Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological invariants. The authors, both of whom have contributed significantly to the field, develop the classification theory for integrable systems with two degrees of freedom. This theory allows one to distinguish such systems up to two natural equivalence relations: the equivalence of the associated foliation into Liouville tori and the usual orbital equaivalence. The authors show that in both cases, one can find complete sets of invariants that give the solution of the classification problem. The first part of the book systematically presents the general construction of these invariants, including many examples and applications. In the second part, the authors apply the general methods of the classification theory to the classical integrable problems in rigid body dynamics and describe their topological portraits, bifurcations of Liouville tori, and local and global topological invariants. They show how the classification theory helps find hidden isomorphisms between integrable systems and present as an example their proof that two famous systems--the Euler case in rigid body dynamics and the Jacobi problem of geodesics on the ellipsoid--are orbitally equivalent. Integrable Hamiltonian Systems: Geometry, Topology, Classification offers a unique opportunity to explore important, previously unpublished results and acquire generally applicable techniques and tools that enable you to work with a broad class of integrable systems.

Many of the modern variational problems in topology arise in different but overlapping fields of scientific study: mechanics, physics and mathematics. In this work, Professor Fomenko offers a concise and clean explanation of some of these problems (both solved and unsolved), using current methods and analytical topology. The author's skillful exposition gives an unusual motivation to the theory expounded, and his work is recommended reading for specialists and nonspecialists alike, involved in the fields of physics and mathematics at both undergraduate and graduate levels.

This volume considers the theory and applications of integrable Hamiltonian systems. Basic elements of Liouville functions and their singularities is systematically described and a classification of such systems for the case of integrable Hamiltonian systems with two degrees of freedom is presented. Nontrivial connections between the theory of integrable Hamiltonian systems with two degrees of freedom and three-dimensional topology is described and a topological description of the behaviour of integral trajectories under Liouville tori bifurcation is given. The book is divided into two parts, describing theory and applications respectively, and is well illustrated. It will be of use to graduate students of mathematics and mathematicians working in the theory of dynamical systems and their applications.

This easy-to-follow book offers a statistico-geometrical approach for dating ancient star catalogs. The authors' scientific methods reveal statistical properties of ancient catalogs and overcome the difficulties of their dating originated by the low accuracy of these catalogs. Methods are tested on reliably dated medieval star catalogs and applied to the star catalog of the Almagest. Here, the dating of Ptolemy's famous star catalog is reconsidered and recalculated using modern mathematical techniques.The text provides necessary information from astronomy and astrometry. It also covers the history of observational equipment and methods for measuring coordinates of stars. Many chapters are devoted to the Almagest, from a preliminary analysis to a global statistical processing of the catalog and its basic parts. Mathematics are simplified in this book for easy reading. This book will prove invaluable for mathematicians, astronomers, astrophysicists, specialists in natural sciences, historians interested in mathematical and statistical methods, and second-year mathematics students.Features:

Today the methods of applied statistics have penetrated very different fields of knowledge, including the investigation oftexts ofvarious origins. These "texts" may be considered as signal sequences of different kinds, long genetic codes, graphic representations (which may be coded and represented by a "text"), as well as actual narrative texts (for example, historical chronicles, originals, documents, etc. ). One ofthe most important problems arising here is to recognize dependent text, i. e. , texts which have a measure of "resemblance", arising from some kind of "common origin". For instance, in pattern-recognition problems, it is essential to identify from a large set of "patterns" a pattern that is "closest" to a given one; in studying long signal sequences, it is important to recognize "homogeneous subsequences" and the places of their junction. This includes, in particular, the well-known change-point prob lern, which is given considerable attention in mathematical statistics and the theory of stochastic processes. As applied to the study of narrative texts, the problern of recognizing depen dent and independent texts ( e . g. , chronicles) Ieads to the problern offinding texts having a common source, i. e. , the sameoriginal (such texts are naturally called dependent), or, on the contrary, having different sources (such texts are natu rally called independent). Clearly, such problems are exceedingly complicated, and therefore the appearance of new empirico-statistical recognition methods which, along with the classical approaches, may prove useful in concrete studies (e. g. , source determination) is welcome.

We present certain empirico-statistical methods for the analysis of narrative and nu merical data extracted from different texts of historical character such as chronicles or annals. They are based on several statistical principles worked out by the author, and originally reported at the Third International Vilnius Conference on Probability Theory and Mathematical Statistics in 1981. The principal results were published in the papers [15]-[32], [293]-[299], [304]-[319] and in the book: A. T. Fomenko, Methods for Statistical Analysis of Narrative Texts and Applications to Chronol ogy, Moscow Univ. Press, Moscow, 1990 (in Russian). See also Part 1. The methods are applied to the problem of correct dating of the events in ancient and medieval history. These results induce conjectures on the redating of some important ancient historical events. Generally speaking, we might say that the commonly accepted "Modern Text book" of ancient and medieval European, Mediterranean, Egyptian and Middle Eastern history is a fibered (layered) chronicle obtained by gluing together four nearly identical copies of a shorter "original" chronicle. The other three chronicles are obtained from the "original" chronicle by redating and renaming the events de scribed in them; we rigidly move the "original" chronicle in its entirety backwards in time by approximately 333, 1053 and 1778 years. Thus, the full "Modern Textbook" can be reconstructed from its smaller part, namely from the "original" chronicle for the 9-17th cc. A.D. See Appendix 1, Figs. 101-104.

Geodesic flows of Riemannian metrics on manifolds are one of the classical objects in geometry. A particular place among them is occupied by integrable geodesic flows. We consider them in the context of the general theory of integrable Hamiltonian systems, and in particular, from the viewpoint of a new topological classification theory, which was recently developed for integrable Hamiltonian systems with two degrees of freedom. As a result, we will see that such a new approach is very useful for a deeper understanding of the topology and geometry of integrable geodesic flows. The main object to be studied in our paper is the class of integrable geodesic flows on two-dimensional surfaces. There are many such flows on surfaces of small genus, in particular, on the sphere and torus. On the contrary, on surfaces of genus 9 > 1, no such flows exist in the analytical case. One of the most important and interesting problems consists in the classification of integrable flows up to different equivalence relations such as (1) an isometry, (2) the Liouville equivalence, (3) the trajectory equivalence (smooth and continuous), and (4) the geodesic equivalence. In recent years, a new technique was developed, which gives, in particular, a possibility to classify integrable geodesic flows up to these kinds of equivalences. This technique is presented in our paper, together with various applications. The first part of our book, namely, Chaps.

Up until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) have come gradually to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say, which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education, and what might be the appropriate level of abstractness of their exposition. The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstractness of its exposition were dictated by the view that, in addition to the geometry of curves and surfaces, the following topics are certainly useful in the various areas of application of mathematics (especially in elasticity and relativity, to name but two), and are therefore essential: the theory of tensors (including covariant differentiation of them); Riemannian curvature; geodesics and the calculus of variations (including the conservation laws and Hamiltonian formalism); the particular case of skew-symmetric tensors (i. e.

Over the past fifteen years, the geometrical and topological methods of the theory of manifolds have as- sumed a central role in the most advanced areas of pure and applied mathematics as well as theoretical physics. The three volumes of Modern Geometry - Methods and Applications contain a concrete exposition of these methods together with their main applications in mathematics and physics. This third volume, presented in highly accessible languages, concentrates in homology theory. It contains introductions to the contemporary methods for the calculation of homology groups and the classification of manifesto. Both scientists and students of mathematics as well as theoretical physics will find this book to be a valuable reference and text.

This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised.

One service mathematics has rendered the 'Eot moi, ..., si j'avait JU comment en revenir. human race. h has put common sense back je n'y serais point aUe:' Jules Verne where it belongs, 011 the topmost shelf nen to the dusty canister labeUed 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. H es viside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other pans and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. 1hen one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin' . * 1111 Oulik'. n. . Chi" *. * ~ Mm~ Mu,d. ", Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

One service mathematics has rendered the 'Et moi, ..., si j'avait su comment en revenir, je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Matht"natics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics seNe as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

One service mathematics has rendered the human race. It has put common sense back where it belongs. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled discarded nonsense. Eric TBell Every picture tells a story. Advenisement for for Sloan's backache and kidney oils, 1907 The book you have in your hands as you are reading this, is a text on3-dimensional topology. It can serve as a pretty comprehensive text book on the subject. On the other hand, it frequently gets to the frontiers of current research in the topic. If pressed, I would initially classify it as a monograph, but, thanks to the over three hundred illustrations of the geometrical ideas involved, as a rather accessible one, and hence suitable for advanced classes. The style is somewhat informal; more or less like orally presented lectures, and the illustrations more than make up for all the visual aids and handwaving one has at one's command during an actual presentation.

Geodesic flows of Riemannian metrics on manifolds are one of the classical objects in geometry. A particular place among them is occupied by integrable geodesic flows. We consider them in the context of the general theory of integrable Hamiltonian systems, and in particular, from the viewpoint of a new topological classification theory, which was recently developed for integrable Hamiltonian systems with two degrees of freedom. As a result, we will see that such a new approach is very useful for a deeper understanding of the topology and geometry of integrable geodesic flows. The main object to be studied in our paper is the class of integrable geodesic flows on two-dimensional surfaces. There are many such flows on surfaces of small genus, in particular, on the sphere and torus. On the contrary, on surfaces of genus 9 > 1, no such flows exist in the analytical case. One of the most important and interesting problems consists in the classification of integrable flows up to different equivalence relations such as (1) an isometry, (2) the Liouville equivalence, (3) the trajectory equivalence (smooth and continuous), and (4) the geodesic equivalence. In recent years, a new technique was developed, which gives, in particular, a possibility to classify integrable geodesic flows up to these kinds of equivalences. This technique is presented in our paper, together with various applications. The first part of our book, namely, Chaps.

One service mathematics has rendered the human race. It has put common sense back where it belongs. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled discarded nonsense. Eric TBell Every picture tells a story. Advenisement for for Sloan's backache and kidney oils, 1907 The book you have in your hands as you are reading this, is a text on3-dimensional topology. It can serve as a pretty comprehensive text book on the subject. On the other hand, it frequently gets to the frontiers of current research in the topic. If pressed, I would initially classify it as a monograph, but, thanks to the over three hundred illustrations of the geometrical ideas involved, as a rather accessible one, and hence suitable for advanced classes. The style is somewhat informal; more or less like orally presented lectures, and the illustrations more than make up for all the visual aids and handwaving one has at one's command during an actual presentation.

Today the methods of applied statistics have penetrated very different fields of knowledge, including the investigation oftexts ofvarious origins. These "texts" may be considered as signal sequences of different kinds, long genetic codes, graphic representations (which may be coded and represented by a "text"), as well as actual narrative texts (for example, historical chronicles, originals, documents, etc. ). One ofthe most important problems arising here is to recognize dependent text, i. e. , texts which have a measure of "resemblance", arising from some kind of "common origin". For instance, in pattern-recognition problems, it is essential to identify from a large set of "patterns" a pattern that is "closest" to a given one; in studying long signal sequences, it is important to recognize "homogeneous subsequences" and the places of their junction. This includes, in particular, the well-known change-point prob lern, which is given considerable attention in mathematical statistics and the theory of stochastic processes. As applied to the study of narrative texts, the problern of recognizing depen dent and independent texts ( e . g. , chronicles) Ieads to the problern offinding texts having a common source, i. e. , the sameoriginal (such texts are naturally called dependent), or, on the contrary, having different sources (such texts are natu rally called independent). Clearly, such problems are exceedingly complicated, and therefore the appearance of new empirico-statistical recognition methods which, along with the classical approaches, may prove useful in concrete studies (e. g. , source determination) is welcome.

We present certain empirico-statistical methods for the analysis of narrative and nu merical data extracted from different texts of historical character such as chronicles or annals. They are based on several statistical principles worked out by the author, and originally reported at the Third International Vilnius Conference on Probability Theory and Mathematical Statistics in 1981. The principal results were published in the papers [15]-[32], [293]-[299], [304]-[319] and in the book: A. T. Fomenko, Methods for Statistical Analysis of Narrative Texts and Applications to Chronol ogy, Moscow Univ. Press, Moscow, 1990 (in Russian). See also Part 1. The methods are applied to the problem of correct dating of the events in ancient and medieval history. These results induce conjectures on the redating of some important ancient historical events. Generally speaking, we might say that the commonly accepted "Modern Text book" of ancient and medieval European, Mediterranean, Egyptian and Middle Eastern history is a fibered (layered) chronicle obtained by gluing together four nearly identical copies of a shorter "original" chronicle. The other three chronicles are obtained from the "original" chronicle by redating and renaming the events de scribed in them; we rigidly move the "original" chronicle in its entirety backwards in time by approximately 333, 1053 and 1778 years. Thus, the full "Modern Textbook" can be reconstructed from its smaller part, namely from the "original" chronicle for the 9-17th cc. A.D. See Appendix 1, Figs. 101-104.

A collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.

This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised.

One service mathematics has rendered the 'Et moi, ..., si j'avait su comment en revenir, je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Matht"natics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics seNe as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

Over the past fifteen years, the geometrical and topological methods of the theory of manifolds have as- sumed a central role in the most advanced areas of pure and applied mathematics as well as theoretical physics. The three volumes of Modern Geometry - Methods and Applications contain a concrete exposition of these methods together with their main applications in mathematics and physics. This third volume, presented in highly accessible languages, concentrates in homology theory. It contains introductions to the contemporary methods for the calculation of homology groups and the classification of manifesto. Both scientists and students of mathematics as well as theoretical physics will find this book to be a valuable reference and text.

One service mathematics has rendered the 'Eot moi, ..., si j'avait JU comment en revenir. human race. h has put common sense back je n'y serais point aUe: ' Jules Verne where it belongs, 011 the topmost shelf nen to the dusty canister labeUed 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. H es viside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other pans and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. 1hen one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin' . * 1111 Oulik'. n. . Chi" *. * ~ Mm~ Mu,d. ", Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.