Results of Associated Implication Algebra on a Partial Ordered Set

Some sufficient and necessary conditions that implication algebra on a partial ordered set is associated implication algebra are obtained, and the relation between lattice H implication algebra and associated implication algebra is discussed. Also, the concept of filter is proposed with some basic properties being studied.

From Optimal Stopping Problems over Tree Sets to Optimal Stopping Problems over Partially Ordered Sets

In this paper, we discuss relations between optimal stopping problems over tree sets and partially ordered sets, prove that there is a 1-1 correspondence between them and so every optimal strategy can be obtained in the set of optimal control variables.

Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations

We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first-order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution.

Special Relationship among Decision Making Units based on Partially Ordered Set and New Evaluation and Projection Methods

This paper provides data envelopment analysis methods based on partially ordered set theory.These methods reveal the special relationships between two decision making units from the perspective of mathematical theory and offer the classification,projection and improvement methods of decision making units.It is proved that an efficient decision making unit must be a maximal element of the related poset,and the maximal element may not be efficient.For this,we introduce the concepts of minimum envelope and efficiency envelope which further reveal the special relationship among efficient and inefficient decision making units.Compared with the previous methods,this method not only reveals theoretically the complex relationship among decision making units and the causes of the ineffectiveness,but also gives a new importance and competitiveness measurement method to each decision making unit.Finally,related algorithm and examples are given for the application of these methods to complex decision making problems.

New method in information processing for maintaining an efficient dynamic ordered set

This paper investigates how to maintain an efficient dynamic ordered set of bit strings, which is an important problem in the field of information search and information processing. Generally, a dynamic ordered set is required to support 5 essential operations including search, insertion, deletion, max-value retrieval and next-larger-value retrieval. Based on previous research fruits, we present an advanced data structure named rich binary tree （RBT）, which follows both the binary-search-tree property and the digital-search-tree property. Also, every key K keeps the most significant difference bit （MSDB） between itself and the next larger value among K＇s ancestors, as well as that between itself and the next smaller one among its ancestors. With the new data structure, we can maintain a dynamic ordered set in O（L） time. Since computers represent objects in binary mode, our method has a big potential in application. In fact, RBT can be viewed as a general-purpose data structure for problems concerning order, such as search, sorting and maintaining a priority queue. For example, when RBT is applied in sorting, we get a linear-time algorithm with regard to the key number and its performance is far better than quick-sort. What is more powerful than quick-sort is that RBT supports constant-time dynamic insertion/deletion.

SOME EQUAL DIVISIONS ON ORDERED TREE SET

We say a divisien (S = SA or SB, g) is equal if it divides set S into two subsets SA and SB satisfying f (SA) = f (SB), where f is som attribute function on these sets and SA or SB need not be empty. In the present paper, we have obtained some equal divisions on set of ordered tree with n modes: two equal divisions on tree set; two equal divisions on node set; two divisions on leaf set, one is equal, another one is ''asymptotically'' equal. We conclude that, ordered tree set is on its good behavior in equal dividing.

The Counting Problem of an Order N-group of Set

In this paper, we discuss the counting prob lem of an order n-group of set (A 1,A 2,…,A n) which satisfies ∪ni=1A i=｛a 1,a 2,…,a m｝ and one of the following: (1) ∩ni=1A i=Φ; (2) ∩ni=1A i=｛b 1,b 2,…,b k｝;(3) ∩ni=1A 1｛b 1,b 2,…,b k｝; (4) A i≠Φ (i=1,2,…,k). We solve these problems by element analytical meth od.

EKELAND’S VARIATIONAL PRINCIPLE AND CARISTI’S COINCIDENCE THEOREM FOR SET-VALUED MAPPINGS IN PROBABILISTIC METRIC SPACES

By using the partial ordering method,a more general type,of Ekeland’s ariational principle and a set-valued Caristi’s coincidence theorem in probabilistic metric spaces are obtained.In addition,a direct simple proof of the equivalence between these two theorems in probabilistic metric spaces is given.

The Fractal Structures of the Sample Path of a General Subordinator and the Random Re-orderings of the Cantor Set

In this paper we have found a general subordinator, X, whose range up to time 1, X([0,1)), has similar structure as random re orderings of the Cantor set K(ω).X([0,1)) and K(ω) have the same exact Hausdorff measure function and the integal test of packing measure.

Logical Properties of Lattice Filter of Lattice Implication Algebra

Some new properties of lattice filters are presented based on the order-preserving mapping and lattice homomorphism, and two necessary and sufficient conditions for lattice filters under the chain type are given. Then, the relations between lattice filter and lattice implication algebras （LIAs）, i. e., the relations between lattice filter and LIA-filters, and the related properties are investigated. In addition, three necessary and sufficient conditions for LIA-filters are discussed. The obtained results may serve as some theoretical supports to lattice-valued logical system.

On Relationships Between Semi- antichains andUnichain Coverings in Discrete Mathematics

The paper explores the relationships between the largest cardinality of a semi antichain and the smallest cardinality of its unichain covering on the direct product space induced by two partially ordered sets,through studying on partially ordered sets. A sufficient condition under which they are equal is obtained.

Fuzzy TOPSIS method to primary crusher selection for Golegohar Iron Mine(Iran)

Selection of the crusher required a great deal of design regarding to the mine planning. Selection of suitable primary crusher from all of available primary crushers is a multi-criterion decision making(MCDM) problem. The present work explores the use of technique for order performance by similarity to ideal solution(TOPSIS) with fuzzy set theory to select best primary crusher for Golegohar Iron Mine in Iran. Gyratory, double toggle jaw, single toggle jaw, high speed roll crusher, low speed sizer, impact crusher, hammer mill and feeder breaker crushers have been considered as alternatives. Also, the capacity, feed size, product size, rock compressive strength, abrasion index and application of primary crusher for mobile plants were considered as criteria for solution of this MCDM problem. To determine the order of the alternatives, closeness coefficient is defined by calculating the distances to the fuzzy positive ideal solution(FPIS) and fuzzy negative ideal solution(FNIS). Results of our work based on fuzzy TOPSIS method show that the gyratory is the best primary crusher for the studied mine.

CLOSURE AXIOMS FOR POSET MATROIDS

An excellent introduction to the topic of poset matroids is due to M, Barnabei, G, Nicoletti and L. Pezzoli. In this paper, we extensively study the closure operators of poset matroids and obtain the closure axioms for poset matroids; thereby we can characterize poset matroids in terms of the closure axioms. Some corresponding properties of combinatorial schemes are also obtained.

Measures of Semi Noncompactness and Set Valued AM Mappings

In this paper we define measures of semi noncompactness in a locally convex topological linear space with respect to a given seminorm. Then we get a fixed point theorem for a class of condensing set valued mappings and apply it to differential inclusions.

A Generalization of Caristi's Fixed Point Theorem and Its Applications

The paper generalizes the classical Caristi＇s fixed point theorem. As an application. the classical Ekeland variational principle is generalized. In addition, it is proved that the generalized Caristi＇s fixed point theorem is equivalent to the generalized Ekeland variational principle.

DATA PREORDERING IN GENERALIZED PAV ALGORITHM FOR MONOTONIC REGRESSION

Monotonic regression （MR） is a least distance problem with monotonicity constraints induced by a partiaily ordered data set of observations. In our recent publication [In Ser. Nonconvex Optimization and Its Applications, Springer-Verlag, （2006） 83, pp. 25-33], the Pool-Adjazent-Violators algorithm （PAV） was generalized from completely to partially ordered data sets （posets）. The new algorithm, called CPAV, is characterized by the very low computational complexity, which is of second order in the number of observations. It treats the observations in a consecutive order, and it can follow any arbitrarily chosen topological order of the poset of observations. The CPAV algorithm produces a sufficiently accurate solution to the MR problem, but the accuracy depends on the chosen topological order. Here we prove that there exists a topological order for which the resulted CPAV solution is optimal. Furthermore, we present results of extensive numerical experiments, from which we draw conclusions about the most and the least preferable topological orders.

The Characterization of Finite Simple Groups,L_3(3^(2m-1))(m 2),by Their Element Orders

In this paper the following theorem is proved： Every group L3（q） for q = 3^（2m-1）（m≥2） is characterized by its set of element orders.