ALBERT

Notes: The origin of and distinction between the two deformation techniques currently in use for single particle paraoscillators is made clear. For paraoscillators deformed using the more familiar method, algebra deformation, (i) a single deformation function unifying parabosons and parafermions is provided and (ii) Fock space representations of all single paraoscillator systems as bilinear combinations of deformed oscillators are obtained. For the (distinct) Fock space deformation method, the deformed paraoscillator algebras are worked out in detail and their Casimir operators are given. Difficulties concerning the relationship between the single particle Calogero–Vasiliev oscillator and the single particle parabose system are resolved. © 1995 American Institute of Physics.

Notes: An optimization approach to a three-dimensional acoustic inverse problem is considered in the time-domain. The velocity and the density are reconstructed by minimizing an objective functional. By introducing a dual function, the gradient of the objective functional is found with an explicit expression. The parameters are then reconstructed by an iterative algorithm (the conjugate gradient method). The uniqueness of the solution is also proved. © 1995 American Institute of Physics.

Notes: Using the noncommuting nature of fermionic fields we obtain a nonperturbative method to calculate the high temperature limit of the grand canonical partition function of interacting fermionic models. This nonperturbative approach is applied to the Hubbard model in d=2(1+1) and we recover the known results in the limit T → ∞. © 1995 American Institute of Physics.

Notes: In this article, we prove that the solution of the forced nonlinear Schrödinger equation (1.1) below for u0∈H1 and Q(t)∈C1 with u0(0)=Q(0) exists globally if and only if ∫T0||Q'(t)||2 dt〈∞. This result positively answers the conjecture of Q. Y. Bu ["On well-posedness of the forced NLS equation,'' Appl. Anal. 46, 219–239 (1992)]. © 1995 American Institute of Physics.

Notes: In this second of two articles (designated I and II), the bilinear transformation method is used to obtain stationary periodic solutions of the partially integrable regularized long-wave (RLW) equation. These solutions are expressed in terms of Riemann theta functions, and this approach leads to a new and compact expression for the important dispersion relation. The periodic solution (or cnoidal wave) can be represented as an infinite sum of sech2 "solitary waves'': this remarkable property may be interpreted in the context of a nonlinear superposition principle. The RLW cnoidal wave approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes, respectively. Analytic approximations and error estimates are given which shed light on the character of the cnoidal wave in the different parameter regimes. Similar results are presented in brief for the related RLW Boussinesq (RLWB) equation. © 1995 American Institute of Physics.

Notes: An important example of a multi-dimensional integrable system is the anti-self-dual Einstein equations. By studying the symmetries of these equations, a recursion operator is found and the associated hierarchy constructed. Owing to the properties of the recursion operator one may construct a hierarchy of symmetries and find the algebra generated by them. In addition, the Lax pair for this hierarchy is constructed. © 1995 American Institute of Physics.

Notes: Two exact solutions of Einstein's field equations corresponding to a cylinder of dust with net zero angular momentum are considered. In one of the cases, the dust distribution is homogeneous, whereas in the other, the angular velocity of dust particles is constant [Nuovo Cimento B 21, 64 (1974)]. For both solutions the junction conditions to the exterior static vacuum Levi–Civita space–time were studied. From this study we find an upper limit for the energy density per unit length σ of the source equals 1/4 for both cases. Thus the homogeneous cluster provides another example [Class. Quant. Grav. 8, 727 (1991); J. Math. Phys. 27, 152 (1980)] where the limit of σ is 1/4. It was also found that the cluster of homogeneous dust has a superior limit for its radius, depending on the constant volumetric energy density ρ0. © 1995 American Institute of Physics.

Notes: Using similarity methods, the Einstein field equations coupled to two oppositely directed null fluids for a spherically symmetric space–time are reduced to an autonomous system of three ordinary differential equations. The space of solutions is studied in some detail and solutions are found that represent: (i) the backscattering of an initially outgoing thick null fluid shell in a background gravitational field with a central naked singularity, (ii) the formation of strong space–time singularities by the interaction of thick null fluid shells, (iii) the interaction of a core of null radiation with an incoming shell of null fluid, and (iv) cosmological models of Kantowski–Sachs type with initial and final singularities clothed by apparent horizons. © 1995 American Institute of Physics.

Notes: The initial value problem for a compact, spherically symmetric star with a rather general equation of state is discussed. For any given time it is possible to calculate the area of the spheres of symmetry which intersect the boundary, if space–time is C5 and the initial energy density is smooth (i.e., C∞) and satisfies some minor technical conditions. The physical relevance of this result is also discussed. © 1995 American Institute of Physics.

Notes: The notion of *-derivation on an algebra of unbounded operators is extended to partial O*-algebras, and the corresponding notion of spatiality is investigated. Special emphasis is given to *-derivations associated to one-parameter groups of *-automorphisms and to *-derivations of partial GW*-algebras. © 1995 American Institute of Physics.

Notes: An algebraic interpretation for two classes of continuous q-polynomials is provided. Rogers' continuous q-Hermite polynomials and continuous q-ultraspherical polynomials are shown to realize, respectively, bases for representation spaces of the q-Heisenberg algebra and a q-deformation of the Euclidean algebra in these dimensions. A generating function for the continuous q-Hermite polynomials and a q-analog of the Fourier–Gegenbauer expansion are naturally obtained from these models. © 1995 American Institute of Physics.

Notes: A noncommutative integration calculus arising in the mathematical description of Schwinger terms of fermion–Yang–Mills systems is discussed. The differential complexes of forms u0[cursive-epsilon,u1]...[cursive-epsilon,un] with cursive-epsilon a grading operator on a Hilbert space H and ui bounded operators on H which naturally contains the compactly supported de Rham forms on Rd (i.e., cursive-epsilon is the sign of the free Dirac operator on Rd and H is a L2-space on Rd) are considered. An elementary proof is presented in which the integral of d-forms ∫RdtrN(X0dX1...dXd) for Xi∈C0∞(Rd;glN) is equal, up to a constant, to the conditional Hilbert space trace of ΓX0[cursive-epsilon,X1]...[cursive-epsilon,Xd] where Γ=1 for d odd and Γ=γd+1 (‘γ5-matrix') a spin matrix anticommuting with cursive-epsilon for d even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes' noncommutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace. © 1995 American Institute of Physics.

Notes: The exponential decay (or growth) of resonances provides an arrow of time which is described as the semigroup time evolution of Gamow vector in a new formulation of quantum mechanics. Another direction of time follows from the fact that a state must first be prepared before observables can be measured in it. Applied to scattering experiments, this produces another quantum mechanical arrow of time. The mathematical statements of these two arrows of time are shown to be equivalent. If the semigroup arrow is interpreted as microphysical irreversibility and if the arrow of time from the prepared in-state to its effect on the detector of a scattering experiment is interpreted as causality, then the equivalence of their mathematical statements implies that causality and irreversibility are interrelated. © 1995 American Institute of Physics.

Notes: In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra @Fg@Fl(N) under a certain homomorphism from the center of U(@Fg@Fl(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine @Fg@Fl(N) at the critical level k=−N defines a correspondence between higher Sugawara operators and quantum integrals of the Hamiltonian of the quantum many-body problem with the potential equal to constant times the Weierstrass function. This allows one to give a new proof of the Olshanetsky–Perelomov theorem stating that this Hamiltonian defines a completely integrable quantum system. A new expression is also given for eigenfunctions of the quantum integrals of the Sutherland operator as traces of intertwining operators between certain representations of @Fg@Fl(N). © 1995 American Institute of Physics.

Notes: A deformation of quantum mechanics obtained by replacing commutators by q-commutators (where q is close enough to unity to be compatible with experiment) is studied. This idea is explored by studying the q-harmonic oscillator in both configuration space and momentum space. The complete set of states in both representations as well as the transformation between x and p space is obtained. The states as well as the matrix elements lie in the SUq(2) algebra. To obtain expectation values and transition probabilities one must average over the SUq(2) algebra. © 1995 American Institute of Physics.

Notes: The problem under consideration in this article is the inverse boundary spectral problem for the Schrödinger operator with magnetic field. The operator is considered on a compact Riemannian manifold (with nonzero boundary) so that the corresponding unperturbed operator is the Laplace–Beltrami operator on the manifold. It is shown that boundary spectral data determine the potential and the boundary impedance uniquely, while the magnetic field may be found to within the group of gauge transformations. © 1995 American Institute of Physics.

Notes: The general theory of deformed commutation relations for creation and annihilation operators of particles equipped with arbitrary braid statistics (Ψ-statistics) is studied in terms of Wick algebras. The construction and examples of such algebras are given. © 1995 American Institute of Physics.

Notes: Extension of the standard construction of the Kadomtsev–Petviashvili (KP) hierarchy by the use of Riemann–Liouville integral is given. In consequence we obtain the new classes of integer as well as fractional graded KP hierarchies, which are further investigated. The fractional calculus leads to the new generalization of the w∞ algebra. © 1995 American Institute of Physics.

Notes: The formal series symmetry approach is applied to the conditionally integrable nonlinear evolution equation. A set of infinitely many symmetries for a 3+1-dimensional model, Jimbo–Miwa equation with a conditional equation (Kadomtsev–Petviashvilli equation), are obtained by an explicitly constructable formula. Similar to the usual KP equation, these symmetries constitute a generalized W∞ symmetry algebra. © 1995 American Institute of Physics.

Notes: A finite-dimensional involutive system is presented, and the Wadati–Konno–Ichikawa (WKI) hierarchy of nonlinear evolution equations and their commutator representations are discussed in this article. By this finite-dimensional involutive system, it is proven that under the so-called Bargmann constraint between the potentials and the eigenfunctions, the eigenvalue problem (called the WKI eigenvalue problem) studied by Wadati, Konno, and Ichikawa [J. Phys. Soc. Jpn. 47, 1698 (1979)] is nonlinearized as a completely integrable Hamiltonian system in the Liouville sense. Moreover, the parametric representation of the solution of each equation in the WKI hierarchy is obtained by making use of the solution of two compatible systems. © 1995 American Institute of Physics.

Notes: A diffeomorphism f: R2→R2 possessing a simple heteroclinic cycle [J. Franks, Publ. Math. IHES 71, 105 (1990)] bounding a disk D is first considered. Using the Poincaré index argument, one gets an easy result that f has a fixed point in D. If f is area preserving, there is a sequence of points in each of the stable and unstable manifolds of the simple heteroclinic cycle converging to the boundary of any connected component of the maximal invariant set in D. This result is applied to an area-preserving twist map. It is shown that each of the stable and unstable manifolds of any p/q-homoclinic Birkhoff saddle in a zone of instability of the map has a sequence of points converging, for example, to the boundary of the connected invariant set containing any second p'/q'-Birkhoff point. Finally, the relation between a homoclinic Birkhoff saddle and a second Birkhoff periodic point is briefly discussed. © 1995 American Institute of Physics.

Notes: Using the elementary symmetric functions of n Euclidean coordinates xn as symmetry variables and the isovector approach, the generalized forms of nonlinear Klein–Gordon and Liouville equations in n-dimensional Euclidean space are analyzed herein. The construction of the components of the associated isovector field via transport property under Lie derivative leads to orbital equations whose solutions yield invariant groups of transformations. These invariant groups reduce the equations under consideration to nonlinear ordinary differential equations (ODEs) involving arbitrary functions of the new dependent and independent variables. Finally, exact solutions of these resulting ODEs are tabulated for different forms of these arbitrary functions. Beside recovering the available results for particular forms of the said arbitrary functions involved in the ordinary differential equations, some new exact solutions are reported. © 1995 American Institute of Physics.

Notes: States on free *-algebras expressed in terms of ordered partitions are studied. To this class belong twisted limit states that are derived from a convolution type q- analog of the quantum central limit theorem (qclt) for twisted free *-bialgebras Cq, studied previously in the case of quantum groups. A special form of the GNS representation in terms of generalized creation and annihilation operators, A(v) and A(v*), respectively, living in the infinite tensor product of Hilbert spaces is given leading to a generalization of the Fock space. It is also shown that another family of states of similar combinatorics can be obtained from the convolution series. © 1995 American Institute of Physics.

Notes: The field equations governing Bianchi type I space–times having elastic media as sources are constructed. Such equations turn out to be a system of three ordinary differential equations for the three independent components of the gravitational field. Plane symmetric and Robertson–Walker space–times are also discussed, and some examples of solutions are given. In the Robertson–Walker case, it turns out that it is possible to "tune'' the elastic constitutive equations in such a way that any "required'' behavior of the scale factor may be obtained. This feature closely resembles the scalar-field dominated case. © 1995 American Institute of Physics.

Notes: For any non-negative integer p we calculate the number of CPn-instantons of charge np+n over a Riemann surface of genus np+p. © 1995 American Institute of Physics.

Notes: Vector fields on a disk satisfying two types of boundary conditions are studied. These boundary conditions are selected by BRST-invariance in electrodynamics. They also appear in the de Rham complex. The construction of the harmonic expansion is presented. The eigenfunctions of the vector Laplace operator are expressed in terms of fields satisfying pure Dirichlet or Robin boundary conditions. For the case of four-dimensional disk several first coefficients of the heat kernel expansion are computed. An error in the analytical expression by Branson and Gilkey is corrected. © 1995 American Institute of Physics.

Notes: The theory of oscillatory integrals on Hilbert spaces (the mathematical version of "Feynman path integrals'') is extended to cover more general integrable functions, preserving the property of the integrals to have converging finite dimensional approximations. An application to the representation of solutions of the time-dependent Schrödinger equation with a scalar and a magnetic potential by oscillatory integrals on Hilbert spaces is given. A relation with Ramer's functional in the corresponding probabilistic setting is found. © 1995 American Institute of Physics.

Notes: Localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories are reviewed. These are the functional integral counterparts of the Mathai–Quillen formalism, the Duistermaat–Heckman theorem, and the Weyl integral formula, respectively. In each case, the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups) is introduced, and the finite dimensional integration formulae described. Then some applications to path integrals are discussed and an overview of the relevant literature is given. The applications include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang–Mills theory. © 1995 American Institute of Physics.

Notes: A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given which will serve as a quick reference for solving path integrals. Explicit formulas for the so-called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based. © 1995 American Institute of Physics.

Notes: In the article, the complete investigation is given of the large times behavior of the solutions of the Belavkin quantum filtering equation describing a quantum particle with continuously observed coordinate. It turns out that these solutions have an extraordinary property, namely: as t → ∞, any solution will tend to a Gaussian function (for almost all realizations of innovating Wiener process). As a consequence, it follows that the dispersion of coordinate will always tend to the same limit, as t → ∞, not depending on initial data. © 1995 American Institute of Physics.

Notes: The inverse scattering problem on the line is studied for the generalized Schrödinger equation (d2ψ/dx2)+k2H(x)2ψ=Q(x)ψ, where H(x) is a positive, piecewise continuous function with positive limits H± as x → ±∞. This equation, in the frequency domain, describes the wave propagation in a nonhomogeneous medium, where Q(x) is the restoring force and 1/H(x) is the variable wave speed changing abruptly at various interfaces. A related Riemann–Hilbert problem is formulated, and the associated singular integral equation is obtained and proved to be uniquely solvable. The solution of this integral equation leads to the recovery of H(x) in terms of the scattering data consisting of Q(x), a reflection coefficient, either of H±, and the bound state energies and norming constants. Some explicitly solved examples are provided. © 1995 American Institute of Physics.

Notes: The Moses curl eigenfunctions are used to describe linear force-free magnetic fields, enabling a proof that such fields are defined entirely by the value of their curl transform on the unit hemisphere in transform space. This change of viewpoint suggests an orderly approach to the exploration and classification of the properties of such fields, which is briefly sketched. The simplest force-free fields defined on zero- , one- , and two-dimensional sets on the transform sphere are exhibited, and the possibility of fields with fractal support sets suggested. Connections with other mathematical descriptions are pointed out, as well as several promising directions for further exploration. All results apply equally well to the description of the Trkalian subset of Beltrami fields in fluid dynamics. © 1995 American Institute of Physics.

Notes: The core of this article is a general theorem with a large number of specializations. Given a manifold N and a finite number of one-parameter groups of point transformations on N with generators Y,X(1),...,X(d), we obtain, via functional integration over spaces of pointed paths on N (paths with one fixed point), a one-parameter group of functional operators acting on tensor or spinor fields on N. The generator of this group is a quadratic form in the Lie derivatives LX(α) in the X(α)-direction plus a term linear in LY. The basic functional integral is over L2,1 paths x:T→N (continuous paths with square integrable first derivative). Although the integrator is invariant under time translation, the integral is powerful enough to be used for systems which are not time translation invariant. Seven nontrivial applications of the basic formula are given, and the semiclassical expansion is computed. The methods of proof are rigorous and combine Albeverio–Høegh-Krohn oscillatory integrals with Elworthy's parametrization of paths in a curved space. Unlike other approaches, Schrödinger type equations are solved directly, rather than solving first diffusion equations and then using analytic continuation.

Notes: Dissipation causes distortions in wave propagations. A new scheme for restoring such distorted waveforms based on a generalized Kac's method of solving wave equation with dissipation is proposed. From only the component parts of the wave received over time at a point in space, and without the knowledge of them at other points, the wave that would have been received at the point in the absence of dissipation is then restored. The new scheme can be adapted to function as a waveform designer for designing the to-be-transmitted waveform such that the distorted distorted-waveform to be received becomes the intended one. The details of such a mathematical construction of the waveform restorer and designer are given. © 1995 American Institute of Physics.

Notes: Quantum kinematics are considered for two classes of systems: (i) the coordinate space is a homogeneous Riemannian manifold and the kinetic energy is the Laplace–Beltrami operator, (ii) the phase-space is a homogeneous Kählerian manifold where the algebra of observables is the universal enveloping algebra of the symmetry group. In both the cases, the path integrals are derived from more fundamental principles. The derivation enables a definite description of the classes of trajectories which constitute the domain of integration in the path integrals. The specification of the classes of functions is important for consistency of the approach. The standard action functional and the variational principle appear in the steepest descent method corresponding to the semiclassical approximation. © 1995 American Institute of Physics.

Notes: It is demonstrated for one-dimensional Hamiltonians, and under a special hypothesis on the ground state wave function, that it is possible to build an analytical overcomplete basis that generalizes the coherent state basis of the harmonic oscillator, without any reference to group theory. The semi-classical consequences of this formalism are developed and the usual ideas of normal and antinormal expansions for operators are extended. © 1995 American Institute of Physics.

Notes: Using the coherent states representation and the Schwinger boson model, the propagator of a spinning particle in interaction with an arbitrary magnetic field is derived in the framework of the path integral formalism. To show the relevance of the formalism to some applications, the Green function of spinning particle s=1/2 in special configuration of magnetic field is calculated. The transition amplitude and Berry's phase for an arbitrary spin and time dependent magnetic field are deduced. The case of half integer spin is also considered. © 1995 American Institute of Physics.

Notes: A model for multichannel parallel relaxation is suggested based on the following assumptions: (a) an individual channel is characterized by a set of continuous state variables; the corresponding relaxation rate is a function of the state variables as well as of the time interval for which the channel is open; (b) the number of channels is a random variable described by a correlated point process defined in the space of state parameters of an individual channel. Analytical expressions for the generating functional of the overall relaxation rate and for the average survival function are derived in terms of the generating functional of the point process. The general formalism is applied to the problem of direct energy transfer from excited donors to acceptors in fractal systems with dynamic disorder. It is assumed that the number of acceptors obeys a Poissonian distribution law with a constant average density in a df-dimensional fractal structure embedded in a ds-dimensional Euclidean space (ds=1,2,3) and that an individual relaxation rate is an inverse power function of the distance between the acceptor and the donor molecules.The dynamic disorder is described in terms of three different functions: the rate ω(t) of opening of a channel at time t, the attenuation function cursive-phi(t) of the reactivity of an individual channel at time t, and the probability density ψ(t) of the time interval within which a channel is open. Several particular cases corresponding to different functions ω(t), cursive-phi(t), and ψ(t) are investigated. The static disorder corresponds to a survival function of the stretched exponential type exp[−(Ωt)β] with 1(approximately-greater-than)β(approximately-greater-than)0. For very strong dynamic disorder there is no attenuation of reactivity, the opening time is infinite and the survival function is given by a compressed exponential exp[−const.t1+β], 1(approximately-greater-than)β(approximately-greater-than)0. The other cases analyzed correspond to a slowly decreasing attenuation function and to an exponential distribution of the opening time, respectively; for them the efficiency of relaxation is between the ones corresponding to the two extreme cases of static and very strong dynamic disorder. The general conclusion is that the passage from static to the dynamic disorder results in an increase of the efficiency of the relaxation process. © 1995 American Institute of Physics.

Notes: The machinery of braided geometry introduced previously is used now to construct the ε "totally antisymmetric tensor'' on a general braided vector space determined by R-matrices. This includes natural q-Euclidean and q-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as SOq(4) or SOq(1,3). The Hodge * operator and differentials are also constructed in this approach. © 1995 American Institute of Physics.

Notes: Coherent state operators (CSOs) are defined as operator valued functions on G=SL(n,C) being homogeneous with respect to right multiplication by lower triangular matrices. They act on a model space containing all holomorphic finite dimensional representations of G with multiplicity 1. CSOs provide an analytic tool for studying G invariant 2- and 3-point functions, which are written down in the case of SU3. The quantum group deformation of the construction gives rise to a noncommutative coset space. A "standard'' polynomial basis is introduced in this space (related to but not identical with the Lusztig canonical basis) that is appropriate for writing down Uq(sl3) invariant 2-point functions for representations of the type (λ,0) and (0,λ). General invariant 2-point functions are written down in a mixed Poincaré–Birkhoff–Witt type basis. © 1995 American Institute of Physics.

Notes: Improved rigorous high-energy upper bounds are established on the absorptive parts of the elastic scattering amplitudes for the momentum-transfer-squared variable t positive and within the Lehmann–Martin ellipse. These bounds are used to set upper bounds on Regge trajectories and also to extend the region in the complex t plane where the amplitudes cannot have zeros. No assumption is made about the high-energy behavior of the total cross sections. © 1995 American Institute of Physics.

Notes: A Lagrangian treatment of the quantization of first class Hamiltonian systems with constraints and Hamiltonian linear and quadratic in the momenta, respectively, is performed. The "first reduce and then quantize'' and the "first quantize and then reduce'' (Dirac's) methods are compared. A source of ambiguities in this latter approach is pointed out and its relevance on issues concerning self-consistency and equivalence with the "first reduce'' method is emphasized. One of the main results is the relation between the propagator obtained à la Dirac and the propagator in the full space. As an application of the formalism developed, quantization on coset spaces of compact Lie groups is presented. In this case it is shown that a natural selection of a Dirac quantization allows for full self-consistency and equivalence. Finally, the specific case of the propagator on a two-dimensional sphere S2 viewed as the coset space SU(2)/U(1) is worked out. © 1995 American Institute of Physics.

Notes: Poincaré invariance of the Maxwell equations in Minkowski space enables one to generate in a usual way new solutions (corresponding to new sources) from a given one by means of the pullback operation. A method is proposed generalizing this to a wider class of diffeomorphisms f:M → M of a Lorentzian manifold (M,g). It results in the simple explicit formulas for obtaining the new field Ff and its source jf from a given pair (F,j). Some applications, e.g., a uniformly rotating or radially pulsating source, are presented. © 1995 American Institute of Physics.

Notes: In this article, the stationary Kuramoto–Sivashinsky (KS) equation is studied. It is shown, first, that this equation can be transformed into a one-dimensional Hamiltonian system describing the motion of a particle in a time-dependent potential. Second, working with this formalism and coming back, then, to the initial KS equation, a subset of initial conditions is derived for which the corresponding solutions u(x) will behave like the dominant pole 120/(x−x0)3, deduced from the Painlevé analysis, and will explode in a finite "time.'' Finally, special attention is given to the well-known Kuramoto–Tsuzuki solution and it is shown that provided the independent variable x is initialized in a specified domain, this solution explodes also like 120/(x−x0)3. © 1995 American Institute of Physics.

Notes: The tensor product of three irreducible representations of the quantum superalgebra Uq(osp(1||2)) is studied and the super-q analogs of Racah coefficients and 6−j symbols for the quantum superalgebra Uq(osp(1||2)) are defined. Racah coefficients and 6−j symbols depend on the superspin l and the parity λ which characterize the irreducible representations of Uq(osp(1||2)) but the dependence on parities can be factored out so that one can define parity independent 6−j symbols. It is shown that the 6−j symbols for the quantum superalgebra Uq(osp(1||2)) satisfy symmetry properties and orthogonality relations similar to those of the 6−j symbols for quantum algebra Uq(su(2)). © 1995 American Institute of Physics.

Notes: The development of p-adic quantum mechanics has made it necessary to construct a probability theory in which the probabilities of events are p-adic numbers. The foundations of this theory are developed here. The frequency definition of probability is used. A general principle of statistical stabilization of relative frequencies is formulated. By virtue of this principle, statistical stabilization of relative frequencies, which are, like all experimental data, rational numbers, can be considered not only in the real topology but also in p-adic topologies. © 1995 American Institute of Physics.

Notes: K-matrix theory is an effective algebraic way of determining inner products and constructing orthonormal bases vectors for the carrier spaces of induced representations of Lie groups and Lie algebras. The theory is applied in this paper to give canonical resolutions of missing label problems. © 1995 American Institute of Physics.

Notes: The semiclassical asymptotics of the unitary quantum propagator applied to the Wigner functions, which are associated with the initial quantum state belonging to Schwartz class, are considered herein. The limit (h-dash-bar) → 0 of the semiclassical asymptotics of these Wigner functions is investigated. As a consequence of this result, there exists a simple semiclassical asymptotic, which obeys the unitary conservation law over any finite time interval [0,T]. © 1995 American Institute of Physics.

Notes: The symmetries associated with the closed bosonic string partition function are examined so that the integration region in Teichmuller space can be determined. The conditions on the period matrix defining the fundamental region can be translated to relations on the parameters of the uniformizing Schottky group. The growth of the lower bound for the regularized partition function is derived through integration over a subset of the fundamental region. © 1995 American Institute of Physics.

Notes: An alternative approach to the Hurwitz (H) transformation reducing Euclidean space E8 to Euclidean space E5 is developed. It is shown how refusal of the bilinearity condition leads to the replacement of the H-transformation by its modified nonbilinear (left and right) version: HL- and HR-transformations. The HL(HR)-transformation of the specific type eight-dimensional hyperspherical coordinates is investigated. The radial coordinate u and the polar angle θ/2, in this approach, transform into u2 and θ, respectively, like in the H-case. Action of these transformations on the remaining hyperspherical coordinates, unlike in the H-case, is equivalent to the invariance (shutting) of one angular triplet and the shutting (invariance) of another triplet. The connection of HL and HR with H is established and the structure of the H-transformation itself is revealed on this basis. © 1995 American Institute of Physics.

Notes: Sufficient conditions are given so that the solutions of the initial-boundary value problem for the nonlinear Klein–Gordon equation do not exist for all t(approximately-greater-than)0. © 1995 American Institute of Physics.

Notes: Low-energy dynamics in the unit-charge sector of the CP1 model on spherical space (space-time S2×R) is treated in the approximation of geodesic motion on the moduli space of static solutions, a six-dimensional manifold with nontrivial topology and metric. The structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full field theory. Evaluation of the metric is then reduced to finding five functions of one coordinate, which may be done explicitly. Some totally geodesic submanifolds are found, and the qualitative features of motion on these described. © 1995 American Institute of Physics.

Notes: The principle of equivalence says that at any given space–time point, there exists a local coordinate system with respect to which the three-acceleration of a freely falling test body vanishes regardless of its three-velocity. In this article, a more intrinsic and geometric criterion for free fall motion is provided. The criterion is much simpler than a recently proposed criterion that is based on the Desargues property. For space–times of dimension greater than 2, the Desargues property is a theorem. It is shown that it suffices to require that a version of Pasch's axiom is satisfied; that is, two paths in a plane intersect up to corrections of order cursive-epsilon3, where cursive-epsilon is the scale parameter of a shrinking process. © 1995 American Institute of Physics.

Notes: Relations between determining equations for nonclassical symmetries of Bluman and Cole and integrability conditions of differential equations supplemented by invariance conditions are studied. It is shown that every system of equations with p independent variables admits p-dimensional groups of such symmetries. Nonclassical symmetries of the eikonal equation in a curved space are investigated. They are related to shear-free congruences of null geodesics. © 1995 American Institute of Physics.

Notes: Essential prerequisites for the study of q-deformed physics are particle states in position and momentum representation. In order to relate x and p space by Fourier transformations the appropriate q-exponential series related to orthogonal quantum symmetries is constructed. It turns out to be a new q-special function giving rise to q-plane wave solutions that transform with a noncommutative phase under translations. © 1995 American Institute of Physics.

Notes: A new derivation of the algebra of functions on the two-dimensional Euclidean Poincaré group is proposed. It is based on a contraction of the Hopf algebra Fun(SOq(3)), where the deformation parameter q is sent to one. In this geometric construction, the action of Fun(SOq(3)) on the noncommutative Euclidean space is a key ingredient. © 1995 American Institute of Physics.

Notes: In the first article a method for finding a mass spectrum from a given field equation was developed. The method is extended here to provide for the self-electromagnetic effect on the rest mass. The process is described by example: the method is applied here for the equations of a complex scalar field interacting with the electromagnetic field. A spectrum of charged-particle masses is obtained. A quantum number for the charges of the mass levels appears. After some special assumptions about the dependence on the vector potential are made, there are finite electromagnetic contributions to the rest masses of the particle states. © 1995 American Institute of Physics.

Notes: A generalization of magnetic monopoles is given over an odd dimensional contact manifold and we discuss whether the Yang–Mills–Higgs functional attains at generalized monopoles the absolute minimal value, the topological invariant. © 1995 American Institute of Physics.

Notes: A reduction of the self-dual Yang–Mills (SDYM) equations is studied by imposing two space–time symmetries and by requiring that the connection one-form belongs to a Lie algebra of formal matrix-valued differential operators in an auxiliary variable. In this article, the scalar case and the canonical cases for 2×2 matrices are examined. In the scalar case, it is shown that the field equations can be reduced to the forced Burgers equation. In the matrix case, several well-known 2+1 integrable equations are obtained. Also examined are certain transformation properties between the solutions of some of these 2+1 equations. © 1995 American Institute of Physics.

Notes: An eigenvalue problem is considered. New hierarchies of bi-Hamiltonian systems are constructed. Some examples of these systems and their reductions are presented. © 1995 American Institute of Physics.

Notes: A class of nonlinear perturbations of the Newtonian uniform expansion background is studied, for which only the velocity field is assumed to be perturbed. The general closed form solution to the corresponding equations of motion is found, and several particular families of solutions are analyzed. All the flows of this class are proved to be essentially rotational. A major part of liquid particles constituting the flows have a nonzero angular momentum and move along parabolas, the rest move along straight lines. The analysis is carried out by means of both the Euler and Lagrange variables. © 1995 American Institute of Physics.

Notes: There are infinitely many topological solitons in any given complex affine Toda field theory and most of them have complex-energy density. When we require the energy density of the solitons to be real, we find that the reality condition is related to a simple "pairing condition.'' Unfortunately, rather few soliton solutions in these theories survive the reality constraint, especially if one also demands positivity. The resulting implications for the physical applicability of these theories are briefly discussed. © 1995 American Institute of Physics.

Notes: It is shown that the number of bound states of coupled systems of radial Schrödinger equations can be found by computing the number of zeros of a suitable determinant. This is related to the concept of conjugate points from Morse theory and is the basis for a numerically efficient method. © 1995 American Institute of Physics.

Notes: Analysis of the Wentzel–Kramers–Brillouin (WKB) exactness in some homogeneous spaces is attempted. CPN as well as its noncompact counterpart DN,1 is studied. U(N+1) or U(N,1) based on the Schwinger bosons leads us to CPN or DN,1 path integral expression for the quantity tr e−iHT, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made. © 1995 American Institute of Physics.

Notes: The Dirac equation for a particle of mass m submitted to a spherically symmetric vector potential is considered. Similar to the Schrödinger case, a nodal characterization of solutions for energies in the spectral gap (−m,m) is shown. As a consequence some bounds on the number of bound states which reduce in the nonrelativistic limit to the classical bounds of Bargmann and Calogero are proven. © 1995 American Institute of Physics.

Notes: The basic ingredients of the "consistent histories'' approach to a generalized quantum theory are "histories'' and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham ["Quantum logic and the histories approach to quantum theory,'' J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space UP of propositions about histories with an orthoalgebra or lattice. When UP is given by the lattice of projectors P(V) in some Hilbert space V, consistent sets correspond to certain partitions of the unit operator in V into mutually orthogonal projectors {α1,α2,...}, such that the function d(α,α) is a probability distribution on the boolean algebra generated by {α1,α2,...}. Using the classification theorem for decoherence functionals proven in Isham et al. ["The classification of decoherence functionals: An analog of Gleason's Theorem,'' J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where V is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α,α)=p(α) holds. © 1995 American Institute of Physics.

Notes: The stability analysis for spherically symmetric stellar equilibrium models with respect to "small'' adiabatic Lagrangian perturbations leads to the consideration of a class of densely defined, linear symmetric operators in Hilbert space, which are induced by certain singular vector–integro–partial differential operator. The extension properties of these operators as well as the spectral properties of the linear self-adjoint extensions which are chosen by physical boundary conditions are investigated. For this, the equilibrium models are assumed to be polytropic, with a constant adiabatic index only near the center and near the boundary of the star. Among others it is shown that the operators of the class having a polytropic index near the boundary which is ≥1 are in particular essentially self-adjoint and have a closure with a pure point spectrum. © 1995 American Institute of Physics.

Notes: In the following we introduce a new way to solve the Cauchy problem for non- vanishing potentials at infinity for the spectral problem of the Zakharov and Shabat, Ablowitz, Kaup, Newell, and Segur (ZS-AKNS) hierarchy. Then we apply the results to solve a class of coupled systems, with this nonvanishing condition on the potential and with quite general boundary conditions on the other fields. Finally we present an application to a physical system governed by a three waves interaction type model. © 1995 American Institute of Physics.

Notes: The solutions to the sine–Gordon equation which are derived from an ansatz based on the methods of twistor theory are shown to correspond to Darboux transformations for a certain linear equation. The linear problem is shown to arise in several different ways, and a relationship between the Darboux transformation for linear equations of this type and the Bäcklund transformation is examined. A possible extension to well known monopole ansatze is briefly considered. © 1995 American Institute of Physics.

Notes: Some exact solutions for the Einstein field equations corresponding to inhomogeneous G2 cosmologies with an exponential-potential scalar field, which generalize solutions obtained previously, are considered. Several particular cases are studied and the properties related to generalized inflation and asymptotic behavior of the models are discussed. © 1995 American Institute of Physics.

Notes: Recently there has been developed a reformulation of general relativity (GR)—referred to as the null surface version of GR—where instead of the metric field as the basic variable of the theory, families of three-surfaces in a four-manifold become basic. From these surfaces themselves, a conformal metric, conformal to an Einstein metric, can be constructed. A choice of conformal factor turns it into an Einstein metric. The surfaces are then automatically characteristic surfaces of this metric. In the present paper we explore the linearization of this null surface theory and compare it with the standard linear GR. This allows a better understanding of many of the subtle mathematical issues and sheds light on some of the obscure points of the null surface theory. It furthermore permits a very simple solution generating scheme for the linear theory and the beginning of a perturbation scheme for the full theory. © 1995 American Institute of Physics.

Notes: The continuous big q-Hermite polynomials are shown to realize a basis for a representation space of an extended q-oscillator algebra. An expansion formula is algebraically derived using this model. © 1995 American Institute of Physics.

Notes: We show that the unitary irreducible representations (UIRs) of the proper homogeneous Lorentz group SO(3,1) can be realized in a manifestly covariant form, using Hilbert spaces of functions on suitable geometrical configurations of vectors in space time. The concept and properties of isotropic lines, which are generators of spacelike hyperboloids, and of stability groups of lightlike vectors, are crucial to the constructions. Both principal and supplementary Series UIRs are treated. © 1995 American Institute of Physics.

Notes: Different statistics generalizing the Fermi–Dirac one, i.e., statistics associated with parafermions of order p, orthofermions, deformed parafermions, and genons, are studied in connection with Lie superalgebras. The class spl(m/n) appears as the fundamental one in this context. A specific relation between parafermions and orthofermions is also pointed out. © 1995 American Institute of Physics.

Notes: A general discussion of the construction of free fields based on Weinberg anszatz is provided and various applications appearing in the literature are considered. © 1995 American Institute of Physics.

Notes: Some insight is offered into the dimensional dependence of the Wentzel–Kramers–Brillouin (WKB) and improved-WKB approximations introduced by Steiner in connection with hyperspherical (originally circular) quantum billiards. The accuracy of every new D-dimensional picture is estimated by examining the residues of the spectral zeta-function poles. © 1995 American Institute of Physics.

Notes: Asymptotic properties of classical field electrodynamics are considered. Special attention is paid to the long-range structure of the electromagnetic field. It is shown that conserved Poincaré quantities may be expressed in terms of the asymptotic fields. Long-range variables are shown to be responsible for an angular momentum contribution which mixes Coulomb and infrared free field characteristics; otherwise angular momentum and energy-momentum separate into electromagnetic and matter fields contributions. © 1995 American Institute of Physics.

Notes: The information entropy of the harmonic oscillator potential V(x)=1/2λx2 in both position and momentum spaces can be expressed in terms of the so-called "entropy of Hermite polynomials,'' i.e., the quantity Sn(H):= −∫−∞+∞H2n(x)log H2n(x) e−x2dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x)=exp(−||x||m), m(approximately-greater-than)0. Here, a very precise and general result of the entropy of Freud polynomials recently established by Aptekarev et al. [J. Math. Phys. 35, 4423–4428 (1994)], specialized to the Hermite kernel (case m=2), leads to an important refined asymptotic expression for the information entropies of very excited states (i.e., for large n) in both position and momentum spaces, to be denoted by Sρ and Sγ, respectively. Briefly, it is shown that, for large values of n, Sρ+1/2logλ(approximately-equal-to)log(π(square root of)2n/e)+o(1) and Sγ−1/2log λ(approximately-equal-to)log(π(square root of)2n/e)+o(1), so that Sρ+Sγ(approximately-equal-to)log(2π2n/e2)+o(1) in agreement with the generalized indetermination relation of Byalinicki-Birula and Mycielski [Commun. Math. Phys. 44, 129–132 (1975)]. Finally, the rate of convergence of these two information entropies is numerically analyzed. In addition, using a Rakhmanov result, we describe a totally new proof of the leading term of the entropy of Freud polynomials which, naturally, is just a weak version of the aforementioned general result. © 1995 American Institute of Physics.

Notes: The discrete Painlevé III equation is investigated based on the bilinear formalism. It is shown that it admits the solutions expressed by the Casorati determinant whose entries are given by the discrete Bessel functions. Moreover, based on the observation that these discrete Bessel functions are transformed to the q-Bessel functions by a simple variable transformation, we present a q-difference analog of the Painlevé III equation. © 1995 American Institute of Physics.

Notes: A new spectral transform system is introduced to solve the initial-value problem for the intermediate nonlinear Schrödinger (INLS) equation describing envelope waves in a deep stratified fluid. The spectral system is a combination of the Zakharov–Shabat linear system and a local Riemann–Hilbert problem in a strip of the complex plane. The inverse scattering transform technique is developed and the Bäcklund–Darboux transformation, soliton solutions and an infinite number of conservation laws are constructed. It is shown that all these properties of the INLS equation are closely related to those of the intermediate long-wave equation. © 1995 American Institute of Physics.

Notes: We present cylindrically symmetric solutions of the Einstein–Maxwell electrovacuum equations, considering a δ=2 Tomimatsu–Sato solution, describing a cylindrical symmetric space–time. The solutions exhibit an angle deficit, are asymptotically flat, smooth everywhere without curvature singularities and for various values of the parameters may be interpreted as open cosmic strings with gravity and electromagnetism coupled in a relativistic manner. © 1995 American Institute of Physics.

Notes: It is shown rigorously that any static symmetric solution of the Einstein–Yang–Mills (YM) equations with SU(2) gauge group that is well behaved in the far field is one of three types: black hole, particlelike, or Riessner–Nordström-like (RN) solution. (In particular, any solution with finite ADM mass is well behaved in the far field.) Black-hole solutions are proven to be analytic at the event horizon and thus coincides with Bartnik–McKinnon (BM) black holes. Furthermore, the singularity in the metric at the event horizon can be transformed away by a Kruskal-like change of coordinates in which the YM field remains well behaved. Particlelike solutions are shown to satisfy the same initial conditions as the BM solutions at r=0. RN-like solutions will be considered elsewhere. © 1995 American Institute of Physics.

Notes: *-structures on quantum and braided spaces of the type defined via an R-matrix are studied. These include q-Minkowski and q-Euclidean spaces as additive braided groups. The duality between the *-braided groups of vectors and covectors is proved and some first applications to braided geometry are made. © 1995 American Institute of Physics.

Notes: Based on the quantized universal enveloping (QUE) algebras, a quantization of the automorphism group of some nonassociative algebras is given in the formulation employing noncommuting matrix entries. A quantum group of G2 type included in this scheme is studied in detail in the Faddeev–Reshetikhin–Takhtajan (FRT) approach. Also in the formulation employing noncommuting matrix entries, the QUE-algebra of G2 type is reconstructed through the pairing induced by the R-matrix between the quantum group and the QUE-algebra. © 1995 American Institute of Physics.

Notes: Using an approach based on the canonical formalism, the Yang–Mills theories on a cylinder are rigorously analyzed. In this way the moduli space A/G, can be explicitly described with A being the space of connections and G the group of gauge transformations. In particular A/G0, G0 being the group of the pointed gauge transformations, is diffeomorphic to the structure group of the theory G, whereas A/G is G modulo the group of inner automorphisms. It is also proven that A → G is a principal fiber bundle with structure group G0. © 1995 American Institute of Physics.

Notes: In the present paper the problem of a relativistic Dirac electron is analyzed in the presence of a combination of a Coulomb field, a 1/r scalar potential, as well as a Dirac magnetic monopole and an Aharonov–Bohm potential. Using the algebraic method of separation of variables, the Dirac equation expressed in the local rotating diagonal gauge is completely separated in spherical coordinates, and exact solutions are obtained. The energy spectrum is computed and its dependence on the intensity of the Aharonov–Bohm and the magnetic monopole strengths is analyzed. © 1995 American Institute of Physics.

Notes: In this article the existence of generalized solutions to the modified Korteweg–de Vries equation ut−6σu2ux+uxxx=0 is studied. The solutions are found in certain algebras of new generalized functions containing spaces of distributions. © 1995 American Institute of Physics.

Notes: It is shown that the integrable subclasses of the equations q,t=f(x,t)q,3 +H(x,t,q,q,1) are the same as the integrable subclasses of the equations q,t=q,3 +F(q,q,1). © 1995 American Institute of Physics.

Notes: Hitherto, the bilinear transformation method has been used to obtain analytic solutions of completely integrable (or nonlinear evolution) equations. In this first of two studies (designated I and II), it is shown that the method may apply equally well to those nonlinear wave equations dubbed partially integrable. Here the regularized long-wave (RLW) and RLW Boussinesq (RLWB) equations are considered as examples. In each case, the bilinear form is found to have an "extra'' equation which would appear to signal their nonintegrability. Their solitary-wave solutions are reconstructed and the nonexistence of multisoliton solutions is demonstrated. The present work provides the basis for the construction of periodic solutions in II. © 1995 American Institute of Physics.

Notes: The two-dimensional Hamiltonian system with a quartic potential is known to be integrable in four cases, that is, for four sets of parameter values. In three of these cases the system is also known to be separable. In this paper, separating variables for the fourth integrable case are obtained, allowing its solution in terms of elliptic functions. Lax pairs with a spectral parameter for the four integrable cases are also obtained. © 1995 American Institute of Physics.

Notes: Topology changing space–times are investigated by breaking them into their constituent parts. It is shown that, for a large class of space–times, topology change between kink zero (e.g., spacelike) surfaces must produce singularities regardless of the existence of closed timelike curves. A relationship between the kink number and causality is also described. © 1995 American Institute of Physics.

Notes: A sheaf of functions on a topological space is called a differential structure if it satisfies an axiom of a closure with respect to composition with the Euclidean functions. A differential structure on a nonempty set is called a structured space. It is a generalization of the smooth manifold concept and of an earlier concept of differential space. Differential geometry on structured spaces is developed (tangent space, vector fields, differential forms, exterior algebra, linear connection, curvature, and torsion). Some of its techniques are applied to the classical singularity problem in general relativity. It turns out that Einstein's equations can be defined on space–times with singularities. This can have important consequences for the search of the quantum theory of gravity. © 1995 American Institute of Physics.

Notes: The spinor Green's function and the twisted spinor Green's function in a Euclidean space with a conical-type line singularity are generally determined. In particular, in the neighborhood of the point source, they are expressed as a sum of the usual Euclidean spinor Green's function and a regular term. In four dimensions, these determinations can be used to calculate the vacuum energy density and the twisted one for a massless spinor field in the space–time of a straight cosmic string. In the Minkowski space–time, the vacuum energy density for a massive twisted spinor field is explicitly determined. © 1995 American Institute of Physics.

Notes: Light cone cuts are the intersection of the light cones of space–time events with an initial data surface which is usually taken to be null infinity. These are determined by a "cut function,'' a function of the space–time coordinates and the sphere of generators of null infinity. The Bach equations and the conformal vacuum equations are shown to lead to a single scalar "main'' equation on the cut function. In simplified cases, i.e., when the Weyl curvature is either small or self-dual, this equation can be integrated to give a simple scalar conformally invariant elliptic equation on the sphere of null directions. It has the remarkable property that the general solution of the Einstein vacuum equations in these cases can be derived from the solution to this auxiliary equation in two dimensions. In general, it will not be possible to reduce it to an auxilliary equation on the sphere. Nevertheless, the main equation is still a necessary condition and it is conjectured that it is also sufficient to imply the Bach equations in general and supporting arguments are given. It is also shown how to extend the Kozameh–Newman framework to the case of light cone cuts of a spacelike Cauchy hypersurface. The analogous procedures for Yang–Mills are also discussed. A conformally invariant formalism for calculations on null infinity is described in an Appendix. This is used for all the calculations which are only described in brief form in the main text. © 1995 American Institute of Physics.

Notes: In this paper explicitly natural (from the geometrical point of view) Fock-space representations (contragradient Verma modules) of the quantized enveloping algebras are constructed. In order to do so, one starts from the Gauss decomposition of the quantum group and introduces the differential operators on the corresponding q-deformed flag manifold (assumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, the representatives of the elements of the quantized enveloping algebra corresponding to the left-invariant vector fields on the quantum group are expressed as first-order differential operators on the q-deformed flag manifold. © 1995 American Institute of Physics.

Notes: The classification of the finite-dimensional irreducible representations of the super Yangian associated with the Lie superalgebra gl(1||1) is presented in detail. © 1995 American Institute of Physics.