Volume 24, Issue 6, June 1983
Index of content:

Normal forms of elements of classical real and complex Lie and Jordan algebras
View Description Hide DescriptionElements of the classical complex and real Lie and Jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical Lie group. Normal forms of representatives of each conjugacy class are chosen so as to resemble the Jordan normal forms of n×n complex matrices. For completeness similar results are given for g l(n,C), g l(n,R), and g l(n,H).

Versal deformations of elements of classical Jordan algebras
View Description Hide DescriptionEvery involution α defining, by α(X)=−X, the elements of one of the classical Lie algebraso(n, C) and s p(2n, C) over C, and o(p,q), s p(2n,R), u(p,q),o*(2n), and s p(2p,2q) over R, also defines α‐symmetric matrices Y=α(Y) spanning the classical Jordan algebras. The paper contains an explicit description of all perturbations of α‐symmetric matrices, considered up to equivalence under the action of the corresponding classical Lie group. Such exhaustive perturbations are called versal deformations.

The fundamental invariants of inhomogeneous classical groups
View Description Hide DescriptionThe Casimir operators of the following groups are explicitly constructed :R^{ n }⧠Gl(n,R), R^{ n }⧠Sl(n,R), R^{ n }⧠O(p,q),C^{ n }⧠U(p,q),R^{2n }⧠Sp(n,R), C^{ n }⧠Gl(n,C),C^{ n }⧠Sl(n,C), C^{ n }⧠O(n,C), C^{2n }⧠Sp(n,C), H^{ n }⧠U*(2n), H^{ n }⧠SU*(2n), H^{ n }⧠O*(2n), H^{ n }⧠Sp( p,q), and R⧠(R^{2n }⧠Sp(n,R)). The method is based on a particular fiber bundle structure of the generic orbits generated by the co‐adjoint representation of a semidirect product.

Representation matrices for U(4)
View Description Hide DescriptionWe propose an algorithm for the numerical calculation of matrix elements of general U(4) group elements, applicable to large totally symmetric representations of U(4). A possible generalization to the U(6) case is pointed out.

Sufficient conditions for weak minimum of a functional depending on n functions and their derivatives
View Description Hide DescriptionThe sufficient condition for positive definiteness of a quadratic form involving unsymmetric matrix operators is given in terms of a global criterion. This determines the sufficient condition for the weak minimum of a functional depending on n‐functions and their derivatives. The criterion is formulated in terms of the absence of conjugate points in the interval of the variational integral and is a generalization of a similar criterion due to Newcomb in problems of hydromagnetic stability in plasma physics.

The fundamental identity for iterated spherical means and the inversion formula for diffraction tomography and inverse scattering
View Description Hide DescriptionA uniform derivation of the inversion formula for diffraction tomography and inverse scattering from the fundamental identity for iterated spherical means is presented.

Exact solution of a nonlinear Langevin equation with applications to photoelectron counting and noise‐induced instability
View Description Hide DescriptionA stochastic differential equation of the Langevin type involving a linear and a quadratic noise is considered and an exact solution for the stochastic average of this equation is obtained. This solution is employed to calculate the effect of amplitude fluctuations of an electromagnetic field on the photoelectron counting, and an explicit expression is obtained for the photon counting probability. Finally, the steady state limit is studied and the possibility of an instability induced by the noise is discussed.

The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative
View Description Hide DescriptionIn this paper we investigate the Painlevé property for partial differential equations. By application to several well‐known partial differential equations (Burgers, KdV, MKdV, Bousinesq, higher‐order KdV and KP equations) it is shown that consideration of the ‘‘singular manifold’’ leads to a formulation of these equations in terms of the ‘‘Schwarzian derivative.’’ This formulation is invariant under the Moebius group (acting on dependent variables) and is shown to obtain the appropriate Lax pair (linearization) for the underlying nonlinear pde.

The Hamiltonian structure of the nonabelian Toda hierarchy
View Description Hide DescriptionWe show that a subset of the whole class of nonlinear differential‐difference equations, associated with the discrete analog of the matrix Schrödinger operator, is endowed with a Hamiltonian structure and exhibits an infinite number of integrals of motion in involution. We also establish the relation between these integrals of motion and the transmission coefficient of the underlying linear problem, and show that such a relation implies that, for the whole class previously introduced, there exists an infinite number of conserved quantities.

On the expansion of integrals containing Fermi distributions
View Description Hide DescriptionWe derive an expansion of integrals containing a general function multiplied by a Fermi function raised to an arbitrary power ν. When ν is an integer, direct expressions for the expansion coefficients are given. The expansion is found to converge quite rapidly if the diffuseness of the Fermi distribution is small, and when ν≳1.

Conformal geometry of flows in n dimensions
View Description Hide DescriptionFlows generated by smooth vector fields are considered from the point of view of conformal geometry. A flow is defined to be conformally geodesic if it preserves the distribution of vector spaces orthogonal to the lines of the flow. It is shear‐free if, moreover, it preserves the conformal structure on these vector spaces. Differential equations characterizing such flows are derived for the general case of an n‐dimensional conformal space of arbitrary signature. In the special case of null flows in spacetime, one obtains a refined version of the theorem connecting null solutions of Maxwell’sequations with null flows that are geodesic and shear‐free.

Homogeneous quadratic invariants for one‐dimensional classical homogeneous quadratic Hamiltonians: Some properties of the auxiliary equation
View Description Hide DescriptionUsing a previously derived characterization of the homogeneous quadratic invariants for an arbitrary one‐dimensional classical homogeneous quadratic Hamiltonian we obtain various ‘‘nonlinear superposition laws.’’ We establish a close connection between one of these (the quadratic superposition law) and the original characterization.

Wobbling kinks in φ^{4} and sine‐Gordon theory
View Description Hide DescriptionWhen the φ^{4} model admits a kinksolution, it also admits a wobbling kink, which satisfies the boundary conditions of a kink, but possesses an internal degree of freedom. In this paper we develop a formal perturbation series for the wobbling kink in φ^{4}theory, and give the first two terms in the series explicitly. Then we prove that the formal series actually is asymptotic for a rather long time [O ( K ln(1/ε)) for a certain K]. Finally, we construct an exact 3‐soliton solution of the sine‐Gordon equation that also has the properties of a wobbling kink. For the sine‐Gordon equation, the wobbling kink seems to be mildly unstable.

Generalized linear inversion and the first Born theory for acoustic media
View Description Hide DescriptionA procedure is derived which incorporates a generalized linear inverse viewpoint within a multidimensional Born inversion method. The method we present is a more general Born theory which can accommodate insufficient and inaccurate data. This general method reduces to the ordinary Born procedure when the data requirements of the latter technique are satisfied.

On the Radon–Nikodym property for σ‐classes
View Description Hide DescriptionWe show that a σ‐class with the Radon–Nikodym property need not be a Boolean σ‐algebra. This disproves the conjecture of Gudder and Zerbe.

Compatibility of the uncorrelated joint distribution with the classical limit
View Description Hide DescriptionIt is shown that the uncorrelated joint distribution, when adopted for quantum mechanics, is not (despite initial appearances to the contrary) in conflict with the known correlations between position and momentum in the classical limit. It is therefore not excluded on these grounds as a possible joint distribution expression for quantum mechanics.

Continuity statements and counterintuitive examples in connection with Weyl quantization
View Description Hide DescriptionWe use the properties of an integral transform relating a classical function f with the matrix elements between coherent states of its quantal counterpart Q f, to derive continuity properties of the Weyl transform from classes of distributions to classes of quadratic forms. We also give examples of pathological behavior of the Weyl transform with respect to other topologies (e.g., bounded functions leading to unbounded operators).

A rigged Hilbert space of Hardy‐class functions: Applications to resonances
View Description Hide DescriptionThe explicit construction of a dense subspace Φ of square integrable functions on the positive half of the real line is given. This space Φ has the properties that: (1) it is endowed with a metrizable nuclear topology, (2) it is stable under multiplication by x, and (3) the functions in Φ have suitable analytical continuation to a half plane. The space Φ* of functions which are conjugate to elements of Φ is also considered. Then the triplets Φ⊆ L^{2} (0,∞)⊆Φ′ and Φ*⊆ L^{2} (0,∞)⊆Φ*′ are used to give a description of resonances.

Operator theory on the WKB method and Bremmer series
View Description Hide DescriptionThe well‐known WKB method and Bremmer series in mathematical physics have various modifications and applications. Basically, they are all limited to the finite‐dimensional case. Such methods have been extended to the infinite‐dimensional case in this paper, i.e., operators on a general normed linear space. This enables us to apply the methods to solve the problem of transport of radiation through a layer of cloud. The derivation is based on input‐output states and linear operators. Some elementary properties are exploited, along with physical interpretations. The convergence of the series is established. An example is given to demonstrate the methods and the rate of convergence.

Large coupling behavior of the energy eigenvalues of two anharmonic oscillators
View Description Hide DescriptionWe give two methods for calculating the behavior of the energy levels of H≡(1/2){−∂^{2} _{ x }−∂^{2} _{ y } +g[(x ^{2}−a ^{2})^{2}+(y ^{2}−a ^{2})^{2}]+b(x−y)^{2}} as the coupling constant g goes to infinity.