L─Fuzzy拓扑空间中几种正则及关系

本文我们将介绍三种不同于文献［１］中的正则．分别记作［正则］１、［正则］２、［正则］３。并讨论它们之间以及与［１］中正则的关系，我们将证明［正则］２不仅拓宽了原正则［１］。而且还具有相应好的性质。同时还将证明［正则］３在可拓扑生成条件下与［１］中正则是等价的。     

中止规则的平均延迟时间及其应用

本文以平均延迟时间为度量，对适用于连续抽样方案的四种中止规则，即规则［Ｓ］，［Ｒ］，［Ｎ，ｃ］及［Ｒ，ｄ］的中止“速度”进行了比较。结果表明：［Ｒ］优于［Ｓ］，而［Ｎ，ｃ］与［Ｒ，ｄ］均优于［Ｒ］。这些结论及方法可被用来适当地选择中止规则，以提高连续型生产的质量控制水平     

关于Fuzzy Drazin子半群

本文引进了ＦｕｚｚｙＤｒａｚｉｎ子半群的概念,并讨论了它们的一些代数性质,改进并推广了［５］、［６］、［８］、［１０］及［１１］中的许多重要结果     

一类高阶线性微分方程解的增长性

该文研究了一类高阶线性微分方程解的增长性，推广并完善了文献［３］［４］［５］［７ ］的结果.     

关于对Fuzzy弱自反关系证明的注记

陈贻源［１］和张文修［２］分别引入了Ｆｕｚｚｙ弱自反关系的定义和有关的证明。本文在［１］和［２］给出Ｆｕｚｚｙ弱自反关系定义的基础上，对［１］和［２］的有关Ｆｕｚｚｙ弱自反关系证明提出了异议，并给出了新的证明。     

也谈费马─斯坦勒尔问题

也谈费马─斯坦勒尔问题刘凯年（重庆师范学院数学系６３００４７）文献［１］（见本刊１９９４年第３期）、［２］、［３］用不同方法解决了费马一斯坦勒尔问题．但［３］为实验解，未给出严格的数学证明，［１］、［２］的方法又比较繁难，且［１］的方法很难为中学生所...     

再谈欧氏空间的变换和线性变换

文［３］推广了文［１］～［２］中的全部定理,文［４］又推广了文［３］的全部定理,本文推广了文［４］的全部定理,指出了文［４］中的一个错误论断,并给出了线性变换的一个等价条件．为简便,文中的Ｖ表示欧氏空间,ＶＶ表示Ｖ的所有变换的集合,Ｌ（Ｖ）表示Ｖ的所...     

关于积分第一中值定理的补充说明

关于积分第一中值定理的补充说明邓波（贵州织金煤勘一七四队子校５５２１００）在数学分析教材［１］、［２］及［３］中，积分第一中值定理被叙述为：“若ｆ（ｘ）在［ａ，ｂ］上连续，ｇ（ｘ）在［ａ，ｂ］上不变号，且在［ａ，ｂ］上可积，则在［ａ，ｂ］中存在一点ξ...     

关于J.Vukman的一个问题

关于Ｊ．Ｖｕｋｍａｎ的一个问题黄允宝（杭州教育学院）本文Ｒ始终表示一个中心为Ｃ的结合环，我们将用符号［ｘ，ｙ］表示ｘｙ－ｙｘ并使用恒等式［ｘｙ，ｚ］＝［ｘ，ｚ］ｙ＋ｘ［ｙ，ｚ］，［ｘ，ｙｚ］＝［ｘ，ｙ］ｚ＋ｙ［ｘ，ｚ］．称环Ｒ是素环如果ａＲｂ＝０蕴含...     

有约束错误指定模型下的优化问题

文［１］ 利用文［3］的结果证明了一般有约束线性模型下参数的最优估计的方差协方差阵与有约束错误指定模型下最优估计的方差协方差阵间差是非负定的充分条件［２］也是必要的。更进一步，文［4］将［1］中的定理1推广到奇异线性模型上。该文的主要目的是证明了［２］的猜想在奇异线性模型中也是正确的，同时，推广了［１］中的定理2。     

Asymptotical properties of distributions of isotropic Lévy processes

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic Lévy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1.The asymptotic expressions are given in terms of the radial part of characteristic exponent $\psi$ and its derivative. In particular, when $\psi \left(\lambda \right)?\frac{\lambda }{2}{\psi }^{\prime }\left(\lambda \right)$ varies regularly, as $\frac{t\psi {\left(r^{?1}\right)}^2}{\psi \left(r^{?1}\right)?{\left(2r\right)}^{?1}{\psi }^{\prime }\left(r^{?1}\right)}\to 0$ the tail probability $–(\left|X_t\right|\ge r)$ is asymptotically equal to a constant times $t(\psi \left(r^{?1}\right)?{\left(2r\right)}^{?1}{\psi }^{\prime }\left(r^{?1}\right)).$     

Problems on matchings and independent sets of a graph

Let $G$ be a finite simple graph. For $X?V\left(G\right)$, the difference of $X$, $d\left(X\right)?\left|X\right|?\left|N\left(X\right)\right|$ where $N\left(X\right)$ is the neighborhood of $X$ and $max\{d\left(X\right):X?V\left(G\right)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d\left(X\right)$ equals the critical difference and $\ker \left(G\right)$ is the intersection of all critical sets. $\mathrm{diadem}\left(G\right)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with$d\left(S\right)>0$ if no proper subset of $S$ has positive difference.A graph $G$ is called a König–Egerváry graph if the sum of its independence number $\alpha \left(G\right)$ and matching number $\mu \left(G\right)$ equals $\left|V\left(G\right)\right|$.In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set $S$ with $d\left(S\right)>0$ is at least the critical difference of the graph.We also give a new short proof of the inequality $\left|\ker \left(G\right)\right|+\left|\mathrm{diadem}\left(G\right)\right|\le 2\alpha \left(G\right)$.A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha \left(G\right)?\mu \left(G\right)$, is proved.We also make an observation about $\ker \left(G\right)$ using Edmonds–Gallai Structure Theorem as a concluding remark.     

Decomposability index of tournaments

Given a tournament $T$, a module of $T$ is a subset $X$ of $V\left(T\right)$ such that for $x,y\in X$ and $v\in V\left(T\right)?X$, $(x,v)\in A\left(T\right)$ if and only if $(y,v)\in A\left(T\right)$. The trivial modules of $T$ are $?$, $\left\{u\right\}$ $(u\in V\left(T\right))$ and $V\left(T\right)$. The tournament $T$ is indecomposable if all its modules are trivial; otherwise it is decomposable. The decomposability index of $T$, denoted by $\delta \left(T\right)$, is the smallest number of arcs of $T$ that must be reversed to make $T$ indecomposable. For $n\ge 5$, let $\delta \left(n\right)$ be the maximum of $\delta \left(T\right)$ over the tournaments $T$ with $n$ vertices. We prove that $?\frac{n+1}{4}?\le \delta \left(n\right)\le ?\frac{n?1}{3}?$ and that the lower bound is reached by the transitive tournaments.     

Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution $(Y^{t,\xi },Z^{t,\xi },H^{t,\xi })$, $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ of the split equations are proved. The first and the second order derivatives of the process $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ with respect to $x$, the derivative of the process $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ with respect to the measure $P_{\xi }$, and the derivative of the process $({?}_{\mu }{Y}^{t,x,P_{\xi }}\left(y\right),{?}_{\mu }{Z}^{t,x,P_{\xi }}\left(y\right),{?}_{\mu }{H}^{t,x,P_{\xi }}\left(y\right))$ with respect to $y$ are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in $L^2$. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function $V(t,x,P_{\xi })?Y_t^{t,x,P_{\xi }}$ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.