Optimal average approximations for functions mapping in quasi-Banach spaces

In 1994, M.M. Popov [6] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the _?p${?}_p$-spaces for 0<p<1$0<p<1$, and the question arose whether such a significant result might hold in some other non-Banach spaces. In this article we completely settle the problem by proving that the fundamental theorem of calculus breaks down in the context of any non-locally convex quasi-Banach space. Our approach introduces the tool of Riemann-integral averages of continuous functions, and uses it to bring out to light the differences in behavior of their approximates in the lack of local convexity. As a by-product of our work we solve a problem raised in [1] on the different types of spaces of differentiable functions with values on a quasi-Banach space.     

On the notion of derivo-periodicity

A De Blasi-like differentiable multivalued function is shown to have a periodic derivative (i.e., to be derivo-periodic) if and only if it is a sum of a function of a continuous (single-valued) periodic function, linear function and a bounded interval (a multivalued constant). At the same time, the single-valued part is derivo-periodic a.e. in the usual sense. In the single-valued case, a characterization of a more general class of derivo-periodic ACG_∗-functions is given. Derivo-periodicity in terms of the Clarke subdifferentials and an impossibility of an almost-periodic analogy are also discussed. The obtained results are finally applied to differential equations and inclusions.     

《例说循环论证之谬误》注

指出《例说循环论证之谬误》中的第二个例子并非循环论证,并给出例子说明相关问题中更有可能产生循环论证之处。     

积分中值定理的改进

本改进了(第一)积分中值定理的结论，证明了定理中的中间值ζ属于开区间(α，b)．在将定理条件适当加强后，(第一)积分中值定理可由微分中值定理证得，揭示了它们之间的内在联系。     

Denjoy integral and Henstock-Kurzweil integral in vector lattices,I

In this paper we define the derivative and the Denjoy integral of mappings from a vector lattice to a complete vector lattice and show the fundamental theorem of calculus.      

Denjoy integral and Henstock-Kurzweil integral in vector lattices,II

In a previous paper we defined a Denjoy integral for mappings from a vector lattice to a complete vector lattice. In this paper we define a Henstock-Kurzweil integral for mappings from a vector lattice to a complete vector lattice and consider the relation between these two integrals.      

微积分学中中值定理的统一证明

由一个定理的结论,给出Lagrange中值定理,Cauchy中值定理,积分中值定理和Taylor中值定理的统一证明及一个计算待定型极限的方法.     

Solvability of mixed type operator equations

In this paper we prove the existence of monotonic solution of the mixed type operator equations. We discuss also the existence of solutions of nonlinear integral equations of fractional orders. The technique rely on the concept of measure of noncompactness and its associated Darbo fixed point theorem.     

On Pfaff systems with coefficients and their applications in differential geometry

We prove that a Pfaff system with coefficients in , p>2, in a simply-connected open subset Ω of has at least a nontrivial solution of class provided that its coefficients satisfies a compatibility condition in the distributional sense. If in addition the set Ω is connected, the Cauchy problem associated with the Pfaff system has a unique solution. An application of this result is that the fundamental theorem of surface theory holds under the assumption that the first and second fundamental forms are respectively of class and , with p>2, and satisfy together the Gauss and Codazzi–Mainardi equations in the distributional sense.     

Global monotonic solutions of multi term fractional differential equations

Based on the Leray-Schauder principle, a fixed point theorem is established to study the existence of a global monotonic solution for some multi term differential equations of fractional type. Some existence result for the inclusion problem is proved.     

关于可约代数体函数的基本定理

通常所说的代数体函数都是由不可约方程确定出来的.最近,孙与高~([1-2])研究代数体函数的四则运算时把这一要求去掉了.该文对可约方程确定出来的代数体函数进行了研究,探讨了其单值解析分支及分支点,论证了代数体函数的第一、二基本定理及对数导数引理对于可约代数体函数同样成立.     

含极限变量的定积分的极限

根据极限变量在定积分中位置不同对定积分的极限进行分类,并给出相应类型极限的求解方法。     

从积分概念的发展看数学分析基本概念的发展（待续）

全文分四部分概述了积分概念的发展历史,此为第一部分,介绍古希腊到牛顿-莱布尼兹积分学概念的生成.主要有穷竭法和不可分量的出现,牛顿和莱布尼兹的贡献,以及贝克莱关于微积分学基础的批判.     

D'Alembert's proof of the fundamental theorem of algebra

D'Alembert's proof of the fundamental theorem of algebra (FTA), the first published, is still widely misunderstood. Typical of d'Alembert, his work is bold and imaginative but in need of significant repair. The proof is examined in detail, in both the 1746 and 1754 versions, along with commentary over 250 years and recent efforts to revive d'Alembert's reputation. A particular challenge is to work with algebraic equations while avoiding dependence on the FTA itself. A repaired version is offered.