A new approach for power system black-start decision-making with vague set theory

Power system restoration after a large area outage involves many factors, and the procedure is usually very complicated. A decision-making support system could then be developed so as to find the optimal black-start strategy. In order to evaluate candidate black-start strategies, some indices, usually both qualitative and quantitative, are employed. However, it may not be possible to directly synthesize these indices, and different extents of interactions may exist among these indices. In the existing black-start decision-making methods, qualitative and quantitative indices cannot be well synthesized, and the interactions among different indices are not taken into account. The vague set, an extended version of the well-developed fuzzy set, could be employed to deal with decision-making problems with interacting attributes. Given this background, the vague set is first employed in this work to represent the indices for facilitating the comparisons among them. Then, a concept of the vague-valued fuzzy measure is presented, and on that basis a mathematical model for black-start decision-making developed. Compared with the existing methods, the proposed method could deal with the interactions among indices and more reasonably represent the fuzzy information. Finally, an actual power system is served for demonstrating the basic features of the developed model and method.

Descriptive set theory and the ergodic theory of countable groups

The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel reducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of residually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence. One of the main results answers a question of Abért and Elek by showing that within any free weak equivalence class the isomorphism relation does not admit classification by countable structures. The proof relies on affirming a conjecture of Ioana by showing that the product of a free action with a Bernoulli shift is weakly equivalent to the original action. Chapter 5 studies the relationship between mixing and freeness properties of measure preserving actions. Chapter 6 studies how approximation properties of ergodic actions and unitary representations are reflected group theoretically and also operator algebraically via a group's reduced C^*-algebra. Chapter 7 is an appendix which includes various results on mixing via filters and on Gaussian actions.

Free algebras in Von Neumann-Bernays-Gӧdel set theory and positive elementary inductions in reasonable structures

This thesis consists of two independent chapters. The first chapter deals with universal algebra. It is shown, in von Neumann-Bernays-Gӧdel set theory, that free images of partial algebras exist in arbitrary varieties. It follows from this, as set-complete Boolean algebras form a variety, that there exist free set-complete Boolean algebras on any class of generators. This appears to contradict a well-known result of A. Hales and H. Gaifman, stating that there is no complete Boolean algebra on any infinite set of generators. However, it does not, as the algebras constructed in this chapter are allowed to be proper classes. The second chapter deals with positive elementary inductions. It is shown that, in any reasonable structure ᶆ, the inductive closure ordinal of ᶆ is admissible, by showing it is equal to an ordinal measuring the saturation of ᶆ. This is also used to show that non-recursively saturated models of the theories ACF, RCF, and DCF have inductive closure ordinals greater than _ω.

Additions and corrections to “Terminal coalgebras in well-founded set theory”

This note is to correct certain mistaken impressions of the author's that were in the original paper, “Terminal coalgebras in well-founded set theory”, which appeared in Theoretical Computer Science 114 (1993) 299–315.

Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory

In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory (including the Diagonal Argument, the Continuum Hypothesis and Cantor’s Theorem) and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing (completed) infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems.

Some Lattice Problems In Fuzzy Set Theory And Fuzzy Topology

It is believed that every fuzzy generalization should be formulated in such a way that it contain the ordinary set theoretic notion as a special case. Therefore the definition of fuzzy topology in the line of C.L.CHANG E9] with an arbitrary complete and distributive lattice as the membership set is taken. Almost all the results proved and presented in this thesis can, in a sense, be called generalizations of corresponding results in ordinary set theory and set topology. However the tools and the methods have to be in many of the cases, new. Here an attempt is made to solve the problem of complementation in the lattice of fuzzy topologies on a set. It is proved that in general, the lattice of fuzzy topologies is not complemented. Complements of some fuzzy topologies are found out. It is observed that (L,X) is not uniquely complemented. However, a complete analysis of the problem of complementation in the lattice of fuzzy topologies is yet to be found out

Fuzzy set Theory and Quantum Chemistry

Es discuteixen breument algunes consideracions sobre l'aplicació de la Teoria dels Conjunts difusos a la Química quàntica. Es demostra aqui que molts conceptes químics associats a la teoria són adequats per ésser connectats amb l'estructura dels Conjunts difusos. També s'explica com algunes descripcions teoriques dels observables quàntics es potencien tractant-les amb les eines associades als esmentats Conjunts difusos. La funció densitat es pren com a exemple de l'ús de distribucions de possibilitat al mateix temps que les distribucions de probabilitat quàntiques

An email classification model based on rough set theory

The communication via email is one of the most popular services of the Internet. Emails have brought us great convenience in our daily work and life. However, unsolicited messages or spam, flood our email boxes, which results in bandwidth, time and money wasting. To this end, this paper presents a rough set based model to classify emails into three categories - spam, no-spam and suspicious, rather than two classes (spam and non-spam) in most currently used approaches. By comparing with popular classification methods like Naive Bayes classification, the error ratio that a non-spam is discriminated to spam can be reduced using our proposed model.

An illustration of variable precision rough set theory : The gender classification of the European barn swallow (Hirundo rustica)

This paper introduces a new technique in the investigation of object classification and illustrates the potential use of this technique for the analysis of a range of biological data, using avian morphometric data as an example. The nascent variable precision rough sets (VPRS) model is introduced and compared with the decision tree method ID3 (through a ‘leave n out’ approach), using the same dataset of morphometric measures of European barn swallows (Hirundo rustica) and assessing the accuracy of gender classification based on these measures. The results demonstrate that the VPRS model, allied with the use of a modern method of discretization of data, is comparable with the more traditional non-parametric ID3 decision tree method. We show that, particularly in small samples, the VPRS model can improve classification and to a lesser extent prediction aspects over ID3. Furthermore, through the ‘leave n out’ approach, some indication can be produced of the relative importance of the different morphometric measures used in this problem. In this case we suggest that VPRS has advantages over ID3, as it intelligently uses more of the morphometric data available for the data classification, whilst placing less emphasis on variables with low reliability. In biological terms, the results suggest that the gender of swallows can be determined with reasonable accuracy from morphometric data and highlight the most important variables in this process. We suggest that both analysis techniques are potentially useful for the analysis of a range of different types of biological datasets, and that VPRS in particular has potential for application to a range of biological circumstances.