THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS

For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation hay been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.

A Note on the Mean Value Theorems

The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point P on the graph such that the tangent at P is parallel to the secant through the two endpoints. The mean value theorem for definite integrals says that the area under the function is equal to the area of a rectangle whose base is the length of the interval and height of some point Q on the graph. These two theorems have been studied and utilized extensively and they form the backbone of many important theorems in different branches of mathematics. In this note, we pose the question: for what functions do the two points P and Q always coincide? We find that the only analytic functions satisfying this condition are linear or exponential functions.

柯西中值定理的证明及其应用探索

为促进高职学生深入理解柯西中值定理的内容及应用,探索了两种证明柯西中值定理的方法,并阐释了三个中值定理之间的关系.介绍了柯西中值定理在等式证明、不等式证明、函数单调性判断及极限计算中的应用,并进一步应用柯西中值定理证明了洛必达法则、积分中值定理及泰勒定理.对柯西中值定理的证明、内涵及其在解题中的应用进行探索,为微分中值定理的学习和应用提供参考.

基于换相角预测计算的高压直流输电双桥共同换相预判方法

高压直流输电系统交流故障下出现双桥共同换相会增大换相失败的风险,提前判断双桥共同换相的发生有利于及时采取措施避免换相失败。为获取对双桥共同换相进行预判所需的高精度换相角预测计算结果,利用拉格朗日中值定理建立换相角与中值时刻的关系,并构造虚拟换相过程用于确定中值时刻。仿真结果表明,所提换相角预测计算方法具有良好的鲁棒性和精度,用于双桥共同换相预判中能有效预判双桥共同换相的发生。

几类新型中值定理中值点的渐近性

研究几类新型中值定理中值点的渐近性,利用比较函数和引理,在一定条件下,建立了几类新型中值定理中值点更广泛的渐近估计式,所得结果推广和改进了有关文献中的相应结果。

基于一阶偏导数判定多元函数极值的一个充分条件

在一元函数极值充分条件的基础上,用类比的方法给出二元函数极值的一阶充分条件,并推广到多元函数,且该充分条件适用于驻点和偏导数不存在点的判断,其证明过程只涉及到拉格朗日中值定理和偏导数的计算.

微分中值定理中思政元素的探讨

数学分析课程不仅传授数学知识,而且蕴含着丰富的思政元素。以微分中值定理教学为例,通过细致解读三大中值定理的核心内容,并结合实际问题中的应用,巧妙地融入哲学思想、价值观教育和科学家精神等思政元素。在教学实践中,通过将思想政治教育与数学分析教学的有机结合,旨在培养学生的数学思维品质,使其树立正确的世界观、人生观和价值观,同时激发学生对数学分析课程的学习兴趣和热情。

Observer design for induction motor： an approach based on the mean value theorem

In this paper, observer design for an induction motor has been investigated. The peculiarity of this paper is the synthesis of a mono-Luenberger observer for highly coupled system. To transform the nonlinear error dynamics for the induction motor into the linear parametric varying （LPV） system, the differential mean value theorem combined with the sector nonlinearity transformation has been used. Stability conditions based on the Lyapunov function lead to solvability of a set of linear matrix inequalities. The proposed observer guarantees the global exponential convergence to zero of the estimation error. Finally, the simulation results are given to show the performance of the observer design.

Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros

The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.

A Different Version of Flett's Mean Value Theorem and an Associated Functional Equation

In this paper we present a mean value theorem derived from Flett's mean value theorem. It turns out that cubic polynomials have the midpoint of the interval as their mean value point.To answer what class of functions have this property,we consider a functional equation associated with this mean value theorem.This equation is then solved in a general setting on abelian groups.

Generation of a Problem about Mean Value Theorem

This paper presents a generalized form and its application to a problem, which was proposed by F.P. Callham.

Classical state feedback controller for nonlinear systems using mean value theorem： closed Ioop-FOC of PMSM motor application

The problem of state feedback controllers for a class of Takagi-Sugeno （T-S） Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the differential mean value theorem （DMVT） and convex theory. The proposed design approach is based on the mean value theorem （MVT） to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients as linear parameter varying （LPV） systems. Using the Lyapunov theory, stability conditions are obtained and expressed in terms of linear matrix inequalities （LMIs）. The controller gains are then obtained by solving linear matrix inequalities. The effectiveness of the proposed approach for closed loop-field oriented control （CL-FOC） of permanent magnet synchronous machine （PMSM） drives is demonstrated through an illustrative simulation for the proof of these approaches. Furthermore, an extension for controller design with parameter uncertainties and perturbation performance is discussed.

Nonlinear unknown input observer using mean value theorem and genetic algorithm

In this note,we consider a new unknown input observer design for nonlinear systems.The main idea consists in determining the estimation error and mean value theorem parameters(β)to introduce them into proposed observer structure.This process is designed on the basis of mean value theorem and genetic algorithm.The stability study relies on the use of a classical quadratic Lyapunov function.The observer’s gains are determined systematically.For the validation of theoretical development proposed in this paper,we consider two practical realizations that deals with the secure communication problem.

A Bombieri-type mean-value theorem concerning exponential sums over primes

In several classical problems in number theory,like the Goldbach problem or the twinprime problem,one has to treat the exponential

平方反比有心力作用下质点椭圆轨道运动的若干均值关系

基于平方反比有心力作用下的质点的束缚态椭圆轨道运动的开普勒运动规律,以及机械能守恒律与角动量守恒定律,计算了质点的椭圆轨道运动的一些特征参量(速率、极半径、转动惯量、角速度、动能、势能等)的周期平均值,并证明了若干均值关系,讨论了2个典型的均值特征点.根据计算所得的椭圆轨道的动能、势能等特征量的周期平均值,直接证实了位力定理,最后讨论和强调了这些均值关系之于物理教学的参考意义.

一类导数不等式证明题的一般形式

给出了一类导数不等式证明题的一般形式,通过主要结果的证明统一解决了此类问题,揭示了控制导数值可以限制函数值增长的内涵.

幂指函数求导探析

幂指函数求导是常用对数求导法,除了对数求导法外,探索了偏导数求导、应用指数函数和幂函数求导等多种幂指函数求导方法,进而进一步推导出一元幂指函数和多元幂指函数的求导公式,为幂指函数求导提供参考。

广义Lagrange型和Cauchy型Taylor公式

通过引入适当的辅助函数,并以单侧导数替换双侧导数,以高阶导数替换一阶导数,证明了一类广义的Lagrange型和Cauchy型Taylor公式.所得结论既是Lagrange中值定理和Cauchy中值定理的推广,同时也是一种新型的Taylor公式.

微分方程法在中值定理问题辅助函数构造中的应用

本文基于微分方程理论结合罗尔中值定理,给出一类二阶中值问题辅助函数构造的通用思路和方法,将原本复杂的二阶中值问题转化为简单的求解微分方程问题.通过实例说明本文所提出的方法具有很高的普适性.

一种带有嵌入式复合函数不定式极限的计算方法

本文利用拉格朗日中值定理和夹逼准则相结合的方法,将一类嵌入式复合函数不定式的极限计算问题简化,并通过实例展示这一方法.