Extension Theory of Formally Normal and Symmetric Subspaces
Extension Theory of Formally Normal and Symmetric Subspaces

Author: Earl A. Coddington

Publisher: American Mathematical Soc.

Published: 1973

Total Pages: 87

ISBN-13: 0821818341

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Let [italic]H be a Hilbert space. Formally normal, normal, symmetric, selfadjoint, and semibounded subspaces of [italic]H2=[italic]H2[circled plus][italic]H are defined by means of the corresponding properties of the graphs of operators in H which are formally normal, normal, symmetric, selfadjoint, or semibounded, respectively. The author gives a complete description of all formally normal and normal subspace extensions in [italic]H2 of a given formally normal subspace [italic]N of [italic]H2. Those extensions which are graphs of operators are explicitly characterized. The symmetric and selfadjoint extensions of a given symmetric subspace are also classified; this result extends the well-known result of von Neumann characterizing the selfadjoint extensions of a (densely defined) symmetric operator. The construction of the "Friedrichs extension'' of a semibounded symmetric subspace is outlined. The existence of formally normal and symmetric extensions in a larger Hilbert space is also studied. A formally normal subspace need not have any normal subspace extension in a bigger subspace. But (as is known for operators), every symmetric subspace has selfadjoint extensions in suitable larger spaces; these extensions are completely characterized.

Yang-Mills Fields and Extension Theory
Yang-Mills Fields and Extension Theory

Author: Robert Pool

Publisher: American Mathematical Soc.

Published: 1987

Total Pages: 74

ISBN-13: 0821824228

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In this memoir, Yang-Mills field curvature and current are described as elements of cohomology groups on ambitwistor space, and it is shown that the Yang-Mills current corresponds to the third-order obstruction to extending the corresponding vector bundle. There is also a description of the Yang-Mills field's action density. In addition, several groups of solutions to various differential equations on Miskowski space are characterized as elements of various cohomology classes over ambitwistor space.

Extenics and Innovation Methods
Extenics and Innovation Methods

Author: Wen Cai

Publisher: CRC Press

Published: 2013-07-15

Total Pages: 350

ISBN-13: 1138000493

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Society forges ahead in the process of solving various contradictory problems and it is ceaselessly innovating. It is the desire of mankind to use computers and computing networks to help deal with contradictory problems and to conduct innovative activities. Using formal models to discuss object extension and the possibility of change, as well as the rules and methods for innovation, Extenics is applied to solving contradictory problems and has become the basic theory, method and instrument to achieve this goal. In the 30 years since the foundation of Extenics, researchers have built relatively complete theoretical systems —‘extension theory’, studied formal and modeling innovation methods —‘extension innovation methods’, and launched the applications in various fields such as information, design, automation and management etc. —‘extension engineering’. Extension theory, the extension innovation method and extension engineering jointly constitute the new discipline—Extenics. At the same time, the practical activities of engineering technology and management promote the integration of various innovation methods such as TRIZ and brainstorming etc. This book collects together, from scholars in various fields, the research achievements in Extenics and innovation methods, in order to facilitate and promote the development of Extenics and the various innovation theories and methods, as well as to improve its innovative capacity in academic and business circles.

Dimension and Extensions
Dimension and Extensions

Author: J.M. Aarts

Publisher: Elsevier

Published: 1993-01-28

Total Pages: 347

ISBN-13: 0080887619

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Two types of seemingly unrelated extension problems are discussed in this book. Their common focus is a long-standing problem of Johannes de Groot, the main conjecture of which was recently resolved. As is true of many important conjectures, a wide range of mathematical investigations had developed, which have been grouped into the two extension problems. The first concerns the extending of spaces, the second concerns extending the theory of dimension by replacing the empty space with other spaces.The problem of de Groot concerned compactifications of spaces by means of an adjunction of a set of minimal dimension. This minimal dimension was called the compactness deficiency of a space. Early success in 1942 lead de Groot to invent a generalization of the dimension function, called the compactness degree of a space, with the hope that this function would internally characterize the compactness deficiency which is a topological invariant of a space that is externally defined by means of compact extensions of a space. From this, the two extension problems were spawned.With the classical dimension theory as a model, the inductive, covering and basic aspects of the dimension functions are investigated in this volume, resulting in extensions of the sum, subspace and decomposition theorems and theorems about mappings into spheres. Presented are examples, counterexamples, open problems and solutions of the original and modified compactification problems.

Hermann Graßmann
Hermann Graßmann

Author: Hans-Joachim Petsche

Publisher: Springer Science & Business Media

Published: 2009-12-30

Total Pages: 320

ISBN-13: 3764388609

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Hermann Günther Graßmann was one of the most remarkable personalities in 19th-century science. A "small-town genius", he developed a groundbreaking n-dimensional algebra of space and contributed to a revolution in the understanding of mathematics. His work fascinated great mathematicians such as W. R. Hamilton, J. W. Gibbs and A. N. Whitehead. This intellectual biography traces Graßmann’s steps towards scientific brilliance by untangling a complicated web of influences: the force of unsolved problems in mathematics, Friedrich Schleiermacher’s Dialectic, German Romanticism and life in 19th-century Prussia. The book also introduces the reader to the details of Graßmann’s mathematical work without neglecting his achievements in Sanskrit philology and physics. And, for the first time, it makes many original sources accessible to the English-language reader.

Artificial Intelligence Applications and Innovations II
Artificial Intelligence Applications and Innovations II

Author: Daoliang Li

Publisher: Springer Science & Business Media

Published: 2005-08-25

Total Pages: 858

ISBN-13: 0387283188

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Artificial Intelligence is one of the oldest and most exciting subfields of computing, covnering such areas as intelligent robotics, intelligent planning and scheduling, model-based reasoning, fault diagnosis, natural language processing, maching translation, knowledge representation and reasoning, knowledge-based systems, knowledge engineering, intelligent agents, machine learning, neural nets, genetic algorithms and knowledge management. The papers in this volume comprise the refereed proceedings of the Second International Conference on Artificial Intelligence Applications and Innovations,held in Beijing, China in 2005. A very promising sign of the growing importance of Artificial Intelligence techniques in practical applications is the large number of submissions received for the conference - more than 150. All papers were reviewed by at least two members of the Program Committee and the test 93 were selected for the conference and are included in this volume. The international nature of IFIP is amply reflected in the large number of countries represented here.

The Law of Non-Contradiction
The Law of Non-Contradiction

Author: Graham Priest

Publisher: Clarendon Press

Published: 2006-11-30

Total Pages: 456

ISBN-13: 0191548065

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The Law of Non-Contradiction-that no contradiction can be true-has been a seemingly unassailable dogma since the work of Aristotle, in Book Gamma of the Metaphysics. It is an assumption challenged from a variety of angles in this collection of original papers. Twenty-three of the world's leading experts investigate the 'law', considering arguments for and against it and discussing methodological issues that arise whenever we question the legitimacy of logical principles. The result is a balanced inquiry into a venerable principle of logic, one that raises questions at the very centre of logic itself. The aim of this volume is to present a comprehensive debate about the Law of Non-Contradiction, from discussions as to how the law is to be understood, to reasons for accepting or re-thinking the law, and to issues that raise challenges to the law, such as the Liar Paradox, and a 'dialetheic' resolution of that paradox. One of the editors contributes an introduction which surveys the issues and serves to frame the debate. This collection will be of interest to anyone working on philosophical logic, and to anyone who has ever wondered about the status of logical laws and about how one might proceed to mount arguments for or against them.

Mathematical Challenges of Zero-Range Physics
Mathematical Challenges of Zero-Range Physics

Author: Alessandro Michelangeli

Publisher: Springer Nature

Published: 2021-02-04

Total Pages: 331

ISBN-13: 3030604535

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Since long over the decades there has been a large transversal community of mathematicians grappling with the sophisticated challenges of the rigorous modelling and the spectral and scattering analysis of quantum systems of particles subject to an interaction so much localised to be considered with zero range. Such a community is experiencing fruitful and inspiring exchanges with experimental and theoretical physicists. This volume reflects such spirit, with a diverse range of original contributions by experts, presenting an up-to-date collection of most relevant results and challenging open problems. It has been conceived with the deliberate two-fold purpose of serving as an updated reference for recent results, mathematical tools, and the vast related literature on the one hand, and as a bridge towards several key open problems that will surely form the forthcoming research agenda in this field.

Emerging Intelligent Computing Technology and Applications
Emerging Intelligent Computing Technology and Applications

Author: De-Shuang Huang

Publisher: Springer Science & Business Media

Published: 2009-08-28

Total Pages: 1156

ISBN-13: 3642040691

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This book - in conjunction with the volume LNAI 5755 - constitutes the refereed proceedings of the 5th International Conference on Intelligent Computing, ICIC 2009, held in Ulsan, South Korea in September 2009. The 214 revised full papers of these two volumes were carefully reviewed and selected from a total of 1082 submissions. The papers are organized in topical sections on Supervised & Semi-supervised Learning, Machine Learning Theory and Methods, Biological and Quantum Computing, Intelligent Computing in Bioinformatics, Intelligent Computing in Computational Biology and Drug Design, Computational Genomics and Proteomics, Intelligent Computing in Signal Processing, Intelligent Computing in Pattern Recognition, Intelligent Computing in Image Processing, Intelligent Computing in Communication and Computer Networks, Intelligent Computing in Robotics, Intelligent Computing in Computer Vision, Intelligent Agent and Web Applications, Intelligent Sensor Networks, Intelligent Fault Diagnosis & Financial Engineering, Intelligent Control and Automation, Intelligent Data Fusion and Security, Intelligent Prediction & Time Series Analysis, Natural Language Processing and Expert Systems, Intelligent Image/Document Retrievals, Computational Analysis and Data Mining in Biological Systems, Knowledge-Based Systems and Intelligent Computing in Medical Imaging, Applications of Intelligent Computing in Information Assurance & Security, Computational Analysis and Applications in Biomedical System, Intelligent Computing Algorithms in Banking and Finance, and Network-Based Intelligent Technologies.