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| 1 | Analytic proof of dual variational formula for the first eigenvalue in dimension one显示文摘The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well-known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case are proposed and certain extension is made. | 陈木法 | 1999 | Science China Mathematics1999，42，8： | 25 |
| 2 | A generalized variational principle and theoretical model for magnetoelastic interaction of ferromagnetic bodies显示文摘The quantitative analysis shows that no theoretical model for 3-d magnetoelastic bodies, in literatures to date, can commonly simulate two kinds of distinct experimental phenomena on magnetoelastic interaction of ferromagnetic structures. This makes it difficult to effectively discribe the magnetoelastic mechanical behavior of structures with complex geometry, such as shells. Therefore, it is a key step for simulating magnetoelastic mechanical characteristics of structures with complex geometry to establish a 3-d model which also can commonly characterize the two distinct experimental phenomena. A theoretical model for three dimension magnetizable elastic bodies, which is commonly suitable for the two kinds of experimental phenomena on magnetoelastic interaction of ferromagnetic plates, is presented by the variational principle for the total energy functional of the coupling system of the 3-d ferromagnetic bodies. It is found that for the case of linear isotropic magnetic materials, the magnetic | 周又和 郑晓静 | 1999 | Science China Mathematics1999，42，6： | 26 |
| 3 | Theoretical analyses and numerical experiments of variational assimilation for one-dimensional ocean temperature model with techniques in inverse problems显示文摘In the present work, the data assimilation problem in meteorology and physical oceanography is re-examined using the variational optimal control approaches in combination with regularization techniques in inverse problem. Here the estimations of the initial condition, boundary condition and model parameters are performed simultaneously in the framework of variational data assimilation. To overcome the difficulty of ill-posedness, especially for the model parameters distributed in space and time, an additional term is added into the cost functional as a stabilized functional. Numerical experiments show that even with noisy observations the initial conditions and model parameters are recovered to an acceptable degree of accuracy. | HUANG Sixun1,2, HAN Wei1 & WU Rongsheng2 1. P. O. Box 003, Nanjing 211101,China 2. Key Laboratory of Meso-Scale Severe Weather /MOE, Department of Atmospheric Sciences, Nanjing University, Nanjing 210093, China Correspondence should be addressed to Huang Sixun | 2004 | Science China Earth Sciences2004，47，7： | 18 |
| 4 | On the Controllability of an Advection-diffusion Equation with Respect to the Diffusion Parameter:Asymptotic Analysis and Numerical Simulations显示文摘The advection-diffusion equation y_t~ε-εy_(xx)~ε+ M y_x~ε= 0,(x, t) ∈(0, 1) ×(0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient ε goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to ε if T is greater than a minimal time T_M, with T_M in the interval [1, 2×3^(1/2)]/M for M > 0 and in the interval [2×2^(1/2), 2(1 +3^(1/2))]/|M | for M < 0. The exact value of TM is however unknown.We investigate in this work the determination of the minimal time T_M employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and ε. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the advection-diffusion equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter ε, of the optimality system associated to the control of minimal L^2-norm. The matched asymptotic expansion method is used to describe the multiple boundary layers. | Youcef AMIRAT Arnaud MNCH | 2019 | Acta Mathematicae Applicatae Sinica2019，35，1： | 18 |
| 5 | An emission source inversion model based on satellite data and its application in air quality forecasts显示文摘This paper aims at constructing an emission source inversion model using a variational processing method and adaptive nudging scheme for the Community Multiscale Air Quality Model (CMAQ) based on satellite data to investigate the applicability of high resolution OMI (Ozone Monitoring Instrument) column concentration data for air quality forecasts over the North China. The results show a reasonable consistency and good correlation between the spatial distributions of NO2 from surface and OMI satellite measurements in both winter and summer. Such OMI products may be used to implement integrated variational analysis based on observation data on the ground. With linear and variational corrections made, the spatial distribution of OMI NO2 clearly revealed more localized distributing characteristics of NO2 concentration. With such information, emission sources in the southwest and southeast of North China are found to have greater impacts on air quality in Beijing. When the retrieved emission source inventory based on high-resolution OMI NO2 data was used, the coupled Weather Research Forecasting CMAQ model (WRF-CMAQ) performed significantly better in forecasting NO2 concentration level and its tendency as reflected by the more consistencies between the NO2 concentrations from surface observation and model result. In conclusion, satellite data are particularly important for simulating NO2 concentrations on urban and street-block scale. High-resolution OMI NO2 data are applicable for inversing NOx emission source inventory, assessing the regional pollution status and pollution control strategy, and improving the model forecasting results on urban scale. | CHENG XingHong XU XiangDe DING GuoAn | 2010 | Science China Earth Sciences2010，53，5： | 17 |
| 6 | A class of discounted models for singular diffusion control显示文摘A kind of discounted problems for singular diffusion control have been studied. The drift and diffusion coefficients of state process are nonlinear. The class of models has been basically extended from a corresponding one established by Karatzes et al. before. By applying some analysis methods different from earlier works, the sufficient and necessary conditions of the existence of optimal control have been obtained. If an optimal control exists, it is a 'transient reflection' of state process for two lines. | LIU Kunhui, QIN Mingda & LU Chuanlai1. College of Sciences, Northern Jiaotong University, Beijing 100044, China 2. Department of Mathematics and Mechanics, Beijing University of Science and Technology, Beijing 100083 .China 3. Department of Information Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China Correspondence should be addressed to Lu Chuanlai | 2000 | Chinese Science Bulletin2000，45，5： | 17 |
| 7 | Variational Principle of Instability of Atmospheric Motions显示文摘Problems of instability of rotating atmospheric motions are investigated by using nonlinear governing equations and the variational principle. The method suggested in this paper is universal for obtaining criteria of instability in all models with all possible basic flows. For example, the model can be barotropic or baroclinic, layer or continuous, quasi-geostrophic or primitive equations; the basic flow can be zonal or nonzonal, steady or unsteady.Although the basic flows possess a great deal of variety, they all are the stationary points in the functional space determined by an appropriate invariant functional. The basic flow is an unsteady one if the conservation of angular momentum is included in the associated functional.The second variation, linear or nonlinear, gives the criteria of instability. Especially, the general criteria of instability for unsteady basic flow, orographically disturbed flow as well as nongeostrophic flow are first obtained by the method described in this paper.It is also | 曾庆存 | 1989 | Advances in Atmospheric Sciences1989，6，2： | 17 |
| 8 | Studies on the analytical potential energy function of diatomic molecular ion XY^+ using variational method显示文摘A new analytical potential energy function for diatomic molecular ion XY+ is proposed based on the energy consistent method (ECM). The Coulomb potential included in the new ionic potential contains multipole corrections, converges quickly and is variationally changeable. The new potential and the ECM are applied to variationally studying the potential energies of eight electronic states of several diatomic molecular ions: the A2P state of CO+, the X2+g state of 2+Li, the X2+g state of 2,+Hethe 12Pu state of ,+2Nathe A2Pu state of ,+2N the X1S+ state of KrH+, the X2S+ state of SiO+ and the A2P state of SO+ ion. The present results agree excellently with the experiment-based Rydberg-Klein-Rees (RKR) potentials, and are superior to the commonly used Huxley-Murrell-Sorbie (HMS) analytical potentials, and are better in some cases than some quantum mechanical ab initio potentials in the ionic asymptotic and dissociation regions. | LIU Guoyue1,3, SUN Weiguo1,2 & FENG Hao1 1. Institute of Physics, Sichuan University, Chengdu 610065, China 2. College of Chemical Engineering, Sichuan University, Chengdu 610065, China 3. Mianyang Teacher抯 College, Mianyang 621000, China Correspondence should be addressed to Sun Weiguo (e-mail: weiguosun@x263.net or guoyueliu@ sohu.com) | 2004 | Science China(Physics,Mechanics & Astronomy)2004，47，2： | 15 |
| 9 | Highly accurate symplectic element based on two variational principles显示文摘For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element(NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses. | Guanghui Qing Jia Tian | 2018 | Acta Mechanica Sinica2018，34，1： | 15 |
| 10 | Boundary value problems of discrete generalized Emden-Fowler equation显示文摘By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established. | YU Jianshe GUO Zhiming | 2006 | Science China Mathematics2006，49，10： | 14 |
| 11 | Virtual screening of inorganic materials synthesis parameters with deep learning显示文摘Virtual materials screening approaches have proliferated in the past decade,driven by rapid advances in first-principles computational techniques,and machine-learning algorithms.By comparison,computationally driven materials synthesis screening is still in its infancy,and is mired by the challenges of data sparsity and data scarcity:Synthesis routes exist in a sparse,highdimensional parameter space that is difficult to optimize over directly,and,for some materials of interest,only scarce volumes of literature-reported syntheses are available.In this article,we present a framework for suggesting quantitative synthesis parameters and potential driving factors for synthesis outcomes.We use a variational autoencoder to compress sparse synthesis representations into a lower dimensional space,which is found to improve the performance of machine-learning tasks.To realize this screening framework even in cases where there are few literature data,we devise a novel data augmentation methodology that incorporates literature synthesis data from related materials systems.We apply this variational autoencoder framework to generate potential SrTiO_(3) synthesis parameter sets,propose driving factors for brookite TiO_(2) formation,and identify correlations between alkali-ion intercalation and MnO_(2) polymorph selection. | Edward Kim Kevin Huang Stefanie Jegelka Elsa Olivetti | 2017 | npj Computational Materials2017，，1： | 14 |
| 12 | The generalized quasi-variational principles of non-conservative systems with two kinds of variables显示文摘According to the corresponding relations between general forces and general displacements, the balancing and geometrical equations of elasticity are multiplied by the corresponding virtual quantities, integrated with volume and area, and then added algebraically. Proceeding to the next step, by substituting constitutive relation and considering that body force and surface force are both fellow forces, the generalized quasi-variational principles with the two kinds of variables of the first type are established in non-conservative systems. Through substituting another constitutive relation, using similar methods as above, the generalized quasi-variational principles with the two kinds of variables of the second type are established in non-conservative systems. By using the generalized quasi-complementary energy principles with the two kinds of variables of the first type, a method for solving two kinds of variables (internal force and deformation) is given for non-conservative systems of the typical fellow forces. | LIANG Lifu LIU Diankui SONG Haiyan | 2005 | Science China(Physics,Mechanics & Astronomy)2005，48，5： | 12 |
| 13 | Generalized mixed finite element method for 3D elasticity problems显示文摘Without applying any stable element techniques in the mixed methods, two simple generalized mixed element(GME) formulations were derived by combining the minimum potential energy principle and Hellinger–Reissner(H–R) variational principle. The main features of the GME formulations are that the common C0-continuous polynomial shape functions for displacement methods are used to express both displacement and stress variables, and the coefficient matrix of these formulations is not only automatically symmetric but also invertible. Hence, the numerical results of the generalized mixed methods based on the GME formulations are stable. Displacement as well as stress results can be obtained directly from the algebraic system for finite element analysis after introducing stress and displacement boundary conditions simultaneously. Numerical examples show that displacement and stress results retain the same accuracy. The results of the noncompatible generalized mixed method proposed herein are more accurate than those of the standard noncompatible displacement method. The noncompatible generalized mixed element is less sensitive to element geometric distortions. | Guanghui Qing Junhui Mao Yanhong Liu | 2018 | Acta Mechanica Sinica2018，34，2： | 11 |
| 14 | Variational principles and governing equations in nano-dielectrics with the flexoelectric effect显示文摘The flexoelectric effect is very strong and coupled with large strain gradients for nanoscale dielectrics. At the nanoscale, the electrostatic force cannot be ignored. In this paper, we have established the electric enthalpy variational principle for nanosized dielectrics with the strain gradient and the polarization gradient effect, as well as the effect of the electrostatic force. The complete governing equations, which include the effect of the electrostatic force, are derived from this variational principle, and based on the principle the generalized electrostatic stress is obtained, the generalized electrostatic stress contains the Maxwell stress corresponding to the polarization and strain, and stress related to the polarization gradient and strain gradient. This work provides the basis for the analysis and computations for the electromechanical problems in nanosized dielectric materials. | HU ShuLing & SHEN ShengPing* MOE Key Laboratory for Strength and Vibration, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China | 2010 | Science China(Physics,Mechanics & Astronomy)2010，53，8： | 11 |
| 15 | Noether-type theorem for discrete nonconservative dynamical systems with nonregular lattices显示文摘We investigate Noether symmetries and conservation laws of the discrete nonconserved systems with nonregular lattices. The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems. Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the nonconserved forces under the infinitesimal transformations with respect to the time and generalized coordinates, we give the discrete analog of generalized variational formula. From this formula we derive the discrete analog of generalized Noether-type identity, and then we present the generalized quasi-extremal equations and properties of these equations for the systems. We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems. We discuss an example to illustrate these results. | FU JingLi1, CHEN LiQun2 & CHEN BenYong3 1 Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China 2 Department of Mechanics, Shanghai University, Shanghai 200072, China 3 Faculty of Mechanical-Engineering & Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China | 2010 | Science China(Physics,Mechanics & Astronomy)2010，53，3： | 11 |
| 16 | GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA显示文摘Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media. | Yang Xiao Cheng Changjun Department o,f Mechanics, Shanghai Institute of Applied Mathematics and Mechanics,Shanghai University,Shanghai 200436,China) | 2003 | Acta Mechanica Solida Sinica2003，16，1： | 10 |
| 17 | Wind power prediction based on variational mode decomposition multi-frequency combinations显示文摘Because of the uncertainty and randomness of wind speed, wind power has characteristics such as nonlinearity and multiple frequencies. Accurate prediction of wind power is one effective means of improving wind power integration. Because the traditional single model cannot fully characterize the fluctuating characteristics of wind power, scholars have attempted to build other prediction models based on empirical mode decomposition(EMD) or ensemble empirical mode decomposition(EEMD) to tackle this problem. However, the prediction accuracy of these models is affected by modal aliasing and illusive components. Aimed at these defects, this paper proposes a multi-frequency combination prediction model based on variational mode decomposition(VMD). We use a back propagation neural network(BPNN),autoregressive moving average(ARMA)model, and least square support vector machine(LS-SVM) to predict high, intermediate,and low frequency components,respectively. Based on the predicted values of each component, the BPNN is applied to combine them into a final wind power prediction value.Finally,the prediction performance of the single prediction models(ARMA,BPNN and LS-SVM)and the decomposition prediction models(EMD and EEMD) are used to compare with the proposed VMD model according to the evaluation indices such as average absolute error, mean square error,and root mean square error to validate its feasibility and accuracy. The results show that the prediction accuracy of the proposed VMD model is higher. | Gang ZHANG Hongchi LIU Jiangbin ZHANG Ye YAN Lei ZHANG Chen WU Xia HUA Yongqing WANG | 2019 | Journal of Modern Power Systems and Clean Energy2019，7，2： | 10 |
| 18 | Quasi-variational principles of single flexible body dynamics and their applications显示文摘The reasons for studying single flexible body dynamics are that on one hand,it is the basis of flexible multi-body dynamics.If the theory of the single flexible body dynamics has been deeply studied,the theory of flexible multi-body dynamics will be researched easily.On the other hand,it has its unique and important applications.Quasi-variational principle of non-conservative single flexible body dynamics is established under the cross-link of particle rigid body mechanics and deformable body mechanics.Taking the interceptor as an example,this paper has explained the physical meaning of the quasi-stationary value condition of the quasi-variational principle in non-conservative single flexible body dynamics.Taking the launch of rocket as an example,it has illustrated the features of'one force for two effects'in a single flexible body dynamics.With an example of the extending flexible beam coupled with the spacecraft attitude,it has shown the transition from the single flexible body dynamics to the flexible multi-body dynamics.Finally,a number of related problems are discussed. | LIANG LiFu LIU ShiQuan ZHOU JianSheng | 2009 | Science China(Physics,Mechanics & Astronomy)2009，52，5： | 9 |
| 19 | Unconventional Hamilton-type variational principles for nonlinear coupled thermoelastodynamics显示文摘According to the basic idea of classical yin-yang complementarity and modem dual-com plementarity, in a simple and unified new way proposed by Luo, the unconventional Hamilton-type vari ational principles for geometrically nonlinear coupled thermoelastodynamics can be established system atically. The new unconventional Hamilton-type variational principle can fully characterize the initia boundary-value problem of this dynamics. In this paper, an important integral relation is given, which can be considered as the expression of the generalized principle of virtual work for geometrically nonlin ear coupled thermodynamics. Based on this relation, it is possible not only to obtain the principle of vir tual work in geometrically nonlinear coupled thermodynamics, but also to derive systematically the complementary functionals for eight-field, six-field, four-field and two-field unconventional Hamilton type variational principles by the generalized Legendre transformations given in this paper. Further more, with this approach, the intrinsic relationship among various principles can be explained clearly. | 罗恩 黄伟江 邝君尚 罗志国 | 2002 | Science China Mathematics2002，45，6： | 9 |
| 20 | Solution and applications of a class of general linear variational inequalities显示文摘Many problems in mathematical programming can be described as a general linear variational inequality of the following form: find a vector u*, such thatSome iterative methods for solving a class of general linear variational inequalities have been presented. It is pointed out that the methods can be used to solve some practical extended programming problems. | 何炳生 | 1996 | Science China Mathematics1996，39，4： | 8 |
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