A piecewise-linearized algorithm based on the Krylov subspace for solving stiff ODEs

Numerical methods for solving Ordinary Differential Equations (ODEs) have received considerable attention in recent years. In this paper a piecewise-linearized algorithm based on Krylov subspaces for solving Initial Value Problems (IVPs) is proposed. MATLAB versions for autonomous and non-autonomous ODEs of this algorithm have been implemented. These implementations have been compared with other piecewise-linearized algorithms based on Padé approximants, recently developed by the authors of this paper, comparing both precisions and computational costs in equal conditions. Four case studies have been used in the tests that come from stiff biology and chemical kinetics problems. Experimental results show the advantages of the proposed algorithms, especially when the dimension is increased in stiff problems.     

多复变全纯映射正规族关于连续移动超平面的一个充要条件

证明了从C~n的一个区域到P~N(C)中的多复变全纯映射族为正规族的一个充要条件.这个充要条件与P~N(C)中的逐点处于一般位置的连续移动超平面族有关.证明的主要方法来自于微分方程.     

一阶线性模糊微分方程的解

利用文[8]中的扩展原理求解了一阶线性模糊微分方程,给出了解可表示的条件,讨论了同其他求解方法之间的关系,最后给出了具体算例。     

有关中值定理的探究性教学

通过对中值定理教学思路的设计,给出探究性教学方法的一个实例,即通过导数概念的物理意义导出Lagrange中值定理,经特殊化后推出Rolle定理,再经化归思想给出Lagrange定理的证明,最后推广得到Cauchy中值定理,并借助类比或化归思想分别给出Cauchy定理的证明．     

Taylor expansions of the value function associated with a bilinear optimal control problem

A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton–Jacobi–Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker–Planck equation is also provided.     

Student Quantifications as Meanings for Quantified Variables in Complex Mathematical Statements

This study investigates undergraduate students’ meanings for quantified variables in mathematical statements involving multiple quantifiers. Clinical interviews with nine undergraduate students were conducted to explore students’ meanings for quantified variables. More specifically, students were asked to interpret and evaluate the Intermediate Value Theorem (IVT) and three other statements with similar logical structure to the IVT. In this paper, we provide our definitions for a quantified variable and student quantification as theoretical constructs from a constructivist viewpoint. Using these theoretical constructs, we interpret our main results, five categories of students’ meanings for quantified variables, which emerged from our qualitative analysis. We also discuss how our findings of students’ various meanings for quantified variables contribute to literature on theories of meaning and address issues in curriculum and instruction.     

Partial Orderings in the Category of Matrices over the Quaternion Field

ＰａｒｔｉａｌＯｒｄｅｒｉｎｇｓｉｎｔｈｅＣａｔｅｇｏｒｙｏｆＭａｔｒｉｃｅｓｏｖｅｒｔｈｅＱｕａｔｅｒｎｉｏｎＦｉｅｌｄＺｈｕａｎｇＷａｊｉｎ（庄瓦金）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＺｈａｎｇＮｏｒｍａｌＣｏｌｌｅｇｅ，Ｆｕｊｉ...     

一类三阶拟线性发展方程的整体解

本文研究一类三阶拟线性发展方程ｕｔｔ－Δｕｔ＋ｕｔ＝ｎｉ＝１ｘｉσｉ（ｕｘｉ），（ｘ，ｔ）∈Ω×（０，Ｔ）的初边值问题，其中ΩＲｎ（ｎ≥１）为一有界域．证明了只要σｉ（ｓ）∈Ｃ１，σ′ｉ（ｓ）（ｉ＝１，…，ｎ）有界并且初始函数满足一定的条件，则上述问题存在唯一的整体弱解．     

LIPSCHITZ柱形域上具Lp,q边值的抛物方程边值问题

设Ω为Ｒｎ中—Ｌｉｐｓｃｈｉｔｚ区域，ＳＴ为柱形域的边界Ω×（０，Ｔ）本文研究了此区域上的一般二阶常系数抛物方程具Ｌｐｑ边值的抛物方程边值问题．采用Ｒ．Ｂｒｏｗｎ和Ｚ．Ｓｈｅｎ的方法，我们证明了所涉及的双层位势和单层位势算子的可逆性，从而证明了初边值问题的唯一可解性，而且这种解可由位势算子表示．     

一类半线性卷积积分微分方程的初边值问题

本文考虑具线性粘弹性杆受粘性阻尼的横振动所引出的积分微分方程的初边值问题，在某些条件下，得到了解的存在性、唯一性、稳定性和正则性的结论。