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■,satisfying the normalization constraint f_(R)^N u^2dx=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.

Normalized Ground State Solutions of Nonlinear Choquard Equations with Nonconstant Potential

In this paper,we mainly focus on the following Choquard equation-{△u-V(x)(I_(a*)|u|^(p))|u|^(p-2)u=λu,x∈R^(N),u∈H^(1)(R^(N))where N≥1,λ∈R will arise as a Lagrange multiplier,0<a<N and N+a/N<p<N+a+2/N Under appropriate hypotheses on V(x),we prove that the above Choquard equation has a normalized ground state solution by utilizing variational methods.

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.

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.

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].

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.

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.

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.

Relationship between drought and soil erosion based on the normalized differential water index(NDWI)and revised universal soil loss equation(RUSLE)model

The Langat River Basin in Malaysia is vulnerable to soil erosion risks because of its exposure to intensive land use activities and its topography,which primarily consists of steep slopes and mountainous areas.Furthermore,climate change frequently exposes this basin to drought,which negatively affects soil and water conservation.However,recent studies have rarely shown how soil reacts to drought,such as soil erosion.Therefore,the purpose of this study is to evaluate the relationship between drought and soil erosion in the Langat River Basin.We analyzed drought indices using Landsat 8 satellite images in November 2021,and created the normalized differential water index(NDWI)via Landsat 8 data to produce a drought map.We used the revised universal soil loss equation(RUSLE)model to predict soil erosion.We verified an association between the NDWI and soil erosion data using a correlation analysis.The results revealed that the southern and northern regions of the study area experienced drought events.We predicted an average annual soil erosion of approximately 58.11 t/(hm^(2)·a).Analysis of the association between the NDWI and soil erosion revealed a strong positive correlation,with a Pearson correlation coefficient of 0.86.We assumed that the slope length and steepness factor was the primary contributor to soil erosion in the study area.As a result,these findings can help authorities plan effective measures to reduce the impacts of drought and soil erosion in the future.

The analysis and application of simplified two-parameter moveout equation for C-waves in VTI anisotropy media

Several parameters are needed to describe the converted-wave （C-wave） moveout in processing multi-component seismic data, because of asymmetric raypaths and anisotropy. As the number of parameters increases, the converted wave data processing and analysis becomes more complex. This paper develops a new moveout equation with two parameters for C-waves in vertical transverse isotropy （VTI） media. The two parameters are the C-wave stacking velocity （Vc2） and the squared velocity ratio （7v,i） between the horizontal P-wave velocity and C-wave stacking velocity. The new equation has fewer parameters, but retains the same applicability as previous ones. The applicability of the new equation and the accuracy of the parameter estimation are checked using model and real data. The form of the new equation is the same as that for layered isotropic media. The new equation can simplify the procedure for C-wave processing and parameter estimation in VTI media, and can be applied to real C-wave processing and interpretation. Accurate Vc2 and Yvti can be deduced from C-wave data alone using the double-scanning method, and the velocity ratio model is suitable for event matching between P- and C-wave data.

Entire solutions of a certain type of functional-differential equations

In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonlinear differential equations.

THE CONSTITUTIVE EQUATION RESEMBLANCE OF ELASTIC-PLASTIC MULTIPHASE SOLID

Based on composite materials, the equivalent elastic-plastic constitutive equations of multiphase solid are researched. According to the suggested definition of. constitutive equivalence it is demonstrated that the multiphase solid, composed of several kinds of homogeneous elastic-plastic media that conform to the generalized normality rule, has the same type of constitutive equations as its constituents have that also conform to the generalized normality rule.

New conception in continuum theory of constitutive equation for anisotropic crystalline polymer liquids

A new continuum theory of the constitutive equation of co-rotational derivative type is developed for anisotropic viscoelastic fluid—liquid crystalline (LC) polymers. A new concept of simple anisotropic fluid is introduced. On the basis of principles of anisotropic simple fluid, stress behaviour is described by velocity gradient tensor and spin tensor instead of the velocity gradient tensor in the classic Leslie—Ericksen continuum theory. Analyzing rheological nature of the fluid and using tensor analysis a general form of the constitutive equ- ation of co-rotational type is established for the fluid. A special term of high order in the equation is introduced by author to describe the sp- ecial change of the normal stress differences which is considered as a result of director tumbling by Larson et al. Analyzing the experimental results by Larson et al., a principle of Non- oscillatory normal stress is introduced which leads to simplification of the problem with relaxation times. The special behaviour of non- symmetry of the shear stress is predicted by using the present model for LC polymer liquids. Two shear stresses in shear flow of LC polymer liquids may lead to vortex and rotation flow, i.e. director tumbling in the flow. The first and second normal stress differences are calculated by the model special behaviour of which is in agree- ment with experiments. In the research, the com- putational symbolic manipulation such as computer software Maple is used. For the anisotropic viscoelastic fluid the constitutive equation theory is of important fundamental significance.

On the Order of the Solutions of Systems of Complex Algebraic Differential Equations

This paper is concerned with the order of the solutions of systems of high-order complex algebraic differential equations.By means of Zalcman Lemma,the systems of equations of[1]is extended to more general form.

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.

KAM TORI FOR DEFOCUSING KDV-MKDV EQUATION

In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u^2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.