It has been shown by Sierpinski that a compact, Hausdorff, connected topological space (otherwise known as a continuum) cannot be decomposed into either a finite number of two or more disjoint, nonempty, closed sets o...It has been shown by Sierpinski that a compact, Hausdorff, connected topological space (otherwise known as a continuum) cannot be decomposed into either a finite number of two or more disjoint, nonempty, closed sets or a countably infinite family of such sets. In particular, for a closed interval of the real line endowed with the usual topology, we see that we cannot partition it into a countably infinite number of disjoint, nonempty closed sets. On the positive side, however, one can certainly express such an interval as a union of c disjoint closed sets, where c is the cardinality of the real line. For example, a closed interval is surely the union of its points, each set consisting of a single point being closed. Surprisingly enough, except for a set of Lebesgue measure 0, these closed sets can be chosen to be perfect sets, i.e., closed sets every point of which is an accumulation point. They even turn out to be nowhere dense (containing no intervals). Such nowhere dense, perfect sets are sometimes called Cantor sets.展开更多
In this paper, the authors introduce and study the concept of （1, 2）^＊-generalized closed sets with respect to an ideal in a bitopological space. Also, some characterizations and applications of（1, 2）^＊-generali...In this paper, the authors introduce and study the concept of （1, 2）^＊-generalized closed sets with respect to an ideal in a bitopological space. Also, some characterizations and applications of（1, 2）^＊-generalized closed sets are given.展开更多
In this paper, we introduce the notion of intuitionistic fuzzy α-generalized closed sets in intuitionistic fuzzy minimal structure spaces and investigate some of their properties. Further, we introduce and study the ...In this paper, we introduce the notion of intuitionistic fuzzy α-generalized closed sets in intuitionistic fuzzy minimal structure spaces and investigate some of their properties. Further, we introduce and study the concept of intuitionistic fuzzy α-generalized minimal continuous functions.展开更多
This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the F...This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.展开更多
Based on rough similarity degree of rough sets and close degree of fuzzy sets, the definitions of rough similarity degree and rough close degree of rough fuzzy sets are given, which can be used to measure the similar ...Based on rough similarity degree of rough sets and close degree of fuzzy sets, the definitions of rough similarity degree and rough close degree of rough fuzzy sets are given, which can be used to measure the similar degree between two rough fuzzy sets. The properties and theorems are listed. Using the two new measures, the method of clustering in the rough fuzzy system can be obtained. After clustering, the new fuzzy sample can be recognized by the principle of maximal similarity degree.展开更多
The purpose of this paper is to introduce the notions of <em>m</em>-asymmetric semiopen sets and <em>M</em>-asymmetric semicontinuous multifunctions defined between asymmetric sets satisfying c...The purpose of this paper is to introduce the notions of <em>m</em>-asymmetric semiopen sets and <em>M</em>-asymmetric semicontinuous multifunctions defined between asymmetric sets satisfying certain minimal conditions in the framework of bitopological spaces. Some new characterizations of <em>m</em>-asymmetric semiopen sets and <em>M</em>-asymmetric semicontinuous multifunctions will be investigated and several fundamental properties will be obtained.展开更多
Objective： To evaluate the efficiency of an implanted chip system for the control of heart rate （HR）. Methods： The HR was recorded in six conscious Sprague-Dawley （SD） rats. An implanted chip system was designed...Objective： To evaluate the efficiency of an implanted chip system for the control of heart rate （HR）. Methods： The HR was recorded in six conscious Sprague-Dawley （SD） rats. An implanted chip system was designed to regulate the HR by stimulating the right cervical vagus nerve according to the feedback of real time HR. Each rat was subjected to 30-min regulation and 30-min recovery. The change of HR during the regulation period was compared with the control. The ECG was recorded during the experiment for 24 h. Results： The ECG signals were successfully recorded during the experiment. The HR was significantly decreased during the period of regulation compared with control （-79.3 ± 34.5, P 〈 0.01, n = 6） and then recovered to normal after regulation. Conclusion： The described implanted chip system can regulate the HR to a designated set point.展开更多
In this article, under quite weaker conditions, theorems of Helly’s type for(not necessarily finite) family of partially closed half-spaces are presented in both analytic and geometric forms. Examples are also provid...In this article, under quite weaker conditions, theorems of Helly’s type for(not necessarily finite) family of partially closed half-spaces are presented in both analytic and geometric forms. Examples are also provided to show that these conditions cannot be omitted in general.展开更多
In this paper we prove three equivalent conditions of bounded closed convexset K in Banach space to have the drop and weak drop properties. We also give fourequivalent conditions of Banach space and its dual space to ...In this paper we prove three equivalent conditions of bounded closed convexset K in Banach space to have the drop and weak drop properties. We also give fourequivalent conditions of Banach space and its dual space to have the drop and weak dropproperties.展开更多
We give some extensions of Monch-Harton inequalities with respect to measures of noncompactness. As an example of the application, we obtain two existencetheorems of solutions for Cauchy problems of differential equat...We give some extensions of Monch-Harton inequalities with respect to measures of noncompactness. As an example of the application, we obtain two existencetheorems of solutions for Cauchy problems of differential equations on closed setsunder weaker compactness conditions.展开更多
We study the number of solutions N（B,F） of the diophantine equation n1n2 = n3n4,where 1 ≤ n1≤B, 1≤ n3 ≤B, n2, n4 ∈ F and F C [1, B] is a factor closed set. We study more particularly the case when F = {m = p1^...We study the number of solutions N（B,F） of the diophantine equation n1n2 = n3n4,where 1 ≤ n1≤B, 1≤ n3 ≤B, n2, n4 ∈ F and F C [1, B] is a factor closed set. We study more particularly the case when F = {m = p1^ε1…pk^εk ,εj∈ {0, 1}, 1≤ j ≤ k}, p1,… ,pk being distinct prime numbers.展开更多
文摘It has been shown by Sierpinski that a compact, Hausdorff, connected topological space (otherwise known as a continuum) cannot be decomposed into either a finite number of two or more disjoint, nonempty, closed sets or a countably infinite family of such sets. In particular, for a closed interval of the real line endowed with the usual topology, we see that we cannot partition it into a countably infinite number of disjoint, nonempty closed sets. On the positive side, however, one can certainly express such an interval as a union of c disjoint closed sets, where c is the cardinality of the real line. For example, a closed interval is surely the union of its points, each set consisting of a single point being closed. Surprisingly enough, except for a set of Lebesgue measure 0, these closed sets can be chosen to be perfect sets, i.e., closed sets every point of which is an accumulation point. They even turn out to be nowhere dense (containing no intervals). Such nowhere dense, perfect sets are sometimes called Cantor sets.
文摘In this paper, the authors introduce and study the concept of （1, 2）^＊-generalized closed sets with respect to an ideal in a bitopological space. Also, some characterizations and applications of（1, 2）^＊-generalized closed sets are given.
文摘In this paper, we introduce the notion of intuitionistic fuzzy α-generalized closed sets in intuitionistic fuzzy minimal structure spaces and investigate some of their properties. Further, we introduce and study the concept of intuitionistic fuzzy α-generalized minimal continuous functions.
基金supported by the Mathematical Tianyuan Foundation (No. 10826078)the National Natural Science Foundation of China (No. 60874006)
文摘This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.
基金the Fujian Provincial Natural Science Foundation of China (Z0510492006J0391)
文摘Based on rough similarity degree of rough sets and close degree of fuzzy sets, the definitions of rough similarity degree and rough close degree of rough fuzzy sets are given, which can be used to measure the similar degree between two rough fuzzy sets. The properties and theorems are listed. Using the two new measures, the method of clustering in the rough fuzzy system can be obtained. After clustering, the new fuzzy sample can be recognized by the principle of maximal similarity degree.
文摘The purpose of this paper is to introduce the notions of <em>m</em>-asymmetric semiopen sets and <em>M</em>-asymmetric semicontinuous multifunctions defined between asymmetric sets satisfying certain minimal conditions in the framework of bitopological spaces. Some new characterizations of <em>m</em>-asymmetric semiopen sets and <em>M</em>-asymmetric semicontinuous multifunctions will be investigated and several fundamental properties will be obtained.
基金supported by grant from National Nature Science Found (30670767)
文摘Objective： To evaluate the efficiency of an implanted chip system for the control of heart rate （HR）. Methods： The HR was recorded in six conscious Sprague-Dawley （SD） rats. An implanted chip system was designed to regulate the HR by stimulating the right cervical vagus nerve according to the feedback of real time HR. Each rat was subjected to 30-min regulation and 30-min recovery. The change of HR during the regulation period was compared with the control. The ECG was recorded during the experiment for 24 h. Results： The ECG signals were successfully recorded during the experiment. The HR was significantly decreased during the period of regulation compared with control （-79.3 ± 34.5, P 〈 0.01, n = 6） and then recovered to normal after regulation. Conclusion： The described implanted chip system can regulate the HR to a designated set point.
文摘In this article, under quite weaker conditions, theorems of Helly’s type for(not necessarily finite) family of partially closed half-spaces are presented in both analytic and geometric forms. Examples are also provided to show that these conditions cannot be omitted in general.
基金the Natural Science Foundation of China(11361042,11071108)the Natural Science Foundation of Jiangxi Province of China(2010GZS0147)the Youth Foundation of theEducation Department of Jiangxi(GJJ13012)
文摘In this paper we prove three equivalent conditions of bounded closed convexset K in Banach space to have the drop and weak drop properties. We also give fourequivalent conditions of Banach space and its dual space to have the drop and weak dropproperties.
文摘We give some extensions of Monch-Harton inequalities with respect to measures of noncompactness. As an example of the application, we obtain two existencetheorems of solutions for Cauchy problems of differential equations on closed setsunder weaker compactness conditions.
文摘We study the number of solutions N（B,F） of the diophantine equation n1n2 = n3n4,where 1 ≤ n1≤B, 1≤ n3 ≤B, n2, n4 ∈ F and F C [1, B] is a factor closed set. We study more particularly the case when F = {m = p1^ε1…pk^εk ,εj∈ {0, 1}, 1≤ j ≤ k}, p1,… ,pk being distinct prime numbers.
