bluuebunny/crossref_metadata_2025 huggingface dataset:
Second and fourth indices of plethysms
Authors: McKay J. , Patera J. , Sharp R. T.
Published: March 2003

The direct product of several copies of a representation decomposes into a direct sum of components each with a definite permutation symmetry. The decomposition of any of the components into a direct sum of irreducible representations is the computation of a plethysm. The decomposition is often simply effected when the dimension and its analogs, the second and fourth indices of the plethysm, are known. The paper contains formulas for second and fourth indices of many specific plethysms as well as a prescription for the general plethysm. The same formula is valid for the plethysm based on any finite representation of any semisimple Lie algebra. Applications are illustrated by decomposition of all plethysms of degree 3 based on the E8 representation of dimension 3875; all fourth-degree E8-scalars are enumerated.
Irreducible representations of the central extension of Sl(2) ΛT2
Authors: Hughes J. W. B.
Published: March 2003

Using shift operator techniques a classification is given of the irreducible star representations of the central extension algebra C(Sl(2)ΛT2). It is found to possess two generic series of such representations, together with an isolated representation which is just the metaplectic representation of Sl(2). This is the only representation it possesses in common with the superalgebra Osp(2, 1).
Current responses of first and second order in a collisionless plasma. I. Stationary plasma
Authors: Larsson Jonas
Published: March 2003

The current responses of first and second order, due to an electromagnetic perturbation of a stationary but otherwise arbitrary solution of the relativistic Vlasov–Maxwell equations, are studied. In particular the symmetries leading to the (approximative, due to wave-particle interaction) conservation of wave energy for an inhomogeneous plasma are considered. Thereby we clarify the mathematical structure of certain previously derived formulas for the response operators and also make these more readily accessible for applications.
On the classification of Clifford algebras and their relation to spinors in n dimensions
Authors: Salingaros Nikos
Published: March 2003

A classification of all the Clifford algebras is given in terms of Kronecker products of the quaternion and dihedral groups. The relationship to spinors in n dimensions is explicitly determined. We show that the real Clifford algebra in Minkowski spacetime is distinct from both the algebra of Dirac matrices and the algebra of Majorana matrices, and cannot be realized by the spinor framework. The matrix representations of Clifford algebras are discussed, and are utilized to give a classification of the real forms of Lie algebras. We are thus able to relate Clifford, Lie, and spinor algebras in an intrinsic geometrical setting.
Gribov ambiguities from the bifurcation theory viewpoint
Authors: Malec Edward
Published: March 2003

The connection between singularities of the Faddeev–Popov determinant and the local gauge degeneracy is discussed. A criterion relating the two phenomena is given. It is proved that a whole neighborhood of the local gauge copy is filled with copies of transverse potentials.
A new mathematical formulation of accelerated observers in general relativity. II
Authors: Retzloff D. G. , DeFacio B. , Dennis P. W.
Published: March 2003

The observation of a general vector field based on exp* is employed to obtain formulas for the coordinate velocity and coordinate acceleration of a geodesic particle. Our results are shown to reduce to those based on a parallel transport definition of observation in special relativity. In general relativity the difference between the expressions for the coordinate velocity and coordinate acceleration derived from the two definitions of observation is given in terms of the Riemann curvature tensor.
A Schwarzschild-like interior solution for charged spheres
Authors: Krori K. D. , Paul B. B.
Published: March 2003

A Schwarzschild-like interior solution for charged spheres is presented in this paper. The solution is regular everywhere.
Stochastic quantization of the linearized gravitational field
Authors: Davidson Mark
Published: March 2003

Stochastic field equations for linearized gravity are presented. The theory is compared with the usual quantum field theory and questions of Lorentz covariance are discussed. The classical radiation approximation is also presented.
The continuum limit of a classical 3-component, one-dimensional Heisenberg model is Brownian motion on the surface of a sphere
Authors: Isaacson D.
Published: March 2003

We show by explicit computation that when the temperature T is chosen as a function of the lattice spacing ε so that the correlation length stays fixed at one as ε→0, then the correlation functions of a classical one-dimensional Heisenberg model converge as ε→0 to the correlation functions of Brownian motion on the surface of a sphere. We remark that this provides a ’’statistical mechanical’’ construction of Brownian motion on the surface of a sphere.
Lippmann–Schwinger equations for a scalar macroscopic field in an anisotropic stratified medium
Authors: Iosilevskii Ya. A.
Published: March 2003

We develop a general approach to solve the transmission problem for a scalar macroscopic field in an anisotropic stratified medium. The method is based on a chainlike set of functional equations of the Lippmann−Schwinger type. A typical example of the fields under consideration is the wave field of a particle in the effective mass tensor approximation.
Some topics in pre-Hilbert space
Authors: Epifanio G. , Trapani C.
Published: March 2003

We examine some questions concerning pre-Hilbert spaces and operators defined in them. Some classical results, which hold true in Hilbert space, are extended, under particular conditions, to noncomplete space.
Statistics with number operator C
C*
Authors: Robbins Steven
Published: March 2003

The number operator a*a, where a is an annihilation operator, plays a fundamental role in the statistics of bosons and fermions. However, it is possible for other statistics to have the same number operator. We have previously shown that for one degree of freedom there is a type of statistics having this number operator corresponding to each p which is a positive integer or ∞. Fermions are obtained when p = 1 and bosons are obtained when p = ∞. No state can have more than p particles. In order to treat many degrees of freedom it is necessary to first consider the case of one degree of freedom differently. We show here that for more degrees of freedom a similar situation occurs but for each case other than bosons and fermions there is a positive integer q, such that no state can have more than q particles, even when the number of degrees of freedom is infinite. Thus these statistics are probably not physically realizable except in an approximate way.
The three magnon bound-state equation in one dimension
Authors: Majumdar Chanchal K. , Banerjee S.
Published: March 2003

The mathematical analogy of the three magnon bound-state equation with other momentum-space integral equations is studied. It is shown that a variable transformation, similar to Wick’s transformation in the Bethe–Salpeter equation, leads to alternative methods of complete analytic solutions of this equation.
The quantum pendulum in the WKBJ approximation
Authors: Kesarwani R. N. , Varshni Y. P.
Published: March 2003

Eigenvalues are determined for the plane pendulum problem by the WKBJ method in one- and four-term approximations, and the results are compared. It is found that at high quantum numbers, the four-term WKBJ approximation can yield eigenvalues of eight-significant-figure accuracy, but for low quantum numbers the results continue to be poor.
A new mathematical formulation of accelerated observers in general relativity. I
Authors: Retzloff D. G. , DeFacio B. , Dennis P. W.
Published: March 2003

Invariant methods of modern differential geometry are used to formulate exact closed form expressions for the coordinate velocity and coordinate acceleration of a geodesic particle in the tangent space of a general relativistic accelerating rotating observer. The observation of a general vector field is shown to be definable in two ways from presymmetry and covariance arguments. Our results for the parallel translation definition of observation are shown to subsume existing work in both special and general relativity on accelerated observers.
Rigorous iterated solutions to a nonlinear integral evolution problem in particle transport theory
Authors: Boffi V. C. , Spiga G.
Published: March 2003

After a preliminary functional study of the operator associated with the relevant Boltzmann equation, which is shown to be a contraction operator, a nonlinear integral evolution problem occurring in the diffusion of the particles of a mixture is solved by resorting to a rigorous iterative scheme, in the case without removal. According to this scheme, an explicit recursive representation for the general iterated solution of order n is developed. Structure and behavior of the solution so obtained are investigated and commented on.
Some spectral properties of the Kostant complex
Authors: Puta Mircea
Published: March 2003

In this paper we establish some spectral properties of an elliptic complex introduced by Kostant in the context of geometric quantization.
Analytic continuation from data points with unequal errors
Authors: Ciulli S. , Spearman T. D.
Published: March 2003

The problem solved in this paper is that of constructing zero-free holomorphic functions which will (a) assume specified values at a set of discrete data points in a data region Γ1 inside the holomorphy domain, and which at the same time will (b) provide the optimum solution to various stabilizing (boundedness or smoothness) conditions on the boundaries ΓR. The immediate motivation to this problem arose from the need to renormalize data with unequal errors by using a holomorphic weight function to bring all the errors to the same value: this was a preliminary step before making an analytic continuation off the data region Γ1. Since a stabilizing condition has to be imposed on the boundaries ΓR, the weight function must be chosen so as to introduce the minimum additional instability on ΓR. Although this was the specific motivation, other interesting applications suggest themselves and some of these are discussed. The stability conditions on ΓR which are treated may all be expressed in terms of the real parts, or of the normal derivative of the real parts. Γ1 is taken to be on the real axis and the functions considered satisfy a reflection principle which means that the data values are real. It follows that the results obtained may be expressed in terms of the real parts alone—in other words the problem solved here is in fact that of obtaining harmonic functions which take specific values inside their harmonicity domain and which satisfy the appropriate extremum condition on the boundary.
Some comments about the tensor virial theorem and orthogonal linear transformations
Authors: Fernández Francisco M. , Castro Eduardo A.
Published: March 2003

The tensor virial theorem is analyzed in relation to orthogonal linear transformations. The physical implications are discussed.
Gradient theorem for completely integrable Hamiltonian systems
Authors: Arthur M. D. , Case K. M.
Published: March 2003

For evolution equations which can be written in Hamiltonian form two ways, there exists a relation between two functions Q(1) and Q(2), both of which are gradients of conserved functionals. The relation can be extended to define (recursively) functions Q(n). It is shown that the Q(n) corresponding to the general evolution equation associated with the Zakharov–Shabat eigenvalue problem are all gradients of conserved functionals. This in turn implies all these functionals are in involution.
Demiański-type metric in Brans–Dicke theory
Authors: Krori K. D. , Bhattacharjee D. R.
Published: March 2003

A Demiański-like metric is obtained by means of a complex coordinate transformation in the Brans–Dicke theory.
On the relativistic Bose–Einstein integrals
Authors: Haber Howard E. , Weldon H. Arthur
Published: March 2003

Two integrals which appear in the study of the relativistic Bose gas are analyzed. The complete low-temperature and high-temperature expansions are computed.
On the inconsistency of a photon creation mechanism in an expanding universe
Authors: Ceccatto H. , Foussats A. , Giacomini H. , Zandron O.
Published: March 2003

We show here that if the quantum equivalence principle (QEP), as it was formulated in previous papers, is applied to the massless vector field, an inconsistent unphysical photon creation is found. This timelike and longitudinal photon creation is obtained when the 4-potential Aμ is quantized in a covariant way.
Nonperturbative confinement in quantum chromodynamics. II. Mandelstam’s gluon propagator
Authors: Atkinson D. , Johnson P. W. , Stam K.
Published: March 2003

It is shown that Mandelstam’s approximate equation for the gluon propagator has a solution with very singular infrared behavior. At the origin in the squared momentum variable there are a double pole, a branch-point, and an accumulation of complex first-sheet branch-points. Although the double pole is suggestive of confinement, the existence of acausal complex singularities indicates a deficiency in this first step of the approximation scheme.
Local conformal invariance of the wave equation for finite-component fields. II. Classification of relevant indecomposable fields
Authors: Bracken A. J. , Jessup Barry
Published: March 2003

It has been shown in Part I that the requirement of local conformal invariance of the wave equation for finite-component fields focuses attention on fields whose index spaces carry a certain type of finite-dimensional, indecomposable representation of the nonsemisimple Lie algebra ((k4⊕d)⊕sl(2, C)). All representations of this type are here described in complete detail, in each case in an sl(2, C) basis. Although indecomposable, these representations are in general not fully reducible.
Dynamical group of microscopic collective states. II. Boson representations in d dimensions
Authors: Deenen J. , Quesne C.
Published: March 2003

The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of 𝒮𝓅c(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ℱc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in ℱc, classified according to the chain 𝒮𝓅c(2d,R)⊇𝒰c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain 𝔘(ν)⊇𝔘(d), in a Bargmann Hilbert space ℬ of analytic functions in ν complex variables representing ν-dimensional boson states. By equating the complex variables of ℱc and their conjugate momenta with those of ℬ, we get the non-Hermitian Dyson representation of 𝒮𝓅c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a 𝔘(ν) group, which in the physical three-dimensional case reduces to 𝔘(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.
G
2 van der Waerden invariant
Authors: Gaskell R. , Sharp R. T.
Published: March 2003

The G2 van der Waerden invariant is given. It solves the external labeling problem connected with direct products of irreducible representations of G2.
A closed formula for the product of irreducible representations of SU(3)
Authors: O’Reilly Michael F.
Published: March 2003

We determine a closed formula in terms of p, q, r, and s for the decomposition of the product [p, q] [r, s] of finite-dimensional irreducible representations of SU(3). We also determine in terms of p, q, r, s, m, and n necessary and sufficient conditions that a term [m, n] appears in this decomposition and its multiplicity.
Addition formula for the Z
N×Z
N symmetric solutions of the factorization equations
Authors: Doria Mauro M.
Published: March 2003

An addition formula is derived which contains the addition relations for theta functions with characteristic proportional to 1/N, N integer.
Discontinuous solutions for first-order systems through a limiting process
Authors: Gravel Pierre
Published: March 2003

The possibility for physically general quasilinear differential systems to have discontinuous solutions or solutions with discontinuous derivatives is investigated using the method of asymptotic regularizations.
Laplace asymptotic expansions of conditional Wiener integrals and generalized Mehler kernel formulas
Authors: Davies Ian , Truman Aubrey
Published: March 2003

Imitating Schilder’s results for Wiener integrals rigorous Laplace asymptotic expansions are proven for conditional Wiener integrals. Applications are given for deriving generalized Mehler kernel formulas, up to arbitrarily high orders in powers of ℏ, for exp{−TH(ℏ)/ℏ}(x, y), T>0 where H(ℏ)=[(−ℏ2/2)Δ1+V], Δ1 being the one-dimensional Laplacian, V being a real-valued potential V∈C∞(R), bounded below, together with its second derivative.
Trace identities in the inverse scattering transform method associated with matrix Schrödinger operators
Authors: Alonso L. Martínez , Olmedilla E.
Published: March 2003

Trace identities arising in the scattering theory of one-dimensional matrix Schrödinger operators are deduced. They derive from the properties of an asymptotic expansion of the trace of the resolvent kernel in inverse powers of the spectral parameter. Applications of these trace identities for characterizing infinite families of conservation laws for nonlinear evolution equations are given.
Formal solutions of inverse scattering problems. IV. Error estimates
Authors: Prosser Reese T.
Published: March 2003

The formal solutions of certain three-dimensional inverse scattering problems presented in papers I–III in this series [J. Math. Phys. 10, 1819 (1969); 17, 1175 (1976); 21, 2648 (1980)] are employed here to obtain quantitative estimates on the error resulting from the use of the Born approximations in both direct and inverse potential scattering problems. These estimates are uniformly valid at all energies, and for all sufficiently weak potentials.
Perturbation theory of inelastic resonances
Authors: Bosanac S.
Published: March 2003

Perturbation theory of inelastic resonances is developed. From the fact that the resonances appear as the roots of the Jost function, we show that perturbation coefficients are obtained without use of the complete set. The theory is generalized to the case when the unperturbed Hamiltonian is p-fold degenerate. The near degenerate case is also discussed, and the radius of convergence for the perturbation series is estimated. We also treat the perturbation theory of residues both in the nondegenerate and degenerate case.
Orthogonal transitivity, invertibility and null geodesic separability in type D vacuum solutions of Einstein’s field equations with cosmological constant
Authors: Czapor S. R. , McLenaghan R. G.
Published: March 2003

It is shown without explicit integration that all Petrov type-D vacuum solutions with cosmological constant admit at least a two-parameter, abelian, orthogonally transitive group of local isometries. In the case when the orbits are non-null the group is invertible and a symmetric null tetrad is shown to exist in which it is manifest that the principal null congruences defined by the type-D Weyl tensor are indistinguishable. General forms for the metrics are given for both the case of non-null and null group orbits. It is also demonstrated that there always exists a system of coordinates for these solutions in which the Hamilton–Jacobi equation for the null geodesics is solvable by separation of variables, a fact that explains the existence of a conformal Killing tensor therein. A partial invariant characterization of Kinnersley’s type-D vacuum solutions is given from which it follows that all his metrics except those of Case III admit a (0,2) Yano–Killing tensor and hence a full (0,2) Killing tensor.
Impact parameter representation for scattering involving arbitrary spins
Authors: Bhakar B. S.
Published: March 2003

An impact-parameter representation that is valid at all angles and all energy has been obtained for systems with arbitrary spins. In contrast to the spinless case, this representation gives rise to generalized profile functions that are distributions even in the physical region. This proves kinematical equivalence between partial wave representation and impact parameter representation for the case of arbitrary spin systems.
On the mean motion and some statistical properties of a quasiperiodic observable in a fermion–boson model
Authors: Hioe F. T.
Published: March 2003

We study properties of an atomic observable of a fermion–boson model in the region after a long time, compared to its initial ‘‘collapse’’ and ‘‘revival’’ times, had elapsed. We determine its mean motion which characterizes its mean oscillation. We also determine its average frequency of achieving a particular value. Its behavior in the small average boson number limit is shown to be particularly interesting.
Fermion clustering in a soluble one-dimensional many-body system
Authors: Aguilera-Navarro V. C. , Ley-Koo E. , de Llano M. , Peltier S. M. , Plastino A.
Published: March 2003

A one-dimensional gas of N fermions with ν intrinsic degrees of freedom, such that N/ν is integer, with the particles interacting via attractive delta pair potentials is considered in its ground state. In particular, the ‘‘clustering’’ into N/ν entities is shown to appear and studied comparatively from both the Schrödinger and Hartree–Fock approximation viewpoints in terms of the energy and wavefunction overlap. The Schrödinger ground state energy for all nonzero densities is calculated numerically and compared to the nonzero density Hartree–Fock results for ν=2.
Mean field approximations for U(N) and SU(N) lattice gauge theories
Authors: Müller V. F. , Rühl W.
Published: March 2003

A particular parametrization of the groups U(N) and SU(N) is constructed by observing that the sphere S2N−1 is homeomorphic to the factor spaces U(N)/U(N−1) or SU(N)/SU(N−1) and continuing the corresponding fibration. The spheres are naturally embedded into Euclidean spaces and thus allow an extension of the mean field approximation by a saddle-point method to U(N) and SU(N) lattice gauge theories. It differs from the standard variational approach, the result of which can also be obtained by embedding the group into the Euclidean space of matrices. For both approaches the phase transition points are calculated and compared with the results of Monte-Carlo simulations. The best agreement is obtained for the standard variational approach with axial gauge fixing.
An embedding of some Stiefel bundles
Authors: Hogan P. A.
Published: March 2003

We present an embedding of Stiefel manifolds fn : V2,1(F)→Vn+1,n(F), (n≥2) with F=C or R having the property that the pullback via fn of the curvature 2-form on Vn+1,n(F) to V2,1(F) is an integer multiple n times the curvature 2-form on V2,1(F). We thus obtain the field of a magnetic pole of strength g=n/2.
Angular momentum operator in a nonabelian gauge field of an N-dimensional monopole and an SO(N) top operator
Authors: Hou Bo-Yu , Hou Bo-Yuan , Wang Pei
Published: March 2003

This paper discusses the structure of the angular momentum operator in a nonabelian gauge field of an N-dimensional monopole. Besides the finite rotation operators with a general gauge group, infinitesimal operators in various moving frames are introduced. By using these the transformation of frames may be easily operated recursively, so the applied relation between the transitive and stationary parts in the top is displayed, and the explicit formulation of angular momentum operators including different gauge potential of a monopole are obtained quite conveniently. Hence the gauge potential of an N-dimensional monopole in the Schwinger gauge and the Wu– Yang overlapping gauge are given explicitly. In addition, we obtain also the result that the self-rotation operators (Nth axis is stationary) in a fixed frame under geodesic gauge equal to the sum or difference of isospin and orbital momentum. Finally we have discussed the general gauge group of an N-dimensional monopole.
Scattering by slightly nonshperical particles in the high frequency limit. I. Impenetrable particles
Authors: Krebs Jerome
Published: March 2003

The scattering of a scalar plane wave by a slightly nonspherical impenetrable particle in the high frequency limit is investigated using a combination of boundary perturbation theory and the modified Watson transformation. Boundary perturbation theory is used to find the perturbed S-matrix for the problem. The modified Watson transformation is then used to sum the perturbed partial wave expansion in the short wavelength limit. This method is applied to backscattering by a slightly dilated sphere and a sphere whose shape is perturbed by making the radius a function of θ given by rp(θ)=a[1+δP2(cosθ)], where P2 is a Legendre function, a is the radius of the unperturbed sphere, and δ is the perturbation parameter. This method will be valid for (ka)1/3≫1, where k is the wavenumber.
Inverse scattering: determination of inhomogeneities in sound speed
Authors: Balanis George N.
Published: March 2003

The one-dimensional wave equation is examined for the purpose of developing a method applicable to indirect probing of propagation speed inhomogeneities. The examination utilizes the theory of inverse scattering and results in the development of a new method. The method consists of an algorithm which derives the inhomogeneities through processing in a special way the backscattered waves generated by an incident δ-function.
Semiclassical statistical mechanics of a two-dimensional fluid
Authors: Sinha S. K. , Sainger Y. S. , Singh Y.
Published: March 2003

Expansions are obtained for the radial distribution function and free energy for a two-dimensional hard-core fluid in the semiclassical limit, using the ‘‘modified’’ Wigner–Kirkwood expansion method. These results are used to obtain expressions for the density-independent part of the radial distribution function and the first-order density correction to it. Quantum corrections to the second and third virial coefficients are discussed in detail. Explicit results are given for the Sutherland, Yukawa-tail, Wood–Saxan, square-well, and Lennard-Jones (12-6) pair potential models.
Exact theory for the self-similarity and decay of homogeneous turbulence
Authors: Rosen Gerald
Published: March 2003

It is shown that space-time dilatation invariance (x→ξ−1x, t→ξ−2t, with concomitant transformations for dependent variables) and linearity of the Φ-equation engender an exact, time-explicit generic form for the solution applicable to freely-decaying homogeneous incompressible fluid turbulence. This solution features a summation over mutually independent dynamical modes labeled by the dilatation scaling-index n(>1). Without the assumption of isotropy nor introduction of a closure approximation procedure, the theory provides an explanation for the experimentally observed self-similarity of the correlation tensors and the decay laws 〈‖u(x, t)‖2〉∝t−n for the different types and decay stages of homogeneous turbulence.
Global solution to a nonlinear integral evolution problem in particle transport theory
Authors: Boffi V. C. , Spiga G.
Published: March 2003

Existence and uniqueness of the solution to a nonlinear integral evolution equation, arising in particle transport theory, is discussed and proved for any time interval [0,T]. This is pursued by a suitable application of the contracting mapping principle to the study of the nth power An of the relevant nonlinear inhomogeneous integral operator A.
On a generalized Hilbert problem
Authors: Newton Roger G.
Published: March 2003

The problem analyzed is to find functions f±, meromorphic in C±, respectively, with values that are linear operators on a Banach space, and such that their boundary values on R satisfy the equation f−=ω f+, where the operator-valued function ω as well as the positions of the poles of f± and the ranges of their residues are given. Uniqueness results are obtained, under certain conditions an index is proved to exist, and the determination of f± is reduced to the solution of a generalization of Marchenko’s fundamental equation. The results are applied to inverse scattering and inverse spectral problems.
Carleman embedding and Lyapunov exponents
Authors: Andrade R. F. S.
Published: March 2003

We investigate the solutions of those autonomous systems with quadratic nonlinearities in a N-dimensional vector space together with the solutions of their first variational equation systems by means of the Carleman embedding. An iterative procedure based on this result is developed to evaluate the Lyapunov exponents of the considered systems. We test the method by giving some results for the Lyapunov exponents of the Lorenz model.
Nonclassical fields with singularities on a moving surface
Authors: Betounes David E.
Published: March 2003

Fields with singularities on a moving surface S with boundary ∂S can be represented as distributions which have their support concentrated on S and ∂S. This paper considers such fields of the form F={ f }+λδS̃, where { f } is the distribution determined by a field f and λδS̃ is a Dirac delta distribution with density λ concentrated on the tube S̃ swept out by the moving surface. A straightforward calculation of the distributional gradient, curl, divergence, and time derivative of such fields yields fields of the following general form: G={ g } +αδS̃ +βδ∂S̃ +γ∇n(⋅)δS̃. The density α is shown to contain all the information which is customarily presented in the jump conditions for fields with singularities at a moving interface. Examples from electromagnetic field theory are presented to show the significance of the other terms { g }, βδ∂S̃, and γ∇n(⋅)δS̃.
The inverse problem of the calculus of variations applied to continuum physics
Authors: Bampi F. , Morro A.
Published: March 2003

Necessary and sufficient conditions for a differential system of equations to admit a variational formulation are established by having recourse to Vainberg’s theorem which provides also a systematic method for producing the sought functional. An application of the method to the Lagrangian description of fluid dynamics leads to a new variational principle which, while being fully general, reveals a hierarchy between variational approaches to fluid dynamics. Next, the method is applied in an attempt to obtain new variational formulations in various areas of research pertaining to continuum physics: water wave models, elasticity, heat conduction in solids, dynamics of anharmonic crystals, and electromagnetism. Owing to the power of the method, relevant variational formulations are found whenever the given system allows them. The paper places particular emphasis on equations which have, or are supposed to have, soliton solutions.
The complexification of a nonrotating sphere: An extension of the Newman–Janis algorithm
Authors: Herrera L. , Jiménez J.
Published: March 2003

A procedure given by Newman and Janis, to obtain the exterior Kerr metric from the exterior Schwarzschild metric by performing a complex coordinate transformation, is applied to an interior spherically symmetric metric. The resulting metric can be matched to the exterior Kerr metric on the boundary of the source which is chosen to be an oblate spheroid. A specific example of an interior solution for which the energy density is positive is given in detail.
Geodesic deviation and first integrals of motion
Authors: Caviglia Giacomo , Salmistraro Franco , Zordan Clara
Published: March 2003

Generalized Killing tensors (GKT) and generalized conformal Killing tensors (GCKT) are defined as the class of totally symmetric tensor fields that generate solutions of a suitable inhomogeneous equation of geodesic deviation along arbitrary and null geodesics, respectively. A geometric interpretation of these fields as generators of Jacobi fields along arbitrary and null geodesics is also given. It is shown that well known fields such as Killing tensors, conformal Killing tensors, and geodesic collineations belong to the above classes. Finally, first integrals of geodesic motion concomitant with the existence of GKT’s and GCKT’s are determined.
Cascaded Poisson processes
Authors: Matsuo Kuniaki , Saleh Bahaa E. A. , Teich Malvin Carl
Published: March 2003

We investigate the counting statistics for stationary and nonstationary cascaded Poisson processes. A simple equation is obtained for the variance-to-mean ratio in the limit of long counting times. Explicit expressions for the forward-recurrence and inter-event-time probability density functions are also obtained. The results are expected to be of use in a number of areas of physics.
A generating integral for matrix elements of the Coulomb Green’s function
Authors: Hill Robert Nyden , Huxtable Barton D.
Published: March 2003

The generating integral Il(λ,λ′;E) =∫d3r d3r′ exp(−λr−λ′r′)(rr′)l−1 ×Ȳl,m(̄θ̄,̄φ̄)̄G(r,r′;E) ×Yl,m(θ′,φ′) is evaluated, where the Coulomb Green’s function G is the inverse (E−H)−1 with H the hydrogenic Hamiltonian −(1/2) ∇2−Zr−1 and Yl,m is a spherical harmonic. The result can be used for the evaluation of matrix elements of G with respect to wave functions of the form Yl,m(θ,φ)f(α)l,p(r) where f(α)l,p(r) =N(α)l,p ×exp(−ar/2)(ar)lL(α)p (ar) with L(α)p a Laguerre polynomial and N(α)l,p a normalizing factor. For general E the result is given in terms of a special case of the hypergeometric function which satisfies an inhomogeneous linear first-order ordinary differential equation. For E=En where En=−Z2/(2n2) is a hydrogenic bound state energy, G is replaced by the generalized Green’s function (generalized inverse) and the results are given in closed form in terms of elementary functions.
A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians
Authors: Lewis H. Ralph , Leach P. G. L.
Published: March 2003

For a classical Hamiltonian H=(1/2) p2+V(q,t) with an arbitrary time-dependent potential V(q,t), exact invariants that can be expressed as series in positive powers of  p, I(q,p,t)=∑∞n=0pnfn(q,t), are examined. The method is based on direct use of the equation dI/dt=∂I/∂t +[I,H] =0. A recursion relation for the coefficients fn(q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in p are determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.
Linear Hamiltonian systems are integrable with quadratics
Authors: Kocak Huseyin
Published: March 2003

A new proof of a theorem of Williamson on the complete integrability of time-independent, real, linear Hamiltonian differential equations with quadratic integrals is given. The sets where these integrals are functionally dependent are explicitly found.
Measurements, Hilbert space and quantum logics
Authors: Gudder S. P. , Rüttimann G. T. , Greechie R. J.
Published: March 2003

We consider single and multiple measurements on a quantum logic (P,S) as well as states and propositions conditioned by a measurement. We show that corresponding to any measurement A, there is a canonically associated Hilbert space HA. Algebraic and statistical properties of (P,S) that are preserved in HA are found. We then study the problem of embedding a quantum logic in Hilbert space.
Generalized second-order Coulomb phase shift functions
Authors: Rosendorff S.
Published: March 2003

Some specific properties and the evaluation of the generalized second-order Coulomb phase shift functions (two-dimensional integrals of four spherical cylinder functions) are discussed. The dependence on the three momenta k1,k̄,k2, corresponding to the final, intermediate, and initial states is illustrated.
The unitary irreducible representations of SL(2, R) in all subgroup reductions
Authors: Basu Debabrata , Wolf Kurt Bernardo
Published: March 2003

We use the canonical transform realization of SL(2, R) in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group. Explicit results are given for all mixed bases and subgroup reductions. These provide the full multiparameter set of integral transforms and series expansions associated to SL(2, R).
Clifford algebra approach to twistors
Authors: Abl/amowicz R. , Oziewicz Z. , Rzewuski J.
Published: March 2003

Local particle interpretation or, equivalently, an enlargement of a structure group to the Poincaré group at each point of a Riemannian space-time manifold naturally results in a complexification of the Clifford algebra for the tangent Minkowski space. Following Crumeyrolle, twistor space is identified with an appropriate one-sided ideal of this algebra. Every antiautomorphism of the latter provides a unique projection from the complexified Clifford algebra onto the affine complex Minkowski space. This projection commutes with the action of the Poincaré group. Using the above approach, three projections (the cases of symmetric, antisymmetric, and Hermitian tensors) are derived. The projection in terms of the antisymmetric, decomposable tensors is shown to give the Penrose projection.
Timelike infinity
Authors: Persides S.
Published: March 2003

A new formulation is presented for analyzing the structure of a space–time at timelike infinity. An asymptotically simple space–time is defined as a space–time (ℳ,g) which can be imbedded in a space (ℳ̄,ĝ) with boundary 𝒯,a C∞ metric ĝ and a C∞ scalar field Ω, such that Ω = 0 on 𝒯, Ω≳0 on ℳ−𝒯 andĝμν−ĝμλĝνρΩ‖λ Ω‖ρ = Ω−2gμν−Ω−4gμλ gνρΩ;λΩ;ρ in a neighborhood of 𝒯. Demanding that 𝒯 = 𝒯−U 𝒯+, where each one of 𝒯− and 𝒯+ is isometric to the unit spacelike hyperboloid, and ĝμν Ω‖μ Ω‖ν = Ω−4gμν Ω;μ Ω;ν = 1 on 𝒯, we have an almost asymptotically flat (at timelike infinity) space–time. The group of asymptotic symmetries of (ℳ,g) at timelike infinity is found to be isomorphic to the Lorentz group. Some properties of the space–time near 𝒯 are shown.
A unified formulation of timelike, null and spatial infinity
Authors: Persides S.
Published: March 2003

A formulation is presented for studying simultaneously the timelike, null, and spatial infinities of space-times which resemble asymptotically Minkowski’s space-time. For this a relation of the form f(g,ĝ,Ω) = 0 is determined so that given a space-time (ℳ,g) a space (ℳ,g) with boundary ℬ can be found with ℳ imbedded in ℳ̂ and ĝ and ΩC∞ fields on ℳ̂. A space-time (ℳ,g) for which this is possible is called (globally) asymptotically simple. Then the conditions for ℬ to resemble the boundary of Minkowski’s space-time are determined. Thus the concept of a (globally) almost asymptotically flat space-time is defined.
Tensor fields on crystals
Authors: Litvin D. B.
Published: March 2003

A new method is presented to determine the irreducible representations of the space group of a crystal contained in the representation whose basis functions are the components of a tensor field defined on the atoms of a crystal. This reducible representation is the direct product of a tensor representation, dependent only on the tensor, and a permutation representation dependent only on how the atoms permute under elements of the space group. The permutation representation is first separately reduced prior to the reduction of the direct product. The permutation representation is shown to be an induced representation and its reduction is facilitated using the theory of induced representations. Examples and tables of results of applying this method are given in the case of a polar vector tensor field, applicable to lattice vibrational problems, and crystals, as the diamond structure, of space group symmetry 07h .
Feynman path integral and Poisson processes with piecewise classical paths
Authors: Combe Ph. , Hoegh-Krohn R. , Rodriguez R. , Sirugue M. , Sirugue-Collin M.
Published: March 2003

We prove the existence of a Feynman integral formula for gentle perturbations of the harmonic oscillator. This result is extended to Bose relativistic theory.
The Riemann tensor, the metric tensor, and curvature collineations in general relativity
Authors: McIntosh C. B. G. , Halford W. D.
Published: March 2003

The equation xμνRμ λαβ+xμλRμ ναβ = 0, where xμν and Rμ ναβ are the components of an arbitrary symmetric tensor and of the Riemann tensor formed from the metric tensor gμν, is trivially satisfied by xμν = φgμν. Nontrivial solutions are important in various areas of general relativity such as in the study of curvature collineations, and also in the study of algebraic methods given by Hlavatý and Ihrig for the determination of gμν, from a given set of Rμ ναβ. We have found all Rμ ναβ for which there exist nontrivial solutions of the above equation, and we have given the form of the xμν in each case. Various examples of space–times for explicit nontrivial solutions are discussed.
Complementary energy principles in dissipative fluids
Authors: Laedke E. W. , Spatschek K. H.
Published: March 2003

Necessary and sufficient conditions for exponential stability of equilibria in dissipative fluids are discussed on the basis of energy principles. First, a known maximum principle is reformulated in a manner which is more appropriate for evaluation of actual growth rates. Secondly, complementary variational principles are presented. The latter are quite useful for qualitative estimates and numerical computations since they lead to upper and lower bounds for the exact growth rates. The results are applicable to ideal magnetohydrodynamic (MHD) cases as well as resistive plasmas and also solitary waves.
Singular symmetries of integrable curves and surfaces
Authors: Kupershmidt B. A.
Published: March 2003

If w = w(x,t) is a solution of wt = 6wwx−wxxx+6ε2w2wx then ? = −w−ε−2 is also a solution. In general, integrable families and their members admit discrete symmetries whereas original systems may not.
An identity in Riemann–Cartan geometry
Authors: Nieh H. T. , Yan M. L.
Published: March 2003

We derive a new Gauss–Bonnet type identity in Riemann-Cartan geometry: (−g)1/2εμνλρ (Rμνλρ + (1/2) CαμνCαλν) = ∂μ (−(−g)1/2εμνλρCμνλρ), where Cαμν is the torsion tensor.
String mechanics based on 2-forms
Authors: Mitra P.
Published: March 2003

Nambu invented a mechanics for strings by replacing the fundamental 1-form pidqi−H dt of Hamiltonian mechanics by a certain 2-form. We study the mechanics corresponding to a more general 2-form applicable to weighted strings. Our equations of motion are fully deterministic, unlike those of Nambu, which need a supplementary condition. We set up a Hamilton–Jacobi formalism closely paralleling ordinary mechanics.
Global theorems for boundary-free viscous incompressible fluid flows of finite energy
Authors: Rosen Gerald
Published: March 2003

It is shown that the action for boundary-free incompressible fluid flow (i.e., the time-integral of the kinetic energy of the motion) is an absolute minimum with respect to all velocity-field transformations u→u* if u* is structured suitably in terms of u and an arbitrary solenoidal test field f. As suggested by this physical minimum principle, inequality analysis is applied to obtain an upper bound on the time derivative of the dissipation integral, from which there follow sufficient conditions for a monotone-decreasing dissipation integral and a monotone-decreasing global Reynolds number. The latter result provides an experimentally consistent necessary condition for passage from laminar to turbulent flow. Finally, inequality analysis is employed to derive a time-dependent lower bound on the maximum velocity gradient in a generic boundary-free flow of finite energy.
Mathematical problems of irreversible statistical mechanics for quantum systems. I: Analytic continuation of the collision and destruction operators by spectral deformation method
Authors: Courbage M.
Published: March 2003

We study some mathematical problems posed in nonequilibrium statistical mechanics and subdynamics theory developed by Prigogine and coworkers. We study in the superspace of the Hilbert–Schmidt operators the solution of the Liouville–von Neumann equation. The application in this frame of the spectral deformation methods yields expression of analytic continuations of a class of matrix elements of the resolvent of the Liouville–von Neumann operator, and this allows the analytic continuation of the collision and destruction operators. However, it is impossible to obtain simultaneously analytic continuation of the creation operator in this frame. The above results will be used in the second part of the article in order to study the pseudo-Markovian equation.
Analytic structure of the Henon–Heiles Hamiltonian in integrable and nonintegrable regimes
Authors: Chang Y. F. , Tabor M. , Weiss J.
Published: March 2003

The solutions of the Henon–Heiles Hamiltonian are investigated in the complex time plane. The use of the ’’Painlevé property,’’ i.e., the property that the only movable singularities exhibited by the solution are poles, enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable. Special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes. These boundaries have a remarkable self-similar structure whose form changes as a function of the nonlinear coupling.
Classical mechanics of nonspherical bodies. II. Boltzmann equation and H-theorem in two dimensions
Authors: Elskens Yves , Speiser David
Published: March 2003

An H-theorem is proved for a gas of two-dimensional rigid bodies which are not spherically symmetric but possess only a symmetry axis.
Time-dependent quadratic constants of motion, symmetries, and orbit equations for classical particle dynamical systems with time-dependent Kepler potentials
Authors: Katzin Gerald H. , Levine Jack
Published: March 2003

It is shown there are only two classes of time-dependent Kepler potentials [V2≡λ0(at2+bt+c)−1/2/r, (b2−4ac≠0), and V3≡λ0(αt+β)−1/r] for which the associated classical dynamical equations will admit quadratic first integrals more general than quadratic functions of the angular momentum. In addition to the angular momentum the system defined by V2 admits only a ’’generalized time-dependent energy integral,’’ while the system defined by V3 admits in addition to these a time-dependent vector first integral that is a generalization of the Laplace–Runge–Lenz vector constant of motion (associated with the time-independent Kepler system). For the V3 system the time-dependent vector first integral is employed to obtain in a simple manner the orbit equations in completely integrated form. The complete group of (velocity-independent) symmetry mappings is obtained for each of these two classes of dynamical systems and used to show that the generalized energy integral is expressible as a Noether constant of motion.