A Vector Space Approach to Geometry

A Vector Space Approach to Geometry PDF Author: Melvin Hausner
Publisher: Courier Dover Publications
ISBN: 0486835391
Category : Mathematics
Languages : en
Pages : 417

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Book Description
A fascinating exploration of the correlation between geometry and linear algebra, this text also offers elementary explanations of the role of geometry in other branches of math and science. 1965 edition.

A Vector Space Approach to Geometry

A Vector Space Approach to Geometry PDF Author: Melvin Hausner
Publisher: Courier Dover Publications
ISBN: 0486835391
Category : Mathematics
Languages : en
Pages : 417

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Book Description
A fascinating exploration of the correlation between geometry and linear algebra, this text also offers elementary explanations of the role of geometry in other branches of math and science. 1965 edition.

Space and Geometry in the Light of Physiological, Psychological and Physical Inquiry

Space and Geometry in the Light of Physiological, Psychological and Physical Inquiry PDF Author: Ernst Mach
Publisher:
ISBN:
Category : Geometry
Languages : en
Pages : 164

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Book Description


Conceptual Spaces

Conceptual Spaces PDF Author: Peter Gardenfors
Publisher: MIT Press
ISBN: 9780262572194
Category : Psychology
Languages : en
Pages : 324

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Book Description
Within cognitive science, two approaches currently dominate the problem of modeling representations. The symbolic approach views cognition as computation involving symbolic manipulation. Connectionism, a special case of associationism, models associations using artificial neuron networks. Peter Gärdenfors offers his theory of conceptual representations as a bridge between the symbolic and connectionist approaches. Symbolic representation is particularly weak at modeling concept learning, which is paramount for understanding many cognitive phenomena. Concept learning is closely tied to the notion of similarity, which is also poorly served by the symbolic approach. Gärdenfors's theory of conceptual spaces presents a framework for representing information on the conceptual level. A conceptual space is built up from geometrical structures based on a number of quality dimensions. The main applications of the theory are on the constructive side of cognitive science: as a constructive model the theory can be applied to the development of artificial systems capable of solving cognitive tasks. Gärdenfors also shows how conceptual spaces can serve as an explanatory framework for a number of empirical theories, in particular those concerning concept formation, induction, and semantics. His aim is to present a coherent research program that can be used as a basis for more detailed investigations.

The Geometry of Domains in Space

The Geometry of Domains in Space PDF Author: Steven G. Krantz
Publisher: Springer Science & Business Media
ISBN: 1461215749
Category : Mathematics
Languages : en
Pages : 311

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Book Description
The analysis of Euclidean space is well-developed. The classical Lie groups that act naturally on Euclidean space-the rotations, dilations, and trans lations-have both shaped and guided this development. In particular, the Fourier transform and the theory of translation invariant operators (convolution transforms) have played a central role in this analysis. Much modern work in analysis takes place on a domain in space. In this context the tools, perforce, must be different. No longer can we expect there to be symmetries. Correspondingly, there is no longer any natural way to apply the Fourier transform. Pseudodifferential operators and Fourier integral operators can playa role in solving some of the problems, but other problems require new, more geometric, ideas. At a more basic level, the analysis of a smoothly bounded domain in space requires a great deal of preliminary spadework. Tubular neighbor hoods, the second fundamental form, the notion of "positive reach", and the implicit function theorem are just some of the tools that need to be invoked regularly to set up this analysis. The normal and tangent bundles become part of the language of classical analysis when that analysis is done on a domain. Many of the ideas in partial differential equations-such as Egorov's canonical transformation theorem-become rather natural when viewed in geometric language. Many of the questions that are natural to an analyst-such as extension theorems for various classes of functions-are most naturally formulated using ideas from geometry.

Geometry

Geometry PDF Author: John Tabak
Publisher: Infobase Publishing
ISBN: 0816068763
Category : Electronic books
Languages : en
Pages : 241

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Book Description
Greek ideas about geometry, straight-edge and compass constructions, and the nature of mathematical proof dominated mathematical thought for about 2,000 years.

Art and Geometry

Art and Geometry PDF Author: William M. Ivins
Publisher: Courier Corporation
ISBN: 0486143589
Category : Art
Languages : en
Pages : 130

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Book Description
This highly stimulating study observes many historical interrelationships between art and mathematics. It explores ancient and Renaissance painting and sculpture, the development of perspective, and advances in projective geometry.

Designing Learning Environments for Developing Understanding of Geometry and Space

Designing Learning Environments for Developing Understanding of Geometry and Space PDF Author: Richard Lehrer
Publisher: Routledge
ISBN: 0805819487
Category : Education
Languages : en
Pages : 520

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Book Description
This volume reflects an appreciation of the interactive roles of subject matter, teacher, student, and technologies in designing classrooms that promote understanding of geometry and space. Although these elements of geometry education are mutually constituted, the book is organized to highlight, first, the editors' vision of a general geometry education; second, the development of student thinking in everyday and classroom contexts; and third, the role of technologies. Rather than looking to high school geometry as the locus--and all too often, the apex--of geometric reasoning, the contributors to this volume suggest that reasoning about space can and should be successfully integrated with other forms of mathematics, starting at the elementary level and continuing through high school. Reintegrating spatial reasoning into the mathematical mainstream--indeed, placing it at the core of K-12 mathematics environments that promote learning with understanding--will mean increased attention to problems in modeling, structure, and design and reinvigoration of traditional topics such as measure, dimension, and form. Further, the editors' position is that the teaching of geometry and spatial visualization in school should not be compressed into a characterization of Greek geometry, but should include attention to contributions to the mathematics of space that developed subsequent to those of the Greeks. This volume is essential reading for those involved in mathematics education at all levels, including university faculty, researchers, and graduate students.

Mathematizing Space

Mathematizing Space PDF Author: Vincenzo De Risi
Publisher: Birkhäuser
ISBN: 3319121022
Category : Mathematics
Languages : en
Pages : 320

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Book Description
This book collects the papers of the conference held in Berlin, Germany, 27-29 August 2012, on 'Space, Geometry and the Imagination from Antiquity to the Modern Age'. The conference was a joint effort by the Max Planck Institute for the History of Science (Berlin) and the Centro die Ricerca Matematica Ennio De Giorgi (Pisa).

Space, Number, and Geometry from Helmholtz to Cassirer

Space, Number, and Geometry from Helmholtz to Cassirer PDF Author: Francesca Biagioli
Publisher: Springer
ISBN: 3319317792
Category : Philosophy
Languages : en
Pages : 258

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Book Description
This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism.

Symmetry, Shape and Space

Symmetry, Shape and Space PDF Author: L.Christine Kinsey
Publisher: Springer Science & Business Media
ISBN: 9781930190092
Category : Mathematics
Languages : en
Pages : 524

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Book Description
This book will appeal to at least three groups of readers: prospective high school teachers, liberal arts students, and parents whose children are studying high school or college math. It is modern in its selection of topics, and in the learning models used by the authors. The book covers some exciting but non-traditional topics from the subject area of geometry. It is also intended for undergraduates and tries to engage their interest in mathematics. Many innovative pedagogical modes are used throughout.