Convergence Problems of Orthogonal Series

Convergence Problems of Orthogonal Series

Author: G. Alexits

Publisher: Elsevier

Published: 2014-07-23

Total Pages: 362

ISBN-13: 1483222772

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Convergence Problems of Orthogonal Series deals with the theory of convergence and summation of the general orthogonal series in relation to the general theory and classical expansions. The book reviews orthogonality, orthogonalization, series of orthogonal functions, complete orthogonal systems, and the Riesz-Fisher theorem. The text examines Jacobi polynomials, Haar's orthogonal system, and relations to the theory of probability using Rademacher's and Walsh's orthogonal systems. The book also investigates the convergence behavior of orthogonal series by methods belonging to the general theory of series. The text explains some Tauberian theorems and the classical Abel transform of the partial sums of a series which the investigator can use in the theory of orthogonal series. The book examines the importance of the Lebesgue functions for convergence problems, the generalization of the Walsh series, the order of magnitude of the Lebesgue functions, and the Lebesgue functions of the Cesaro summation. The text also deals with classical convergence problems in which general orthogonal series have limited significance as orthogonal expansions react upon the structural properties of the expanded function. This reaction happens under special assumptions concerning the orthogonal system in whose functions the expansion proceeds. The book can prove beneficial to mathematicians, students, or professor of calculus and advanced mathematics.


Book Synopsis Convergence Problems of Orthogonal Series by : G. Alexits

Download or read book Convergence Problems of Orthogonal Series written by G. Alexits and published by Elsevier. This book was released on 2014-07-23 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convergence Problems of Orthogonal Series deals with the theory of convergence and summation of the general orthogonal series in relation to the general theory and classical expansions. The book reviews orthogonality, orthogonalization, series of orthogonal functions, complete orthogonal systems, and the Riesz-Fisher theorem. The text examines Jacobi polynomials, Haar's orthogonal system, and relations to the theory of probability using Rademacher's and Walsh's orthogonal systems. The book also investigates the convergence behavior of orthogonal series by methods belonging to the general theory of series. The text explains some Tauberian theorems and the classical Abel transform of the partial sums of a series which the investigator can use in the theory of orthogonal series. The book examines the importance of the Lebesgue functions for convergence problems, the generalization of the Walsh series, the order of magnitude of the Lebesgue functions, and the Lebesgue functions of the Cesaro summation. The text also deals with classical convergence problems in which general orthogonal series have limited significance as orthogonal expansions react upon the structural properties of the expanded function. This reaction happens under special assumptions concerning the orthogonal system in whose functions the expansion proceeds. The book can prove beneficial to mathematicians, students, or professor of calculus and advanced mathematics.

Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems

Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems

Author: Harold Kushner

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 245

ISBN-13: 146124482X

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The book deals with several closely related topics concerning approxima tions and perturbations of random processes and their applications to some important and fascinating classes of problems in the analysis and design of stochastic control systems and nonlinear filters. The basic mathematical methods which are used and developed are those of the theory of weak con vergence. The techniques are quite powerful for getting weak convergence or functional limit theorems for broad classes of problems and many of the techniques are new. The original need for some of the techniques which are developed here arose in connection with our study of the particular applica tions in this book, and related problems of approximation in control theory, but it will be clear that they have numerous applications elsewhere in weak convergence and process approximation theory. The book is a continuation of the author's long term interest in problems of the approximation of stochastic processes and its applications to problems arising in control and communication theory and related areas. In fact, the techniques used here can be fruitfully applied to many other areas. The basic random processes of interest can be described by solutions to either (multiple time scale) Ito differential equations driven by wide band or state dependent wide band noise or which are singularly perturbed. They might be controlled or not, and their state values might be fully observable or not (e. g. , as in the nonlinear filtering problem).


Book Synopsis Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems by : Harold Kushner

Download or read book Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems written by Harold Kushner and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book deals with several closely related topics concerning approxima tions and perturbations of random processes and their applications to some important and fascinating classes of problems in the analysis and design of stochastic control systems and nonlinear filters. The basic mathematical methods which are used and developed are those of the theory of weak con vergence. The techniques are quite powerful for getting weak convergence or functional limit theorems for broad classes of problems and many of the techniques are new. The original need for some of the techniques which are developed here arose in connection with our study of the particular applica tions in this book, and related problems of approximation in control theory, but it will be clear that they have numerous applications elsewhere in weak convergence and process approximation theory. The book is a continuation of the author's long term interest in problems of the approximation of stochastic processes and its applications to problems arising in control and communication theory and related areas. In fact, the techniques used here can be fruitfully applied to many other areas. The basic random processes of interest can be described by solutions to either (multiple time scale) Ito differential equations driven by wide band or state dependent wide band noise or which are singularly perturbed. They might be controlled or not, and their state values might be fully observable or not (e. g. , as in the nonlinear filtering problem).

Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems

Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems

Author: Larisa Beilina

Publisher: Springer Science & Business Media

Published: 2012-03-09

Total Pages: 420

ISBN-13: 1441978054

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Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems is the first book in which two new concepts of numerical solutions of multidimensional Coefficient Inverse Problems (CIPs) for a hyperbolic Partial Differential Equation (PDE) are presented: Approximate Global Convergence and the Adaptive Finite Element Method (adaptivity for brevity). Two central questions for CIPs are addressed: How to obtain a good approximations for the exact solution without any knowledge of a small neighborhood of this solution, and how to refine it given the approximation. The book also combines analytical convergence results with recipes for various numerical implementations of developed algorithms. The developed technique is applied to two types of blind experimental data, which are collected both in a laboratory and in the field. The result for the blind backscattering experimental data collected in the field addresses a real world problem of imaging of shallow explosives.


Book Synopsis Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems by : Larisa Beilina

Download or read book Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems written by Larisa Beilina and published by Springer Science & Business Media. This book was released on 2012-03-09 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems is the first book in which two new concepts of numerical solutions of multidimensional Coefficient Inverse Problems (CIPs) for a hyperbolic Partial Differential Equation (PDE) are presented: Approximate Global Convergence and the Adaptive Finite Element Method (adaptivity for brevity). Two central questions for CIPs are addressed: How to obtain a good approximations for the exact solution without any knowledge of a small neighborhood of this solution, and how to refine it given the approximation. The book also combines analytical convergence results with recipes for various numerical implementations of developed algorithms. The developed technique is applied to two types of blind experimental data, which are collected both in a laboratory and in the field. The result for the blind backscattering experimental data collected in the field addresses a real world problem of imaging of shallow explosives.

From Conflict to Convergence: Coming Together to Solve Tough Problems

From Conflict to Convergence: Coming Together to Solve Tough Problems

Author: Robert Fersh

Publisher: John Wiley & Sons

Published: 2024-07-30

Total Pages: 263

ISBN-13: 1394198566

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Strategies to achieve meaningful and lasting conflict resolution In From Conflict to Convergence: Coming Together to Solve Tough Problems, two expert collaborative problem solvers deliver an incisive, hands-on guide to de-escalating conflict and constructively engaging with those you disagree with to find better solutions to problems. In this book full of real-life stories and examples, you'll find a collection of tried and tested strategies you can employ immediately as you negotiate and navigate your most seemingly intractable conflicts. You'll learn how finding what the authors call “higher ground” can advance your interests even when facing people and groups you think you have little in common with and how this can set the stage for longer term cooperation. The authors explain how to improve your ability to understand how other people think, feel, and perceive the world around you, and how to use that knowledge to develop mutually beneficial solutions that help advance your interests and the interests of the people you're dealing with. You'll also find: Strategies for distinguishing the message from the messenger, so you can appreciate the arguments and intentions of imperfectly-presented positions Techniques for responding to emotional and powerful conflicts and disagreements without getting lost in argument Ways to find breakthrough solutions to long-term conflicts that have failed to respond to previous attempts at resolution Perfect for business and organizational leaders, board members, community and religious leaders, public servants, mediators, and anyone else looking to find common ground with people with differing views and perspectives, From Conflict to Convergence also speaks to concerned citizens looking for concrete pathways to lessen troubling divides in their workplaces, their communities, and society at large. From Conflict to Convergence is a must-read resource for an increasingly combative and conflicted world.


Book Synopsis From Conflict to Convergence: Coming Together to Solve Tough Problems by : Robert Fersh

Download or read book From Conflict to Convergence: Coming Together to Solve Tough Problems written by Robert Fersh and published by John Wiley & Sons. This book was released on 2024-07-30 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: Strategies to achieve meaningful and lasting conflict resolution In From Conflict to Convergence: Coming Together to Solve Tough Problems, two expert collaborative problem solvers deliver an incisive, hands-on guide to de-escalating conflict and constructively engaging with those you disagree with to find better solutions to problems. In this book full of real-life stories and examples, you'll find a collection of tried and tested strategies you can employ immediately as you negotiate and navigate your most seemingly intractable conflicts. You'll learn how finding what the authors call “higher ground” can advance your interests even when facing people and groups you think you have little in common with and how this can set the stage for longer term cooperation. The authors explain how to improve your ability to understand how other people think, feel, and perceive the world around you, and how to use that knowledge to develop mutually beneficial solutions that help advance your interests and the interests of the people you're dealing with. You'll also find: Strategies for distinguishing the message from the messenger, so you can appreciate the arguments and intentions of imperfectly-presented positions Techniques for responding to emotional and powerful conflicts and disagreements without getting lost in argument Ways to find breakthrough solutions to long-term conflicts that have failed to respond to previous attempts at resolution Perfect for business and organizational leaders, board members, community and religious leaders, public servants, mediators, and anyone else looking to find common ground with people with differing views and perspectives, From Conflict to Convergence also speaks to concerned citizens looking for concrete pathways to lessen troubling divides in their workplaces, their communities, and society at large. From Conflict to Convergence is a must-read resource for an increasingly combative and conflicted world.

Variational Convergence And Stochastic Homogenization Of Nonlinear Reaction-diffusion Problems

Variational Convergence And Stochastic Homogenization Of Nonlinear Reaction-diffusion Problems

Author: Omar Anza Hafsa

Publisher: World Scientific

Published: 2022-06-21

Total Pages: 321

ISBN-13: 9811258503

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A substantial number of problems in physics, chemical physics, and biology, are modeled through reaction-diffusion equations to describe temperature distribution or chemical substance concentration. For problems arising from ecology, sociology, or population dynamics, they describe the density of some populations or species. In this book the state variable is a concentration, or a density according to the cases. The reaction function may be complex and include time delays terms that model various situations involving maturation periods, resource regeneration times, or incubation periods. The dynamics may occur in heterogeneous media and may depend upon a small or large parameter, as well as the reaction term. From a purely formal perspective, these parameters are indexed by n. Therefore, reaction-diffusion equations give rise to sequences of Cauchy problems.The first part of the book is devoted to the convergence of these sequences in a sense made precise in the book. The second part is dedicated to the specific case when the reaction-diffusion problems depend on a small parameter ∊ₙ intended to tend towards 0. This parameter accounts for the size of small spatial and randomly distributed heterogeneities. The convergence results obtained in the first part, with additionally some probabilistic tools, are applied to this specific situation. The limit problems are illustrated through biological invasion, food-limited or prey-predator models where the interplay between environment heterogeneities in the individual evolution of propagation species plays an essential role. They provide a description in terms of deterministic and homogeneous reaction-diffusion equations, for which numerical schemes are possible.


Book Synopsis Variational Convergence And Stochastic Homogenization Of Nonlinear Reaction-diffusion Problems by : Omar Anza Hafsa

Download or read book Variational Convergence And Stochastic Homogenization Of Nonlinear Reaction-diffusion Problems written by Omar Anza Hafsa and published by World Scientific. This book was released on 2022-06-21 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: A substantial number of problems in physics, chemical physics, and biology, are modeled through reaction-diffusion equations to describe temperature distribution or chemical substance concentration. For problems arising from ecology, sociology, or population dynamics, they describe the density of some populations or species. In this book the state variable is a concentration, or a density according to the cases. The reaction function may be complex and include time delays terms that model various situations involving maturation periods, resource regeneration times, or incubation periods. The dynamics may occur in heterogeneous media and may depend upon a small or large parameter, as well as the reaction term. From a purely formal perspective, these parameters are indexed by n. Therefore, reaction-diffusion equations give rise to sequences of Cauchy problems.The first part of the book is devoted to the convergence of these sequences in a sense made precise in the book. The second part is dedicated to the specific case when the reaction-diffusion problems depend on a small parameter ∊ₙ intended to tend towards 0. This parameter accounts for the size of small spatial and randomly distributed heterogeneities. The convergence results obtained in the first part, with additionally some probabilistic tools, are applied to this specific situation. The limit problems are illustrated through biological invasion, food-limited or prey-predator models where the interplay between environment heterogeneities in the individual evolution of propagation species plays an essential role. They provide a description in terms of deterministic and homogeneous reaction-diffusion equations, for which numerical schemes are possible.

The Convergence Problem for Dissipative Autonomous Systems

The Convergence Problem for Dissipative Autonomous Systems

Author: Alain Haraux

Publisher: Springer

Published: 2015-09-05

Total Pages: 147

ISBN-13: 3319234072

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The book investigates classical and more recent methods of study for the asymptotic behavior of dissipative continuous dynamical systems with applications to ordinary and partial differential equations, the main question being convergence (or not) of the solutions to an equilibrium. After reviewing the basic concepts of topological dynamics and the definition of gradient-like systems on a metric space, the authors present a comprehensive exposition of stability theory relying on the so-called linearization method. For the convergence problem itself, when the set of equilibria is infinite, the only general results that do not require very special features of the non-linearities are presently consequences of a gradient inequality discovered by S. Lojasiewicz. The application of this inequality jointly with the so-called Liapunov-Schmidt reduction requires a rigorous exposition of Semi-Fredholm operator theory and the theory of real analytic maps on infinite dimensional Banach spaces, which cannot be found anywhere in a readily applicable form. The applications covered in this short text are the simplest, but more complicated cases are mentioned in the final chapter, together with references to the corresponding specialized papers.


Book Synopsis The Convergence Problem for Dissipative Autonomous Systems by : Alain Haraux

Download or read book The Convergence Problem for Dissipative Autonomous Systems written by Alain Haraux and published by Springer. This book was released on 2015-09-05 with total page 147 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book investigates classical and more recent methods of study for the asymptotic behavior of dissipative continuous dynamical systems with applications to ordinary and partial differential equations, the main question being convergence (or not) of the solutions to an equilibrium. After reviewing the basic concepts of topological dynamics and the definition of gradient-like systems on a metric space, the authors present a comprehensive exposition of stability theory relying on the so-called linearization method. For the convergence problem itself, when the set of equilibria is infinite, the only general results that do not require very special features of the non-linearities are presently consequences of a gradient inequality discovered by S. Lojasiewicz. The application of this inequality jointly with the so-called Liapunov-Schmidt reduction requires a rigorous exposition of Semi-Fredholm operator theory and the theory of real analytic maps on infinite dimensional Banach spaces, which cannot be found anywhere in a readily applicable form. The applications covered in this short text are the simplest, but more complicated cases are mentioned in the final chapter, together with references to the corresponding specialized papers.

The Master Equation and the Convergence Problem in Mean Field Games

The Master Equation and the Convergence Problem in Mean Field Games

Author: Pierre Cardaliaguet

Publisher: Princeton University Press

Published: 2019-08-13

Total Pages: 224

ISBN-13: 0691190712

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This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.


Book Synopsis The Master Equation and the Convergence Problem in Mean Field Games by : Pierre Cardaliaguet

Download or read book The Master Equation and the Convergence Problem in Mean Field Games written by Pierre Cardaliaguet and published by Princeton University Press. This book was released on 2019-08-13 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

Convergence

Convergence

Author: National Research Council

Publisher: National Academies Press

Published: 2014-06-16

Total Pages: 234

ISBN-13: 0309301645

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Convergence of the life sciences with fields including physical, chemical, mathematical, computational, engineering, and social sciences is a key strategy to tackle complex challenges and achieve new and innovative solutions. However, institutions face a lack of guidance on how to establish effective programs, what challenges they are likely to encounter, and what strategies other organizations have used to address the issues that arise. This advice is needed to harness the excitement generated by the concept of convergence and channel it into the policies, structures, and networks that will enable it to realize its goals. Convergence investigates examples of organizations that have established mechanisms to support convergent research. This report discusses details of current programs, how organizations have chosen to measure success, and what has worked and not worked in varied settings. The report summarizes the lessons learned and provides organizations with strategies to tackle practical needs and implementation challenges in areas such as infrastructure, student education and training, faculty advancement, and inter-institutional partnerships.


Book Synopsis Convergence by : National Research Council

Download or read book Convergence written by National Research Council and published by National Academies Press. This book was released on 2014-06-16 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convergence of the life sciences with fields including physical, chemical, mathematical, computational, engineering, and social sciences is a key strategy to tackle complex challenges and achieve new and innovative solutions. However, institutions face a lack of guidance on how to establish effective programs, what challenges they are likely to encounter, and what strategies other organizations have used to address the issues that arise. This advice is needed to harness the excitement generated by the concept of convergence and channel it into the policies, structures, and networks that will enable it to realize its goals. Convergence investigates examples of organizations that have established mechanisms to support convergent research. This report discusses details of current programs, how organizations have chosen to measure success, and what has worked and not worked in varied settings. The report summarizes the lessons learned and provides organizations with strategies to tackle practical needs and implementation challenges in areas such as infrastructure, student education and training, faculty advancement, and inter-institutional partnerships.

Convergence of Stochastic Processes

Convergence of Stochastic Processes

Author: D. Pollard

Publisher: David Pollard

Published: 1984-10-08

Total Pages: 223

ISBN-13: 0387909907

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Functionals on stochastic processes; Uniform convergence of empirical measures; Convergence in distribution in euclidean spaces; Convergence in distribution in metric spaces; The uniform metric on space of cadlag functions; The skorohod metric on D [0, oo); Central limit teorems; Martingales.


Book Synopsis Convergence of Stochastic Processes by : D. Pollard

Download or read book Convergence of Stochastic Processes written by D. Pollard and published by David Pollard. This book was released on 1984-10-08 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: Functionals on stochastic processes; Uniform convergence of empirical measures; Convergence in distribution in euclidean spaces; Convergence in distribution in metric spaces; The uniform metric on space of cadlag functions; The skorohod metric on D [0, oo); Central limit teorems; Martingales.

Governance challenges for disaster risk reduction and climate change adaptation convergence in agriculture - Guidance for analysis

Governance challenges for disaster risk reduction and climate change adaptation convergence in agriculture - Guidance for analysis

Author: Food and Agriculture Organization of the United Nations

Publisher: Food & Agriculture Org.

Published: 2019-07-19

Total Pages: 68

ISBN-13: 925131649X

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This discussion paper aims to help practitioners work in a more informed and politically sensitive way to integrate actions on disaster risk reduction (DRR) and climate change adaptation (CCA) in agriculture. It illustrates some typical governance and political economy-related barriers that may hamper convergence or integration of DRR and CCA actions. It also provides guidance for in-depth governance analysis, putting the analytical focus at national and subnational levels, while considering the international context as an important factor for convergence. The FAO Governance and Policy Support Discussion Paper series provides perspectives and concepts on critical governance and policy issues that are relevant to FAO work at country, regional and global levels. Discussion Papers are often based on work in progress, and we welcome suggestions and ideas by email at: [email protected]. The series is available at: http://www.fao.org/policy-support/resources/


Book Synopsis Governance challenges for disaster risk reduction and climate change adaptation convergence in agriculture - Guidance for analysis by : Food and Agriculture Organization of the United Nations

Download or read book Governance challenges for disaster risk reduction and climate change adaptation convergence in agriculture - Guidance for analysis written by Food and Agriculture Organization of the United Nations and published by Food & Agriculture Org.. This book was released on 2019-07-19 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt: This discussion paper aims to help practitioners work in a more informed and politically sensitive way to integrate actions on disaster risk reduction (DRR) and climate change adaptation (CCA) in agriculture. It illustrates some typical governance and political economy-related barriers that may hamper convergence or integration of DRR and CCA actions. It also provides guidance for in-depth governance analysis, putting the analytical focus at national and subnational levels, while considering the international context as an important factor for convergence. The FAO Governance and Policy Support Discussion Paper series provides perspectives and concepts on critical governance and policy issues that are relevant to FAO work at country, regional and global levels. Discussion Papers are often based on work in progress, and we welcome suggestions and ideas by email at: [email protected]. The series is available at: http://www.fao.org/policy-support/resources/