Sets and Supersets

Peer reviewed

Sets and Supersets

Peer reviewed

Insights from ecological psychology and dynamical systems theory can underpin a philosophy of coaching

The aim of this paper is to show how principles of ecological psychology and dynamical systems theory can underpin a philosophy of coaching practice in a nonlinear pedagogy. Nonlinear pedagogy is based on a view of the human movement system as a nonlinear dynamical system. We demonstrate how this perspective of the human movement system can aid understanding of skill acquisition processes and underpin practice for sports coaches. We provide a description of nonlinear pedagogy followed by a consideration of some of the fundamental principles of ecological psychology and dynamical systems theory that underpin it as a coaching philosophy. We illustrate how each principle impacts on nonlinear pedagogical coaching practice, demonstrating how each principle can substantiate a framework for the coaching process.

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.

Colour Sensation and Empathy: Arthur Schopenhauer's Negative Philosophy and Colour Theory as the Basis of an Elemental Colour Architecture

Arthur Schopenhauer proposed a theory of colour as a consequence of his first hand knowledge of J.W. Goethe’s experiments with color phenomena. This colour theory can be used to explore an interesting proposition Schopenhauer made about architecture. For Schopenhauer, architecture is about feelings, not about functions or forms, its purpose as an art is to reveal the principles of primitive forces, specifically gravity and rigidity. For Schopenhauer, architecture expresses these forces in the poised equilibrium of massive structures built out of stone. Schopenhauer was inclined to believed that architecture had already achieved its most perfect expression in Greek temple architecture. However; he did offer one possibility for architectural research: this was the suggestion that architecture was also concerned with the expression of light. It seems never to have occurred to Schopenhauer to use his colour theory to speculate about light in architecture. This paper explores some of the implications of Schopenhauer’s theory of colour for his aesthetics of architecture?

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.

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.

Logic and The Open Society : revising the place of Tarski’s theory of truth within Popper’s political philosophy

This chapter retraces the way in which the Austrian philosopher Sir Karl Popper came to accept a Correspondence Theory of Truth from the work of the Polish logician and mathematician Alfred Tarski. It is argued that Popper’s use of Tarski’s semantic theory of truth reveals crucial insights into the fundamental characteristics of Popper’s social philosophy.  Quite deceptively, arguments based upon Tarski’s theory of truth appear implicitly throughout the text of The Open Society and Its Enemies (1945). It is then demonstrated how Popper integrated a correspondence theory of truth into a theory of the functions of communicative language that he received from Karl Bühler.

Applications of random set theory in econometrics

In recent years, the econometrics literature has shown a growing interest in the study of partially identified models, in which the object of economic and statistical interest is a set rather than a point. The characterization of this set and the development of consistent estimators and inference procedures for it with desirable properties are the main goals of partial identification analysis. This review introduces the fundamental tools of the theory of random sets, which brings together elements of topology, convex geometry, and probability theory to develop a coherent mathematical framework to analyze random elements whose realizations are sets. It then elucidates how these tools have been fruitfully applied in econometrics to reach the goals of partial identification analysis.

A dictionary of chemistry, exhibiting the present state of the theory and practice of that science, its application to natural philosophy, the processes of manufactures, metallurgy, and numerous other arts dependant on the properties and habitudes of bodies, in the mineral, vegetable, and animal kingdoms. With a considerable number of tables, expressing the elective attractions, specific gravities, comparative heats, component parts, combinations, and other affections of the objects of chemical research. Illustrated with engravings.

Paged continuously.

Phytologia; or, The philosophy of agriculture and gardening. With the theory of draining morasses and with an improved construction of the drill plough.

Each plate accompanied by leaf with descriptive letterpress.

Fuzzy Set Theory Approach as the Basis of Analysis of Financial Flows in the Economical Security System

The problems of formalization of the process of matching different management subjects’ functioning characteristics obtained on the financial flows analysis basis is considered. Formal generalizations for gaining economical security system knowledge bases elements are presented. One of feedback directions establishment between knowledge base of the system of economical security and financial flows database analysis is substantiated.

The Development of the Generalization Algorithm Based on the Rough Set Theory

This paper considers the problem of concept generalization in decision-making systems where such features of real-world databases as large size, incompleteness and inconsistence of the stored information are taken into account. The methods of the rough set theory (like lower and upper approximations, positive regions and reducts) are used for the solving of this problem. The new discretization algorithm of the continuous attributes is proposed. It essentially increases an overall performance of generalization algorithms and can be applied to processing of real value attributes in large data tables. Also the search algorithm of the significant attributes combined with a stage of discretization is developed. It allows avoiding splitting of continuous domains of insignificant attributes into intervals.

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.