The error and perturbation bounds for the absolute value equations with some applications

To our knowledge, so far, the error and perturbation bounds for the general absolute value equations are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error bounds and perturbation bounds for two types of absolute value equations (AVEs): Ax-B|x|=b and Ax-|Bx|=b. Some useful error bounds and perturbation bounds for the above two types of absolute value equations are presented. By applying the absolute value equations, we also obtain the error and perturbation bounds for the horizontal linear complementarity problem (HLCP). In addition, a new perturbation bound for the LCP without constraint conditions is given as well, which are weaker than the presented work in [SIAM J. Optim., 2007, 18: 1250-1265] in a way. Besides, without limiting the matrix type, some computable estimates for the above upper bounds are given, which are sharper than some existing results under certain conditions. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds

Palatini formulation of f(R,T)f(R,T) gravity theory, and its cosmological implications

We consider the Palatini formulation of $f(R,T)$ gravity theory, in which a nonminimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similarly to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra-force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the $f(R,T)$ gravity in the Palatini formulation. Cosmological models with Lagrangians of the type $f=R-\alpha ^2/R+g(T)$ and $f=R+\alpha ^2R^2+g(T)$ are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.Comment: 22 pages, no figures, accepted for publication in EPJC; references adde

Palladium and silver abundances in stars with [Fe/H] > -2.6

Palladium (Pd) and silver (Ag) are the key elements for probing the weak component in the rapid neutron-capture process (r-process) of stellar nucleosynthesis. We performed a detailed analysis of the high-resolution and high signal-to-noise ratio near-UV spectra from the archive of HIRES on the Keck telescope, UVES on the VLT, and HDS on the Subaru Telescope, to determine the Pd and Ag abundances of 95 stars. This sample covers a wide metallicity range with -2.6 $\lesssim$ [Fe/H] $\lesssim$ +0.1, and most of them are dwarfs. The plane-parallel LTE MAFAGS-OS model atmosphere was adopted, and the spectral synthesis method was used to derive the Pd and Ag abundances from Pd I {\lambda} 3404 {\AA} and Ag I {\lambda} 3280/3382 {\AA} lines. We found that both elements are enhanced in metal-poor stars, and their ratios to iron show flat trends at -0.6 < [Fe/H] < +0.1. The abundance ratios of [Ag/H] and [Pd/H] are well correlated over the whole abundance range. This implies that Pd and Ag have similar formation mechanisms during the Galactic evolution.Comment: 15 pages, 12 figures, accepted to A&

A General Theorem Relating the Bulk Topological Number to Edge States in Two-dimensional Insulators

We prove a general theorem on the relation between the bulk topological quantum number and the edge states in two dimensional insulators. It is shown that whenever there is a topological order in bulk, characterized by a non-vanishing Chern number, even if it is defined for a non-conserved quantity such as spin in the case of the spin Hall effect, one can always infer the existence of gapless edge states under certain twisted boundary conditions that allow tunneling between edges. This relation is robust against disorder and interactions, and it provides a unified topological classification of both the quantum (charge) Hall effect and the quantum spin Hall effect. In addition, it reconciles the apparent conflict between the stability of bulk topological order and the instability of gapless edge states in systems with open boundaries (as known happening in the spin Hall case). The consequences of time reversal invariance for bulk topological order and edge state dynamics are further studied in the present framework.Comment: A mistake corrected in reference

Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors

Journal ArticleWe propose models of two-dimensional paramagnetic semiconductors where the intrinsic spin Hall effect is exactly quantized in integer units of a topological charge. The model describes a topological insulator in the bulk and a "holographic metal" at the edge, where the number of extended edge states crossing the Fermi level is dictated by (exactly equal to) the bulk topological charge. We also demonstrate the spin Hall effect explicitly in terms of the spin accumulation caused by the adiabatic flux insertion