Author

# Ian Stewart

Other affiliations: Maynooth University, University of Connecticut, University of Houston ...read more

Bio: Ian Stewart is an academic researcher from National University of Ireland, Galway. The author has contributed to research in topics: Relational frame theory & Equivariant map. The author has an hindex of 64, co-authored 527 publications receiving 23624 citations. Previous affiliations of Ian Stewart include Maynooth University & University of Connecticut.

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01 Jan 1985

TL;DR: Singularities and groups in bifurcation theory as mentioned in this paper have been used to solve the problem of finding a group of singularities in a set of problems with multiple solutions.

Abstract: This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob- lems and we hope to convert the reader to this view In this preface we will discuss what we feel are the strengths of the singularity theory approach This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V I Arnold In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence In this volume our emphasis is on singularity theory, with group theory playing a subordinate role In Volume II the emphasis will be more balanced Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation As we use the term, bifurcation theory is the study of equations with multiple solutions

3,214 citations

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31 Oct 1974

TL;DR: In this article, the authors consider a class of Lie algebras in which every subalgebra is a subideal, and they show that it is possible to construct a locally coalescent class of these classes.

Abstract: 1. Basic concepts.- 1. Preliminaries.- 2. Nilpotency and solubility.- 3. Subideals.- 4. Derivations.- 5. Classes and closure operations.- 6. Representations and modules.- 7. Chain conditions.- 8. Series.- 2. Soluble subideals.- 1. The circle product.- 2. The Derived Join Theorems.- 3. Coalescent classes of Lie algebras.- 1. An example.- 2. Coalescence of classes with minimal conditions.- 3. Coalescence of classes with maximal conditions.- 4. The local coalescence of D.- 5. A counterexample.- 4. Locally coalescent classes of Lie algebras.- 1. The algebra of formal power series.- 2. Complete and locally coalescent classes.- 3. Acceptable subalgebras.- 5. The Mal'cev correspondence.- 1. The Campbell-Hausdorff formula.- 2. Complete groups.- 3. The matrix version.- 4. Inversion of the Campbell-Hausdorff formula.- 5. The general version.- 6. Explicit descriptions.- 6. Locally nilpotent radicals.- 1. The Hirsch-Plotkin radical.- 2. Baer, Fitting, and Gruenberg radicals.- 3. Behaviour under derivations.- 4. Baer and Fitting algebras.- 5. The Levi?-Tokarenko theorem.- 7. Lie algebras in which every subalgebra is a subideal.- 1. Nilpotent subideals.- 2. The key lemma and some applications.- 3. Engel conditions.- 4. A counterexample.- 5. Unsin's algebras.- 8. Chain conditions for subideals.- 1. Classes related to Min-si.- 2. The structure of algebras in Min-si.- 3. The case of prime characteristic.- 4. Examples of algebras with Min-si.- 5. Min-si in special classes of algebras.- 6. Max-si in special classes of algebras.- 7. Examples of algebras satisfying Max-si.- 9. Chain conditions on ascendant abelian subalgebras.- 1. Maximal conditions.- 2. Minimal conditions.- 3. Applications.- 10. Existence theorems for abelian subalgebras.- 1. Generalised soluble classes.- 2. Locally finite algebras.- 3. Generalisations of Witt algebras.- 11. Finiteness conditions for soluble Lie algebras.- 1. The maximal condition for ideals.- 2. The double chain condition.- 3. Residual finiteness.- 4. Stuntedness.- 12. Frattini theory.- 1. The Frattini subalgebra.- 2. Soluble algebras: preliminary reductions.- 3. Proof of the main theorem.- 4. Nilpotency criteria.- 5. A splitting theorem.- 13. Neoclassical structure theory.- 1. Classical structure theory.- 2. Local subideals.- 3. Radicals in locally finite algebras.- 4. Semisimplicity.- 5. Levi factors.- 14. Varieties.- 1. Verbal properties.- 2. Invariance properties of verbal ideals.- 3. Ellipticity.- 4. Marginal properties.- 5. Hall varieties.- 15. The finite basis problem.- 1. Nilpotent varieties.- 2. Partially well ordered sets.- 3. Metabelian varieties.- 4. Non-finitely based varieties.- 5. Class 2-by-abelian varieties.- 16. Engel conditions.- 1. The second and third Engel conditions.- 2. A non-locally nilpotent Engel algebra.- 3. Finiteness conditions on Engel algebras.- 4. Left and right Engel elements.- 17. Kostrikin's theorem.- 1. The Burnside problem.- 2. Basic computational results.- 3. The existence of an element of order 2.- 4. Elements which generate abelian ideals.- 5. Algebras generated by elements of order 2.- 6. A weakened form of Kostrikin's theorem.- 7. Sketch proof of Kostrikin's theorem.- 18. Razmyslov's theorem.- 1. The construction.- 2. Proof of non-nilpotence.- Some open questions.- References.- Notation index.

2,499 citations

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885 citations

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TL;DR: Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.

Abstract: Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

39,297 citations

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[...]

TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.
Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

33,785 citations

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TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.

Abstract: Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering. © 2005 Elsevier B.V. All rights reserved.

9,441 citations

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01 Jan 1979

TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.

Abstract: I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions with two parameters VII. Zonal polynomials

8,730 citations

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TL;DR: This work aims to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture.

Abstract: The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.

7,665 citations