Rein Taagepera
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199534661
- eISBN:
- 9780191715921
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534661.003.0018
- Subject:
- Political Science, Comparative Politics, Political Economy
Society needs more from social sciences than they have delivered, and this book offers openings. To the society at large, quantitative social scientists presently seem no better at prediction than ...
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Society needs more from social sciences than they have delivered, and this book offers openings. To the society at large, quantitative social scientists presently seem no better at prediction than qualitative historians, philosophers, and journalists — they just look more boring. computers could be a boon to social sciences, but they have turned out a curse in disguise, by enabling people with insufficient understanding of scientific process to use canned computer programs and grind out reams of numbers parading as “results,” to be printed — and hardly ever used again. One may discard this book on the basis of errors of detail, but the problems it points out will still be there. Unless corrected, they will lead to a Ptolemaic dead end.Less
Society needs more from social sciences than they have delivered, and this book offers openings. To the society at large, quantitative social scientists presently seem no better at prediction than qualitative historians, philosophers, and journalists — they just look more boring. computers could be a boon to social sciences, but they have turned out a curse in disguise, by enabling people with insufficient understanding of scientific process to use canned computer programs and grind out reams of numbers parading as “results,” to be printed — and hardly ever used again. One may discard this book on the basis of errors of detail, but the problems it points out will still be there. Unless corrected, they will lead to a Ptolemaic dead end.
Sergey Dorogovtsev
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199548927
- eISBN:
- 9780191720574
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199548927.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book provides a concise modern introduction to the science of complex networks. The book is based on lectures for university students and non-specialists. The aim is to introduce the world of ...
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This book provides a concise modern introduction to the science of complex networks. The book is based on lectures for university students and non-specialists. The aim is to introduce the world of networks to the subject without need for a serious background in mathematics or physics. The introductory chapters fill the existing gap between popular science books and comprehensive reference volumes on complex networks (including the Internet, World Wide Web, etc.). The book provides the shortest path to the world of networks and discusses the main directions of modern research in this active field as well as the history of network studies. The book describes the current state of the art in complex networks and includes recent results.Less
This book provides a concise modern introduction to the science of complex networks. The book is based on lectures for university students and non-specialists. The aim is to introduce the world of networks to the subject without need for a serious background in mathematics or physics. The introductory chapters fill the existing gap between popular science books and comprehensive reference volumes on complex networks (including the Internet, World Wide Web, etc.). The book provides the shortest path to the world of networks and discusses the main directions of modern research in this active field as well as the history of network studies. The book describes the current state of the art in complex networks and includes recent results.
Sergey N. Dorogovtsev
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199548927
- eISBN:
- 9780191720574
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199548927.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
In this concluding section of the text, the three major milestones marking the history of the exploration of networks are indicated. These are: Leonhard Euler's work (1735), the introduction of ...
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In this concluding section of the text, the three major milestones marking the history of the exploration of networks are indicated. These are: Leonhard Euler's work (1735), the introduction of random graphs (1950s), and the launch of the large-scale study of complex networks (the end of the 1990s). A list of a few particularly hot and prospective topics in complex networks is presented.Less
In this concluding section of the text, the three major milestones marking the history of the exploration of networks are indicated. These are: Leonhard Euler's work (1735), the introduction of random graphs (1950s), and the launch of the large-scale study of complex networks (the end of the 1990s). A list of a few particularly hot and prospective topics in complex networks is presented.
Stephen M. Kosslyn
- Published in print:
- 2006
- Published Online:
- March 2012
- ISBN:
- 9780195311846
- eISBN:
- 9780199847075
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195311846.001.0001
- Subject:
- Psychology, Cognitive Psychology
Graphs have become a fixture of everyday life, used in scientific and business publications, in magazines and newspapers, on television, on billboards, and even on cereal boxes. Nonetheless, ...
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Graphs have become a fixture of everyday life, used in scientific and business publications, in magazines and newspapers, on television, on billboards, and even on cereal boxes. Nonetheless, surprisingly few graphs communicate effectively, and most graphs fail because they do not take into account the goals, needs, and abilities of the viewers. This book addresses these problems by presenting eight psychological principles for constructing effective graphs. Each principle is solidly rooted both in the scientific literature on how we perceive and comprehend graphs and in general facts about how our eyes and brains process visual information. The author uses these eight psychological principles as the basis for hundreds of specific recommendations that serve as a concrete, step-by-step guide to deciding whether a graph is an appropriate display to use, choosing the correct type of graph for a specific type of data and message, and then constructing graphs that will be understood at a glance. The book includes a complete review of the scientific literature on graph perception and comprehension, appendices that provide a quick tutorial on basic statistics, and a checklist for evaluating computer-graphics programs.Less
Graphs have become a fixture of everyday life, used in scientific and business publications, in magazines and newspapers, on television, on billboards, and even on cereal boxes. Nonetheless, surprisingly few graphs communicate effectively, and most graphs fail because they do not take into account the goals, needs, and abilities of the viewers. This book addresses these problems by presenting eight psychological principles for constructing effective graphs. Each principle is solidly rooted both in the scientific literature on how we perceive and comprehend graphs and in general facts about how our eyes and brains process visual information. The author uses these eight psychological principles as the basis for hundreds of specific recommendations that serve as a concrete, step-by-step guide to deciding whether a graph is an appropriate display to use, choosing the correct type of graph for a specific type of data and message, and then constructing graphs that will be understood at a glance. The book includes a complete review of the scientific literature on graph perception and comprehension, appendices that provide a quick tutorial on basic statistics, and a checklist for evaluating computer-graphics programs.
Mark Newman
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199206650
- eISBN:
- 9780191594175
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199206650.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet ...
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The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet and the wide availability of inexpensive computers have made it possible to gather and analyze network data on a large scale, and the development of a variety of new theoretical tools has allowed us to extract new knowledge from many different kinds of networks. The study of networks is broadly interdisciplinary and important developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. This book brings together the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Subjects covered include the measurement and structure of networks in many branches of science, methods for analyzing network data, including methods developed in physics, statistics, and sociology, the fundamentals of graph theory, computer algorithms, and spectral methods, mathematical models of networks, including random graph models and generative models, and theories of dynamical processes taking place on networks.Less
The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet and the wide availability of inexpensive computers have made it possible to gather and analyze network data on a large scale, and the development of a variety of new theoretical tools has allowed us to extract new knowledge from many different kinds of networks. The study of networks is broadly interdisciplinary and important developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. This book brings together the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Subjects covered include the measurement and structure of networks in many branches of science, methods for analyzing network data, including methods developed in physics, statistics, and sociology, the fundamentals of graph theory, computer algorithms, and spectral methods, mathematical models of networks, including random graph models and generative models, and theories of dynamical processes taking place on networks.
Sergey V. Krivovichev
- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780199213207
- eISBN:
- 9780191707117
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213207.001.1
- Subject:
- Physics, Crystallography: Physics
This book deals with the structural crystallography of inorganic oxysalts in general. A special emphasis is placed upon structural topology and the methods of its description. The latter include ...
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This book deals with the structural crystallography of inorganic oxysalts in general. A special emphasis is placed upon structural topology and the methods of its description. The latter include graph theory, nets, 2-D and 3-D tilings, polyhedra, etc. The structures considered range from minerals to organically templated oxysalts, for all of which this book provides a unified approach to structure interpretation and classification. Most of the structures are analysed and it is shown that they possess the same topological genealogy and relationships, sometimes despite their obvious chemical differences. In order to expand the range of oxysalts considered, the book offers traditional schemes and also alternative approaches such as anion topologis, anion-centered polyhedra and cation arrays. It also looks into the amazingly complex and diverse world of inorganic oxysalts.Less
This book deals with the structural crystallography of inorganic oxysalts in general. A special emphasis is placed upon structural topology and the methods of its description. The latter include graph theory, nets, 2-D and 3-D tilings, polyhedra, etc. The structures considered range from minerals to organically templated oxysalts, for all of which this book provides a unified approach to structure interpretation and classification. Most of the structures are analysed and it is shown that they possess the same topological genealogy and relationships, sometimes despite their obvious chemical differences. In order to expand the range of oxysalts considered, the book offers traditional schemes and also alternative approaches such as anion topologis, anion-centered polyhedra and cation arrays. It also looks into the amazingly complex and diverse world of inorganic oxysalts.
Pavol Hell and Jaroslav Nešetřil
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198528173
- eISBN:
- 9780191713644
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528173.003.0004
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter focuses on the structure, as opposed to just the existence, of the family of homomorphisms among a set of graphs. The difference is noticeable with even just one graph. For instance, a ...
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This chapter focuses on the structure, as opposed to just the existence, of the family of homomorphisms among a set of graphs. The difference is noticeable with even just one graph. For instance, a graph having only the identity homomorphisms to itself is called rigid; rigid graphs are the building blocks of many constructions. Many useful constructions of rigid graphs are provided, and it is shown that asymptotically almost all graphs are rigid; infinite rigid graphs with arbitrary cardinality are also constructed. The homomorphisms among a set of graphs impose the algebraic structure of a category, and every finite category is represented by a set of graphs. This is the generalization of the theorem of Frucht from Chapter 1. Also, as in the case studied by Frucht, it is shown that the representing graphs can be required to have any of a number of graph theoretic properties. However, these properties cannot include having bounded degrees — somewhat surprisingly, since Frucht proved that cubic graphs represent all finite groups.Less
This chapter focuses on the structure, as opposed to just the existence, of the family of homomorphisms among a set of graphs. The difference is noticeable with even just one graph. For instance, a graph having only the identity homomorphisms to itself is called rigid; rigid graphs are the building blocks of many constructions. Many useful constructions of rigid graphs are provided, and it is shown that asymptotically almost all graphs are rigid; infinite rigid graphs with arbitrary cardinality are also constructed. The homomorphisms among a set of graphs impose the algebraic structure of a category, and every finite category is represented by a set of graphs. This is the generalization of the theorem of Frucht from Chapter 1. Also, as in the case studied by Frucht, it is shown that the representing graphs can be required to have any of a number of graph theoretic properties. However, these properties cannot include having bounded degrees — somewhat surprisingly, since Frucht proved that cubic graphs represent all finite groups.
Laura E. Chávez Lomelí and Luis A. Goddyn
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0002
- Subject:
- Mathematics, Probability / Statistics
This chapter focuses on extending the characterizations of Eulerian graphs via orientations. An Eulerian tour of a graph G induces an orientation with the property that every cocircuit (minimal edge ...
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This chapter focuses on extending the characterizations of Eulerian graphs via orientations. An Eulerian tour of a graph G induces an orientation with the property that every cocircuit (minimal edge cut) in G is traversed an equal number of times in each direction. In this sense, the orientation can be considered balanced. Applying duality to planar graphs, these notions produce characterizations of bipartite graphs. These notions are further extended to oriented matroids.Less
This chapter focuses on extending the characterizations of Eulerian graphs via orientations. An Eulerian tour of a graph G induces an orientation with the property that every cocircuit (minimal edge cut) in G is traversed an equal number of times in each direction. In this sense, the orientation can be considered balanced. Applying duality to planar graphs, these notions produce characterizations of bipartite graphs. These notions are further extended to oriented matroids.
Stefanie Gerke, Colin McDiarmid, Angelika Steger, and Andreas Weißl
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0006
- Subject:
- Mathematics, Probability / Statistics
This chapter investigates the behaviour of the function γ(q). In particular, it considers γ(q) as q approaches 1 from below and above and 3 from below, and explains the discontinuity as it approach 1 ...
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This chapter investigates the behaviour of the function γ(q). In particular, it considers γ(q) as q approaches 1 from below and above and 3 from below, and explains the discontinuity as it approach 1 from below by changing scale appropriately. New properties of Rn,q, for example in terms of the number of automorphisms, are investigated. Lower bounds are given on the maximum node degree and the maximum face size of a plane embedding. The relation between the growth constants of labelled and unlabelled planar graphs with given average degree is studied.Less
This chapter investigates the behaviour of the function γ(q). In particular, it considers γ(q) as q approaches 1 from below and above and 3 from below, and explains the discontinuity as it approach 1 from below by changing scale appropriately. New properties of Rn,q, for example in terms of the number of automorphisms, are investigated. Lower bounds are given on the maximum node degree and the maximum face size of a plane embedding. The relation between the growth constants of labelled and unlabelled planar graphs with given average degree is studied.
Andrew Goodall
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0007
- Subject:
- Mathematics, Probability / Statistics
This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in ...
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This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in terms of cosets of the Q-flows of G. Other applications to spanning trees of Cayley graphs and to group-valued models on phylogenetic trees are also presented to illustrate methods.Less
This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in terms of cosets of the Q-flows of G. Other applications to spanning trees of Cayley graphs and to group-valued models on phylogenetic trees are also presented to illustrate methods.
Rolf Niedermeier
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566076
- eISBN:
- 9780191713910
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566076.003.0010
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter provides an introduction to tree decomposition and treewidth, important concepts from modern graph theory. Treewidth is one of the best studied and most significant structural ...
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This chapter provides an introduction to tree decomposition and treewidth, important concepts from modern graph theory. Treewidth is one of the best studied and most significant structural parameters. The construction of tree decompositions is briefly discussed, followed by special considerations applying to planar graphs. The main focus of the chapter is on dynamic programming on tree decompositions, here demonstrated for the problems Vertex Cover and Dominating Set. The chapter closes by sketching the relationship to monadic second-order logic and some related graph width parameters.Less
This chapter provides an introduction to tree decomposition and treewidth, important concepts from modern graph theory. Treewidth is one of the best studied and most significant structural parameters. The construction of tree decompositions is briefly discussed, followed by special considerations applying to planar graphs. The main focus of the chapter is on dynamic programming on tree decompositions, here demonstrated for the problems Vertex Cover and Dominating Set. The chapter closes by sketching the relationship to monadic second-order logic and some related graph width parameters.
Rolf Niedermeier
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566076
- eISBN:
- 9780191713910
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566076.003.0015
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter briefly illustrates numerous case studies for parameterized complexity analysis. It begins with problems on planar and more general graphs, then discusses graph modification and other ...
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This chapter briefly illustrates numerous case studies for parameterized complexity analysis. It begins with problems on planar and more general graphs, then discusses graph modification and other graph problems, exhibits computational biology as an important application field, and concludes with logic and miscellaneous problems.Less
This chapter briefly illustrates numerous case studies for parameterized complexity analysis. It begins with problems on planar and more general graphs, then discusses graph modification and other graph problems, exhibits computational biology as an important application field, and concludes with logic and miscellaneous problems.
László Lovász
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0012
- Subject:
- Mathematics, Probability / Statistics
This chapter introduces a matrix, the kth connection matrix, for a given graph parameter and integer k ≥ 0. The properties of these matrices are closely connected with properties of the parameter. ...
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This chapter introduces a matrix, the kth connection matrix, for a given graph parameter and integer k ≥ 0. The properties of these matrices are closely connected with properties of the parameter. For example, the rank of this matrix is considered the minimum number of real numbers that has to be communicated across a node-cut of size k in order to evaluate the invariant. It is shown that other properties of this matrix, like whether it is semidefinite, also turn out to have graph theoretic significance.Less
This chapter introduces a matrix, the kth connection matrix, for a given graph parameter and integer k ≥ 0. The properties of these matrices are closely connected with properties of the parameter. For example, the rank of this matrix is considered the minimum number of real numbers that has to be communicated across a node-cut of size k in order to evaluate the invariant. It is shown that other properties of this matrix, like whether it is semidefinite, also turn out to have graph theoretic significance.
Steven D. Noble
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0013
- Subject:
- Mathematics, Probability / Statistics
This chapter examines the complexity of evaluating graph polynomials, related to the Tutte polynomial, for various classes of matroids. It begins with a short introduction to matroids, complexity, ...
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This chapter examines the complexity of evaluating graph polynomials, related to the Tutte polynomial, for various classes of matroids. It begins with a short introduction to matroids, complexity, and the Tutte polynomial. The intractability results for the Tutte polynomial are then discussed, including proof of the most often quoted result of classifying the hard points for evaluation of the Tutte polynomial in the graphic case. Contrasting results are presented, giving polynomial time algorithms for restricted classes of input, particularly graphs with bounded tree-width. Finally, the complexity results for various classes of matroids are presented.Less
This chapter examines the complexity of evaluating graph polynomials, related to the Tutte polynomial, for various classes of matroids. It begins with a short introduction to matroids, complexity, and the Tutte polynomial. The intractability results for the Tutte polynomial are then discussed, including proof of the most often quoted result of classifying the hard points for evaluation of the Tutte polynomial in the graphic case. Contrasting results are presented, giving polynomial time algorithms for restricted classes of input, particularly graphs with bounded tree-width. Finally, the complexity results for various classes of matroids are presented.
Rein Taagepera
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199534661
- eISBN:
- 9780191715921
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534661.003.0015
- Subject:
- Political Science, Comparative Politics, Political Economy
Take seriously the introductory advice by most introductory texts of statistics: graph the data and look at the graph so as to make sure linear regression makes sense from a statistical viewpoint. ...
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Take seriously the introductory advice by most introductory texts of statistics: graph the data and look at the graph so as to make sure linear regression makes sense from a statistical viewpoint. Graph more than the data – graph the entire conceptually allowed area and anchor points so as to make sure linear regression makes sense from a substantive viewpoint. If using linear regression, report not only the regression coefficients and the intercept but also the ranges, mean values, and medians of all input variables. While symmetric regression has advantages over OLS, fully reported symmetric regression is still merely regression.Less
Take seriously the introductory advice by most introductory texts of statistics: graph the data and look at the graph so as to make sure linear regression makes sense from a statistical viewpoint. Graph more than the data – graph the entire conceptually allowed area and anchor points so as to make sure linear regression makes sense from a substantive viewpoint. If using linear regression, report not only the regression coefficients and the intercept but also the ranges, mean values, and medians of all input variables. While symmetric regression has advantages over OLS, fully reported symmetric regression is still merely regression.
Roger D. Roger and Miles A. Whittington
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780195342796
- eISBN:
- 9780199776276
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195342796.003.0010
- Subject:
- Neuroscience, Molecular and Cellular Systems, Development
VFO occurs in in vitro models when chemical receptors are blocked. In particular, VFO does not require GABAA receptors, even though interneurons fire at high rates during in vivo very fast ...
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VFO occurs in in vitro models when chemical receptors are blocked. In particular, VFO does not require GABAA receptors, even though interneurons fire at high rates during in vivo very fast oscillations. VFO can be accounted for by a model in which neuronal spiking percolates through a sparse network of electrically coupled axons. This model predicts that VFO frequency depends on gap junction conductance, mediated by an effect on crossing time (i.e. the time it takes for a spike in one axon to elicit a spike in a coupled axon, estimated to be of order 0.2 ms). VFO in cerebellar slices also depends on gap junctions, but the physical principles are slightly different: cerebellar VFO appears to depend on many:one propagation of spiking, in effect a form of axonal coincidence detection.Less
VFO occurs in in vitro models when chemical receptors are blocked. In particular, VFO does not require GABAA receptors, even though interneurons fire at high rates during in vivo very fast oscillations. VFO can be accounted for by a model in which neuronal spiking percolates through a sparse network of electrically coupled axons. This model predicts that VFO frequency depends on gap junction conductance, mediated by an effect on crossing time (i.e. the time it takes for a spike in one axon to elicit a spike in a coupled axon, estimated to be of order 0.2 ms). VFO in cerebellar slices also depends on gap junctions, but the physical principles are slightly different: cerebellar VFO appears to depend on many:one propagation of spiking, in effect a form of axonal coincidence detection.
Thomas Koshy
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0008
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter presents delightful occurrences of Catalan numbers in the theory of trees. They include binary trees, full binary trees, planted trivalent binary trees, triangulations and prefix ...
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This chapter presents delightful occurrences of Catalan numbers in the theory of trees. They include binary trees, full binary trees, planted trivalent binary trees, triangulations and prefix expressions, chessboard, ordered rooted trees, forest, and ordered forests.Less
This chapter presents delightful occurrences of Catalan numbers in the theory of trees. They include binary trees, full binary trees, planted trivalent binary trees, triangulations and prefix expressions, chessboard, ordered rooted trees, forest, and ordered forests.
Remco van der Hofstad
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0006
- Subject:
- Mathematics, Geometry / Topology
In this chapter, we define percolation and random graph models, and survey the features of these models.
In this chapter, we define percolation and random graph models, and survey the features of these models.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0003
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter shows that the boxing and circling decorations of the BZL patterns are in a sense dual to each other. The circling and boxing rules seem quite different from each other, but they are ...
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This chapter shows that the boxing and circling decorations of the BZL patterns are in a sense dual to each other. The circling and boxing rules seem quite different from each other, but they are actually closely related, and the involution also sheds light on this fact. The chapter describes a natural box-circle duality: a bijection between the bᵢ and the lᵢ in which bᵢ is circled if and only if the corresponding lᵢ is boxed. It also considers a striking property of the crystal graph and proceeds by obtaining two BZL patterns in which circled entries in one correspond to boxed entries in the other, and illustrates this with an example.Less
This chapter shows that the boxing and circling decorations of the BZL patterns are in a sense dual to each other. The circling and boxing rules seem quite different from each other, but they are actually closely related, and the involution also sheds light on this fact. The chapter describes a natural box-circle duality: a bijection between the bᵢ and the lᵢ in which bᵢ is circled if and only if the corresponding lᵢ is boxed. It also considers a striking property of the crystal graph and proceeds by obtaining two BZL patterns in which circled entries in one correspond to boxed entries in the other, and illustrates this with an example.
Charles P. Boyer and Krzysztof Galicki
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0010
- Subject:
- Mathematics, Geometry / Topology
This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The ...
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This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The differential topology of links is a beautiful piece of mathematics, and the chapter offers a hands-on ‘user's guide’ approach with much emphasis on the famous work of Brieskorn in determining the difieomorphism types of certain homotopy spheres. This includes a presentation of the well known Brieskorn graph theorem as well as the geometry of Brieskorn-Pham links. When the singularities arise from weighted homogeneous polynomials, the links have a natural Sasakian structure with either definite (positive or negative) or null basic first Chern class. Emphasis is given to the positive case which corresponds to having positive Ricci curvature.Less
This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The differential topology of links is a beautiful piece of mathematics, and the chapter offers a hands-on ‘user's guide’ approach with much emphasis on the famous work of Brieskorn in determining the difieomorphism types of certain homotopy spheres. This includes a presentation of the well known Brieskorn graph theorem as well as the geometry of Brieskorn-Pham links. When the singularities arise from weighted homogeneous polynomials, the links have a natural Sasakian structure with either definite (positive or negative) or null basic first Chern class. Emphasis is given to the positive case which corresponds to having positive Ricci curvature.
