A note on property (gb) and perturbations

An operator $T \in \mathcal{B}(X)$ defined on a Banach space $X$ satisfies property $(gb)$ if the complement in the approximate point spectrum $\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\sigma_{SBF_{+}^{-}}(T)$ coincides with the set $\Pi(T)$ of all poles of the resolvent of $T$. In this note we continue to study property $(gb)$ and the stability of it, for a bounded linear operator $T$ acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators commuting with $T$. Two counterexamples show that property $(gb)$ in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.Comment: 10 page

First-Principles Study on the Structural and Electronic Properties of N Atoms Doped-Rutile TiO 2

For the propose of considering the actual situation of electronic neutral, a simulation has been down on the basis of choosing the position of dual N and researching the oxygen vacancy. It is found that the reason why crystal material gets smaller is due to the emergence of impurity levels. By introducing the oxygen vacancy to the structure, the results show that while the oxygen vacancy is near the two nitrogen atoms which have a back to back position, its energy gets the lowest level and its structure gets the most stable state. From its energy band structure and density, the author finds that the impurity elements do not affect the migration of Fermi level while the oxygen vacancy has been increased. Instead of that, the conduction band of metal atoms moves to the Fermi level and then forms the N-type semiconductor material, but the photocatalytic activity is not as good as the dual N-doping state

The structure of f(R)f(R)-brane model

Recently, a family of interesting analytical brane solutions were found in $f(R)$ gravity with $f(R)=R+\alpha R^2$ in Ref. [Phys. Lett. B 729, 127 (2014)]. In these solutions, inner brane structure can be turned on by tuning the value of the parameter $\alpha$. In this paper, we investigate how the parameter $\alpha$ affects the localization and the quasilocalization of the tensorial gravitons around these solutions. It is found that, in a range of $\alpha$, despite the brane has an inner structure, there is no graviton resonance. However, in some other regions of the parameter space, although the brane has no internal structure, the effective potential for the graviton KK modes has a singular structure, and there exists a series of graviton resonant modes. The contribution of the massive graviton KK modes to the Newton's law of gravity is discussed shortly.Comment: v2: 10 pages, 8 figures, to be published in EPJ

Preparation of MAl2O4: Eu2+, Sm3+ (M = Ca, Sr, Ba) Phosphors by the Combustion Method and their Luminescent Properties

A series of MAl2O4: Eu2+, Sm3+ (M = Ca, Sr, Ba) phosphors was prepared by the combustion method, and the influence of these alkaline earth metals on the structure and luminescent performances for these phosphors was investigated. A relationship was established between their composition, crystallization capacity and luminescent properties. The results show that the CaAl2O4: Eu2+, Sm3+ and SrAl2O4: Eu2+, Sm3+ phosphors exhibit monoclinic crystal systems and that the BaAl2O4: Eu2+, Sm3+ phosphor exhibits a hexagonal structure. The alkaline earth metal has a positive effect on the crystallization process, the heat treatment procedure and the luminescent properties of the corresponding phosphors. According to the order Ca > Sr > Ba, the maximum emission wavelengths of the corresponding phosphors are found at 505 nm, 490 nm and 478 nm, and the relevant colours of emitted light are between blue and green, which indicates that there is a clear blue shift in the emission spectra, but their excitation spectra show almost no difference. The decay curve shows that the afterglow intensity and time are not only related to the activating processes and the hole transport, but also to the phase structure and the grain size,which is in the order Sr>Ca>Ba.Keywords: Long afterglow, alkaline earth aluminates, spectra, combustion metho

Theory of Critical Phenomena with Long-Range Temporal Interaction

We develop a systematic theory for the critical phenomena with memory in all spatial dimensions, including $dd_c$, the upper critical dimension. We show that the Hamiltonian plays a unique role in dynamics and the dimensional constant $\mathfrak{d}_t$ that embodies the intimate relationship between space and time is the fundamental ingredient of the theory. However, its value varies with the space dimension continuously and vanishes exactly at $d=4$, reflecting reasonably the variation of the amount of the temporal dimension that is transferred to the spatial one with the strength of fluctuations. Such variations of the temporal dimension save all scaling laws though the fluctuation-dissipation theorem is violated. Various new universality classes emerge.Comment: 32 pages, 1 figure. Version 2: 41 pages, 1 figure. Clarify the origin of the memor