Volume 25, Issue 7, July 1984
Index of content:

A unified theory of point groups. VI. The projective corepresentations of the magnetic point groups of infinite order
View Description Hide DescriptionThis paper provides all the p‐inequivalent projective irreducible unitary corepresentations of all the magnetic point groups of infinite order with full use of their isomorphisms.

Signatures of finite SU( p,q) representations
View Description Hide DescriptionThe signature S of a finite‐dimensional representation of SU( p,q) is the difference between the number of positive and negative signs in the bilinear invariant in its diagonal form. An expression for S is derived starting from the Weyl character formula for U( p,q) representations.

The representation matrix elements of the group O^{+}(2,2)
View Description Hide DescriptionMaking use of the isomorphism of O^{+}(2,2) with the direct product O^{+}(2,1)×O^{+}(2,1), the matrix elements of O^{+}(2,2) in its unitary irreducible representations are explicitly calculated in terms of Euler angles introduced in a previous paper. The expressions so obtained consist of infinite sums of product of Clebsch–Gordan coefficients and Bargmann’s v functions, both for the group O^{+}(2,1).

Some geometrical consequences of physical symmetries
View Description Hide DescriptionInvariant submanifolds of the linear representation space C^{4m } of the physical symmetry group SU(2,2)×SU(m) and its subgroup P×SU(m) are studied in some detail. It is shown that there exists only one such manifold admitting unique projection onto Minkowski space. The structure of this manifold is investigated by using proper local coordinate systems.

Proof of an algorithm for the evaluation of the branching multiplicity SO(2n)→SO(2n−2)⊗U(1)
View Description Hide DescriptionThe proof of an algorithm, previously proposed by us, for the evaluation of the branching multiplicity SO(2n)→SO(2n−2)⊗U(1) is given. This proof is based on explicit construction of lowering shift operators for the class D _{ n } of Cartan.

Fixed symmetry and fixed class generating functions
View Description Hide DescriptionA fixed symmetry (or fixed plethysm) generating function enumerates all representations R _{ b } of a compact Lie groupG contained in that part of the direct product of p copies of any irreducible representation R _{ a } of G that has a particular exchange symmetry under the symmetric group S _{ p }. Fixed symmetry generating functions are conveniently given as linear combinations of the simpler fixed class generating functions. We give a systematic procedure for their construction and some examples for SU(2), SU(3), and SO(5). For SU(3) the examples include plethysms of up to three boxes; for SO(5) we treat two‐box plethysms in general and give the scalar content of three‐box plethysms; the SU(2) examples include up to six boxes.

Comments on superposition rules for nonlinear coupled first‐order differential equations
View Description Hide DescriptionSome comments are made on the classification of finite‐dimensional subalgebras of the Lie algebra of vector fields in n variables and of the related nonlinear ordinary differential equations with superposition principles. In particular for n=2 a very natural requirement of indecomposability implies that only two types of equations need be considered.

The summation of Bessel products
View Description Hide DescriptionWe produce in what follows the closed forms for the summation of certain products of Bessel functions pertinent to a number of distinct fields of research, such as the theory of plasma waves, charged particle beaminteraction with plasma, and density wavetheory in galactic dynamics.

Poisson reduction and quantization for the n+1 photon
View Description Hide DescriptionFor a dynamical system in which the constraints are given by the vanishing of a singular momentum map J, reduction in the usual group‐theoretic sense may not be possible. Nonetheless, one may still ‘‘reduce’’ J ^{−} ^{1}(0), at least on the level of Poisson algebras. An example of such a singular constrained system is the ‘‘n+1 photon,’’ that is, a massless, spinless particle in (n+1)‐dimensional Minkowski space‐time. We apply the generalized reduction procedure to the n+1 photon, explicitly constructing the Poisson algebra of gauge invariant observables. This technique also enables us to completely analyze the effects of the singularities in J ^{−} ^{1}(0) on the system. We then quantize, obtaining results which are in agreement with a quantization of the extended phase space and the subsequent imposition of the constraint.

Estimation of inverse temperature and other Lagrange multipliers: The dual distribution
View Description Hide DescriptionIt is shown that the problem of parameter estimation for distributions of the exponential type, has a unique consistent Bayesian solution: The requirement that Bayes’ rule and maximum entropy lead to the same inverse distribution determines the loss function. Similarly, the demand that the best estimate for a random variable, given an observed value of that variable, coincides with the observed value, determines the prior distribution for the corresponding conjugate parameter. Properties of the dual distribution thus determined are investigated. In particular, the symmetrical role of parameter and constraint as a pair of conjugate variables is shown to imply an inherent uncertainty principle. Possible applications to temperature fluctuations and to an imbedding of classical mechanics in a statistical background are indicated.

Decoupling of a system of partial difference equations with constant coefficients and application
View Description Hide DescriptionConsider D multi‐variable functions, A _{ j }(n), j=1 to D, where n stands for the evaluation point in the associated multi‐dimensional space of coordinates (n _{1},n _{2},...). Let us assume that the A _{ j }’s satisfy a system of D linearly coupled finite difference equations: the value of each function A _{ i } at the evaluation point n is given as a linear combination of the values of this function and others at shifted evaluation points. By introducing D suitable generating functions, G _{ j }, j=1 to D, one is able to replace the D coupled difference equations by a system of Dlinear equations where the G _{ j }’s play the role of the D unknowns. After solving this new system of equations, it is then possible to construct a difference equation for each of the A _{ j }’s relating the value of A _{ i } at the evaluation point n to the values of A _{ i } itself at shifted arguments. The solution of such a decoupled equation can then be handled using the multi‐dimensional combinatorics function technique.

Poisson branching point processes
View Description Hide DescriptionWe investigate the statistical properties of a special branching point process. The initial process is assumed to be a homogeneous Poisson point process (HPP). The initiating events at each branching stage are carried forward to the following stage. In addition, each initiating event independently contributes a nonstationary Poisson point process (whose rate is a specified function) located at that point. The additional contributions from all points of a given stage constitute a doubly stochastic Poisson point process (DSPP) whose rate is a filtered version of the initiating point process at that stage. The process studied is a generalization of a Poissonbranching process in which random time delays are permitted in the generation of events. Particular attention is given to the limit in which the number of branching stages is infinite while the average number of added events per event of the previous stage is infinitesimal. In the special case when the branching is instantaneous this limit of continuous branching corresponds to the well‐known Yule–Furry process with an initial Poisson population. The Poisson branching point process provides a useful description for many problems in various scientific disciplines, such as the behavior of electron multipliers, neutronchain reactions, and cosmic ray showers.

Orthogonal polynomials with exponential weight in a finite interval and application to the optical model
View Description Hide DescriptionA quadrature procedure is developed which makes the construction of momentum‐space meson‐nucleus optical potentials more accurate. We deal with numerical evaluation of integrals with finite t‐integration range which contain exp(D t) explicitly, where D is a parameter. The Gaussian rule is used with abscissas determined as roots of orthogonal polynomials with exponential weight function in the interval [−1,1]. Recurrence relations and inequalities for these polynomials are obtained. A nonlinear recursion is derived, which permits the evaluation of abscissas and weights without accumulation of roundoff error. The nonlinear recursion is solved by means of an iterative procedure, the convergence properties of which are established. The quadrature procedure is summarized as an easily implementable algorithm. The rate of convergence is demonstrated for several test integrals.

Two‐dimensional time‐dependent Hamiltonian systems with an exact invariant
View Description Hide DescriptionWe present a direct approach to investigate the existence of an exact invariant for two‐dimensional Hamiltonians, in which the potential depends explicitly on time. The method is based on an expansion of the invariant in the velocities. The problem is solved completely for invariants linear and quadratic in the momenta. Our results contain as a particular case the results of Lewis and Leach on one‐dimensional systems.

Integrals of motion for Toda systems with unequal masses
View Description Hide DescriptionWe present new integrals of motion for the Toda lattice (chain of particles in one dimension with exponential interaction) for two special cases of boundary conditions: the free‐end lattice with three non‐equal‐mass particles and the fixed‐end lattice for two particles. In both cases, we use two distinct approaches in order to identify the integrable cases: direct search of the integral of motion and group theoretical methods. Our results are in agreement with the predictions of Painlevé analysis.

The Liouville–Bäcklund transformation for the two‐dimensional SU(N) Toda lattice
View Description Hide DescriptionWe describe the Liouville–Bäcklund transformation for the two‐dimensional SU(N) Toda lattice with free end points. Integration of this transformation gives us the general solution of this equation, which depends on the N arbitrary solutions of the two‐dimensional Laplace equation.

Dynamical invariants for two‐dimensional time‐dependent classical systems
View Description Hide DescriptionGeneral equations are formulated to determine all potentials for two‐dimensional systems of the type L= (1)/(2) ( p ^{2} _{1} +p ^{2} _{2}) −V(q _{1},q _{2},t), which admits invariants of the form I=a _{0}+a _{ i }ξ_{ i } + (1)/(2) a _{ i j }ξ_{ i }ξ_{ j }, i, j=1,2, where ξ_{1} =ż=q̇_{1}+i q̇_{2}, ξ_{2}=z̄=q̇_{1}−i q̇_{2}, a _{0}, a _{ i }, a _{ i j } are arbitrary functions of t, z=q _{1}+i q _{2}, and z̄=q _{1}−i q _{2}. Simplifying restrictions reduce the general equation to a tractable form. The resulting equations are solved for a special class of time‐separable potentials and derive (i) the van der Waals‐type long‐range potential, V(r,t)=β(t)(b/r ^{4}+d) and (ii) the quark‐confining logarithmic potential, V(r,t)=β(t)λ(ln r+b _{1}/r ^{4}+d _{1}). Invariants I for the resulting dynamical systems are found. Some observations on the present method in the context of Katzin and Levine and of Lewis and Leach analyses have also been made.

General prolongations and (x, t)‐depending pseudopotentials for the KdV equation
View Description Hide DescriptionGiven an exterior differential system on a manifoldM, we study general prolongations of the system on a locally trivial fiber bundle (M̃, π̃, M) by a Cartan–Ehresmann connection. We characterize such prolongations for the system associated with the KdV equation without any assumption of ‘‘(x, t) independence.’’ The partial Lie algebra discovered by Wahlquist–Estabrook [J. Math. Phys. 1 6, 1 (1975)] appears by this way as an intrinsic tool. Simple analytic pseudopotentials are classified up to diffeomorphism.

The sine‐Gordon equations: Complete and partial integrability
View Description Hide DescriptionThe sine–Gordon equation in one space‐one time dimension is known to possess the Painlevé property and to be completely integrable. It is shown how the method of ‘‘singular manifold’’ analysis obtains the Bäcklund transform and the Lax pair for this equation. A connection with the sequence of higher‐order KdV equations is found. The ‘‘modified’’ sine–Gordon equations are defined in terms of the singular manifold. These equations are shown to be identically Painlevé. Also, certain ‘‘rational’’ solutions are constructed iteratively. The double sine–Gordon equation is shown not to possess the Painlevé property. However, if the singular manifold defines an ‘‘affine minimal surface,’’ then the equation has integrable solutions. This restriction is termed ‘‘partial integrability.’’ The sine–Gordon equation in (N+1) variables (N space, 1 time) where N is greater than one is shown not to possess the Painlevé property. The condition of partial integrability requires the singular manifold to be an ‘‘Einstein space with null scalar curvature.’’ The known integrable solutions satisfy this constraint in a trivial manner. Finally, the coupled KdV, or Hirota–Satsuma, equations possess the Painlevé property. The associated ‘‘modified’’ equations are derived and from these the Lax pair is found.

An exact nonghost solution for a plane‐symmetric cosmology containing a classical spinor field
View Description Hide DescriptionAn exact solution (up to quadratures) of the Einstein–Dirac system is presented for cosmological models that depend only on one temporal and one space coordinate. Four solutions to the Dirac equation, all with zero helicity, are given.