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Volume 26, Issue 9, September 1985

A generalization and refinement of Madivanane’s theorem on general involutional transformations
View Description Hide DescriptionA result of Madivanane on ‘‘quasianticommuting’’ complex square matrices, which are roots of the identity matrix, is generalized to linear operators on infinite‐dimensional complex vector spaces. A constant which appears in the result is shown to be nonzero by giving a simple formula for its absolute value. Also, a simple formula for the constant itself is given, which is valid for some cases.

Nontrivial zeros of weight 1 3j and 6j coefficients: Relation to Diophantine equations of equal sums of like powers
View Description Hide DescriptionThe nontrivial zeros of weight 1 3j and 6j coefficients given previously are shown to be the set of all such zeros. The relation of these zeros to the solutions of well‐known Diophantine equations is also discussed.

Locally operating realizations of transformation Lie groups
View Description Hide DescriptionUsing the Mackey theory of induced representations, a systematic study of the locally operating multiplier realizations of a connected Lie groupG that acts transitively on a space‐time manifold is presented. We obtain a mathematical characterization of the locally operating multiplier realizations and a reduction of the problem of multiplier locally operating realizations to linear ones via a splitting group G‘;m for G. In this way the locally operating multiplier realizations are obtained by induction from finite‐dimensional linear representations of a well‐determined subgroup of Ḡ. Some examples, such as the two‐dimensional Euclidean group, the Galilei group, and the one‐dimensional Newton–Hooke group, are given.

Boson realization of sp(4). I. The matrix formulation
View Description Hide DescriptionHolstein and Primakoff derived long ago the boson realization of a su(2) Lie algebra for an arbitrary irreducible representation (irrep) of the SU(2) group. The corresponding result for su(1,1)≅sp(2) is also well known. This raises the question of whether it is possible to obtain in an explicit, analytic, and closed form, and for any integer d, the boson realization of a sp(2d) Lie algebra for an arbitrary irrep of the Sp(2d) group, which is a problem of considerable physical interest. The case d=2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp(4). The Dyson realization is well known, and the passage to bosons satisfying the appropriate Hermiticity conditions can be done by a similarity transformation through an operator K. What we want, though, is an explicit boson realization for sp(2d) similar to the one that exists for sp(2). In Sec. VI we show how we can get it for sp(4) if the operator K is known. Unfortunately while the matrix form of K ^{2} can be explicitly derived from definite recursion relations, the same cannot be said of K as it involves, in general, the solution of algebraic equations of high degree. Thus the conclusion, corroborated also by a classical analysis where K does not appear, is that an explicit, analytic, and closed boson realization of sp(4), and thus also of sp(2d), is only possible for particular irreps of the corresponding groups.

A set of commuting missing label operators for SO(5)⊃SO(3)
View Description Hide DescriptionThe two‐missing label problem for basis vectors of an SO(5) irreducible representation reduced according to the principal SO(3) subalgebra is considered. A pair of commuting Hermitian operators which are scalars with respect to the SO(3) subalgebra are explicitly constructed within the SO(5) enveloping algebra. One label generating operator is of fourth order and the other of sixth order in the SO(5) basis elements.

Branching index sum rules for simple Lie algebras
View Description Hide DescriptionLet L and L _{0} be a simple Lie algebra and its sub‐Lie algebra, respectively. Then, a given irreducible representation ω of L decomposes into a direct sum of irreducible components of L _{0}, which is called the branching rule. The general Dynkin indices introduced earlier satisfy many sum rules for the branching rule. These are found to be strong enough to uniquely determine the branching rule for many cases we have studied. The sum rules are especially useful for cases of exceptional Lie algebras.

Evaluation of the self‐energy of a droplet interacting via a Yukawa force
View Description Hide DescriptionThe binding energy of a homogeneously charged, classical droplet is considered, whose charge elements interact via a Yukawa force. The six‐dimensional self‐energy integral is reduced to a three‐dimensional integral for axially symmetric, but otherwise arbitrarily shaped, droplets. This integral is brought into a form particularly convenient for numerical calculations. Two integrals involving products of Bessel functions are evaluated, which are either not listed in standard tables of integrals or given in an erroneous form.

On the phase retrieval problem in two dimensions
View Description Hide DescriptionThe paper contains a discussion of the phase retrieval problem in two dimensions and proposes criteria to select those resolutions of the discrete ambiguity of the zero trajectories which are compatible with the analyticity in two variables of the scattered field.

Some nodal theorems for noncentral forces
View Description Hide DescriptionTwo lemmas are proved for local noncentral forces in multidimensional space. First, the lowest partial wave for the ground state is nodeless. Second, the lowest partial wave for the first excited state has at least one node. Ballot–Fabre de la Ripelle perturbation theory is also used to show that higher partial waves for the ground state have nodes near the positions of nodes (if any) in the corresponding element of the matrix element of the noncentral potential.

Exact symmetries of unidimensional self‐similar flow
View Description Hide DescriptionIn addition to the symmetries that are known to apply to arbitrary flow, the self‐similar equations may present other symmetries of their own. We present here such a symmetry of the self‐similar unidimensional flow of an adiabatic inviscid fluid, with arbitrary polytropic index and arbitrary power‐law entropy distribution. The new symmetry can be extended to the non‐self‐similar case if the flow is assumed isentropic. A connection with the theory of Riemann invariants is also discussed.

Generalized Stäckel matrices
View Description Hide DescriptionStäckel and differential‐Stäckel matrices are generalized so that the matrix elements may be functions of the derivatives of the dependent variable as well as the independent variable. The inverses of these matrices are characterized and it is shown that for significant classes of linear and nonlinear partial differential equations, variable separation is accomplished via this generalized Stäckel mechanism.

Modified equations, rational solutions, and the Painlevé property for the Kadomtsev–Petviashvili and Hirota–Satsuma equations
View Description Hide DescriptionWe propose a method for finding the Lax pairs and rational solutions of integrable partial differential equations. That is, when an equation possesses the Painlevé property, a Bäcklund transformation is defined in terms of an expansion about the singular manifold. This Bäcklund transformation obtains (1) a type of modified equation that is formulated in terms of Schwarzian derivatives and (2) a Miura transformation from the modified to the original equation. By linearizing the (Ricati‐type) Miura transformation the Lax pair is found. On the other hand, consideration of the (distinct) Bäcklund transformations of the modified equations provides a method for the iterative construction of rational solutions. This also obtains the Lax pairs for the modified equations. In this paper we apply this method to the Kadomtsev–Petviashvili equation and the Hirota–Satsuma equations.

Geometrical interpretation of the solutions of the sine–Gordon equation
View Description Hide DescriptionThe sine–Gordon equation is known to possess solutions that correspond to solitons, that is, localized entities that maintain their shape after collisions, and have certain properties characteristic of elementary particles. Although the algebraic structure of these solutions is well known, their geometric interpretation as surfaces of constant negative curvature has not been previously illuminated. We discuss these surfaces herein. Curves drawn on these surfaces along the asymptotic directions at each point simulate solutions of the nonlinear waveequation φ_{ x x }−φ_{ t t }=sin φ.

Harmonic analysis of the Euclidean group in three‐space. II
View Description Hide DescriptionWe develop the harmonic analysis for spinor functions which are defined by the matrix elements of the unitary irreducible representations of E(3) with the representation space on the translation subgroup.

Dynamic systems driven by Markov processes
View Description Hide DescriptionConsider a differential equationẎ=V ( X(t)) Y(t), where X(t) is a random function. Sufficient conditions for asymptotic stability of the solution in terms of a generator of the stochastic processX(t) are given. The results are illustrated by several examples.

Noncoordinated basis and Schild’s solution
View Description Hide DescriptionIn this paper we show the paradoxical consequences which appear in Schild’s solution, as seen from a noninertial frame (NIF). We propose a treatment for the noninertial frame which resolves the ambiguities.

Splitting methods for time‐independent wave propagation in random media
View Description Hide DescriptionTime‐independent wave propagation is treated in media where the index of refraction contains a random component, but its mean is invariant with respect to translation in some direction distinguishing the wave propagation. Abstract splitting operators are used to decompose the wave field into forward and backward traveling components satisfying a coupled pair of equations. Mode‐coupled equations follow directly from these after implementing a specific representation for the abstract splitting operators. Here we indicate a formal solution to these equations, concentrating on the diffusion regime, where we estimate the forward‐ and backscattering contributions to the mode specific diffusion coefficients. We consider, in detail, random media with uniform (random atmosphere) and square law (stochastic lense) mean refractive indices.

Singularities of the continuation of fields and validity of Rayleigh’s hypothesis
View Description Hide DescriptionTo formulate general results concerning the validity of the Rayleigh hypothesis, we first introduce a definition of the foci and antifoci of an analytic curve. Then, we state two lemmas on the properties of an analytic or harmonic function satisfying given conditions on an analytic curve. This allows us to predict the behavior of the analytic continuation of the field in electrostatics. The use of a conformal mapping permits the generalization of this method in electromagnetics and acoustics. As a consequence, we are able to predict the limit of validity of the Rayleigh hypothesis.

Comments on the perturbed sine–Gordon equation
View Description Hide DescriptionWe examine the sine–Gordon equation with a perturbation λΔV. We derive necessary conditions on ΔV such that the perturbed equation has solutions with finite energy, analytic in λ, and which reduce to the static soliton when the perturbation is removed (λ→0). Several examples illustrating these conditions are presented.

Wavelength‐dependent electromagnetic parameters for coherent propagation in correlated distributions of small‐spaced scatterers
View Description Hide DescriptionEarlier results for coherent propagation of electromagnetic waves in pair‐correlated random distributions of scatterers (of radius a and physical parameters ε’,μ’) with minimum separation of centers b≥2a small compared to wavelength (2π/k) are generalized to obtain polarization, refraction, and absorption terms to order k ^{2}. The development includes multiple scattering and multipole coupling by electric and magnetic dipoles, as well as quadrupoles to appropriate order. The correlation aspects are determined by simple integrals of the statistical mechanics radial distribution functionf for impenetrable particles (spheres, cylinders, and slabs) of diameter b. For slab scatterers, in terms of the exact Zernike–Prins f, the correlation integrals are expressed as algebraic functions of the volume fraction w; the resultant bulk values reduce to those of one particle at full packing, w=1. Similar results are obtained for spheres in terms of the Wertheim–Thiel solution of the Percus–Yevick approximation of f at the unrealizable bound w=1.