ALBERT

Notes: Canonical Monte Carlo results are presented for mixtures of primitive model electrolytes with a common ion. Both symmetric mixtures, where the ions differ only in size, and charge asymmetric mixtures were studied for ionic strengths ranging from 0.003 to 1.0 M. The hypernetted chain (HNC) equation theory and the simple "exponential'' (EXP) approximation are both applied to the same electrolyte models. The electrolyte mixing coefficients w0 and w1 were calculated from osmotic coefficients. Comparison with Monte Carlo data indicates that the HNC equation yields accurate predictions for the zeroth mixing coefficient w0, while the simple EXP approximation yields qualitatively correct results. The Monte Carlo results for the first mixing coefficient, w1, are not precise enough to allow a quantitative comparison with other theories. However, a strong concentration dependence of w1 for dilute solutions, found previously for the nonprimitive models in the HNC approximation, is confirmed by the Monte Carlo results.

Notes: A new model electron–ammonia pseudopotential parameterized to ab initio quantum chemistry calculations on small lithium ammonia clusters, Li(NH3)n(n=1,4), is studied in a variety of environments. For Li(NH3)n clusters, n=16,32,64,128 the valence electron of the lithium is found to exist in a surface state far from the cation which is localized near the center of mass of the cluster. No bulk states were stabilized. Cluster anions (NH3)−n were also studied and the electron centroid-cluster center of mass probability distribution for (NH3)−64 calculated using umbrella sampling. In the present model, there is apparently no barrier to the dissociation of the surface states. No bulk states were found. This set of results appears to disagree with experiments which have been interpreted to indicate bulk states for cluster anions and the clusters containing lithium. Bulk properties of both a single excess electron and the lithium atom in solution are also reported. The solvation energy agrees well with experiment but the spectrum of the excess electron remains somewhat blue shifted as in earlier calculations. However, the valence electron of the lithium atom is found to spontaneously dissociate; a property not reproduced in previous work.

Notes: The geometrical structures of M+(Ar)n ions, with n=1–14, have been studied by the minimization of a many-body potential surface with a simulated annealing procedure. The minimization method is justified for finite systems through the use of an information theory approach. It is carried out for eight potential-energy surfaces constructed with two- and three-body terms parametrized from experimental data and ab initio results. The potentials should be representative of clusters of argon atoms with first-row transition-metal monocations of varying size. The calculated geometries for M+=Co+ and V+ possess radial shells with small (ca. 4–8) first-shell coordination number. The inclusion of an ion-induced-dipole–ion-induced-dipole interaction between argon atoms raises the energy and generally lowers the symmetry of the cluster by promoting incomplete shell closure. Rotational constants as well as electric dipole and quadrupole moments are quoted for the Co+(Ar)n and V+(Ar)n predicted structures.

Notes: Dynamic light scattering (depolarized Rayleigh and polarized Rayleigh–Brillouin) and quasielastic neutron scattering are employed to study the dynamics of the glass-forming liquid di-2-ethylhexyl phthalate (DOP) (Tg=184 K). The depolarized Rayleigh scattering measurements were made in the temperature range from 303 to 433 K, the polarized Rayleigh–Brillouin measurements in the range from 263 to 433 K, and the quasielastic neutron-scattering measurements in the range from 37 to 312 K and in the Q range from 0.33 to 1.84 A(ring)−1. The orientation times for DOP, obtained from a single Lorentzian fit to the experimental depolarized spectra at high T, are in good agreement with recent dielectric data for the primary (α) relaxation. However, at lower T, the viscosity increases more strongly than the orientation times and the Stokes–Einstein–Debye equation which can adequately describe the dynamics in the high-T range is insufficient at temperatures close to Tg. The relaxation time obtained from the Rayleigh–Brillouin experiment is about 1 order of magnitude faster than the orientation times. In the neutron-scattering experiment we find a strong decrease of the elastic intensity and a corresponding increase of the quasielastic intensity around Tg.The data analysis with respect to the dynamics (from a two Lorentzian fit) revealed the existence of three processes affecting the high-frequency range: (i) a "fast'' (τ2∼10 ps) Q-independent motion with weak T dependence (E2=1.54 kcal/mol), (ii) a "slow'' Q-dependent motion, and (iii) a flat background increasing with T and Q. The fast process is discussed in terms of a very localized motion of the phenyl group (β relaxation) and, as such, as a precursor of the the primary (α) relaxation. The relaxation time of this process (τ2) was found to compare nicely with the time τmax from the Rayleigh–Brillouin (RB) experiment suggesting that the latter is caused by fast localized motions. The slow process is discussed in terms of the jump-diffusion model. The activation energy associated with the jump-diffusion times is 6.1 kcal/mol and it is associated with large-scale diffusional motion of the DOP molecule. The relaxation times obtained from this process are compared with the relaxation times obtained from the depolarized and dielectric techniques for the primary relaxation. Finally, the background can be identified with fast local motions and/or low-frequency excitations relaxing outside the energy window of our experiment.

Notes: We consider the effect of shear velocity gradients on the size (L) of rodlike micelles in dilute and semidilute solution. A kinetic equation is introduced for the time-dependent concentration of aggregates of length L, consisting of "bimolecular'' combination processes L+L' →(L+L') and "unimolecular'' fragmentations L→L'+(L−L'). The former are described by a generalization (from spheres to rods) of the Smoluchowski mechanism for shear-induced coalesence of emulsions, and the latter by incorporating the tension-deformation effects due to flow. Steady-state solutions to the kinetic equation are obtained, with the corresponding mean micellar size (L¯) evaluated as a function of the Peclet number P, i.e., the dimensionless ratio of flow rate γ(overdot) and rotational diffusion coefficient Dr. For sufficiently dilute solutions, we find only a weak dependence of L¯ on P. In the semidilute regime, however, an apparent divergence in L¯ at P(approximately-equal-to)1 suggests a flow-induced first-order gelation phenomenon.

Notes: The methods developed in our earlier papers, dealing with Raman scattering from small isolated, spherical particles, uniformly coated with radially uniaxial molecules, are extended to include scattering driven by applied multipolar fields. Such fields are generated by the action of the incident electromagnetic radiation on extraneous material, such as other particles belonging to the same high-density dispersion. In this paper we treat inelastic scattering associated with the Raman dipole matrix elements of the adsorbed molecules ignoring other contributions. Analytical expressions are developed for calculating the primary Raman–Stokes moments given the amplitudes of the applied multipolar fields. As a test of the practicability of the procedure, a full calculation for parallel polarization has been performed for two identical contacting, CO-coated Ag particles oriented with their line of centers parallel to the direction of polarization of the incident radiation. For this case, the effect of interparticle coupling is to broaden and intensify the excitation spectrum, extending the region of intense Raman scattering to much lower frequencies. Intensity enhancements approaching 106 are calculated for frequencies as low as 80% of the single-particle surface-plasmon resonance frequency.

Notes: We have simulated the photodissociation of CH3Br adsorbed at a variety of surfaces. We have considered photodissociation at a smooth LiF (001) substrate and at three rough LiF surfaces which were constructed by removing atoms from the smooth surface. We have also considered photodissociation from several surfaces which have the same structure as the β phase of solid CH3Br to simulate dissociation from high coverages of the adsorbate where CH3Br ice is formed. The simulations were performed using the stochastic classical trajectory method. The asymptotic photofragment kinetic energy and angular distributions were determined and compared with the experimental results of Harrison et al. [J. Chem Phys. 89, 1475 (1988)] and Tabares et al. [J. Chem. Phys. 86, 738 (1987)]. When CH3Br is oriented with CH3 toward a surface, the CH3 kinetic energy distributions are shifted to much lower energies due to energy loss from multiple collisions with Br and the surface; the angular distributions are also significantly broadened. Much of the energy loss in these collisions goes into the translational mode of the Br fragments, causing the Br kinetic energy distributions to have a high-energy tail. When the molecule is in this orientation in a restricted geometry, collisions from the CH3 fragment lead to more effective energy transfer causing the peak of the Br kinetic energy distributions to be shifted to much higher energies and the corresponding angular distributions to become narrower. The main features of the experimental results from photodissociation of CH3Br adsorbed on LiF can be qualitatively explained using the results of the classical trajectory simulations.

Notes: The microscopic properties of the inner surface of the cavities of Anopore and Nuclepore membranes are investigated with scanning electron microscopy (SEM), nitrogen adsorption isotherms, and deuterium nuclear magnetic resonance (2H-NMR). Useful information about the cavity orientation and morphology, and internal surface area is obtained. Analysis of SEM photographs yields estimates of the surface area and the porosity of these membranes which complements the adsorption results. The orientation of liquid crystals permeated in the cavities of Anopore and Nuclepore membranes is probed with the 2H-NMR technique. It is found that the orientation of the liquid crystal molecules is governed by the confining volume and surface conditions. The 2H-NMR spectral line shape of the confined liquid crystal also provides information on the substrate morphology and roughness that is consistent with SEM and adsorption experiments.

Notes: We develop a simple but general three-variable model skeleton to describe complex nonlinear behaviors in electrochemical processes taking place at either a hanging mercury drop electrode (HMDE) or a rotating-disk electrode (RDE). We apply our formalism to the reduction of indium(III) at a HMDE in the presence of thiocyanate, a reaction known to exhibit complex mixed-mode and chaotic oscillations. Besides the role of the negative Faradaic impedance in destabilizing the electrochemical system, mass transport appears to be crucial as the model explicitly takes into account, in a truncated fashion, the time-dependent relaxation of the concentration profile. We study in detail the nonlinear dynamic behavior of our model of the indium/thiocyanate system and a RDE model. The models support mixed-mode sequences that appear either as incomplete Farey sequences or as periodic-chaotic sequences, which we discuss in terms of an incomplete homoclinic scenario whose definition and properties are worked out here. Our results compare very well to the experimental observations in the indium/thiocyanate system and the electrodissolution of a rotating copper disk in phosphoric acid. This satisfactory agreement strongly suggests that diffusion relaxation is an important phenomenon in electrochemical oscillations and could be the essential third variable in many dynamical electrochemical processes.

Notes: The electronic structure of BaH is investigated using a 10-electron relativistic pseudopotential on Ba, frozen core configuration interaction (CI) with three active electrons and core-polarization potential. Fine structure is taken into account with a semiempirical spin–orbit operator. The electronic properties of all states dissociating into Ba(6s2,6s15d1,6s16p1) +H(1s) are obtained and generally found in agreement with experiment within 0.03 A(ring) for equilibrium distances, 300 cm−1 for transition energies, 30 cm−1 for vibrational frequencies, except for the D 2Σ+ state. The methodology used in this work is discussed in the light of the results presented here.

Notes: We study the phase diagram and scattering functions of a ternary mixture of A and B homopolymers, in equal concentrations, and AB diblock copolymer. We locate the disorder line, which marks the appearance of damped oscillatory decaying correlation functions in the system, and also the Lifshitz and equimaxima lines of the structure functions. At these lines, the oscillatory component becomes dominant in a given correlation so that the peak in scattering intensity is no longer at the zero wave vector. We find that while the Lifshitz line of the structure function of all A monomers is quite close to the disorder line, that of A monomers in the homopolymer only is far from it. This shows that the copolymer orders the homopolymers inefficiently, a behavior which is also reflected in the copolymer's weak ability to solubilize them, and which contrasts with the amphiphilic solubilizers of oil and water. The disorder line and all Lifshitz lines meet at a Lifshitz tricritical point. As these ternary mixtures provide a rare opportunity to study this unusual point, we discuss its effects on the structure functions and surface tensions.

Notes: The introduction of a quadratic composition dependent χ into Flory–Huggins theory of binary blends is shown to create three new types of phase behavior in which two pairs of coexisting phases may be present in the two-phase region. This leads, for instance, to an unstable A-rich mixture separating into two A-rich phases instead of A- and B-rich phases.

Notes: The anharmonic potential energy surface of water has been computed ab initio using an augmented coupled cluster method and various basis sets. Whereas the Pople 6–311 G family is manifestly unsatisfactory, Huzinaga–Dunning basis sets perform quite well. The [5s4p2d1f,3s2p] surface reproduces harmonic frequencies and anharmonicity constants to better than about 2 and 1 cm−1, respectively. For quantitative agreement with experiment, both f functions on oxygen and inclusion of core correlation seem to be prerequisite. Comparison with various experimentally derived force fields reveals that the ab initio force field is of comparable quality. From the best computed force field, a set of spectroscopic constants has been derived for all important isotopomers of water. Using a hybrid analytic/direct summation method recently developed by the present authors, the thermodynamic functions gef(T), hcf(T), S0, and Cp are computed including exact account of anharmonicity and rovibrational coupling, and very good analytical approximations to centrifugal distortion and quantum rotation effects. The computed functions substantially revise previous literature results at high temperatures. Differences between thermodynamic functions from various computed force fields are an order of magnitude smaller than these errors. Thermodynamic tables in JANAF style from 100 to 3000 K, as well as a full set of rovibrational spectroscopic constants, are presented. It is concluded that obtaining force fields of near-spectroscopic accuracy, and thermodynamic tables of very high accuracy, is presently feasible for small polyatomic molecules.

Notes: The collective dynamics of liquid methanol-d4 is studied by means of molecular-dynamics simulation. The model potential is validated by means of lattice energy calculations and shows a very good agreement with the experimentally obtained crystal structure. Center-of-mass density and momentum fluctuations are investigated in the (Q,ω) region which is also accessible to inelastic neutron-scattering (INS) techniques. A simple viscoelastic model previously used for the analysis of INS data is tested against the dynamic structure factor computed from the simulation. A direct comparison with the INS results themselves is also made and qualitative agreement is found. Also, a tentative assignment of the peaks appearing in the current–current correlations is made on the basis of lattice-dynamics calculations for the polycrystalline low-temperature α phase.

Notes: A small-angle light-scattering experiment probing the time-dependent behavior of a semidilute polymer solution under simple shear is presented. Specifically, this paper investigates time-dependent anisotropic growth and orientation of concentration fluctuations in a semidilute solution of polystyrene in dioctyl phthalate above the quiescent cloud-point temperature. Two flow cells, which permitted investigation of the structure and dynamics of concentration fluctuations in both the flow-shear and the flow-vorticity planes, were used. Concentration fluctuations were found to be enhanced anisotropically by the shear flow in both planes. In the flow-vorticity plane, shear-enhanced concentration fluctuations orient along the vorticity direction, while in the flow-shear plane, the axis of average orientation is at an angle between 135° and 45° to the direction of flow. Qualitative trends of this anisotropic growth of concentration fluctuations are in agreement with the predictions of the Helfand–Fredrickson model in the flow-shear plane. Furthermore, the Onuki–Doi theory relating the scattering dichroism to the structure factor has been verified experimentally by comparing scattering dichroism calculated from scattering patterns with previous experimentally measured dichroism. Scattering dichroism, both calculated from structure factors and experimentally determined over a range of shear rates and temperatures, is demonstrated to depend only on the Weissenberg number. Relaxation of shear-enhanced fluctuations on cessation of flow allowed for the calculation of a diffusion coefficient.

Notes: A nonlocal nonlinear evolution equation is proposed that describes pulse formation in a dissipative system. A novel feature of the equation is that it can be solved exactly through a linearization procedure. The solutions are constructed under appropriate initial and boundary conditions and their properties are investigated in detail. Of particular interest is pulse formation, which is caused by a balance between nonlinearity and dissipation. The asymptotic behavior of the solution for large time is then represented by a train of moving pulses with equal amplitudes. The corresponding position of each pulse is shown to be characterized by the zero of the Hermite polynomial, irrespective of initial conditions.

Notes: In the construction of a tensor product of quaternion Hilbert modules, given in a previous work (real, complex, and quaternionic), inner products were defined in the vector spaces formed from the tensor product of quaternion algebras H modulo an appropriate left ideal in each case. Under conditions that are necessary for the definition of a scalar product in the quaternionic Hilbert modules and a natural condition on the algebraic structure, it is proven that the scalar products which are defined are unique.

Notes: The concept of (full) polarization subalgebra in a Group Approach to Quantization on a Lie group G˜ as a generalization of the analogous concept in geometric or standard quantization is discussed. The lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition. A generalization of the subalgebra of (full) polarization is then provided, made out of higher-order differential operators in the enveloping algebra of G˜. Higher-order polarizations can also be used to quantize nonanomalous theories in different "representations.'' Numerous examples are analyzed, including the finite-dimensional dynamics associated with the Schrödinger group, which presents an anomaly, and an infinite-dimensional anomalous system associated with the Virasoro group. In the last example, the operators in the higher-order polarization are in one-to-one correspondence with the null vectors in the Verma module approach.

Notes: The vacuum energy for a massless scalar field defined on a hyperbolic manifold H3/Γ is evaluated using zeta-function and heat kernel regularization techniques and Selberg trace formula for cocompact group Γ. A negative contribution to the vacuum energy associated with trivial line bundle related to character χ=1 is found.

Notes: The regulated propagator G'ε(ξ,ξ') is considered for points ξ and ξ' on the two-dimensional flat torus and with a regularization analogous to the proper-time method with cut-off ε. The inequivalent tori are labeled by a modular parameter τ. In this regularization, the propagator has an expansion in eigenmodes which is a particular Jacobi–Riemann function with beautiful properties and which can be transformed to an integral representation by a Sommerfeld–Watson method. This latter method allows one to derive exact analytic expressions in several domains of the parameters ||ξ−ξ'||, ε, and of the parameter τ. For generic value of τI≡Im(τ) and small value of the cut-off ε we recover the known results relative to short-distance phenomena. At large value of τI(approximately-equal-to)O(cst/4πε), the propagator diverges as τI and the coefficient is computed. A general formula for the Weyl variation of the propagator is derived; again the behavior of the Weyl variation at fixed τI is reproduced, while the behavior at large τI is novel. As a result we find that to O((square root of)1/τI) the Weyl variation of the propagator at large τI is local in the Weyl variation δcursive-phi(ξ).

Notes: While a light wave travels in a nonvacuum medium with speed less than the speed of light in vacuum, its energy-momentum four-vector is spacelike. Thus the little group for this light wave is locally isomorphic to O(2,1). It is shown that the transformation of this little group produces observable effects on the light wave, in contrast to the case of the E(2)-like little group for photons in vacuum. It is also shown that the little group can be defined for a localized light wave consisting of a superposition of waves with different frequencies traveling in the same direction.

Notes: The two-dimensional Euclidean algebra provides an algebraic setting for q-Bessel functions. Using a two-variable realization of this algebra, q analogs of the Lommel and Graf addition formulas for the q-Bessel functions are obtained.

Notes: The main result of this paper is the complete classification of the third-order ordinary differential equations according to their symmetries. The same classification was done for second order by Tresse ["Détermination des invariants ponctuels de l'équation différentielle ordinaire du second ordre y‘=ω(x,y,y'),'' Gekrönte Preisschrift, Hirzel, Leipzig (1896)], and recently for arbitrary order linear ordinary differential equations. The sections preceding the classification consist of a brief description of the concepts and methods along the lines of Krause and Michel [Lecture Notes Phys. 382, 251 (1991)]. These sections also contain some definitions and a table listing the prolongations of a few vector fields. Finally, two appendices give additional information relevant to equations in real variables and describe how some of the results can be easily generalized to higher orders.

Notes: The classical hypervirial and Hellmann–Feynman theorems are used to formulate a "perturbation theory without Fourier series'' that can be used to generate canonical series expansions for the energies of perturbed periodic orbits for separable classical Hamiltonians. As in the case where these theorems are used to generate quantum mechanical Rayleigh–Schrödinger perturbation series, the method is very efficient and may be used to generate expansions to large order either numerically or in algebraic form. Here, the method is applied to one-dimensional anharmonic oscillators and radial Kepler problems. In all cases, the classical series for energies and expectation values are seen to correspond to the expansions associated with their quantum mechanical counterparts through an appropriate action preserving classical limit as discussed by Turchetti, Graffi, and Paul. This "action fixing'' is inherent in the classical Hellmann–Feynman theorem applied to periodic orbits.

Notes: A pure jump Bernstein process whose probability density is the product of the solution of the (imaginary time) Schrödinger equation and its adjoint equation, associated to a class of time dependent Hamiltonians describing spin-1/2 particles, is constructed. A path integral representation of these solutions, in terms of this process, is obtained as well as a calculus over path space for the associated Euclidean quantum observables.

Notes: An SU(1,1) Lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems. The Liouville space formulation, thermo field dynamics, and the disentanglement theorem of SU(1,1) Lie algebra play essential roles in this formulation. In the Liouville space, the time-evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of SU(1,1) Lie algebra and the thermal state condition of thermo field dynamics. The presented formulation is used for investigating a dissipative nonlinear oscillator, the quantum mechanical model of phase modulation, and the photon echo in the localized electron–phonon system. This algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems.

Notes: In theories involving gravity, including QCD-generated gravitylike effects in hadrons, SA(4,R) plays a role. Its single Casimir invariant and that of its SA(2,R) and SA(3,R) subgroups are evaluated. The group orbits are studied and the unitary irreducible representations are classified.

Notes: In this paper, the relation between the Painlevé property for ordinary differential equations and some coordinate transformations in the complex plane is investigated. Two sets of transformations are introduced for which the transformations of singularities structure in the complex plane are explicitly computed. The first set acts only on the dependent variables while the second one acts also on the independent variables. It allows for defining an extended Painlevé test which is coordinates invariant. Our results are applied on various problems arising in the singularity analysis such as the problem of negative resonances and the weak Painlevé conjecture. The method for finding new first integrals for dynamical systems is also shown.

Notes: The first and the second Hamiltonian structures of the supersymmetric KdV equation are derived, following the conventional approach of Gel'fand and Dickey. In particular, it is shown that the nonlocal nature of the first bracket is a direct consequence of the constrained nature of the Lax operator.

Notes: The physical channel with a cubic pole in the D5 affine Toda field theory is studied, showing the consistency with the S matrix result up to β4 order. It is found that the interplay between the Landau singularity and renormalization plays a very crucial role in the physical channels.

Notes: A recursion is derived that yields, exactly, the composite nearest-neighbor degeneracies when several kinds of simple particles [indistinguishable within each kind] are distributed on a two-dimensional, square-cell lattice in such a way as to create prescribed numbers of the various possible types of nearest-neighbor pairs.

Notes: A model of the Markovian chain of N atoms is constructed with the help of extension theory. Behavior of averaged wave functions of random Dirac operator is studied for finite and large N with respect to parameters of the model and Markovian process. Study of a random Dirac operator is connected with the study of nonrandom double-dimensional non-self-adjoint Dirac operator. Dynamics of bands of effective periodic operator with respect to the Markovian parameter is presented.

Notes: Two new results are derived for the scattering theory of waves in nondissipative elastic media. These results are (1) a Newton–Marchenko equation, and (2) a generalized optical theorem. The Newton–Marchenko equation is an integral equation for the wavefield in terms of scattering amplitude data. It is supposed that it may play an important role in the development of exact inverse scattering methods for elastodynamics. The generalized optical theorem provides explicit relations between the scattering amplitudes, and is expected to be useful in the analysis of both the forward and inverse scattering problem. The results are presented within the context of a localized, possibly anisotropic, inhomogeneity contained in an otherwise uniform isotropic elastic medium of infinite extent in R3.

Notes: The supergroup UOSp(1,2) is used to construct geometrically a twice graded supersymmetric supersigma model. In an appropriate contraction limit, the model reduces to the conventional supersymmetric sigma model.

Notes: Weyl orbits of affine rank 2 and 3 algebras are obtained analytically, along with the depths of their weights, in terms of an integrity basis consisting of sets of compatible (or adjacent) weights, one from each fundamental orbit. The fundamental orbits and others of the same level and congruence class are then decomposed into irreducible representations (IR). Since the orbit →IR matrix is triangular, it will be easy to invert it to get orbit multiplicities and subalgebra branching rules.

Notes: Coset spaces of the Euclidean group of rigid motions in three-space are used as a model of the rays in geometric polarization optics. The Haar measure invariance leads to canonicity of the phase space transformations. The phase space of optical rays undergoes a canonical map also under the effect of smooth refracting or reflecting surfaces between two optical media that is governed by Snell's law. This is a conservation law between two irreducible representations of the Euclidean group labeled by the refractive indices n, n' of the media. These surface transformations are canonical and, furthermore, factorize into two root transformations that are also canonical. This factorization applies equally to the transformation of the polarization vector. The root transformation permits the computation of an aberration expansion of the polarization field over the object screen. The generators of infinitesimal surface transformations are found, i.e., those that transform between media n and n+dn. It is shown that, together with displacements, they yield the Hamilton equations of optics for inhomogeneous media.

Notes: A method of constructing Lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed. It is shown that under some simple but essential conditions, the corresponding Lax operators may constitute infinite-dimensional Lie algebras with respect to the binary operation (large-closed-square)⋅,⋅(large-closed-square) given in this paper. Furthermore, the detailed analyses for the KdV couple, the AKNS couple, and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples.

Notes: In a series of papers Fordy and his collaborators have studied families of integrable models in (1+1) dimensions associated with Hermitian symmetric spaces. These models were also generalized to (2+1) dimensions. However, the generalization used is not unique, and in this paper a different generalization is considered, resulting in new families of integrable models in (2+1) dimensions.

Notes: A generalization of the Adler–Gel'fand–Dikii scheme is used to generate bi-Hamiltonian structures in two spatial dimensions. In order to implement this scheme, a Hamiltonian theory is built over a noncommutative ring, namely the ring of formal pseudodifferential operators. Bi-Hamiltonian structures generated in this way can be used for the Kadomtsev–Petviashvili equation as well as other integrable equations in 2+1.

Notes: Shifted vertex operator algebras and superalgebras are defined, both in bosonic and in fermionic pictures. Their isomorphism (Boson–Fermion correspondence) is shown. Partition functions of bosonic shifted vertex operator algebras are realized analytically and are interpreted as string path integrals over elliptic curves. Their modular properties follow, and elliptic systems are constructed based on fermionic shifted vertex operator algebras.

Notes: Einstein's equations are written in the form of covariant gauge theory in two-dimensional space with the binomial solvable gauge group, with respect to two noncommutative Killing vectors. In the one-dimensional case the theory is exactly integrable and in the two-dimensional case, the series of partial exact solutions is constructed.

Notes: This paper presents several new aspects of discrete kinetic theory (DKT). First a hierarchy of d-dimensional (d=1,2,3) models is proposed with (2d+3) velocities and three moduli speeds: 0, 2, and a third one that can be arbitrary. It is assumed that the particles at rest have an internal energy which, for microscopic collisions, supplies for the loss of the kinetic energy. In a more general way than usual, collisions are allowed that mix particles with different speeds. Second, for the (1+1)-dimensional restriction of the systems of PDE for these models which have two independent quadratic collision terms we construct different exact solutions. The usual types of exact solutions are studied: periodic solutions and shock wave solutions obtained from the standard linearization of the scalar Riccati equations called Riccatian shock waves. Then other types of solutions of the coupled Riccati equations are found called non-Riccatian shock waves and they are compared with the previous ones. The main new result is that, between the upstream and downstream states, these new solutions are not necessarily monotonous. Further, for the shock problem, a two-dimensional dynamical system of ODE is solved numerically with limit values corresponding to the upstream and downstream states. As a by-product of this study two new linearizations for the Riccati coupled equations with two functions are proposed.

Notes: In a recent article in this journal, asymptotic considerations were used to define appropriate boundary conditions for diffusive approximations to linear kinetic equations. A variational treatment was invoked to carry out the details of the analysis. Here these details are carried out exactly, and it is shown that the earlier variational results are accurate to a few percent.

Notes: A spectral parameter-dependent solution of the graded Yang–Baxter equation is obtained, which is universal in the sense that it lives in Uq(gl(m/n))⊗End(V) with V the vector module of Uq(gl(m/n)). The invariants of this quantum supergroup are constructed using this solution.

Notes: It is shown that a regularized determinant based on Hilbert's approach (which we call the "p determinant'') of a quotient of elliptic operators defined on a manifold with boundary is equal to the "p determinant'' of a quotient of pseudodifferential operators. The last ones are entirely expressible in terms of boundary values of solutions of the original differential operators. It is argued that, in the context of quantum field theory, these boundary values also determine the subtractions (i.e., the counterterms) to which this regularization scheme gives rise.

Notes: The character summation formula analogous to Dixon's theorem for the hypergeometric function 3F2 is derived for R and Qp.

Notes: Since the notion of angular momentum is defined in any dimension by using the exterior product in Rn, one would guess that central force problems in any dimension are completely integrable, as it is known for n=2 or 3. This is proved explicitly in this paper, by constructing n first integrals independent and involution: the energy and some combinations of the angular momentum components. It is shown that these problems are always reduced to a two-dimensional plane, and the invariant manifolds are topologically, the Cartesian product of those of the reduced problem times n−2 circle factors. It is also proved that for n(approximately-greater-than)2 these problems are degenerate in the sense that their Hamiltonians do not satisfy one of the hypotheses of the Kolmogorov–Arnold–Moser theorem for persistence of invariant tori, when one considers small perturbations.

Notes: The resonance properties of certain arrays of potentials defined by a self-adjoint integral operator of finite rank are studied. It is shown that a uniformly spaced infinite linear array of such potentials can admit zero-width resonances, and that a uniformly spaced circular array can admit resonances whose widths tend to zero exponentially, as the number of potentials tends to infinity while the arc-length spacing remains constant−properties that have previously been studied for the Fermi pseudopotential (point interaction). Additionally, it is demonstrated that the pseudopotential can be derived from such an integral-operator potential as an appropriate limit. The relevance of these calculations to the study of arrays of cylindrical dipoles in classical electromagnetism is discussed.

Notes: It is shown that conditions of causal simplicity and pseudoconvexity are closely related. It is proved that under certain reasonable conditions space-time is maximally null pseudoconvex. It is indicated that pseudoconvexity may play the role of global hyperbolicity even though it is a weaker condition. Consequently, obtained results provide some tentative evidence for the validity of the cosmic censorship hypothesis.

Notes: Ricci collineations for Gödel-type space-times are studied. The members of this family of metrics admitting these symmetries are identified. It is shown that these space-times do not admit Homothetic vector fields. Also, the unique Affine Collineation, associated to Gödel-type space-times, is exhibited.

Notes: A new method of integration of the Dirac equation on curved space-time is presented. The method is based on diagonalization and on separation of variables. The diagonalization allows the use of the well-known method of variables separation of scalar equations for integration of the Dirac equation. All conditions of diagonalization are found. All metrics and potentials admitting the method are presented.

Notes: A method is given for obtaining new exact solutions of nonlinear Schrödinger equation describing the propagation of pulses in optical fibers for positive group-velocity dispersion. The method is based on the construction of a certain complete integrable finite-dimensional dynamical system, whose solutions determine the exact solutions of nonlinear Schrödinger equation. The set of exact analytic solutions contains dark solitary waves, dark soliton, periodic, and stationary solutions which are very important from a physical point of view.

Notes: A formally exact analytical solution of the time-dependent rototranslational diffusion equation (RTDE) with mixed boundary conditions applied over both the translational and rotational variables has been derived. The approximation of isotropic translational motion has been imposed and no further approximations have been used in the calculations. The solution has been obtained in the form of a set of homogeneous linear algebraic equations.

Notes: The SL(3,R) theory of projective transformations of the plane is applied to the Lagrangians of all one-dimensional Newtonian linear systems. Noether and non-Noether equivalent Lagrangians, as well as the associated Noether and non-Noether constants of motion, are thus obtained in a completely general and systematic way. Complete unification is achieved by this group-theoretic approach to Lagrangians of one-dimensional linear systems.

Notes: It is shown that the sets of tensor superoperators for the super-rotation algebra can be used to build explicit bases for the representations of several superalgebras. The representations built in this way are the fundamental representations of the special linear superalgebras sl(2j+1||2j) and of the orthosymplectic superalgebras osp(2j+1||2j) and the (4j+1)-dimensional representations of osp(1||2) and sl(1||2) (Stavraki) superalgebras. It is shown that the chain osp(1||2)⊆osp(2j+1||2j) or osp(2j||2j+1) explains the existence of a series of nontrival zeros for the super-rotation 6-j symbol (SR6-j symbols).

Notes: Complete orbit–orbit branching rules, or equivalently, reduction of Weyl group orbits, between each simple algebra of rank 3 or less, and its equal-rank subalgebras and between F4 and each of its equal-rank subalgebras are given. The generic case An&supuline;An−1×U(1), the subjoining F4(approximately-greater-than)B3×U(1), and E8&supuline;E6×A2 (first and seventh highest weight labels nonzero) are also treated. Branching rules between rank 4 and 5 simple algebras and E6 and their equal-rank subalgebras are available in a depository.

Notes: The theory of generalized functions on the p-adic superspace Qn,mp over the (super)commutative Banach superalgebra Λ=Λ0⊕Λ1 with the trivial Λ1 annihilator is proposed. The p-adic Gaussian "supermeasure'' is defined as a generalized function on the p-adic superspace. The connection between the p-adic Gaussian integration and the p-adic gamma function of Morita is considered

Notes: Classical nonholonomic mechanical systems are studied whose evolution space is a fiber bundle τ: M→R. The framework is that of jet spaces in which the geometrical meaning of the theory emerges clearly. For systems with linear constraints a new interpretation is given of Appell's equations as first-order differential equations associated with a suitable vector field. The d'Alembert principle is formulated in an appropriate way to be generalized to systems with nonlinear constraints. The equivalence between the equations of motion arising from this generalization, the ones set up on Gauss' principle of least constraint and Hertz's principle of least curvature is established.

Notes: A new asymptotic formula in terms of elementary functions is demonstrated for the Laguerre polynomials Lαn(x) in the limit when all three parameters (n,α,x) become large. This asymptotic limit appears in the problem of obtaining the low-field semiclassical approximation from the full quantum mechanical behavior of a two-dimensional electron gas in a magnetic field.

Notes: The phase-space representation of supersymmetric quantum mechanics is given. The super Wigner function is constructed and its transformation property is investigated. In the case of a supersymmetric oscillator, it is explicitly shown how the invariance under the supersymmetry transformation determines the form of the super Wigner function.

Notes: The classical solution is used as a variable for the one-dimensional quantum-mechanical oscillator, and this leads to a separable differential equation solvable in closed form which yields a set of new quantum mechanical solutions, not energy eigenfunctions, that have the property of being localized. These new solutions are shown to be eigenfunctions of a time-independent position operator. The operator algebra of that operator and an analogous momentum operator is given, as well as the relationship of the eigenfunctions to the energy eigenfunction set. The new eigenfunctions, which are not square integrable but possess a δ-function normalization, themselves represent wavepackets that at times corresponding to when the classical position is at an extremum collapse to Dirac δ functions.

Notes: Various applications of the real trajectories in the complex-time method are discussed. It is shown that the proper way of summation over semiclassical orbits removes existing doubts and leads to the more general formulas than the standard WKB-type one in the energy eigenvalue problem. This method is also used in the description of explicitly time-dependent phenomena. In the problem of escaping from a metastable state, the estimation of a lifetime is in a good agreement with the inverse width of a false ground state energy.

Notes: Nondipersive solutions of the Dirac equation, that is, solutions depending on an arbitrary function F(u), where u is itself a solution of the characteristic equation (∇u)2−(c−1∂tu)2=0, are discussed.

Notes: This work continues investigations on the C*-algebra of bosonic strings started in Comm. Math. Phys. 136, 369 (1991). There a canonical section of a bundle was used implicitly in order to pass to an algebra of functions. In this article an algebra of sections is worked with directly. This clarifies the discussion on associativity of the algebra.

Notes: The "dyon'' system of D'Hoker and Vinet, consisting of a spin 1/2-particle with anomalous gyromagnetic ratio 4 in the combined field of a Dirac monopole plus a Coulomb plus a suitable 1/r2 potential (which arises in the long-range limit of a self-dual monopole), is studied following Biedenharn's approach to the Dirac–Coulomb problem: the explicit solution is obtained using the "Biedenham–Temple operator,'' Γ, and the extra twofold degeneracy is explained by the subtle supersymmetry generated by the "monopole helicity'' or generalized "Johnson–Johnson–Lippmann'' operator R. The new supersymmetry anticommutes with the chiral one discussed in previous papers.

Notes: Morse theory is used to rigorously obtain counting formulas and lower bounds for the total number of images of a background point source, not on a caustic, undergoing lensing by a single-plane microlens system having compact bodies plus either subcritical or supercritical continuously distributed matter. An image-counting formula is also found for the case when external shear is added. In addition, it is proven that a microlens system consisting of k lens planes will generate N = 2M− + Πki=1(1 − gi) images of a background point source not on a caustic, where M− is the total number of critical points of odd index of the time-delay map and gi is the number of stars on the ith lens plane. Morse theoretic tools also yield that the smallest value N can have is Πi=1k(1+gi).

Notes: The exact matrix elements for transitions from the ground state to some excited state of a nonrelativistic hydrogenic atom, accompanied by the emission or absorption of a real or virtual photon, are here obtained in closed form. The matrix elements are simple rational functions of the photon energy if the excited state is bound, or have a simple transcendental factor if the excited state is unbound. It follows from the form of these elements that the contribution of each such process to the second-order radiative correction to the ground state energy level is always finite, without renormalization. It also follows that the sum of these contributions over all bound intermediate states is finite, while the sum over all unbound intermediate states diverges at most logarithmically without renormalization.

Notes: Eigenfunctions of the curl derivatives are used in many different fields of theoretical physics. For example, in plasma physics, an eigenfunction of curl represents a certain eigenstate of a plasma that has the force-free property. The present theory proves that the spectral resolution of the curl operator gives a complete eigenfunction expansion of a solenoidal vector field with homogeneous boundary and flux conditions. Useful decomposition theories and a previous abstract result [Math. Z. 204, 235 (1990)] are concisely summarized. Explicit eigenfunctions have been studied in cylindrical geometries. Chandrasekhar–Kendall functions [Astrophys. J. 126, 457 (1957)], which are eigenfunctions of the curl derivatives in a cylindrical domain, give an orthogonal complete basis of the Hilbert space that is the orthogonal complement of the space of irrotational vector fields.

Notes: A new class of Gaussian ("Bernstein'') diffusion processes associated with semiclassical quantum dynamics is introduced and studied according to a new Euclidean strategy for the quantization of systems of particles in a potential ("Euclidean quantum mechanics''). All the measures known by other means, up to now, are included as particular cases of the present framework. It is shown that the theory may be regarded as a well-defined probabilistic version of Feynman's space-time approach distinct, in particular, from the one due to Kac, and with much closer relations with classical Lagrangian dynamics. Many explicit examples are considered.

Notes: The metaplectic representation of the quantum algebra suq(1,1) is shown to provide a group-theoretic setting for certain basic orthogonal polynomials generalizing the usual Gegenbauer polynomials.

Notes: Uq(sl(2)) covariant Bose and Fermi oscillators are studied systematically and applied to constructing vertex operators and Uq invariant blocks. For qp=−1 (p=2,3,...) a p2-dimensional unitary Fock space representation of the Bose oscillator algebra A−(q) is constructed. This representation does not allow, however, for the construction of all Uq invariant blocks associated with an indecomposable representation of Uq. An extension of A− is needed in this case, which involves operators creating states of q spin projection ±p/2 from the Fock vacuum. The extended algebra only has infinite-dimensional indefinite metric space representations. An algebraic notion of trace is introduced (provided q2≠1 in the Bose case). This allows the introduction of a family of (in general, mixed) Uq invariant "temperature'' states, such that the vacuum expectation value appears in the zero temperature limit.

Notes: Solutions of the coupled Einstein conformally invariant massless scalar field equations are derived under the assumption that the metric admits a four-parameter group of isometries with spacelike generators, where further, the three-parameter subgroup of isometries acts multiply transitive on two-dimensional surfaces. The general solution of the system is obtained and some properties concerning the singularity structure are briefly discussed. Results for all three cases are presented, i.e., for the case where the three-dimensional group of isometries acts on two-dimensional orbits of positive, negative, or zero curvature. The first two classes belong to the Kantowski–Sachs class of metrics, while the third one is of Bianchi type I with additional rotational symmetry.

Notes: Solutions of the coupled Einstein massless scalar field equation are discussed under the assumption that the metric belongs to the "Kantowski–Sachs'' class of metrics. More precisely, the general solution of the coupled Einstein massless scalar field equations is derived under the following assumptions: (a) The scalar field is massless minimally coupled to gravity and obeys the Klein–Gordon equation; and (b) the metric belongs to the spherical Kantowski–Sachs class of metrics. The main feature of the results is a tractable, very simple, compact form of the space-time metric. This is accomplished by expressing the solution in a special coordinate gauge. An analysis of the geometry shows the presence of curvature singularities which consist of a spacelike part and a null part. The spacelike singularity appears to "squeeze'' the SO(3) orbits to zero proper area while simultaneously stretching their proper separation to infinity length. The null part appears to squeeze to zero values both SO(3) orbits and their proper separation as well. There also exists a subclass of metrics for which the singularity structure is different. The null singularity is replaced by a spacelike part. Both "initial'' and "final'' singularities exhibit the same qualitative features and they are of finite proper time away from each other.

Notes: Under the assumption of weak spatial and temporal gradients as well as weak absorption and sources, it is well known that the classic kinetic equation describing linear particle transport has a diffusion equation as an asymptotic limit. Considered here is such particle transport in a stochastic Markovian mixture of two immiscible fluids. It is shown that a variety of asymptotic diffusive limits exist under the scaling indicated above, depending upon the additional scaling introduced for the Markov transition lengths. These lengths are indicative of the mean chord lengths of the fluid packets in the stochastic mixture, and it is thus found that the details of the diffusive limits are strongly dependent upon the presumed sizes of the fluid packets. Asymptotically consistent initial and boundary conditions are also derived for each diffusion description obtained.

Notes: WKB equivalent potentials (WKB-EP's) giving the same WKB spectrum as the q-deformed harmonic oscillator are determined in analytic form. For q being complex the WKB-EP resembles the Pöschl–Teller, Morse, or Woods–Saxon potentials, widely used in nuclear, hypernuclear, and molecular physics. For q real, the WKB-EP still goes to infinity at large x, while at small x it resembles a sextic oscillator. The WKB-EP's of the anharmonic oscillators with Uq(2) or SUq(1,1) symmetries, used for the description of vibrational molecular spectra, are also calculated. They correspond to q-deformed versions of the modified Pöschl–Teller potential.

Notes: The new type of reflection principle applicable to the Laplace transform of the arrival probability for biased walks developed by Khantha and Balakrishnan is extended to the cases of biased correlated walks with absorbing and/or reflecting walls. An exact expression for the mean escape time 〈t(approximately-greater-than) for the linear discrete walk bounded by absorbing walls at 0 and N is obtained as follows: 〈t(approximately-greater-than)=N(1−δ)[2(N−1)(a−d)]−1[(aN+dN)/ (aN−dN)−(1+δ)/(a−d)] +δ[(N−1)(a−d)]−1{1−[N(a−d)dN−1]/ (aN−dN)}, where a=1/2(a+δ+ε) and d=1/2(1+δ−ε), and δ and ε are the degrees of correlation and anisotropy, respectively.

Notes: A formalism comprising and extending quasifree second quantization of charged bosons and fermions is presented. The second quantization of one-particle observables leads to current superalgebras, and a super-Schwinger term shows up. Anticommuting parameters are introduced in order to construct super-Bogoliubov transformations mixing bosons and fermions. As an application, representations of Lie superalgebras are given which are semidirect products of extensions of affine Kac–Moody algebras and the Virasoro algebra, and of the super-Virasoro algebra.

Notes: A body or collection of bodies made of perfect fluid can be described in general relativity by a solution of the Einstein–Euler system where the mass density has spatially compact support. It is shown that for certain equations of state there exists a wide class of solutions of this type corresponding to appropriate initial data given on a spacelike hypersurface. This class is not constrained by any symmetry requirements. The key element of the proof is to write the equations as a symmetric hyperbolic system which is regular both for nonvanishing density and in vacuum.

Notes: Exact expressions are obtained for Grassmann integrals of exponentials corresponding to the partition functions of systems of lattice fermions with nearest-neighbor interactions. It is achieved by relating to the dimer problem and use of its combinatorial techniques.

Notes: The Zp spin system in two dimensions from the point of view of fractional statistics is studied herein. It is found that, for p≤4, the system represents a gas of particles having a specific fermion number when p=2 corresponds to fermion number 1 and p=3 and 4 corresponds to a fractional fermion number. However, for p≥5, the corresponding system does not represent particles having any specific statistics; rather, it represents a hedgehog behavior. The XY model represented by a Zp spin system at p→∞ limit corresponds to bosonic kinks composed of vortex–antivortex pairs below a certain temperature. The two massive phases observed in a Zp system for 2≤p≤4 is related to their intrinsic fermionic character, and the three phases observed for p≥5 when a massless phase occurs in between two massive phases is related to the hedgehog behavior of the corresponding particles so far as their statistics is concerned. The XY model corresponding to a Zp spin system at p→∞ represents massive bosonic kinks as bound states of vortex–antivortex pairs below a certain temperature when the spin waves are massless and, as the temperature goes up, the bound states dissociate and massless kinks and massive spin waves appear. The nonexistence of an ordered phase for a continuous symmetry model is related to the inherent bosonic character of the corresponding particles.

Notes: It is shown that a relativistic wave equation with a scalar coupling correctly describes the Dirac oscillator in the sense that in the nonrelativistic limit this results in the Schrödinger equation for the harmonic oscillator with an extra σ⋅rˆ term. It is demonstrated that this equation has built-in supersymmetry, and that it guarantees the stability of the Dirac vacuum.

Notes: Using an enlarged space of n-point functions F and fictitious time s linear problems are reformulated in quantum and statistical field theory in such a way that non-negative sesquilinear forms play a crucial role in the description of system. First- and second-order "evolution'' type equations are considered and their connection with weak and strong convergence is shown. Supersymmetrical "evolution'' of systems is presented without increasing the number of fields. The five-dimensional field theory was found to be useful in developing the formalism.

Notes: Based on the renormalized pointwise product version of the operator formalism, the new examples of the quantum projective [sl(2,C)-invariant] field theory, connected with the vertex operators of the discrete spectrum in the model of the Verma modules over the Lie algebra sl(2,C), are constructed.

Notes: The singularity structure of the solutions of the equations of motion for the Sigma and Skyrme model Lagrangians with the hedgehog ansatz is investigated. In both cases the solutions present superposed a polelike term and a logarithmic branch point. The set of solitonic configurations of the Skyrme model is characterized by the sequence of locations of their singularities on the negative real axis of the dimensionless variable z = (eFπr)2, with an accumulation point at z=0. The first few terms of the Laurent series representing the Skyrme soliton profile are shown to reproduce well the exact values in an interval about the origin. Padé approximants to the residual power series obtained after subtraction of the dominant pole term are modified in order to satisfy the constraints imposed by the asymptotic power series expansion, and approximate representations are built for the Skyrme soliton configuration, which are shown to combine simplicity with accuracy.

Notes: A set of coupled higher-order nonlinear Schrödinger equations, which describe electromagnetic pulse propagation in coupled optical waveguides, is formulated in terms of an eigenvalue problem. Using that result, the inverse scattering problem is solved and explicit soliton solutions are found. Additionally, linear coupling terms are studied systematically.

Notes: In this paper the structure of vortex coadjoint orbits pertaining to perfect fluids having smooth vorticities in R3, within the framework set up by J. Marsden and A. Weinstein [Physica D 7, 305–323 (1983)], in terms of an associated Hamiltonian Kähler manifold (the Clebsch manifold, described in terms of the so-called Clebsch variables) is investigated. The topological quantization of Mikhailov and Kuznetsov is related to geometric quantization. Natural candidates for the coherent states on the Clebsch manifold are also exhibited.

Notes: A natural gauge condition is used to select special orthonormal frames and thereby a preferred teleparallel geometry on a Riemannian manifold of any dimension. The condition is equivalent to a nonlinear elliptic system for a rotation. Existence and uniqueness of solutions are discussed. Certain special cases and applications are noted.

Notes: A variational principle for the Nambu dynamics is analyzed. Since the equations of motion single out a distinguished two-form rather then a one-form, the usual construction of the action S[γ] as an integral of a one-form along the curve γ on the extended phase space has to be modified.

Notes: The time domain scattering problem for axially symmetric fields and a dissipative stratified slab is considered. Explicit representations are derived for the wave splitting operators that factorize the dissipative Hankel transformed wave equation. The Green functions approach associated with this splitting is developed. Partial differential equations (PDEs) for the Green functions are derived and used to derive the imbedding equations for the reflection and transmission kernels. Numerical results for the internal fields are given in the direct problem, and the phase velocity is reconstructed in the inverse problem.

Notes: The classical equations of motion on compact coherent state manifolds that have Hermitian symmetric space structure are considered. In the case of Hamiltonians linear in the generators of the group that determines the structure of the manifold of coherent states, the classical motion and the quantum evolution are both described by the same first-order second degree differential equation. The explicit forms of the classical equations of motion as Matrix Riccati equations on the Grassmann manifold GA(Cn) and the coset manifold SO (2n)/U(n), attached to the Hartree–Fock and, respectively, Hartree–Fock–Bogoliubov problem are obtained.

Notes: It is established in this paper that an arbitrary operator algebra of the quantum conformal field theory admits the strict representation by matrices, whose elements belong to the special operator algebra−the algebra of the vertex operators in the model of the Verma modules over the Virasoro algebra.

Notes: Vector coherent state (VCS) theory is applied to the group chain SOn+2&supuline;SOn×SO2. Matrix elements of SOn+2 generators in the SOn+2&supuline;SOn×SO2 basis are derived. A new formula for the evaluation of some isoscalar factors for SOn+2&supuline;SOn×SO2 with branching multiplicity is derived in the VCS framework. As a simple example, a new expression of some isoscalar factors for SO5&supuline;SO3×SO2, which involves only 6j coefficients and K-normalization factors, are obtained by using this formula.

Notes: Multiple series generating relations for the generalized continuous Hahn polynomials are obtained. Special cases consist of results involving products of several confluent hypergeometric functions.

Notes: By performing the most general Born–Oppenheimer procedure, the (non-Abelian) Berry connection for quantum systems in a quantum environment is derived. This method is then applied to the rapid rotation of a particle about a slowly changing axis, as exemplified by the electronic motion of a diatomic molecule. The angular part of the resulting dynamics for the quantum environment is equivalent to that of a monopole.

Notes: Quantum algebras are represented by functions on phase space in terms of *-product, and explicit examples for Uη(sl(n+↑ 1), R) with n≥1 are given.

Notes: The nature of primary operators of coset conformal field theories is discussed. Some primary states of typical coset conformal field theory are explicitly calculated that have correct conformal dimensions and dynamics. The nature of modular invariant partition functions from the diagonal sum of absolute squares of characters of coset primary states is studied.

Notes: Octonionic root system of E8 is decomposed as the 9 sets of Hurwitz integers, each set satisfying the binary tetrahedral group structure, and the 12 sets of quaternionic units, each set obeying the dicylic group structure of order 12. This fact is used to prove that the automorphism group of the octonionic root system of E7 is the finite subgroup of G2, of order 12 096, an explicit 7×7 matrix realization of which is constructed. Possible use of the octonionic representation of the E6 root system as orbifolds and the relevance of the binary tetrahedral structures with the statistical mechanics models are suggested.

Notes: In the classification of finite-dimensional modules of Lie superalgebras, Kac distinguished between typical and atypical modules. Kac introduced an induced module, the so-called Kac module V¯(Λ) with highest weight Λ, which was shown to be simple if Λ is a typical highest weight. If Λ is an atypical highest weight, the Kac module is indecomposable and the simple module V(Λ) can be identified with a quotient module of V¯(Λ). In the present paper the problem of determining the composition factors of the Kac modules for the Lie superalgebra sl(m/n) is considered. An algorithm is given to determine all these composition factors, and conversely, an algorithm is given to determine all the Kac modules containing a given simple module as a composition factor. The two algorithms are presented in the form of conjectures, and illustrated by means of detailed examples. Strong evidence in support of the conjectures is provided. The combinatorial way in which the two algorithms are intertwined is both surprising and interesting, and is a convincing argument in favor of the solution to the composition factor problem presented here.