Nonconvex Optimization In Mechanics

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Nonconvex Optimization in Mechanics

Author : E.S. Mistakidis
Publisher : Springer Science & Business Media
ISBN 13 : 1461558298
Total Pages : 295 pages
Book Rating : 4.4/5 (615 download)

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Book Synopsis Nonconvex Optimization in Mechanics by : E.S. Mistakidis

Download or read book Nonconvex Optimization in Mechanics written by E.S. Mistakidis and published by Springer Science & Business Media. This book was released on 2013-11-21 with total page 295 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonconvexity and nonsmoothness arise in a large class of engineering applica tions. In many cases of practical importance the possibilities offered by opti mization with its algorithms and heuristics can substantially improve the per formance and the range of applicability of classical computational mechanics algorithms. For a class of problems this approach is the only one that really works. The present book presents in a comprehensive way the application of opti mization algorithms and heuristics in smooth and nonsmooth mechanics. The necessity of this approach is presented to the reader through simple, represen tative examples. As things become more complex, the necessary material from convex and nonconvex optimization and from mechanics are introduced in a self-contained way. Unilateral contact and friction problems, adhesive contact and delamination problems, nonconvex elastoplasticity, fractal friction laws, frames with semi rigid connections, are among the applications which are treated in details here. Working algorithms are given for each application and are demonstrated by means of representative examples. The interested reader will find helpful references to up-to-date scientific and technical literature so that to be able to work on research or engineering topics which are not directly covered here.

Nonsmooth/Nonconvex Mechanics

Author : David Yang Gao
Publisher : Springer Science & Business Media
ISBN 13 : 1461302757
Total Pages : 505 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Nonsmooth/Nonconvex Mechanics by : David Yang Gao

Download or read book Nonsmooth/Nonconvex Mechanics written by David Yang Gao and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 505 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonsmooth and nonconvex models arise in several important applications of mechanics and engineering. The interest in this field is growing from both mathematicians and engineers. The study of numerous industrial applications, including contact phenomena in statics and dynamics or delamination effects in composites, require the consideration of nonsmoothness and nonconvexity. The mathematical topics discussed in this book include variational and hemivariational inequalities, duality, complementarity, variational principles, sensitivity analysis, eigenvalue and resonance problems, and minimax problems. Applications are considered in the following areas among others: nonsmooth statics and dynamics, stability of quasi- static evolution processes, friction problems, adhesive contact and debonding, inverse problems, pseudoelastic modeling of phase transitions, chaotic behavior in nonlinear beams, and nonholonomic mechanical systems. This volume contains 22 chapters written by various leading researchers and presents a cohesive and authoritative overview of recent results and applications in the area of nonsmooth and nonconvex mechanics. Audience: Faculty, graduate students, and researchers in applied mathematics, optimization, control and engineering.

Optimization in Mechanics

Author : P. Brousse
Publisher : Elsevier
ISBN 13 : 148329014X
Total Pages : 292 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Optimization in Mechanics by : P. Brousse

Download or read book Optimization in Mechanics written by P. Brousse and published by Elsevier. This book was released on 2013-10-22 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimization in Mechanics: Problems and Methods investigates various problems and methods of optimization in mechanics. The subjects under study range from minimization of masses and stresses or displacements, to maximization of loads, vibration frequencies, and critical speeds of rotating shafts. Comprised of seven chapters, this book begins by presenting examples of optimization problems in mechanics and considering their application, as well as illustrating the usefulness of some optimizations like those of a reinforced shell, a robot, and a booster. The next chapter outlines some of the mathematical concepts that form the framework for optimization methods and techniques and demonstrates their efficiency in yielding relevant results. Subsequent chapters focus on the Kuhn Tucker theorem and duality, with proofs; associated problems and classical numerical methods of mathematical programming, including gradient and conjugate gradient methods; and techniques for dealing with large-scale problems. The book concludes by describing optimizations of discrete or continuous structures subject to dynamical effects. Mass minimization and fundamental eigenvalue problems as well as problems of minimization of some dynamical responses are studied. This monograph is written for students, engineers, scientists, and even self-taught individuals.

Duality Principles in Nonconvex Systems

Author : David Yang Gao
Publisher : Springer Science & Business Media
ISBN 13 : 9780792361459
Total Pages : 476 pages
Book Rating : 4.3/5 (614 download)

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Book Synopsis Duality Principles in Nonconvex Systems by : David Yang Gao

Download or read book Duality Principles in Nonconvex Systems written by David Yang Gao and published by Springer Science & Business Media. This book was released on 2000-01-31 with total page 476 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this book is the first to provide, within a unified framework, a self-contained comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics. Topics covered include the classical (minimax) mono-duality of convex static equilibria, the beautiful bi-duality in dynamical systems, the interesting tri-duality in non-convex problems and the complicated multi-duality in general canonical systems. A potentially powerful sequential canonical dual transformation method for solving fully nonlinear problems is developed heuristically and illustrated by use of many interesting examples as well as extensive applications in a wide variety of nonlinear systems, including differential equations, variational problems and inequalities, constrained global optimization, multi-well phase transitions, non-smooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and non-convex dynamical systems. With exceptionally coherent and lucid exposition, the work fills a big gap between the mathematical and engineering sciences. It shows how to use formal language and duality methods to model natural phenomena, to construct intrinsic frameworks in different fields and to provide ideas, concepts and powerful methods for solving non-convex, non-smooth problems arising naturally in engineering and science. Much of the book contains material that is new, both in its manner of presentation and in its research development. A self-contained appendix provides some necessary background from elementary functional analysis. Audience: The book will be a valuable resource for students and researchers in applied mathematics, physics, mechanics and engineering. The whole volume or selected chapters can also be recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the notions of optimization and variational methods are employed.

Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics

Author : Vladimir F. Demyanov
Publisher : Springer Science & Business Media
ISBN 13 : 1461541131
Total Pages : 362 pages
Book Rating : 4.4/5 (615 download)

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Book Synopsis Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics by : Vladimir F. Demyanov

Download or read book Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics written by Vladimir F. Demyanov and published by Springer Science & Business Media. This book was released on 2013-11-21 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonsmooth energy functions govern phenomena which occur frequently in nature and in all areas of life. They constitute a fascinating subject in mathematics and permit the rational understanding of yet unsolved or partially solved questions in mechanics, engineering and economics. This is the first book to provide a complete and rigorous presentation of the quasidifferentiability approach to nonconvex, possibly nonsmooth, energy functions, of the derivation and study of the corresponding variational expressions in mechanics, engineering and economics, and of their numerical treatment. The new variational formulations derived are illustrated by many interesting numerical problems. The techniques presented will permit the reader to check any solution obtained by other heuristic techniques for nonconvex, nonsmooth energy problems. A civil, mechanical or aeronautical engineer can find in the book the only existing mathematically sound technique for the formulation and study of nonconvex, nonsmooth energy problems. Audience: The book will be of interest to pure and applied mathematicians, physicists, researchers in mechanics, civil, mechanical and aeronautical engineers, structural analysts and software developers. It is also suitable for graduate courses in nonlinear mechanics, nonsmooth analysis, applied optimization, control, calculus of variations and computational mechanics.

Canonical Dual Finite Element Method for Solving Nonconvex Mechanics and Topology Optimization

Author : Elaf Jaafar Ali
Publisher :
ISBN 13 :
Total Pages : 266 pages
Book Rating : 4.:/5 (11 download)

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Book Synopsis Canonical Dual Finite Element Method for Solving Nonconvex Mechanics and Topology Optimization by : Elaf Jaafar Ali

Download or read book Canonical Dual Finite Element Method for Solving Nonconvex Mechanics and Topology Optimization written by Elaf Jaafar Ali and published by . This book was released on 2018 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Canonical duality theory (CDT) is a newly developed, potentially powerful methodological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m ≤ n and without a duality gap. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving general nonconvex variational problems. This thesis, first, presents a detailed study of large deformation problems in 2-D structural system. Based on the canonical duality theory, a canonical dual finite element method is applied to find a global minimization to the general nonconvex optimization problem using a new primal-dual semi-definite programming algorithm. Applications are illustrated by numerical examples with different structural designs and different external loads. Next, a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam is investigated. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using the canonical dual finite element method, a new primal-dual semi-definite programming algorithm is presented, which can be used to obtain all possible post-buckled solutions. In order to verify the triality theory, mixed meshes of different dual stress interpolation are applied to obtain the closed dimensions between discretized displacement and discretized stress. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to the external load, thickness of the beam, numerical precision, and the size of finite elements. Finally, a mathematically rigorous and computationally powerful method for solving 3-D topology optimization problems is demonstrated. This method is based on CDT developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard Knapsack problem in topology optimization can be solved deterministically in polynomial-time via a canonical penalty-duality (CPD) method to obtain precise global optimal 0-1 density distribution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Finally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly addressed." -- Abstract.

Introduction to Global Optimization

Author : R. Horst
Publisher : Springer Science & Business Media
ISBN 13 : 9780792367567
Total Pages : 376 pages
Book Rating : 4.3/5 (675 download)

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Book Synopsis Introduction to Global Optimization by : R. Horst

Download or read book Introduction to Global Optimization written by R. Horst and published by Springer Science & Business Media. This book was released on 2000-12-31 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: A textbook for an undergraduate course in mathematical programming for students with a knowledge of elementary real analysis, linear algebra, and classical linear programming (simple techniques). Focuses on the computation and characterization of global optima of nonlinear functions, rather than the locally optimal solutions addressed by most books on optimization. Incorporates the theoretical, algorithmic, and computational advances of the past three decades that help solve globally multi-extreme problems in the mathematical modeling of real world systems. Annotation copyright by Book News, Inc., Portland, OR

From Convexity to Nonconvexity

Author : R.P. Gilbert
Publisher : Springer Science & Business Media
ISBN 13 : 1461302870
Total Pages : 395 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis From Convexity to Nonconvexity by : R.P. Gilbert

Download or read book From Convexity to Nonconvexity written by R.P. Gilbert and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of papers is dedicated to the memory of Gaetano Fichera, a great mathematician and also a good friend to the editors. Regrettably it took an unusual amount of time to bring this collection out. This was primarily due to the fact that the main editor who had collected all of the materials, for this volume, P. D. Panagiotopoulos, died unexpectedly during the period when we were editing the manuscript. The other two editors in appreciation of Panagiotopoulos' contribution to this field, believe it is therefore fitting that this collection be dedicated to his memory also. The theme of the collection is centered around the seminal research of G. Fichera on the Signorini problem. Variants on this idea enter in different ways. For example, by bringing in friction the problem is no longer self-adjoint and the minimization formulation is not valid. A large portion of this collection is devoted to survey papers concerning hemivariational methods, with a main point of its application to nonsmooth mechanics. Hemivariational inequali ties, which are a generalization of variational inequalities, were pioneered by Panagiotopoulos. There are many applications of this theory to the study of non convex energy functionals occurring in many branches of mechanics. An area of concentration concerns contact problems, in particular, quasistatic and dynamic contact problems with friction and damage. Nonsmooth optimization methods which may be divided into the main groups of subgradient methods and bundle methods are also discussed in this collection.

Topological Methods in Complementarity Theory

Author : G. Isac
Publisher : Springer Science & Business Media
ISBN 13 : 1475731418
Total Pages : 691 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Topological Methods in Complementarity Theory by : G. Isac

Download or read book Topological Methods in Complementarity Theory written by G. Isac and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 691 pages. Available in PDF, EPUB and Kindle. Book excerpt: Complementarity theory is a new domain in applied mathematics and is concerned with the study of complementarity problems. These problems represent a wide class of mathematical models related to optimization, game theory, economic engineering, mechanics, fluid mechanics, stochastic optimal control etc. The book is dedicated to the study of nonlinear complementarity problems by topological methods. Audience: Mathematicians, engineers, economists, specialists working in operations research and anybody interested in applied mathematics or in mathematical modeling.

Advances in Applied Mathematics and Global Optimization

Author : David Y. Gao
Publisher : Springer Science & Business Media
ISBN 13 : 0387757147
Total Pages : 520 pages
Book Rating : 4.3/5 (877 download)

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Book Synopsis Advances in Applied Mathematics and Global Optimization by : David Y. Gao

Download or read book Advances in Applied Mathematics and Global Optimization written by David Y. Gao and published by Springer Science & Business Media. This book was released on 2009-04-09 with total page 520 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles that comprise this distinguished annual volume for the Advances in Mechanics and Mathematics series have been written in honor of Gilbert Strang, a world renowned mathematician and exceptional person. Written by leading experts in complementarity, duality, global optimization, and quantum computations, this collection reveals the beauty of these mathematical disciplines and investigates recent developments in global optimization, nonconvex and nonsmooth analysis, nonlinear programming, theoretical and engineering mechanics, large scale computation, quantum algorithms and computation, and information theory.

Duality Principles in Nonconvex Systems

Author : David Yang Gao
Publisher : Springer Science & Business Media
ISBN 13 : 1475731760
Total Pages : 463 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Duality Principles in Nonconvex Systems by : David Yang Gao

Download or read book Duality Principles in Nonconvex Systems written by David Yang Gao and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 463 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this book is the first to provide, within a unified framework, a self-contained comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics. Topics covered include the classical (minimax) mono-duality of convex static equilibria, the beautiful bi-duality in dynamical systems, the interesting tri-duality in non-convex problems and the complicated multi-duality in general canonical systems. A potentially powerful sequential canonical dual transformation method for solving fully nonlinear problems is developed heuristically and illustrated by use of many interesting examples as well as extensive applications in a wide variety of nonlinear systems, including differential equations, variational problems and inequalities, constrained global optimization, multi-well phase transitions, non-smooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and non-convex dynamical systems. With exceptionally coherent and lucid exposition, the work fills a big gap between the mathematical and engineering sciences. It shows how to use formal language and duality methods to model natural phenomena, to construct intrinsic frameworks in different fields and to provide ideas, concepts and powerful methods for solving non-convex, non-smooth problems arising naturally in engineering and science. Much of the book contains material that is new, both in its manner of presentation and in its research development. A self-contained appendix provides some necessary background from elementary functional analysis. Audience: The book will be a valuable resource for students and researchers in applied mathematics, physics, mechanics and engineering. The whole volume or selected chapters can also be recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the notions of optimization and variational methods are employed.

Convex Optimization

Author : Stephen P. Boyd
Publisher : Cambridge University Press
ISBN 13 : 9780521833783
Total Pages : 744 pages
Book Rating : 4.8/5 (337 download)

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Book Synopsis Convex Optimization by : Stephen P. Boyd

Download or read book Convex Optimization written by Stephen P. Boyd and published by Cambridge University Press. This book was released on 2004-03-08 with total page 744 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.

Deterministic Global Optimization

Author : Christodoulos A. Floudas
Publisher : Springer Science & Business Media
ISBN 13 : 9780792360148
Total Pages : 774 pages
Book Rating : 4.3/5 (61 download)

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Book Synopsis Deterministic Global Optimization by : Christodoulos A. Floudas

Download or read book Deterministic Global Optimization written by Christodoulos A. Floudas and published by Springer Science & Business Media. This book was released on 2000 with total page 774 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a unified and insightful treatment of deterministic global optimization. It introduces theoretical and algorithmic advances that address the computation and characterization of global optima, determine valid lower and upper bounds on the global minima and maxima, and enclose all solutions of nonlinear constrained systems of equations. Among its special features, the book: Introduces the fundamentals of deterministic global optimization; Provides a thorough treatment of decomposition-based global optimization approaches for biconvex and bilinear problems; Covers global optimization methods for generalized geometric programming problems Presents in-depth global optimization algorithms for general twice continuously differentiable nonlinear problems; Provides a detailed treatment of global optimization methods for mixed-integer nonlinear problems; Develops global optimization approaches for the enclosure of all solutions of nonlinear constrained systems of equations; Includes many important applications from process design, synthesis, control, and operations, phase equilibrium, design under uncertainty, parameter estimation, azeotrope prediction, structure prediction in clusters and molecules, protein folding, and peptide docking. Audience: This book can be used as a textbook in graduate-level courses and as a desk reference for researchers in all branches of engineering and applied science, applied mathematics, industrial engineering, operations research, computer science, economics, computational chemistry and molecular biology.

Global Optimization with Non-Convex Constraints

Author : Roman G. Strongin
Publisher : Springer Science & Business Media
ISBN 13 : 9780792364900
Total Pages : 742 pages
Book Rating : 4.3/5 (649 download)

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Book Synopsis Global Optimization with Non-Convex Constraints by : Roman G. Strongin

Download or read book Global Optimization with Non-Convex Constraints written by Roman G. Strongin and published by Springer Science & Business Media. This book was released on 2000-10-31 with total page 742 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a new approach to global non-convex constrained optimization. Problem dimensionality is reduced via space-filling curves. To economize the search, constraint is accounted separately (penalties are not employed). The multicriteria case is also considered. All techniques are generalized for (non-redundant) execution on multiprocessor systems. Audience: Researchers and students working in optimization, applied mathematics, and computer science.

Lectures on Modern Convex Optimization

Author : Aharon Ben-Tal
Publisher : SIAM
ISBN 13 : 0898714915
Total Pages : 500 pages
Book Rating : 4.8/5 (987 download)

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Book Synopsis Lectures on Modern Convex Optimization by : Aharon Ben-Tal

Download or read book Lectures on Modern Convex Optimization written by Aharon Ben-Tal and published by SIAM. This book was released on 2001-01-01 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here is a book devoted to well-structured and thus efficiently solvable convex optimization problems, with emphasis on conic quadratic and semidefinite programming. The authors present the basic theory underlying these problems as well as their numerous applications in engineering, including synthesis of filters, Lyapunov stability analysis, and structural design. The authors also discuss the complexity issues and provide an overview of the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming. The book's focus on well-structured convex problems in conic form allows for unified theoretical and algorithmical treatment of a wide spectrum of important optimization problems arising in applications.

Nonsmooth Approach to Optimization Problems with Equilibrium Constraints

Author : Jiri Outrata
Publisher : Springer Science & Business Media
ISBN 13 : 1475728255
Total Pages : 281 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Nonsmooth Approach to Optimization Problems with Equilibrium Constraints by : Jiri Outrata

Download or read book Nonsmooth Approach to Optimization Problems with Equilibrium Constraints written by Jiri Outrata and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the early fifties, applied mathematicians, engineers and economists started to pay c10se attention to the optimization problems in which another (lower-Ievel) optimization problem arises as a side constraint. One of the motivating factors was the concept of the Stackelberg solution in game theory, together with its economic applications. Other problems have been encountered in the seventies in natural sciences and engineering. Many of them are of practical importance and have been extensively studied, mainly from the theoretical point of view. Later, applications to mechanics and network design have lead to an extension of the problem formulation: Constraints in form of variation al inequalities and complementarity problems were also admitted. The term "generalized bi level programming problems" was used at first but later, probably in Harker and Pang, 1988, a different terminology was introduced: Mathematical programs with equilibrium constraints, or simply, MPECs. In this book we adhere to MPEC terminology. A large number of papers deals with MPECs but, to our knowledge, there is only one monograph (Luo et al. , 1997). This monograph concentrates on optimality conditions and numerical methods. Our book is oriented similarly, but we focus on those MPECs which can be treated by the implicit programming approach: the equilibrium constraint locally defines a certain implicit function and allows to convert the problem into a mathematical program with a nonsmooth objective.

Convex Analysis and Variational Problems

Author : Ivar Ekeland
Publisher : SIAM
ISBN 13 : 9781611971088
Total Pages : 414 pages
Book Rating : 4.9/5 (71 download)

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Book Synopsis Convex Analysis and Variational Problems by : Ivar Ekeland

Download or read book Convex Analysis and Variational Problems written by Ivar Ekeland and published by SIAM. This book was released on 1999-12-01 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains different developments of infinite dimensional convex programming in the context of convex analysis, including duality, minmax and Lagrangians, and convexification of nonconvex optimization problems in the calculus of variations (infinite dimension). It also includes the theory of convex duality applied to partial differential equations; no other reference presents this in a systematic way. The minmax theorems contained in this book have many useful applications, in particular the robust control of partial differential equations in finite time horizon. First published in English in 1976, this SIAM Classics in Applied Mathematics edition contains the original text along with a new preface and some additional references.