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This book presents the various algebraic techniques for solving
partial differential equations to yield exact solutions, techniques
developed by the author in recent years and with emphasis on
physical equations such as: the Maxwell equations, the Dirac
equations, the KdV equation, the KP equation, the nonlinear
Schrodinger equation, the Davey and Stewartson equations, the
Boussinesq equations in geophysics, the Navier-Stokes equations and
the boundary layer problems. In order to solve them, I have
employed the grading technique, matrix differential operators,
stable-range of nonlinear terms, moving frames, asymmetric
assumptions, symmetry transformations, linearization techniques and
special functions. The book is self-contained and requires only a
minimal understanding of calculus and linear algebra, making it
accessible to a broad audience in the fields of mathematics, the
sciences and engineering. Readers may find the exact solutions and
mathematical skills needed in their own research.
Vertex algebra was introduced by Boreherds, and the slightly
revised notion "vertex oper- ator algebra" was formulated by
Frenkel, Lepowsky and Meurman, in order to solve the problem of the
moonshine representation of the Monster group - the largest
sporadie group. On the one hand, vertex operator algebras ean be
viewed as extensions of eertain infinite-dimensional Lie algebras
such as affine Lie algebras and the Virasoro algebra. On the other
hand, they are natural one-variable generalizations of commutative
associative algebras with an identity element. In a certain sense,
Lie algebras and commutative asso- ciative algebras are reconciled
in vertex operator algebras. Moreover, some other algebraie
structures, such as integral linear lattiees, Jordan algebras and
noncommutative associa- tive algebras, also appear as subalgebraic
structures of vertex operator algebras. The axioms of vertex
operator algebra have geometrie interpretations in terms of Riemman
spheres with punctures. The trace functions of a certain component
of vertex operators enjoy the modular invariant properties. Vertex
operator algebras appeared in physies as the fundamental algebraic
structures of eonformal field theory, whieh plays an important role
in string theory and statistieal meehanies.
Moreover,eonformalfieldtheoryreveals
animportantmathematiealproperty,the so called "mirror symmetry"
among Calabi-Yau manifolds. The general correspondence between
vertex operator algebras and Calabi-Yau manifolds still remains
mysterious. Ever since the first book on vertex operator algebras
by Frenkel, Lepowsky and Meur- man was published in 1988, there has
been a rapid development in vertex operator su- peralgebras, which
are slight generalizations of vertex operator algebras.
This book presents the various algebraic techniques for solving
partial differential equations to yield exact solutions, techniques
developed by the author in recent years and with emphasis on
physical equations such as: the Maxwell equations, the Dirac
equations, the KdV equation, the KP equation, the nonlinear
Schrodinger equation, the Davey and Stewartson equations, the
Boussinesq equations in geophysics, the Navier-Stokes equations and
the boundary layer problems. In order to solve them, I have
employed the grading technique, matrix differential operators,
stable-range of nonlinear terms, moving frames, asymmetric
assumptions, symmetry transformations, linearization techniques and
special functions. The book is self-contained and requires only a
minimal understanding of calculus and linear algebra, making it
accessible to a broad audience in the fields of mathematics, the
sciences and engineering. Readers may find the exact solutions and
mathematical skills needed in their own research.
Vertex algebra was introduced by Boreherds, and the slightly
revised notion "vertex oper- ator algebra" was formulated by
Frenkel, Lepowsky and Meurman, in order to solve the problem of the
moonshine representation of the Monster group - the largest
sporadie group. On the one hand, vertex operator algebras ean be
viewed as extensions of eertain infinite-dimensional Lie algebras
such as affine Lie algebras and the Virasoro algebra. On the other
hand, they are natural one-variable generalizations of commutative
associative algebras with an identity element. In a certain sense,
Lie algebras and commutative asso- ciative algebras are reconciled
in vertex operator algebras. Moreover, some other algebraie
structures, such as integral linear lattiees, Jordan algebras and
noncommutative associa- tive algebras, also appear as subalgebraic
structures of vertex operator algebras. The axioms of vertex
operator algebra have geometrie interpretations in terms of Riemman
spheres with punctures. The trace functions of a certain component
of vertex operators enjoy the modular invariant properties. Vertex
operator algebras appeared in physies as the fundamental algebraic
structures of eonformal field theory, whieh plays an important role
in string theory and statistieal meehanies.
Moreover,eonformalfieldtheoryreveals
animportantmathematiealproperty,the so called "mirror symmetry"
among Calabi-Yau manifolds. The general correspondence between
vertex operator algebras and Calabi-Yau manifolds still remains
mysterious. Ever since the first book on vertex operator algebras
by Frenkel, Lepowsky and Meur- man was published in 1988, there has
been a rapid development in vertex operator su- peralgebras, which
are slight generalizations of vertex operator algebras.
The transformation of China's economy has involved major changes in
the financial sector. This book offers a detailed and authoritative
guide to financial reform in China since 1979. Bank loans replaced
budgetary grants as the most important source of funds for
investment. A two-tier financial structure, consisting of a central
bank and a newly created specialised commercial bank, developed.
Nonbank financial institutions also mushroomed. The book outlines
the process of change, compares these changes to the earlier
mono-banking system, and shows the problems which remained -
including the lack of a proper financial control mechanism. There
is a detailed case-study of the Shanghai financial markets.
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