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Recent developments in mathematical physics
Book reviews
261
Chapter 4 is devoted to the study of the stability of multilayer layer and multilayer
schemes are reduced to an equivalent
some auxiliary space. The necessary and sufficient
difference schemes. Three-
two-layer difference scheme, defined in
conditions
of stability with respect to the initial
data and right side are obtained for multilayer difference schemes. The application of the theory of stability of multilayer schemes to specific problems of mathematical physics is illustrated by a number of examples. Chapter 5 is of an expository
nature. Here the theory of the stability of difference schemes is
applied to the study of difference schemes for the multivariate Economic factorized and additive schemes are considered.
equations
of mathematical
The investigation
factorized schemes is reduced to a check of the stability conditions
physics.
of the stability of
obtained in chapters 2-4. The
study of the stability of additive schemes is based on special prior estimates. Then the maximum principle is explained and prior estimates of the solution of difference schemes in a uniform norm are given. Chapter 5 concludes with a detailed review of the work of Soviet and foreign authors on the theory of the stability of difference schemes. The book is distinguished
by the generality
of its results and its methodical
unity. The authors
have succeeded in explaining fairly completely and at the same time intelligibly the important and complex problems of the theory of the stability of difference schemes, which are illustrated by a large number of examples contributing The monograph
will be extremely
the field of computational problems of mathematical
mathematics, physics.
to a deeper understanding
of the subject.
useful to students and postgraduate and also to scientists interested
students specializing in
in the numerical solution of
A. D. Lyashko Translated by J. Berry
P. URBAN, Editor. Recent developments in mathematical physics. vi + 610 p. Springer, Wien-New York, 1973. (Nekotorye novye rezul’taty v matematicheskoi fizike. Red. P. Urban.) The book represents the proceedings of the Twelfth International
University Weeks of Nuclear
Physics, held in February 1973 at Graz (Austria). It consists of papers read over the course of two weeks by scientists from various countries. The lectures of M. Bdhm, H. Joos and M. Krammer were devoted to the dynamic problems of the relativistic quark model in terms of standard field theory. Commencing from an equation of the Bete-Salpeter type, the authors develop a method of describing the meson as a connected quark-antiquark state. The problems of mass spectra, lepton discharge, annihilation of an electron and a positron etc. are considered in terms of specific phknomenological models. In lectures of H. Leutwyler the concepts of the parton model are discussed in terms of the “physics of the light cone” intensively developed at the present time. In particular, the question of how far the fundamental ideas of the parton model agree with the hypothesis of light-cone dominance was considered. The lectures of H. Lehrnann were devoted to kiral symmetry and its consequences on lowenergy n-n-scattering. We notice, however, the non-covariance of the approximation scheme used. The technique developed by A. Slavnov and L. Faddeev seems more natural.
262
Book reviews
0. Steinmann in his lectures proposed a new way of solving the problem of the infrared divergence in charged-particle theories. This method proposes that the scattering phenomena be regarded as localized in space-time, thereby processes in which photons are not detectable are considered. The lectures of K. Symanzik were devoted to similar problems, in which infrared singularities in the massless @theory and in theories with added massive particles are investigated by the renormalized group method. The lectures of I. Todorov give a review of conformal-invariant theories, the main attention being given to the Euclidean formulation of conformal invariance and the selective theory of perturbations. Exact expressions for conformally-invariant two-point and three-point Schwinger functions are derived. J. Klauder in his paper records a curious phenomenon: in theories with a sufficiently singular potential the approach to zero of the coupling constant does not in general lead to the free theory. If the function g(x) is sufficiently singular, an interaction of the form 5 : ‘q”(z) : g(x) dx, can be used as an example. In the lectures of F. Jegerlehner and B. Schroer critical phenomena of statistical mechanics are studied by the use of the methods of quantum field theory, in particular, the formalism of normal products. The paper of K. Hepp is devoted to some problems of the irreversible processes of systems which are far from thermal equilibrium. Some questions of quantum-mechanical many-body systems were considered in the paper by F. Thirring. Particular attention was paid to estimating the lower limit of the spectra of the Hamiltonians. The mathematical rigour distinguishing this paper from the majority of the others must be mentioned. In the paper of H. Narnhofer problems of the return of statistical systems to equilibrium are considered within the limits of the algebraic approach. The paper of A. Barut was devoted to the dynamics of particles possessing a magnetic charge. Using simple physical models containing such particles, it is shown that it is possible to obtain definite information about the internal properties of hadrons, for example, about the form factors, the structural functions etc. The book ends with two short r&urn& (for the first week by Strait and for the second week of the conference by Todorov). These short summaries are of a semi-humorous nature. Yu. Drozhzhinov and B. Zav’yalov Translated by J. Berry
261
Chapter 4 is devoted to the study of the stability of multilayer layer and multilayer
schemes are reduced to an equivalent
some auxiliary space. The necessary and sufficient
difference schemes. Three-
two-layer difference scheme, defined in
conditions
of stability with respect to the initial
data and right side are obtained for multilayer difference schemes. The application of the theory of stability of multilayer schemes to specific problems of mathematical physics is illustrated by a number of examples. Chapter 5 is of an expository
nature. Here the theory of the stability of difference schemes is
applied to the study of difference schemes for the multivariate Economic factorized and additive schemes are considered.
equations
of mathematical
The investigation
factorized schemes is reduced to a check of the stability conditions
physics.
of the stability of
obtained in chapters 2-4. The
study of the stability of additive schemes is based on special prior estimates. Then the maximum principle is explained and prior estimates of the solution of difference schemes in a uniform norm are given. Chapter 5 concludes with a detailed review of the work of Soviet and foreign authors on the theory of the stability of difference schemes. The book is distinguished
by the generality
of its results and its methodical
unity. The authors
have succeeded in explaining fairly completely and at the same time intelligibly the important and complex problems of the theory of the stability of difference schemes, which are illustrated by a large number of examples contributing The monograph
will be extremely
the field of computational problems of mathematical
mathematics, physics.
to a deeper understanding
of the subject.
useful to students and postgraduate and also to scientists interested
students specializing in
in the numerical solution of
A. D. Lyashko Translated by J. Berry
P. URBAN, Editor. Recent developments in mathematical physics. vi + 610 p. Springer, Wien-New York, 1973. (Nekotorye novye rezul’taty v matematicheskoi fizike. Red. P. Urban.) The book represents the proceedings of the Twelfth International
University Weeks of Nuclear
Physics, held in February 1973 at Graz (Austria). It consists of papers read over the course of two weeks by scientists from various countries. The lectures of M. Bdhm, H. Joos and M. Krammer were devoted to the dynamic problems of the relativistic quark model in terms of standard field theory. Commencing from an equation of the Bete-Salpeter type, the authors develop a method of describing the meson as a connected quark-antiquark state. The problems of mass spectra, lepton discharge, annihilation of an electron and a positron etc. are considered in terms of specific phknomenological models. In lectures of H. Leutwyler the concepts of the parton model are discussed in terms of the “physics of the light cone” intensively developed at the present time. In particular, the question of how far the fundamental ideas of the parton model agree with the hypothesis of light-cone dominance was considered. The lectures of H. Lehrnann were devoted to kiral symmetry and its consequences on lowenergy n-n-scattering. We notice, however, the non-covariance of the approximation scheme used. The technique developed by A. Slavnov and L. Faddeev seems more natural.
262
Book reviews
0. Steinmann in his lectures proposed a new way of solving the problem of the infrared divergence in charged-particle theories. This method proposes that the scattering phenomena be regarded as localized in space-time, thereby processes in which photons are not detectable are considered. The lectures of K. Symanzik were devoted to similar problems, in which infrared singularities in the massless @theory and in theories with added massive particles are investigated by the renormalized group method. The lectures of I. Todorov give a review of conformal-invariant theories, the main attention being given to the Euclidean formulation of conformal invariance and the selective theory of perturbations. Exact expressions for conformally-invariant two-point and three-point Schwinger functions are derived. J. Klauder in his paper records a curious phenomenon: in theories with a sufficiently singular potential the approach to zero of the coupling constant does not in general lead to the free theory. If the function g(x) is sufficiently singular, an interaction of the form 5 : ‘q”(z) : g(x) dx, can be used as an example. In the lectures of F. Jegerlehner and B. Schroer critical phenomena of statistical mechanics are studied by the use of the methods of quantum field theory, in particular, the formalism of normal products. The paper of K. Hepp is devoted to some problems of the irreversible processes of systems which are far from thermal equilibrium. Some questions of quantum-mechanical many-body systems were considered in the paper by F. Thirring. Particular attention was paid to estimating the lower limit of the spectra of the Hamiltonians. The mathematical rigour distinguishing this paper from the majority of the others must be mentioned. In the paper of H. Narnhofer problems of the return of statistical systems to equilibrium are considered within the limits of the algebraic approach. The paper of A. Barut was devoted to the dynamics of particles possessing a magnetic charge. Using simple physical models containing such particles, it is shown that it is possible to obtain definite information about the internal properties of hadrons, for example, about the form factors, the structural functions etc. The book ends with two short r&urn& (for the first week by Strait and for the second week of the conference by Todorov). These short summaries are of a semi-humorous nature. Yu. Drozhzhinov and B. Zav’yalov Translated by J. Berry