-the-art portfolios.Comment: 20 pages, 11 Postscript figure

Different steady states for spin currents in noncollinear multilayers

We find there are at least two different steady states for transport across noncollinear magnetic multilayers. In the conventional one there is a discontinuity in the spin current across the interfaces which has been identified as the source of current induced magnetic reversal; in the one advocated herein the spin torque arises from the spin accumulation transverse to the magnetization of a magnetic layer. These two states have quite different attributes which should be discerned by current experiments.Comment: 8 pages, no figure. Accepted for publication in Journal of Physics: Condensed Matte

The steady state in noncollinear magnetic multilayers

There are at least two different putative steady state solutions for current across noncollinear magnetic multilayers; one has a discontinuity in the spin current at the interface the other is continuous. We compare the resistance of the two and find the solution with the continuous spin currents is lower. By using the entropic principle we can state that this solution is a better estimate of the resistance for a noncollinear magneticComment: 14 pages, 4 figures,Submitted to Physical Review

An annular lithium-drifted germanium detector for studying nuclear reaction gamma-rays

Fabrication and development of annular lithium drifted germanium detector for studying nuclear reaction gamma ray