罗尔中值定理的一些新证法

给出了罗尔微分中值定理的三种新的证明方法,其中第二种很简便的方法仅依赖于大家熟知的Heine-Borel有限覆盖定理.由此可见罗尔微分中值定理可以是实数的完备性的直接推论.     

基于曲线导数的二元函数微分中值定理

给定二元函数,文献[1]定义了其在光滑曲线上的方向导数(简称为曲线导数).本文主要利用曲线导数建立二元函数的微分中值定理,比如罗尔定理,拉格朗日中值定理,柯西中值定理.这些中值定理可视作一元函数微分中值定理在二维情形的推广.     

微分中值定理的另类证明与推广

通常教科书中,微分中值定理的证明建立在罗尔(Rolle)定理之上.本文以实数连续性中的重要定理———区间套定理为依据,给出了拉格朗日微分中值定理的另类证明.此外,还给出了中值定理的若干推广形式.     

微分中值定理历史与发展

本文简述了罗尔微分中值定理、拉格朗日中值定理和柯西中值定理产生的历史背景；详细总结了这些中值定理在各种情形下的推广和进一步发展     

利用微分方程求解微分中值问题的逆向思维方法

有关微分中值定理的证明题的证题关键是构造辅助函数.为了找到构造辅助函数的通用方法,本文基于罗尔中值定理和微分方程理论,给出通过求解微分方程证明此类题型的逆向思维方法.实例表明本文提出的逆向思维方法在求证微分中值问题中具有一定的普适性.     

微分中值定理的应用及推广

介绍应用微分中值定理时,构造所需辅助函数的两种有效方法：观察法和解微分方程法。并通过变量代换法化无限为有限,将罗尔定理的应用推广到无限区间上。     

关于辅助函数的几种构造方法——谈微分中值命题的证明

构造函数法是一种重要的数学方法,在教学中有意识地培养学生掌握这种方法,对于开阔学生思路、提高分析问题和解决问题的能力有着重要的意义．在高等数学中,微分中值定理的证明就是通过构造适当辅助函数,由这个函数满足罗尔定理而得到要证的结论．本文主要介绍证明微分中值命题时常用的构造辅助函数的几种方法．一、几何直观法构造辅助函数例1（拉格朗日定理）设连续,在内可导,则存在各分析该命题条件不满足罗尔定理中从图1可见满足罗尔定理的条件,其中直线AB的函数地从而可作辅助函数证明本题．同理,对于平行于AB且过原点的直线C…     

基于罗尔定理的两个积分中值问题

以变上限积分函数为纽带,建立了微分中值问题与积分中值问题的联系,构造相应辅助多项式,应用罗尔定理,证明两个带有二阶导数的积分中值问题.     

利用行列式构造辅助函数证明微分中值命题

利用行列式的性质及行列式函数的求导公式的特点构造辅助函数，把一些典型的微分中值命题归结为罗尔定理的情形来证明。     

罗尔定理证明一类存在性问题

提出罗尔定理证明一类存在性问题的方法,采用拉格朗日中值定理或柯西中值定理来证明这类问题往往需要构造精巧的辅助函数,我们还指出了这种方法的一般性.     

Rolle's Theorem for Polynomials of Degree Four in a Hilbert Space

In an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem.     

A version of Rolle's theorem and applications

This paper deals with global injectivity of vector fields defined on euclidean spaces. Our main result establishes a version of Rolle's Theorem under generalized Palais-Smale conditions. As a consequence of this, we prove global injectivity for a class of vector fields defined on n-dimensional spaces.Research partially supported by CNPq/Brazil under grant # 307014/89-4Research partially supported by CNPq/Brazil under grant # 301251/78-9     

Approximate Rolle's theorems for the proximal subgradient and the generalized gradient

We establish approximate Rolle's theorems for the proximal subgradient and for the generalized gradient. We also show that an exact Rolle's theorem for the generalized gradient is completely false in all infinite-dimensional Banach spaces (even when they do not possess smooth bump functions).     

On an extension of Rolle's theorem to locally convex spaces

This work concerns the extension of a weak form of the Rolle's theorem to locally convex spaces that satisfy an axiom of separation. The result provides a condition for asserting the uniqueness of a solution to nonlinear functional equations, including nonlinear integro-differential equations. We use the extended Rolle's theorem to prove the uniqueness of a solution to a nonlinear, fractional differential equation.     

The Additivity Theorem in K-Theory

We present a method for converting Theorem B style proofs in algebraic K-theory to Theorem A style proofs and apply it to the additivity theorem.     

The affine separation problem revisited

The aim of this note is to characterize in terms of inequalities those pairs of real functions (acting on a convex subset of a vector space) that possess an affine separator. The main result is originally due to Behrends and Nikodem. Their method is based on the Hahn–Banach Theorem and a variant of the Helly Theorem. In our approach, a direct and independent proof is presented via the Radon and the Helly Theorems.     

Fibred Frobenius theorem

We give a version of Frobenius Theorem for fibred manifolds whose proof is shorter than the “short proofs” of the classical Frobenius Theorem. In fact, what shortens the proof is the fibred form of the statement, since it permits an inductive process which is not possible from the standard statement.     

Other Proofs of Old Results

We transform the proof of the second incompleteness theorem given in [3] to a proof-theoretic version, avoiding the use of the arithmetized completeness theorem. We give also new proofs of old results: The Arithmetical Hierarchy Theorem and Tarski's Theorem on undefinability of truth; the proofs in which the construction of a sentence by means of diagonalization lemma is not needed.     

UNCONDITIONAL CAUCHY SERIES AND UNIFORM CONVERGENCE ON MATRICES

The authors obtain new characterizations of unconditional Cauchy series in termsof separation properties of subfamilies of P(N), and a generalization of the Orlicz-PettisTheorem is also obtained. New results on the uniform convergence on matrices anda new version of the Hahn-Schur summation theorem are proved. For matrices whoserows define unconditional Cauchy series, a better sufficient condition for the basicMatrix Theorem of Antosik and Swartz, new necessary conditions and a new proof ofthat theorem are given.