Attribute reduction based on generalized fuzzy evidence theory in fuzzy decision systems

Attribute reduction is viewed as an important issue in data mining and knowledge representation. This paper studies attribute reduction in fuzzy decision systems based on generalized fuzzy evidence theory. The definitions of several kinds of attribute reducts are introduced. The relationships among these reducts are then investigated. In a fuzzy decision system, it is proved that the concepts of fuzzy positive region reduct, lower approximation reduct and generalized fuzzy belief reduct are all equivalent, the concepts of fuzzy upper approximation reduct and generalized fuzzy plausibility reduct are equivalent, and a generalized fuzzy plausibility consistent set must be a generalized fuzzy belief consistent set. In a consistent fuzzy decision system, an attribute set is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy plausibility reduct. But in an inconsistent fuzzy decision system, a generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct in general.     

L.P.L. a fuzzy programming language: 1. Syntactic aspects

The L.P.L. language (Linguistic oriented Programming Language) that we shall present in this paper and in a forthcoming one is a new language aiming at an implementation of the concepts of the fuzzy sets theory. In the first paper, after an introduction to the basic concepts. we shall describe the syntactic aspects of L.P.L., i.e., the data declarations structure, the statements structure, the general structure of L.P.L. models. The second paper will deal first with the basic semantic and the semantics of logical expressions, then the semantics of basic statements as well as the semantics of the control structure of programs.     

Semantics and pragmatics of fuzzy sets and systems

A philosophical formalism of a new methodological aspect of humanistic systems design and evaluation is given. A requisite concept of context-dependency is highlighted, and some approaches to fuzzy sets and linguistics subsequently extended. It is consequently shown that a mathematical theory of pragmatic fuzzy subsets is not only conceptually possible but practically implementable in man-machine studies as well. Thus, the important context-dependent implication of the subjective nature of Zadeh's theory of fuzzy sets can be better exploited.As a pragmatic theory, the approach here is a seeming connection between pragmatism and ontology, concepts that are traditionally diametrically opposed to each other. The attitude adopted has been the equation of pragmatism and psychophysical measurements of ontological objects (noumena). Pragmatism is tacitly defined as a form of empiricism whereby linguistic constructs (i.e., linguistic-variable denotions) that represent any aspect(s) of a humanistic system are nothing more than an operational procedure used to achieve psychophysical measurements of the aspect(s). In this fashion, pragmatism would enable the contents of assertions, which are made through declarative propositions, about humanistic systems to be deciphered within relevant contexts. For pragmatism, direct sense experience provides both the meaning and the criterion of reality judgements. The context-dependent nature of the physical reality of ontological entities is, therefore, better understood pragmatically vis-a-vis the appropriate evaluative criteria and interpretative conventions.By emphasizing the distinction between a fuzzy system (naturally fuzzified) and a fuzzified mathematical structure (meta-mathematically fuzzified), the use of the standard fuzzy topologies is justified although it is conceptually possible to develop a yet more general topology or perhaps an alternative one especially in the case of a meta-mathematically fuzzified structure. However, for the express purpose of machine implementation, a naturally fuzzified system, such as a (complex) humanistic system, is more amenable to an initial test of the philosophy of pragmatic fuzzy subsets. Consequently, the fuzzy topology and spaces employed are intended to be devoid of extensive generalities, in this instance.     

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

In this paper we study extensions of the arithmetic operators +, -, ·, ÷ to the lattice of closed subintervals of the unit interval. Starting from a minimal set of axioms that these operators must fulfill, we investigate which properties they satisfy. We also investigate some classes of t-norms on which can be generated using these operators; these classes provide natural extensions of the Łukasiewicz, product, Frank, Schweizer–Sklar and Yager t-norms to .     

Optimal and perfect difference systems of sets

Difference systems of sets (DSS) were introduced in 1971 by Levenstein for the construction of codes for synchronization, and are closely related to cyclic difference families. In this paper, algebraic constructions of difference systems of sets using functions with optimum nonlinearity are presented. All the difference systems of sets constructed in this paper are perfect and optimal. One conjecture on difference systems of sets is also presented.     

Global periodicity and openness of the set of solutions for discrete dynamical systems

In this paper, we find additional conditions to be satisfied by a globally periodic discrete dynamical system, so that its good set (the set of initial conditions providing well-defined solutions) is an open set of ? ^k or ? ^k . We will pay especial attention to the rational case and several examples will be given.     

Generalized average dwell time approach to stability and input-to-state stability of hybrid impulsive stochastic differential systems

This paper is concerned with the stability properties of a class of impulsive stochastic differential systems with Markovian switching. Employing the generalized average dwell time (gADT) approach, some criteria on the global asymptotic stability in probability and the stochastic input-to-state stability of the systems under consideration are established. Two numerical examples are given to illustrate the effectiveness of the theoretical results, as well as the effects of the impulses and the Markovian switching on the systems stability.     

概率推理系统与证据理论对应方法的改进

通过研究R.Haenni的概率推理系统和D-S理论的联系,提出了一种新的构造信任势的方法,使得概率推理系统与D-S信任势相对应,有效减少了信任势焦元的数量并给出理论证明.     

Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems

In this paper, we consider the existence and multiplicity of solutions of second-order Hamiltonian systems. We propose a generalized asymptotically linear condition on the gradient of Hamiltonian function, classify the linear Hamiltonian systems, prove the monotonicity of the index function, and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear Hamiltonian systems by global analysis methods such as the Leray-Schauder degree theory, the Morse theory, the Ljusternik-Schnirelman theory, etc.     

Generalized Fenchel’s Conjugation Formulas and Duality for Abstract Convex Functions

In this paper, we present a generalization of Fenchel’s conjugation and derive infimal convolution formulas, duality and subdifferential (and ε-subdifferential) sum formulas for abstract convex functions. The class of abstract convex functions covers very broad classes of nonconvex functions. A nonaffine global support function technique and an extended sum-epiconjugate technique of convex functions play a crucial role in deriving the results for abstract convex functions. An additivity condition involving global support sets serves as a constraint qualification for the duality. Work of Z.Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.     

The optimal control of just-in-time-based production and distribution systems and performance comparisons with optimized pull systems

In just-in-time (JIT) production systems, there is both input stock in the form of parts and output stock in the form of product at each stage. These activities are controlled by production-ordering and withdrawal kanbans. This paper discusses a discrete-time optimal control problem in a multistage JIT-based production and distribution system with stochastic demand and capacity, developed to minimize the expected total cost per unit of time. The problem can be formulated as an undiscounted Markov decision process (UMDP); however, the curse of dimensionality makes it very difficult to find an exact solution. The author proposes a new neuro-dynamic programming (NDP) algorithm, the simulation-based modified policy iteration method (SBMPIM), to solve the optimal control problem. The existing NDP algorithms and SBMPIM are numerically compared with a traditional UMDP algorithm for a single-stage JIT production system. It is shown that all NDP algorithms except the SBMPIM fail to converge to an optimal control.Additionally, a new algorithm for finding the optimal parameters of pull systems is proposed. Numerical comparisons between near-optimal controls computed using the SBMPIM and optimized pull systems are conducted for three-stage JIT-based production and distribution systems. UMDPs with 42 million states are solved using the SBMPIM. The pull systems discussed are the kanban, base stock, CONWIP, hybrid and extended kanban.     

The method of lower and upper solutions for damped elastic systems in Banach spaces

In this paper, we are concerned with the initial value problem of a class of damped elastic systems in an order Banach spaces $E$. By employing the method of lower and upper solutions, we discuss the existence of extremal mild solutions between lower and upper mild solutions for such problem with the associated semigroup is equicontinuous. In addition, two examples are given to illustrate our results.     

Factoring and decomposing a class of linear functional systems

Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, over-determined, under-determined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M′, where M (resp., M′) is a module intrinsically associated with the linear functional system Ry = 0 (resp., R′z = 0). These morphisms define applications sending solutions of the system R′z = 0 to solutions of R y = 0. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a non-injective endomorphism of the module M is equivalent to the existence of a non-trivial factorization R = R₂R₁ of the system matrix R. The corresponding system can then be integrated “in cascade”. Under certain conditions, we also show that the system Ry = 0 is equivalent to a system R′z = 0, where R′ is a block-triangular matrix of the same size as R. We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system Ry = 0 to the integration of two independent systems R₁y₁ = 0 and R₂y₂ = 0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system Ry = 0, i.e., they allow us to compute an equivalent system R′z = 0, where R′ is a block-diagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in the Maple package Morphisms based on the library oremodules.     

Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems

This paper is devoted to inequalities of Lieb-Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues (λi(V))_i∈N^∗. We prove that there exists a positive constant C(γ), such that, if γ>d/2, then

(∗)     

New lower bound for the Hilbert number in piecewise quadratic differential systems

We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by $H_p(n)$ the extension of the Hilbert number to degree n piecewise polynomial differential systems, then $H_p(2)\ge 16$. As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, in the studied cases, all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center.     

A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

The concept of acombinatorial(W; P; U)-geometryfor a Coxeter groupW, a subsetPof its generating involutions and a subgroupUofWwithP  Uyields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even Δ-matroids. The concept of a (W, P)-matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes ofWandPone gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial (W; P; U)-geometries and suggest two equivalent definitions of a (W; P; U)-matroid with coefficients in a fuzzy ringK. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that (W; P; U)-matroids with coefficients encompass matroids with coefficients and Δ-matroids with coefficients.     

Generalized intersection bodies are not equivalent

In [A. Koldobsky, A functional analytic approach to intersection bodies, Geom. Funct. Anal. 10 (2000) 1507-1526], A. Koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent. The structures of these two types of generalized intersection bodies have been studied by the author in [E. Milman, Generalized intersection bodies, J. Funct. Anal. 240 (2) (2006) 530-567], providing substantial evidence for a positive answer to this question. The purpose of this note is to construct a counter-example, which provides a surprising negative answer to this question in a strong sense. This implies the existence of non-trivial non-negative functions in the range of the spherical Radon transform, and the existence of non-trivial spaces which embed in Lp for certain negative values of p.     

非线性中立型控制系统的镇定与控制

本文利用Riccati矩阵方程的对称正定解构造正定二次型V函数,运用L yapunov分解等价法,给出了非线性中立型定常控制系统的镇定与次优控制。同时得到了滞后量与非线性项的界限的估计公式。     

Generalized spherical and simplicial coordinates

Elementary trigonometric quantities are defined in l2,p analogously to that in l2,2, the sine and cosine functions are generalized for each p>0 as functions sinp and cosp such that they satisfy the basic equation p|cosp(φ)|+p|sinp(φ)|=1. The p-generalized radius coordinate of a point ξ∈Rn is defined for each p>0 as . On combining these quantities, ln,p-spherical coordinates are defined. It is shown that these coordinates are nearly related to ln,p-simplicial coordinates. The Jacobians of these generalized coordinate transformations are derived. Applications and interpretations from analysis deal especially with the definition of a generalized surface content on ln,p-spheres which is nearly related to a modified co-area formula and an extension of Cavalieri's and Torricelli's indivisibeln method, and with differential equations. Applications from probability theory deal especially with a geometric interpretation of the uniform probability distribution on the ln,p-sphere and with the derivation of certain generalized statistical distributions.