MINIMAL AND MAXIMAL FIXED POINT THEOREMS AND ITERATIVE TECHNIQUE FOR NONLINEAR OPERATORS IN PRODUCT SPACES

In this paper, we study minimal and maximal fixed point theorems and iterative technique for nonlinear operators in product spaces. As a corollary of our result, some coupled fixed point theorems are obtained, which generalize the coupled fixed point theorems obtained by Guo Da-jun and Lankshmikantham[21 and the results obtained by Lan in [4], and [6].

A Simple Method to Derive Minimal Cut Sets for a Non-coherent Fault Tree

Minimal cut sets （or prime implicants： minimal combinations of basic event conditions leading to system failure） are important information for reliability/safety analysis and design. To obtain minimal cut sets for general non-coherent fault trees, including negative basic events or multi-valued basic events, a special procedure such as the consensus rule must be applied to the results obtained by logical operations for coherent fault trees, which will require more steps and time. This paper proposes a simple method for a non-coherent fault tree, whose top event is represented as an AND combination of monotonic sub-trees. A ＂monotonic＂ sub-tree means that it does not have both positive and negative representations for each basic event. It is proven that minimal cut sets can be obtained by a conventional method for coherent fault trees. An illustrative example of a simple event tree analysis shows the detail and characteristics of the proposed method.

Approximations by Ideal Minimal Structure with Chemical Application

The theory of rough set represents a non-statistical methodology for analyzing ambiguity and imprecise information.It can be characterized by two crisp sets,named the upper and lower approximations that are used to determine the boundary region and accurate measure of any subset.This article endeavors to achieve the best approximation and the highest accuracy degree by using the minimal structure approximation space MSAS via ideal J.The novel approach(indicated by JMSAS)modifies the approximation space to diminish the bound-ary region and enhance the measure of accuracy.The suggested method is more accurate than Pawlak’s and EL-Sharkasy techniques.Via illustrated examples,several remarkable results using these notions are obtained and some of their properties are established.Several sorts of near open(resp.closed)sets based on JMSAS are studied.Furthermore,the connections between these assorted kinds of near-open sets in JMSAS are deduced.The advantages and disadvan-tages of the proposed approach compared to previous ones are examined.An algorithm using MATLAB and a framework for decision-making problems are verified.Finally,the chemical application for the classification of amino acids(AAs)is treated to highlight the significance of applying the suggested approximation.

A minimal axiom group for rough set based on quasi-ordering

Rough set axiomatization is one aspect of rough set study to characterize rough set theory using dependable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. The classic rough set theory is based on equivalent relation, but rough set theory based on reflexive and transitive relation (called quasi-ordering) has wide applications in the real world. To characterize topological rough set theory, an axiom group named RT, consisting of 4 axioms, is proposed. It is proved that the axiom group reliability in characterizing rough set theory based on similar relation is reasonable. Simultaneously, the minimization of the axiom group, which requires that each axiom is an equation and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods.

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.

MINIMAL ESSENTIAL SETS AND ESSENTIAL COMPONENTS OF THE EQUILIBRIA OF PRODUCTION ECONOMIES

The purpose of this paper is to give a further study on the stability of production economies. The new results were given by considering the 'set-valued' stability of equilibria. It is proved that there exists at least one minimal essential set of equilibrium points of the economy and every minimal essential set is connected. Based on these results, it is easy to prove that there is at least one essential component of the set of equilibrium points.

Minimal feasible sets in variable resource constrained projects

To determine the reasonable resource dependent relations between activities for the purpose of exactly computing the total floats and the free floats of activities, correctly identifying critical activities and critical sequences in a project schedule with variable resource constraints, the concept of the minimal feasible set （MFS） is proposed and the properties of MFS are discussed. The methods to identify optimal MFSs and resource links are then studied. Furthermore, MFS is generalized to the situation that the preconditions of MFS are not satisfied. Contrastive results show that in establishing resource links and resolving floats, MFS is at least not inferior to other methods in all cases and is superior in most situations.

The Number of Maximal Independent Sets in Quasi-Tree Graphs and Quasi-Forest Graphs

A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex x ∈V(G) such that G −x?is a tree (respectively, forest). In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are also given.

An Alternative Proof of the Largest Number of Maximal Independent Sets in Connected Graphs Having at Most Two Cycles

G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.

Calculation of Minimal Dominating Set in Wireless Sensor Network with Host Switch-on/off

This paper proposes a simple and efficient distributed algorithm for calculating minimal dominating set in wireless sensor network. This method can avoid maintaining the connectivities between backbone hosts. Considering that the hosts in mobile networks have different characteristics, this paper proposes a method of calculating minimal dominating set with weight. The nodes can be chosen to form a minimal dominating set when the network topology changes. For the host switch on/off operation, the updating algorithm was provided. The change in the status of a hostaffects only the status of hosts in the restricted vicinity. Simulation results show that the proposed method can ensure fewer dominators but with higher weight to form the minimal dominating set and the nodes can be adaptive to the changes of network topology.

MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID

This article is concerned with the existence of maximal attractors in Hi （i = 1, 2, 4） for the compressible Navier-Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Ωn in Rn（n = 2,3）. One of the important features is that the metric spaces H（1）, H（2）, and H（4） we work with are three incomplete metric spaces, as can be seen from the constraints θ 〉 0 and u 〉 0, with θand u being absolute temperature and specific volume respectively. For any constants δ1, δ2……,δ8 verifying some conditions, a sequence of closed subspaces Hδ（4） H（i） （i = 1, 2, 4） is found, and the existence of maximal （universal） attractors in Hδ（i） （i = 1.2.4） is established.

Attributes reduct and decision rules optimization based on maximal tolerance classification in incomplete information systems with fuzzy decisions

A new approach to knowledge acquisition in incomplete information system with fuzzy decisions is proposed. In such incomplete information system, the universe of discourse is classified by the maximal tolerance classes, and fuzzy approximations are defined based on them. Three types of relative reducts of maximal tolerance classes are then proposed, and three types of fuzzy decision rules based on the proposed attribute description are defined. The judgment theorems and approximation discernibility functions with respect to them are presented to compute the relative reduct by using Boolean reasoning techniques, from which we can derive optimal fuzzy decision rules from the systems. At last, three types of relative reducts of the system and their computing methods are given.

3D Multiphase Piecewise Constant Level Set Method Based on Graph Cut Minimization

Segmentation of three-dimensional(3D) complicated structures is of great importance for many real applications.In this work we combine graph cut minimization method with a variant of the level set idea for 3D segmentation based on the Mumford-Shah model.Compared with the traditional approach for solving the Euler-Lagrange equation we do not need to solve any partial differential equations.Instead,the minimum cut on a special designed graph need to be computed.The method is tested on data with complicated structures.It is rather stable with respect to initial value and the algorithm is nearly parameter free.Experiments show that it can solve large problems much faster than traditional approaches.

Maximal sequence length of exact match between members from a gene family during early evolution

Mutation (substitution, deletion, insertion, etc.) in nucleotide acid causes the maximal sequence lengths of exact match (MALE) between paralogous members from a duplicate event to become shorter during evolution. In this work, MALE changes between members of 26 gene families from four representative species (Arabidopsis thaliana, Oryza sativa, Mus mus- culus and Homo sapiens) were investigated. Comparative study of paralogous’ MALE and amino acid substitution rate (dA<0.5) indicated that a close relationship existed between them. The results suggested that MALE could be a sound evolutionary scale for the divergent time for paralogous genes during their early evolution. A reference table between MALE and divergent time for the four species was set up, which would be useful widely, for large-scale genome alignment and comparison. As an example, de- tection of large-scale duplication events of rice genome based on the table was illustrated.

Maximal elements and generalized games involving condensing mappings in locally FC-uniform spaces and applications(Ⅰ)

First, the notions of the measure of noncompactness and condensing setvalued mappings are introduced in locally FC-uniform spaces without convexity structure. A new existence theorem of maximal elements of a family of set-valued mappings involving condensing mappings is proved in locally FC-uniform spaces. As applications, some new equilibrium existence theorems of generalized game involving condensing mappings are established in locally FC-uniform spaces. These results improve and generalize some known results in literature to locally FC-uniform spaces. Some further applications of our results to the systems of generalized vector quasi-equilibrium problems will be given in a follow-up paper.

Solving the k-Independent Sets Problem of Graphs by Gröbner Bases

The aim of this paper is to given an algebraic computational method for finding maximal independent sets as well as the independent number of an arbitrary finite graph of n vertices G by strengthening the problem of finding maximal independent sets of G to the problem of finding k-independent sets in G for. It is shown that the existence of k-independent sets in G is equivalent to the existence of solutions of a system of multivariate polynomial equations. It follows that the problem of finding k-independent sets can be realized by using Gröbner bases of polynomial ideals. Since the number of k-independent sets is finite, the triangular equations composed by Gröbner bases are easier to be solved. Consequently, the maximal independent sets and the independent number of G are obtained after solving at most n such equations. Finally, the numerical example is presented to illustrate the effectiveness of this algebraic computational method.

Error reachable set based stabilization of switched linear systems with bounded peak disturbances

This paper investigates the error reachable set based stabilization problem for a class of discrete-time switched linear systems with bounded peak disturbances under persistent dwell-time(PDT)constraint.A double-clockdependent control scheme is presented that can split the disturbed switched system into a nominal system and an error system,and assign to each system a controller scheduled by a clock.A necessary and sufficient convex stability criterion is presented for the nominal system,and is further extended to the stabilization controller design with a nominal clock.In the presence of bounded peak disturbances,another stabilization controller with an error clock is developed for the error system,with the purpose of‘‘minimizing’’the reachable set of the error system by the ellipsoidal techniques.It is demonstrated that the disturbed system is also globally exponentially stable in the sense of converging to an over approximation of the reachable set of the error system,i.e.,a union of a family of bounding ellipsoids,that can also be regarded as the cross section of a tube containing the trajectories of the disturbed system.Two numerical examples are provided to verify the effectiveness of the developed results.