Volume 31, Issue 5, May 1990
Index of content:

Additive isometries on a quaternionic Hilbert space
View Description Hide DescriptionA systematic study of additive isometries on a quaternionic Hilbert space is presented. A number of new results describing the properties of such operators are proved. The work culminates in the first mathematical proof of Wigner’s theorem for quaternionic Hilbert spaces of dimension other than 2 which asserts that any operator which preserves the absolute value of the inner product on a quaternionic Hilbert space is equivalent, in the sense of differing pointwise by a mere phase factor, to a linear isometry. A complete and concise description of the exceptional situation in a two‐dimensional quaternionic Hilbert space is given.

Some examples of the algebra of flows
View Description Hide DescriptionThe algebra of the group of smooth maps from a manifoldM to a compact simple Lie groupG is studied for two cases. The first is when M is the double coset SO (d,R)/SO(d+1,R)/SO (d,R), the corresponding maps are those from a d sphere to G that are invariant under left translations by elements from SO (d, R). In the second example, M is a two‐dimensional torus. The problem of central extension of these algebras is solved. For the first example, no central extension is possible. For the second, the number of independent central extensions is infinite.

The Jordan–Schwinger representations of Cayley–Klein groups. I. The orthogonal groups
View Description Hide DescriptionThe Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the orthogonal groups. The Jordan–Schwinger representations of Cayley–Klein groups are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent and Fock states.

The Jordan–Schwinger representations of Cayley–Klein groups. II. The unitary groups
View Description Hide DescriptionThe unitary Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

The Jordan–Schwinger representations of Cayley–Klein groups. III. The symplectic groups
View Description Hide DescriptionThe symplectic Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the classical symplectic groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

Application of the eigenfunction method to the icosahedral group
View Description Hide DescriptionThe group table for the icosahedral group I is constructed by using the isomorphism between the group I and a subgroup of the permutation group S _{1} _{2}. The single‐valued irreducible representations and Clebsch–Gordan (CG) coefficients of I are calculated by a computer code based on the eigenfunction method. The irreducible matrix elements for all the 60 group elements are given explicitly in the form of (m/n)^{1/2}[exp(iφ)]^{ p } [2 cos φ]^{ q } [2 cos 2φ]^{ r }, where m, n, p, q, and r are integers and φ=2π/5. The Clebsch–Gordan coefficients of I are all real under a new phase convention for time reverse states and tabulated in the form of (m/n)^{1/2}.

Decomposition of the enveloping algebra of sl_{3}
View Description Hide DescriptionThe adjoint representation of sl_{3} on its universal enveloping algebraU is explicitly decomposed. The result is expressed as a presentation by generators and relations of the commutant in U of the raising operators in sl_{3}. Application is made to the analysis of the representations End(W) for finite‐dimensional simple W.

Highest weight irreducible unitarizable representations of Lie algebras of infinite matrices. The algebra A _{∞}
View Description Hide DescriptionIn this paper all unitarizable irreducible highest weight representations of the infinite‐dimensional Lie algebraA _{∞}, which is a completion and central extension of the general linear Lie algebra gl_{∞}, are considered. Within each representation space a basis is introduced and expressions for the transformations of the basis under the action of the Chevalley generators are written.

A simple derivation of the quantum Clebsch–Gordan coefficients for SU(2)_{ q }
View Description Hide DescriptionTo SU(2)_{ q }, the quantum deformation of SU(2), the van der Waerden method for calculating the Clebsch–Gordan (CG) coefficients is genaralized. The polynomial basis for irreducible representations of SU(2)_{ q }, the relevant polynomial invariants, and the deduction of the q analog of the CG coefficients are given.

Infinite‐dimensional algebras and a trigonometric basis for the classical Lie algebras
View Description Hide DescriptionThis paper explores features of the infinite‐dimensional algebras that have been previously introduced. In particular, it is shown that the classical simple Lie algebras (A _{ N }, B _{ N }, C _{ N }, D _{ N }) may be expressed in an ‘‘egalitarian’’ basis with trigonometric structure constants. The transformation to the standard Cartan–Weyl basis, and the particularly transparent N→∞ limit that this formulation allows is provided.

Nonlinear equations invariant under the Poincaré, similitude, and conformal groups in two‐dimensional space‐time
View Description Hide DescriptionAll realizations of the Lie algebras p(1,1), sim(1,1), and conf(1,1) are classified under the action of the group of local diffeomorphisms of R^{3}. The result is used to obtain all second‐order scalar differential equations, invariant under the corresponding Poincaré, similitude, and conformal groups. The invariant equations are, in general, nonlinear, and the requirement of linearity turns out to be very restrictive. Group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation.

New method for deriving nonlinear integrable systems
View Description Hide DescriptionA new approach is proposed to derive nonlinear integrable systems. It is used to obtain several new nonlinear integrable systems. The above results are relevant to some problems of hydrodynamics,plasma physics, solid‐state physics, etc.

Comment on ‘‘Some results from a Mellin transform expansion for the heat kernel’’ [J. Math. Phys. 3 0, 1226 (1989)]
View Description Hide DescriptionIn this note the inaccuracies contained in the above mentioned paper are corrected and some of the results therein are modified and extended. In particular, formulae are given for directly calculating the coefficients of the heat kernel expansion, not making appeal to recurrence in the heat equation. The nonexistence of a class of anomalies in odd dimension is proved.

Averaging principle and systems of singularly perturbed stochastic differential equations
View Description Hide DescriptionBy developing certain auxiliary results, a modified version of the stochastic averaging principle is developed to investigate dynamical systems consisting of fast and slow phenomena. Moreover, an attempt is made to establish a relationship between the averaging assumption and certain ergodic‐type properties of the random process determined by an auxiliary system of stochastic differential equations. Finally, an example is given to illustrate the scope of the results.

A triangular property of the associated Legendre functions
View Description Hide DescriptionThe property of the associated Legendre functions with non‐negative integer indices, P ^{ m } _{ n }(z), described by the formula: P ^{ m } _{ n } (cos β)=(−1)^{ m }(a/c)^{ n } ∑^{ n−m } _{ k=0} (^{ n+m } _{ k }) (−b/a)^{ n−k } ×P ^{ m } _{ n−k }(cos γ), where a,b,c are the sides of an assigned triangle and α,β,γ the respective opposite angles, is introduced. A useful application of this series in simplifying the calculation of collisional electron–atom cross sections higher than the dipole is mentioned. A proof of the stated identity by use of the Gegenbauer polynomials and of their generating function is given.

Bäcklund transformations, soliton solutions and wave functions of Kaup–Newell and Wadati–Konno–Ichikawa systems
View Description Hide DescriptionUsing the Bargmann–Darboux method, the Bäcklund transformations, n‐soliton solutions and corresponding wave functions of the Kaup–Newell and Wadati–Konno–Ichikawa systems are obtained. These results culminate in an algebraic recursive procedure for the determination of multisoliton solutions and their wave functions of the derivative and mixed derivative nonlinear Schrödinger equations i Q _{ t }+Q _{ x x }∓iα(‖Q‖^{2} Q)_{ x } ±β‖Q‖^{2} Q=0, α>0, β≥0.

A Hamiltonian‐free description of single particle dynamics for hopelessly complex periodic systems
View Description Hide DescriptionA picture of periodic systems that does not rely on the Hamiltonian of the system, but on maps between a finite number of time locations, is developed. Moser or Deprit‐like normalizations are done directly on the maps, thereby avoiding the complex time‐dependent theory. Linear and nonlinear Floquet variables are redefined entirely in terms of maps. This approach relies heavily on the Lie representation of maps introduced by Dragt and Finn [J. Math. Phys. 2 0, 2649 (1979); J. Geophys. Res. 8 1, 13 (1976)]. One might say that although the Hamiltonian is not used in the normalization transformation, Lie operators are used, which are themselves, in some sense, pseudo‐Hamiltonians for the maps they represent. The techniques find application in accelerator dynamics or in any field where the Hamiltonian is periodic, but hopelessly complex, such as magnetic field design in stellarators.

Lorentz action on the space of relative velocities and relativity on a three‐manifold
View Description Hide DescriptionThe space of relative velocities in special relativity has a three‐dimensional hyperbolic structure. This provides not only a geometric interpretation of the Einstein velocity addition law, but also a purely three‐dimensional reformulation of both special and general relativity on a three‐manifold whose tangent bundle is endowed with a hyperbolic distance function on each fiber. Here the basic concepts are that of local physical observers and local time in terms of nonsingular vector fields and their local flows. The hyperbolic structure on the tangent space enables one to define a relative velocity function between two physical observers, as well as space and time measurements and inertial physical observers. It is possible to rederive Lorentz time dilation and gravitational and cosmological redshifts and reformulate Maxwell (or Yang–Mills) equations in a purely three‐dimensional framework instead of the traditional four‐dimensional space‐time approach.

Solutions of an equation that lead to solutions of a modified Emden equation and a coupled Korteweg–deVries equation
View Description Hide DescriptionIn a recent paper, Moreira [Res. Rep. IF/UFRJ/83/25, Universidade Federal do Rio de Janeiro Inst. de Fisica, Cidade Univ., Ilha do Fundao, Rio de Janeiro, Brazil] obtained a nonlinear second‐order differential equation that leads to the first integral of a modified Emden equation. He also obtained two particular solutions of his equation. This paper completely integrates Moreira’s equation and uses it to get a class of solutions of a coupled Korteweg–deVries (KdV) equation, recently studied by Guha Ray, Bagchi, and Sinha [J. Math. Phys. 2 7, 2558 (1986)].

Remarks on a system of coupled nonlinear wave equations
View Description Hide DescriptionIt is shown that a generalized system of coupled KdV‐MKdV equation exhibits solitary wave solutions. It has been shown that an increase in nonlinearity in one variable in a particular fashion does not affect the existence of solitary wave solutions. The periodic traveling wave solutions of the system have also been investigated.