Abstract:
In the present paper, we shall give an extension of the well known Pecaric-Rajic inequality in a quasi-Banach space, we establish the generalized inequality for an arbitrary number of finitely many nonzero elements of a quasi-Banach space, and obtain the corresponding upper and lower bounds. As a result, we get some more general inequalities.

Abstract:
In this paper, we present an extension of the so-called classical Sherman-Morrison-Woodbury (for short SMW) formula for bounded homogeneous generalized inverse in Banach spaces. Some particular cases and applications will be also considered. Our results generalize the results of many authors for finite dimensional matrices and Hilbert space operators in the literature.

Abstract:
We calculated the proportion of cancers attributable to alcohol use to estimate the burden of alcohol-related cancer. The population attributable fraction was calculated based on the assumption of no alcohol drinking. Data on alcohol drinking prevalence were from two large-scale national surveys of representative samples of the Chinese population. Data on relative risk were obtained from meta-analyses and large-scale studies.We found that a total of 78,881 cancer deaths were attributable to alcohol drinking in China in 2005, representing 4.40% of all cancers (6.69% in men, 0.42% in women). The corresponding figure for cancer incidence was 93,596 cases (3.63% of all cancer cases). Liver cancer was the main alcohol-related cancer, contributing more than 60% of alcohol-related cancers.Particular attention needs to be paid to the harm of alcohol as well as its potential benefits when making public health recommendations on alcohol drinking.Cancer constitutes a serious burden of disease worldwide as well as in China. Ranked as the second leading cause of death in China, cancer accounts for 22.32% of deaths, behind only cerebrovascular disease (22.45% of total deaths)[1]. Alcohol consumption is causally associated with the increased risk of certain cancers [2]. In China, alcohol drinking has a long history and continues to be an important part of life and culture. Alcohol consumption by Chinese citizens has increased in recent decades following rapid economic development that entailed cultural and behavioral changes [3,4]. As a strategic program of "Healthy China 2020" is being developed [5], an estimate of the burden of alcohol-related cancer is imperative for guiding policymakers on issues of cancer prevention and control. The aim of our study is to estimate the cancer burden which is attributable to alcohol drinking in the year 2005.Our study is an evidence-based and consistent evaluation and is part of a project called "attributable causes of cancer in China", aimed a

Abstract:
The interaction of a weak probe laser with an inverted-Y type four-level atomic system driven by two additional coherent fields is investigated theoretically. Under the influence of the coherent coupling fields, the steady-state linear susceptibility of the probe laser shows that the system can have single or double electromagnetically induced transparency windows depending on the amplitude and the detuning of the coupling lasers. The corresponding index of refraction associated with the group velocity of the probe laser can be controlled at both transparency windows by the coupling fields. The propagation of the probe field can be switched from superluminal near the resonance to subluminal on resonance within the single transparency window when two coupling lasers are on resonance. This provides a potential application in quantum information processing. We propose an atomic $^{87}Rb$ system for experimental observation.

Abstract:
We investigate the spontaneous emission from an inverted Y-type atomic system coupled by three coherent fields. We use the Schr\"{o}dinger equation to calculate the probability amplitudes of the wave function of the system and derive an analytical expression of the spontaneous emission spectrum to trace the origin of the spectral features. Quantum interference effects, such as the spectral line narrowing, spectrum splitting and dark resonance are observed. The number of spectral components, the spectral linewidth, and relative heights can be very different depending on the physical parameters. A variety of spontaneous emission spectral features can be controlled by the amplitudes of the coupling fields and the preparation of the initial quantum state of the atom. We propose an ultracold atomic $^{87}Rb$ system for experimental observation.

Abstract:
To minimize the excessive vibration and prolong the fatigue life of the offshore wind turbine systems, it is of value to control the vibration that is induced within the structure by implementing certain kinds of dampers. In this paper, a ball vibration absorber (BVA) is experimentally investigated through a series of shake table tests on a 1/13 scaled wind turbine model. The reductions in top displacement, top acceleration, bottom stress and platform stress of the wind turbine tower system subjected to earthquakes and equivalent wind-wave loads, respectively, with a ball absorber are examined. Cases of the tower with rotating blades are also investigated to validate the efficacy of this damper in mitigating the vibration of an operating wind turbine. The experimental results indicate that the dynamic performance of the tested wind turbine model with a ball absorber is significantly improved compared with that of the uncontrolled structure in terms of the peak response reduction.

Abstract:
Let and be Banach spaces, and let be a bounded linear operator. In this paper, we first define and characterize the quasi-linear operator (resp., out) generalized inverse (resp., ) for the operator , where and are homogeneous subsets. Then, we further investigate the perturbation problems of the generalized inverses and . The results obtained in this paper extend some well-known results for linear operator generalized inverses with prescribed range and kernel. 1. Introduction and Preliminaries Let and be Banach spaces, let be a mapping, and let be a subset of . Recall from [1, 2] that a subset in is called to be homogeneous if for any and , we have . If for any and , we have , then we call as a homogeneous operator on , where is the domain of ; is called a bounded homogeneous operator if maps every bounded set in into bounded set in . Denote by the set of all bounded homogeneous operators from to . Equipped with the usual linear operations for , and for , the norm is defined by , and then similar to the space of all bounded linear operators from to , we can easily prove that is a Banach space (cf. [2, 3]). Throughout this paper, we denote by , , and the domain, the null space, and the range of a bounded homogeneous operator , respectively. Obviously, we have . For an operator , let and be closed subspaces of and , respectively. Recall that the out inverse with prescribed range and kernel is the unique operator satisfying . It is well known that the important kinds of generalized inverses, the Moore-Penrose inverse, the Drazin inverse, the group inverse, and so on, are all generalized inverse (cf. [4, 5]). Researches on the generalized inverse of operators or matrices have been actively ongoing for many years (see [5–12], e. g.). Let and let and be two homogeneous subsets in and , respectively. Motivated by related work on in the literature mentioned above and by our own recent research papers [13, 14], in this paper, we will establish the definition of the quasi-linear operator outer generalized inverse with prescribed range and kernel . We give the necessary and sufficient conditions for the existence of the generalized inverses , and we will also study the perturbation problems of the generalized inverse . Similar results on the generalized inverse are also given. 2. Definitions and Some Characterizations of and We first give the concepts of quasi-additivity and quasi-linear projectors in Banach spaces, which are important for us to present the main results in this paper. Definition 1. Let be a subset of and let be a mapping. Ones calls as

Abstract:
In this paper, we investigate the various different generalized inverses in a Banach algebra with respect to prescribed two idempotents $p$ and $q$. Some new characterizations and explicit representations for these generalized inverses, such as $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$ and $a^{(2,l)}_{p,q}$ will be presented. The obtained results extend and generalize some well--known results for matrices or operators.

Abstract:
In this paper, we first study the perturbations and expressions for the generalized inverses $a^{(2)}_{p,q}$, $a^{(1, 2)}_{p,q}$, $a^{(2, l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate the general perturbation analysis and error estimate for some of these generalized inverses when $p,\,q$ and $a$ also have some small perturbations.