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P-Distances, Q-Distances and a Generalized Ekeland's Variational Principle in Uniform Spaces

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Acta Mathematica Sinica,English Series 2012年第2期28卷 235-254页

作者： jing hui QIU Fei HE School of Mathematical Sciences Soochow UniversitySuzhou 215006 P.R.China School of Mathematical Sciences Inner Mongolia UniversityHohhot 010021P.R.China

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland＇s variational principle. In the framework of uniiorm spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance （particularly, a p-distance）, which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi＇s nonconvex minimization theorem, a generalized Ekeland＇s variational principle and a generalized Caristi＇s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland＇s variational principle, we deduce a number of particular versions of Ekeland＇s principle, which include many known versions of the principle and their improvements.

关键词： Ekeland's variational principle Takahashi's nonconvex minimization theorem Caristi'sfixed point theorem uniform space locally convex space p-distance q-distance

A General Vectorial Ekeland's Variational Principle with a P-distance

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Acta Mathematica Sinica,English Series 2013年第9期29卷 1655-1678页

作者： jing hui QIU Fei HE School of Mathematical Sciences Soochow University School of Mathematical Sciences Soochow University School of Mathematical Sciences Inner Mongolia University

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle （in short EVP）, where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space （which is a uniform space） has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi＇s fixed point theorem and a general vectorial Takahashi＇s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.

关键词： Vectorial Ekeland’s variational principle vectorial Caristi’s fixed point theorem vectorial Takahashi’s minimization theorem p-distance Gerstewitz’s function

Sequentially Lower Complete Spaces and Ekeland's Variational Principle

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Acta Mathematica Sinica,English Series 2015年第8期31卷 1289-1302页

作者： Fei HE jing-hui QIU School of Mathematical Sciences Inner Mongolia University School of Mathematical Sciences Soochow University

By using sequentially lower complete spaces（see [Zhu, J., Wei, L., Zhu, C. C.： Caristi type coincidence point theorem in topological spaces. J. Applied Math., 2013, ID 902692（2013）]）, we give a new version of vectorial Ekeland＇s variational principle. In the new version, the objective function is defined on a sequentially lower complete space and taking values in a quasi-ordered locally convex space, and the perturbation consists of a weakly countably compact set and a non-negative function p which only needs to satisfy p（x, y） = 0 iff x = y. Here, the function p need not satisfy the *** the new Ekeland＇s principle, we deduce a vectorial Caristi＇s fixed point theorem and a vectorial Takahashi＇s non-convex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. By considering some particular cases, we obtain a number of corollaries,which include some interesting versions of fixed point theorem.

关键词： Vectorial Ekeland variational principle vectorial Caristi's fixed point theorem vectorial Takahashi's non-convex minimization th

Vectorial Variational Principle with Variable Set-Valued Perturbation

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Acta Mathematica Sinica,English Series 2015年第4期31卷 595-614页

作者： Jian ZHANG jing hui QIU School of Mathematical Sciences Soochow University

We give a general vectorial Ekeland＇s variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered real linear space. Being quite different from the previous versions of vectorial Ekeland＇s variational principle, the perturbation in our version is no longer only dependent on a fixed positive vector or a fixed family of positive vectors. It contains a family of set-valued functions taking values in the positive cone and a family of subadditive functions of topology generating quasi-metrics. Hence, the direction of the perturbation in the new version is a family of variable subsets which are dependent on the objective function values. The general version includes and improves a number of known versions of vectorial Ekeland＇s variational principle. From the general Ekeland＇s principle, we deduce the corresponding versions of Caristi-Kirk＇s fixed point theorem and Takahashi＇s nonconvex minimization theorem. Finally, we prove that all the three theorems are equivalent to each other.

关键词： Vectorial Ekeland's variational principle F-type topological space locally convex space,pre-ordered linear space direction of perturbation

Generalized Two-State Theory for an Atom Laser with Nonlinear Couplings

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Communications in Theoretical Physics 2002年第7期38卷 35-40页

作者： jing hui TIAN Li-Jun Laboratory for Quantum Optics Shanghai Institute of Optics and Fine Mechanics Shanghai 201800 China Theoretical Physics Division Nankai Institute of Mathematics Nankai University Tianjin 300071 China

We present a generalized two-state theory to investigate the quantum dynamics and statistics of an atom laser with nonlinear couplings. The rotating wave approximate Hamiltonian of the system is proved to be analytically solvable. The fraction of output atoms is then showed to exhibit an interesting collapse and revival phenomenon with respect to the evolution time, a sign of nonlinear couplings. Several nonclassical effects, such as sub-Poissonian distribution, quadrature squeezing effects, second-order cross-correlation and accompanied violation of Cauchy-Schwartz inequality are also revealed for the output matter wave. The initial global phase of the trapped condensate, in weak nonlinear coupling limits, is found to exert an interesting impact on the quantum statistical properties of the propagating atom laser beam.

关键词： atom laser, nonlinear couplings, nonclassical effects

Quench sensitivity of Al-Cu-Mg alloy thick plate

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Rare Metals 2023年第9期42卷 3161-3169页

作者： Yuan Yin Bing-hui Luo Zhen-Hai Bai hui-Bo jing School of Materials Science and Engineering Central South UniversityChangsha410083China

The quench sensitivity of Al-Cu-Mg alloy was investigated at different thicknesses of the thick *** quenching process was simulated via finite element analysis(FEA);time-temperature-property(TTP)curves and time-temperature-transformation(TTT)curves were obtained through hardness test and differential scanning calorimetry(DSC)test;and the microstructural observation was carried out by scanning electron microscopy(SEM)and transmission electron microscopy(TEM).Experimental results exhibit that the quench cooling rate decreases dramatically from the surface to the center of the plate,and the inhomogeneous quenching causes the difference in *** the decrease in quench cooling rate,constituent particles are coarsening gradually;the quantity of T-phase(Al_(20)Cu_(2)Mn_(3))increases and the S-phase(Al_(2)Cu Mg)*** to the precipitation kinetics analysis,the decrease in S-phase is caused by the increase in precipitate activation *** that the center of the plate shows the highest quenching sensitivity,which is consistent with the analysis of time-temperature-property curves and time-temperature-transformation curves.

关键词： Al-Cu-Mg alloy Quench sensitivity Precipitation kinetics Time-temperature-property curves Microstructure

Superradiance-Driven Phonon Laser

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Chinese Physics Letters 2018年第4期35卷 49-52页

作者： Ya-jing Jiang Hao Lu hui jing Department of Physics Henan Normal University Xinxiang 453007 Key Laboratory for Quantum Optics Shanghai Institute of Optics and Fine Mechanics Chinese Academy of Sciences Shanghai 201800 Key Laboratow of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education Department of Physics and Synergetic hmovation Center for Quantum Effects and Applications Hunan Normal University Changsha 410081

We propose to enhance the generation of a phonon laser by exploiting optical superradiance. In our scheme, the optomechanical cavity contains a movable membrane, which supports a mechanical mode, and the superradiance cavity can generate the coherent collective light emissions by applying a transverse pump to an ultracold intracavity atomic gas. The superradiant emission turns out to be capable of enhancing the phonon laser *** indicates a new way to operate a phonon laser with the assistance of coherent atomic gases trapped in a cavity or lattice potentials.

关键词： Superradiance-Driven Phonon Laser

Resonant Activation Induced by Four-Value Noise

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Communications in Theoretical Physics 2008年第2期49卷 385-390页

作者： HAN Yin-Xia LI jing-hui P.O. Box 58 Faculty of Science Ningbo University Ningbo 315211 China

The phenomenon of the resonant activation （RA） of a particle over a fluctuating potential barrier with a four-value noise is investigated. It is shown that the mean first passage time （MFPT） displays six minima as the function of the transition rates γ1, γ2, γ3, γ4, γ5, and 76 of the four-value noise, respectively. In addition, the effect of other parameters of the system, such as the noise strength D of the additive Gaussian white noise and the parameter value a, b, c, and d of the four-value noise, on the RAs is also investigated.

关键词： resonant activation four-value noise mean first passage time

Identifying indigenous tree species for land reforestation,forest restoration, and plantation transformation on Hainan Island, China

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Journal of Mountain Science 2018年第11期15卷 2433-2444页

作者： WANG Jian-jun MENG jing-hui Key Laboratory for Forest Resources and Ecosystem Processes of Belling Belling Forestry University Beijing 100083 China

Selecting suitable species is the most important issue for bare land reforestation, degraded secondary forest restoration, and single-species plantation transformation. However, little information has been documented related to tree species selection for these silvicultural endeavors on tropical Hainan Island of Southern China. The present study employed Baisha County, the ecological core area of Hainan Island, as a case study area. We initially inventoried a slightly disturbed primary forest and attempted to produce diameter distribution curves for each tree species. Second, the tree species were classified into shade intolerant, opportunist, and shade tolerant species based on shape of their diameter distribution curves. Third, market value was determined for each tree species based on published literature and on-site investigations at local wood trading companies. Totally 118 tree species were encountered in the inventoried forest and 13 tree species present were finally identified as potential tree species for our silvicultural endeavor on Hainan Island, of which 3 species are shade intolerant, 5 species are opportunist and 5 species are shade tolerant. Additionally, we also selected 12 tree species that were not in the inventoried forest but were extremely economically valuable and ecologically important. This study should contribute to the formulation of a sustainable forest management strategy on Hainan Island and the methodology might be replicated in other tropical region where suitable species also need to be identified for silvicultural endeavor.

关键词： Diameter distribution curves Primary forest Shade intolerant Opportunist species Shade tolerant Forest management