Shape Theory

Categorical Methods of Approximation

Author: J. M. Cordier,T. Porter

Publisher: Courier Corporation

ISBN: 0486783472

Category: Science

Page: 208

View: 3495

This in-depth treatment uses shape theory as a "case study" to illustrate situations common to many areas of mathematics, including the use of archetypal models as a basis for systems of approximations. It offers students a unified and consolidated presentation of extensive research from category theory, shape theory, and the study of topological algebras. A short introduction to geometric shape explains specifics of the construction of the shape category and relates it to an abstract definition of shape theory. Upon returning to the geometric base, the text considers simplical complexes and numerable covers, in addition to Morita's form of shape theory. Subsequent chapters explore Bénabou's theory of distributors, the theory of exact squares, Kan extensions, the notion of a stable object, and stability in an Abelian context. The text concludes with a brief description of derived functors of the limit functor theory—the concept that leads to movability and strong movability of systems—and illustrations of the equivalence of strong movability and stability in many contexts.
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Mathematics Learning in Early Childhood

Paths Toward Excellence and Equity

Author: National Research Council,Division of Behavioral and Social Sciences and Education,Center for Education,Committee on Early Childhood Mathematics

Publisher: National Academies Press

ISBN: 9780309147439

Category: Education

Page: 398

View: 4247

Early childhood mathematics is vitally important for young children's present and future educational success. Research demonstrates that virtually all young children have the capability to learn and become competent in mathematics. Furthermore, young children enjoy their early informal experiences with mathematics. Unfortunately, many children's potential in mathematics is not fully realized, especially those children who are economically disadvantaged. This is due, in part, to a lack of opportunities to learn mathematics in early childhood settings or through everyday experiences in the home and in their communities. Improvements in early childhood mathematics education can provide young children with the foundation for school success. Relying on a comprehensive review of the research, Mathematics Learning in Early Childhood lays out the critical areas that should be the focus of young children's early mathematics education, explores the extent to which they are currently being incorporated in early childhood settings, and identifies the changes needed to improve the quality of mathematics experiences for young children. This book serves as a call to action to improve the state of early childhood mathematics. It will be especially useful for policy makers and practitioners-those who work directly with children and their families in shaping the policies that affect the education of young children.
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Axiomatic Method and Category Theory

Author: Andrei Rodin

Publisher: Springer Science & Business Media

ISBN: 3319004042

Category: Philosophy

Page: 285

View: 5530

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.
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A Generative Theory of Shape

Author: Michael Leyton

Publisher: Springer

ISBN: 3540454888

Category: Computers

Page: 549

View: 7150

The purpose of this book is to develop a generative theory of shape that has two properties we regard as fundamental to intelligence –(1) maximization of transfer: whenever possible, new structure should be described as the transfer of existing structure; and (2) maximization of recoverability: the generative operations in the theory must allow maximal inferentiability from data sets. We shall show that, if generativity satis?es these two basic criteria of - telligence, then it has a powerful mathematical structure and considerable applicability to the computational disciplines. The requirement of intelligence is particularly important in the gene- tion of complex shape. There are plenty of theories of shape that make the generation of complex shape unintelligible. However, our theory takes the opposite direction: we are concerned with the conversion of complexity into understandability. In this, we will develop a mathematical theory of und- standability. The issue of understandability comes down to the two basic principles of intelligence - maximization of transfer and maximization of recoverability. We shall show how to formulate these conditions group-theoretically. (1) Ma- mization of transfer will be formulated in terms of wreath products. Wreath products are groups in which there is an upper subgroup (which we will call a control group) that transfers a lower subgroup (which we will call a ?ber group) onto copies of itself. (2) maximization of recoverability is insured when the control group is symmetry-breaking with respect to the ?ber group.
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Sets for Mathematics

Author: F. William Lawvere,Robert Rosebrugh

Publisher: Cambridge University Press

ISBN: 9780521010603

Category: Mathematics

Page: 261

View: 5905

In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.
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Exceptions are the rule

an inquiry into methods in the social sciences

Author: Joel H. Levine

Publisher: Westview Pr

ISBN: N.A

Category: Social Science

Page: 312

View: 7163

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Subject Guide to Books in Print

An Index to the Publishers' Trade List Annual

Author: N.A

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 9130

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Mind and Nature

Selected Writings on Philosophy, Mathematics, and Physics

Author: Hermann Weyl

Publisher: Princeton University Press

ISBN: 9781400833320

Category: Mathematics

Page: 272

View: 3145

Hermann Weyl (1885-1955) was one of the twentieth century's most important mathematicians, as well as a seminal figure in the development of quantum physics and general relativity. He was also an eloquent writer with a lifelong interest in the philosophical implications of the startling new scientific developments with which he was so involved. Mind and Nature is a collection of Weyl's most important general writings on philosophy, mathematics, and physics, including pieces that have never before been published in any language or translated into English, or that have long been out of print. Complete with Peter Pesic's introduction, notes, and bibliography, these writings reveal an unjustly neglected dimension of a complex and fascinating thinker. In addition, the book includes more than twenty photographs of Weyl and his family and colleagues, many of which are previously unpublished. Included here are Weyl's exposition of his important synthesis of electromagnetism and gravitation, which Einstein at first hailed as "a first-class stroke of genius"; two little-known letters by Weyl and Einstein from 1922 that give their contrasting views on the philosophical implications of modern physics; and an essay on time that contains Weyl's argument that the past is never completed and the present is not a point. Also included are two book-length series of lectures, The Open World (1932) and Mind and Nature (1934), each a masterly exposition of Weyl's views on a range of topics from modern physics and mathematics. Finally, four retrospective essays from Weyl's last decade give his final thoughts on the interrelations among mathematics, philosophy, and physics, intertwined with reflections on the course of his rich life.
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Category Theory in Context

Author: Emily Riehl

Publisher: Courier Dover Publications

ISBN: 0486820807

Category: Mathematics

Page: 272

View: 2843

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.
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Science's First Mistake

Delusions in Pursuit of Theory

Author: Ian O. Angell,Dionysios Demetis

Publisher: A&C Black

ISBN: 1780932332

Category: Reference

Page: 256

View: 2440

This book seeks to deconstruct the process of scientific knowledge discovery and theory construction by scrutinizing the circumstances under which all scientific hypotheses are conceived. It concentrates on the interrelatedness of observation, paradox, delusion and self reference in scientific theory and method.
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Algorithms for Approximation

Proceedings of the 5th International Conference, Chester, July 2005

Author: Armin Iske,Jeremy Levesley

Publisher: Springer Science & Business Media

ISBN: 3540465510

Category: Mathematics

Page: 389

View: 8432

Approximation methods are vital in many challenging applications of computational science and engineering. This is a collection of papers from world experts in a broad variety of relevant applications, including pattern recognition, machine learning, multiscale modelling of fluid flow, metrology, geometric modelling, tomography, signal and image processing. It documents recent theoretical developments which have lead to new trends in approximation, it gives important computational aspects and multidisciplinary applications, thus making it a perfect fit for graduate students and researchers in science and engineering who wish to understand and develop numerical algorithms for the solution of their specific problems. An important feature of the book is that it brings together modern methods from statistics, mathematical modelling and numerical simulation for the solution of relevant problems, with a wide range of inherent scales. Contributions of industrial mathematicians, including representatives from Microsoft and Schlumberger, foster the transfer of the latest approximation methods to real-world applications.
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Cultural Techniques

Grids, Filters, Doors, and Other Articulations of the Real

Author: Bernhard Siegert

Publisher: Meaning Systems (Fup)

ISBN: 0823263754

Category: Literary Criticism

Page: 265

View: 9589

"This volume designates a shift within posthumanistic media studies, that dissolves the concept of media into a network of operations, that reproduce, process and reflect the distinctions that are fundamental for a given culture, e.g. the anthropological difference, the distinctions between natural object and cultural sign, noise and information, eye and gaze"--
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Shape in Picture

Mathematical Description of Shape in Grey-level Images

Author: O Ying Lie,Alexander Toet,David Foster,Henk J.A.M. Heijmans,Peter Meer

Publisher: Springer Science & Business Media

ISBN: 366203039X

Category: Computers

Page: 682

View: 9169

The fields of image analysis, computer vision, and artificial intelligence all make use of descriptions of shape in grey-level images. Most existing algorithms for the automatic recognition and classification of particular shapes have been devel oped for specific purposes, with the result that these methods are often restricted in their application. The use of advanced and theoretically well-founded math ematical methods should lead to the construction of robust shape descriptors having more general application. Shape description can be regarded as a meeting point of vision research, mathematics, computing science, and the application fields of image analy sis, computer vision, and artificial intelligence. The NATO Advanced Research Workshop "Shape in Picture" was organised with a twofold objective: first, it should provide all participants with an overview of relevant developments in these different disciplines; second, it should stimulate researchers to exchange original results and ideas across the boundaries of these disciplines. This book comprises a widely drawn selection of papers presented at the workshop, and many contributions have been revised to reflect further progress in the field. The focus of this collection is on mathematical approaches to the construction of shape descriptions from grey-level images. The book is divided into five parts, each devoted to a different discipline. Each part contains papers that have tutorial sections; these are intended to assist the reader in becoming acquainted with the variety of approaches to the problem.
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A budget of paradoxes

Author: Augustus De Morgan

Publisher: N.A

ISBN: N.A

Category: Philosophy

Page: 402

View: 492

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Shape

Talking about Seeing and Doing

Author: George Stiny

Publisher: MIT Press

ISBN: 0262195313

Category: Computers

Page: 422

View: 3078

How design is calculating with shapes: formal details and design applications.
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Linear Models with R, Second Edition

Author: Julian J. Faraway

Publisher: CRC Press

ISBN: 1439887349

Category: Mathematics

Page: 286

View: 2639

A Hands-On Way to Learning Data Analysis Part of the core of statistics, linear models are used to make predictions and explain the relationship between the response and the predictors. Understanding linear models is crucial to a broader competence in the practice of statistics. Linear Models with R, Second Edition explains how to use linear models in physical science, engineering, social science, and business applications. The book incorporates several improvements that reflect how the world of R has greatly expanded since the publication of the first edition. New to the Second Edition Reorganized material on interpreting linear models, which distinguishes the main applications of prediction and explanation and introduces elementary notions of causality Additional topics, including QR decomposition, splines, additive models, Lasso, multiple imputation, and false discovery rates Extensive use of the ggplot2 graphics package in addition to base graphics Like its widely praised, best-selling predecessor, this edition combines statistics and R to seamlessly give a coherent exposition of the practice of linear modeling. The text offers up-to-date insight on essential data analysis topics, from estimation, inference, and prediction to missing data, factorial models, and block designs. Numerous examples illustrate how to apply the different methods using R.
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Analysis of Multivariate and High-Dimensional Data

Author: Inge Koch

Publisher: Cambridge University Press

ISBN: 0521887933

Category: Business & Economics

Page: 526

View: 478

This modern approach integrates classical and contemporary methods, fusing theory and practice and bridging the gap to statistical learning.
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Categorical Homotopy Theory

Author: Emily Riehl

Publisher: Cambridge University Press

ISBN: 1139952633

Category: Mathematics

Page: N.A

View: 5537

This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.
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