Function S-rough sets and mining-discovery of rough law in systems

Function S-rough sets （function singular rough sets） is defined on a -function equivalence class [u]. Function S-rough sets is the extension form of S-rough sets. By using the function S-rough sets, this paper gives rough law generation model of a-function equivalence class, discussion on law mining and law discovery in systems, and application of law mining and law discovery in communication system. Function S-rough sets is a new theory and method in law mining research.

On Spanning Wide Diameter of Graphs

A t-container Ct(u,v)is a set of t internally disjoint paths between two distinct vertices u and v in a graph G,i.e.,Ct(u,v)={P_(1),P_(2),···,Pt}.Moreover,if V(P_(1))∪V(P_(2))∪···∪V(Pt)=V(G)then Ct(u,v)is called a spanning t-container,denoted by C_(t)^(sc)(u,v).The length of C_(t)^(sc)(u,v)={P_(1),P_(2),···,Pt}is l(C_(t)^(sc)(u,v))=max{l(P_(i))|1≤i≤t}.A graph G is spanning t-connected if there exists a spanning t-container between any two distinct vertices u and v in G.Assume that u and v are two distinct vertices in a spanning t-connected graph G.Let D_(t)^(sc)(u,v)be the collection of all C_(t)^(sc)(u,v)’s.Define the spanning t-wide distance between u and v in G,d_(t)^(sc)(u,v)=min{l(C_(t)^(sc)(u,v))|C_(t)^(sc)(u,v)∈D_(t)^(sc)(u,v)},and the spanning t-wide diameter of G,D_(t)^(sc)(G)=max{d_(t)^(sc)(u,v)|u,v∈V(G)}.In particular,the spanning wide diameter of G is D_(κ)^(sc)(G),whereκis the connectivity of G.In the paper we provide the upper and lower bounds of the spanning wide diameter of a graph,and show that the bounds are best possible.We also determine the exact values of wide diameters of some well known graphs including Harary graphs and generalized Petersen graphs et al..

Security transmission and recognition of F-knowledge

The concept of F-knowledge is presented by employing S-rough sets. By engrafting and penetrating between the F-knowledge generated by S-rough sets and the RSA algorithm, the security transmission and recognition of multi-agent F-knowledge are proposed, which includes the security transmission of multi-agent F-knowledge with positive direction secret key and the security transmission of multi-agent F-knowledge with reverse direction secret key. Finally, the recognition criterion and the applications of F-knowledge are presented. The security of F-knowledge is a new application research direction of S-rough sets in information systems.

F-Law collision and system state recognition

Using function one direction S-rough sets （function one direction singular rough sets）, f-law and F- law and the concept of law distance and the concept of system law collided by F-law are given. Using these concepts, state characteristic presented by system law collided by F-law and recognition of these states characteristic and recognition criterion and applications are given. Function one direction S-rough sets is one of basic forms of function S-rough sets （function singular rough sets）. Function one direction S-rough sets is importance theory and is a method in studying system law collision.

ON SOME RESULTS OF YI AND THEIR APPLICATIONS IN THE FUNCTIONAL EQUATIONS

In this article, we establish some uniqueness theorems that improves some results of H. X. Yi for a family of meromorphic functions, and as applications, we give some results about the non-existence of meromorphic solutions of Fermat type functional equations.

■-law collision and system state recognition

Using dual function one direction S-rough sets, this article gives the f-law, the F-law, law distance and the concept of system law collided by the F-law. The characteristics presented by the system law collided by the F-law, the recognition of these characteristics and recognition criterion are also proposed. The dual function one direction S-rough sets is one of the basic forms of function S-rough sets. Its basic theory and application in the study of system law collision are reviewed.

Stochastic Maximum Principle for a Kind of Risk-sensitive Optimal Control Problem and Application to Portfolio Choice

在这份报纸，我们主要学习一种公事包选择问题在某些金融市场激发的一种风险敏感的最佳的控制问题。用古典凸的变化技术，我们为这种问题获得最大的原则。最大的原则的形式类似于它的风险中立的对应物。但是伴随方程和变化不平等重重地取决于风险敏感的参数。这是从风险中立的案例的主要差别之一。我们使用这结果解决一种最佳的公事包选择问题。鸣钟者获得的最佳的公事包策略当投资者仅仅投资家契约和股票时，动态编程原则是我们的结果的一种特殊情况。计算结果和图明确地说明在最大的期望的实用程序和模型的参数之间的关系。

Symmetric solutions of singular nonlocal boundary value problems for systems of differential equations

In this paper,we investigate the existence of symmetric solutions of singular nonlocal boundary value problems for systems of differential equations.Our analysis relies on a nonlinear alternative of Leray-schauder type.Our results presented here unify,generalize and significantly improve many known results in the literature.

Reciprocal Distance Laplacian Eigenvalue Distribution Based on Graph Parameters

Let G be a connected graph of order n and m_(RD)^(L)_(G)I denote the number of reciprocal distance Laplacian eigenvaluesof G in an interval I.For a given interval I,we mainly present several bounds on m_(RD)^(L)_(G)I in terms of various structuralparameters of the graph G,including vertex-connectivity,independence number and pendant vertices.

Function S-Rough sets and its applications

Based on S-rough sets（singular rough sets）, this paper presents function S-rough sets （function singular rough sets）and its mathematical structures and features. Function S-rough sets has two forms： function one direction S-rough sets （function one direction singular rough sets） and function two direction S-rough sets （function two direction singular rough sets）. This paper advances the relationship theorem of function S-rough sets and S-rough sets. Function S-rough sets is the general form of S-rough sets, and S-rough sets is the special ease of function S-rough sets. In this paper, applications of function S-rough sets in rough law mining-discovery of system are given. Function S-rough sets is a new research direction of rough sets and rough system.

The Maximum Principle for Fully Coupled Forward-backward Stochastic Control System

The maximum principle for fully coupled forward-backward stochastic control system in the global form is proved, under the assumption that the forward diffusion coefficient does not contain the control variable, but the control domain is not necessarily convex.

F-interference law generation and its feature recognition

By using two-directional S-rough sets, the concepts of （f, f）-interference generation and （f, f）- interference law generation of knowledge, F-interference generation and F-interference law generation of two-directional S-rough sets are proposed. Based on the concepts above, the relation theoreras between F-interference loss and F-interference degree, the relation theorems between F-interference loss law and F-interference degree law, the dis- cernibility theorems between F-interference and F-interference law are presented. At last, the recognition criterion of F-interference law and its application are given.

Simulated annealing algorithm for the optimal translation sequence of the jth agent in rough communication

In rough communication, because each agent has a different language and cannot provide precise communication to each other, the concept translated among multi-agents will loss some information and this results in a less or rougher concept. With different translation sequences, the problem of information loss is varied. To get the translation sequence, in which the jth agent taking part in rough communication gets maximum information, a simulated annealing algorithm is used. Analysis and simulation of this algorithm demonstrate its effectiveness.

THE OPTIMAL STRATEGY FOR INSURANCE COMPANY UNDER THE INFLUENCE OF TERMINAL VALUE

This paper considers a model of an insurance company which is allowed to invest a risky asset and to purchase proportional reinsurance. The objective is to find the policy which maximizes the expected total discounted dividend pay-out until the time of bankruptcy and the terminal value of the company under liquidity constraint. We find the solution of this problem via solving the problem with zero terminal value. We also analyze the influence of terminal value on the optimal policy.

S-Rough communication and its characteristics

In view of certain defects of common rough communication, using the S-rough sets, this article presents a S-rough communication model. The S-rough communication model is the extension of the common rough communication model. S-rough communication has two kinds of forms： one-direction S-rough communication and two-direction S-rough communication. The mathematical structure and characteristics of the one-direction S-rough communication and the two-direction S-rough communication, the relationship theorem between the one-direction S-rough communication and the two-direction S-rough communication are also presented. The S-rough communication is a dynamic communication method, and it is a novel research direction in rough sets field.

Application of rough graph in relationship mining

Based on the definition of class shortest path in weighted rough graph, class shortest path algorithm in weighted rough graph is presented, which extends classical shortest path algorithm. The application in relationship mining shows effectiveness of it.

Optimal proportional reinsurance and dividend payments with transaction costs and internal competition

We study the dividend optimization problem for an insurance company under the consideration of internal competition between different units inside company and transaction costs when dividends occur. The management of the company controls the reinsurance rate, the timing and the amount of dividends paid out to maximize the expected total dividends paid out to the shareholders until ruin time. By solving the corresponding quasi-variational inequality, we obtain the optimal return function and the optimal strategy.

Some Further Results on the Unique Range Sets of Meromorphic Functions

This paper studies the problem of uniqueness of meromorphic functions which share three common sets with weight, and improves a result of M. L. Fang.

On the Formation of Abstract Prime Number Theorem

In this paper we prove a zero-free region for L-functions LG（z,Х）. As an application, an abstract prime number theorem with sharp error-term for formations is established.

New judging model of fuzzy cluster optimal dividing based on rough sets theory

To investigate the judging problem of optimal dividing matrix among several fuzzy dividing matrices in fuzzy dividing space, correspondingly, which is determined by the various choices of cluster samples in the totality sample space, two algorithms are proposed on the basis of the data analysis method in rough sets theory： information system discrete algorithm （algorithm 1） and samples representatives judging algorithm （algorithm 2）. On the principle of the farthest distance, algorithm 1 transforms continuous data into discrete form which could be transacted by rough sets theory. Taking the approximate precision as a criterion, algorithm 2 chooses the sample space with a good representative. Hence, the clustering sample set in inducing and computing optimal dividing matrix can be achieved. Several theorems are proposed to provide strict theoretic foundations for the execution of the algorithm model. An applied example based on the new algorithm model is given, whose result verifies the feasibility of this new algorithm model.