Methods Of Qualitative Theory In Nonlinear Dynamics Part Ii

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Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ii)

Author : Leon O Chua
Publisher : World Scientific
ISBN 13 : 9814494291
Total Pages : 591 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ii) by : Leon O Chua

Download or read book Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ii) written by Leon O Chua and published by World Scientific. This book was released on 2001-09-27 with total page 591 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bifurcation and chaos has dominated research in nonlinear dynamics for over two decades, and numerous introductory and advanced books have been published on this subject. There remains, however, a dire need for a textbook which provides a pedagogically appealing yet rigorous mathematical bridge between these two disparate levels of exposition. This book has been written to serve that unfulfilled need.Following the footsteps of Poincaré, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many of the qualitative methods and tools presented in the book have been developed only recently and have not yet appeared in textbook form.In keeping with the self-contained nature of the book, all the topics are developed with introductory background and complete mathematical rigor. Generously illustrated and written at a high level of exposition, this invaluable book will appeal to both the beginner and the advanced student of nonlinear dynamics interested in learning a rigorous mathematical foundation of this fascinating subject.

Methods of Qualitative Theory in Nonlinear Dynamics

Author : Leonid P Shilnikov
Publisher : World Scientific
ISBN 13 : 9814496421
Total Pages : 416 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Methods of Qualitative Theory in Nonlinear Dynamics by : Leonid P Shilnikov

Download or read book Methods of Qualitative Theory in Nonlinear Dynamics written by Leonid P Shilnikov and published by World Scientific. This book was released on 1998-12-08 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bifurcation and Chaos has dominated research in nonlinear dynamics for over two decades and numerous introductory and advanced books have been published on this subject. There remains, however, a dire need for a textbook which provides a pedagogically appealing yet rigorous mathematical bridge between these two disparate levels of exposition. This book is written to serve the above unfulfilled need. Following the footsteps of Poincaré, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many of the qualitative methods and tools presented in this book were developed only recently and have not yet appeared in a textbook form. In keeping with the self-contained nature of this book, all topics are developed with an introductory background and complete mathematical rigor. Generously illustrated and written with a high level of exposition, this book will appeal to both beginners and advanced students of nonlinear dynamics interested in learning a rigorous mathematical foundation of this fascinating subject. Contents:Basic ConceptsStructurally Stable Equilibrium States of Dynamical SystemsStructurally Stable Periodic Trajectories of Dynamical SystemsInvariant ToriCenter Manifold. Local CaseCenter Manifold. Non-Local Case Readership: Engineers, students, mathematicians and researchers in nonlinear dynamics and dynamical systems. Keywords:Bifurcations;Dynamical Systems;Qualitative Theory;Chaos;Strange Attractors;Nonlinear DynamicsReviews: “It is well-written and clearly organized with excellent figures … This rigorous book, with its emphasis on mathematical technique, would form an excellent basis for an engineering course if supplemented with applications.” Applied Mechanics Reviews “Short remarks concerning various, not only mathematical, aspects of the theory add an extra flavour to the text. I recommend the book for all persons interested in the qualitative theory of differential equations.” Mathematical Reviews

Methods of Qualitative Theory in Nonlinear Dynamics

Author : L. P. Shilʹnikov
Publisher :
ISBN 13 : 9789810233822
Total Pages : 957 pages
Book Rating : 4.2/5 (338 download)

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Book Synopsis Methods of Qualitative Theory in Nonlinear Dynamics by : L. P. Shilʹnikov

Download or read book Methods of Qualitative Theory in Nonlinear Dynamics written by L. P. Shilʹnikov and published by . This book was released on 1998 with total page 957 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bifurcation and Chaos has dominated research in nonlinear dynamics for over two decades and numerous introductory and advanced books have been published on this subject. There remains, however, a dire need for a textbook which provides a pedagogically appealing yet rigorous mathematical bridge between these two disparate levels of exposition. This book is written to serve the above unfulfilled need. Following the footsteps of Poincare, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many of the qualitative methods and tools presented in this book were developed only recently and have not yet appeared in a textbook form. In keeping with the self-contained nature of this book, all topics are developed with an introductory background and complete mathematical rigor. Generously illustrated and written with a high level of exposition, this book will appeal to both beginners and advanced students of nonlinear dynamics interested in learning a rigorous mathematical foundation of this fascinating subject

Methods of Qualitative Theory in Nonlinear Dynamics

Author : Leonid P. Shilnikov
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (179 download)

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Book Synopsis Methods of Qualitative Theory in Nonlinear Dynamics by : Leonid P. Shilnikov

Download or read book Methods of Qualitative Theory in Nonlinear Dynamics written by Leonid P. Shilnikov and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Methods of Qualitative Theory in Nonlinear Dynamics

Author : Leonid P. Shilnikov
Publisher : World Scientific
ISBN 13 : 9789810233822
Total Pages : 420 pages
Book Rating : 4.2/5 (338 download)

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Book Synopsis Methods of Qualitative Theory in Nonlinear Dynamics by : Leonid P. Shilnikov

Download or read book Methods of Qualitative Theory in Nonlinear Dynamics written by Leonid P. Shilnikov and published by World Scientific. This book was released on 1998 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bifurcation and Chaos has dominated research in nonlinear dynamics for over two decades and numerous introductory and advanced books have been published on this subject. There remains, however, a dire need for a textbook which provides a pedagogically appealing yet rigorous mathematical bridge between these two disparate levels of exposition. This book is written to serve the above unfulfilled need. Following the footsteps of Poincare, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many of the qualitative methods and tools presented in this book were developed only recently and have not yet appeared in a textbook form. In keeping with the self-contained nature of this book, all topics are developed with an introductory background and complete mathematical rigor. Generously illustrated and written with a high level of exposition, this book will appeal to both beginners and advanced studentsof nonlinear dynamics interested in learning a rigorous mathematical foundation of this fascinating subject.

'A Nonlinear Dynamics Perspective of Wolfram''s New Kind of Science'

Author : Leon O Chua
Publisher : World Scientific
ISBN 13 : 9814478687
Total Pages : 396 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis 'A Nonlinear Dynamics Perspective of Wolfram''s New Kind of Science' by : Leon O Chua

Download or read book 'A Nonlinear Dynamics Perspective of Wolfram''s New Kind of Science' written by Leon O Chua and published by World Scientific. This book was released on 2006-06-08 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: ' This novel book introduces cellular automata from a rigorous nonlinear dynamics perspective. It supplies the missing link between nonlinear differential and difference equations to discrete symbolic analysis. A surprisingly useful interpretations of cellular automata in terms of neural networks is also given. The book provides a scientifically sound and original analysis, and classifications of the empirical results presented in Wolfram''s monumental “New Kind of Science.” Readership: Graduate students, academics and researchers in nonlinear dynamics, computer science and complexity theory.Keywords:Cellular Automata;Nonlinear Dynamics;Wolfram;Neural Networks;Cellular Neural Networks;CNN;Universal Computation;Turing Machine;Chaos;Nonlinear Science;Complexity;Emergence "This book is a colourful presentation with fresh ideas and attractive illustrations … those studying non-linear sciences, electronic engineering, mathematics and logics, complexity and emergent phenomena, and possibly even chemistry and biology will certainly discover exciting concepts, analogies and research tools in this refreshing text. Anyone from freshmen to elderly academics will find parts interesting to them. The volumes are somewhat special and exciting because they possess a unique ''Chua brand'' and show gradual development of ideas and concepts in an educational and entertaining hence mathematically rigorous manner." Journal of Cellular Automata "There is much of interest here, and in particular many interesting examples presented in novel ways." Zentralblatt MATH '

Nonlinear Dynamics New Directions

Author : Hernán González-Aguilar
Publisher : Springer
ISBN 13 : 3319098640
Total Pages : 240 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Nonlinear Dynamics New Directions by : Hernán González-Aguilar

Download or read book Nonlinear Dynamics New Directions written by Hernán González-Aguilar and published by Springer. This book was released on 2015-02-10 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, along with its companion volume, Nonlinear Dynamics New Directions: Theoretical Aspects, covers topics ranging from fractal analysis to very specific applications of the theory of dynamical systems to biology. This second volume contains mostly new applications of the theory of dynamical systems to both engineering and biology. The first volume is devoted to fundamental aspects and includes a number of important new contributions as well as some review articles that emphasize new development prospects. The topics addressed in the two volumes include a rigorous treatment of fluctuations in dynamical systems, topics in fractal analysis, studies of the transient dynamics in biological networks, synchronization in lasers, and control of chaotic systems, among others. This book also: · Develops applications of nonlinear dynamics on a diversity of topics such as patterns of synchrony in neuronal networks, laser synchronization, control of chaotic systems, and the study of transient dynamics in biological · Includes a study of self-organized regularity in long-range systems · Explains use of Levenstein's distance for measuring lexical evolution rates

Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems

Author : Vasile Marinca
Publisher : Springer Nature
ISBN 13 : 303075653X
Total Pages : 479 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems by : Vasile Marinca

Download or read book Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems written by Vasile Marinca and published by Springer Nature. This book was released on 2021-07-14 with total page 479 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the optimal auxiliary functions method and applies it to various engineering problems and in particular in boundary layer problems. The cornerstone of the presented procedure is the concept of “optimal auxiliary functions” which are needed to obtain accurate results in an efficient way. Unlike other known analytic approaches, this procedure provides us with a simple but rigorous way to control and adjust the convergence of the solutions of nonlinear dynamical systems. The optimal auxiliary functions are depending on some convergence-control parameters whose optimal values are rigorously determined from mathematical point of view. The capital strength of our procedure is its fast convergence, since after only one iteration, we obtain very accurate analytical solutions which are very easy to be verified. Moreover, no simplifying hypothesis or assumptions are made. The book contains a large amount of practical models from various fields of engineering such as classical and fluid mechanics, thermodynamics, nonlinear oscillations, electrical machines, and many more. The book is a continuation of our previous books “Nonlinear Dynamical Systems in Engineering. Some Approximate Approaches”, Springer-2011 and “The Optimal Homotopy Asymptotic Method. Engineering Applications”, Springer-2015.

Visions of Nonlinear Science in the 21st Century

Author : Jose L. Huertas
Publisher : World Scientific
ISBN 13 : 9789810233372
Total Pages : 874 pages
Book Rating : 4.2/5 (333 download)

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Book Synopsis Visions of Nonlinear Science in the 21st Century by : Jose L. Huertas

Download or read book Visions of Nonlinear Science in the 21st Century written by Jose L. Huertas and published by World Scientific. This book was released on 1999 with total page 874 pages. Available in PDF, EPUB and Kindle. Book excerpt: Authoritative and visionary, this festschrift features 12 highly readable expositions of virtually all currently active aspects of nonlinear science. It has been painstakingly researched and written by leading scientists and eminent expositors, including L Shilnikov, R Seydel, I Prigogine, W Porod, C Mira, M Lakshmanan, W Lauterborn, A Holden, H Haken, C Grebogi, E Doedel and L Chua; each chapter addresses a current and intensively researched area of nonlinear science and chaos, including nonlinear dynamics, mathematics, numerics and technology. Handsomely produced with high resolution color graphics for enhanced readability, this book has been carefully written at a high level of exposition and is somewhat self-contained. Each chapter includes a tutorial and background information, as well as a survey of each area's main results and state of the art. Of special interest to both beginners and seasoned researchers is the identification of future trends and challenging yet tractable problems thatare likely,to be solved before the end of the 21st century. The visionary and provocative nature of this book makes it a valuable and lasting reference.

Chua Lectures, The: From Memristors And Cellular Nonlinear Networks To The Edge Of Chaos - Volume Iii. Chaos: Chua's Circuit And Complex Nonlinear Phenomena

Download Chua Lectures, The: From Memristors And Cellular Nonlinear Networks To The Edge Of Chaos - Volume Iii. Chaos: Chua's Circuit And Complex Nonlinear Phenomena PDF Online Free

Author : Leon O Chua
Publisher : World Scientific
ISBN 13 : 981121591X
Total Pages : 244 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Chua Lectures, The: From Memristors And Cellular Nonlinear Networks To The Edge Of Chaos - Volume Iii. Chaos: Chua's Circuit And Complex Nonlinear Phenomena by : Leon O Chua

Download or read book Chua Lectures, The: From Memristors And Cellular Nonlinear Networks To The Edge Of Chaos - Volume Iii. Chaos: Chua's Circuit And Complex Nonlinear Phenomena written by Leon O Chua and published by World Scientific. This book was released on 2020-08-19 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This 4-volume compendium contains the verbatim hard copies of all color slides from the Chua Lecture Series presented at HP in Palo Alto, during the period from September 22 to November 24, 2015. Each lecture consists of 90 minutes, divided into a formal lecture, a discussion session, and an Encore of special trivia that the audience found mesmerizing.These lectures share some unique features of the classic Feynman Lectures on Physics, as much of the materials are presented in the unique style of the author, and the content is original as discovered or invented by the author himself. Unlike most technical books that suffer a notoriously short life span as their features could be superseded by superior models, this series of Chua lectures are intended to never be obsolete — many concepts and principles introduced are in fact new laws of nature, written in the language of sophomore-level mathematics, providing the foundation and the elan vital for initiating and nurturing future concepts and inventions.Volume III — presents an overview of the fascinating phenomenon called chaos, while immersing the audience with the sights and sound of chaos from the Chua Circuit, invented in 1984 by Leon Chua, and has now become the standard textbook example of chaos exhibited by a real nonlinear electronic circuit, and not by computer simulations.

Qualitative Methods in Nonlinear Dynamics

Author : A.A. Martynyuk
Publisher : CRC Press
ISBN 13 : 9780824707354
Total Pages : 326 pages
Book Rating : 4.7/5 (73 download)

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Book Synopsis Qualitative Methods in Nonlinear Dynamics by : A.A. Martynyuk

Download or read book Qualitative Methods in Nonlinear Dynamics written by A.A. Martynyuk and published by CRC Press. This book was released on 2001-11-05 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Presents new approaches to qualitative analysis of continuous, discreteptime, and impulsive nonlinear systems via Liapunov matrix-valued functions that introduce more effective tests for solving problems of estimating the domains of asymptotic stability."

Chua Lectures, The: From Memristors And Cellular Nonlinear Networks To The Edge Of Chaos (In 4 Volumes)

Author : Leon O Chua
Publisher : World Scientific
ISBN 13 : 9811228930
Total Pages : 1008 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Chua Lectures, The: From Memristors And Cellular Nonlinear Networks To The Edge Of Chaos (In 4 Volumes) by : Leon O Chua

Download or read book Chua Lectures, The: From Memristors And Cellular Nonlinear Networks To The Edge Of Chaos (In 4 Volumes) written by Leon O Chua and published by World Scientific. This book was released on 2020-08-19 with total page 1008 pages. Available in PDF, EPUB and Kindle. Book excerpt: This 4-volume compendium contains the verbatim hard copies of all color slides from the Chua Lecture Series presented at HP in Palo Alto, during the period from September 22 to November 24, 2015. Each lecture consists of 90 minutes, divided into a formal lecture, a discussion session, and an Encore of special trivia that the audience found mesmerizing.These lectures share some unique features of the classic Feynman Lectures on Physics, as much of the materials are presented in the unique style of the author, and the content is original as discovered or invented by the author himself. Unlike most technical books that suffer a notoriously short life span as their features could be superseded by superior models, this series of Chua lectures are intended to never be obsolete — many concepts and principles introduced are in fact new laws of nature, written in the language of sophomore-level mathematics, providing the foundation and the elan vital for initiating and nurturing future concepts and inventions.Volume I — covers everything that a researcher may want to know about memristors but is too afraid to ask.Volume II — shows that memristors can be either volatile or non-volatile, and effectively proving that synapses are non-volatile memristors, while action potentials are generated by locally-active memristors.Volume III — presents an overview of the fascinating phenomenon called chaos, while immersing the audience with the sights and sound of chaos from the Chua Circuit, invented in 1984 by Leon Chua, and has now become the standard textbook example of chaos exhibited by a real nonlinear electronic circuit, and not by computer simulations.Volume IV — surprises the audience with a new law of nature — dubbed the local activity principle, as discovered and proved mathematically in 1996 by Leon Chua. In particular, a Corollary of Chua's local activity theorem, dubbed the edge of chaos, is shown via insightful examples to be the originator of most complex phenomena, including intelligence, creativity, and deep learning. The edge of chaos is Mother Nature's tool for overcoming the tyranny of the second law of thermodynamics by providing an escape hatch for entropy to decrease over time. Indeed, the local activity principle which is profusely illustrated in the final volume, is widely recognized as a new law of thermodynamics, and is identified as the sine qua non of all complex phenomena, including life itself.Exclusive Access to the accompanying Video and Audio materials comes with the purchase of this book.

Synchronization

Author : Arkady Pikovsky
Publisher : Springer Science & Business Media
ISBN 13 : 9401002177
Total Pages : 258 pages
Book Rating : 4.4/5 (1 download)

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Book Synopsis Synchronization by : Arkady Pikovsky

Download or read book Synchronization written by Arkady Pikovsky and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: Synchronization is a universal phenomenon that is encountered in nature, science and engineering. The book presents a broad view of modern theoretical and experimental approaches to synchronization, especially in complex and chaotic systems, and its applications in life sciences and engineering. Contributors include applied mathematicians, physicists, biologists, and specialists in communications and control theory. The study of synchronization is presented in its many aspects: basic mathematical theory, numerical simulation of complex systems, applications of methods in theoretical physics, experimental implementation, and applications in engineering and life sciences.

Principles of Brain Dynamics

Author : Mikhail I. Rabinovich
Publisher : MIT Press
ISBN 13 : 0262549905
Total Pages : 371 pages
Book Rating : 4.2/5 (625 download)

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Book Synopsis Principles of Brain Dynamics by : Mikhail I. Rabinovich

Download or read book Principles of Brain Dynamics written by Mikhail I. Rabinovich and published by MIT Press. This book was released on 2023-12-05 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: Experimental and theoretical approaches to global brain dynamics that draw on the latest research in the field. The consideration of time or dynamics is fundamental for all aspects of mental activity—perception, cognition, and emotion—because the main feature of brain activity is the continuous change of the underlying brain states even in a constant environment. The application of nonlinear dynamics to the study of brain activity began to flourish in the 1990s when combined with empirical observations from modern morphological and physiological observations. This book offers perspectives on brain dynamics that draw on the latest advances in research in the field. It includes contributions from both theoreticians and experimentalists, offering an eclectic treatment of fundamental issues. Topics addressed range from experimental and computational approaches to transient brain dynamics to the free-energy principle as a global brain theory. The book concludes with a short but rigorous guide to modern nonlinear dynamics and their application to neural dynamics.

Nonlinear Dynamics Perspective Of Wolfram's New Kind Of Science, A (In 2 Volumes) - Volume Ii

Author : Leon O Chua
Publisher : World Scientific
ISBN 13 : 9814478695
Total Pages : 600 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Nonlinear Dynamics Perspective Of Wolfram's New Kind Of Science, A (In 2 Volumes) - Volume Ii by : Leon O Chua

Download or read book Nonlinear Dynamics Perspective Of Wolfram's New Kind Of Science, A (In 2 Volumes) - Volume Ii written by Leon O Chua and published by World Scientific. This book was released on 2007-07-10 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: This novel book introduces cellular automata from a rigorous nonlinear dynamics perspective. It supplies the missing link between nonlinear differential and difference equations to discrete symbolic analysis. A surprisingly useful interpretations of cellular automata in terms of neural networks is also given. The book provides a scientifically sound and original analysis, and classifications of the empirical results presented in Wolfram's monumental ';New Kind of Science.';/a

Approaches to the Qualitative Theory of Ordinary Differential Equations

Author : Tong-Ren Ding
Publisher : World Scientific
ISBN 13 : 981270468X
Total Pages : 394 pages
Book Rating : 4.8/5 (127 download)

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Book Synopsis Approaches to the Qualitative Theory of Ordinary Differential Equations by : Tong-Ren Ding

Download or read book Approaches to the Qualitative Theory of Ordinary Differential Equations written by Tong-Ren Ding and published by World Scientific. This book was released on 2007 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an ideal text for advanced undergraduate students and graduate students with an interest in the qualitative theory of ordinary differential equations and dynamical systems. Elementary knowledge is emphasized by the detailed discussions on the fundamental theorems of the Cauchy problem, fixed-point theorems (especially the twist theorems), the principal idea of dynamical systems, the nonlinear oscillation of Duffing's equation, and some special analyses of particular differential equations. It also contains the latest research by the author as an integral part of the book.

Global Bifurcation Theory and Hilbert’s Sixteenth Problem

Author : V. Gaiko
Publisher : Springer Science & Business Media
ISBN 13 : 1441991689
Total Pages : 199 pages
Book Rating : 4.4/5 (419 download)

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Book Synopsis Global Bifurcation Theory and Hilbert’s Sixteenth Problem by : V. Gaiko

Download or read book Global Bifurcation Theory and Hilbert’s Sixteenth Problem written by V. Gaiko and published by Springer Science & Business Media. This book was released on 2013-11-27 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt: On the 8th of August 1900 outstanding German mathematician David Hilbert delivered a talk "Mathematical problems" at the Second Interna tional Congress of Mathematicians in Paris. The talk covered practically all directions of mathematical thought of that time and contained a list of 23 problems which determined the further development of mathema tics in many respects (1, 119]. Hilbert's Sixteenth Problem (the second part) was stated as follows: Problem. To find the maximum number and to determine the relative position of limit cycles of the equation dy Qn(X, y) -= dx Pn(x, y)' where Pn and Qn are polynomials of real variables x, y with real coeffi cients and not greater than n degree. The study of limit cycles is an interesting and very difficult problem of the qualitative theory of differential equations. This theory was origi nated at the end of the nineteenth century in the works of two geniuses of the world science: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare. A. M. Lyapunov set forth and solved completely in the very wide class of cases a special problem of the qualitative theory: the problem of motion stability (154]. In turn, H. Poincare stated a general problem of the qualitative analysis which was formulated as follows: not integrating the differential equation and using only the properties of its right-hand sides, to give as more as possi ble complete information on the qualitative behaviour of integral curves defined by this equation (176].