Linear and sigmoidal fuzzy cognitive maps: An analysis of fixed points

Fuzzy cognitive mapping is commonly used as a participatory modelling technique whereby stakeholders create a semi-quantitative model of a system of interest. This model is often turned into an iterative map, which should (ideally) have a unique stable fixed point. Several methods of doing this have been used in the literature but little attention has been paid to differences in output such different approaches produce, or whether there is indeed a unique stable fixed point. In this paper, we seek to highlight and address some of these issues. In particular we state conditions under which the ordering of the variables at stable fixed points of the linear fuzzy cognitive map (iterated to) is unique. Also, we state a condition (and an explicit bound on a parameter) under which a sigmoidal fuzzy cognitive map is guaranteed to have a unique fixed point, which is stable. These generic results suggest ways to refine the methodology of fuzzy cognitive mapping. We highlight how they were used in an ongoing case study of the shift towards a bio-based economy in the Humber region of the UK.     

A dual problem to least fixed points

After a simple and convenient generalization of the notion of continuous functions and continuous lattices we answer the following question: when for a given element x of a complete lattice there is a least continuous function having x as a least fixed point? The minimal continuous functions having x as a least fixed point are characterized through a correspondence with maximal ascending sequences converging to x.     

A new fixed point theorem in domain theory

Natural Computing - We generalize the standard fixed point theorem in domain theory: a dcpo has the fixed point property iff the image of its least deflation has the fixed point property, provided...     

Existence and uniqueness of common fixed points for two multivalued operators in ordered metric spaces

In this paper, we discuss some new fixed point theorems for a pair of multivalued operators which satisfy weakly generalized contractive conditions. Our results are the extension and improvement of corresponding results of [J. Harjani and K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Analysis, 71 (2009) 3403–3410] and [X. Zhang, Common fixed point theorems for some new generalized contractive type mappings, J. Math. Anal. Appl. 333 (2007) 780–786]. Finally, some examples are given to illustrate the usability of our results.     

固定结构约束下并行计算关键路径不变的可扩展方法

在固定结构的需求下,并行计算无法通过规模扩展提升其计算性能。针对此类并行计算可扩展问题,分析影响可扩展性的并行任务因素及体系结构因素,采用带权图对并行任务及体系结构进行建模,并提出一种关键路径不变的可扩展方法,其核心思想是固定图结构,仅合理地调整图节点权值和边权值。通过进一步推导,得出一些关于新扩展方法的性质结论。应用网格计算模拟工具SimGrid开展的实验结果表明,所提出的扩展方法能有效解决此类扩展问题,且保持速度效率不变。     

On the Computation of Natural Observers in Discrete-Event Systems

Natural projections with the observer property have proved effective in reducing the computational complexity of nonblocking supervisory control design, and the state sizes of the resulting controllers. In this paper we present an algorithm to verify this property, or if necessary to achieve it. A natural projection is a special type of general causal reporter map; for the latter an algorithm is already known for verification and modification. This algorithm could be used to verify the observer property of a natural projection, but if the natural projection is not an observer the algorithm is not applicable to modify it to an observer. Also, while a general reporter map always admits a unique smallest refinement with the observer property, a natural projection does not. Indeed there may exist several minimal extensions to the original observable event set of a natural projection. We show that the problem of finding a minimal extension is NP-hard, but propose a polynomial-time algorithm that always finds an acceptable extension. While not guaranteed to be minimal, it is in practice often reasonably small.     

Shape sensitivity analysis using a fixed basis function finite element approach

An approach is presented for the determination of solution sensitivity to changes in problem domain or shape. A finite element displacement formulation is adopted and the point of view is taken that the finite element basis functions and grid are fixed during the sensitivity analysis; therefore, the method is referred to as a “fixed basis function” finite element shape sensitivity analysis. This approach avoids the requirement of explicit or approximate differentiation of finite element matrices and vectors and the difficulty or errors resulting from such calculations. Effectively, the sensitivity to boundary shape change is determined exactly; thus, the accuracy of the solution sensitivity is dictated only by the finite element mesh used. The evaluation of sensitivity matrices and force vectors requires only modest calculations beyond those of the reference problem finite element analysis; that is, certain boundary integrals and reaction forces on the reference location of the moving boundary are required. In addition, the formulation provides the unique family of element domain changes which completely eliminates the inclusion of grid sensitivity from the shape sensitivity calculation. The work is illustrated for some one-dimensional beam problems and is outlined for a two-dimensional C⁰ problem; the extension to three-dimensional problems is straight-forward. Received December 5, 1999?Revised mansucript received July 6, 2000     

Least fixed points revisited

Parameter mechanisms for recursive procedures are investigated. Contrary to the view of Manna et al..it is argued that both call-by-value and call-by-name mechanisms yield the least fixed points of the functionals determined by the bodies of the procedures concerned. These functionals differ, however, according to the mechanism chosen. A careful and detailed presentation of this result is given, along the lines of a simple typed lambda calculus, with interpretation rules modelling program execution in such a way that call-by-value determines a change in the environment and call-by-name a textual substitution in the procedure body.     

三类不动点与一类随机动力系统的稳定性

不动点理论已被成功地应用于随机动力系统零解稳定性的研究,但Krasnoselskii不动点方法使用的较少.本文在采用Banach和Schauder不动点方法研究的基础上进一步采用Krasnoselskii不动点方法研究了一类随机动力系统零解的指数均方稳定性,得出了使得该系统零解指数均方稳定的充分条件.通过实例与现有文献结论的比较表明,相比于Banach和Schauder等不动点方法,Krasnoselskii不动点方法的应用更加灵活和简便.本文的结论在一定程度上改进和拓展了相关文献的结果,完善了不动点理论在研究随机动力系统零解稳定性上的应用.     

Positive fixed points for nonlinear operators

This paper is concerned with the existence and uniqueness of positive fixed points for nonlinear decreasing operators with sublinear perturbation operators in ordered Banach spaces. A new fixed-point theorem, without the assumption of compactness and continuity of operators, is established, and an iterative sequence yielding the fixed point is also given. In addition, an application to nonlinear integral equations is shown.     

Some ordered fixed point results and the property (P)

In 2010, Kadelburg et al. ([7]) by providing an example showed that a contraction in an ordered metric space is not necessarily a contraction (in the classical sense). Thus fixed point results in ordered metric spaces are generalizations of ones in metric spaces in a sense. In this paper, we give some ordered fixed point results for convex contractions and special mappings which satisfy some contraction conditions. Also, we give some results concerning the property (P).     

A Mezei-Wright theorem for categorical algebras

The main result of this paper is a generalization of the Mezei-Wright theorem, a result on solutions of a system of fixed point equations. In the typical setting, one solves a system of fixed point equations in an algebra equipped with a suitable partial order; there is a least element, suprema of ω-chains exist, the operations preserve the ordering and least upper bounds of ω-chains. In this setting, one solution of this kind of system is provided by least fixed points. The Mezei-Wright theorem asserts that such a solution is preserved by a continuous, order preserving algebra homomorphism.In several settings such as (countable) words or synchronization trees there is no well-defined partial order but one can naturally introduce a category by considering morphisms between the elements. The generalization of this paper consists in replacing ordered algebras by “categorical algebras”; the least element is replaced by an initial element, and suprema of ω-chains are replaced by colimits of ω-diagrams. Then the Mezei-Wright theorem for categorical algebras is that initial solutions are preserved by continuous morphisms. We establish this result for initial solutions of parametric fixed point equations.One use of the theorem is to characterize an “algebraic” element as one that can arise as a solution of some system of fixed point equations. In familiar examples, an algebraic element is one that is context-free, regular or rational. Then, if h:A→B is a continuous morphism of categorical algebras, the algebraic objects in B are those isomorphic to h-images of algebraic objects in A.     

The fixed point theory of unbounded non-determinism

This paper presents a fixed point theorem for the infinite traces model of CSP. Unlike any other model of CSP, there is no complete partial order over the infinite traces model whose fixed point theory agrees with the operational semantics (A. W. Roscoe, Oxford University Computing Laboratory Technical Monograph PRG-67, 1988). This arises from the introduction of unbounded non-determinism. However, the subset of pre-deterministic processes, that is those which describe the behaviour of a process on some run of its implementation, do form a subset for which the usual order is complete. By requiring that each CSP operator has a monotonic implementation which preserves pre-determinism, it is possible to show that all CSP operators have a least fixed point. In effect, it is the requirement that all operators have a methodical implementation.The author carried out the research for this paper on the Espirit Genesis project at the Programming Research Group, Oxford University.     

A double-sequence iteration process for fixed points of continuous pseudocontractions

Let C be a bounded closed convex nonempty subset of a (real) Hilbert space H. The idea of a double-sequence iteration is introduced, and it is proved that a Mann-type double-sequence iteration process converges strongly to a fixed point of a continuous pseudocontractive map T which maps C into C. Related results deal with the strong convergence of the iteration process to fixed points of nonexpansive maps.     

A note on the average number of RSA fixed points

Recently, Chmielowiec (2010) [3] studied an upper bound for the average number of fixed points in RSA encryption and asserted that it is on the order of (logn)² for randomly chosen RSA parameters (n,e). In this paper, we point out some error in his estimation and present detailed procedures for correct evaluation. It is shown that the expected number of RSA fixed points is in fact O((logn)³).     

An energy function-based design method for discrete hopfield associative memory with attractive fixed points

An energy function-based autoassociative memory design method to store a given set of unipolar binary memory vectors as attractive fixed points of an asynchronous discrete Hopfield network (DHN) is presented. The discrete quadratic energy function whose local minima correspond to the attractive fixed points of the network is constructed via solving a system of linear inequalities derived from the strict local minimality conditions. The weights and the thresholds are then calculated using this energy function. If the inequality system is infeasible, we conclude that no such asynchronous DHN exists, and extend the method to design a discrete piecewise quadratic energy function, which can be minimized by a generalized version of the conventional DHN, also proposed herein. In spite of its computational complexity, computer simulations indicate that the original method performs better than the conventional design methods in the sense that the memory can store, and provide the attractiveness for almost all memory sets whose cardinality is less than or equal to the dimension of its elements. The overall method, together with its extension, guarantees the storage of an arbitrary collection of memory vectors, which are mutually at least two Hamming distances away from each other, in the resulting network.     

Regional observability for distributed semi-linear hyperbolic systems

The objective of this paper is to study the notion of regional observability of hyperbolic semilinear systems. For the state reconstruction in a given subregion, we give a first approach which combines the extension of the HUM method and the fixed point techniques. The analytical case is then explored using sectorial property of the considered dynamic operator and converted to a fixed point problem. The two approaches lead to algorithms which are successfully implemented numerically and illustrated with examples and simulations.     

Generic local computation

Many problems of artificial intelligence, or more generally, many problems of information processing, have a generic solution based on local computation on join trees or acyclic hypertrees. There are several variants of this method all based on the algebraic structure of valuation algebras. A strong requirement underlying this approach is that the elements of a problem decomposition form a join tree. Although it is always possible to construct covering join trees, if the requirement is originally not satisfied, it is not always possible or not efficient to extend the elements of the decomposition to the covering join tree. Therefore in this paper different variants of an axiomatic framework of valuation algebras are introduced which prove sufficient for local computation without the need of an extension of the factors of a decomposition. This framework covers the axiomatic system proposed by Shenoy and Shafer (1990) [1]. A particular emphasis is laid on the important special cases of idempotent algebras and algebras with some notion of division. It is shown that all well-known architectures for local computation like the Shenoy–Shafer architecture, Lauritzen–Spiegelhalter and HUGIN architectures may be adapted to this new framework. Further a new architecture for idempotent algebras is presented. As examples, in addition to the classical instances of valuation algebras, semiring-based valuation algebras, Gaussian potentials and the relational algebra are presented.     

Convergence of chaotic iterative least fixed point computations

Iterative methods for computing fixed points of functions could usually be accommodated with some kinds of chaotic mechanisms for decreasing computational complexity. In this paper, chaotic iterative least fixed point computations of multidimensional functions on partially ordered sets are investigated, representative functions are devised to describe these computation processes, and based on this representation, a necessary and sufficient condition judging the convergence of all these computations is given.     

The complexity of fixed point models of trust in distributed networks

In this paper we consider the complexity of some problems arising in a fixed point model of trust in large-scale distributed systems, based on the notion of trust structures introduced by Carbone, Nielsen and Sassone; a set of trust levels with two distinct partial orderings. In the trust model, a global trust state exists as the least fixed point of a collection of local policy functions of nodes in the network.