Fuzzy映射与F基数

本文定义了从一个Ｆｕｚｚｙ集到另外一个Ｆｕｚｚｙ集的映射，称之为Ｆｕｚｚｙ映射，它不同于以往人们习惯用的“模糊映射”；给出了Ｆｕｚｚｙ映射的等价条件并研究了Ｆｕｚｚｙ映射的性质；基于这样的Ｆｕｚｚｙ映射定义了Ｆｕｚｚｙ映集的基数简称为Ｆ基数，讨论了它的基本性质；最后说明了Ｆ基数对于连续统假设的影响。     

提取合并同位积数法（续）

基数加减定数法中，包括有基数加定法和基数减定法两种，这两种方法，只适应于乘6或乘6以后的数，否则，效果不佳，所以，本节中的基数加定法和基数减定法都从乘6开始。     

结构Q_kλ上的优序关系

我们构造了结构Ｑｋλ上的优序关系，．证明了：若Ｑκλ有弱划分性质，则（１）对Ｑκλ上的任何优序，ＬＰ都相同；（２）κ是弱紧基数     

广义循环Fuzzy矩阵半群的格林关系等价类

研究了广义循环Fuzzy矩阵半群Cn（F）上的格林关系．得到的主要结果是：（1）给出了任意一个o-循环Fuzzy矩阵所在的格林关系各等价类及其基数；（2）给出任意一个，一循环Fuzzy矩阵所在的-等价类及其基数．     

大基数与Chang氏模型C

本文研究了一些基数在Ｃｈａｎｇ氏模型Ｃ中的存在性，证明发如下结果：（１）如果Ｋ是弱紧基数，则Ｋ在Ｃ中也是弱紧基数，（２）如果Ｋ是ｉｎｅｆｆａｂｌｅ基数，则Ｋ在Ｃ中也是ｉｎｅｆｆａｂｌｅ基数。（３）如果Ｋ是完全ｉｎｅｆｆａｂｌｅ基数，则Ｋ在Ｃ中也是完全ｉｎｅｆｆａｂｌｅ基数。（４）设Ｊ：Ｃ→Ｃ为初等嵌入，Ｋ为最小变动的基数，则Ｋ在Ｃ中完全ｉｎｅｆｆａｂｌｅ基数，且是完全Ｒａｍｓｅｙ基数。     

Fuzzy集的基数

关于Fuzzy集的基数，[1]中曾对有限支集的Fuzzy集以及极苛刻条件下的无限Fuzzy集给出过一种定义。但是，正如本文将要指出的那样，该定义是不合理的。本文将从研究Fuzzy映射入手，给出Fuzzy集基数的一般性定义。基于这一定义，不但得到有关基数的大部分结论，而且有其自身的特殊性。     

提取合并同位积数法(续)

第三节 基数加减定数法 基数加减定数法中,包括有基数加定法和基数减定法两种,这两种方法,只适应于乘6或乘6以后的数,否则,效果不佳,所以,本节中的基数加定法和基数减定法都从乘6开始。     

Fuzzy幂群的基数定理

文（１）提出了幂群的概念，给出了幂群中各元素是等势的基数定理，文（２）提出了Ｆｕｚｚｙ幂群的概念，但没研究其中各元素的基数问题，本文深入研究这一问题，得到了由Ｄ．Ｄｕｂｏｉｓ等在文（３）中提出的和由李洪兴等在文（４）中提出的两种Ｆｕｚｚｙ集基数形式下的Ｆｕｚｚｙ幂群的基数定理，并给出了Ｆｕｚｚｙ幂群中与基数有关的若干结果。     

定身加准基数的应用

定身加是种古老的算法，我们把这种古老的算法应用到现代珠算，来个珠脑结合，有许多数字相乘，根本用不着逐位实乘，只凭眼看用心脑算，就可很快得出答数来。方法就是定身加准基数。首先我们要了解，什么是准基数呢，凡两数相乘，如能利用数与数之间的特殊关系，利用数的运算规律和数的性质，采用     

简易珠算优选乘除法

现介绍一种简易珠算优选乘、除法。它的特点是：1、简单易学，青少年和成年人只要会记数、会并数，会九九就可以在短期内学会，容易普及推广，能大面积培训。见数识积，直观明了，易学易记，鉴定、比赛和实际工作都用得上，能够巩固；2、因数利宜，根据不同基数得积；3、适当结合复数、补数、过大商法(正、负商)；4、易教、易学、易用是优选的着眼点。     

闭模糊拟阵模糊圈的极值问题

本文主要方法是通过基本序列、导出拟阵序列和模糊集分解定理,将模糊圈的研究转化为对圈子集套和数组的研究。在闭模糊拟阵中,我们得出三个结论:以同一集合为支撑集的模糊圈的最大模糊圈总是存在;以同一子集串为圈子集套的模糊圈的最大模糊圈不一定存在。但是,找到了存在最大模糊圈的充要条件;以同一集合为支撑集的模糊圈的最小模糊圈,以同一子集串为圈子集套的模糊圈的最小模糊圈都是不存在的。但它们的最小模糊势是存在的,而且找出了计算最小模糊势的公式。我们构造了两个算法:一是构造支撑集最大模糊圈算法。通过这个算法可构造出支撑集最大模糊圈,同时计算出其最大模糊势;二是判断和构造圈子集套最大模糊圈算法。通过这个算法首先判断最大模糊圈是否存在,如果存在就可以找出圈子集套最大模糊圈同时计算出最大模糊势。     

Fuzzy cardinality and the modeling of imprecise quantification

The concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners. A systematic investigation of this notion is performed which unifies and improves previous attempts. The cardinality of a fuzzy set, viewed as a fuzzy integer, is related to scalar cardinality indices. The closely related question of the probability of a fuzzy event is dealt with. Lastly, the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized.     

Fuzzy turnover rate chance constraints portfolio model

One concern of many investors is to own the assets which can be liquidated easily. Thus, in this paper, we incorporate portfolio liquidity in our proposed model. Liquidity is measured by an index called turnover rate. Since the return of an asset is uncertain, we present it as a trapezoidal fuzzy number and its turnover rate is measured by fuzzy credibility theory. The desired portfolio turnover rate is controlled through a fuzzy chance constraint. Furthermore, to manage the portfolios with asymmetric investment return, other than mean and variance, we also utilize the third central moment, the skewness of portfolio return. In fact, we propose a fuzzy portfolio mean–variance–skewness model with cardinality constraint which combines assets limitations with liquidity requirement. To solve the model, we also develop a hybrid algorithm which is the combination of cardinality constraint, genetic algorithm, and fuzzy simulation, called FCTPM.     

Fuzzy cardinality based evaluation of quantified sentences

Quantified statements are used in the resolution of a great variety of problems. Several methods have been proposed to evaluate statements of types I and II. The objective of this paper is to study these methods, by comparing and generalizing them. In order to do so, we propose a set of properties that must be fulfilled by any method of evaluation of quantified statements, we discuss some existing methods from this point of view and we describe a general approach for the evaluation of quantified statements based on the fuzzy cardinality and fuzzy relative cardinality of fuzzy sets. In addition, we discuss some concrete methods derived from the mentioned approach. These new methods fulfill all the properties proposed and, in some cases, they provide an interpretation or generalization of existing methods.     

Generalized fuzzy matrices

We give a systematic development of fuzzy matrix theory. Many of our results generalize to matrices over the two element Boolean algebra, over the nonnegative real numbers, over the nonnegative integers, and over the semirings, and we present these generalizations. Our first main result is that while spaces of fuzzy vectors do not have a unique basis in general they have a unique standard basis, and the cardinality of any two bases are equal. Thus concepts of row and column basis, row and column rank can be defined for fuzzy matrices. Then we study Green's equivalence classes of fuzzy matrices. New we give criteria for a fuzzy matrix to be regular and prove that the row and column rank of any regular fuzzy matrix are equal. Various inverses are also studied. In the next section, we obtain bounds for the index and period of a fuzzy matrix.     

Questions of cardinality of finite fuzzy sets

In this paper we focus our attention on finite fuzzy sets. A complete, simple and easily applicable cardinality theory for them is presented. Questions of equipotency and non-classically understood cardinal numbers of finite fuzzy sets are discussed in detail. Also, problems of arithmetical operations (addition, subtraction, multiplication, division, and exponentiation) on as well as ordering relation for those cardinals are carefully investigated.     

Fuzzy幂群的基数及表示

研究Fuzzy幂群^[2]中元素的基数,证明这种Fuzzy幂群并非Zadeh意义的Fuzzy子群,而且可以用一分明群表示,给出有限群上Fuzzy幂群的正则表示,并讨论Fuzzy幂群的同态性质。     

Prototypical knowledge for interpreting fuzzy concepts and quantifiers

The ability to understand truly natural language expressions which involve fuzzy concepts and quantifiers (like many, few, most, etc.) presents many problems, some of which are worth mentioning: cardinality of a fuzzy set, extensions of the classical syllogisms to fuzzy syllogisms, dispositions, etc. Apart from these problems, which have been discussed in the literature, the main difficulty in evaluating such expressions is the strong interaction between the definition of the fuzzy concept and the domain knowledge.In this paper we will try to make such a claim apparent and describe some initial solutions, which provide an intelligent system with the capability of representing and understanding fuzzy concepts and quantifiers by taking into account domain knowledge.     

Cardinality of fuzzy sets via bags

We discuss the concept of a bag. We then investigate the use of these structures for the representation of cardinality of a fuzzy set.     

Fuzzy portfolio selection model with real features and different decision behaviors

In the ever changing financial markets, investor’s decision behaviors may change from time to time. In this paper, we consider the effect of investor’s different decision behaviors on portfolio selection in fuzzy environment. We present a possibilistic mean-semivariance model for fuzzy portfolio selection by considering some real investment features including proportional transaction cost, fixed transaction cost, cardinality constraint, investment threshold constraints, decision dependency constraints and minimum transaction lots. To describe investor’s different decision behaviors, we characterize the return rates on securities by LR fuzzy numbers with different shape parameters in the left- and right-hand reference functions. Then, we design a novel hybrid differential evolution algorithm to solve the proposed model. Finally, we provide a numerical example to illustrate the application of our model and the effectiveness of the designed algorithm.