The Information Manifold
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Author | : Antonio Badia |
Publisher | : MIT Press |
Total Pages | : 353 |
Release | : 2019-11-05 |
Genre | : Social Science |
ISBN | : 0262355183 |
An argument that information exists at different levels of analysis—syntactic, semantic, and pragmatic—and an exploration of the implications. Although this is the Information Age, there is no universal agreement about what information really is. Different disciplines view information differently; engineers, computer scientists, economists, linguists, and philosophers all take varying and apparently disconnected approaches. In this book, Antonio Badia distinguishes four levels of analysis brought to bear on information: syntactic, semantic, pragmatic, and network-based. Badia explains each of these theoretical approaches in turn, discussing, among other topics, theories of Claude Shannon and Andrey Kolomogorov, Fred Dretske's description of information flow, and ideas on receiver impact and informational interactions. Badia argues that all these theories describe the same phenomena from different perspectives, each one narrower than the previous one. The syntactic approach is the more general one, but it fails to specify when information is meaningful to an agent, which is the focus of the semantic and pragmatic approaches. The network-based approach, meanwhile, provides a framework to understand information use among agents. Badia then explores the consequences of understanding information as existing at several levels. Humans live at the semantic and pragmatic level (and at the network level as a society), computers at the syntactic level. This sheds light on some recent issues, including “fake news” (computers cannot tell whether a statement is true or not, because truth is a semantic notion) and “algorithmic bias” (a pragmatic, not syntactic concern). Humans, not computers, the book argues, have the ability to solve these issues.
Author | : Antonio Badia |
Publisher | : |
Total Pages | : 408 |
Release | : 2019 |
Genre | : Communication |
ISBN | : 9780262355179 |
Author | : Loring W. Tu |
Publisher | : Springer Science & Business Media |
Total Pages | : 426 |
Release | : 2010-10-05 |
Genre | : Mathematics |
ISBN | : 1441974008 |
Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.
Author | : Shun-ichi Amari |
Publisher | : Springer |
Total Pages | : 378 |
Release | : 2016-02-02 |
Genre | : Mathematics |
ISBN | : 4431559787 |
This is the first comprehensive book on information geometry, written by the founder of the field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide range of applications, covering information science, engineering, and neuroscience. It consists of four parts, which on the whole can be read independently. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. This part (Part I) can be apprehended without any knowledge of differential geometry. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry. Information geometry of statistical inference, including time series analysis and semiparametric estimation (the Neyman–Scott problem), is demonstrated concisely in Part III. Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. This book is highly recommended to graduate students and researchers who seek new mathematical methods and tools useful in their own fields.
Author | : A. L. Besse |
Publisher | : Springer Science & Business Media |
Total Pages | : 271 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3642618766 |
X 1 O S R Cher lecteur, J'entre bien tard dans la sphere etroite des ecrivains au double alphabet, moi qui, il y a plus de quarante ans deja, avais accueilli sur mes terres un general epris de mathematiques. JI m'avait parle de ses projets grandioses en promettant d'ailleurs de m'envoyer ses ouvrages de geometrie. Je suis entiche de geometrie et c'est d'elle dontje voudrais vous parler, oh! certes pas de toute la geometrie, mais de celle que fait l'artisan qui taille, burine, amene, gauchit, peaufine les formes. Mon interet pour le probleme dont je veux vous entretenir ici, je le dois a un ami ebeniste. En effet comme je rendais un jour visite il cet ami, je le trouvai dans son atelier affaire a un tour. Il se retourna bientot, puis, rayonnant, me tendit une sorte de toupie et me dit: {laquo}Monsieur Besse, vous qui calculez les formes avec vos grimoires, que pensez-vous de ceci?)) Je le regardai interloque. Il poursuivit: {laquo}Regardez! Si vous prenez ce collier de laine et si vous le maintenez fermement avec un doigt place n'importe ou sur la toupie, eh bien! la toupie passera toujours juste en son interieur, sans laisser le moindre espace.)) Je rentrai chez moi, fort etonne, car sa toupie etait loin d'etre une boule. Je me mis alors au travail ...
Author | : Frank W. Warner |
Publisher | : Springer Science & Business Media |
Total Pages | : 283 |
Release | : 2013-11-11 |
Genre | : Mathematics |
ISBN | : 1475717997 |
Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.
Author | : John M. Lee |
Publisher | : Springer |
Total Pages | : 447 |
Release | : 2019-01-02 |
Genre | : Mathematics |
ISBN | : 3319917552 |
This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.
Author | : John M. Lee |
Publisher | : Springer Science & Business Media |
Total Pages | : 232 |
Release | : 2006-04-06 |
Genre | : Mathematics |
ISBN | : 0387227261 |
This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.
Author | : D. E. Blair |
Publisher | : Springer |
Total Pages | : 153 |
Release | : 2006-11-14 |
Genre | : Mathematics |
ISBN | : 3540381546 |
Author | : Shun-ichi Amari |
Publisher | : American Mathematical Soc. |
Total Pages | : 220 |
Release | : 2000 |
Genre | : Computers |
ISBN | : 9780821843024 |
Information geometry provides the mathematical sciences with a fresh framework of analysis. This book presents a comprehensive introduction to the mathematical foundation of information geometry. It provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, and convex analysis.