关于复合函数求导法则的证明问题

举例阐明同济大学应用数学系所编《高等数学》第六版对复合函数求导法则的传统证法是不严格的．并给出一种严格证法．     

关于复合函数求导法则的探讨

对复合函数求导法则的证明作了讨论，并指出该法则在特殊极限求法中的应用。     

关于复合函数求导法则的探

对复合函数求导法则的证明作了讨论,并指出该法则在特殊极限求法中的应用.     

三种复合函数与导函数问题的讲解

针对学生在复合函数求导过程中容易混淆df[g(x)]/dx、f′[g(x)]和f′[g′(x)]三种类型求导的问题,通过两道典型例题,帮助学生理解df[g(x)]/dx、f′[g(x)]和f′[g′(x)]的确切含义.     

一阶线性微分方程与求导计算

利用一阶线性齐次微分方程的求解公式,建立了两类重要函数的求导公式,从而揭示了线性微分方程与函数导数之间的紧密联系.     

分段函数在分界点求导的一个方法

在求分段函数的导函数时,学生们往往借助其分界点的左右导函数的极限来讨论它在分界点的可导性.由于教材上没有相应的定理可依,学生常出一些错误.本文就此问题给予诠释.     

隐函数存在定理的新证明

给出数学分析中一元隐函数存在定理的一个新的证明,与以前的证明相比,本文给出的证明更易于理解和掌握.     

速率函数零点定理的一个新证明

本给出了速率函数零点定理的一种简捷证明方法。     

函数积求导法则的逆用技巧

借助实例说明逆用函数积求导法则解决微分方程问题及微分中值问题的方法与技巧.     

A Chain Rule for Upper Semicontinuous (CF)-Mappings

A chain rule for calculating convexificators of composite functions of the type f = h∘g, with the inner factor g being a transformation of , is proposed. The proof is based on a double application of a mean value theorem for (CF)-mappings due to V.F. Demyanov and V. Jeyakumar (see [4]), along with a stability property for the support of a certain ɛ-perturbation of (CF)-mappings.     

Proofs and surfaces

A formal sequent system dealing with Menelaus' configurations is introduced in this paper. The axiomatic sequents of the system stem from 2-cycles of Δ-complexes. The Euclidean and projective interpretations of the sequents are defined and a soundness result is proved. This system is decidable and its provable sequents deliver incidence results. A cyclic operad structure tied to this system is presented by generators and relations.     

拉格朗日中值定理的证明方法

分类总结拉格朗日中值定理的各种证明方法,并加以分析讨论,以求深化对微分中值定理的理解．     

Chain recurrence rates and topological entropy

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.     

Constructive Proofs of the Brouwer Type Coincidence Theorems

In 1 966,Schirmer[1 ] proved the following interesting Brouwer type coincidencetheorem for the n dimensional closed unit disc Dn.　 Theorem1　 Suppose that the continuous mapping f:Dn→ Dn maps the boundary Sn- 1essentially onto itself.Then for any continuous mapping g:Dn→Dn,the equationf(x) =g(x)has a solution in Dn.　 This result is closely related to Brouwer' s celebrated fixed point theorem. Someimportant generalizations and applications of it can be found in [2— 4 ] .　 In this not…     

一道极限题的四种证法

分别利用泰勒公式、不动点理论结合牛顿迭代法,给出一道关于极限问题的四种证明方法．     

Monotone Proofs of the Pigeon Hole Principle

We study the complexity of proving the Pigeon Hole Principle (PHP)in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We prove a size‐depth trade‐off upper bound for monotone proofs of the standard encoding of the PHP as a monotone sequent. At one extreme of the trade‐off we get quasipolynomia ‐size monotone proofs, and at the other extreme we get subexponential‐size bounded‐depth monotone proofs. This result is a consequence of deriving the basic properties of certain monotone formulas computing the Boolean threshold functions. We also consider the monotone sequent expressing the Clique‐Coclique Principle (CLIQUE) defined by Bonet, Pitassi and Raz [9]. We show that monotone proofs for this sequent can be easily reduced to monotone proofs of the one‐to‐one and onto PHP, and so CLIQUE also has quasipolynomia ‐size monotone proofs. As a consequence of our results, Resolution, Cutting Planes with polynomially bounded coefficients, and Bounded‐Depth Frege are exponentially separated from the monotone Gentzen Calculus. Finally, a simple simulation argument implies that these results extend to the Intuitionistic Gentzen Calculus. Our results partially answer some questions left open by P. Pudlák.     

Rule learning: Ordinal prediction based on rough sets and soft-computing

This work promotes a novel point of view in rough set applications: rough sets rule learning for ordinal prediction is based on rough graphical representation of the rules. Our approach tackles two barriers of rule learning. Unlike in typical rule learning, we construct ordinal prediction with a mathematical approach, rough sets, rather than purely rule quality measures. This construction results in few but significant rules. Moreover, the rules are given in terms of ordinal predictions rather than as unique values. This study also focuses on advancing rough sets theory in favor of soft-computing. Both theoretical and a designed architecture are presented. The features of our proposed approach are illustrated using an experiment in survival analysis. A case study has been performed on melanoma data. The results demonstrate that this innovative system provides an improvement of rule learning both in computing performance for finding the rules and the usefulness of the derived rules.     

销量预测的相对误差马尔柯夫链模糊修正模型研究

在销量预测的改进型灰色预测GM(1,1)模型中,针对传统的相对误差修正的马尔柯夫链预测法,在相对误差微小变化的情况下,存在状态跃变的不合理性,提出应用马尔柯夫链模糊模型的方法.并开展实证研究,得出改进的销量预测值为103.5095万箱的结论.     

低碳环境下供应链差异化定价与协调机制研究

以低碳和普通产品两个制造商和一个零售商组成的两级供应链系统为研究对象,探讨了供应链的差异化定价与协调机制问题。求解得到了两制造商和零售商的最优定价策略及可行的低碳产品生产成本范围。研究发现:1)分散决策的差异化定价策略无法达到供应链协调;2)在可行的低碳产品生产成本范围内,低碳产品制造商、零售商和整个供应链系统的利润都随低碳产品生产成本的增加而减少,只有普通产品制造商的利润随低碳产品生产成本的增加而增加。因此,降低低碳产品生产成本是促进低碳产品推广的关键。针对分散决策造成供应链效率损失的情况,采用Shapley值法进行协调,并给出了契约协调机制。最后,通过算例分析了消费者价格敏感性变化和低碳产品生产成本上升对最优决策和供应链利润的影响。