bluuebunny/crossref_metadata_2025 huggingface dataset:
Weight Lowering Operators and the Multiplicity-Free Isoscalar Factors for the Group R5
Authors: Ališauskas S. J. , Jucys A. P.
Published: February 2004

Expressions for the matrix elements of powers of infinitesimal operators of R5 are obtained in a basis reduced according to R5⊃(SU2×SU2). Operators producing the basis functions of any weight from those of highest weight are considered. The coupling and recoupling procedures for the semistretched and the other multiplicity-free cases are considered. By use of projection operators the expressions for the isoscalar factors of both kinds of semistretched Clebsch-Gordan coefficients of R5, as well as a new formula for the 9j-coefficient of SU2 involving only three summation parameters, are obtained. Methods of obtaining the normalized isoscalar factor coupling the basis functions of two symmetrized representations of R5 are discussed.
Horizons and Analytic Extensions in Static Two-Dimensional Space-Times
Authors: Godfrey Brendan B.
Published: February 2004

Necessary and sufficient conditions are derived for the analytic extendability of a static two-dimensional space-time. For all allowed cases, explicit analytic extensions are determined together with their corresponding Penrose-Carter diagrams. Extensions are classified and further discussed in terms of these diagrams, with special consideration given to the question of bifurcate Killing horizons. The application of these results to four-dimensional relativistic space-times is illustrated with specific examples.
Geometric Approach to Invariance Groups and Solution of Partial Differential Systems
Authors: Harrison B. Kent , Estabrook Frank B.
Published: February 2004

Methods are discussed for discovery of physically or mathematically special families of exact solutions of systems of partial differential equations. Such systems are described geometrically using equivalent sets of differential forms, and the theory derived for obtaining the generators of their invariance groups-vector fields in the space of forms. These isovectors then lead naturally to all the special solutions discussed, and it appears that other special ansätze must similarly be capable of geometric description. Application is made to the one-dimensional heat equation, the vacuum Maxwell equations, the Korteweg-de Vries equation, one-dimensional compressible fluid dynamics, the Lambropoulos equation, and the cylindrically symmetric Einstein-Maxwell equations.
Finite Transformations in Various Representations of SU(3)
Authors: Rosen S. P.
Published: February 2004

A projection operator expansion is used to obtain the explicit forms of finite transformations in the triplet, octet, and decuplet representations of SU(3). The projection operators are obtained from the characteristic equations for the matrices in these representations. These characteristic equations are developed from known properties of the λi, Fi, and decuplet matrices; where multiple eigenvalues appear, it is shown that the relevant matrices also satisfy reduced equations in which no eigenvalue appears twice or more. General features of the method are discussed.
Relation between the Singularities of the S Matrix and the L2 Class of Solutions in Potential Theory
Authors: Nelson C. A. , Rajagopal A. K. , Shastry C. S.
Published: February 2004

Ma has shown that, in the case of an exponential potential, in the s-wave S matrix there exist poles that do not contribute to the completeness, even though they appear in the same part of the k plane as the bound state poles. These poles were called ``redundant'' poles. Subsequently, other examples of redundant poles have appeared in the literature. Recently their importance with regard to the concept of ``shadow'' states has been stressed by Sudarshan. If we construct the Green's function for the Schrödinger equation with the boundary conditions of regularity at the origin and the outgoing spherical waves at large distances, the singularities of it in the k plane completely determine the L2 class of eigensolutions. From this Green's function, the T matrix (and hence the S matrix) is constructed explicitly. This S matrix is found to be the same as the one defined through the usual Jost solution, and it is shown that the singularities of the S matrix, besides corresponding to those appearing in the Green's function, also contain the ``redundant'' singularities. Thus it is shown that the wavefunctions associated with the ``redundant'' singularities do not belong to the L2 class. By a careful derivation, we resolve the Ma paradox concerning the Heisenberg identity. In the particular cases of the exponential potential and the Eckart potential, the redundant solutions correspond to the vanishing of a Wronskian pointing to the breakdown of the linear independence of the starting wavefunctions in defining the S matrix through the Jost functions. It is stressed that when singularities appear in the S matrix other than those corresponding to the bound and scattering states of the given problem, the ``redundant'' singularities can only be understood by a ``dynamical'' equation such as the Schrödinger equation.
Exactly Solvable Cell Model with a Melting Transition
Authors: Gates D. J.
Published: February 2004

A new cell model for classical particle systems is presented and analyzed. In this model the particles are confined to congruent, interconnected, cubic cells of volume ω centered on the points of a cubic lattice with lattice spacing 1/γ. The particles interact via a 2-body potential of the form q(r) + ω−1K(γr). The paper deals with the limiting form of this model in which the cells are very large but their separation is much larger. The free energy density is defined by a(ρ,T)≡lim lim ω→∞lim lim γ→0ã(ρ,T,γ,ω),where ã(ρ, T, γ, ω) is the free energy density at density ρ, temperature T, and arbitrary γ and ω. For a very general class of functions q and K, it is proved that a(ρ, T) is given by a variational principle. For a certain class of functions K (including K ≤ 0), a(ρ, T) is given by the Lebowitz-Penrose generalization of the van der Waals-Maxwell theory. For a different class of functions K the system has crystalline states. When K is chosen so that only particles in nearest-neighbor cells interact and K is isotropic, it is proved that the most general crystalline state of the system has a density distribution with two values ρ+ and ρ− arranged in a checkerboard (sodium chloride) pattern. For the special case with K repulsive, K(0) = 0 and q = 0, the system has a second-order melting transition from a crystalline to a fluid state, with no critical temperature. Various correlation functions are defined and evaluated. In the 1-dimensional nearest-neighbor case, the results include exact versions of the Ornstein-Zernike theory for both fluid and crystalline states. Magnetic systems are also considered. Different special cases of the model yield precisely the Weiss theory of ferromagnetism and the Néel-van Vleck theory of antiferromagnetism.
Multiplicities in the Classical Groups. II
Authors: Gruber B. , Han B. H. , Saldanha J. A.
Published: February 2004

In a previous paper, a method has been developed which allows a calculation in the same manner of all multiplicities occurring in the theory of linear representations of the classical compact Lie groups. This method is applied to the group SO(9) and some of its subgroups which are of interest in physics. It is demonstrated with the group SO(9) as example that this method of calculating the multiplicities allows us to accumulate in a condensed way a large amount of information and, moreover, that this method can be easily applied to any other (classical, compact) rank-4 group and its semisimple subgroups.
Geometric Properties of Neutrino Fields in Curved Space-Time
Authors: Wainwright J.
Published: February 2004

Classical neutrino fields in curved space-time are studied subject to the condition that the neutrino energy tensor Tab satisfies Tabuaub ≠ 0 for all timelike vectors ua. It is shown that the principal null congruence of these neutrino fields is geodesic and that its shear and twist are restricted. In addition, there exists a canonical null tetrad with respect to which Tab assumes a simple form. These conditions, in fact, characterize this class of neutrino fields. In addition, it is shown that if Tab satisfies the stronger condition that Tabub be a timelike or null vector for all timelike vectors ua, then the principal null congruence is also shear-free. Comparisons are made with the well-known properties of the electromagnetic energy tensor.
Arbitrary-Spin Wave Equations and Lorentz Invariance
Authors: Seetharaman M. , Jayaraman J. , Mathews P. M.
Published: February 2004

In an earlier paper it was shown that invariance of the arbitrary-spin wave equation i(∂ψ/∂t) = Hψ [with ψ transforming according to the representation D(0,s)⊕D(s,0) under rotations, T, C, and P, and boosts in the direction of the momentum, permits four infinite classes of solutions for H. Here we show that when invariance under boosts transverse to the momentum also is imposed, just two possible forms for H survive (in the c-number formalism). When we go over to the q-number theory, one of these forms leads to a consistent theory for half-integer spins only and the other for integer spins.
SO(4, 2)-Formulation of the Symmetry Breaking in Relativistic Kepler Problems with or without Magnetic Charges
Authors: Barut A. O. , Bornzin G. L.
Published: February 2004

The relativistic Kepler problems in Dirac and Klein-Gordon forms are solved by dynamical group methods for particles having both electric and magnetic charges (dyons). The explicit forms of the O(4, 2)-algebra and two special O(2, 1)-algebras (which coincide in the symmetry limit) are given, and a new group-theoretical form of the symmetry breaking is pointed out. The Klein-Gordon O(2, 1)-algebra also solves the dynamics in the case of very strong coupling constants (attractive singular potential), if the principal series of representations is used instead of the discrete series.
SL(2, C) Representations in Explicitly ``Energy-Dependent'' Basis. II
Authors: Szegö K. , Tóth K.
Published: February 2004

A systematic treatment is presented for the transformation matrix elements between different type of basis sets in the SL(2, C) representation space and for the boost matrix elements. The treatment is based on the theory of invariant bilinear functionals and makes use of the knowledge of explicit basis functions. New expressions are given for the SU(2) ↔ E(2) and SU(1, 1) ↔ E(2) overlap functions, and it is shown that they can be considered as a Fourier transform of a function, which is also known. Some relations are proved between boost functions and transformation matrix elements. In the Appendix a new and relatively simple expression is derived for the boost matrix elements djmj′j0σ(ξ) in SU(2) basis.
Lattice Green's Function. Introduction
Authors: Katsura Shigetoshi , Morita Tohru , Inawashiro Sakari , Horiguchi Tsuyoshi , Abe Yoshihiko
Published: February 2004

Physical, analytical, and numerical properties of the lattice Green's functions for the various lattices are described. Various methods of evaluating the Green's functions, which will be developed in the subsequent papers, are mentioned.
Lattice Green's Function for the Simple Cubic Lattice in Terms of a Mellin-Barnes Type Integral
Authors: Katsura Shigetoshi , Inawashiro Sakari , Abe Yoshihiko
Published: February 2004

The lattice Green's function for the simple cubic lattice I(a)=1π3[triple intergral operator]0πdx dy dza+iε−cos x−cos y−cos zfor a > 3 is expressed as a Mellin-Barnes type integral. The analytic continuation gives simple and useful expressions in series for the numerical calculation of the real part IR(a) and the imaginary part II(a) of the integral for 0 < a < 1 and 1 < a < 3. The values at a = 1, a = 0, and a=√5 are obtained exactly: IR(1) = (π/2)[Γ(5/8)Γ(7/8)]−2, II(1)/(−i)=(√2)IR(1), and II(0) = 3·2−11/3π−4[(Γ(1/3)]6.
Lorentz Basis for the 3 + 2 de Sitter Group
Authors: Bose S. K. , Parker R.
Published: February 2004

Matrix elements of the Lie algebra generators of the de Sitter group O(3, 2) are explicitly constructed on a basis in which the Casimir operators of the homogeneous Lorentz group are diagonal. As an application, a class of Hermitian representations of the algebra are obtained.
Critical-Point Behavior of a Class of Model Hamiltonians
Authors: Scott H. L.
Published: February 2004

A rigorous method for obtaining the thermodynamic functions for a class of model Hamiltonians is examined near points where fluctuations in certain order parameters become large. It is shown that if a phase transition occurs at such a point in a model, then the transition is classical in nature. That is, the free energy is analytic in the relevant order parameter.
Evaluation Techniques for Quantum Mechanical Integrals Involving Orthogonal Polynomials
Authors: Chen Tien Chi
Published: February 2004

Quantum mechanical matrix elements often involve orthogonal polynomials, whose properties can be exploited for evaluation. Nullity conditions are considered for hydrogenic and harmonic oscillator radial integrals, and Gaunt's triangular condition for integrals over triple products of associated Legendre functions is generalized so that the superscripts form a triangle of even perimeter. Where the integral is nonzero, Gaussian quadratures are exact for polynomial integrands and converge under very general conditions. When the integrand is a product of three functions, one of which is a linear polynomial, the quadrature is equivalent to first developing summation-orthogonal expansions of the component functions and then integrating the product exactly.
Wavefunction Expansion in Terms of Spherical Bessel Functions
Authors: Gersten A.
Published: February 2004

A formalism is derived for expanding the solutions of the Schrödinger equation in terms of spherical Bessel functions. The regular and the irregular solutions are treated. A relation between the expansion coefficients and the phase shifts is derived. As an application, the expansion coefficients of both the irregular and regular Coulomb wavefunctions are given in a form of a simple recurrence relation. The expansions have been checked numerically and found to be very suitable for calculating the regular Coulomb wavefunction in a very large region of the coordinate and the Coulomb parameter.
Sum Rule Functions
Authors: Barnsley Michael
Published: February 2004

In many cases the quantum mechanical sum rules S(k), as k ranges over all real values such that S(k) is convergent, form a particular example from a class of functions called sum rule functions. If a set of up to five values taken by a sum rule function is given, then, by the use of another class of functions called N-sum rule functions, it is shown how one can impose the very best possible bounds on the sum rule function for arbitrary k, on the basis of the given information. In particular, these results apply to quantum mechanical sum rules.
AC Susceptibility of Biased One-Dimensional Stochastic Ising Model
Authors: Schmidt V. Hugo
Published: February 2004

The ac susceptibility for the one-dimensional Ising model is obtained for arbitrary coupling strength in the presence of a dc bias field strong enough to align most of the dipoles in one direction. The dipole flip probability is assumed proportional to the Boltzmann factor corresponding to half the energy change resulting from the flip. The general expression for ac susceptibility is analyzed in three limiting cases: weak coupling with strong bias, strong coupling with strong bias, and strong coupling with weak bias. In the latter case, relatively long chains of anti-aligned dipoles exist and give rise to large susceptibility.
Simplified Intrinsic Proof of the Rainich Differential Relation
Authors: Luehr C. P. , Rosenbaum M.
Published: February 2004

The necessity and sufficiency of the Rainich differential relation, a basic equation of the ``already unified field theory,'' is proved in a simplified way by intrinsic techniques. Some tensor identities of interest in their own right are derived and used in this proof. The results in this paper generalize those given in a previous paper [Ann. Phys. (N.Y.) 60, 384 (1970)], where only the necessity of the Rainich differential condition was proved.
Analytic Continuation of the Regular Solution of the Schrödinger Equation in the Complex Angular Momentum Plane
Authors: Kelemen P. J.
Published: February 2004

In this paper we develop a method of performing the analytic continuation of functions that are Stieltjes or Laplace transforms. We apply the method to find the regular solution and its asymptotic behavior in the left half-plane. The irregular solution and the S matrix are briefly discussed.
Construction of Relativistic Potentials When the Energy Is Fixed
Authors: Coudray Christiane , Coz Marcel
Published: February 2004

We use generalized translation operators for solving the relativistic inverse problem at fixed energy and study the extension of Newton's method. Both the Klein-Gordon equation and the Dirac equation are considered. The determination of the so-called coefficients of interpolation remains a crucial point of the solution. In the case of the Klein-Gordon equation, these coefficients are obtained by the same system of equations as the system obtained for nonrelativistic spinless particles. Therefore, the same singular problem is encountered whether the spinless particles are relativistic or not. In the case of the Dirac equation, the problem with two potentials differs from the problem where only one potential is present. When there are two potentials, a generalized translation operator exists, and the inverse problem can be solved by inverting a singular matrix equation for the coefficients of interpolation. When only one potential is present, the solution of the inverse problem is restricted by a compatibility requirement.
Radiation and Energy in an Asymptotically Friedmann Space-Time
Authors: Glass Edward N.
Published: February 2004

A bounded gravitating system in an asymptotically hyperbolic Friedmann space is examined. The Bondi-Metzner-Sachs (BMS) group is shown to be the asymptotic symmetry group. The BMS generators and the Weyl tensor are used to construct an asymptotic invariant which allows the energy of the gravitating system to be calculated without infinite contributions from the cosmological dust. It is shown that a bounded gravitating system in a cosmological space can be understood in much the same manner as a similar physical system in an asymptotically flat space-time.
Representations of States of Infinite Systems in Statistical Mechanics
Authors: Shen C. Y. , Carter D. S.
Published: February 2004

This paper deals with the representation of states of infinite systems in classical statistical mechanics in terms of probability densities, correlation functions, and zero-density correlation functions. The class of states considered includes equilibrium states treated previously by Ruelle, and is believed to contain low-density nonequilibrium states as well. The theory is based on Carter's exponential construction for measure spaces, the representations being in terms of functions on the exponential (i.e., union of symmetrized direct powers) of one-particle phase space. The main result, which establishes the connections between these three representations, essentially extends a result of Ruelle, but is based on L1−convergence rather than uniform convergence on compacts.
Invariant Imbedding and the Variational Treatment of Fredholm Integral Equations with Displacement Kernels
Authors: Casti J. , Kalaba R. , Ueno S.
Published: February 2004

Fredholm integral equations with displacement kernels play a significant role in such areas as radiative transfer and optimal filtering. Frequently, these equations are studied by using the fact that their solutions minimize certain quadratic functionals, which opens the way to the employment of the Rayleigh-Ritz method. The aim presented in this paper is radically different. It is shown that the minimizer of the quadratic functional satisfies a Cauchy problem, no use being made of the integral equation. This notion is of analytic interest and computational utility.
Radiative Transfer in Spherical Shell Atmospheres with Radial Symmetry
Authors: Ueno S. , Kagiwada H. , Kalaba R.
Published: February 2004

In recent years, the need to allow for the effect of curvature on the formation of spectral lines has been recognized not only in the fields of planetary and stellar atmospheric physics but also in the field of neutron transport in spherical reactors. This paper is of interest in these and related fields. Initial value systems are obtained for the scattering and transmission functions and emergent intensities for inhomogeneous, anisotropically scattering spherical media both with internal and external sources and with various types of cores.
Distribution Functions for an Exactly Soluble Quantum Statistical System
Authors: Storer R. G.
Published: February 2004

Exact expressions are obtained for the distribution functions, in configuration and momentum space, for a system of bosons or spinless fermions interacting via pairwise harmonic forces.
New Tool for Scattering Studies
Authors: Sabatier Pierre C.
Published: February 2004

In potential scattering, from the transformation kernel, a function K(ρ) is constructed whose Bessel transforms are the Jost functions. K(ρ) contains the whole information on the potential V(ρ). There is a one-to-one correspondence between the two functions, and they are related, in both senses of this correspondence, through integral equations. Since the relation between K(ρ) and the phase shift is very direct, it is a useful tool for all analyses of the relations between the information contained in the dynamics of the problem (viz., the potential) and the measurable information (viz., the phase shifts). This tool will be applied to the inverse problem in forthcoming publications. Besides, the derivation of K(ρ) makes clear that a similar study can be done in all cases to which the Gel'fand-Levitan scheme applies and therefore in most scattering problems in physics.
Nonlinear Spinor Equation and Asymmetric Connection in General Relativity
Authors: Hehl F. W. , Datta B. K.
Published: February 2004

In order to take full account of spin in general relativity, it is necessary to consider space-time as a metric space with torsion, as was shown elsewhere. We treat a Dirac particle in such a space. The generalized Dirac equation turns out to be of a Heisenberg-Pauli type. The nonlinear terms induced by torsion express a universal spin-spin interaction of range zero.
Micromagnetism and Superconductivity
Authors: Eringen A. Cemal
Published: February 2004

The balance laws of microelectromagnetic theory developed by Eringen and Kafadar [J. Math. Phys. 11, 1984 (1970)] are shown to contain, as special cases, the classical theories of micromagnetism and Londons' equation of superconductivity. Various generalizations are indicated.
Vacuum Expectation Values as Indicatrices of Maximal Regularity
Authors: Pfaffelhuber E.
Published: February 2004

Matrix elements and propagators of renormalizable field theories are defined in terms of indicatrices of maximal regularity. The relation of the indicatrix approach with analytic regularization and with the subtraction formalism is established. It is shown that the indicatrix formalism gives rise to well-defined propagators and that the self-energy of a particle vanishes on the mass shell so that no mass renormalization is necessary. The renormalization constant Z3 equals one without the theory becoming a free one. This implies that, within the frame of the indicatrix approach, convergence of the theory is assured and renormalization actually is superfluous.
Off-Shell T Matrix Corresponding to a Sum of Coulomb and Separable Potentials by Expansion in O(4) Harmonics
Authors: Zachary W. W.
Published: February 2004

Using a group-theoretic approach similar to methods previously used in the pure Coulomb problem, we derive representations for the off-shell ``nuclear'' T matrix-the difference between the complete T matrix and the Coulomb T matrix-corresponding to the sum of Coulomb and separable potentials, in particular, for the sum of Coulomb and Yamaguchi potentials. These representations have some analogous properties to those representations already known for the pure Coulomb T matrix, especially in the on-shell limit.
Exact Results for a Body-Centered Cubic Lattice Green's Function with Applications in Lattice Statistics. I
Authors: Joyce G. S.
Published: February 2004

In this paper the body-centered cubic lattice Green's functionP(I,z)=1π3∫ ∫ 0π∫(1−zcosx1cosx2cosx3)−1cosl1x1cosl2x2cosl3x3 dx1 dx2 dx3,where l1, l2, and l3 are all even, or all odd, is studied. A complete analytic continuation for P(z) ≡ P(0, z) is derived of the form P(z)= ∑ n=0∞Bn(1−z2)n−(1−z2)12 ∑ n=0∞Cn(1−z2)n,where |1 − z2| < 1. Explicit formulas, recurrence relations, and asymptotic expansions are established for the coefficients Bn and Cn. A similar analytic continuation in powers of 1 − z is also investigated. The generalized Watson integral I(m,n)=1π3∫ ∫ 0π∫(1−cosx1cosx2cosx3)−1cos2mx1cos2nx2 dx1 dx2 dx3,where m ≥ 0 and n ≥ 0, is evaluated in closed form. Using this result, we show that P(I, 1) can, in principle, be evaluated for arbitrary I. Exact expressions and numerical values for P(I, 1) are given for 0 ≤ l1 ≤ l2 ≤ l3 ≤ 8. Detailed applications of the above results are made in the theory of random walks on a body-centered cubic lattice. In particular, a new asymptotic expansion for the expected number of distinct lattice sites visited during an n-step random walk is obtained. The closely related Green's functionlim lim ε→0+1π3∫ ∫ 0π∫(ξ0−iε−cosx1cosx2cosx3)−1 dx1 dx2 dx3,where ξ0 is real, is expressed in terms of complete elliptic integrals for all ξ0 > 0, and evaluated numerically in the range 0 < ξ0 ≤ 1. The behavior of this Green's function in the neighborhood of the singularities at ξ0 = 0 and 1 is also discussed. No attempt is made, in the present paper, to discuss P(I, z) for the general case I ≠ 0 and z ≠ 1.
Quadratic Fermion Interaction Hamiltonian
Authors: Gidas Basilis
Published: February 2004

The interaction Hamiltonian λ∫:ψ̄(0)(x)ψ(0)(x):g(x)dsx,g(x)∈S(Rs) is studied. An ultraviolet cutoff is introduced. We remove this cutoff, and take the limit g → 1 in S(Rs), by working with the Heisenberg fields. The limiting fields are well defined on the Fock space associated with the bare mass m0. In the limit we get a new representation of the canonical anticommutation relations which is given by a (generalized) Bogoliubov transformation. The new representation is not always unitarily equivalent to the bare mass Fock representations.
Poincaré-Irreducible Tensor Operators for Positive-Mass One-Particle States. II
Authors: Steiger Arno D.
Published: February 2004

The present communication studies relations between the two classes of Poincaré-irreducible tensor operators which were constructed in a previous paper. The tensor operators of the first class transform according to those representations of the Poincaré group that are induced by the one-valued irreducible unitary representations of SU(1, 1) which belong to the continuous and the discrete principal series. The tensor operators of the second class transform according to the Poincaré group representations that are induced by the irreducible unitary principal series representations of SL(2, C). The Poincaré-irreducible tensor operators of the second class are decomposed into Lorentz-irreducible tensor operators which transform by the irreducible unitary SL(2, C) representations of the principal series. The expansions of a spin density matrix describing a statistical ensemble of wavepackets into Poincaré- and Lorentz-irreducible components are derived. The infinite-dimensional associative algebra for the Poincaré-irreducible tensor operators of the second class is constructed. No such algebra exists for the Poincaré-irreducible tensor operators of the first class. The operator product of these tensors can only be represented as a linear superposition of the first class tensors in the limit of vanishing 4-momentum. In this limit, however, the first class tensors are no longer irreducible.
Unit Adjoint Tensor Operators in SO(n)
Authors: Wong M. K. F.
Published: February 2004

Unit adjoint tensor operators in SO(n) have been obtained, and the eigenvalues of the invariants connected with them evaluated. These adjoint tensor operators are keys to the solution of the multiplicity problem of the Wigner coefficients of SO(n). These adjoint tensor operators are used in the calculation of the Wigner coefficients of SO(5) in the direct product of the 10-dimensional representation with an arbitrary representation. The results agree with Hecht [K. T. Hecht, Nucl. Phys. 63, 177 (1965)].
Kinematic Dynamo Theory: The Dyson Equation and the Large-Scale Field; the Bethe-Salpeter Equation and the Fluctuation Intensity
Authors: Lerche Ian
Published: February 2004

An exact statistical set of kinematic dynamo equations is given representing turbulent generation of both a large-scale magnetic field and a small-scale turbulent field. These equations do not rely on approximately solving the fluctuation equations and using the results in the ordered field equations as do most treatments of statistical kinematic dynamos. Instead they treat both the fluctuation equations and the ordered field equation exactly. The results obtained indicate several points. First: The fluctuation intensity equation has the character of the Bethe-Salpeter equation. Depending on whether one uses the long-slow or the short-sudden approximation there may, or may not, be an upper limit on the velocity turbulence in order that the energy density stored in the magnetic field fluctuations remain finite. No such restriction is found using approximate kinematic dynamo equations. Second: The normal modes of the large-scale field (which obeys a Dyson equation) may, or may not, be mirrored in the singular eigenmodes of the fluctuation intensity equation. Third: the structure of the exact statistical kinematic dynamo equations is very different from the structure of the approximate kinematic dynamo equations-particularly in the equation describing the fluctuation intensity. We have done this problem in order to demonstrate that the exact solution of at least one problem in statistical kinematic dynamo theory introduces new and interesting phenomena which are not brought to light in approximate treatments of the same phenomena.
Korteweg-de Vries Equation and Generalizations. IV. The Korteweg-de Vries Equation as a Hamiltonian System
Authors: Gardner Clifford S.
Published: February 2004

It is shown that if a function of x and t satisfies the Korteweg-de Vries equation and is periodic in x, then its Fourier components satisfy a Hamiltonian system of ordinary differential equations. The associated Poisson bracket is a bilinear antisymmetric operator on functionals. On a suitably restricted space of functionals, this operator satisfies the Jacobi identity. It is shown that any two of the integral invariants discussed in Paper II of this series have a zero Poisson bracket.
Glauber Amplitudes in Charged Particle Hydrogen Atom Collisions
Authors: Thomas Brian K. , Gerjuoy E.
Published: February 2004

The Glauber scattering amplitudes for the excitation of the ns and np levels of atomic hydrogen by electrons (or protons) incident upon the ground state have been obtained in closed form. In contradistinction to previously quoted results, these new expressions require no numerical integration. For small n, the amplitudes reduce to simple sums of hypergeometric functions. Hence the Glauber amplitudes for these transitions now can be computed with scarcely more difficulty than the Born amplitudes. A selection rule for arbitrary transitions is derived.
Heat Flow in Regular and Disordered Harmonic Chains
Authors: Casher A. , Lebowitz J. L.
Published: February 2004

We investigate the steady state heat flux J in a large harmonic crystal containing different masses whose ends are in contact with heat baths at different temperatures. Calling ΔT the temperature difference and L the distance between the ends, we are interested in the behavior of J/ΔT as L→∞. For a perfectly periodic harmonic crystal, J/ΔT approaches a fixed positive value as L→∞ corresponding to an infinite heat conductivity. We show that this will be true also for a general one-dimensional harmonic chain (arbitrary distribution of different masses) if the spectral measure of the infinite chain contains an absolutely continuous part. We also show that for an infinite chain containing two different masses, the cumulative frequency distribution is continuous and that the spectrum is not exhausted by a denumerable number of points, i.e., the spectrum cannot consist entirely of point eigenvalues with a denumerable number of limit points. Using a theorem of Matsuda and Ishii, we show that for a random chain, corresponding to the mass at each site being an independent random variable, the heat flux approaches zero as L→∞, with probability one. This implies that the spectrum of a disordered chain has, with probability one, no absolutely continuous part.
Polynomial Bases for Representations of Semisimple Lie Algebras
Authors: Devi V. Syamala
Published: February 2004

It is shown that the polynomial bases for representations of a semisimple Lie algebra are just the various terms of typical concomitants of the Lie algebra. Consequently, the construction of polynomial bases reduces to a problem in the theory of invariants.
The Sound Wave Boundary Value Problem in Kinetic Theory. I
Authors: Richardson T. G. , Sirovich L.
Published: February 2004

A detailed investigation of the generation of sound waves from an oscillating piston is considered within the framework of kinetic theory. Two kinetic models appropriate to the problem are developed. Both of these are exactly solved by the Wiener-Hopf method. Under a certain special limit, gas dynamics is shown to hold. A variety of other special limits are also considered.
Some Exact Statistical Properties of Turbulent Kinematic Dynamo Equations
Authors: Lerche I.
Published: February 2004

The exact statistical properties of solutions to two restricted turbulent kinematic dynamo problems are given and discussed in some detail. In view of the fact that all discussion and solutions of the turbulent kinematic dynamo equations given so far in the literature are approximate, we believe that the present paper, containing two exactly soluble turbulent dynamo problems, is of more than academic interest. The method of solution is rather general and may, perhaps, be of basic interest. Further, the exact statistical solutions, which admit of regenerative dynamo action, allow approximate solutions to be compared and contrasted with the exact solutions, thus outlining the regime of applicability of the approximate solutions.
Eigenvalue Problems for Lagrangian Systems. V
Authors: Barston E. M.
Published: February 2004

Sufficient conditions for exponential instability are obtained for a class of infinite-dimensional gyroscopic Lagrangian systems. Applications are made to the problems of the stability of a rotating thin annular elastic disk and the stability of the steady flow of a one-dimensional warm electron plasma.
Reduction of Group Representations. V
Authors: Gabriel John R.
Published: February 2004

This paper shows how to use information which can be obtained by use of the Todd Coxeter algorithm and related techniques, to reduce a group representation. The method reduces a representation of G given a reduction for a subgroup H of finite index in G. If the character table for G is given, only rational algebraic processes and square roots are used. If the character table for G is not known it can be found by solution of eigenvalue equations, using the class sums, which are obtained in the course of the calculation without scanning the whole group.
Statistically Exact Kinematic Dynamo Action in a Box
Authors: Lerche I.
Published: February 2004

A set of statistically exact equations is set up to describe dynamo action brought about by velocity turbulence confined to a finite spatial domain (a ``box''). Using a variation of a limit-theorem employed elsewhere by Kac, we demonstrate that in large enough boxes a large-scale magnetic field is regenerated by such velocity turbulence.
Lattice Green's Function for the Orthorhombic Lattice in Terms of the Complete Elliptic Integral
Authors: Horiguchi Tsuyoshi , Yamazaki Yoshitake , Morita Tohru
Published: February 2004

The real and imaginary parts of the orthorhombic lattice Green's function at the origin are expressed as a sum of simple integrals of the complete elliptic integrals of the first kind. In order to give the expressions for all values of the variable from − ∞ to + ∞, use is made of the method of analytic continuation. The results of the numerical computations are shown by figures.
On Representations of the Conformal Group Which When Restricted to Its Poincaré or Weyl Subgroups Remain Irreducible
Authors: Mickelsson J. , Niederle J.
Published: February 2004

Unitary irreducible representations of the conformal group which, when restricted to its Poincaré or Weyl (= Poincaré group extended by dilatations) subgroups, remain irreducible are found. In particular it is proved that the continuous spin representations (pμ·pμ = 0) of the Poincaré group cannot be extended to the conformal group and that, on the other hand, a known extension in the discrete spin case is unique (up to a unitary equivalence). Similar results hold for Weyl group for which, in addition, extensions also exist in the case pμ·pμ ≠ 0. Namely, each unitary irreducible representation of the Weyl group characterized by the sign of pμ·pμ (≠0) and by invariants of the corresponding little group can be extended to a one-parameter family of irreducible representations of the conformal group. Finally, it is shown that, besides the above mentioned extensions of unitary irreducible representations of the Weyl group, there are no others.
Wave Propagation in Certain One-Dimensional Random Media
Authors: Molyneux John E.
Published: February 2004

In this paper we investigate the stochastic ordinary differential equation u″+k02[1+εy(t)]u=0 with y(t) a random process. Two specific types of process y(t) are considered. Both of these arise from a bounded mapping y(t) = f(x(t)) of a countable state space Markov process x(t). Exact equations are derived for the statistical moments of u(t), and the behavior of the first two moments is discussed in the limit of small ε. A description of the layered media to which our results apply is given and a comparison of our exact results with certain perturbation methods is made.
Electromagnetic Radiation in a Uniformly Moving, Homogeneous Medium
Authors: Johannsen Günther
Published: February 2004

A new method of treating radiation problems in a uniformly moving, homogeneous medium is presented. A certain transformation technique in connection with the four-dimensional Green's function method makes it possible to elaborate the Green's functions of the governing differential equations in the rest system of the medium, whereas the final integrals determining the field may be calculated in the rest system of the source.
The U
n,1 and IUn Representation Matrix Elements
Authors: Wolf Kurt Bernardo
Published: February 2004

We construct a realization of the Un,1 and IUn groups as multiplier representations of the space of functions on the Un group manifold. Making use of the orthogonality and completeness of the Un unitary irreducible representation matrix elements (UIRME's), we are able to express the Un,1 boost and IUn translation matrix elements (the generalized Wigner d-functions) of the principal series of UIR's as an integral over a compact domain (unit disc) of two Un d-functions, phases, and the multiplier. This is an extension to the unitary groups of a method previously used [J. Math. Phys. 12, 197 (1971)] to find the SOn, SOn,1, and ISOn UIRME's in a recursive fashion. We establish a number of symmetry properties, the asymtotic (Regge-like) and contraction (Un,1 → IUn) behavior of these functions.
Operational Statistics. I. Basic Concepts
Authors: Foulis D. J. , Randall C. H.
Published: February 2004

A generalized notion of a ``sample space'' is developed which allows for the simultaneous representation of the outcomes of a set of related ``random experiments.'' Affiliated with each such generalized sample space is a so-called ``logic,'' the elements of which are propositions that can be confirmed or refuted by observing the outcomes of the random experiments. Stochastic models for the experimental situation represented by a given sample space are introduced, and it is shown that such stochastic models induce generalized probability measures on the logic of this sample space.
Polarized Elastic Materials with Electronic Spin-A Relativistic Approach
Authors: Maugin Gérard A. , Eringen A. Cemal
Published: February 2004

In this article, we derive the balance laws and constitutive equations of polarized elastic solids with electronic spin by use of a relativistic (special) variational principle. The theory is fully dynamical and nonlinear. It is shown that this approach encompasses several previous works in micromagnetism (magnetoelastic interactions) and elastic dielectric theory.
Relativistic dynamical symmetries and dilatations
Authors: Roman P. , Aghassi J. J. , Huddleston P. L.
Published: February 2004

Characterizing the state of a relativistic particle by a pair (xμ,ξμ) of 4-vectors, we are led, in a natural way, to a group H5 of canonical transformations which includes the Poincaré group and dilatations. The structure of the group and its induced irreducible unitary representations are explored. It is shown that H5 has a semisimple noncompact subgroup which permits a systematic treatment of exact and of broken dilatation symmetry. The relevance of these ideas to scale dimension and to a new symmetry, scale conjugation, is discussed. As an application, a mass formula is derived from broken dilatation symmetry.
A kinetic theory for power transfer in stochastic systems
Authors: Papanicolaou George C.
Published: February 2004

In the asymptotic limit of weak inhomogeneities and long times or distances, we obtain a system of kinetic equations governing the power transfer among the modes of oscillation of certain stochastic dynamical systems. We include applications to coupled oscillators, waveguides, beams, the quantized motion of a particle in a random potential, and the Klein-Gordon equation with random plasma frequency.
Diffraction by an infinite array of parallel strips
Authors: Kent W. H. , Lee S. W.
Published: February 2004

The problem of diffraction of a plane wave by an infinite array of parallel strips is attacked by the newly developed modified residue calculus method. The solution is found in terms of an infinite set of zeros of an analytic function. The asymptotic behavior of the set of zeros is specified by the edge condition, while the first several zeros are determined from a matrix equation. The rapid convergence of these zeros to their asymptotic values is demonstrated through numerical examples. For a given array of strips, it is shown that there exists a total reflection phenomenon at a critical frequency and incident angle. This fact suggests the possibility of constructing an open resonator with an extremely sparse resonance frequency.
On the inverse problem for a hyperbolic dispersive partial differential equation
Authors: Weston V. H.
Published: February 2004

The inverse problem for a two-dimensional (space-time) hyperbolic partial differential equation, with coefficients, functions of the spatial variable only, is considered. Exterior to a region of compact support in the spatial variable, the equation reduces to the wave equation, and, from knowledge of the solution in the exterior region (namely in terms of reflected and transmitted waves for a prescribed incident wave), the problem is to deduce the coefficients in the interior region. This is achieved by treating the problem as a Cauchy initial value problem and using the Riemann function to deduce a dual set of integral equations. The coefficients or linear combinations of them are deduced from the solutions of the integral equations. The question of uniqueness is partially answered, by estimating the domain of convergence of the Neumann series. The application of the analysis to electromagnetic scattering from a slab of varying conductivity and permitivity is indicated.
A Statistical Mechanical Approach to the Problem of a Fluid in an External Field
Authors: Millard Kenneth
Published: February 2004

A rigorous, equilibrium statistical mechanical treatment of a fluid in a weak external field is given. The technique involves a cell division which leads to upper and lower bounds for the free energy density. Under a suitable double limiting procedure these limits coalesce, yielding a free energy consisting of a field-free term plus a field-dependent term. The cell division allows a direct physical definition of the local pressure p(s) and the local density ρ(s). This treatment provides a rigorous derivation of the thermodynamics of a fluid in a weak external field and, in particular, the hydrostatic equation gradp = − ρ gradφ.
Nonlinear Initial-Boundary-Value Problem for Convection, Diffusion, Ionization, and Recombination Processes
Authors: Wilhelm H. E.
Published: February 2004

The time-dependent convection-diffusion equation with ionization and recombination reactions is reduced by means of a nonlinear transformation to a differential equation, in which the nonlinear term represents a small perturbation. The general procedure of solution for the corresponding nonlinear initial-boundary-value problem is then established by means of the method of successive approximations. Uniqueness and convergence of the analytical solution are discussed. As applications, the temporal change of an initial distribution of electrons and ions is discussed for a finite box system and an infinitely extended system, respectively.
Colliding Plane Gravitational Waves
Authors: Szekeres P.
Published: February 2004

The equations governing the collision of two plane gravitational waves are derived. The general exact solution representing this situation when both waves are linearly polarized are found, and some special solutions of possible physical interest are discussed in detail.
Scattering from a Periodic Corrugated Structure. II. Thin Comb with Hard Boundaries
Authors: DeSanto John A.
Published: February 2004

The scattered field is calculated for plane wave incidence on a periodic rectangularly corrugated surface (thin comb grating) with hard (Neumann) boundary conditions. Except for the hard boundary (and the consequent representation of the field in the comb wells), the formalism is similar to that of a previous paper [J. Math. Phys. 12, 1913 (1971)]. Reflection coefficients are plotted, grating anomalies illustrated, and a correspondence between reflection coefficients and amplitude phases (as a function of corrugation depth) is illustrated.
Relativistic Spin-Zero Wave Equation
Authors: Lepore Joseph V. , Riddell Robert J.
Published: February 2004

We have studied the solutions of a wave equation which describes a spin-zero particle in the Coulomb field of a nucleus. An interesting feature of this equation is that the kernel is not of the Fredholm type. The behavior of the momentum space wavefunction for large momentum is not determined solely by the angular momentum state but, as in the cases of the Dirac and Klein-Gordon equations, it depends on the electric charge as well. Our analysis of the asymptotic properties is based on a Mellin transformation of the momentum space equation. This leads to a singular integral equation with a Cauchy-type kernel which may be treated by standard methods. The equation is shown to have unique solutions.