An unification of general theory of relativity with Dirac's large number hypothesis

Taking a hint from Dirac's large number hypothesis, we note the existence of cosmologically combined conservation laws that work to cosmologically long time. We thus modify Einstein's theory of general relativity with fixed gravitation constant $G$ to a theory for varying $G$, with a tensor term arising naturally from the derivatives of $G$ in place of the cosmological constant term usually introduced \textit{ad hoc}. The modified theory, when applied to cosmology, is consistent with Dirac's large number hypothesis, and gives a theoretical Hubble's relation not contradicting the observational data. For phenomena of duration and distance short compared with that of the universe, our theory reduces to Einstein's theory with $G$ being constant outside the gravitating matter, and thus also passes the crucial tests of Einstein's theory.Comment: 9 pages, 1 figur

Surface friction of rock in terrestrial and simulated lunar environments

The conventional probe-on-the rotating-disk concept was used to determine the surface friction in mineral probe/specimen interfaces. Nine rocks or minerals and two stainless steels were tested in both new (NT) and same track (ST) tests under three different pressure environments-atmospheric, UHV, and dry nitrogen. Each environment was further subdivided into two testing conditions, that is, ambient and elevated (135 C) temperatures. In NT tests, friction was the lowest in an atmospheric pressure condition for all rock types and increased to the largest in UHV ambient condition except for pyroxene and stainless steel. Friction values measured in dry nitrogen ambient condition lie between the two extremes. Heating tends to increase friction in atmospheric and dry nitrogen environment but decreases in UHV environment with the exception of stainless steel, basalt, and pyroxene. In ST tests, friction was the lowest in the first run and increased in subsequent runs except for stainless steel where the reverse was true. The increases leveled off after a few runs ranging from the second to the seventh depending on rock types

Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information

We consider variants of trust-region and cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve $\epsilon$-approximate second-order optimality which have shown to be tight. Our Hessian approximation conditions constitute a major relaxation over the existing ones in the literature. Consequently, we are able to show that such mild conditions allow for the construction of the approximate Hessian through various random sampling methods. In this light, we consider the canonical problem of finite-sum minimization, provide appropriate uniform and non-uniform sub-sampling strategies to construct such Hessian approximations, and obtain optimal iteration complexity for the corresponding sub-sampled trust-region and cubic regularization methods.Comment: 32 page

Globular cluster formation efficiencies from black-hole X-ray binary feedback

We investigate a scenario in which feedback from black-hole X-ray binaries (BHXBs) sometimes begins inside young star clusters before strong supernova feedback. Those BHXBs could reduce the gas fraction inside embedded young clusters whilst maintaining virial equilibrium, which may help globular clusters (GCs) to stay bound when supernova-driven gas ejection subsequently occurs. Adopting a simple toy model with parameters guided by BHXB population models, we produce GC formation efficiencies consistent with empirically-inferred values. The metallicity dependence of BHXB formation could naturally explain why GC formation efficiency is higher at lower metallicity. For reasonable assumptions about that metallicity dependence, our toy model can produce a GC metallicity bimodality in some galaxies without a bimodality in the field-star metallicity distribution.Comment: Accepted to ApJ Letters on 19th July. 6 pages. The definitive version is available from: http://iopscience.iop.org/2041-8205/809/1/L16