What is The Rolle’s Theorem?(1)

All the textbooks about calculus are shown the Rolle’s theorem at the first,then the Mean Value Theorem(MVT).

What is The Rolle’s Theorem?(3)

Now we will talk about howthe Rolle’s theorem works?Example 1:Find the local extremums of the function f(x)=x4-2x2on[-2,2].SOLUTION:

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Whole Perfect Vectors and Fermat’s Last Theorem

A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.

Investigation of Probability Generating Function in an Interdependent <i>M/M/</i>1:(∞;GD) Queueing Model with Controllable Arrival Rates Using Rouche’s Theorem

Present paper deals a M/M/1:(∞;GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration;which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.

Rothe’s Fixed Point Theorem and the Controllability of the Benjamin-Bona-Mahony Equation with Impulses and Delay

For many control systems in real life, impulses and delays are intrinsic phenomena that do not modify their controllability. So we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system do not destroy its controllability. There are many practical examples of impulsive control systems with delays, such as a chemical reactor system, a financial system with two state variables, the amount of money in a market and the savings rate of a central bank, and the growth of a population diffusing throughout its habitat modeled by a reaction-diffusion equation. In this paper we apply the Rothe’s Fixed Point Theorem to prove the interior approximate controllability of the following Benjamin Bona-Mohany(BBM) type equation with impulses and delay where and are constants, Ω is a domain in , ω is an open non-empty subset of Ω , denotes the characteristic function of the set ω , the distributed control , are continuous functions and the nonlinear functions are smooth enough functions satisfying some additional conditions.

Ground roll attenuation using a time-frequency dependent polarization filter based on the S transform

The ground roll and body wave usually show significant differences in arrival time, frequency content, and polarization characteristics, and conventional polarization filters that operate in either the time or frequency domain cannot consider all these elements. Therefore, we have developed a time-frequency dependent polarization filter based on the S transform to attenuate the ground roll in seismic records. Our approach adopts the complex coefficients of the S transform of the multi-component seismic data to estimate the local polarization attributes and utilizes the estimated attributes to construct the filter function. In this study, we select the S transform to design this polarization filter because its scalable window length can ensure the same number of cycles of a Fourier sinusoid, thereby rendering more precise estimation of local polarization attributes. The results of applying our approach in synthetic and real data examples demonstrate that the proposed polarization filter can effectively attenuate the ground roll and successfully preserve the body wave.

用区间套定理证明Rolle定理、Lagrange定理

Rolle定理和Lagarange定理是两个重要的微分中值定理,它们是Cauchy定理的基础,进一步为L'Hospital法则求极限提供了理论依据.它们还是研究函数增减性、凹凸性的基础.它在整个微分学中起着把微分的概念和方法应用于许多数学物理问题的桥梁作用.本文用区间套定理给出它的另一种证明.

关于罗尔(Rolle)中值定理条件的研究

罗尔中值定理指出,当函数f(x)满足三个特定条件时,在区间内部至少存在一点ξ,使得f′(ξ)=0,本文针对在区间[a,b]端点处不连续的函数以及无穷区间上的可导函数的相关问题作了进一步研究,所得结论推广和完善了文献中相应的定理。

Rolle定理的推广和Rota问题

本文将经典的Rolle定理推广到一类微分算子，并对哑演算中Rota提出的至今尚未解决的广义Taylor展开式的余项估计问题所派生的问题作了研究．

Rolle中值定理的一个新证明

本文给出了Rolle中值定理的一种新的证明方法

多项式插值与Rolle定理

本文利用多项式插值 ,对微分学和积分学中的一些典型题目 ,给出了统一的处理方法 .这种方法易于模仿 ,有一定的适用性 .

Rolle定理的推广

本文对一元函数的可微分性的Rolle中值定理进行了推广。

Rolle定理证明及应用的探讨

对Rolle定理的两种不同证明方法,即闭区间套定理、最大值与最小值定理证明进行了深入的探讨,并对定理的推广及应用作了进一步的分析研究。

关于Rolle定理的评注

Rolle定理是微分中值的基本定理 ,在微积分学中起着重要作用。关于 Rolle定理有着多种形式的推广。本文得到了如下有意义的结论。设函数 y=f( x)在区间 I的内部可导 ,在区间 I上连续 ,若在区间 I上存在三个点 x1 ,x2 ,x3 ,x1 <x2 <x3 ,使f( x2 )≥ max{ f( x1 ) ,f( x3 ) }或者 f( x2 )≤ min{ f( x1 ) ,f( x3 ) } ,则在区间 I内至少存在一点ξ,使 f (ξ) =0。此结论在证明关于“ξ等式”时十分方便 ,简洁。不论在理论上 。

Rolle定理的可视化检验及证明

Rolle中值定理是重要的微分中值定理,它是Lagarange定理和Cauchy定理基础.本文先利用Maple10对罗尔定理作可视化检验然后探索用实数连续性公理证明Rolle定理.

层次化的K.Fan’s Theorem

在L-拓扑空间中,对任意LF集,我们利用层次化的K.Fan’s Theorem定义了一种层次连通并研究了其基本性质.

Rolle定理引出的反例

微分中值定理是微分学的重要理论基础,也是利用导数的局部性研究函数整体性的重要工具.Rolle定理是微分中值定理的基石,该文就Rolle定理的条件和结论进行深入分析,给出了Rolle定理条件的改变导出的多种反例.

Rolle定理的推广及应用

微分中值定理是数学分析中很重要的基本定理,在数学分析中有着广泛的应用。它是沟通函数及其导数之间的桥梁,是应用导数研究函数在某点的局部性质和在某个区间上的整体性质的重要工具。利用微分中值定理可以论证方程的根的存在问题、方程根的个数问题以及根的存在区间问题,也经常用于证明一些含有导数的等式。在形式结构上,Rolle定理是中值定理的基础,一方面它包含在其它中值定理之中,另一方面其它中值定理的证明又往往通过Rolle定理来实现,但该定理要求自变量的范围是闭区间,这就使某些问题的解决受到了限制。主要将Rolle定理推广到有限开区间和无穷区间,用两种方法进行证明,并且举例说明其应用。