02818nam a2200361 i 4500991003225059707536160728s2015 sz b 001 0 eng d9783319193328b14299811-39ule_instBibl. Dip.le Aggr. Matematica e Fisica - Sez. Matematicaeng512AMS 20F65AMS 03C20AMS 03C65AMS 37A15LC QA174.2.C37Capraro, Valerio717976Introduction to sofic and hyperlinear groups and connes' embedding conjecture /Valerio Capraro, Martino Lupini ; with an appendix by Vladimir PestovCham [Switzerland] :Springer,c2015viii, 151 p. ;24 cmLecture notes in mathematics,0075-8434 ;2136Includes bibliographical references and indexIntroduction ; Sofic and hyperlinear groups ; Connes' embedding conjecture ; ConclusionsThis monograph presents some cornerstone results in the study of sofic and hyperlinear groups and the closely related Connes' embedding conjecture. These notions, as well as the proofs of many results, are presented in the framework of model theory for metric structures. This point of view, rarely explicitly adopted in the literature, clarifies the ideas therein, and provides additional tools to attack open problems. Sofic and hyperlinear groups are countable discrete groups that can be suitably approximated by finite symmetric groups and groups of unitary matrices. These deep and fruitful notions, introduced by Gromov and Radulescu, respectively, in the late 1990s, stimulated an impressive amount of research in the last 15 years, touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and quantum information theory. Several long-standing conjectures, still open for arbitrary groups, are now settled for sofic or hyperlinear groups. The presentation is self-contained and accessible to anyone with a graduate-level mathematical background. In particular, no specific knowledge of logic or model theory is required. The monograph also contains many exercises, to help familiarize the reader with the topics presentGroup theoryOperator theoryLupini, Martinoauthorhttp://id.loc.gov/vocabulary/relators/aut721405.b1429981122-11-1628-07-16991003225059707536LE013 20F CAP11 (2015)12013000293837le013pE36.39-l- 01010.i1578905622-11-16Introduction to sofic and hyperlinear groups and Connes' embedding conjecture1413246UNISALENTOle01328-07-16ma -engsz 0004322oam 2200517 450 991082586680332120190911112724.01-908977-10-8(OCoLC)844311083(MiFhGG)GVRL8RCY(EXLCZ)99267000000036000120130827h20132013 uy 0engurun|---uuuuatxtccrLocal activity principle /Klaus Mainzer, Technische Universitat Munchen, Germany, Leon Chua, University of California, Berkeley, USALondon Imperial College Pressc2013London :Imperial College Press,[2013]�20131 online resource (xii, 443 pages) illustrations (chiefly color)Gale eBooksDescription based upon print version of record.1-908977-09-4 Includes bibliographical references and index.Contents; Preface; 1. The Local Activity Principle and the Emergence of Complexity; 1.1 Mathematical Definition of Local Activity; 1.2 The Local Activity Theorem; 1.3 Local Activity is the Origin of Complexity; 2. Local Activity and Edge of Chaos in Computer Visualization; 2.1 Local Activity and Edge of Chaos of the Brusselator Equations; 2.2 Local Activity and Edge of Chaos of the Gierer-Meinhardt Equations; 2.3 Local Activity and Edge of Chaos of the FitzHugh-Nagumo Equations; 2.4 Local Activity and Edge of Chaos of the Hodgkin-Huxley Equations2.5 Local Activity and Edge of Chaos of the Oregonator Equations3. The Local Activity Principle and the Expansion of the Universe; 3.1 Mathematical Definition of Symmetry; 3.2 Symmetries in the Quantum World; 3.3 Global and Local Symmetries; 3.4 Local Gauge Symmetries and Symmetry Breaking; 4. The Local Activity Principle and the Dynamics of Matter; 4.1 The Local Activity Principle of Pattern Formation; 4.2 The Local Activity Principle and Prigogine's Dissipative Structures; 4.3 The Local Activity Principle and Haken's Synergetics; 5. The Local Activity Principle and the Evolution of Life5.1 The Local Activity Principle of Turing's Morphogenesis5.2 The Local Activity Principle in Systems Biology; 5.3 The Local Activity Principle in Brain Research; 6. The Local Activity Principle and the Co-evolution of Technology; 6.1 The Local Activity Principle of Cellular Automata; 6.2 The Local Activity Principle of Neural Networks; 6.3 The Local Activity Principle of Memristors; 6.4 The Local Activity Principle of Global Information Networks; 7. The Local Activity Principle and Innovation in the Economy and Society; 7.1 The Local Activity Principle in Sociodynamics7.2 The Local Activity Principle and Emerging Risks7.3 The Local Activity Principle in Financial Dynamics; 7.4 The Local Activity Principle in Innovation Dynamics; 7.5 The Local Activity Principle of Sustainable Entrepreneurship; 8. The Message of the Local Activity Principle; 8.1 The Local Activity Principle in Culture and Philosophy; 8.2 What can we Learn from the Local Activity Principle in the Age of Globalization?; References; Author Index; Subject IndexThe principle of local activity explains the emergence of complex patterns in a homogeneous medium. At first defined in the theory of nonlinear electronic circuits in a mathematically rigorous way, it can be generalized and proven at least for the class of nonlinear reaction-diffusion systems in physics, chemistry, biology, and brain research. Recently, it was realized by memristors for nanoelectronic device applications. In general, the emergence of complex patterns and structures is explained by symmetry breaking in homogeneous media, which is caused by local activity. This book argues that Computational complexityMathematical physicsBroken symmetry (Physics)Computational complexity.Mathematical physics.Broken symmetry (Physics)515.9Mainzer Klaus45836Chua Leon O.1936-MiFhGGMiFhGGBOOK9910825866803321Local activity principle3920610UNINA