ALBERT

Notes: We consider Ising systems where all the many-spin couplings JA are positive. We show that the absolute value of all the many-spin correlations does not increase when the value of any of the couplings is reduced, taking any value in the interval [−JA,JA]. Results of this type are motivated by work in systems such as random field Ising models. © 2000 American Institute of Physics.

Notes: It is shown that when the initial condition and the forcing term of the periodic Navier–Stokes or magnetohydrodynamics equations have Fourier coefficients which vanish outside a certain semigroup of frequencies, the same happens to the solutions for all time. Subgroups of frequencies correspond to solutions possessing certain symmetries. By taking as a semigroup the frequencies whose Fourier components are non-negative integers, we get a class of solutions for which the higher modes do not influence the evolution of the lower ones; therefore, the phenomenon of inverse cascading cannot occur for them. © 2000 American Institute of Physics.

Notes: In this article we begin a systematic investigation via multiscale expansions of nonlinear evolution PDEs (partial differential equations). In this first article we restrict consideration to a single, autonomous, but otherwise generic, PDE in 1+1 variables (space+time), of first order in time, whose linear part is dispersive, and to solutions dominated by a single plane wave satisfying the linear part of the PDE. The expansion parameter is an, assumedly small, coefficient multiplying this plane wave. The main (indeed, asymptotically exact) effect of the (weak) nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics, which is generally described, in (appropriately defined) coarse-grained time and space variables, by evolution equations of nonlinear Schrödinger type. A systematic analysis of such equations is presented, corresponding to various assumptions on the "resonances" occurring for the first few harmonics. © 2000 American Institute of Physics.

Notes: Let a differential 4D-manifold with a smooth coframe field be given. Consider the operators on it that are linear in the second order derivatives and quadratic in the first order derivatives of the coframe, both with coefficients that depend on the coframe variables. The article exhibits the class of operators that are invariant under a general change of coordinates, and, also, invariant under the global SO(1,3)-transformation of the coframe. A general class of field equations is constructed. We display two subclasses in it. The subclass of field equations that are derivable from action principles by free variations and the subclass of field equations for which spherical-symmetric solutions, Minkowskian at infinity, exist. Then, for the spherical-symmetric solutions, the resulting metric is computed. Invoking the geodesic postulate, we find all the equations that are experimentally (by the three classical tests) indistinguishable from Einstein field equations. This family also includes, of course, Einstein equations. Moreover, it is shown, explicitly, how to exhibit it. The basic tool employed in the article is an invariant formulation reminiscent of Cartan's structural equations. The article sheds light on the possibilities and limitations of the coframe gravity. It may also serve as a general procedure to derive covariant field equations. © 2000 American Institute of Physics.

Notes: The zeros of the eigenfunctions of self-adjoint Sturm–Liouville eigenvalue problems interlace. For these problems interlacing is crucial for completeness. For the complex Sturm–Liouville problem associated with the Schrödinger equation for a non-Hermitian PT-symmetric Hamiltonian, completeness and interlacing of zeros have never been examined. This paper reports a numerical study of the Sturm–Liouville problems for three complex potentials, the large-N limit of a −(ix)N potential, a quasiexactly-solvable −x4 potential, and an ix3 potential. In all cases the complex zeros of the eigenfunctions exhibit a similar pattern of interlacing and it is conjectured that this pattern is universal. Understanding this pattern could provide insight into whether the eigenfunctions of complex Sturm–Liouville problems form a complete set. © 2000 American Institute of Physics.

Notes: We present a new generalized topological current in terms of the order parameter field φ to describe the arbitrary dimensional topological defects. By virtue of the φ-mapping method, we show that the topological defects are generated from the zero points of the order parameter field φ, and the topological charges of these topological defects are topological quantized in terms of the Brouwer degrees of φ-mapping under the condition that the Jacobian J(φ/v)≠0. When J(φ/v)=0, it is shown that there exist the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, we detail the bifurcation of generalized topological current and find different directions of the bifurcation. The arbitrary dimensional topological defects are found splitting or merging at the degenerate point of field function φ but the total charge of the topological defects is still unchanged. © 2000 American Institute of Physics.

Notes: Nonstandard analysis is used to construct a measure over paths for the path integral solution to the Dirac equation in 3+1 dimension. Paths are considered in momentum space, because the Green function in the configuration space contains a derivative of δ function which keeps us from assigning a measure over paths. The solution is obtained not only as the standard part of a nonstandard path sum with respect to a nonstandard measure, but also as a standard path integral with respect to a standard measure extracted from the nonstandard one. The result is an extension of Gaveau's work [B. Gaveau, J. Funct. Anal. 58, 310–319 (1984)]. © 2000 American Institute of Physics.

Notes: We present the general method to introduce the generalized Chern–Simons form and the descent equation which contain the scalar field in addition to the gauge fields. It is based on the technique in a noncommutative differential geometry (NCG) which extends the four-dimensional Minkowski space M4 to the discrete space such as M4×Z2 with two point space Z2. However, the resultant equations are not only dependent on NCG, but also are justified by the algebraic rules in the ordinary differential geometry. © 2000 American Institute of Physics.

Notes: Schrödinger's equation is generalized to a space–time four-manifold, using standard concepts from differential geometry and operator replacement. This fourth-order equation, which reduces and specializes to the Klein–Gordon equation in the flat space limit, can also be obtained from a variational principle, and must be solved in tandem with the Einstein field equations with suitable stress energy. The propagator, for large momenta, varies like 1/p4. A further attractive feature is that no external currents or stress energies need be imposed: these arise naturally. A generalization to fields with arbitrary spin is proposed. Solving the equation would lead to a determination of the mass, just as energies are found in solving Schrödinger's equation. Flat-space plane wave solutions consist of the superposition of two independent waves, which can be interpreted as propagating strings. © 2000 American Institute of Physics.

Notes: Given an Ehresmann connection γ on a fibered manifold p:E→M, a covariant Hamiltonian density Hγ is then associated to each Lagrangian density L on J1E. Assume E is the bundle of connections of a principal bundle and that L is gauge invariant. Our goal in this paper is to determine conditions on γ under which Hγ is also gauge invariant. The general conclusion is that there is no gauge-invariant Ehresmann connection but there is plenty of such connections providing gauge-invariant covariant Hamiltonians. The relevant cases of U(1) bundles and SU(2) bundles are discussed in detail. © 2000 American Institute of Physics.

Notes: We reconsider the theory of hidden variables under the assumption that the conjecture on the ensemble (experiment run) independence of the distribution of hidden variables (which was indirectly used by J. Bell and his followers) is violated. Ensemble fluctuations imply perturbations of Bell's inequality and its generalizations. We study (by experimental reasons) CHSH (Clauser, Horne, Shimony, Holt) inequality and obtain its modification. This modified inequality is not in disaccord with the predictions of quantum formalism. The deviation from the standard CHSH inequality depends on the magnitude of ensemble fluctuations. We find these magnitude for fluctuating families of Gaussian distributions. We found that if the dimension of the space of hidden variables is very high, then to obtain a contradiction between the local realism and quantum formalism, we must be sure there is no even negligibly small deviations in probability distributions of hidden variables corresponding to different runs of the experiment (in particular, the efficiency of detectors must be equal to one). © 2000 American Institute of Physics.

Notes: A new approach to the theory of anisotropic exciton based on Fock transformation, i.e., on a stereographic projection of the momentum to the unit four-dimensional (4D) sphere, is developed. Hyperspherical functions are used as a basis of the perturbation theory. The binding energies, wave functions and oscillator strengths of elongated as well as flattened excitons are obtained numerically. It is shown that with an increase of the anisotropy degree the oscillator strengths are markedly redistributed between optically active and formerly inactive states, making the latter optically active. An approximate analytical solution of the anisotropic exciton problem taking into account the angular momentum conserving terms is obtained. This solution gives the binding energies of moderately anisotropic exciton with a good accuracy and provides a useful qualitative description of the energy level evolution. © 2000 American Institute of Physics.

Notes: A general mathematical theory covering higher-order mechanical systems subject to constraints of arbitrary order (i.e., depending on time, positions, velocities, accellerations, and higher derivatives) is presented, including higher-order holonomic systems as a particular case. Within differential geometric setting on higher-order jet bundles, the concept of a mechanical system (not necessarily regular, or Lagrangian) is introduced to be a class of 2-forms equivalent with a dynamical form. Dynamics are then represented by means of corresponding exterior differential systems. Higher-order constraint structure on a fibered manifold is defined to be a submanifold endowed with a distribution (canonical distribution, higher-order Chetaev bundle). With help of a constraint structure a constraint force is naturally introduced. Higher-order mechanical systems subject to different kinds of higher-order constraints are then geometrically characterized and their dynamics are studied from a geometrical point of view. Regular and Lagrangian systems appear as important particular cases within the general scheme. © 2000 American Institute of Physics.

Notes: Within the framework of a nonlinear quantum kinetic theory for dissipative far-from-equilibrium systems, based on a nonequilibrium ensemble formalism, a rigorous derivation of the Markovian limit is given. This is done in the framework of the nonequilibrium statistical operator method, and resorting to Zubarev's approach. © 2000 American Institute of Physics.

Notes: Free quantum motion on group manifolds is considered. The Hamiltonian is given by the Laplace–Beltrami operator on the group manifold, and the purpose is to get the (Feynman's) evolution kernel Kt. The spectral expansion, which produced a series of the representation characters for Kt in the compact case, does not exist for noncompact group, where the spectrum is not bounded. In this work real analytical groups are investigated, some of which are of interest for physics. An integral representation for Kt is obtained in terms of the Green's function, i.e., the solution to the Helmholz equation on the group manifold. The alternative series expressions for the evolution operator are reconstructed from the same integral representation, the spectral expansion (when exists) and the sum over classical paths. For noncompact groups, the latter can be interpreted as the (exact) semiclassical approximation, like in the compact case. The explicit form of Kt is obtained for a number of noncompact groups. © 2000 American Institute of Physics.

Notes: The quantum mechanical transition between a free particle Lagrangian and the Klein–Gordon field description of a free particle (particle-wave duality) is conjectured to extend to an analogous construction of relativistically invariant wave equations associated with strings and branes. Electromagnetic interactions in the two systems are discussed. It is emphasized that all integrable free field theories, including those of Dirac–Born–Infeld type, are associated with Lagrangians equivalent to divergences on the space of solutions of the equations of motion. © 2000 American Institute of Physics.

Notes: Three different methods to quantize the spherically symmetric sector of electromagnetism are presented: First, it is shown that this sector is equivalent to Abelian BF-theory in four spacetime dimensions with suitable boundary conditions. This theory, in turn, is quantized by both a reduced phase space quantization and a spin network quantization. Finally, the outcome is compared with the results obtained in the recently proposed general quantum symmetry reduction scheme. In the magnetically uncharged sector, where all three approaches apply, they all lead to the same quantum theory. © 2000 American Institute of Physics.

Notes: We present a new topological tensor current of p˜-branes by making use of the φ-mapping theory. It is shown that the current is identically conserved and behaves as δ(φ(vector)), and every isolated zero of the vector field φ(vector)(x) corresponds to a "magnetic" p˜-brane. Using this topological current, the generalized Nambu action for multi p˜-branes is given, and the field strength F corresponding to this topological tensor current is obtained. It is also shown that the magnetic charges carried by p˜-branes are topologically quantized and labeled by Hopf index and Brouwer degree, the winding number of the φ mapping. © 2000 American Institute of Physics.

Notes: We give a possible generalization to the example in the paper of Zanardi and Rasetti [Phys. Lett. A 264, 94 (1999)]. For this, explicit forms of adiabatic connection, curvature, etc., are given. We also discuss the possibility of another generalization of their model. © 2000 American Institute of Physics.

Notes: Continuing our attempt to explain the degeneracy of the genetic code using basic classical Lie superalgebras, we present the branching schemes for the typical codon representations (typical 64-dimensional irreducible representations) of basic classical Lie superalgebras and find three schemes that do reproduce the degeneracies of the standard code, based on the orthosymplectic algebra osp(5|2) and differing only in details of the symmetry breaking pattern during the last step. © 2000 American Institute of Physics.

Notes: We show that it is possible to define Majorana (s)pinor fields on M-branes which have been identified under the action of the antipodal map on the adS factor of the throat geometry, or which have been wrapped on two-cycles of arbitrary genus. This is an important consistency check, since it means that one may still take the generators of supertranslations in superspace to transform as Majorana fermions under the adjoint action of Spin(10,1) even though the antipodally identified M2-brane is not space-orientable. We point out that similar conclusions hold for any p-branes which have the generic (adS)×(Sphere) throat geometry. © 2000 American Institute of Physics.

Notes: Physically relevant gauge and gravitational theories can be seen as special members of hierarchies of more elaborate systems. The Yang–Mills (YM) system is the first member of a hierarchy of Lagrangians which we will index by p1, and the Einstein–Hilbert (EH) system of general relativity is the first member of another hierarchy which we index by p2. In this paper, we study the classical equations of the p1=1,2 YM hierarchy considered in the background of special geometries (Schwarzschild, deSitter, anti-deSitter) of the p2=1,2,3 EH hierarchy. Solutions are obtained in various dimensions and lead to several examples of nonself-dual YM fields. When p1=p2 self-dual solutions exist in addition. Their action is equal to the Chern–Pontryagin charge and can be compared with that of the nonself-dual solutions. © 2000 American Institute of Physics.

Notes: The equations describing the hydrostatic equilibrium (mass continuity and Tolman–Oppenheimer–Volkoff) of a static anisotropic general relativistic fluid sphere are obtained in D (D≥4) space–time dimensions in the presence of a cosmological constant. The formalism thus developed is used to study homogeneous anisotropic constant density charged fluid spheres and homogeneous anisotropic charged spheres with a neutral isotropic core in higher dimensions. For these configurations and with a particular choice of the proper charge density a complete solution of the coupled Einstein–Maxwell equations is obtained. © 2000 American Institute of Physics.

Notes: The thermodynamical properties of toroidal black holes in the grand canonical ensemble are investigated using York's formalism. The black hole is enclosed in a cavity with finite radius where the temperature and electrostatic potential are fixed. The boundary conditions allow one to compute the relevant thermodynamical quantities, e.g., thermal energy, entropy, and specific heat. This black hole is thermodynamically stable and dominates the grand partition function. This means that there is no phase transition, such as the one encountered for spherical black holes. © 2000 American Institute of Physics.

Notes: Local solutions of the static, spherically symmetric Einstein–Yang–Mills (EYM) equations with SU(2) gauge group are studied using dynamical systems methods. This approach enables us to classify EYM solutions in a neighborhood of the origin, to prove the existence of solutions with an oscillating metric as well as the existence of local solutions for all known formal power series expansions, to study the extendibility of solutions, and to find some new local singular solutions. © 2000 American Institute of Physics.

Notes: This paper gives the Darboux transformations for differential–difference integrable systems and their applications to differential–difference principal chiral equation. We also give some exact solutions of principal chiral equation corresponding to the harmonic map R1+1→SU(n) by using two kinds of limits. © 2000 American Institute of Physics.

Notes: We recently introduced two new sixth-order partial differential equations (PDEs) associated with third-order scattering problems. Here we extend our study of these PDEs by considering the construction of exact solutions both by using the method of symmetry reduction due to Lie, and by using their Darboux transformations (DTs). Amongst the ordinary differential equations (ODEs) obtained by symmetry reduction is an ODE due to Cosgrove that is believed to define a new Painlevé transcendent. This ODE provides soliton solutions for our integrable PDEs that include arbitrary functions of time. The DTs for our PDEs allow the recovery of these solutions and in addition provide other solutions which are not associated with Lie symmetries (either classical or nonclassical). We also consider the iteration of the corresponding Bäcklund transformations (BTs) for these PDEs. The theorem of permutability allows us to reduce this process of iterating the DT from one of solving a third-order linear equation (the spatial part of the Lax pair) to that of solving either a second-order linear equation (for one PDE), or quite remarkably to that of solving a first-order linear equation (for the other PDE). These linear differential equations have coefficients involving three previous solutions of the PDE, and are a natural extension of the linear algebraic equation found by applying the theorem of permutability to the Korteweg–de Vries equation. © 2000 American Institute of Physics.

Notes: We show that α〈0, β〉0, γ=δ=0 Painlevé III equation arises as a zero-curvature condition in the Belinskii–Zakharov inverse scattering formulation for Bianchi cosmological models. For special values of the parameters this Painlevé III equation becomes the dynamical equation for Bianchi I, II, VI0, and VII0 models. © 2000 American Institute of Physics.

Notes: We develop a systematic method to construct induced representations of quantum algebras. The procedure makes use of two Hopf algebras with a nondegenerate pairing and a pair of dual bases for them. We apply the method on three different quantum deformations of the Galilei algebra in (1+1) dimensions. We obtain several families of induced representations including some results already known. © 2000 American Institute of Physics.

Notes: We obtain in this paper a precise estimate of upper bound of the Hausdorff dimension of the global attractor for a damped semilinear wave equation with critical exponent. The obtained Hausdorff dimension decreases as the damping grows and is uniformly bounded for large damping, which conforms to physical intuition. The obtained results generalize and correct the corresponding results obtained by Zhou recently. © 2000 American Institute of Physics.

Notes: We construct a Kac–Moody superalgebra L as the minimal graded Lie superalgebra with local part V*⊕ge⊕V, where g is a "smaller" Lie superalgebra inside L, V is an irreducible highest weight g-module, and V* is the contragredient of V. We show that the weight multiplicities of irreducible highest weight modules over Kac–Moody superalgebras of finite type and affine type [more precisely, Kac–Moody superalgebras of type B(0,r), B(1)(0,r), A(4)(2r,0), A(2)(2r−1,0), and C(2)(r+1)] are given by polynomials in the rank r. The degree of these weight multiplicity polynomials are less than or equal to the depth of weights. © 2000 American Institute of Physics.

Notes: In this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with constructive methods well into the 21st century. © 2000 American Institute of Physics.

Notes: In fundamental physics, this has been the century of quantum mechanics and general relativity. It has also been the century of the long search for a conceptual framework capable of embracing the astonishing features of the world that have been revealed by these two "first pieces of a conceptual revolution." The general requirements on the mathematics and some specific developments toward the construction of such a framework are discussed. Examples of covariant constructions of (simple) generally relativistic quantum field theories have been obtained as topological quantum field theories, in nonperturbative zero-dimensional string theory and its higher-dimensional generalizations, and as spin foam models. A canonical construction of a general relativistic quantum field theory is provided by loop quantum gravity. Remarkably, all these diverse approaches have turned out to be related, suggesting an intriguing general picture of general relativistic quantum physics. © 2000 American Institute of Physics.

Notes: In this paper we are concerned with singularly perturbed variational problems involving the curl functional, which arise in the mathematical theory of liquid crystals. The asymptotic behavior of the minimizers in the singular limiting process is discussed, which is closely related to the variational problems for curl functional under various constraints. © 2000 American Institute of Physics.

Notes: We prove that an m-dimensional unit ball Dm in the Euclidean space Rm cannot be isometrically embedded into a higher-dimensional Euclidean ball Brd⊂Rd of radius r〈1/2 unless one of two conditions is met: (1) the embedding manifold has dimension d≥2m; (2) the embedding is not smooth. The proof uses differential geometry to show that if d〈2m and the embedding is smooth and isometric, we can construct a line from the center of Dm to the boundary that is geodesic in both Dm and in the embedding manifold Rd. Since such a line has length 1, the diameter of the embedding ball must exceed 1. © 2000 American Institute of Physics.

Notes: In a study of finite-dimensional modules of simple Lie superalgebras, Kac introduced certain indecomposable modules, now known as Kac-modules V¯(Λ), which are simple if and only if Λ is typical. For Λ atypical, Hughes et al. presented an algorithm to determine all the composition factors of the Kac-module; they conjectured that there exists a bijection between the composition factors of a Kac-module and so-called permissible codes. The aim in this paper is to contribute to the proof of this conjecture. By constructing explicitly the primitive vector, we prove that for any unlinked code there corresponds a composition factor of the Kac-module. It will be proved in another paper that to any linked code there also corresponds a composition factor of the Kac-module. Thus the proof of the Hughes et al. conjecture will be reduced to the problem whether or not each composition factor corresponds to a linked or unlinked code. © 2000 American Institute of Physics.

Notes: © 2000 American Institute of Physics.

Notes: © 2000 American Institute of Physics.

Notes: Z2-graded Schwinger terms for neutral particles in 1 and 3 space dimensions are considered. © 2000 American Institute of Physics.

Notes: We prove that for any two commuting von Neumann algebras of infinite type, the open set of Bell correlated states for the two algebras is norm dense. We then apply this result to algebraic quantum field theory—where all local algebras are of infinite type—in order to show that for any two spacelike separated regions, there is an open dense set of field states that dictate Bell correlations between the regions. We also show that any vector state cyclic for one of a pair of commuting non-Abelian von Neumann algebras is entangled (i.e., nonseparable) across the algebras—from which it follows that every field state with bounded energy is entangled across any two spacelike separated regions. © 2000 American Institute of Physics.

Notes: For the model formally expressed as H{R}=HD+∑m=1Nαmδ(|x|−Rm); x∈R3,αm∈R, Rm〉0, we give a precise mathematical definition, provide basic properties and main scattering elements. © 2000 American Institute of Physics.

Notes: We analyze the proof of Bell's inequality and demonstrate that this inequality is related to one particular model of probability theory, namely Kolmogorov measure-theoretical axiomatics from 1933. We found a (numerical) statistical correction to Bell's inequality. Such an additional term εφ on the right-hand side of Bell's inequality can be considered as a probability invariant of a quantum state φ. This is a measure of nonreproducibility of hidden variables in different runs of experiments. Experiments to verify Bell's inequality can be considered as just experiments to estimate the constant εφ. It seems that Bell's inequality could not be used as a crucial reason to deny local realism. We consider deterministic as well as stochastic hidden variables models. © 2000 American Institute of Physics.

Notes: Motivated by a recent analysis that presents explicitly the general solution, we consider the eigenvalue problem of the spinless Salpeter equation with a ("hard-core amended") Coulomb interaction potential in one dimension. We prove the existence of a critical coupling constant (which contradicts the assertions of the previous analysis) and give analytic upper bounds on the energy eigenvalues. These upper bounds seem to disprove the previous explicit solution. © 2000 American Institute of Physics.

Notes: Noncommutative geometry (NCG) on a discrete space successfully reproduces the Higgs mechanism of the spontaneously broken gauge theory, in which the Higgs boson field is regarded as a kind of gauge field on the discrete space. People also know how to construct the generalized differential geometry (a GDG) as one version of NCG on a discrete space M4×ZN where ZN denotes the N points discrete space. A GDG is a direct generalization of the differential geometry on the ordinary manifold into the discrete one. In this paper, we attempt to construct the Becchi–Rouet–Stora–Tyutin (BSRT) invariant formulation of spontaneously broken gauge theory based on a GDG and obtain the BSRT invariant Lagrangian with the 't Hooft–Feynman gauge fixing term. © 2000 American Institute of Physics.

Notes: The group theoretical quantization scheme is reconsidered by means of elementary systems. Already the quantization of a particle on a circle shows that the standard procedure has to be supplemented by an additional condition on the admissibility of group actions. A systematic strategy for finding admissible group actions for particular subbundles of cotangent spaces is developed, two-dimensional prototypes of which are T*R+ and S=S1×R+ (interpreted as restrictions of T*R and T*S1 to positive coordinate and momentum, respectively). In this framework (and under an additional, natural condition) an SO↑(1,2)-action on S results as the unique admissible group action. Furthermore, for symplectic manifolds which are (specific) parts of phase spaces with known quantum theory a simple "projection method" of quantization is formulated. For T*R+ and S equivalent results to those of more established (but more involved) quantization schemes are obtained. The approach may be of interest, e.g., in attempts to quantize gravity theories where demanding nondegenerate metrics of a fixed signature imposes similar constraints. © 2000 American Institute of Physics.

Notes: In light of the φ-mapping method and the topological tensor current theory, the topological structure and the topological quantization of topological defects are obtained under the condition that Jacobian J(φ/v)≠0. When J(φ/v)=0, it is shown that there exists the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the generation, annihilation, and bifurcation of the linear defects are detailed in the neighborhoods of the limit points and bifurcation points of φ-mapping, respectively. © 2000 American Institute of Physics.

Notes: It is shown how W-algebras emerge from very peculiar canonical transformations with respect to the canonical symplectic structure on a compact Riemann surface. The action of smooth diffeomorphisms of the cotangent bundle on suitable generating functions is written in the BRS framework while a W-symmetry is exhibited. Subsequently, the complex structure of the symmetry spaces is studied and the related BRS properties are discussed. The specific example of the so-called W3-algebra is treated in relation to some other different approaches. © 2000 American Institute of Physics.

Notes: A new method of the constructing of the basic functions for spaces of tensor representations of the Lie groups, with the help of the generalized Casimir operator, has been proposed. In the definition of the operator there we used the Lie derivatives instead of the corresponding infinitesimal operators. When introducing the generalized Casimir operator we used the metric for which a group being considered will be isometry that follows from the invariance condition for the generalized Casimir operator. This allows us to formulate the eigenvalue and eigenfunction problems correctly. The invariant projection operators have been constructed in order to separate irreducible components. The cases of the Bianchi type G3IX and G3II groups are considered as examples. © 2000 American Institute of Physics.

Notes: The theory of dynamical but non-Cartan (or non-Noether) symmetries and the existence of bi-Hamiltonian structures is studied using the symplectic formalism approach. The results are applied to the study of superintegrable systems. It is shown that certain families of n=2 superintegrable systems related with the harmonic oscillator (as, e.g., the so-called Smorodinsky–Winternitz system) are bi-Hamiltonian systems endowed with dynamical symmetries of non-Cartan class. © 2000 American Institute of Physics.

Notes: In the present paper we consider a system of degenerate Davey–Stewartson equations. We prove the global existence of weak solutions and blow-up of solutions for appropriate initial data. © 2000 American Institute of Physics.

Notes: This is the first paper of a series in which we give some generalizations of the Dereli–Obukhov–Tucker–Wang theorem in the Tucker–Wang approach to metric-affine gravity in which we consider more general actions containing scalar and in general fields which do not depend on the metric or connection. © 2000 American Institute of Physics.

Notes: A simple expression is derived for the terms in the Baker–Campbell–Hausdorff series. One formulation of the result involves a finite number of operations with matrices of rational numbers. Generalizations are discussed. © 2000 American Institute of Physics.

Notes: We consider the mixing properties of the Swendsen–Wang process for the two-state Potts model or Ising model, on the complete n vertex graph Kn and for the Q-state model on an a×n grid where a is bounded as n→∞. © 2000 American Institute of Physics.

Notes: Schlesinger transformations are discrete monodromy-preserving symmetry transformations of the classical Schlesinger system. Generalizing well-known results from the Riemann sphere we construct these transformations for isomonodromic deformations on genus one Riemann surfaces. Their action on the system's tau-function is computed and we obtain an explicit expression for the ratio of the old and the transformed tau-function. © 2000 American Institute of Physics.

Notes: We introduce a class of currents which allows a new and very explicit form for the Massey product of a third order link as a line integral. The explicit form permits the introduction of an asymptotic Massey product analogous to that introduced previously for Gauss's integral by V. Arnold. The average third order asymptotic Massey product is shown to be equal to Berger's third order helicity for divergence-free vector fields in linked tori. © 2000 American Institute of Physics.

Notes: Recent work by Lam on the decomposition of time-ordered products appearing in the time-evolution operator in terms of sums of products of nested commutators is endowed with a recursive algorithm. A sufficient condition for absolute convergence is subsequently obtained. Connection with Magnus expansion is established. © 2000 American Institute of Physics.

Notes: Statistical properties of eigenvectors in non-Hermitian random matrix ensembles are discussed, with an emphasis on correlations between left and right eigenvectors. Two approaches are described. One is an exact calculation for Ginibre's ensemble, in which each matrix element is an independent, identically distributed Gaussian complex random variable. The other is a simpler calculation using N−1 as an expansion parameter, where N is the rank of the random matrix: this is applied to Girko's ensemble. Consequences of eigenvector correlations which may be of physical importance in applications are also discussed. It is shown that eigenvalues are much more sensitive to perturbations than in the corresponding Hermitian random matrix ensembles. It is also shown that, in problems with time evolution governed by a non-Hermitian random matrix, transients are controlled by eigenvector correlations. © 2000 American Institute of Physics.

Notes: Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:ZN→RM, N≤M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Lévy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between "transformations" and "discretizations" is also investigated for quadrilateral lattices. We finally interpret these results within the ∂¯ formalism. © 2000 American Institute of Physics.

Notes: This paper is concerned with the one-dimensional stationary linear Wigner equation, a kinetic formulation of quantum mechanics. Specifically, we analyze the well-posedness of the boundary value problem on a slab of the phase space with given inflow data for a discrete-velocity model. We find that the problem is uniquely solvable if zero is not a discrete velocity. Otherwise one obtains a differential-algebraic equation of index 2 and, hence, the inflow data make the system overdetermined. © 2000 American Institute of Physics.

Notes: We establish a one-to-one correspondence between the "composite particles" with N particles and the Young tableaux with at most N rows. We apply this correspondence to the models of Calogero–Sutherland and Ruijsenaars–Schneider and we obtain a momentum space representation of the "composite particles" in terms of creation operators attached to the Young tableaux. Using the technique of bosonization, we obtain a position space representation of the "composite particles" in terms of products of vertex operators. In the special case where the "composite particles" are bosons and if we add one extra quasiparticle or quasihole, we construct the ground state wave functions corresponding to the Jain series ν=p/(2np±1) of the fractional quantum Hall effect. © 2000 American Institute of Physics.

Notes: An expression for the curvature of the "covariant" determinant line bundle is given in even-dimensional space–time. The usefulness is guaranteed by its prediction of the covariant anomaly and Schwinger term. It allows a parallel derivation of the consistent anomaly and Schwinger term, and their covariant counterparts, which clarifies the similarities and differences between them. © 2000 American Institute of Physics.

Notes: A recently developed method [Semkat et al., Phys. Rev. E 59, 1557 (1999); Kremp et al., in Progress in Nonequilibrium Green's Functions (World Scientific, Singapore, 2000), p. 34] for incorporating initial binary correlations into the Kadanoff–Baym equations (KBE) is used to derive a generalized T-matrix approximation for the self-energies. It is shown that the T-matrix obtains additional contributions arising from initial correlations. Using these results and taking the time-diagonal limit of the KBE, a generalized quantum kinetic equation in binary collision approximation is derived. This equation is a far-reaching generalization of Boltzmann-type kinetic equations: It self-consistently includes memory effects (retardation, off-shell T-matrices) as well as many-particle effects (damping, in-medium T-matrices) and spin-statistics effects (Pauli-blocking). © 2000 American Institute of Physics.

Notes: A theory is developed of product integrals of the form ∏a〈s〈b ∏c〈t〈d(1+g[h] (ds,dt)). Here [a,b[ and [c,d[ are disjoint finite subintervals of R+, and g[h] is a formal power series in the indeterminate h whose constant term is zero and whose coefficients are elements of L⊗L, where L is the space of basic differentials of a multidimensional quantum stochastic calculus. The product integrals are themselves formal power series in h whose coefficients are finite sums of iterated stochastic integrals against the elements of L. They are symmetrized in such a way that ∏a〈s〈b ∏c〈t〈d(1+g[h](ds,dt)) is the image, obtained by applying the representation J[a,b[⊗J[c,d[ to the coefficients, where J[a,b[ is the representation canonically associated with the interval [a,b[, of a formal power series ∏ ∏(1+dg[h]) whose coefficients lie in U⊗U, where U is the universal enveloping algebra of the Lie algebra L. It is shown that the naturally conjectured multiplication rule, analogous to the multiplication rule for simple product integrals, holds in the commutative case. © 2000 American Institute of Physics.

Notes: To get the similarity solutions of a nonlinear physical equation, one may use the classical Lie group approach, nonclassical Lie group approach and the Clarkson and Kruskal (CK) direct method. In this paper the direct method is modified to get the similarity and conditional similarity reductions of a (2+1) dimensional KdV-type equation. Ten types of usual similarity reductions [including the (1+1)-dimensional shallow water wave equation and the variable KdV equation] and six types of conditional similarity reductions of the (2+1)-dimensional KdV equation are obtained. Some special solutions of the conditional similarity reduction equations are found to show the nontriviality of the conditional similarity reduction approach. The conditional similarity solutions cannot be obtained by using the nonclassical Lie group approach in its present form. How to modify the nonclassical Lie group approach to obtain the conditional similarity solutions is still open. © 2000 American Institute of Physics.

Notes: We obtain all the irreducible not similar central quaternionic projective representations of a finite group G from the knowledge of the vector, complex irreducible representations of suitable extensions of G, and classify them according to a generalized Frobenius–Schur classification. Finally, a definition of equivalence of such representations is proposed. © 2000 American Institute of Physics.

Notes: Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g., 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. We review the concepts of (i) Hadamard "partie finie" of such functions at the location of singular points, (ii) the partie finie of their divergent integral. We present and investigate different expressions, useful in applications, for the latter partie finie. To each singular function, we associate a partie-finie (Pf) pseudo-function. The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. We construct a delta-pseudo-function on the class of singular functions, which reduces to the usual notion of Dirac distribution when applied on smooth functions with compact support. We introduce and analyze a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative. This operator is uniquely defined up to an arbitrary numerical constant. Time derivatives and partial derivatives with respect to the singular points are also investigated. In the course of the paper, all the formulas needed in the application to the physical problem are derived. © 2000 American Institute of Physics.

Notes: We present an elementary geometric method to derive a full quantum mechanical description of the spin of a spin-1/2 particle from the only requirement that no two different orthonormal bases in R3 could be distinguished from each other by any spin measurement. Further, this method is used to obtain a generalization of Wigner's theorem on symmetry operations for the special Hilbert space C2. © 2000 American Institute of Physics.

Notes: We study the Dirac sea in the presence of external chiral and scalar/pseudoscalar potentials. In preparation, a method is developed for calculating the advanced and retarded Green's functions in an expansion around the light cone. For this, we first expand all Feynman diagrams and then explicitly sum up the perturbation series. The light-cone expansion expresses the Green's functions as an infinite sum of line integrals over the external potential and its partial derivatives. The Dirac sea is decomposed into a causal and a noncausal contribution. The causal contribution has a light-cone expansion which is closely related to the light-cone expansion of the Green's functions; it describes the singular behavior of the Dirac sea in terms of nested line integrals along the light cone. The noncausal contribution, on the other hand, is, to every order in perturbation theory, a smooth function in position space. © 2000 American Institute of Physics.

Notes: It is shown that there is a false assumption hidden in the description of a relaxed state with inhomogeneous boundary conditions as the vector sum of a potential field, satisfying the boundary conditions, and a sum of eigenfunctions of the associated eigenvalue problem expanded by certain coefficients. In particular, although the Jensen and Chu formula (1984) can provide the correct expansion coefficients, it contains an implicit paradox in its derivation according to a general vector theorem. The same paradox led Chu et al. (1999) to be concerned about a contradiction obtained by taking the curl of their magnetic field expansion which, if permitted, becomes inconsistent with a current normal to the surface. The assumption that the curl can be commuted across an infinite sum of terms is the mechanism leading to these, apparently paradoxical, conclusions. Two mechanisms for resolving this apparent paradox are possible, one of which will be described in some detail below and the other discussed further in a forthcoming, more theoretical paper (Laurence et al., 2000). The decomposition of the magnetic field above is valid with convergence in the mean squared sense, but a decomposition of the current needs to be reinterpreted in terms of negative Sobolev spaces. To avoid this, and remain in a more easily managable and familiar setting, we derive the expansion coefficients in a way that involves the commuting of the inverse curl (as opposed to the curl) and the series. The resulting series converges in a mean square sense. When this is done the calculation can conform to the general vector theorem and a new gauge-invariant expression for the coefficients is obtained. However the consequence of the non-commutability is nullified in the Jensen and Chu formula, in both simply and multiply connected domains, by the important extra requirement of a boundary condition on the vector potential eigenfunctions; this excludes magnetic field eigenfunctions that carry flux, but there remains a complete set for the expansion and all flux is carried by the potential field. The two formulas are then identical. On a different issue, it is shown that if the general expansion is taken over a half-space, by combining positive and negative eigenvalue terms, then the coefficients are anisotropic, that is they are tensors except when evaluated at the first eigenvalue. A specific example is presented to illustrate the situation and to validate the new method of deriving the coefficients. © 2000 American Institute of Physics.

Notes: By decomposing the Riemann curvature into electric and magnetic parts, we define the gravoelectric duality transformation by interchange of active and passive electric parts which amounts to interchange of the Ricci and Einstein tensors. It turns out that the vacuum equation is duality invariant. We obtain solutions dual to the Kerr solution by writing an effective vacuum equation in such a way that it still admits the Kerr solution but is not duality invariant. The dual equation is then solved to obtain the dual-Kerr solution which can be interpreted as the Kerr black hole sitting in a string dust universe. © 2000 American Institute of Physics.

Notes: We explore the geometry and asymptotics of extended Racah coefficients. The extension is shown to have a simple relationship to the Racah coefficients for the positive discrete unitary representation series of SU(1,1) which is explicitly defined. Moreover, it is found that this extension may be geometrically identified with two types of Lorentzian tetrahedra for which all the faces are timelike. The asymptotic formulas derived for the extension are found to have a similar form to the standard Ponzano–Regge asymptotic formulas for the SU(2) 6j symbol and so should be viable for use in a state sum for three dimensional Lorentzian quantum gravity. © 2000 American Institute of Physics.

Notes: We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorizable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class. © 2000 American Institute of Physics.

Notes: The paper is devoted to the mathematical foundation of quantum tomography using the theory of square-integrable representations of unimodular Lie groups. © 2000 American Institute of Physics.

Notes: Simple phase-integral formulas, not involving wave functions, are derived for quantal matrix elements associated with bound states of a quantal particle in a smooth single-well potential. In these formulas one uses an arbitrary order of the phase-integral approximation generated from an unspecified base function. © 2000 American Institute of Physics.

Notes: The connection between the Schrödinger and confluent hypergeometric equations is discussed. It is shown that the factorization of the confluent hypergeometric equation gives a unifying powerful algebraic tool in order to study some quantum mechanical eigenvalue problems. That description includes the linear and N-dimensional harmonic oscillators, as well as the Coulomb and Morse potentials. © 2000 American Institute of Physics.

Notes: We investigate ideal quantum gases in D-dimensional space and confined in a generic external potential by using the semiclassical approximation. In particular, we derive density of states, density profiles and critical temperatures for Fermions and Bosons trapped in isotropic power-law potentials. From such results, one can easily obtain those of quantum gases in a rigid box and in a harmonic trap. Finally, we show that the Bose–Einstein condensation can set up in a confining power-law potential if and only if D/2+D/n〉1, where D is the space dimension and n is the power-law exponent. © 2000 American Institute of Physics.

Notes: We use a continuous-time path integral to obtain the semiclassical propagator for minimal-spread spin coherent states. We pay particular attention to the "extra phase" discovered by Solari and Kochetov, and show that this correction is related to an anomaly in the fluctuation determinant. We show that, once this extra factor is included, the semiclassical propagator has the correct short time behavior to O(T2), and demonstrate its consistency under dissection of the path. © 2000 American Institute of Physics.

Notes: In this paper we propose a new supersymmetric extension of conformal mechanics. The Grassmannian variables that we introduce are the basis of the forms and of the vector fields built over the symplectic space of the original system. Our supersymmetric Hamiltonian itself turns out to have a clear geometrical meaning being the Lie derivative of the Hamiltonian flow of conformal mechanics. Using superfields we derive a constraint which gives the exact solution of the supersymmetric system in a way analogous to the constraint in configuration space which solved the original nonsupersymmetric model. Besides the supersymmetric extension of the original Hamiltonian, we also provide the extension of the other conformal generators present in the original system. These extensions also have a supersymmetric character being the square of some Grassmannian charge. We build the whole superalgebra of these charges and analyze their closure. The representation of the even part of this superalgebra on the odd part turns out to be integer and not spinorial in character. © 2000 American Institute of Physics.

Notes: In this article we study manifolds with C0-metrics and properties of Lorentzian multiply warped products. We represent the interior Schwarzschild space–time as a multiply warped product space–time with warping functions and we also investigate the curvature of a multiply warped product with C0-warping functions. We give the Ricci curvature in terms of f1, f2 for the multiply warped products of the form M=(0,2m)×f1R1×f2S2. © 2000 American Institute of Physics.

Notes: The significant problem with using point groups is the dependence upon extensively tabulated results. We present simple algebraic expressions for the primitive characters, matrix irreps, symmetry adapted functions (SAFs), and Clebsch–Gordan coefficients for all dihedral groups, Dn, Cnv, Dnd, and Dnh. Those results, for arbitrary n and for single- and double-valued representations, have been derived in a simple manner without using group tables. Previously incomplete tabulated results are now redundant. In particular the parity dependence of the SAFs of the improper dihedral groups is shown analytically. Simple relations are derived between the SAFs of the proper and improper dihedral groups, so that the SAFs of the latter can be easily obtained from the SAFs of Dn. © 2000 American Institute of Physics.

Notes: In this paper, the local existence and blow-up for an initial boundary value problem of the Camassa–Holm equation are obtained. © 2000 American Institute of Physics.

Notes: Quantum stochastic differential equations have been used to describe the dynamics of an atom interacting with the electromagnetic field via absorption/emission processes. Here, by using the full quantum stochastic Schrödinger equation proposed by Hudson and Parthasarathy, we show that such models can be generalized to include other processes into the interaction. In the case of a two-level atom we construct a model on the basis of some physical requirements, the main being a balance equation on the fluxes of the ingoing and outgoing photons; in this model the atom-field interaction turns out to be due either to absorption/emission processes either to direct scattering processes, which simulate the interaction due to virtual transitions to the levels which have been eliminated from the description. To see the effects of the new terms, we consider both direct and heterodyne detection of the fluorescence light emitted by an atom stimulated by a monochromatic coherent laser and we deduce from these two detection schemes the expressions of the total, elastic and inelastic electromagnetic cross sections and the spectral distribution of the fluorescence light. The total cross section, as a function of the frequency of the stimulating laser, can present not only a Lorentzian shape, but the full variety of Fano profiles; intensity dependent widths and shifts are obtained. The fluorescence spectrum can present complicated shapes, according to the values of the various parameters; when the direct scattering is not important the usual symmetric triplet structure of the Mollow spectrum appears (for high intensity of the stimulating laser), while a strong contribution of the direct scattering process can distort such a triplet structure or can even make it to disappear. © 2000 American Institute of Physics.

Notes: Geometric models of quantum relativistic rotating oscillators in arbitrary dimensions are defined on backgrounds with deformed anti-de Sitter metrics. It is shown that these models are analytically solvable, deriving the formulas of the energy levels and corresponding normalized energy eigenfunctions. An important property is that all these models have the same nonrelativistic limit, namely the usual harmonic oscillator. © 2000 American Institute of Physics.

Notes: In this paper we investigate various properties of phase observables which could serve to determine the canonical phase observable among the family of all phase observables. We also show that any contractive weighted shift operator defines a unique phase observable, and we characterize phase observables that give the most accurate phase distribution in coherent states in the classical limit. © 2000 American Institute of Physics.

Notes: In this paper we investigate the kinetic foundation of extended irreversible thermodynamics via the moment method. First we consider the construct of the 1-particle distribution function f in terms of its moments by maximizing the entropy density function. We then project f from its L2 space onto the local thermodynamic variables z=(z1,...,zN) in the thermodynamic base space B(circumflex)N. Thus instead of the Boltzmann equation we consider a set of evolution equations of z in B(circumflex)N. Second, we formulate the laws of thermodynamics governing the variable z in B(circumflex)N. These laws exhibit an intrinsic geometric structure of thermodynamics in the setting of contact geometry. Finally, as an illustration, we discuss the evolution equations for the bulk pressure Pb, heat flux Q, and the symmetric traceless tensor π(two down arrows) corresponding to the viscous and heat conduction irreversible processes. These equations can be formulated as an abstract inhomogeneous hyperbolic evolution equation. By employing the C0 semigroup technique, we discuss the solution of the evolution equation and its asymptotic behavior. We show that thermodynamic stability condition of the system implies asymptotic dynamical stability of the solution and vice versa. © 2000 American Institute of Physics.

Notes: In the jet bundle description of field theories (multisymplectic models, in particular), there are several choices for the multimomentum bundle where the covariant Hamiltonian formalism takes place. As a consequence, several proposals for this formalism can be stated, and, on each one of them, the differentiable structures needed for setting the formalism are obtained in different ways. In this work we make an accurate study of some of these Hamiltonian formalisms, showing their equivalence. In particular, the geometrical structures (canonical or not) needed for the Hamiltonian formalism, are introduced and compared, and the derivation of Hamiltonian field equations from the corresponding variational principle is shown in detail. Furthermore, the Hamiltonian formalism of systems described by Lagrangians is performed, both for the hyper-regular and almost-regular cases. Finally, the role of connections in the construction of Hamiltonian field theories is clarified. © 2000 American Institute of Physics.

Notes: We revisit Lewis solution giving a mechanical interpretation to its field equations. The structure of the equations that the metric coefficients satisfy can be associated to the motion of a classical particle in a central field. © 2000 American Institute of Physics.

Notes: We present simple explicit coordinate transformations which serve to decouple the Schrödinger equation for a pair of (not necessarily identical) harmonic oscillators in the presence of bilinear perturbing potentials. We derive general conditions for the decoupling, and give some examples of physical interest. These include the much studied example with just a static perturbation, the parallel problem with a dynamic coupling term, and the classic example of an isotropic two-dimensional oscillator in a transverse magnetic field, first solved by Fock (1928) by separation of variables. © 2000 American Institute of Physics.

Notes: In this paper we study the Bardeen–Cooper–Schrieffer energy gap equation at finite temperatures. When the kernel is positive representing a phonon-dominant phase in a superconductor, the existence and uniqueness of a gap solution is established in a class which contains solutions obtainable from bounded domain approximations. The critical temperatures that characterize superconducting–normal phase transitions realized by bounded domain approximations and full space solutions are also investigated. It is shown under some sufficient conditions that these temperatures are identical. In this case the uniqueness of a full space solution follows directly. We will also present some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs is also considered. It is shown that, at low temperatures, there exist nonzero gap solutions indicating a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition temperature. © 2000 American Institute of Physics.

Notes: In this article we describe the classification of integrable symmetrically coupled potential KdV and modified KdV-type equations that possess higher symmetries. Restricting our attention to the systems that cannot be decoupled by a change of dependent variables, we obtain 11 previously unknown classes of integrable equations. In some cases we present Hamiltonian or bi-Hamiltonian formulations. © 2000 American Institute of Physics.

Notes: Properties of idealized, two-dimensional current distributions on circular loops are investigated analytically via the solution of a constrained optimization problem. The directivity in the far field is maximized under a fixed C=N/T, where N is the integral of the square of the current magnitude and T is the total radiated power. C enters the ensuing Fourier series for the current implicitly through a Lagrange multiplier α. For non-negative α and large electrical radius ka, the directivity and the current are evaluated approximately via combined use of the Poisson summation formula and the Mellin transform technique. As a result, a geometrical-ray representation for the current is derived for the case of directivities that are slightly larger than that of the uniform distribution. The analysis indicates certain advantages of large radiating structures for moderate values of the constraint C. In the limit C→∞ of Oseen's "Einstein needle radiation," an asymptotic formula for the directivity is obtained. Possible extensions of these results to classes of smooth convex loops are briefly discussed. © 2000 American Institute of Physics.

Notes: A theorem of equivalence regarding the weak discontinuity of the solutions (u˜(1),u˜(2),...,u˜(n),u˜) of an underdetermined system of n quasi-linear partial differential equations in one spatial dimension is proven. The theorem demonstrates the theoretical existence of a smooth wave and a weak-discontinuous wave in a generalized unsteady wave, which consists of a shock wave, new (n+1) elementary waves, and a rarefaction wave. A u˜(j)−u˜(i) path has anomalous characteristics such as a peak, an inflection, and a discontinuity in slope. © 2000 American Institute of Physics.

Notes: In this paper we study the behavior of the moments of a particle distribution as it is transported through a Hamiltonian system. Functions of moments that remain invariant for an arbitrary nonlinear Hamiltonian system are constructed perturbatively. The perturbation expansion is carried out according to the degree of nonlinearity of the symplectic mapping associated with the Hamiltonian. The existence of these moment invariants is entirely a consequence of the symplectic nature of the flow generated by Hamiltonian systems. These invariants may be of use in accelerator design, in the general area of charged particle beam transport, and in other areas of Hamiltonian dynamics. © 2000 American Institute of Physics.

Notes: In this article we use one-dimensional nonlinear Schrödinger equations (NLS) to illustrate chaotic and turbulent behavior of nonlinear dispersive waves. It begins with a brief summary of properties of NLS with focusing and defocusing nonlinearities. In this summary we stress the role of the modulational instability in the formation of solitary waves and homoclinic orbits, and in the generation of temporal chaos and of spatiotemporal chaos for the nonlinear waves. Dispersive wave turbulence for a class of one-dimensional NLS equations is then described in detail—emphasizing distinctions between focusing and defocusing cases, the role of spatially localized, coherent structures, and their interaction with resonant waves in setting up the cycles of energy transfer in dispersive wave turbulence through direct and inverse cascades. In the article we underline that these simple NLS models provide precise and demanding tests for the closure theories of dispersive wave turbulence. In the conclusion we emphasize the importance of effective stochastic representations for the prediction of transport and other macroscopic behavior in such deterministic chaotic nonlinear wave systems. © 2000 American Institute of Physics.

Notes: A method for representing probabilistic aspects of quantum systems by means of a density function on the space of pure quantum states is introduced. In particular, a maximum entropy argument allows us to obtain a natural density function that only reflects the information provided by the density matrix. This result is applied to derive the Shannon entropy of a quantum state. The information theoretic quantum entropy thereby obtained is shown to have the desired concavity property, and to differ from the conventional von Neumann entropy. This is illustrated explicitly for a two-state system. © 2000 American Institute of Physics.

Notes: In this paper we construct generalizations to spheres of the well-known Levi-Civita, Kustaanheimo–Steifel, and Hurwitz regularizing transformations in Euclidean spaces of dimensions two, three, and five. The corresponding classical and quantum mechanical analogs of the Kepler–Coulomb problem on these spheres are discussed. © 2000 American Institute of Physics.

Notes: A method for calculating the relativistic path integral solution via sum over perturbation series is given. As an application the exact path integral solution of the relativistic Aharonov–Bohm–Coulomb system is obtained by the method. Different from the earlier treatment based on the space–time transformation and infinite multiple-valued trasformation of Kustaanheimo–Stiefel in order to perform path integral, the method developed in this contribution involves only the explicit form of a simple Green's function and an explicit path integral is avoided. © 2000 American Institute of Physics.