Philip Hans Franses and Richard Paap
- Published in print:
- 2004
- Published Online:
- August 2004
- ISBN:
- 9780199242023
- eISBN:
- 9780191601286
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019924202X.001.0001
- Subject:
- Economics and Finance, Econometrics
This book considers periodic time series models for seasonal data, characterized by parameters that differ across the seasons, and focuses on their usefulness for out-of-sample forecasting. Providing ...
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This book considers periodic time series models for seasonal data, characterized by parameters that differ across the seasons, and focuses on their usefulness for out-of-sample forecasting. Providing an up-to-date survey of the recent developments in periodic time series, the book presents a large number of empirical results. The first part of the book deals with model selection, diagnostic checking, and forecasting of univariate periodic autoregressive models. It discusses tests for periodic integration, and provides an extensive discussion of the role of deterministic regressors in testing for periodic integration and in forecasting. The second part discusses multivariate periodic autoregressive models. It provides an overview of periodic cointegration models, including single-equation type tests and a full-system approach based on generalized method of moments. All methods are illustrated with extensive examples, and the book will be invaluable to advanced graduate students and researchers in econometrics and to practitioners looking for an understanding of how to approach seasonal data.Less
This book considers periodic time series models for seasonal data, characterized by parameters that differ across the seasons, and focuses on their usefulness for out-of-sample forecasting. Providing an up-to-date survey of the recent developments in periodic time series, the book presents a large number of empirical results. The first part of the book deals with model selection, diagnostic checking, and forecasting of univariate periodic autoregressive models. It discusses tests for periodic integration, and provides an extensive discussion of the role of deterministic regressors in testing for periodic integration and in forecasting. The second part discusses multivariate periodic autoregressive models. It provides an overview of periodic cointegration models, including single-equation type tests and a full-system approach based on generalized method of moments. All methods are illustrated with extensive examples, and the book will be invaluable to advanced graduate students and researchers in econometrics and to practitioners looking for an understanding of how to approach seasonal data.
Aman Ullah
- Published in print:
- 2004
- Published Online:
- August 2004
- ISBN:
- 9780198774471
- eISBN:
- 9780191601347
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774478.001.0001
- Subject:
- Economics and Finance, Econometrics
This book presents a comprehensive and unified treatment of finite sample theory, and its application to estimators and test statistics used in various econometric models. Time series, cross section, ...
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This book presents a comprehensive and unified treatment of finite sample theory, and its application to estimators and test statistics used in various econometric models. Time series, cross section, and panel data models are considered. The results are explored for linear and nonlinear models, as well as models with normal and nonnormal errors. The book contains seven chapters. Chapter 1 presents an introduction to finite sample econometrics. Chapter 2 gives methods of obtaining the moments of econometric statistics. Chapter 3 provides methods for analysing distributions. Finite sample results for various econometric models are discussed in Chapters 4-7.Less
This book presents a comprehensive and unified treatment of finite sample theory, and its application to estimators and test statistics used in various econometric models. Time series, cross section, and panel data models are considered. The results are explored for linear and nonlinear models, as well as models with normal and nonnormal errors. The book contains seven chapters. Chapter 1 presents an introduction to finite sample econometrics. Chapter 2 gives methods of obtaining the moments of econometric statistics. Chapter 3 provides methods for analysing distributions. Finite sample results for various econometric models are discussed in Chapters 4-7.
Manuel Arellano
- Published in print:
- 2003
- Published Online:
- July 2005
- ISBN:
- 9780199245284
- eISBN:
- 9780191602481
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199245282.001.0001
- Subject:
- Economics and Finance, Econometrics
This book reviews some of the main topics in panel data econometrics. It analyses econometric models with non-exogenous explanatory variables, and the problem of distinguishing between dynamic ...
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This book reviews some of the main topics in panel data econometrics. It analyses econometric models with non-exogenous explanatory variables, and the problem of distinguishing between dynamic responses and unobserved heterogeneity in panel data models. The book is divided into three parts. Part I deals with static models. Part II discusses pure time series models. Part III considers dynamic conditional models.Less
This book reviews some of the main topics in panel data econometrics. It analyses econometric models with non-exogenous explanatory variables, and the problem of distinguishing between dynamic responses and unobserved heterogeneity in panel data models. The book is divided into three parts. Part I deals with static models. Part II discusses pure time series models. Part III considers dynamic conditional models.
Søren Johansen
- Published in print:
- 1995
- Published Online:
- November 2003
- ISBN:
- 9780198774501
- eISBN:
- 9780191596476
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774508.001.0001
- Subject:
- Economics and Finance, Econometrics
This monograph is concerned with the statistical analysis of multivariate systems of non‐stationary time series of type I(1). It applies the concepts of cointegration and common trends in the ...
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This monograph is concerned with the statistical analysis of multivariate systems of non‐stationary time series of type I(1). It applies the concepts of cointegration and common trends in the framework of the Gaussian vector autoregressive model. The main result on the structure of cointegrated processes as defined by the error correction model is Grangers representation theorem. The statistical results include derivation of the trace test for cointegrating rank, test on cointegrating relations, and test on adjustment coefficients and their asymptotic distributions.Less
This monograph is concerned with the statistical analysis of multivariate systems of non‐stationary time series of type I(1). It applies the concepts of cointegration and common trends in the framework of the Gaussian vector autoregressive model. The main result on the structure of cointegrated processes as defined by the error correction model is Grangers representation theorem. The statistical results include derivation of the trace test for cointegrating rank, test on cointegrating relations, and test on adjustment coefficients and their asymptotic distributions.
Mark J. Joe
- Published in print:
- 2006
- Published Online:
- October 2011
- ISBN:
- 9780199205301
- eISBN:
- 9780191695612
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205301.003.0022
- Subject:
- Business and Management, Corporate Governance and Accountability, Business History
This chapter examines political change in continental Europe. The recent demand for more pro-shareholder corporate governance in continental Europe and for legal structures that facilitate ownership ...
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This chapter examines political change in continental Europe. The recent demand for more pro-shareholder corporate governance in continental Europe and for legal structures that facilitate ownership separation follows European politics' move rightward, away from its late 20th-century history of strong social democratic politics. Politics, product markets, efforts to improve corporate law's institutions, and new capital-raising methods are all moving simultaneously.Less
This chapter examines political change in continental Europe. The recent demand for more pro-shareholder corporate governance in continental Europe and for legal structures that facilitate ownership separation follows European politics' move rightward, away from its late 20th-century history of strong social democratic politics. Politics, product markets, efforts to improve corporate law's institutions, and new capital-raising methods are all moving simultaneously.
Paul C. Gutjahr
- Published in print:
- 2011
- Published Online:
- May 2011
- ISBN:
- 9780199740420
- eISBN:
- 9780199894703
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199740420.003.0021
- Subject:
- Religion, Church History
Chapter twenty-one tells the story of the changing nature of the Biblical Repertory. Upon his return from Europe, Hodge decides to rename and change the name of this periodical, attempting to make it ...
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Chapter twenty-one tells the story of the changing nature of the Biblical Repertory. Upon his return from Europe, Hodge decides to rename and change the name of this periodical, attempting to make it appeal to a wider audience. Popularizing the journal failed, but the Repertory did become a major theological voice within American Presbyterian circles.Less
Chapter twenty-one tells the story of the changing nature of the Biblical Repertory. Upon his return from Europe, Hodge decides to rename and change the name of this periodical, attempting to make it appeal to a wider audience. Popularizing the journal failed, but the Repertory did become a major theological voice within American Presbyterian circles.
Andrew J. Connolly, Jacob T. VanderPlas, Alexander Gray, Andrew J. Connolly, Jacob T. VanderPlas, and Alexander Gray
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691151687
- eISBN:
- 9781400848911
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691151687.003.0010
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter summarizes the fundamental concepts and tools for analyzing time series data. Time series analysis is a branch of applied mathematics developed mostly in the fields of signal processing ...
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This chapter summarizes the fundamental concepts and tools for analyzing time series data. Time series analysis is a branch of applied mathematics developed mostly in the fields of signal processing and statistics. Contributions to this field, from an astronomical perspective, have predominantly focused on unevenly sampled data, low signal-to-noise data, and heteroscedastic errors. The chapter starts with a brief introduction to the main concepts in time series analysis. It then discusses the main tools from the modeling toolkit for time series analysis. Despite being set in the context of time series, many tools and results are readily applicable in other domains, and for this reason the examples presented will not be strictly limited to time-domain data. Armed with the modeling toolkit, the chapter goes on to discuss the analysis of periodic time series, search for temporally localized signals, and concludes with a brief discussion of stochastic processes.Less
This chapter summarizes the fundamental concepts and tools for analyzing time series data. Time series analysis is a branch of applied mathematics developed mostly in the fields of signal processing and statistics. Contributions to this field, from an astronomical perspective, have predominantly focused on unevenly sampled data, low signal-to-noise data, and heteroscedastic errors. The chapter starts with a brief introduction to the main concepts in time series analysis. It then discusses the main tools from the modeling toolkit for time series analysis. Despite being set in the context of time series, many tools and results are readily applicable in other domains, and for this reason the examples presented will not be strictly limited to time-domain data. Armed with the modeling toolkit, the chapter goes on to discuss the analysis of periodic time series, search for temporally localized signals, and concludes with a brief discussion of stochastic processes.
A. C. Lloyd
- Published in print:
- 1998
- Published Online:
- November 2003
- ISBN:
- 9780198238065
- eISBN:
- 9780191597916
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198238061.001.0001
- Subject:
- Philosophy, Ancient Philosophy
Neoplatonism is traditionally considered a mystical philosophy; on the contrary the aim of this book is to show the importance of a logical and epistemological approach for the understanding of ...
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Neoplatonism is traditionally considered a mystical philosophy; on the contrary the aim of this book is to show the importance of a logical and epistemological approach for the understanding of Neoplatonic basic ontological problems. In doing that, Lloyd considers a very wide range of philosophers from Plotinus up to Byzantine Neoplatonists. After a preliminary discussion of how the Neoplatonic semantics and logical concepts are a result of their understanding of previous philosophers (in particular Aristotle), Lloyd introduces the key point of the book, i.e. his theory of the P‐series. A P‐series is a group of terms ordered according to priority a posteriority in which the first term is universal and common to all the others. This logical theory is applied to explain how the reality is structured (procession and emanation from the One) and how knowledge is constituted. The book ends with an analysis of how mystical apprehension differs from the union with the One.Less
Neoplatonism is traditionally considered a mystical philosophy; on the contrary the aim of this book is to show the importance of a logical and epistemological approach for the understanding of Neoplatonic basic ontological problems. In doing that, Lloyd considers a very wide range of philosophers from Plotinus up to Byzantine Neoplatonists. After a preliminary discussion of how the Neoplatonic semantics and logical concepts are a result of their understanding of previous philosophers (in particular Aristotle), Lloyd introduces the key point of the book, i.e. his theory of the P‐series. A P‐series is a group of terms ordered according to priority a posteriority in which the first term is universal and common to all the others. This logical theory is applied to explain how the reality is structured (procession and emanation from the One) and how knowledge is constituted. The book ends with an analysis of how mystical apprehension differs from the union with the One.
Elinor Scarbrough and Eric Tanenbaum (eds)
- Published in print:
- 1998
- Published Online:
- November 2003
- ISBN:
- 9780198292371
- eISBN:
- 9780191600159
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198292376.001.0001
- Subject:
- Political Science, Reference
This volume is a collection of commissioned articles by 16 experts in social science methodology, each contribution introducing experienced social scientists to more advanced analytic techniques. The ...
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This volume is a collection of commissioned articles by 16 experts in social science methodology, each contribution introducing experienced social scientists to more advanced analytic techniques. The contributions explain the theoretical underpinnings of a particular technique, and illustrate the approach with a worked example. The techniques covered are the basic regression model and its extensions, linear structural equation modelling, log‐linear and latent class models, multi‐level modelling, and three extensions to modelling time series data. In these contributions, statistical notation is kept to a minimum; where necessary, it is consigned to footnotes or an appendix. Three final contributions introduce new developments in rational choice theory and discourse analysis.Less
This volume is a collection of commissioned articles by 16 experts in social science methodology, each contribution introducing experienced social scientists to more advanced analytic techniques. The contributions explain the theoretical underpinnings of a particular technique, and illustrate the approach with a worked example. The techniques covered are the basic regression model and its extensions, linear structural equation modelling, log‐linear and latent class models, multi‐level modelling, and three extensions to modelling time series data. In these contributions, statistical notation is kept to a minimum; where necessary, it is consigned to footnotes or an appendix. Three final contributions introduce new developments in rational choice theory and discourse analysis.
Terryl C. Givens
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780195167115
- eISBN:
- 9780199785599
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195167115.003.0017
- Subject:
- Religion, Religion and Society
Film has largely replaced theater as the dramatic medium for Mormon artists. Film was early employed in the anti-Mormon campaign, and Mormons learned early to use it for their own purposes. ...
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Film has largely replaced theater as the dramatic medium for Mormon artists. Film was early employed in the anti-Mormon campaign, and Mormons learned early to use it for their own purposes. Evangelistic (missionary films) and public relations varieties (Homefront series) have given way to a recent generation of talented filmmakers who are creating two kinds: one is an insider genre, while the other explores ways of informing popular film with a Mormon sensibility.Less
Film has largely replaced theater as the dramatic medium for Mormon artists. Film was early employed in the anti-Mormon campaign, and Mormons learned early to use it for their own purposes. Evangelistic (missionary films) and public relations varieties (Homefront series) have given way to a recent generation of talented filmmakers who are creating two kinds: one is an insider genre, while the other explores ways of informing popular film with a Mormon sensibility.
John E. Jackson
- Published in print:
- 1998
- Published Online:
- November 2003
- ISBN:
- 9780198294719
- eISBN:
- 9780191599361
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198294719.003.0032
- Subject:
- Political Science, Reference
Reviews methodological techniques available across the discipline of political science. Econometrics and political science methods include structural equation estimations, time‐series analysis, and ...
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Reviews methodological techniques available across the discipline of political science. Econometrics and political science methods include structural equation estimations, time‐series analysis, and non‐linear models. Alternative approaches analyse public preferences, political institutions, and path dependence political economy modelling. The drawbacks of these methods are examined by questioning their underlying assumptions and examining their consequences. While there is cause for concern, solace lies in the fact that these problems are also faced across other disciplines.Less
Reviews methodological techniques available across the discipline of political science. Econometrics and political science methods include structural equation estimations, time‐series analysis, and non‐linear models. Alternative approaches analyse public preferences, political institutions, and path dependence political economy modelling. The drawbacks of these methods are examined by questioning their underlying assumptions and examining their consequences. While there is cause for concern, solace lies in the fact that these problems are also faced across other disciplines.
D. A. Bini, G. Latouche, and B. Meini
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0003
- Subject:
- Mathematics, Numerical Analysis
This chapter is concerned with infinite block Toeplitz matrices, their relationships with matrix power series and matrix Laurent power series, and the fundamental problem of solving matrix equations ...
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This chapter is concerned with infinite block Toeplitz matrices, their relationships with matrix power series and matrix Laurent power series, and the fundamental problem of solving matrix equations and computing canonical factorizations. It introduces the concept of a Wiener algebra and provides a natural theoretical framework where the convergence properties of algorithms for solving Markov chains can easily be proved. Furthermore, the notion of canonical factorization provides a powerful tool for solving infinite linear systems, such as the fundamental system involving the stationary probability distribution of structured Markov chains.Less
This chapter is concerned with infinite block Toeplitz matrices, their relationships with matrix power series and matrix Laurent power series, and the fundamental problem of solving matrix equations and computing canonical factorizations. It introduces the concept of a Wiener algebra and provides a natural theoretical framework where the convergence properties of algorithms for solving Markov chains can easily be proved. Furthermore, the notion of canonical factorization provides a powerful tool for solving infinite linear systems, such as the fundamental system involving the stationary probability distribution of structured Markov chains.
Leon Ehrenpreis
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198509783
- eISBN:
- 9780191709166
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509783.003.0010
- Subject:
- Mathematics, Mathematical Physics
This chapter shows how the book's main theory applies to computing the periods of Eisenstein and Poincaré series. Our Poincaré series are eigenfunctions of the Laplacian, and they arise naturally in ...
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This chapter shows how the book's main theory applies to computing the periods of Eisenstein and Poincaré series. Our Poincaré series are eigenfunctions of the Laplacian, and they arise naturally in the theory. In particular, the periods of the Poincaré series are Kloosterman sums.Less
This chapter shows how the book's main theory applies to computing the periods of Eisenstein and Poincaré series. Our Poincaré series are eigenfunctions of the Laplacian, and they arise naturally in the theory. In particular, the periods of the Poincaré series are Kloosterman sums.
A. Hald
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509721
- eISBN:
- 9780191709197
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509721.003.0007
- Subject:
- Mathematics, Probability / Statistics
This chapter presents a reprint of part of Hald (2000a), containing a detailed discussion of Thiele's halfinvariants and their use in series expansions. Section 2 describes how Poisson, Bessel, and ...
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This chapter presents a reprint of part of Hald (2000a), containing a detailed discussion of Thiele's halfinvariants and their use in series expansions. Section 2 describes how Poisson, Bessel, and Bienaymé generalized the Laplacean central limit theorem by including more terms in the expansion of the logarithm of the characteristic function. Section 3 discusses Chebyshev's least squares fitting of a polynomial to the observed values of a function by means of orthogonal polynomials. Section 4 explains how Thiele and Gram introduced the Gram-Charlier series from a completely different point of view.Less
This chapter presents a reprint of part of Hald (2000a), containing a detailed discussion of Thiele's halfinvariants and their use in series expansions. Section 2 describes how Poisson, Bessel, and Bienaymé generalized the Laplacean central limit theorem by including more terms in the expansion of the logarithm of the characteristic function. Section 3 discusses Chebyshev's least squares fitting of a polynomial to the observed values of a function by means of orthogonal polynomials. Section 4 explains how Thiele and Gram introduced the Gram-Charlier series from a completely different point of view.
John C. Lennox and Derek J. S. Robinson
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198507284
- eISBN:
- 9780191709326
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507284.003.0001
- Subject:
- Mathematics, Pure Mathematics
This chapter provides an introduction to soluble and nilpotent groups. It discusses the properties of the upper and lower central series and talks about polycyclic groups and Cernikov groups.
This chapter provides an introduction to soluble and nilpotent groups. It discusses the properties of the upper and lower central series and talks about polycyclic groups and Cernikov groups.
Giovanni Ferraris, Emil Makovicky, and Stefano Merlino
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199545698
- eISBN:
- 9780191712111
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199545698.003.0001
- Subject:
- Physics, Crystallography: Physics
The chapter deals with structures and phenomena in which modularity is connected with incremental changes in chemical composition, potentially yielding the series of structurally related but ...
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The chapter deals with structures and phenomena in which modularity is connected with incremental changes in chemical composition, potentially yielding the series of structurally related but chemically distinct compounds. Modular and non-modular crystal chemistry is interconnected by so-called configuration levels. Enumeration and explanation of non-modular (isotypism, homeotypism) and modular (unit-cell or chemical twinning, crystallographic shear, noncommensurability, archetypes) categories of structural similarity is followed by a description of various types of homologous series: accretional series of the extensive and combinatorial types, and their polysomatic description, variable-fit series, and the series combining these principles. Series with more complex relationships are defined as plesiotypic and merotypic series. Further types are sliding series and ordered derivatives of solid solutions. All these concepts are illustrated by examples. The main principles of prediction of modular structures and families dealt with are the direct-space metric properties, reciprocal space features and the use of chemical information.Less
The chapter deals with structures and phenomena in which modularity is connected with incremental changes in chemical composition, potentially yielding the series of structurally related but chemically distinct compounds. Modular and non-modular crystal chemistry is interconnected by so-called configuration levels. Enumeration and explanation of non-modular (isotypism, homeotypism) and modular (unit-cell or chemical twinning, crystallographic shear, noncommensurability, archetypes) categories of structural similarity is followed by a description of various types of homologous series: accretional series of the extensive and combinatorial types, and their polysomatic description, variable-fit series, and the series combining these principles. Series with more complex relationships are defined as plesiotypic and merotypic series. Further types are sliding series and ordered derivatives of solid solutions. All these concepts are illustrated by examples. The main principles of prediction of modular structures and families dealt with are the direct-space metric properties, reciprocal space features and the use of chemical information.
Raymond L. Chambers and Robert G. Clark
- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780198566625
- eISBN:
- 9780191738449
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566625.003.0011
- Subject:
- Mathematics, Probability / Statistics
Inference for non-linear population parameters develops model-based prediction theory for target parameters that are not population totals or means. The development initially is for the case where ...
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Inference for non-linear population parameters develops model-based prediction theory for target parameters that are not population totals or means. The development initially is for the case where the target parameter can be expressed as a differentiable function of finite population means, and a Taylor series linearisation argument is used to get a large sample approximation to the prediction variance of the substitution-based predictor. This Taylor linearisation approach is then generalised to target parameters that can be expressed as solutions of estimating equations. An application to inference about the median value of a homogeneous population serves to illustrate the basic approach, and this is then extended to the stratified population case.Less
Inference for non-linear population parameters develops model-based prediction theory for target parameters that are not population totals or means. The development initially is for the case where the target parameter can be expressed as a differentiable function of finite population means, and a Taylor series linearisation argument is used to get a large sample approximation to the prediction variance of the substitution-based predictor. This Taylor linearisation approach is then generalised to target parameters that can be expressed as solutions of estimating equations. An application to inference about the median value of a homogeneous population serves to illustrate the basic approach, and this is then extended to the stratified population case.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, ...
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Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.Less
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.
I. Grattan-Guinness
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199231256
- eISBN:
- 9780191710803
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199231256.003.0003
- Subject:
- Mathematics, History of Mathematics
This chapter reviews the early works of William Thomson. Thomson began to publish exceptionally early, in 1841, while still in his late teens; and a steady stream of papers came from him thereafter. ...
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This chapter reviews the early works of William Thomson. Thomson began to publish exceptionally early, in 1841, while still in his late teens; and a steady stream of papers came from him thereafter. Most of them in the 1840s concerned heat diffusion, electricity, or magnetism, and related mathematical methods such as the Fourier series and potential theory. These papers manifest a procedure of working by analogy in which methods were taken from one topic and adapted for use in another one.Less
This chapter reviews the early works of William Thomson. Thomson began to publish exceptionally early, in 1841, while still in his late teens; and a steady stream of papers came from him thereafter. Most of them in the 1840s concerned heat diffusion, electricity, or magnetism, and related mathematical methods such as the Fourier series and potential theory. These papers manifest a procedure of working by analogy in which methods were taken from one topic and adapted for use in another one.
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/9780195334548.003.0011
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter presents interesting applications of Catalan numbers to abstract algebra, linear algebra, and sports. In addition, it presents three interesting number patterns involving Catalan numbers ...
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This chapter presents interesting applications of Catalan numbers to abstract algebra, linear algebra, and sports. In addition, it presents three interesting number patterns involving Catalan numbers to alert the fact that patterns can fail.Less
This chapter presents interesting applications of Catalan numbers to abstract algebra, linear algebra, and sports. In addition, it presents three interesting number patterns involving Catalan numbers to alert the fact that patterns can fail.
