Mathew Penrose
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.001.0001
- Subject:
- Mathematics, Probability / Statistics
This book sets out a body of rigorous mathematical theory for finite graphs with nodes placed randomly in Euclidean d-space according to a common probability density, and edges added to connect ...
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This book sets out a body of rigorous mathematical theory for finite graphs with nodes placed randomly in Euclidean d-space according to a common probability density, and edges added to connect points that are close to each other. As an alternative to classical random graph models, these geometric graphs are relevant to the modelling of real networks having spatial content, arising for example in wireless communications, parallel processing, classification, epidemiology, astronomy, and the internet. Their study illustrates numerous techniques of modern stochastic geometry, including Stein's method, martingale methods, and continuum percolation. Typical results in the book concern properties of a graph G on n random points with edges included for interpoint distances up to r, with the parameter r dependent on n and typically small for large n. Asymptotic distributional properties are derived for numerous graph quantities. These include the number of copies of a given finite graph embedded in G, the number of isolated components isomorphic to a given graph, the empirical distributions of vertex degrees, the clique number, the chromatic number, the maximum and minimum degree, the size of the largest component, the total number of components, and the connectivity of the graph.Less
This book sets out a body of rigorous mathematical theory for finite graphs with nodes placed randomly in Euclidean d-space according to a common probability density, and edges added to connect points that are close to each other. As an alternative to classical random graph models, these geometric graphs are relevant to the modelling of real networks having spatial content, arising for example in wireless communications, parallel processing, classification, epidemiology, astronomy, and the internet. Their study illustrates numerous techniques of modern stochastic geometry, including Stein's method, martingale methods, and continuum percolation. Typical results in the book concern properties of a graph G on n random points with edges included for interpoint distances up to r, with the parameter r dependent on n and typically small for large n. Asymptotic distributional properties are derived for numerous graph quantities. These include the number of copies of a given finite graph embedded in G, the number of isolated components isomorphic to a given graph, the empirical distributions of vertex degrees, the clique number, the chromatic number, the maximum and minimum degree, the size of the largest component, the total number of components, and the connectivity of the graph.
Wilfrid S. Kendall and Ilya Molchanov (eds)
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.001.0001
- Subject:
- Mathematics, Geometry / Topology
Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, ...
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Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.Less
Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.
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.
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/9780199548927.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to ...
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This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to the percolation properties of these networks. The value of the percolation threshold for various degree distributions is estimated, and the resilience of scale-free networks against random failures is described. The hierarchical organization of k-cores in these networks is discussed. A few basic epidemic models on complex networks are introduced, and the evolution of diseases in networks is described.Less
This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to the percolation properties of these networks. The value of the percolation threshold for various degree distributions is estimated, and the resilience of scale-free networks against random failures is described. The hierarchical organization of k-cores in these networks is discussed. A few basic epidemic models on complex networks are introduced, and the evolution of diseases in networks is described.
Maria Deijfen and Olle Häggström
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0002
- Subject:
- Mathematics, Probability / Statistics, Analysis
This chapter provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on a lattice. In its simplest formulation, the ...
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This chapter provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on a lattice. In its simplest formulation, the Richardson model describes the evolution of a single infectious entity on the lattice, but more recently the dynamics have been extended to comprise two competing growing entities. For this version of the model, the main question is whether there is a positive probability for both entities to simultaneously grow to occupy infinite parts of the lattice, the conjecture being that the answer is yes if and only if the entities have the same intensity. In this paper attention focuses on the two-type model, but the most important results for the one-type version are also described.Less
This chapter provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on a lattice. In its simplest formulation, the Richardson model describes the evolution of a single infectious entity on the lattice, but more recently the dynamics have been extended to comprise two competing growing entities. For this version of the model, the main question is whether there is a positive probability for both entities to simultaneously grow to occupy infinite parts of the lattice, the conjecture being that the answer is yes if and only if the entities have the same intensity. In this paper attention focuses on the two-type model, but the most important results for the one-type version are also described.
Akira Sakai
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0006
- Subject:
- Mathematics, Probability / Statistics, Analysis
Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical ...
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Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical behaviour. It is rich and still far from fully understood. The reason why it is so difficult is due to the increase to infinity of the number of strongly correlated variables in the vicinity of the critical point. For example, the Ising model, which is a model for magnets, exhibits critical behaviour as the temperature comes closer to its critical value; the closer the temperature is to criticality, the more spin variables cooperate with each other to attain the global magnetization. In this regime, neither standard probability theory for independent random variables nor naive perturbation techniques work. The lace expansion, which is the topic of this article, is currently one of the few approaches to rigorous investigation of critical behaviour for various statistical-mechanical models. The chapter summarizes some of the most intriguing lace-expansion results for self-avoiding walk (SAW), percolation, and the Ising model.Less
Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical behaviour. It is rich and still far from fully understood. The reason why it is so difficult is due to the increase to infinity of the number of strongly correlated variables in the vicinity of the critical point. For example, the Ising model, which is a model for magnets, exhibits critical behaviour as the temperature comes closer to its critical value; the closer the temperature is to criticality, the more spin variables cooperate with each other to attain the global magnetization. In this regime, neither standard probability theory for independent random variables nor naive perturbation techniques work. The lace expansion, which is the topic of this article, is currently one of the few approaches to rigorous investigation of critical behaviour for various statistical-mechanical models. The chapter summarizes some of the most intriguing lace-expansion results for self-avoiding walk (SAW), percolation, and the Ising model.
Geoffrey Grimmett and Colin McDiarmid (eds)
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.001.0001
- Subject:
- Mathematics, Probability / Statistics
Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and ...
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Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and inspired generations of students and researchers in mathematics. This book summarizes and reviews the consistent themes from his work through a series of articles written by renowned experts. These articles, presented as chapters, contain original research work, set in a broader context by the inclusion of review material.Less
Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and inspired generations of students and researchers in mathematics. This book summarizes and reviews the consistent themes from his work through a series of articles written by renowned experts. These articles, presented as chapters, contain original research work, set in a broader context by the inclusion of review material.
Mathew Penrose
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.003.0009
- Subject:
- Mathematics, Probability / Statistics
This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) ...
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This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) arguments are described for estimating the number of connected sets in the lattice, and elements of (lattice) percolation theory are described. A multiparameter ergodic theorem is given, and the basic theory of continuum percolation is described. Some of the theory of Poisson point processes are recalled, including the superposition, thinning, and scaling theorems.Less
This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) arguments are described for estimating the number of connected sets in the lattice, and elements of (lattice) percolation theory are described. A multiparameter ergodic theorem is given, and the basic theory of continuum percolation is described. Some of the theory of Poisson point processes are recalled, including the superposition, thinning, and scaling theorems.
Darrell Duffie
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691138961
- eISBN:
- 9781400840519
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691138961.003.0005
- Subject:
- Economics and Finance, Financial Economics
This chapter describes a simple model of the “percolation” of information of common interest through an over-the-counter market with many agents. It also includes an explicit solution for the ...
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This chapter describes a simple model of the “percolation” of information of common interest through an over-the-counter market with many agents. It also includes an explicit solution for the cross-sectional distribution of posterior beliefs at each time. It begins with the basic information structure for the economy and the setting for search and random matching. It then shows how to solve the model for the dynamics of the cross-sectional distribution of information. The remainder of the chapter is devoted to market settings and to extensions of the model that handle public releases of information, the receipt of new private information over time, and the release of information among groups of more than two agents at a time.Less
This chapter describes a simple model of the “percolation” of information of common interest through an over-the-counter market with many agents. It also includes an explicit solution for the cross-sectional distribution of posterior beliefs at each time. It begins with the basic information structure for the economy and the setting for search and random matching. It then shows how to solve the model for the dynamics of the cross-sectional distribution of information. The remainder of the chapter is devoted to market settings and to extensions of the model that handle public releases of information, the receipt of new private information over time, and the release of information among groups of more than two agents at a time.
Janko Gravner
- Published in print:
- 2003
- Published Online:
- November 2020
- ISBN:
- 9780195137170
- eISBN:
- 9780197561652
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195137170.003.0010
- Subject:
- Computer Science, Systems Analysis and Design
We illustrate growth phenomena in two-dimensional cellular automata (CA) by four case studies. The first CA, which we call Obstacle Course, describes the effect that ...
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We illustrate growth phenomena in two-dimensional cellular automata (CA) by four case studies. The first CA, which we call Obstacle Course, describes the effect that obstacles have on such features of simple growth models as linear expansion and coherent asymptotic shape. Our next CA is random-walk-based Internal Diffusion Limited Aggregation, which spreads sublinearly, but with a shape which can be explicitly computed due to hydrodynamic effects. Then we propose a simple scheme for characterizing CA according to their growth properties, as indicated by two Larger than Life examples. Finally, a very simple case of Spatial Prisoner’s Dilemma illustrates nucleation analysis of CA. In essence, analysis of growth models is an attempt to study properties of physical systems far from equilibrium (e.g., Meakin [34] and more than 1300 references cited in the latter). Cellular automata (CA) growth models, by virtue of their simplicity and amenability to computer experimentation [25], have become particularly popular in the last 20 years, especially in physics research literature [40, 42]. Needless to say, precise mathematical results are hard to come by, and many basic questions remain completely open at the rigorous level. The purpose of this chapter, then, is to outline some successes of the mathematical approach and to identify some fundamental difficulties. We will mainly address three themes which can be summarized by the terms: aggregation, nucleation, and constraint-expansion transition. These themes also provide opportunities to touch on the roles of randomness, monotonicity, and linearity in CA investigations. We choose to illustrate these issues by particular CA rules, with little attempt to formulate a general theory. Simplicity is often, and rightly, touted as an important selling point of cellular automata. We have, therefore, tried to choose the simplest models which, while being amenable to some mathematical analysis, raise a host of intriguing unanswered questions. The next few paragraphs outline subsequent sections of this chapter. Aggregation models typically study properties of growth from a small initial seed. Arguably, the simplest dynamics are obtained by adding sites on the boundary in a uniform fashion.
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We illustrate growth phenomena in two-dimensional cellular automata (CA) by four case studies. The first CA, which we call Obstacle Course, describes the effect that obstacles have on such features of simple growth models as linear expansion and coherent asymptotic shape. Our next CA is random-walk-based Internal Diffusion Limited Aggregation, which spreads sublinearly, but with a shape which can be explicitly computed due to hydrodynamic effects. Then we propose a simple scheme for characterizing CA according to their growth properties, as indicated by two Larger than Life examples. Finally, a very simple case of Spatial Prisoner’s Dilemma illustrates nucleation analysis of CA. In essence, analysis of growth models is an attempt to study properties of physical systems far from equilibrium (e.g., Meakin [34] and more than 1300 references cited in the latter). Cellular automata (CA) growth models, by virtue of their simplicity and amenability to computer experimentation [25], have become particularly popular in the last 20 years, especially in physics research literature [40, 42]. Needless to say, precise mathematical results are hard to come by, and many basic questions remain completely open at the rigorous level. The purpose of this chapter, then, is to outline some successes of the mathematical approach and to identify some fundamental difficulties. We will mainly address three themes which can be summarized by the terms: aggregation, nucleation, and constraint-expansion transition. These themes also provide opportunities to touch on the roles of randomness, monotonicity, and linearity in CA investigations. We choose to illustrate these issues by particular CA rules, with little attempt to formulate a general theory. Simplicity is often, and rightly, touted as an important selling point of cellular automata. We have, therefore, tried to choose the simplest models which, while being amenable to some mathematical analysis, raise a host of intriguing unanswered questions. The next few paragraphs outline subsequent sections of this chapter. Aggregation models typically study properties of growth from a small initial seed. Arguably, the simplest dynamics are obtained by adding sites on the boundary in a uniform fashion.
Sergei Zuyev
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0016
- Subject:
- Mathematics, Geometry / Topology
Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and ...
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Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and analysis of modern telecommunications systems, in which spatial arrangement is typically a crucial consideration in their performance evaluation, optimization or future development. In this survey we aim to summarize the main stochastic geometry models and tools currently used in studying modern telecommunications. We outline specifics of wired, wireless fixed and ad hoc systems and show how stochastic geometry modelling helps in their analysis and optimization. Point and line processes, Palm theory, shot‐noise processes, random tessellations, Boolean models, percolation, random graphs and networks, spatial statistics and optimization: this is a far from exhaustive list of techniques used in studying contemporary telecommunications systems and which we shall briefly discuss.Less
Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and analysis of modern telecommunications systems, in which spatial arrangement is typically a crucial consideration in their performance evaluation, optimization or future development. In this survey we aim to summarize the main stochastic geometry models and tools currently used in studying modern telecommunications. We outline specifics of wired, wireless fixed and ad hoc systems and show how stochastic geometry modelling helps in their analysis and optimization. Point and line processes, Palm theory, shot‐noise processes, random tessellations, Boolean models, percolation, random graphs and networks, spatial statistics and optimization: this is a far from exhaustive list of techniques used in studying contemporary telecommunications systems and which we shall briefly discuss.
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.
Seth Cable
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780195392265
- eISBN:
- 9780199866526
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195392265.003.0004
- Subject:
- Linguistics, Syntax and Morphology
This chapter begins the argument that the ‘Q-based’ account should be extended to all other wh-fronting languages. It first presents some general typological and learning-theoretic arguments. These ...
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This chapter begins the argument that the ‘Q-based’ account should be extended to all other wh-fronting languages. It first presents some general typological and learning-theoretic arguments. These include (i) the fact that the ‘Q-based’ account is transparently motivated for other wh-fronting languages (such as Edo), and (ii) the ability for the ‘Q-based’ account to provide a unified theory of the ill-formedness of P-stranding and left-branch extractions across languages. Following this, the consequences of the account for the theory of pied-piping structures are examined. In particular, it is shown that extending the ‘Q-based’ account to all other wh-fronting languages entails that the phenomenon dubbed ‘pied-piping’ need not exist at all. Finally, the chapter develops a ‘Q-based’ theory of multiple wh-questions in English and German. It is shown that the analysis predicts the complementary distribution of Superiority Effects and Intervention Effects in these languages. Finally, Intervention Effects in pied-piping structures are examined, and the ‘Q-based’ theory is shown to make an accurate (and surprising) prediction.Less
This chapter begins the argument that the ‘Q-based’ account should be extended to all other wh-fronting languages. It first presents some general typological and learning-theoretic arguments. These include (i) the fact that the ‘Q-based’ account is transparently motivated for other wh-fronting languages (such as Edo), and (ii) the ability for the ‘Q-based’ account to provide a unified theory of the ill-formedness of P-stranding and left-branch extractions across languages. Following this, the consequences of the account for the theory of pied-piping structures are examined. In particular, it is shown that extending the ‘Q-based’ account to all other wh-fronting languages entails that the phenomenon dubbed ‘pied-piping’ need not exist at all. Finally, the chapter develops a ‘Q-based’ theory of multiple wh-questions in English and German. It is shown that the analysis predicts the complementary distribution of Superiority Effects and Intervention Effects in these languages. Finally, Intervention Effects in pied-piping structures are examined, and the ‘Q-based’ theory is shown to make an accurate (and surprising) prediction.
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/9780199548927.003.0014
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter demonstrates how a universal requirement for optimality leads to the complex structural organization of a network. It discusses a long-lasting criticism of the preferential concept and ...
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This chapter demonstrates how a universal requirement for optimality leads to the complex structural organization of a network. It discusses a long-lasting criticism of the preferential concept and describes existing approaches to the optimization-driven evolution of complex networks. In particular, the optimized trade-off model of a growing network is described, as well as models showing the explosive percolation phenomenon.Less
This chapter demonstrates how a universal requirement for optimality leads to the complex structural organization of a network. It discusses a long-lasting criticism of the preferential concept and describes existing approaches to the optimization-driven evolution of complex networks. In particular, the optimized trade-off model of a growing network is described, as well as models showing the explosive percolation phenomenon.
Mark Newman
- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805090
- eISBN:
- 9780191843235
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805090.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are ...
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A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are randomly removed from a network, and of the percolation phase transition at which a giant percolating cluster forms. The properties of percolation on configuration model networks are studied, including networks with power-law degree distributions, and including both uniform and non-uniform removal of nodes. Computer algorithms for simulating percolation on real-world networks are also discussed, and numerical results are given for several example networks, including the internet and a social network.Less
A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are randomly removed from a network, and of the percolation phase transition at which a giant percolating cluster forms. The properties of percolation on configuration model networks are studied, including networks with power-law degree distributions, and including both uniform and non-uniform removal of nodes. Computer algorithms for simulating percolation on real-world networks are also discussed, and numerical results are given for several example networks, including the internet and a social network.
A.F. Borghesani
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199213603
- eISBN:
- 9780191707421
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213603.003.0027
- Subject:
- Physics, Condensed Matter Physics / Materials
Experiments on the mobility of electrons in dense helium gas elucidated how localized electron states develop when the gas density gas is increased. Up to 77 K, the density dependence of the mobility ...
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Experiments on the mobility of electrons in dense helium gas elucidated how localized electron states develop when the gas density gas is increased. Up to 77 K, the density dependence of the mobility clearly shows that the formation of electron bubbles is a continuous phenomenon. Localization of electrons in bubbles also appears at high temperatures if the density is so large that the free energy of the localized state is negative enough. Percolation and hydrodynamic models have been devised to explain the continuous transition from high-mobility states to low-mobility states. It is shown that density-dependent, quantum multiple scattering effects modify the energy of the nearly free electron in a way that can be accurately described by heuristically modifying the kinetic theory prediction.Less
Experiments on the mobility of electrons in dense helium gas elucidated how localized electron states develop when the gas density gas is increased. Up to 77 K, the density dependence of the mobility clearly shows that the formation of electron bubbles is a continuous phenomenon. Localization of electrons in bubbles also appears at high temperatures if the density is so large that the free energy of the localized state is negative enough. Percolation and hydrodynamic models have been devised to explain the continuous transition from high-mobility states to low-mobility states. It is shown that density-dependent, quantum multiple scattering effects modify the energy of the nearly free electron in a way that can be accurately described by heuristically modifying the kinetic theory prediction.
Pierre M. Adler, Jean-François Thovert, and Valeri V. Mourzenko
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199666515
- eISBN:
- 9780191748639
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199666515.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
This book aims to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities. Attention is focused on geological media where ...
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This book aims to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities. Attention is focused on geological media where rocks are necessarily fractured at various scales by the slow but constant motion of continental masses. This book is situated between three disciplines. First, geology and geophysics provide most of the data and most applications; a characteristic feature is that one never has a complete knowledge of the studied objects — such as an oil reservoir — in contrast with a laboratory experiment where every quantity can be measured. Second, engineering develops the main tools of analysis for one- and two-phase flows, and calculations of permeability (absolute and relative). Third, statistical physics plays a major role in concepts such as the excluded volume, dimensionless density, percolation threshold and power laws. In view of this interdisciplinary character, the general results presented in this book may have unexpected applications in many different domains. This book is based on courses which have been taught in several countries at Master and Ph.D. levels in universities, in research centers and at conferences. It should provide, in a compact form, all the necessary tools to achieve the general objective. The mathematical level in this book has been kept as low as possible. The interested reader can always go further, thanks to the references that are provided, where the mathematical level is not restricted. The colloquial aspect of a course has been preserved where one tries to explain abstract concepts in simple terms.Less
This book aims to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities. Attention is focused on geological media where rocks are necessarily fractured at various scales by the slow but constant motion of continental masses. This book is situated between three disciplines. First, geology and geophysics provide most of the data and most applications; a characteristic feature is that one never has a complete knowledge of the studied objects — such as an oil reservoir — in contrast with a laboratory experiment where every quantity can be measured. Second, engineering develops the main tools of analysis for one- and two-phase flows, and calculations of permeability (absolute and relative). Third, statistical physics plays a major role in concepts such as the excluded volume, dimensionless density, percolation threshold and power laws. In view of this interdisciplinary character, the general results presented in this book may have unexpected applications in many different domains. This book is based on courses which have been taught in several countries at Master and Ph.D. levels in universities, in research centers and at conferences. It should provide, in a compact form, all the necessary tools to achieve the general objective. The mathematical level in this book has been kept as low as possible. The interested reader can always go further, thanks to the references that are provided, where the mathematical level is not restricted. The colloquial aspect of a course has been preserved where one tries to explain abstract concepts in simple terms.
V.F. Gantmakher
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198567561
- eISBN:
- 9780191718267
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567561.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter is dedicated to electron transitions between localized states. After a description of the states with the help of localization radius and the general expression for the transition ...
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This chapter is dedicated to electron transitions between localized states. After a description of the states with the help of localization radius and the general expression for the transition probability between two states — the hopping conductivity theory — is presented. The theory is based on Abrahams-Miller network of random resistances modelling an insulator and the percolation theory. Three types of hopping conductivity are presented: nearest-neighbour hopping, variable-range hopping with Coulomb gap at the Fermi level (Efros-Shklovskii law), and variable-range hopping without Coulomb gap (Mott law). They can be distinguished by the temperature dependence of the activated conductance.Less
This chapter is dedicated to electron transitions between localized states. After a description of the states with the help of localization radius and the general expression for the transition probability between two states — the hopping conductivity theory — is presented. The theory is based on Abrahams-Miller network of random resistances modelling an insulator and the percolation theory. Three types of hopping conductivity are presented: nearest-neighbour hopping, variable-range hopping with Coulomb gap at the Fermi level (Efros-Shklovskii law), and variable-range hopping without Coulomb gap (Mott law). They can be distinguished by the temperature dependence of the activated conductance.
Richard Evan Schwartz
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691181387
- eISBN:
- 9780691188997
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691181387.001.0001
- Subject:
- Mathematics, Educational Mathematics
Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane ...
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Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.Less
Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.
J. Klafter and I. M. Sokolov
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.001.0001
- Subject:
- Physics, Soft Matter / Biological Physics
The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same ...
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The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays theory of random walks was proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub‐ and superdiffusive transport processes as well. This book discusses main variants of the random walks and gives the most important mathematical tools for their theoretical description.Less
The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays theory of random walks was proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub‐ and superdiffusive transport processes as well. This book discusses main variants of the random walks and gives the most important mathematical tools for their theoretical description.