Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

Rough Sets in Approximate Solution Space

As a new mathematical theory, Rough sets have been applied to processing imprecise, uncertain and incomplete data. It has been fruitful in finite and non-empty set. Rough sets, however, are only served as the theoretic tool to discretize the real function. As far as the real function research is concerned, the research to define rough sets in the real function is infrequent. In this paper, we exploit a new method to extend the rough set in normed linear space, in which we establish a rough set,put forward an upper and lower approximation definition, and make a preliminary research on the property of the rough set.A new tool is provided to study the approximation solutions of differential equation and functional variation in normed linear space. This research is significant in that it extends the application of rough sets to a new field.

ON INFORMATION THEORY METHOD FOR RELIABILITY ASSESSMENT OF SYSTEMS CONSISTING OF DIFFERENT SUCCESS FAILURE MODEL UNITS

Using the characteristic of addition of information quantity and the principle of equivalence of information quantity, this paper derives the general conversion formulae of the formation theory method conversion (synthesis) on the systems consisting of different success failure model units. According to the fundamental method of the unit reliability assessment, the general models of system reliability approximate lower limits are given. Finally, this paper analyses the application of the assessment method by examples, the assessment results are neither conservative nor radical and very satisfactory. The assessment method can be popularized to the systems which have fixed reliability structural models.

Uncertainty Representation of Ocean Fronts Based on Fuzzy-Rough Set Theory

Analysis of ocean fronts＇ uncertainties indicates that they result from indiscemibility of their spatial position and fuzziness of their intensity. In view of this, a flow hierarchy for uncertainty representation of ocean fronts is proposed on the basis of fuzzy-rough set theory. Firstly, raster scanning and blurring are carried out on an ocean front, and the upper and lower approximate sets, the indiscernible relation in fuzzy-rough theories and related operators in fuzzy set theories are adopted to represent its uncertainties, then they are classified into three sets： with members one hundred percent belonging to the ocean front, belonging to the ocean front＇s edge and definitely not belonging to the ocean front. Finally, the approximate precision and roughness degree are utilized to evaluate the ocean front＇s degree of uncertainties and the precision of the representation. It has been proven that the method is not only capable of representing ocean fronts＇ uncertainties, but also provides a new theory and method for uncertainty representation of other oceanic phenomena.

θ-improved limited tolerance relation model of incomplete information system for evaluation of water conservancy project management modernization

The modernization of water conservancy project management is a complicated engineering system involving a management system, a management method, management personnel, the exertion of social, economic, and ecological effects, and so on. However, indices for evaluating the modernization of water conservancy project management are usually unobtainable in practical applications. Conducting appropriate extension of the classical rough set theory and then applying it to an incomplete information system are the key to the application of the rough set theory Based on analysis of some extended rough set models in incomplete information systems, a rough set model based on the θ-improved limited tolerance relation is put forward. At the same time, upper approximation and lower approximation are defined under this improved relation. According to the evaluation index system and management practices, the threshold for θ is defined. An example study indicates the practicability and maneuverability of the model.

On uniqueness of (fuzzy) relation in some generalized rough set model

In rough set theory, crisp and/or fuzzy binary relations play an important role in both constructive and axiomatic considerations of various generalized rough sets. This paper considers the uniqueness problem of the （fuzzy） relation in some generalized rough set model. Our results show that by using the axiomatic approach, the （fuzzy） relation determined by （fuzzy） approximation operators is unique in some （fuzzy） double-universe model.

Rough Integrals Generated by Function Rough Sets

The concepts of the lower approximation integral,the upper approximation integral and rough integrals are given on the basis of function rough sets.Based on these concepts,the relation of the lower approximation integrals,the relation of the upper approximation integrals,the relation of rough integrals,and the double median theorem of rough integrals are discussed.Rough integrals have finite contraction characteristic and finite extension characteristic.

A New Minimal Rough Set Axiom Group

Rough set axiomatization is one aspect of rough set study, and the purpose is to characterize rough set theory using independable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. To characterize rough set theory, an axiom group named H consisting of 4 axioms, is proposed. That validity of the axiom group in characterizing rough set theory is reasonable, is proved. Simultaneously, the minimization of the axiom group, which requires that each axiom is an inequality and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods. Key words rough set - lower approximation - axioms - minimization CLC number TP 18 Foundation item: Supported by the 973 National Basic Research Program of China (2002CB312106) and Science & Technology Program of Zhejiang Province (2004C31G101003)Biography: DAI Jian-hua (1977-), male, Ph. D, research direction: data mining, artificial intelligence, rough sets, evolutionary computation.

Uncertainty and Reduction in Intuitionistic Fuzzy Covering Based Rough Sets

This paper proposes a novel pair of induced IF covering approximation operators in an IF covering approximation space,and discusses some basic properties about definable IFSs.A measure is defined to describe the uncertainty of IFSs in IF covering approximation spaces.Finally,we study the properties of reductions of an IF covering respectively based on induced IF covering approximation operators and IF covering approximation operators.

The Lower Bounds of Eigenvalues by the Wilson Element in Any Dimension

In this paper,we analyze the Wilson element method of the eigenvalue problem in arbitrary dimensions by combining a new technique recently developed in[10]and the a posteriori error result.We prove that the discrete eigenvalues are smaller than the exact ones.