NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS

In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈ℝ×H^(1)(ℝ^(N))to the general Kirchhoff problem-M\left(\int_{\mathbb{R}^N}\vert\nabla u\vert^2{\rm d}x\right)\Delta u+\lambda u=g(u)~\hbox{in}~\mathbb{R}^N,u\in H^1(\mathbb{R}^N),N\geq 1,satisfying the normalization constraint\int_{\mathbb{R}^N}u^2{\rm d}x=c,where M∈C([0,∞))is a given function satisfying some suitable assumptions.Our argument is not by the classical variational method,but by a global branch approach developed by Jeanjean et al.[J Math Pures Appl,2024,183:44–75]and a direct correspondence,so we can handle in a unified way the nonlinearities g(s),which are either mass subcritical,mass critical or mass supercritical.

A Survey Study on TPACK Framework for Normal Students of English Major in Ethnic Colleges and Universities:Empirical Analysis Based on Structural Equation Modeling

With the continuous advancement of education informatization,Technological Pedagogical Content Knowledge(TPACK),as a new theoretical framework,provides a novel method for measuring teachers’informatization teaching ability.This study takes normal students of English majors from three ethnic universities as the research object,collects relevant data through questionnaires,and uses structural equation modeling to conduct data analysis and empirical research to investigate the differences in the TPACK levels of these students at different grades and the structural relationships among the elements in the TPACK structure.The technological pedagogical knowledge element of the TPACK structure was not obtained by exploratory factors analysis but through path analysis and structural equation modeling,the results show that the one-dimensional core knowledge of technological knowledge(TK),content knowledge(CK),and pedagogical knowledge(PK)have a positive effect on the two-dimensional interaction knowledge of technological content knowledge(TCK)and pedagogical content knowledge(PCK);furthermore,TCK and PCK have a positive effect on TPACK;and TK,CK,and PK indirectly affect TPACK through TCK and PCK.On this basis,suggestions are provided to ethnic colleges and universities to develop the TPACK knowledge competence of normal students of English majors.

THE EXISTENCE AND STABILITY OF NORMALIZED SOLUTIONS FOR A BI-HARMONIC NONLINEAR SCHR?DINGER EQUATION WITH MIXED DISPERSION

In this paper,we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrodinger equation with aμ-Laplacian term(BNLS).Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term.Denoting by Qpthe ground state for the BNLS withμ=0,we prove that in the mass-subcritical regime p∈(1,1+8/d),there exist orbit ally stable ground state solutions for the BNLS when p∈(-λ0,∞)for someλ0=λ0(p,d,‖Qp‖L2)>0.Moreover,in the mass-critical case p=1+8/d,we prove the orbital stability on a certain mass level below‖Q*‖L2,provided thatμ∈(-λ1,0),where■and Q*=Q1+8/d.The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality.Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when p is negative and p∈(1,1+8/d].

High-efciency improved symmetric successive over-relaxation preconditioned conjugate gradient method for solving large-scale finite element linear equations

Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing （CAM）. This paper presents a high-efficiency improved symmetric successive over-relaxation （ISSOR） preconditioned conjugate gradient （PCG） method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.

STABILITY OF NONLINEAR COMPARISON EQUATIONS FOR DISCRETE LARGE-SCALE SYSTEMS

On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparison equations was studied in the past. In this paper, various criteria of stability for discrete nonlinear autonomous comparison equations are completely established. Among them, a criterion for asymptotic stability is not only sufficient, but also necessary, from which a criterion on the function class C, is derived. Both of them can be used to determine the unexponential stability, even in the large, for discrete nonlinear (autonomous or nonautonomous) systems. All the criteria are of simple algebraic forms and can be readily used.

Normalization of Hydrodynamic Coefficients in Morison Equation

The hydrodynamic coefficients C-d and C-m are not only dependent on the size of slender cylinder, its location in water, KC number and Re number, but also vary with environmental conditions, i.e., in regular waves or in irregular waves, in pure waves or in wave-current coexisting field. In this paper, the normalization of hydrodynamic coefficients for various environmental conditions is discussed. When a proper definition of KC number and proper characteristic values of irregular waves are used, a unified relationship between C-d, C-m and KC number for regular waves, irregular waves, pure waves and wave-current coexisting field can be obtained.

TO SOLVE MATRIX EQUATION sum (A^iXBi=c) BY THE SMITH NORMAL FORM

By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation A iXB i=C over a field, and obtains the explicit formulas of general solution or unique solution.

SOLUTIONS TO A CLASS OF PARABOLIC INHOMOGENEOUS NORMALIZED p-LAPLACE EQUATIONS

In this article, we prove that viscosity solutions of the parabolic inhomogeneous equationsn＋p/put-△p^Nu=f（x,t）can be characterized using asymptotic mean value properties for all p ≥ 1, including p = 1 and p = ∞. We also obtain a comparison principle for the non-negative or non-positive inhomogeneous term f for the corresponding initial-boundary value problem and this in turn implies the uniqueness of solutions to such a problem.

On Finding Geodesic Equation of Normal Distribution and Gaussian Curvature

In this paper, we apply two different algorithms to find the geodesic equation of the normal distribution. The first algorithm consists of solving a triply partial differential equation where these equations originated from the normal distribution. While the second algorithm applies the well-known Darboux Theory. These two algorithms draw the same geodesic equation. Finally, we applied Baltzer R.’s finding to compute the Gaussian Curvature.

A Principal Theorem of Normal Discretization Schemes for Operator Equations of the First Kind

The authors announce a newly-proved theorem of theirs. This theorem is of principal significance to numerical computation of operator equations of the first kind.

Assessments of Some Simultaneous Equation Estimation Techniques with Normally and Uniformly Distributed Exogenous Variables

In each equation of simultaneous Equation model, the exogenous variables need to satisfy all the basic assumptions of linear regression model and be non-negative especially in econometric studies. This study examines the performances of the Ordinary Least Square (OLS), Two Stage Least Square (2SLS), Three Stage Least Square (3SLS) and Full Information Maximum Likelihood (FIML) Estimators of simultaneous equation model with both normally and uniformly distributed exogenous variables under different identification status of simultaneous equation model when there is no correlation of any form in the model. Four structural equation models were formed such that the first and third are exact identified while the second and fourth are over identified equations. Monte Carlo experiments conducted 5000 times at different levels of sample size (n = 10, 20, 30, 50, 100, 250 and 500) were used as criteria to compare the estimators. Result shows that OLS estimator is best in the exact identified equation except with normally distributed exogenous variables when . At these instances, 2SLS estimator is best. In over identified equations, the 2SLS estimator is best except with normally distributed exogenous variables when the sample size is small and large, and;and with uniformly distributed exogenous variables when n is very large, , the best estimator is either OLS or FIML or 3SLS.

STUDYING THE FOCAL VALUE OF ORDINARY DIFFERENTIAL EQUATIONS BY NORMAL FORM THEORY

We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.

Application of Neurocomputing in Adaptive Control of Large-Scale Aerospace Systems

We are engaged in solving two difficult problems in adaptive control of the large-scale time-variant aerospace system. One is parameter identification of time-variant continuous-time state-space modei; the other is how to solve algebraic Riccati equation (ARE) of large order efficiently. In our approach, two neural networks are employed to independently solve both the system identification problem and the ARE associated with the optimal control problem. Thus the identification and the control computation are combined in closed-loop, adaptive, real-time control system . The advantage of this approach is that the neural networks converge to their solutions very quickly and simultaneously.

Normalized fatigue properties of asphalt mixture at various temperatures

This study normalized the mixture's fatigue behavior at various temperatures,and the strength and fatigue tests of the mixture were conducted.The stress state of the asphalt mixture includes direct tensile,uniaxial compression,and indirect tensile.The Desai yield surface and fatigue path were proposed.And a normalized fatigue characteristics model of the mixture was established.The following conclusions were obtained.With the increases in the loading rate,the strength of the asphalt mixture increased.As the temperature increases,the strength of the mixture is reduced.At various temperatures and rates,the strength forms a closed curved surface.The Desai strength yield surface was established,which forms a closed curved surface.When the loading rate and temperature are below a certain critical line,the asphalt mixture will not undergo strength damage.At a fixed stress state,the fatigue damage path of the mixture was determined.The stress ratio was determined considering the influence of the loading rate.In this way,a normalized model can be described to express the asphalt mixture fatigue properties at various temperatures and stress levels.For the asphalt mixture in an indirect tensile state,the normalized fatigue equation parameter is 4.09.This model is more suitable for reflecting the viscous-elastic behavior of the mixtures than the fatigue equation determined by the notional stress ratio.

Factorized Smith Method for A Class of High-Ranked Large-Scale T-Stein Equations

We introduce a factorized Smith method(FSM)for solving large-scale highranked T-Stein equations within the banded-plus-low-rank structure framework.To effectively reduce both computational complexity and storage requirements,we develop techniques including deflation and shift,partial truncation and compression,as well as redesign the residual computation and termination condition.Numerical examples demonstrate that the FSM outperforms the Smith method implemented with a hierarchical HODLR structured toolkit in terms of CPU time.

A normalized wavefield separation cross-correlation imaging condition for reverse time migration based on Poynting vector

Prestack reverse time migration（PSTM） is a common imaging method; however low-frequency noises reduce the structural imaging precision. Thus, the suppression of migration noises must be considered. The generation mechanism of low-frequency noises is analyzed and the up-, down-, left-, and right-going waves are separated using the Poynting vector of the acoustic wave equation. The computational complexity and memory capacitance of the proposed method are far smaller than that required when using the conventional separation algorithm of 2D Fourier transform. The normalized wavefield separation crosscorrelation imaging condition is used to suppress low-frequency noises in reverse time migration and improve the imaging precision. Numerical experiments using the Marmousi model are performed and the results show that the up-, down-, left-, and right-going waves are well separated in the continuation of the wavefield using the Poynting vector. We compared the imaging results with the conventional method, Laplacian filtering, and wavefield separation with the 2D Fourier transform. The comparison shows that the migration noises are well suppressed using the normalized wavefield separation cross-correlation imaging condition and higher precision imaging results are obtained.

基于双调和插值的锥束CT金属伪影校正算法

在计算机断层扫描(CT)中,金属植入物会引入严重的伪影,导致图像质量降低影响诊断价值。为了对锥束CT中的金属伪影进行校正,提出了一种基于双调和方程的金属伪影校正算法。首先,对含金属伪影的重建图像进行双边滤波和金属阈值分割,获得金属和非金属图像;然后,对二者进行正向投影,获得金属投影区域和先验投影图像;接下来,利用先验投影图像对原始投影进行归一化,并对金属区域进行双调和方程插值修复,获得修复的投影数据;最后,对修复的投影数据去归一化,并利用FDK算法进行重建,再与金属图像融合获得最终的校正图像。为了验证该算法的性能,采用实际拍摄的数据进行金属伪影校正实验。结果表明,与常用的线性插值校正算法和归一化校正算法相比,所提算法ROI区域内图像的均方根误差分别减少了22%和8%,有效地抑制了金属伪影,优于常用的去金属伪影算法。

Kirchhoff型方程正规化解的多重性及渐近行为

该文研究了如下Kirchhoff型方程{−(a+b∫R^(3)|∇u|^(2)dx)Δu=λu+|u|^(p−2)u,x∈R^(3),∥u∥22=ρ,其中a,b,ρ>0,λ∈R是与质量约束∥u∥22=ρ有关的Lagrange乘子.当p∈(2,103)或者p∈(143,6)时,利用亏格理论证明了上述方程L2-正规化解的多重性.此外,该文证明了上述解关于参数b→0^(+)时的渐近行为.

考虑材料应变率效应的冲击响应模型相似律研究

目的研究考虑材料应变率效应的冲击响应模型相似律,获得一个能够综合材料弹塑性和应变率效应的精确相似解,支撑工程领域的模型试验。方法根据BuckinghamП定理,综合考虑材料的弹性、应变硬化和应变率效应,进一步推导了工程结构在冲击响应的模型相似律表达式。在此基础上,提出改变结构质量和冲击器速度来满足模型相似律的方法。结果相比最近冲击模型相似律的研究中存在近似相似、迭代求解的缺陷,该方法是一个显式、完全精确的相似解,并且能够同时考虑材料的弹性、应变硬化和应变率效应。采用圆板撞击的有限元数值模型进行了验证。结论推导出的模型相似律能够准确地预测原型的动态冲击的响应。相比而言,使用以往的相似方法具有明显的误差。

Normalized Solution for p-Kirchhoff Equation with a L2-supercritical Growth

In this paper,we investigate the following p-Kirchhoff equation{∫R^(N)|u|^(2)dx=ρ,(a+b)∫RN(|Δu}^(p)+|u|^(p))dx)(-Δpu+|u|^(p-2u)=|u|^(s-2)u+μu,x∈R^(N),where a>0,b≥0,p>0 are constants,constants,p*=N-P/Np is the critical Sobolev exponent,μis a Lagrange multiplier,-Δpu=-div(|Δu|_(p-2)u),2<p<N2p,μ∈R,and s∈(2N/N+2p-2,p*).We demonstratethat he p-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.