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.

Existence of Positive Solutions for Systems of Second-order Nonlinear Singular Differential Equations with Integral Boundary Conditions on Infinite Interval

By using cone theory and the MSnch fixed theorem combined with a monotone iterative technique, we investigate the existence of positive solutions for systems of second- order nonlinear singular differential equations with integral boundary conditions on infinite interval and establish the existence theorem of positive solutions and iterative sequence for approximating the positive solutions. The results in this paper improve some known results.

Solution of Singular Integrals in Mathematical Model of Mode I Crack near Strength Mismatched Interface

Characteristics of Mode I crack near the interface of elasticity matched but plasticity and strength mismatched materials differ from those of the crack in a homogenous body. Interface body of different strength influences the plastic or cohesive zone at the crack tip in parent body. The mathematical model for load line opening of the crack near the interface in linear elastic regime involves singular integrals. The paper presents explicit solution of these integrals with the help of Cauchy’s principal value theorem. Cases of thin and thick welds between the materials are investigated. Solutions of the integrals are well substantiated. Final results are provided in a consolidated form.

一个新的积分第二中值定理

在加强了假设条件的基础上,得到了一个新的积分第二中值定理.与原定理相比较,新的定理具有明显的几何意义,更富有数学美感.另外,说明了不同形式的积分第二中值定理的相互等价性.

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.

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.

泰勒公式的一个教学注记

文中利用行列式作工具泰勒公式的余项有多种形式,它们在不同的应用中各有所长.本文汇总了泰勒公式常见的4种余项,并论述了它们之间的关系与特点,同时给出了余项的几个新的变式,以供教学参考.

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.

“一轴双线三全育人”理念下高等数学课程思政探索与实践

该文以高等数学课程为载体,遵循“一轴双线三全育人”的理念,以立德树人为“主轴”,以“理论+技能”培养的知识线和“学科素养+使命担当”传承的思想线为“双线”进行运作,构建全员、全程、全课程育人格局。以高等数学课程体系里相应的知识单元为立足点,以最新的思政理念为引领,以“专业”+“思政”融合为目标,以“润物细无声”为标准,扎实推进高等数学课程承载思政的教学改革,为培养高素质创新型人才奠定坚实的基础。

实Clifford分析中双正则函数的一些性质

研究了含有2个变量的正则函数,所谓的双正则函数.正则函数是单复分析中全纯函数在高维空间的推广,全纯函数的经典函数理论如Morera定理,开拓定理等都可推广到正则函数,同样也可推广到双正则函数.以双正则函数的柯西积分公式为基础,得到了双正则函数的Morera定理,开拓定理,平均值定理.

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。

凸域内的平均随机弦长及其极值

为研究在不同随机意义下平面凸域内的平均随机弦长问题,以圆域、正三角形域、矩形域、正方形域为例,用定义法得到这些凸域内的各平均随机弦长。用弦幂积分及其不等式,讨论了任意凸域内5种平均随机弦长的极值问题,并建立了相应的不等式。在此基础上,提出了2个猜想。

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。

实Clifford分析中双正则函数的最大模原理

在给出了实Clifford分析中双正则函数的柯西积分公式的基础上,讨论了双正则函数的平均值定理和最大模原理以及它的一些推论.

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。

县域城乡融合发展的逻辑理路与路径选择——以英德市为例

城乡融合发展是推进高质量发展的必然要求,也是其中关键的评价指标,而其中县域作为城乡融合发展的桥头堡与突破口,在探索城乡融合发展中有着典型的示范意义。县域作为城市之尾、农村之首,一头连着城、一头带着乡,是城镇体系的重要组成部分,也是衔接城乡的重要活动载体。推进县域城乡融合有着深刻的逻辑理路及价值意涵,以研析英德市在开创县域城乡融合发展大局中的5大举措为探索路径,提取其特色经验,对推动高质量发展具有重大意义。

Clifford分析中超正则函数的几个基本定理

超正则函数是复分析中解析函数的一种推广形式.近年来,超正则函数已得到了较系统的研究.给出了有关超正则函数的几个基本定理,如开拓定理,平均值定理以及区域内的超正则函数与定义在边界上连续函数的关系等,使得对超正则函数的研究更加完善.

Theory of truth degrees of formulas in Lukasiewicz n-valued propositional logic and a limit theorem

The concept of truth degrees of formulas in Lukasiewicz n-valued propositional logic Ln is proposed. A limit theorem is obtained, which says that the truth function τ-n induced by truth degrees converges to the integrated truth function τ when n converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Lukasiewicz logic and the continuous valued Lukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.