The Triangle of Representation
The Triangle of Representation

Author: Christopher Prendergast

Publisher: Columbia University Press

Published: 2000

Total Pages: 217

ISBN-13: 0231120915

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One of our best-known cultural critics examines the concept of representation and its many and varied forms in literature and art.

Understanding the Mathematical Way of Thinking – The Registers of Semiotic Representations
Understanding the Mathematical Way of Thinking – The Registers of Semiotic Representations

Author: Raymond Duval

Publisher: Springer

Published: 2017-07-18

Total Pages: 133

ISBN-13: 3319569104

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In this book, Raymond Duval shows how his theory of registers of semiotic representation can be used as a tool to analyze the cognitive processes through which students develop mathematical thinking. To Duval, the analysis of mathematical knowledge is in its essence the analysis of the cognitive synergy between different kinds of semiotic representation registers, because the mathematical way of thinking and working is based on transformations of semiotic representations into others. Based on this assumption, he proposes the use of semiotics to identify and develop the specific cognitive processes required to the acquisition of mathematical knowledge. In this volume he presents a method to do so, addressing the following questions: • How to situate the registers of representation regarding the other semiotic “theories” • Why use a semio-cognitive analysis of the mathematical activity to teach mathematics • How to distinguish the different types of registers • How to organize learning tasks and activities which take into account the registers of representation • How to make an analysis of the students’ production in terms of registers Building upon the contributions he first presented in his classic book Sémiosis et pensée humaine, in this volume Duval focuses less on theoretical issues and more on how his theory can be used both as a tool for analysis and a working method to help mathematics teachers apply semiotics to their everyday work. He also dedicates a complete chapter to show how his theory can be applied as a new strategy to teach geometry. “Understanding the Mathematical Way of Thinking – The Registers of Semiotic Representations is an essential work for mathematics educators and mathematics teachers who look for an introduction to Raymond Duval’s cognitive theory of semiotic registers of representation, making it possible for them to see and teach mathematics with fresh eyes.” Professor Tânia M. M. Campos, PHD.

Peirce’s Twenty-Eight Classes of Signs and the Philosophy of Representation
Peirce’s Twenty-Eight Classes of Signs and the Philosophy of Representation

Author: Tony Jappy

Publisher: Bloomsbury Publishing

Published: 2016-12-15

Total Pages: 227

ISBN-13: 1474264840

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This book is open access and available on www.bloomsburycollections.com. It is funded by Knowledge Unlatched. The major principles and systems of C. S. Peirce's ground-breaking theory of signs and signification are now generally well known. Less well known, however, is the fact that Peirce initially conceived these systems within a 'Philosophy of Representation', his latter-day version of the traditional grammar, logic and rhetoric trivium. In this book, Tony Jappy traces the evolution of Peirce's Philosophy of Representation project and examines the sign systems which came to supersede it. Surveying the stages in Peirce's break with this Philosophy of Representation from its beginnings in the mid-1860s to his final statements on signs between 1908 and 1911, this book draws out the essential theoretical differences between the earlier and later sign systems. Although the 1903 ten-class system has been extensively researched by scholars, this book is the first to exploit the untapped potential of the later six-element systems. Showing how these systems differ from the 1903 version, Peirce's Twenty-Eight Classes of Signs and the Philosophy of Representation offers an innovative and valuable reinterpretation of Peirce's thinking on signs and representation. Exploring the potential of the later sign-systems that Peirce scholars have hitherto been reluctant to engage with and extending Peirce's semiotic theory beyond the much canvassed systems of his Philosophy of Representation, this book will be essential reading for everyone working in the field of semiotics.

Modular Representation Theory of Finite Groups
Modular Representation Theory of Finite Groups

Author: Peter Schneider

Publisher: Springer Science & Business Media

Published: 2012-11-27

Total Pages: 183

ISBN-13: 1447148320

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Representation theory studies maps from groups into the general linear group of a finite-dimensional vector space. For finite groups the theory comes in two distinct flavours. In the 'semisimple case' (for example over the field of complex numbers) one can use character theory to completely understand the representations. This by far is not sufficient when the characteristic of the field divides the order of the group. Modular Representation Theory of finite Groups comprises this second situation. Many additional tools are needed for this case. To mention some, there is the systematic use of Grothendieck groups leading to the Cartan matrix and the decomposition matrix of the group as well as Green's direct analysis of indecomposable representations. There is also the strategy of writing the category of all representations as the direct product of certain subcategories, the so-called 'blocks' of the group. Brauer's work then establishes correspondences between the blocks of the original group and blocks of certain subgroups the philosophy being that one is thereby reduced to a simpler situation. In particular, one can measure how nonsemisimple a category a block is by the size and structure of its so-called 'defect group'. All these concepts are made explicit for the example of the special linear group of two-by-two matrices over a finite prime field. Although the presentation is strongly biased towards the module theoretic point of view an attempt is made to strike a certain balance by also showing the reader the group theoretic approach. In particular, in the case of defect groups a detailed proof of the equivalence of the two approaches is given. This book aims to familiarize students at the masters level with the basic results, tools, and techniques of a beautiful and important algebraic theory. Some basic algebra together with the semisimple case are assumed to be known, although all facts to be used are restated (without proofs) in the text. Otherwise the book is entirely self-contained.

Theory Of Groups And Symmetries: Representations Of Groups And Lie Algebras, Applications
Theory Of Groups And Symmetries: Representations Of Groups And Lie Algebras, Applications

Author: Alexey P Isaev

Publisher: World Scientific

Published: 2020-07-16

Total Pages: 615

ISBN-13: 9811217424

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This book is a sequel to the book by the same authors entitled Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras.The presentation begins with the Dirac notation, which is illustrated by boson and fermion oscillator algebras and also Grassmann algebra. Then detailed account of finite-dimensional representations of groups SL(2, C) and SU(2) and their Lie algebras is presented. The general theory of finite-dimensional irreducible representations of simple Lie algebras based on the construction of highest weight representations is given. The classification of all finite-dimensional irreducible representations of the Lie algebras of the classical series sℓ(n, C), so(n, C) and sp(2r, C) is exposed.Finite-dimensional irreducible representations of linear groups SL(N, C) and their compact forms SU(N) are constructed on the basis of the Schur-Weyl duality. A special role here is played by the theory of representations of the symmetric group algebra C[Sr] (Schur-Frobenius theory, Okounkov-Vershik approach), based on combinatorics of Young diagrams and Young tableaux. Similar construction is given for pseudo-orthogonal groups O(p, q) and SO(p, q), including Lorentz groups O(1, N-1) and SO(1, N-1), and their Lie algebras, as well as symplectic groups Sp(p, q). The representation theory of Brauer algebra (centralizer algebra of SO(p, q) and Sp(p, q) groups in tensor representations) is discussed.Finally, the covering groups Spin(p, q) for pseudo-orthogonal groups SO↑(p, q) are studied. For this purpose, Clifford algebras in spaces Rp, q are introduced and representations of these algebras are discussed.

Descartes's Dreams
Descartes's Dreams

Author: Ann Scholl

Publisher: Peter Lang

Published: 2005

Total Pages: 164

ISBN-13: 9780820452456

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Ann Scholl revises the traditional understanding of the role of imagination and sensory perception in Descartes's Meditations. Traditionally, Cartesian scholars have focused primarily on sensory perception as the more significant of the two «special» modes of thought. In this work, Ann Scholl describes how a better understanding of Descartes's skepticism and his arguments for dualism are reached when imagination instead is understood as the more primary of the two special modes of thought. The result is a fresh reading and interpretation of Descartes's most influential work.