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Volume III of the Collected Works of V.I. Arnold contains papers
written in the years 1972 to 1979. The main theme emerging in
Arnold's work of this period is the development of singularity
theory of smooth functions and mappings. The volume also contains
papers by V.I. Arnold on catastrophe theory and on A.N.
Kolmogorov's school, his prefaces to Russian editions of several
books related to singularity theory, V. Arnold's lectures on
bifurcations of discrete dynamical systems, as well as a review by
V.I. Arnold and Ya.B. Zeldovich of V.V. Beletsky's book on
celestial mechanics. Vladimir Arnold was one of the great
mathematical scientists of our time. He is famous for both the
breadth and the depth of his work. At the same time he is one of
the most prolific and outstanding mathematical authors.
Volume III of the Collected Works of V.I. Arnold contains papers
written in the years 1972 to 1979. The main theme emerging in
Arnold's work of this period is the development of singularity
theory of smooth functions and mappings. The volume also contains
papers by V.I. Arnold on catastrophe theory and on A.N.
Kolmogorov's school, his prefaces to Russian editions of several
books related to singularity theory, V. Arnold's lectures on
bifurcations of discrete dynamical systems, as well as a review by
V.I. Arnold and Ya.B. Zeldovich of V.V. Beletsky's book on
celestial mechanics. Vladimir Arnold was one of the great
mathematical scientists of our time. He is famous for both the
breadth and the depth of his work. At the same time he is one of
the most prolific and outstanding mathematical authors.
This book is devoted to the phenomenon of quasi-periodic motion in
dynamical systems. Such a motion in the phase space densely fills
up an invariant torus. This phenomenon is most familiar from
Hamiltonian dynamics. Hamiltonian systems are well known for their
use in modelling the dynamics related to frictionless mechanics,
including the planetary and lunar motions. In this context the
general picture appears to be as follows. On the one hand,
Hamiltonian systems occur that are in complete order: these are the
integrable systems where all motion is confined to invariant tori.
On the other hand, systems exist that are entirely chaotic on each
energy level. In between we know systems that, being sufficiently
small perturbations of integrable ones, exhibit coexistence of
order (invariant tori carrying quasi-periodic dynamics) and chaos
(the so called stochastic layers). The Kolmogorov-Arnol'd-Moser
(KAM) theory on quasi-periodic motions tells us that the occurrence
of such motions is open within the class of all Hamiltonian
systems: in other words, it is a phenomenon persistent under small
Hamiltonian perturbations. Moreover, generally, for any such system
the union of quasi-periodic tori in the phase space is a nowhere
dense set of positive Lebesgue measure, a so called Cantor family.
This fact implies that open classes of Hamiltonian systems exist
that are not ergodic. The main aim of the book is to study the
changes in this picture when other classes of systems - or contexts
- are considered.
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VLADIMIR I. ARNOLD-Collected Works - Dynamics, Combinatorics, and Invariants of Knots, Curves, and Wave Fronts 1992-1995 (Hardcover, 1st ed. 2023)
Vladimir I. Arnold; Edited by Alexander B. Givental, Boris A Khesin, Mikhail B. Sevryuk, Victor A. Vassiliev, …
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R3,733
Discovery Miles 37 330
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Ships in 12 - 17 working days
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This volume 6 of the Collected Works comprises 27 papers by
V.I.Arnold, one of the most outstanding mathematicians of all
times, written in 1991 to 1995. During this period Arnold's
interests covered Vassiliev's theory of invariants and knots,
invariants and bifurcations of plane curves, combinatorics of
Bernoulli, Euler and Springer numbers, geometry of wave fronts, the
Berry phase and quantum Hall effect. The articles include a list of
problems in dynamical systems, a discussion of the problem of
(in)solvability of equations, papers on symplectic geometry of
caustics and contact geometry of wave fronts, comments on problems
of A.D.Sakharov, as well as a rather unusual paper on projective
topology. The interested reader will certainly enjoy Arnold's 1994
paper on mathematical problems in physics with the opening by-now
famous phrase "Mathematics is the name for those domains of
theoretical physics that are temporarily unfashionable." The book
will be of interest to the wide audience from college students to
professionals in mathematics or physics and in the history of
science. The volume also includes translations of two interviews
given by Arnold to the French and Spanish media. One can see how
worried he was about the fate of Russian and world mathematics and
science in general.
Volume IV of the Collected Works of V.I. Arnold includes papers
written mostly during the period from 1980 to 1985. Arnold's work
of this period is so multifaceted that it is almost impossible to
give a single unifying theme for it. It ranges from properties of
integral convex polygons to the large-scale structure of the
Universe. Also during this period Arnold wrote eight papers related
to magnetic dynamo problems, which were included in Volume II,
mostly devoted to hydrodynamics. Thus the topic of singularities in
symplectic and contact geometry was chosen only as a "marker" for
this volume.There are many articles specifically translated for
this volume. They include problems for the Moscow State University
alumni conference, papers on magnetic analogues of Newton's and
Ivory's theorems, on attraction of dust-like particles, on
singularities in variational calculus, on Poisson structures, and
others. The volume also contains translations of Arnold's comments
to Selected works of H. Weyl and those of A.N. Kolmogorov. Vladimir
Arnold was one of the great mathematical scientists of our time. He
is famous for both the breadth and the depth of his work. At the
same time he is one of the most prolific and outstanding
mathematical authors.
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