Volume 24, Issue 1, January 1983
Index of content:

Type‐adapted representations of semidirect product groups
View Description Hide DescriptionIrreducible representations of compact groups can be partitioned into three classes (character test +,0,−). This classification is the same for real, complex, and quaternionic representations and in all three cases a peculiar, type‐adapted form of the representation matrices may be chosen (t reps). In this paper it is shown how to construct t reps of semidirect products G s g starting with t reps of G and t reps of some covering groups of subgroups of g. The advantage of using t reps shows up in that the factor system of the little cogroups is real in two of three cases and that real, complex, and quaternionic representations are obtained simultaneously. The method is specialized to direct products and generalized to induction from normal subgroups.

Type‐adapted subduction matrices
View Description Hide DescriptionIf an irreducible representation is restricted to a subgroup it becomes reducible in general. The matrices transforming this reducible representation into a direct sum of irreducible constituents are called subduction matrices. Their structure is discussed for real, complex, and quaternionic representations where all these representations are assumed to show a peculiar structure characteristic for the type of this representation (character test +, 0, −). The choice of these type‐adapted representations, a convention possible for all compact groups, considerably reduces the number of parameters needed to fix a subduction matrix.

A new mathematical function connected with boundary value problems in kinetic transport theory
View Description Hide DescriptionA new mathematical function connected with solving the Maxwell transport equation by applying the bimodal two‐stream relaxing distribution is defined. This new function gives a more correct description of the direct nonequilibrium effect in gas molecular distribution on the macroscopic transferring of moment flux. In this paper, the differential equation satisfied by this function, its recurrence relations, and series or asymptotic expansions in various conditions are formulated. The degrees of approximations for these expansions are discussed.

Null field solutions of the wave equation and certain generalizations
View Description Hide DescriptionThe ordinary wave equation in 3+1 dimensions ⧠φ=0, ⧠≡−∂^{2}/∂t ^{2}+∂^{2}/∂x ^{2}+∂^{2}/∂y ^{2} +∂^{2}/∂z ^{2} admits null field solutions, characterized by ∇φ⋅∇φ=0, ∇φ⋅∇φ≡−(∂φ/∂t)^{2}+(∂φ/∂x)^{2} +(∂φ/∂y)^{2}+(∂φ/∂z)^{2} with ∇φ≠0. It is shown that the general null field solution can be obtained from a knowledge of the ‘‘time‐transported’’ solutions, i.e., those solutions of the form φ=t−ψ(x,y,z), where ψ satisfies both Laplace’s equation and the eikonal equation in a Euclidean space. We obtain all second‐order scalar wave equations of form f(φ,φ_{;i } ^{;i }, φ_{;i; j } φ^{;i; j })=0 (in arbitrary dimension and involving a single potential function φ) for which the above technique applies. These equations are shown to be equivalent to the family of quasilinear third‐order equations ∇φ⋅∇(⧠φ)+K(⧠φ)^{2}=0, where K is a constant. Some null solutions of these equations are considered, and related to previous works. The results are applied to determine all shear‐free hypersurface‐orthogonal null geodesic congruences in Minkowski space–time, and some brief comments are made on complex solutions and on more general wave equations.

Deformation of Lie group representations and linear filters, Part I
View Description Hide Description‘‘Deformation theory’’ is a branch of mathematics which studies the geometry of dependence on p a r a m e t e r s of geometric and physical systems. Material arising from Lie grouptheory and mathematical physics (e.g., in the study of asymptotic behavior of angular momentum) is applied to study the asymptotic behavior of certain linear filters depending on parameters. A mathematical machine which unifies many of these problems will be developed in this series.

The paths of integration for the new generalized Bessel transform
View Description Hide DescriptionIt is shown that the comments made in a recent paper regarding the previously developed ‘‘New generalized Bessel transform and its relationship to the Fourier, Watson, and Kontorowich‐Lebedev transforms’’ are based on erroneous assumptions. The claim that the path of integration parallel to the real axis (below the singularities of the transformfunction on the real axis) cannot be transformed to a contour around the singularities of the transfer function in the lower half‐plane contradicts the very basis of the firmly established Watson transformation and the most advanced theories in radio wave propagation over the Earth’s surface and cylindrical structures.

Thorpe–Hitchin inequality for compact Einstein 4‐manifolds of metric signature (++−−) and the generalized Hirzebruch index formula
View Description Hide DescriptionIt is proved that the Euler characteristic and the Hirzebruch index of a compact oriented Einstein 4‐manifold of metric signature (++−−) satisfy an inequality which is well known as the Thorpe–Hitchin inequality for the case of a Riemannian metric. To derive the inequality, a generalized Hirzebruch formula relating the index to the first pseudo‐Pontrjagin number of the manifold is proved. This formula may be contrasted with Chern’s generalized Gauss–Bonnet formula for a pseudo‐Riemannian manifold.

Critical behavior of the two‐state doubling algorithm
View Description Hide DescriptionWe describe an algorithm which produces K ^{2}×K ^{2} matrix approximations to the low energy part of the Schrödinger operator for Ncoupled oscillators. We carry out the algorithm analytically in the case K=2, for arbitrary N. In particular we show explicitly in this case how the N→∞ limit exhibits critical behavior.

Direct approach to the periodic solutions of the multidimensional sine–Gordon equation
View Description Hide DescriptionA number of identities for multidimensional theta functions and their derivatives are derived. Application to the nonlinear partial differential equations is exemplified for the sine–Gordon equation. In consequence, the multidimensional sine–Gordon equation can be reduced to a functional equation, and then to a set of algebraic equations. Several particular cases are also discussed.

Existence and asymptotic behavior of Padé approximants to the Korteweg–de‐Vries multisoliton solutions
View Description Hide DescriptionThe summation procedure of the Padé type is applied to the perturbation expansion of the solution of the potential Korteweg‐de‐Vries equation (K.d.V.), introduced by Rosales. For the N‐soliton solution without background the [(n−1)/n] Padé approximants are shown to exist for n≤N. Their asymptotic behavior is investigated and it is found that it corresponds to a system of n solitons with the leading velocity parameters. The analogous results for the K.d.V. then follow in agreement with some previous numerical observations.

Manifestly parity invariant electromagnetic theory and twisted tensors
View Description Hide DescriptionWe develop here the calculus of twisted tensors and in particular twisted differential forms, treating them as tensors with complementary orientations. These geometrical objects give us the proper language for electromagnetic theory in a 3‐space plus time representation. The parity properties of the fields are simplified and many graphical illustrations are given.

On the stability problem of a pair of adjoint operators
View Description Hide DescriptionAs an introduction, the eigenvalue problem for a linear operator T having a discrete point spectrum and a complete set of eigenfunctions is studied. The bivariational principle for T and its adjoint operator T ^{°} is derived, and the biorthogonal properties of their eigenfunctions are discussed. The main part of the paper is then concerned with the problem whether these features can be extended also to a general pair of adjoint operators, T and T ^{°}, in which case the eigenvalue problem is replaced by the more general stability problem. The stability problem for a pair of adjoint operators—T and T ^{°}—is first formulated in terms of nonorthogonal projectors—O and O ^{°}—which decompose these operators and satisfy the commutation relations T O=O T and T ^{°} O ^{°}=O ^{°} T ^{°}. In the case of a finite space, these skew‐projectors may be explicitly expressed in product forms derived from the reduced Cayley–Hamilton equation for the operator T. It is shown that, if the stable subspaces defined by these projectors are properly classified by their Segre characteristics, one may explicitly derive the form of the projectors for the irreducible stable subspaces associated with the individual Jordan blocks of the so‐called classical canonical forms of the matrix representations of T and T ^{°}. It is further shown that, in such a case, the biorthonormality property of the generalized eigenfunctions is still valid, and that a bivariational principle may be derived. The extension of these results to infinite spaces is finally briefly discussed.

On the Hartree–Fock scheme for a pair of adjoint operators
View Description Hide DescriptionA generalization of the Hartree–Fock scheme for an arbitrary linear operator—and its adjoint—is derived by using the bivariational principle. It is shown that, if the system operator in the transition value is approximated by two Slater determinants, it is determined by a projector ρ, which corresponds to a generalization of the conventional Fock–Dirac density matrix, but which is no longer self‐adjoint. The effective one‐particle operator then takes the same form as in the conventional theory. The solution of the stability problem for a pair of adjoint effective operators is finally discussed. Numerical applications are performed elsewhere.

When is the Wigner function of multidimensional systems nonnegative?
View Description Hide DescriptionIt is shown that, for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form. This result generalizes the one obtained by Hudson [Rep. Math. Phys. 6, 249 (1974)] for one‐dimensional systems.

A characteristic function approach to the discrete spectrum of electrically charged particles
View Description Hide DescriptionThe purpose of this article is to obtain a scalar function the zeros of which give the discrete energy values for a system of electrically charged particles. The relation between the serial expansion of the characteristic function in powers of the system’s eigenvalue and the Stieltjes series has been revealed not only for the electrically charged particles but also generally for any positive (or negative) definite Hermitian operator which has only a discrete spectrum. The use of Padé approximants to express the characteristic function has offered a rapidly convergent scheme to evaluate the system’s eigenvalues. The first few elements of the Padé Table for the reciprocal of the characteristic function of certain systems have been given to verify the presented idea numerically. The determination of these elements needs the values of certain complicated integrals which we name ‘‘Zeroth Order Hyperspherical Spectral Coefficients’’ [HSC(φ_{0})]. The first two of these coefficients are investigated and their evaluation is realized analytically.

Perturbation expansion of S matrix for background scattering
View Description Hide DescriptionThe S matrix near a pole is parametrized into the contribution from the resonance and the background scattering. We develop a perturbation theory for the background scattering, based on the Jost function formalism. A closed expression is found up to the second order in the coupling strength between the channels. A brief comparison with the other formalism is also made and the advantages of the present theory are shown.

Inverse scattering with coinciding‐pole reflection coefficients
View Description Hide DescriptionIn inverse scatteringtheory, algorithms for solving the Gel’fand–Levitan equation normally break down when poles in the reflection coefficient coincide. Here we present a method for treating an arbitrary number of coinciding poles. We give the first explicit solutions for 3, 4, 5, 6, 8, and 10 poles.

A constructive approach to bundles of geometric objects on a differentiable manifold
View Description Hide DescriptionA constructive approach to bundles of geometric objects of finite rank on a differentiable manifold is proposed, whereby the standard techniques of fiber bundle theory are extensively used. Both the point of view of transition functions (here directly constructed from the jets of local diffeomorphisms of the basis manifold) and that of principal fiber bundles are developed in detail. These, together with the absence of any reference to the current functorial approach, provide a natural clue from the point of view of physical applications. Several examples are discussed. In the last section the functorial approach is also presented in a constructive way, and the Lie derivative of a field of geometric objects is defined.

Exact solutions of strong gravity in generalized metrics
View Description Hide DescriptionWe consider classical solutions for the strong gravity theory of Salam and Strathdee in a class of metrics with positive, zero, and negative curvature. It turns out that such solutions exist and their relevance for quark confinement is explored. Only metrics with positive curvature (spherical symmetry) give a confining potential in a simple picture of the scalar hadron. This supports the idea of describing the hadron as a closed microuniverse of the strong metric.

Gödel‐type universe with a perfect fluid and a scalar field
View Description Hide DescriptionThe paper contains, along with a brief review of solutions of general relativistic fieldequations when the metric is of a particular cylindrically symmetric stationary form, a new solution of the same general category when the energy‐momentum tensor is due to a perfect fluid plus a scalar field. It turns out that under these constraints, the space‐time is completely homogeneous and contains closed timelike lines. There is, however, a nonuniqueness in the interpretation as one can introduce a Maxwellianelectromagnetic field of arbitrary strength along with the perfect fluid and the scalar field.