Volume 25, Issue 2, February 1984
Index of content:

A unified theory of the point groups. V. The general projective corepresentations of the magnetic point groups and their applications to magnetic space groups
View Description Hide DescriptionThis paper presents the general expressions of the projective irreducible corepresentations of the 20 homologous sets of the magnetic double point groups from which follow all those of the remaining groups of finite order through isomorphisms (the icosahedral group is excluded). These are explicitly given in terms of the irreducible representations of the unitary double point groups. Their special cases provide all the irreducible corepresentations of all magnetic space groups of wave vector through simple gauge transformations. A method of determining the gauge transformations is discussed through typical examples.

General irreducibility condition for vector and projective corepresentations of antiunitary groups
View Description Hide DescriptionA general condition for irreducibility of vector and projective corepresentations of an antiunitary group is presented. It depends only on the characters of the unitary halving subgroup of the covering group. It reduces to the well known type criterion of corepresentations when it is specialized to the three types of corepresentations.

Semiregular induction of group representations
View Description Hide DescriptionThe method of inducing an irreducible representation of a group from that of a subgroup is extended. This generalized induction process is illustrated to occur in applications and to account for some occurrences of ‘‘intermediate’’ or ‘‘hidden’’ symmetry. Some general results are proved, including a reciprocity theorem relating general induction and subduction processes.

Identities satisfied by the generators of the Dirac algebra
View Description Hide DescriptionThe geometry of real four‐dimensional spinor space and its symmetry groups are reviewed from the perspective of S̄Ō(̄3̄,̄3̄)̄. Two identities that concern the matrix generators of S̄Ō(̄3̄,̄3̄)̄, and which were first proved by Dirac, are generalized.

Infinitesimal symmetry transformations of some one‐dimensional linear systems
View Description Hide DescriptionThe converse problem of similarity analysis is solved in general for the infinitesimal symmetry transformations of any given inhomogeneous ordinary differential equation of the second order ẍ+f _{2}(t)ẋ+f _{1}(t)x=f _{0}(t). The completely general associated Lie algebra is obtained for equations of this kind, which structure constants depend only on the chosen set of initial values f _{0}(0), f _{1}(0), f _{2}(0), and ḟ_{2}(0). The infinitesimal elements of the dynamical group of a Newtonian one‐dimensional linear system are also discussed, and some miscellaneous examples are given.

General indices of representations and Casimir invariants
View Description Hide DescriptionA modified definition of the index of degree p of a finite‐dimensional representation of a simple Lie algebra is given. The definition applies equally to even and odd p. The correspondence between the earlier definitions of indices (even p) and anomaly numbers (odd p) is pointed out as well as the relation to Casimir invariants of the algebra. A closed formula for the fifth‐order index is derived.

Variational bounds on the temperature distribution
View Description Hide DescriptionUpper and lower stationary or variational bounds are obtained for functions which satisfy parabolic linear differential equations. (The error in the bound, that is, the difference between the bound on the function and the function itself, is of second order in the error in the input function, and the error is of known sign.) The method is applicable to a range of functions associated with equalization processes, including heat conduction,mass diffusion, electric conduction, fluid friction, the slowing down of neutrons, and certain limiting forms of the random walk problem, under conditions which are not unduly restrictive: in heat conduction, for example, we do not allow the thermal coefficients or the boundary conditions to depend upon the temperature, but the thermal coefficients can be functions of space and time and the geometry is unrestricted. The variational bounds follow from a maximum principle obeyed by the solutions of these equations.

On Lie–Bäcklund vector fields of the evolution equations ∂^{2} u/∂x ∂t=f(u) and ∂u/∂t=∂^{2} u/∂x ^{2}+f(u)
View Description Hide DescriptionFor the evolution equations ∂^{2} u/∂x ∂t=f(u) and ∂u/∂t=∂^{2} u/∂x ^{2}+f(u) we derive the analytic functions f where Lie–Bäcklund vector fields are admitted.

Singular solutions of the axially symmetric Bogomolny equations
View Description Hide DescriptionA singular Bäcklund transformation is constructed for the Ernst equation and used to construct singular solutions to the axially symmetric Bogomolny equations.

SU(3) symmetry of the equations of unidimensional gas flow, with arbitrary entropy distribution
View Description Hide DescriptionWe have shown in an earlier work that, assuming a particular class of equations of state, the Euler equations of one‐dimensional gas flow are invariant under an SU(3) group of transformations, and in fact admit of a Lie group of symmetry of infinite order; they, therefore, possess an infinite number of conservation laws. We show in the present work that the SU(3) symmetrical formalism still brings about tremendous simplification and analytical order in the most general case where the equation of state is arbitrary. The six characteristicequations assume a vector form and relate two conjugate, three‐dimensional vectors U and X. The SU(3) symmetry is only broken to a minor extent through the occurrence of a multiplicative factor Γ in the equations. The conservation laws take the form of the Cauchy integrability condition for the elements of a traceless second rank tensor ε_{ i j } and, taken all together, form an SU(3) octet; in the most general case, however, there exist four conservation laws only (five if the gas is monatomic) as a result of the breaking of symmetry. Application of these results to the theory of self‐similar flow is also discussed. Finally, we show the invariance of the equations of monatomic gas flow under Lorentz transformations in a three‐dimensional Minkowski space; that raises the question of whether a geometrical relation may exist between the Minkowski light cones and characteristics.

The continuous Heisenberg chain and constrained harmonic motion
View Description Hide DescriptionIt is shown that the equation describing the evolution of the classical continuous Heisenberg ferromagnet can be regarded as one aspect of quadratically constrained harmonic motion, as it is also the case in a number of other integrable systems. Two ingredients are used: a method to solve the inverse scattering problem for second‐order operators using ordinary differential (rather than integral) equations and the equivalence of the Heisenberg chain with the nonlinear Schrödinger equation.

An exact recursion for the composite nearest‐neighbor degeneracy for a 2×N lattice space
View Description Hide DescriptionA set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest‐neighbor degeneracy for simple, indistinguishable particles distributed on a 2×N lattice space. The associated generating functions, as well as the expectation of the resulting statistics are also treated.

On the construction of state spaces for classical dynamical systems with a time‐dependent Hamilton function
View Description Hide DescriptionThe damped linear and the forced harmonic oscillator are used as standard examples for a dynamical system with a time‐dependent Hamilton function to investigate the problem of constructing a Hilbert state space and evolution operators in this space.

Symplectic approach to nonconservative mechanics
View Description Hide DescriptionThe dynamics of autonomous nonconservative systems is studied in terms of Lagrangian submanifolds of a special symplectic manifold. Both the Hamiltonian and Lagrangian description are taken into consideration and the transition between the two descriptions is established by means of the generating function of a symplectic relation.

The forced Toda lattice: An example of an almost integrable system
View Description Hide DescriptionA method for solving forced integrable systems is presented. The method requires the knowledge of at least one piece of information about the solution. Once this is known, one may then construct the remainder of the solution. In this sense these systems are ‘‘almost integrable.’’ The forced semi‐infinite Toda lattice is used as an example and to illustrate the method.

The soliton birth rate in the forced Toda lattice
View Description Hide DescriptionThe soliton birth rate in the semi‐infinite Toda lattice is studied. The lattice is forced by driving the zeroth particle with a constant velocity into the remainder of the lattice. An approximate solution for the soliton birth rate is derived and it is shown to compare quite favorably with the actual birth rate.

Derivation and application of extended parabolic wave theories. I. The factorized Helmholtz equation
View Description Hide DescriptionThe reduced scalar Helmholtz equation for a transversely inhomogeneous half‐space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial‐value plane defines a direct acoustic propagationmodel. This elliptic formulation admits a factorization and is subsequently equivalent to a first‐order Weyl pseudodifferential equation which is recognized as an extended parabolic propagation model. Perturbation treatments of the appropriate Weyl composition equation result in a systematic development of approximate wavetheories which extend the narrow‐angle, weak‐inhomogeneity, and weak‐gradient ordinary parabolic (Schrödinger) approximation. The analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies. The wavetheories foreshadow computational algorithms, the inclusion of range‐dependent effects, and the extension to (1) the vector formulation appropriate for elastic media and (2) the bilinear formulation appropriate for acoustic field coherence.

Derivation and application of extended parabolic wave theories. II. Path integral representations
View Description Hide DescriptionThe n‐dimensional reduced scalar Helmholtz equation for a transversely inhomogeneous medium is naturally related to parabolic propagation models through (1) the n‐dimensional extended parabolic (Weyl pseudodifferential) equation and (2) an imbedding in an (n+1)‐dimensional parabolic (Schrödinger) equation. The first relationship provides the basis for the parabolic‐based Hamiltonian phase space path integral representation of the half‐space propagator. The second relationship provides the basis for the elliptic‐based path integral representations associated with Feynman and Fradkin, Feynman and Garrod, and Feynman and DeWitt‐Morette. Exact and approximate path integral constructions are derived for the homogeneous and transversely inhomogeneous cases corresponding to both narrow‐ and wide‐angle extended parabolic wave theories. The path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework for dynamical approximations, resolution of the square root operator, and the concept of an underlying stochastic process.

The Doppler effect: Now you see it, now you don’t
View Description Hide DescriptionTwo main classes of problems are identified in the theory of electromagneticscattering in velocity‐dependent systems. The first involves transformation of space and time coordinates and field components from the laboratory system of reference to the comoving system of the scatterer, solution of the scattering problem, and inverse transformations. In general, this method displays the Doppler frequency shifts. The second class involves the substitution of Minkowski’s constitutive relations into Maxwell’sequations for harmonic time variation, heuristically stipulating the absence of Doppler frequency shifts. The interrelation between the two methods is investigated here. It is argued that the second method is a limiting case for very low, as well as very high frequencies, and provided the mean square fluctuation of the dielectric constant is small, and the geometrical boundaries defining the scatterers are fixed. Canonical problems of plane, cylindrical, and spherical stratification are discussed and analytical results for the scattered fields are derived. If the parameters of the problem do not meet the above conditions, the first method should be used, giving rise, in general, to a whole spectrum of frequencies due to the Doppler effect.

The J‐matrix reproducing kernel: Numerical weights at the Harris energy eigenvalues
View Description Hide DescriptionThe restriction of the J‐matrix scattering wave function to the subspace where the potential is nonzero is used to define a reproducing kernel in the energy parameters. The values of the kernel at the positive Harris energy eigenvalues are shown to be related to the numerical weights at these eigenvalues.