Volume 33, Issue 12, December 1992
Index of content:

Multiparameter deformations of the universal enveloping algebras of the simple Lie algebras A _{ l } for all l≥2 and the Yang–Baxter equation
View Description Hide DescriptionA {1+1/2l(l−1)}‐parameter deformation of the universal enveloping algebra of the simple Lie algebraA _{ l } (=sl(l+1, C)) is derived for all l≥2, starting from a certain multiparameter matrix that satisfies the quantum Yang–Baxter equation. It is shown that this Hopf algebra has the same product relations and antipode as the standard one‐parameter deformation U _{ q } (sl(l+1,C)), but has a different coproduct. It is also shown that for all l≥2 there exists a 1/2l(l−1)‐parameter Hopf algebra whose product relations are merely the commutation relations of A _{ l } itself, but whose coproduct is different from the usual one for the universal enveloping algebra of A _{ l }.

Duality rotation as a Lie–Bäcklund transformation
View Description Hide DescriptionIt is shown that the duality rotation group for Maxwell electrodynamics in vacuum can be considered as a general Lie–Bäcklund transformation group acting on Fourier transforms of the electromagnetic potential. Conserved quantity connected with this group is found to be a consequence of the Noether theorem.

Invariance transformations of the class y‘=F(x)y ^{ n } of differential equations
View Description Hide DescriptionUsing a systematic search for symmetries of differential equations, transformations are found that leave equations of the type y‘=F(x)y ^{ n } form‐invariant, in particular, in the case n=2. Given the (general) solution in case of a function F, these transformations can be used for constructing the (general) solution for a class of related functions F̃.

On polynomial solutions of differential equations
View Description Hide DescriptionA general method of obtaining linear differential equations having polynomialsolutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the ‘‘projectivized’’ representation possessing an invariant subspace and the spectral problem for a certain linear differential operator with variable coefficients. It is shown in general that polynomialsolutions of partial differential equations occur; in the case of Lie superalgebras there are polynomialsolutions of some matrix differential equations, quantum algebras give rise to polynomialsolutions of finite‐difference equations. Particularly, known classical orthogonal polynomials will appear when considering SL(2, R) acting on RP _{1}. As examples, some polynomials connected to projectivized representations of sl_{2}(R), sl_{2}(R)_{ q }, osp(2,2), and so_{3} are briefly discussed.

Stability conditions for method of self‐similar approximations
View Description Hide DescriptionA generalization for the method of self‐similar approximations suggested recently by the author is given. Stability conditions for the method are formulated: first, the mapping multipliers for the self‐similar recurrence relations have to be less than unity; second, the Lyapunov exponent for the differential form of the self‐similar relations has to be negative. The fixed‐point conditions defining the governing functions are analyzed from the point of view of stability. It is shown that the fixed‐point condition expressed in the form of the principle of minimal sensitivity provides a contracting mapping contrary to the condition in the form of the principle of minimal difference. This explains why, in general, the former fixed‐point condition yields more accurate results than the latter.

Conformal symmetry inheritance with cosmological constant
View Description Hide DescriptionA study is made of space‐times satisfying the Einstein field equations with a nonzero cosmological constant and admitting a conformal Killing vector (CKV), ξ^{ a }, which is such that the Lie derivative of the energy‐momentum tensor in the direction of ξ^{ a } is zero. This condition is necessary, but not sufficient, to ensure that the physical quantities associated with the energy‐momentum tensor inherit the CKV symmetry. It is shown that this condition leads to two classes of space‐times, each of which may be regarded as generalizations of the de Sitter space‐times. The possible physical interpretations of these models are discussed.

Normal form of a metric admitting a parallel field of planes
View Description Hide DescriptionLocal coordinate normal forms for two classes of metrics are obtained. The first class is comprised of those metrics that have a field of degenerate, but not necessarily null, planes, thereby extending a result of A. G. Walker. The second class of metrics are Artinian metrics that possess a pair of parallel, complementary, null distributions. This latter class is shown to be equivalent to a certain class of symplectic forms.

Spinors and the Dirac operator on hypersurfaces. I. General theory
View Description Hide DescriptionIt is shown that a hypersurface immersed isometrically into the Euclidean spaceR ^{ n+1}, where n=2ν or 2ν+1, has a pin structure such that the associated bundle of 2^{ν}‐component spinors is trivial. This is used to derive a new formula for the Dirac operator on hypersurfaces. The Dirac operator is slightly modified to be compatible with the twisted adjoint representation of the pin group. When R ^{ n+1} is foliated by hypersurfaces, then the Dirac operator in R ^{ n+1} splits into a radial and a tangential part with respect to the foliation. There is a corresponding new formula for the Laplacian.

Rigidity of c _{1}‐self‐dual connections on quaternionic Kähler manifolds
View Description Hide DescriptionIn Galicki and Poon [J. Math. Phys. 32, 1263 (1991)], certain self‐dual connections are defined on quaternionic Kähler manifolds. In this paper, it is proved that there is essentially only one c _{1}‐self‐dual connection over any simply connected quaternionic Kähler manifold with nonzero scalar curvature.

Analytical solutions to the generalized spheroidal wave equation and the Green’s function of one‐electron diatomic molecules
View Description Hide DescriptionThe eigenvalue problem of the angular part of generalized spheroidal wave equations (SWE) for describing the scattering of a charged particle on two Coulomb centers with different charges is cast in a matrix form which is then solved by a standard matrix procedure. This simple mathematical method is shown to be the most efficient and accurate among the others given for obtaining the solution to the ordinary SWE and the generalized SWE. An application to a complete solution for describing the scattering of a charged particle by a dipole is presented. The mathematical method for the analytical solution to the radial part of SWE obtained by series expansion in Coulomb wave functions for the two‐center Coulomb scattering is found to be more efficient and accurate than the numerical methods. In the conclusion this work provides an analytical solution to the generalized spheroidal wave equation and the Green’s function of one‐electron diatomic molecules.

Neutrino in the presence of gravitational fields: Exact solutions
View Description Hide DescriptionThe present paper is a further development of our previous paper [G. V. Shishkin and V. M. Villalba, J. Math. Phys. 33, 2093 (1992)]. Using the algebraic method of separation of variables, the structure of the metric functions is obtained in the presence of gravitational fields associated with a diagonal metric, which permits separation of variables in the Dirac equation. Applying the results obtained in a previous paper, we study the possibilities of finding exact solutions in the Dirac equation for massless neutrinos.

k‐symplectic structures
View Description Hide DescriptionThe basic properties of k‐symplectic structures are introduced and developed in analogy with the well‐known symplectic differential geometry. Examples of such structures related to k‐symplectic Lie algebras are given.

Generalization of Jacobi’s decomposition theorem to the rotation and translation of a solid in a fluid
View Description Hide DescriptionJacobi found that the rotation of a symmetrical heavy top about a fixed point can be decomposed into a product of the torque‐free rotations of two triaxial bodies about their centers of mass. This theorem is generalized to the Kirchhoff’s case of the rotation and translation of a symmetrical solid in a fluid.

Comment on: Hamiltonian formulation of a theory with constraints [U. Kulshreshta, J. Math. Phys. 33, 633 (1992)]
View Description Hide DescriptionThe Hamiltonian formulation of a constrained dynamical system, analyzed by Kulshreshta [J. Math. Phys. 33, 633 (1992)] is discussed. The theory possesses four coordinates and four primary second class constraints and thus two physical degrees of freedom. The correct equations of motion for the system are given. The reduced theory (q _{4}=k) is shown to have three primary and one secondary second class constraints instead of three primary first class ones as claimed by Kulshreshta. The equations of motion are derived using the Dirac bracket.

Direct integration of generalized Lie or dynamical symmetries of three degrees of freedom nonlinear Hamiltonian systems: Integrability and separability
View Description Hide DescriptionIt is shown that by directly integrating the characteristic equation of the infinitesimals of the generalized Lie or dynamical symmetries associated with three degrees of freedom Hamiltonians, one can almost, by inspection, obtain the required involutive integrals of motion, whenever they exist. The method is illustrated for the coupled quartic and cubic oscillators considered earlier. Further, all the separable coordinates can be obtained by integrating a subset of the characteristic equation associated with the coordinate variables alone.

Continuum theory of microstretch liquid crystals
View Description Hide DescriptionA continuum theory is proposed for liquid crystals whose molecular elements can expand and contract, in addition to undergoing translations and rotations. This is the microstretch continuum theory, generalizing microplar theory of liquid crystals.Constitutive equations are developed for arbitrarily shaped molecular elements. Thermodynamical restrictions are studied. The theory is then specialized for breathing rodlike liquid crystals. Equilibrium configuration for the microstretch motion is studied.

Symmetries of variable coefficient Korteweg–de Vries equations
View Description Hide DescriptionThe Lie point symmetries of the equation u _{ t }+f(x, t)uu _{ x }+g(x,t)u _{ xxx }=0 are studied. The symmetry group is shown to be, at most, four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the KdV equation with f=g=1. For nine different classes of functions f and g, the symmetry group turns out to be three dimensional. Two‐dimensional and one‐dimensional symmetry groups occur for 11 and 15 classes of equations, respectively.

A time‐harmonic Green’s function technique and wave propagation in a stratified nonreciprocal chiral slab with multiple discontinuities
View Description Hide DescriptionA transversely polarized time‐harmonic electromagnetic plane wave normally incident on a stratified nonreciprocal chiral (bi‐isotropic) slab with multiple discontinuities in the parameters is considered. Up‐ and down‐going modes are identified and used to rewrite Maxwell’sequations. Two Green’s function matrices are introduced to express the internal modes in terms of the down‐going incident modes. Ordinary differential equations for the Green’s functions are given together with the boundary conditions. Several operators are introduced to treat the multiple discontinuities in the permittivity and permeability. The method provides an efficient way to calculate the internal fields as well as the scattered fields.

Some soliton solutions for the quadratic bundle
View Description Hide DescriptionAn explicit form of the N‐soliton solutions related to the auxiliary linear problem of the quadratic bundle of a special kind is obtained. A recipe to find their gauge equivalent solutions is given. It is shown that the additional reduction of the potential is sufficient but not necessary. The asymptotic behavior of the investigated solutions is studied. Some examples are given in the end.

The Casimir theorem and the inversion problem in electrodynamics at first quantized level
View Description Hide DescriptionThe inversion problem in electrodynamics is analyzed for a nonlocalized source. The model used is the generalized Schrödinger current from continuum to continuum states. The general expressions of the electric and magnetic multipoles in terms of the infinite transitions from which they are generated and the contribution of each transition are derived. It is shown, for a particular example, that the scattering wave functions of a given transition are determined in terms of the electric and magnetic multipole amplitudes to which the transition gives rise. It can be shown (but the proof is not given in this paper) that similar results are valid for every transition and are, therefore, quite general. It is also shown that the couple of wave functions corresponding to each transition satisfies a Schrödinger equation with an energy dependent potential which is completely determined by the same multipole amplitudes. In this way, the most clear connection between scattering and electromagnetic phenomena is displayed.