Volume 33, Issue 3, March 1992
Index of content:

Existence and uniqueness of the universal W algebra
View Description Hide DescriptionA precise definition of the universal (classical) Walgebra for the W _{ n } series is given and its existence and uniqueness are proved. The main observation is that there is a natural reduction from W _{ n } to W _{ n−1} that allows us to define the universal Walgebra as an inverse limit. This universal Walgebra is, in a sense, the smallest Walgebra from which all W _{ n } can be obtained by reduction. These results extend to other Walgebras obtained by reducing the Gel’fand–Dickey brackets, as well as to W superalgebras obtained from the supersymmetric Gel’fand–Dickey brackets.

The Stokes constants associated with the differential equation d ^{2}ψ/dz ^{2}−4A ^{2} z ^{ n−2}(z ^{ n }+c)ψ=0
View Description Hide DescriptionThe solutions of the differential equationd ^{2}ψ/dz ^{2}−4A ^{2} z ^{ n−2}(z ^{ n }+c)ψ=0 are expressed in terms of convenient integral representations of the Laplace type leading to asymptotic expressions that are rewritten as linear combinations of phase‐integral functions. Formulas for the Stokes constants in the Fröman–Fröman phase‐integral method are then derived. For the first‐ and the third‐order approximations, explicit formulas are given, and the limiting case of large absolute values of Ac in particular is considered. A discussion of numerical results is also given.

Lattices and Θ‐function identities. I: Theta constants
View Description Hide DescriptionThe gluing construction of lattices is used to generate and study a number of theta constant identities. It is shown, for example, that the Jacobi identity, a well‐known degree 4 identity, can be derived from a degree 2 identity. A list of all quadratic identities in θ_{3} derivable from this lattice method, containing over 30 algebraically independent identities, and conjectured to yield all quadratic identities in θ_{3} derivable by any means, is included. In contrast, only two independent quadratic identities in θ_{3} have been found elsewhere in the mathematical literature. The theta constants of lattices and their glue classes are investigated, and the fact that the theta constant of each glue class is a root of a polynomial whose coefficients are linear combinations of theta constants of lattices is shown.

Lattices and θ‐function identities. II: Theta series
View Description Hide DescriptionThe gluing construction of lattices is used to generate and study a number of theta function densities. It is shown, for example, that Riemann’s formula, a fundamental degree 4 identity, can be derived from a degree 2 identity. It is proved that all in a large class of identities can be derived from these lattice techniques. A list of 24 independent quadratic identities in the Jacobi functions ϑ_{1} and ϑ_{3}, conjectured to be complete, with all but two of them seeming to be new, is given. The theta series of glue classes is also investigated, and it is shown that all of them, as well as the functions ϑ_{ a, b }, satisfy polynomials whose coefficients are linear combinations of theta series of lattices; these polynomials have some interesting properties.

The Painlevé classification of partial differential equations. III. Rational equations with several poles
View Description Hide DescriptionSemilinear second‐order partial differential equations in two independent variables with rational right‐hand sides are investigated. It is found that the Painlevé admissible partial differential equations can be transformed to the form with maximally three poles and equations with two and three poles are classified. Results of a series of papers on the Painlevé classification are summarized.

Nijenhuis–Bianchi identity and BRST‐like operators
View Description Hide DescriptionLet M be a differentiable manifold with a (1,1) tensor field S _{ν} ^{μ}(x). A notion of distorted torsion tensor is introduced that reduces to the ordinary torsion tensor when S _{ν} ^{μ}=δ_{ν} ^{μ}. It is then shown that the distorted torsion tensor satisfies a generalization of the first Bianchi identity. The new identity involves the Nijenhuis tensor in contrast to the standard Bianchi identity. Using this generalized formula, BRST‐like operators are constructed that satisfy a Lie‐super algebra, when both Nijenhuis and Riemann tensors vanish identically.

On coadjoint orbits of rotational perfect fluids
View Description Hide DescriptionIn this paper the structure of vortex coadjoint orbits pertaining to perfect fluids having smooth vorticities in R^{3}, within the framework set up by J. Marsden and A. Weinstein [Physica D 7, 305–323 (1983)], in terms of an associated Hamiltonian Kähler manifold (the Clebsch manifold, described in terms of the so‐called Clebsch variables) is investigated. The topological quantization of Mikhailov and Kuznetsov is related to geometric quantization. Natural candidates for the coherent states on the Clebsch manifold are also exhibited.

Special orthonormal frames
View Description Hide DescriptionA natural gauge condition is used to select special orthonormal frames and thereby a preferred teleparallel geometry on a Riemannian manifold of any dimension. The condition is equivalent to a nonlinear elliptic system for a rotation. Existence and uniqueness of solutions are discussed. Certain special cases and applications are noted.

Dirac equation in external fields: Separation of variables in curvilinear coordinates
View Description Hide DescriptionThe algebraic method of separation of variables in the Dirac equation proposed by one of the present authors [Gravitation and Electromagnetism (U.P., Minsk, 1989) Issue 4, p. 156 (in Russian)] is developed for the case of the most general interaction of the Dirac particle in an external field, taking into account scalar, vector, tensor, pseudovector, pseudoscalar, and gravitation connections. The present work, which concludes this series of papers entitled ‘‘Dirac equation in external fields’’ [J. Math. Phys. 32, 3184 (1991); 33, XXX (1992)] is dedicated to the investigation of the problem in the case of general orthogonal curvilinear coordinates, and allows the introduction of gravitation through the spinor connection and Lame’s functions. Special consideration is given to the cases, when the generalized Lame’s functions do not separate the variables multiplicatively (e.g., elliptic cylindrical, parabolic cylindrical, and oblate and prolate spheroidal coordinates). All the previous results and numerous results of other authors are particular cases of this investigation.

On a geometrical formulation of the Nambu dynamics
View Description Hide DescriptionA geometrical formulation of the N‐triplet Nambu dynamics is given. It enables one to treat some general issues of the latter, including integral invariants and canonical transformations in a simple way. A possible underlying geometrical structure is discussed.

On a variational principle for the Nambu dynamics
View Description Hide DescriptionA variational principle for the Nambu dynamics is analyzed. Since the equations of motion single out a distinguished two‐form rather then a one‐form, the usual construction of the action S[γ] as an integral of a one‐form along the curve γ on the extended phase space has to be modified.

Relativistic tautochrone
View Description Hide DescriptionThe path joining two points A and B, which a particle falling from rest in a uniform gravitational field must adopt, so that the time of transit from A to B is independent of the location of A is called the tautochrone. In the nonrelativistic case, the path is known to be a cycloid, a standard method associated with the derivation of this result being, for example, the method of Laplace transforms. For the relativistic case that is studied herein, the methods of fractional calculus are shown to be more useful in the derivation of the exact relativistic tautochrone. This latter derivation is then checked out by the Laplace transform approach for the relativistic problem, from the point of view of consistency. Using the same method, the relativistic tautochrone associated with a charged particle of charge q and mass m falling from rest in an uniform electric field is also worked out. The tautochrone turns out to be an incomplete elliptic function of the second kind E(δ,r). As an application, the power radiated by the charged particle as it accelerates along the curve is then computed; it is found to be proportional to (1 − v ^{2}/c ^{2})^{−2}, with v(c) the velocity of the particle (light). Finally, an appendix highlights the utility of fractional calculusvis‐á‐vis the approach of Abel for the relativistic tautochrone.

The R‐matrix theory and the reduction of Poisson manifolds
View Description Hide DescriptionThe relation between the R‐matrix approach to the construction of bi‐Hamiltonian structures and the reduction of Poissonmanifolds is discussed. The crucial role of the reduction process is emphasized by showing, on the basis of specific examples, that the knowledge of a R matrix may not be sufficient. The reduction of the Poissonstructures related with R from the entire algebra to the Lax submanifold may become an obstruction in constructing the bi‐Hamiltonian structures of a specific hierarchy of integrable equations.

Factorization of a dissipative wave equation and the Green functions technique for axially symmetric fields in a stratified slab
View Description Hide DescriptionThe time domain scattering problem for axially symmetric fields and a dissipative stratified slab is considered. Explicit representations are derived for the wave splitting operators that factorize the dissipative Hankel transformed wave equation. The Green functions approach associated with this splitting is developed. Partial differential equations(PDEs) for the Green functions are derived and used to derive the imbedding equations for the reflection and transmission kernels. Numerical results for the internal fields are given in the direct problem, and the phase velocity is reconstructed in the inverse problem.

On the stability of collapse in the critical case
View Description Hide DescriptionCollapsing solutions in the Cauchy problem of the nonlinear Schrödinger equation i ∂_{ t }ψ + ∇^{2}ψ +‖ ψ‖^{ p }ψ = 0 (x∈R^{ d }) are considered in the so‐called critical case pd=4, where d is the spatial dimension. A stability theorem for radial collapse is presented which proves that the formation of the singularity remains ‘‘close’’ to the self‐similar collapsing solution with a spatial profile given by the ground state solitary wave, provided the energy H{ψ}<0. An analogous result is given for the scalar Zakharov equations and a generalized Korteweg–de Vries equation ∂_{ tu } + (p + 1)u ^{ p } ∂_{ xu } + ∇^{2} ∂_{ xu } = 0.

A comment on the generalized Kirchhoff–d’Adhémar formula for massless free fields in Minkowski space‐time
View Description Hide DescriptionA ‘‘spin‐reduction’’ argument is used to obtain the generalized Kirchhoff–d’Adhémar formula for spin≳0 massless free fields in Minkowski space‐time from the corresponding expression for a scalar field satisfying D’Alembert’s equation. A similar result is obtained for the Newman–Penrose constants in flat space‐time.

The Berry connection and Born–Oppenheimer method
View Description Hide DescriptionBy performing the most general Born–Oppenheimer procedure, the (non‐Abelian) Berry connection for quantum systems in a quantum environment is derived. This method is then applied to the rapid rotation of a particle about a slowly changing axis, as exemplified by the electronic motion of a diatomic molecule. The angular part of the resulting dynamics for the quantum environment is equivalent to that of a monopole.

Topological aspects of a fermion, chiral anomaly, and Berry phase
View Description Hide DescriptionThe relationship of the Berry phase with the topological aspects of a fermion and chiral anomaly is studied here. It is observed that this phase reflects the effect of quantum geometry when the quantization procedure of a Fermi field is considered from a classical system and may be considered as the manifestation of the multiply connected nature of quantum space‐time.

On equations of motion on compact Hermitian symmetric spaces
View Description Hide DescriptionThe classical equations of motion on compact coherent statemanifolds that have Hermitian symmetric space structure are considered. In the case of Hamiltonians linear in the generators of the group that determines the structure of the manifold of coherent states, the classical motion and the quantum evolution are both described by the same first‐order second degree differential equation. The explicit forms of the classical equations of motion as Matrix Riccati equations on the Grassmann manifoldG _{ A }(C ^{ n }) and the coset manifold SO (2n)/U(n), attached to the Hartree–Fock and, respectively, Hartree–Fock–Bogoliubov problem are obtained.

Matrix elements and Wigner coefficients for U _{ q }[gl(n)]
View Description Hide DescriptionIn this series of papers the fundamental Wigner coefficients and the matrix elements of the generators for the quantum groupU _{ q }[gl(n)] are derived. The U _{ q }[gl(n)]:U _{ q }[gl(n−1)] reduced Wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain U _{ q }[gl(n−1)] invariants in the quantum groupU _{ q }[gl(n)]. The matrix elements of the elementary quantum group generators, in the Gel’fand–Tsetlin basis, are derived below, while all fundamental Wigner coefficients are derived in the second paper of the series.