ALBERT

Notes: In this paper we address the problem of the reflection coefficient limit in a periodic stratified structure when the number of periods tends to infinity. We show that the graph of the reflected energy admits a superior envelope and we exhibit an explicit formula. © 1998 American Institute of Physics.

Notes: Using successive approximations, we derive a hierarchical set of integro-differential equations for the evolution of the statistical moments of a wave propagating through non-Gaussian longitudinally correlated media. A closure procedure in the spirit of the Ornstein–Uhlenbeck process produces the most general set of equations currently available. The procedure can be applied to a large class of Cauchy problems in mathematical physics. © 1998 American Institute of Physics.

Notes: An analytic-bilinear approach for the construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. A generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy, and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. The resolution of functional equations also leads to the τ function and addition formulas to it. © 1998 American Institute of Physics.

Notes: We construct the supersymmetric extensions of the Darboux–Bäcklund transformations (DBTs) for the Manin–Radul super Korteweg–de Vries (KdV) hierarchy using the super-pseudo-differential operators. The elementary DBTs are triggered by the gauge operators constructed from the wave functions and adjoint wave functions of the hierarchy. Iterating these elementary DBTs, we obtain not only Wronskian-type but also binary-type superdeterminant representations of the solutions. © 1998 American Institute of Physics.

Notes: Gowdy space–times are generalized to admit two commuting spatial local Killing vectors, and some new varieties of them are presented, which are all closely related to Thurston's geometries. Explicit spatial compactifications, as well as the boundary conditions for the metrics, are given in a systematic way. A short comment on an implication of their dynamics toward the initial singularity is made. © 1998 American Institute of Physics.

Notes: We consider quasilinear parabolic equations with gradient diffusivity ut=div(|∇u|σ∇u)+f(u), x∈RN, t〉0, where σ≠−1 is a fixed constant and f(u) is a given smooth function. We also study quasilinear parabolic equations with a gradient-dependent coefficient ut=h(|∇u|)Δu+f(u), with a smooth function h(p). For both classes of equations we derive first-order sign-invariants, i.e., first-order operators preserving their signs on the evolution orbits {u(⋅,t),t〉0}. We give a complete description of (maximal) sign-invariants of prescribed structures. As a consequence, we construct new exact solutions of some quasilinear equations. © 1998 American Institute of Physics.

Notes: A thorough analysis of Lie super-bialgebra structures on Lie super-algebras osp(1;2) and super e(2) is presented. A combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied. In all the cases, Poisson–Lie brackets on supergroups are found. The possibility of quantizing them in order to obtain quantum groups is discussed. It turns out to be straightforward for all but one structure for the super-E(2) group. © 1998 American Institute of Physics.

Notes: The number of bound states of the one-dimensional Schrödinger equation is analyzed in terms of the number of bound states corresponding to "fragments" of the potential. When the potential is integrable and has a finite first moment, the sharp inequalities 1−p+∑j=1p Nj≤N≤∑j=1p Nj are proved, where p is the number of fragments, N is the total number of bound states, and Nj is the number of bound states for the jth fragment. When p=2 the question of whether N=N1+N2 or N=N1+N2−1 is investigated in detail. An illustrative example is also provided. © 1998 American Institute of Physics.

Notes: By means of the geometric algebra the general decomposition of SU(2) gauge potential on the sphere bundle of a compact and oriented four-dimensional manifold is given. Using this decomposition theory the SU(2) Chern density has been studied in detail. It shows that the SU(2) Chern density can be expressed in terms of the δ-function δ(φ). One can also find that the zero points of the vector fields φ are essential to the topological properties of a manifold. It is shown that there exists the crucial case of branch process at the zero points. Based on the implicit function theorem and the Taylor expansion, the bifurcation of the Chern density is detailed in the neighborhoods of the bifurcation points of φ. It is pointed out that, since the Chern density is a topological invariant, the sum topological chargers of the branches will remain constant during the bifurcation process. © 1998 American Institute of Physics.

Notes: Low energy effective actions are computed by integrating out degrees of freedom heavier than some energy scale of physical interest. At the one-loop level, the low energy effective action can be computed in terms of the functional trace of the heat kernel associated with the operator appearing in the part of the action quadratic in the heavy fields. Here, a technique for computing this functional trace is illustrated in the case of heavy scalar fields coupled to a Yang-Mills background. It makes use of the similarity of the functional trace of the heat kernel expressed in momentum space to a Gaussian integral. An explicit computation of the piece of the low energy effective action for the Yang-Mills fields quadratic in derivatives of the field strength is carried out, extending previously known results on the form of the long wavelength limit of the low energy effective action. © 1998 American Institute of Physics.

Notes: The Wheeler–DeWitt equation for the minimally coupled Friedman–Robertson–Walker-massive-scalar-field minisuperspace is written as a two-component Schrödinger equation with an explicitly "time"-dependent Hamiltonian. This reduces the solution of the Wheeler–DeWitt equation to the eigenvalue problem for a nonrelativistic one-dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler–DeWitt equation. Further analysis of the mode expansion shows that in general the solutions of the Wheeler–DeWitt equation for this model are doubly graded, i.e., every solution is a superposition of two definite-parity solutions. Moreover, it is shown that the mode expansion of both even- and odd-parity solutions is always infinite. It may be terminated artificially to construct approximate solutions. This is demonstrated by working out an explicit example which turns out to satisfy DeWitt's boundary condition at initial singularity. © 1998 American Institute of Physics.

Notes: A quantum version is presented for the Anosov system defined by the time evolution implemented by the geodesic coflow on the cotangent bundle of any compact quotient manifold obtained from the Poincaré half-plane. While the canonical Weyl algebra does not close under time evolution, the symplectic structure of these classical systems can be exploited to produce objects akin to the CCR algebras encountered in quantum field theory. This construction allows one to lift both the geodesic and the horocyclic flows to a Weyl algebra describing the quantum dynamics corresponding to the systems under consideration. The Anosov relations as proposed in Ref. 〈citeref RID="R1" STYLE="ONLINE"〉1 are found to be valid for these models. A quantum version of the classical ergodicity of these systems is discussed in the last section. © 1998 American Institute of Physics.

Notes: The integrability of multivector fields in a differentiable manifold is studied. Then, given a jet bundle J1E→E→M, it is shown that integrable multivector fields in E are equivalent to integrable connections in the bundle E→M (that is, integrable jet fields in J1E). This result is applied to the particular case of multivector fields in the manifold J1E and connections in the bundle J1E→M (that is, jet fields in the repeated jet bundle J1J1E), in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections. These results allow us to set the Lagrangian evolution equations for first-order classical field theories in three equivalent geometrical ways (in a form similar to that in which the Lagrangian dynamical equations of nonautonomous mechanical systems are usually given). Then, using multivector fields, we discuss several aspects of these evolution equations (both for the regular and singular cases); namely, the existence and nonuniqueness of solutions, the integrability problem and Noether's theorem, giving insights into the differences between mechanics and field theories. © 1998 American Institute of Physics.

Notes: A supersymmetric extension for an arbitrary number of odd parameters of the stationary KdV equation is studied; we call this an N-extended super KdV. It is shown that it is related to superelliptic curves, and thereafter the complete ring of superdifferential operators commuting with the associated Schrödinger operator is constructed. © 1998 American Institute of Physics.

Notes: For the equations describing second harmonic generation (SHG), we find a linear system different from one found by Kaup (but connected with it by a gauge transformation), which enables us to consider the SHG equations as related to the first negative flow in the coupled KdV hierarchy and to find therefore the Hamiltonian structures and the recursion operator. With the help of the recursion operator we write down the simplest commuting flows and consider their stationary points in order to find particular solutions of the SHG equations. We also investigate some particular invariant solutions of the SHG equations that can be of physical interest, and find their connections with invariant solutions of the Tzitzéica equation. The latter are either connected with Painlevé III or solvable in elliptic functions. © 1998 American Institute of Physics.

Notes: Einstein's general relativity with both metric and vielbein treated as independent fields is considered, demonstrating the existence of a consistent variational principle and deriving a Hamiltonian formalism that treats the spatial metric and spatial vielbein as canonical coordinates. This results in a Hamiltonian in standard form that consists of Hamiltonian and momentum constraints as well as constraints that generate spatial frame transformations—all appearing as primary, first class constraints on phase space. The formalism encompasses the standard coordinate frame and vielbein approaches to general relativity, and the constraint algebra derived herein reproduces known results in either limit. © 1998 American Institute of Physics.

Notes: The purpose of this paper is to provide definitions for, and proofs of, the asymptotic formulas given by Edmonds, which relate the 3j and 6j symbols to rotation matrices. © 1998 American Institute of Physics.

Notes: A concept of quantum martingale measure is introduced and examples are constructed as quantum stochastic spectral integrals in Fock space. These are then utilized as space–time noise to drive a parabolic stochastic partial differential equation (spde). We establish the existence and uniqueness of the solutions as families of densely defined closable operators in Fock space that are jointly continuous in time and space variables and satisfy a Markov property. © 1998 American Institute of Physics.

Notes: Positive discrete series representations of the Lie algebra su(1,1) and the quantum algebra Uq(su(1,1)) are considered. The diagonalization of a self-adjoint operator (the Hamiltonian) in these representations and in tensor products of such representations is determined, and the generalized eigenvectors are constructed in terms of orthogonal polynomials. Using simple realizations of su(1,1), Uq(su(1,1)), and their representations, these generalized eigenvectors are shown to coincide with generating functions for orthogonal polynomials. The relations valid in the tensor product representations then give rise to new generating functions for orthogonal polynomials, or to Poisson kernels. In particular, a group theoretical derivation of the Poisson kernel for Meixner–Pollaczek and Al-Salam–Chihara polynomials is obtained. © 1998 American Institute of Physics.

Notes: It is known that in the Wentzel–Kramers–Brillouin approximation of multicomponent systems like the Dirac equation or Born–Oppenheimer approximation, an additional phase appears apart from the Berry phase. So far, this phase was only examined in special cases, or under certain restrictive assumptions, namely, that the eigenspaces of the matrix or endomorphism valued symbol of the Hamiltonian form trivial bundles. We give a completely global derivation of this phase which does not depend on any choice of local trivializing sections. This is achieved using a star product approach to quantization. Furthermore, we give a systematic and global approach to a reduction of the problem to a problem defined completely on the different "polarizations." Finally, we discuss to what extent it is actually possible to reduce the problem to a really scalar one, and make some comments on obstructions to the existence of global quasiclassical states. © 1998 American Institute of Physics.

Notes: We give the construction of quantum Lax equations for IRF models and the difference version of the Calogero–Moser–Sutherland model introduced by Ruijsenaars. We solve the equations using factorization properties of the underlying face Hopf algebras/elliptic quantum groups. This construction is in the spirit of the Adler–Kostant–Symes method and generalizes our previous work to the case of face Hopf algebras/elliptic quantum groups with dynamical R matrices. © 1998 American Institute of Physics.

Notes: Generalized plasma dispersion functions have recently arisen in the derivation of an exact expression for the quadratic response tensor for a plasma with constituent species having Maxwellian velocity distributions. A range of mathematical properties satisfied by the generalized plasma dispersion functions are derived here, including recursion relations, series expansions, and approximate analytic forms, including Padé approximants. These properties are of central importance to the application of the exact quadratic response tensor to descriptions of second-order processes in weak and strong turbulence theory, such as three-wave interactions in warm plasmas. © 1998 American Institute of Physics.

Notes: In this paper we construct Gamow vectors for resonances given by poles of the analytic continuation of the S matrix of any finite order. We study their modes of decay (or growth). We obtain Jordan block structures for the extended Hamiltonians and evolution operators on the subspaces spanned by these Gamow vectors. We perform this study within the context of the rigged Hilbert space extension of quantum theory. We construct an explicit Friedrichs model with a double pole resonance to illustrate the general formulation. © 1998 American Institute of Physics.

Notes: Highest-weight-type representation theories of the affine sl(2) and N=2 superconformal algebras are shown to be equivalent modulo the respective spectral flows. © 1998 American Institute of Physics.

Notes: The Friedrichs model has often been used in order to obtain explicit formulas for eigenvectors associated to complex eigenvalues corresponding to lifetimes. Such eigenvectors are called Gamow vectors and they acquire meaning in extensions of the conventional Hilbert space of quantum theory to the so-called rigged Hilbert space. In this paper, Gamow vectors are constructed for a solvable model of an unstable relativistic field. As a result, we obtain a time asymmetric relativistic extension of the Fock space. This extension leads to two distinct Poincaré semigroups. The time reversal transformation maps one semigroup to the other. As a result, the usual PCT invariance should be extended. We show that irreversibility as expressed by dynamical semigroups is compatible with the requirements of relativity. © 1998 American Institute of Physics.

Notes: The information entropies of the two-dimensional harmonic oscillator, V(x,y)=1/2λ(x2+y2), and the one-dimensional hydrogen atom, V(x)=−1/|x|, can be expressed by means of some entropy integrals of Laguerre polynomials whose values have not yet been analytically determined. Here, we first study the asymptotical behavior of these integrals in detail by extensive use of strong asymptotics of Laguerre polynomials. Then, this result (which is also important by itself in a context of both approximation theory and potential theory) is employed to analyze the information entropies of the aforementioned quantum-mechanical potentials for the very excited states in both position and momentum spaces. It is observed, in particular, that the sum of position and momentum entropies has a logarithmic growth with respect to the main quantum number which characterizes the corresponding physical state. Finally, the rate of convergence of the entropies is numerically examined. © 1998 American Institute of Physics.

Notes: We study supersymmetric quantum mechanics on RPn, SO(n), G2, and U(2) to examine Witten's Morse theory concretely. We confirm the simple instanton picture of the de Rham cohomology that has been given in a previous paper. We use a reflection symmetry of each theory to select the true vacuums. The number of selected vacuums agrees with the de Rham cohomology for each of the above manifolds. © 1998 American Institute of Physics.

Notes: We analyze the number-theoretical spin chain with partition function Z(β) =ζ(β−1)/ζ(β) using the polymer model technique. The finite (grand) canonical chains give bounds for the limit free energy and internal energy. The correlation functions for inverse temperature β=−1 are products of two-point functions. A combinatorial result for general interval graphs is derived. © 1998 American Institute of Physics.

Notes: In this work we introduce a least-action formulation of the protein folding problem casted within a coarse description of the soft mode dynamics of the peptide chain. Ultimately, we show that this coarse variational approach can be lifted to yield the microscopic long-time torsional dynamics responsible for the actual folding process. As a first step, a binary coding of local topological constraints associated to each structural motif is introduced to coarsely mimic the long-time dynamics. Folding pathways are initially resolved as transitions between patterns of locally encoded structural signals. Our variational approach is aimed at identifying the most economic pathway with respect to the stepwise cost in conformational freedom. Our treatment allows us to account for the expediency of the process in proteins effectively capable of in vitro renaturation. We identify the dominant pathway by introducing a coarse version of Lagrangian microscopic dynamics. The coarse folding pathways are generated by a parallel search for structural patterns in a matrix of local topological constraints (LTM) of the chain. Each local topological constraint represents a coarse description of a local torsional state and each pattern is evaluated, translated, and finally recorded as a contact matrix (CM), an operation that is subject to a renormalization feedback loop. The renormalization operation periodically introduces long-range correlations on the LTM according to the latest CM generated by translation. Local topological constraints may form consensus regions in portions of the chain that translate as secondary structure motifs or tertiary interactions. Nucleation steps and cooperative effects are accounted for by means of the renormalization operation, which warrants the persistence of seeding patterns upon successive LTM evaluations. Relevant folding time scales beyond the realm of molecular dynamics simulations become accessible through the coarsely codified representation of local torsional constraints. The validity of our approach is tested vis à vis experimentally probed folding pathways generating tertiary interactions in proteins that may recover their active structure under in vitro renaturation conditions. We focus on determining significant folding intermediates and the late kinetic bottlenecks that occur within the first 〈fraction SHAPE="CASE"〉1100 s of the renaturation process. After the computational accessibility of this coarse solution of the folding problem becomes apparent, we show how to lift our variational problem to microscopic dynamics of the peptide chain. The consistency of our approach is revealed by an actual generation of Newton's laws at the microscopic level through an inverse projection of the coarse dynamics originally generated through the pattern recognition computation. © 1998 American Institute of Physics.

Notes: We reformulate the theory of simple liquids in a field theoretical way by taking into account the triplet potential wijk, in addition to the external potential ui and the pair potential vij. The innovation here is the inversion method and the on-shell expansion which are the building blocks of a novel use of Legendre transformation developed in field theory. By the inversion method, we renormalize the theory in terms of one-, two-, and three-particle densities, and present a diagrammatic representation for a thermodynamical functional, which is the entropy except for a trivial constant, in terms of renormalized variables. In other words, we present an expression for the entropy in terms of only one-, two-, and three-particle densities: the n-particle density where n≥4 does not appear in the expression. The on-shell condition, which is a starting point of the on-shell expansion, of the thermodynamical functional thus obtained (the entropy) leads to a set of three self-consistent equations for one-, two-, and three-particle densities. Through one of the self-consistent equations, we can systematically improve the Kirkwood's superposition approximation for the three-particle density. The on-shell conditions for other thermodynamical functionals, also obtained in this article, are found to be extentions of various well-known equations in the theory of simple liquids. The formulation presented here is complementary to the conventional re-summation techniques for renormalization of diagrams. In the present formulation, we do not have to care about the topological structure of diagrams, often characterized by the irreducibility of diagrams. Instead, by a perturbative calculation, we can automatically single out the diagrams with the topological structure predicted by the re-summation techniques. © 1998 American Institute of Physics.

Notes: This is the third and last part of a series of three papers. Using the same method and the same coordinates as in papers I and II, rotating dust solutions of Einstein's equations are investigated that possess three-dimensional symmetry groups, under the assumption that each of the Killing vectors is linearly independent of velocity uα and rotation wα at every point of the spacetime region under consideration. The Killing fields are found and the Killing equations are solved for the components of the metric tensor in every case that arises. No progress was made with the Einstein equations in any of the cases, and no previously known solutions were identified. A brief overview of literature on solutions with rotating sources is given. © 1998 American Institute of Physics.

Notes: For any principal bundle P, one can consider the subspace of the space of connections on its tangent bundle TP given by the tangent bundle TA of the space of connections A on P. The tangent gauge group acts freely on TA. Appropriate BRST operators are introduced for quantum field theories that include as fields elements of TA, as well as tangent vectors to the space of curvatures. As the simplest application, the BRST symmetry of the so-called BF-Yang–Mills theory is described and the relevant gauge fixing conditions are analyzed. A brief account on the topological BF theories is also included and the relevant Batalin–Vilkovisky operator is described. © 1998 American Institute of Physics.

Notes: A set of generalized coherent states as eigenstates of the annihilation operator is proposed. These states are analytic functions of a complex variable and admit a resolution of identity with positive measure. Guided by the classical action angle transformation and the correspondence principle a formalism is developed for the construction of the annihilation operator for a given Hamiltonian. © 1998 American Institute of Physics.

Notes: Similarity and symmetry of quantum mechanical operators are the basic theoretical means for systematizing quantum dynamics. The classical counterparts of these concepts are canonical transformations and conservation laws. For operators, a concept called intertwining encompasses both similarity and symmetry. In this paper, we construct pairs of Dirac operators with noncentral potentials that are intertwined by multiplication operators. © 1998 American Institute of Physics.

Notes: The relativistic force exerted by a classical electromagnetic field on an electron is examined from the point of view that associates the differential geometry of the space–time trajectory with a continuous group of transformations. The group relevant to this particular case is the group generated by the electromagnetic field tensor (electrodynamic group). This group is studied in some detail by treating separately the two parts of the Liénard–Wiechert expression for the tensor. The intrinsic differential equations of the space–time trajectories associated with the electrodynamic group are obtained by the Cartan method of the moving frame. These equations show that the curvature of the trajectory is associated only with the generalized Coulomb field and the hypertorsion with the radiation field. According to these equations, the relativistic force may have two components along orthogonal spacelike directions: The first, an apparent force, reduces to the Coulomb force in the nonrelativistic limit. The other is orthogonal to the four-momentum and to the apparent force; this is the radiation reaction force. In the nonrelativistic limit it manifests itself as a frictional force reducing the momentum. The approach also offers a point of view of the conventional treatment of radiation reaction in which the nonlocality in time emerges from the geometry of the problem. The implications pertaining to other areas of physics of the causal connection found between the hypertorsion and the radiation field are briefly considered. © 1998 American Institute of Physics.

Notes: A generic stochastic method is presented that rapidly evaluates numerical bulk flux solutions to the one-dimensional integrodifferential radiative transport equation, for coherent irradiance of optically anisotropic suspensions of nonspheroidal bioparticles, such as blood. As Fermat rays or geodesics enter the suspension, they evolve into a bundle of random paths or trajectories due to scattering by the suspended bioparticles. Overall, this can be interpreted as a bundle of Markov trajectories traced out by a "gas" of Brownian-like point photons being scattered and absorbed by the homogeneous distribution of uncorrelated cells in suspension. By considering the cumulative vectorial intersections of a statistical bundle of random trajectories through sets of interior data planes in the space containing the medium, the effective equivalent information content and behavior of the (generally unknown) analytical flux solutions of the radiative transfer equation rapidly emerges. The fluxes match the analytical diffuse flux solutions in the diffusion limit, which verifies the accuracy of the algorithm. The method is not constrained by the diffusion limit and gives correct solutions for conditions where diffuse solutions are not viable. Unlike conventional Monte Carlo and numerical techniques adapted from neutron transport or nuclear reactor problems that compute scalar quantities, this vectorial technique is fast, easily implemented, adaptable, and viable for a wide class of biophotonic scenarios. By comparison, other analytical or numerical techniques generally become unwieldy, lack viability, or are more difficult to utilize and adapt. Illustrative calculations are presented for blood medias at monochromatic wavelengths in the visible spectrum. © 1998 American Institute of Physics.

Notes: Utilizing the multiplicative formula of universal R matrices, the correspondence between the L operators and Drinfeld's generators is explicitly calculated for quantum group Uq(g) with g=Al(1),Bl(1),Cl(1),Dl(1). © 1998 American Institute of Physics.

Notes: We investigate the graded structure of the tensor product space Vλ2⊗Vλ1, which arises in the coupling of two irreducible representations λ1 and λ2 of a semisimple Lie algebra G, in the limit in which one of the highest weights becomes asymptotically large. The construction of asymptotic coupling coefficients is considered. © 1998 American Institute of Physics.

Notes: An original regular approach to constructing special type symmetries for boundary value problems, namely renormgroup symmetries, is presented. Different methods of calculating these symmetries based on modern group analysis are described. An application of the approach to boundary value problems is demonstrated with the help of a simple mathematical model. © 1998 American Institute of Physics.

Notes: A description of the relative motion of neutrons and protons in strongly deformed even-even nuclei is reduced to that of two coupled triaxial rotors, one representing the neutrons and the other the protons, by using generating function methods. The two rotor theory is interpreted as a generalization of the Elliott SU(3) model with a neutron–proton direct product SUν(3)×SUπ(3) substructure. The second order Casimir operator for this system is constructed and some of its eigenfunctions are found in explicit form in the Fock–Bargmann space representation. Analytical results for isovector M1 transitions probabilities are obtained and the results compared with experimental data and pseudo-SU(3) calculations. Graphs picturing the relative orientation of neutron-proton rotors in 22Ne are presented. © 1998 American Institute of Physics.

Notes: Introducing physical string coordinate's modes as well as physical string coordino's modes, we derive the generating functional of physical vertex operators (GFPVO) of fermionic particles (in superstring), which is just equal to the one already proposed by using superconformal mapping in the Ramond sector. Inversely solving the superconformal mapping problem, we can derive general formulas, which give physical vertex operators of various fermionic particles up to arbitrarily excited mass levels. As an example, we explicitly derive some (Gliozzi–Scherk–Olive-allowed) low-lying physical vertex operators. © 1998 American Institute of Physics.

Notes: The classical derivation of the canonical transformation theory [H. Goldstein, Classical Mechanics, 2nd ed. (Addison–Wesley, Reading, 1981)] is based on Hamilton's principle which is only valid for conservative systems. This paper avoids this principle by using an approach that is basically reversed compared to the classical derivation. The Lagrange equations of motion are formulated in the undefined and general variable set {Q,P}, and the general Hamilton equations of motion are derived from the Lagrange equations by using a variational principle. The undefined general variables {Q,P} are defined through a transformation to a special (defined) variable set {q,p}. The transformation equations connecting the two sets are derived by using the invariants property of the value of the Lagrangian. This approach results in a more general interpretation of the generator function. © 1998 American Institute of Physics.

Notes: We apply recent results in the theory of PDE, specifically in problems with two different time scales, to Einstein's equations near their Newtonian limit. The results imply a justification to post-Newtonian approximations when initialization procedures to different orders are made on the initial data. We determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations. Using the results mentioned, the first post-Newtonian equations and their solutions in terms of Green's functions are presented in order to indicate how to proceed in calculations with this approach. © 1998 American Institute of Physics.

Notes: We characterize, in the case of sl&dbgcaret;(2),the crystal base of the Demazure module Ew(Λ) in terms of extended Young diagrams or paths for any dominant integral weight Λ and Weyl group element w. Its character is evaluated via two expressions, "bosonic" and "fermionic." © 1998 American Institute of Physics.

Notes: Let (M,g) be a strongly causal space–time, dimM≥3. Let f:M→M be a bijection such that images and pre-images of null geodesics (as point sets) are null geodesics. Then f is a conformal transformation. © 1998 American Institute of Physics.

Notes: An expansion with respect to the inverse principal quantum number n−1 has been derived recently for the hydrogen-like wave functions of the discrete spectrum. The basic functions of this expansion were Bessel functions multiplied by a power rk of the distance r to the origin. We show that this expansion remains valid for the continuous spectrum, with minor changes. For the wave function corresponding to the energy E=+1/(2n2) a.u. of the continuum, the pure imaginary number −in (positive real n) must be substituted for the integer n in the expansion for the discrete spectrum, and a normalizing factor must be introduced. © 1998 American Institute of Physics.

Notes: A group-theoretical description of N identical harmonic oscillators on a line is presented. It provides a scheme for labeling the energy eigenstates that are invariant under the permutation group SN and for obtaining the symmetric operators that transform these degenerate eigenfunctions among themselves. The symmetry algebra that these generators form is in general polynomial. The 2- and 3-particle cases are considered in detail. For the simple 2-body problem the invariance algebra is found to be the cubic SU(2) algebra: [J0,J±]=±J±, [J+,J−]=2J0−αJ03. In the 3-body case, the permutational invariant states are characterized with the help of the subgroup chain U(3)⊃U(2)⊃O(2). The labeling and step operators are obtained from determining an integrity basis for the S3 scalar in U(U(3)). Generating functions techniques are used to that end; an eight-dimensional basis is found whose elements span the symmetry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided. © 1998 American Institute of Physics.

Notes: The canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angular-momentum, rotation-angle, and rotation-axis observables. To this end, SU(2) is "quantized" on the compact group manifold, according to the standard procedure of non-Abelian quantum kinematics. Quantum-kinematic invariant operators are introduced, and their commutation relations with the rotation variables are found in an explicit manner. The quantum-kinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure (which one should solve in the applications, in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models). A wide range of applicability of SU(2) quantum kinematics is suggested. © 1998 American Institute of Physics.

Notes: Recently, Schreckenberg investigated symmetries in the context of history quantum theories. In the case that the space of histories is given by the set of projectors on some finite dimensional Hilbert space he obtained, first, a complete characterization of all physical symmetries of history quantum theories — an analog of Wigner's theorem — and, second, a complete mathematical characterization of symmetries of single decoherence functionals. In this paper we extend Schreckenberg's results to the case where the underlying space of histories is given by the set of projectors on some infinite dimensional Hilbert space. © 1998 American Institute of Physics.

Notes: We use Kepler's eccentric anomaly in order to get explicit equations for the planetary orbits in a solar system. By successively modifying the corresponding Lagrangian, we build a Lagrangian for a relativistic two-body problem, the trajectories of which are conics in Minkowski space. The equations of these trajectories, projected on a three-space, exhibit new anomalies. © 1998 American Institute of Physics.

Notes: Using the quantum inverse scattering method for the XXZ model with open boundary conditions, we obtained the determinant formula for the six-vertex model with reflecting end. © 1998 American Institute of Physics.

Notes: A criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the Benjamin–Ono (BO) equation. This bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the BO equation. The asymptotic approximation of an exponentially small new eigenvalue is derived. The method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions. Explicit expressions are obtained for the soliton potentials. © 1998 American Institute of Physics.

Notes: The circularity property of stationary-axisymmetric asymptotically flat solutions of the coupled Einstein conformally invariant scalar field equations is investigated. It is proven that as a result of the coupling of the scalar field to the background scalar curvature, in general such solutions fail to be circular. Consequently, it is argued that the space of stationary, black hole equilibrium states may contain states which are not circular. However, the circular stationary-axisymmetric sector is nonempty and in fact it is proven that: All circular, stationary-axisymmetric asymptotically flat black hole solutions are those and only those black holes states described by the two parameter family of Kerr black holes. © 1998 American Institute of Physics.

Notes: We consider the problem of extending the solution of a particular two-dimensional Poisson equation to a larger domain. This problem is related to the problem of putting a non-neutral plasma into equilibrium by applying a suitable wall potential, and to similar problems in two-dimensional fluid dynamics. While one cannot always find an exact solution, one can always find an approximate solution if the plasma has no holes. © 1998 American Institute of Physics.

Notes: We consider arbitrary splits of field operators into two parts; ψ=ψ++ψ−, and use the corresponding definition of normal ordering introduced earlier [T. S. Evans and D. A. Steer, Nucl. Phys. B 474, 481 (1996)]. In this case the normal ordered products and contractions have none of the special symmetry properties assumed in existing proofs of Wick's theorem. Despite this, we prove that Wick's theorem still holds in its usual form as long as the contraction is a c-number. Wick's theorem is thus shown to be much more general than existing derivations suggest, and we discuss possible simplifying applications of this result. © 1998 American Institute of Physics.

Notes: We present several examples of quasiexactly solvable N-body problems in one, two, and higher dimensions. We study various aspects of these problems in some detail. In particular, we show that in some of these examples the corresponding polynomials form an orthogonal set, and many of their properties are similar to those of the Bender–Dunne polynomials. We also discuss QES problems where the polynomials do not form an orthogonal set. © 1998 American Institute of Physics.

Notes: This work is devoted to the establishment of a Poisson structure for a format of equations known as generalized Lotka–Volterra systems. These equations, which include the classical Lotka–Volterra systems as a particular case, have been deeply studied in the literature. They have been shown to constitute a whole hierarchy of systems, the characterization of which is made in the context of simple algebra. Our main result is to show that this algebraic structure is completely translatable into the Poisson domain. Important Poisson structures features, such as the symplectic foliation and the Darboux canonical representation, rise as a result of rather simple matrix manipulations. © 1998 American Institute of Physics.

Notes: The analytical condition for a trajectory of the billiard system within an ellipsoid of any dimension is given. An interesting property is obtained—every periodical trajectory with the period less than the dimension of the space is placed in a coordinate hyperplane. © 1998 American Institute of Physics.

Notes: We give a proof of the Kolmogorov–Arnold–Moser (KAM) theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring's approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic Hamiltonians. We show that Thirring's proof for nondegenerate Hamiltonians can be adapted to Hamiltonians with degenerate twist. This case, in fact, drastically simplifies Thirring's proof. © 1998 American Institute of Physics.

Notes: We present new and useful "external potential" forms of the two-body Dirac equations of constraint dynamics for combined scalar, vector, pseudoscalar, pseudovector, and tensor interactions. These equations have potential applications in two-body problems for bound states in meson spectroscopy and phase shift analysis in nucleon–nucleon scattering. Toward this end, we derive their coupled Schrödinger-like forms using matrix techniques and obtain the corresponding radial equations to these forms from scalar and vector spherical harmonic decompositions. © 1998 American Institute of Physics.

Notes: Chaos in Robertson–Walker cosmological models where gravity is coupled to one or more scalar fields has been studied by a few authors, mostly using numerical simulations. In this paper we begin a systematic study of the analytical aspect. We consider one conformally coupled scalar field and, using the fact that the model is integrable when the field is massless, we show in detail how homoclinic chaos arises for nonzero masses using a perturbative method. © 1998 American Institute of Physics.

Notes: A new eight-dimensional conformal gauging solves the auxiliary field problem and eliminates unphysical size change from Weyl's electromagnetic theory. We derive the Maurer–Cartan structure equations and find the zero curvature solutions for the conformal connection. By showing that every one-particle Hamiltonian generates the structure equations we establish a correspondence between phase space and the eight-dimensional base space, and between the action and the integral of the Weyl vector. Applying the correspondence to generic flat solutions yields the Lorentz force law, the form and gauge dependence of the electromagnetic vector potential and minimal coupling. The dynamics found for these flat solutions applies locally in generic spaces. We then provide necessary and sufficient curvature constraints for general curved eight-dimensional geometries to be in 1–1 correspondence with four-dimensional Einstein–Maxwell space–times, based on a vector space isomorphism between the extra four dimensions and the Riemannian tangent space. Despite part of the Weyl vector serving as the electromagnetic vector potential, the entire class of geometries has vanishing dilation, thereby providing a consistent unified geometric theory of gravitation and electromagnetism. In concluding, we give a concise discussion of observability of the extra dimensions. © 1998 American Institute of Physics.

Notes: The billiard system within an ellipsoid in three-dimensional space is considered. We describe periodical trajectories of that mechanical system deriving analytical condition of Cayley's type for generalized Poncelet's theorem, using methods of algebrogeometric integration.© 1998 American Institute of Physics.

Notes: A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space–time is considered here. It is found that there is a natural differential calculus over this version of quantized space–time using which the only possible torsion-free, metric-compatible, linear connection has zero curvature. It is then the noncummutative version of Minkowski space–time. Perturbations of this calculus are shown to give rise to nontrivial gravitational fields. © 1998 American Institute of Physics.

Notes: A quantum particle interacting with a thin solenoid and a magnetic flux is described by a five-parameter family of Hamilton operators, obtained via the method of self-adjoint extensions. One of the parameters, the value of the flux, corresponds to the Aharonov–Bohm effect; the other four parameters correspond to the strength of a singular potential barrier. The spectrum and eigenstates are computed and the scattering problem is solved. © 1998 American Institute of Physics.

Notes: We compute the functional determinant for a Dirac operator in the presence of an Abelian gauge field on a bidimensional disk, under global boundary conditions of the type introduced by Atiyah–Patodi–Singer. We also discuss the connection between our result and the index theorem. © 1998 American Institute of Physics.

Notes: We show that Calabi–Yau spaces with certain types of hypersurface-quotient singularities have unobstructed deformations. This applies in particular to all Calabi–Yau orbifolds nonsingular in codimension 2. © 1998 American Institute of Physics.

Notes: The Casimir operators of a Lie algebra are in one-to-one correspondence with the symmetric invariant tensors of the algebra. There is an infinite family of Casimir operators whose members are expressible in terms of a number of primitive Casimirs equal to the rank of the underlying group. A systematic derivation is presented of a complete set of identities expressing nonprimitive symmetric tensors in terms of primitive tensors. Several examples are given including an application to an exceptional Lie algebra. © 1998 American Institute of Physics.

Notes: Dedicated to the memory of Gerda Dorfmeister. Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider D and X to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of [D, X]=Id. © 1998 American Institute of Physics.

Notes: Using the concept of irreducible tensor basis and the properties of irreducible tensor operator, we give all O(N)⊃O(N−1) reduction factors for (m1m2m30⋅0)⊗(100⋅0). © 1998 American Institute of Physics.

Notes: The Demazure character formula is applied to the Verlinde formula for the fusion rules of Wess–Zumino–Witten (WZW) models. We follow Littelmann's derivation of a generalized Littlewood–Richardson rule that computes tensor products for simple Lie algebras. A combinatorial rule for WZW fusions does not result, however. Only a modified version of the Littlewood–Richardson rule is obtained that computes an (old) upper bound on the fusion coefficients of Ar WZW models. We argue that this is because the characters of simple Lie algebras appear in this treatment, instead of the corresponding affine characters. The Bruhat order on the affine Weyl group must be implicated in any combinatorial rule for WZW fusions; the Bruhat order on subgroups of this group (such as the finite Weyl group) does not suffice. © 1998 American Institute of Physics.

Notes: A direct link between a one-loop N-point Feynman diagram and a geometrical representation based on the N-dimensional simplex is established by relating the Feynman parametric representations to the integrals over contents of (N−1)-dimensional simplices in non-Euclidean geometry of constant curvature. In particular, the four-point function in four dimensions is proportional to the volume of a three-dimensional spherical (or hyperbolic) tetrahedron which can be calculated by splitting into birectangular ones. It is also shown that the known formula of reduction of the N-point function in (N−1) dimensions corresponds to splitting the related N-dimensional simplex into N rectangular ones. © 1998 American Institute of Physics.

Notes: We deal with some problems concerning quantum matter fields thermodynamics on static spherosymmetric black hole backgrounds. By means of variable separation, we study the self-adjointness of the Euclidean wave operators relevant in the calculation of the partition function and of the thermal Green functions for the case of scalar quantum fields in thermal equilibrium with a Schwarzschild black hole. In this respect, the role of the horizon surface is analyzed. An approach to this topic, related to completeness of the Euclidean manifold, is also discussed. It is found that, if the horizon surface is not included in the Euclidean manifold (or if it cannot be included, as in the case of a temperature different from the black hole one), essential self-adjointness is missing and boundary conditions on the horizon are required. We also analyze some qualitative spectral properties of the given wave operators after introducing an infrared regularization by means of a spherical box. We find that in the standard Euclidean path integral approach the spectrum is purely discrete. The same analysis is carried out for the noncovariant choice of the functional measure in the Euclidean path integral. An analogous study is carried for an extremal Reissner–Nordström black hole and it is sketched in the nonextremal one. The latter is completely analogous to the Schwarzschild case, whereas in the extremal case it is found that the corresponding Riemannian manifold is complete for every choice of the temperature and that at finite four volume there is a nonvoid continuous spectrum. Physical implications of our results are discussed. © 1998 American Institute of Physics.

Notes: An analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is proposed. It starts with the generalized Hirota identity for the Cauchy–Baker–Akhiezer (CBA) function and leads to a generalized KP hierarchy in the form of compact functional equations containing a special shift operator. A generalized KP hierarchy incorporates the basic KP hierarchy, modified KP hierarchy, KP singularity manifold equation hierarchy, and corresponding hierarchies of linear problems. Different "vertical" levels of the generalized KP hierarchy are connected via invariants of the Combescure symmetry group. The resolution of functional equations also gives rise to the τ function and the addition formulas for it. © 1998 American Institute of Physics.

Notes: The modified Newton–Sabatier method was recently developed for inelastic inverse problems for the scattering of neutral particles. In this paper, the method is extended to the solution of the inelastic inverse scattering problem with charged particles. The bases are radial Schrödinger equations coupled by a local potential matrix known from a certain distance on. The inversion procedure starts with the S-matrix fixing the asymptotic wave functions of the charged scattering system. This S-matrix can be transformed to another one for asymptotic constant potentials instead of Coulomb potentials. The new S-matrix belongs to the same inner potential matrix as the original one and is solved by the modified Newton–Sabatier method similar to the case of neutral particles. An application with a given S-matrix, belonging to a coupled square-well potential matrix with an outer Coulomb potential yields good agreement between the inverted potential matrix and the original one. © 1998 American Institute of Physics.

Notes: We show that, in conformally static spacetimes, Maxwell equations take their simplest form when written in terms of the so-called "optical metric." © 1998 American Institute of Physics.

Notes: The partial differential equation corresponding to the metric function in a five-dimensional flat space describing the perfect fluid distribution has been examined for generalized symmetries via a new approach. It is shown that the approach utilized here has, besides providing generalized symmetries, for some particular values of the parameters involved, yielded new metrics and Petrov type D solutions. Further, for the general case the problem is reduced to the solution of an Abel's equation of second kind, which obviously generalizes the scope for new metrics and new Petrov type D solutions. © 1998 American Institute of Physics.

Notes: It is shown that the Korteweg–de Vries (KdV) equation can be transformed into an ordinary linear partial differential equation in the wave number domain. Explicit solutions of the KdV equation can be obtained by subsequently solving this linear differential equation and by applying a cascade of (nonlinear) transformations to the solution of the linear differential equation. It is also shown that similar concepts apply to the nonlinear Schrödinger equation. The role of symmetry is discussed. Finally, the procedure which is followed in the one-dimensional cases is successfully applied to find special solutions of higher-dimensional nonlinear partial differential equations. © 1998 American Institute of Physics.

Notes: It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical invariants then define topological conserved quantities. Underlying evolution equations are shown to be associated with a triad of linear equations. Our examples include Ishimori equation and Myrzakulov equations which are shown to be geometrically equivalent to Davey-Stewartson and Zakharov-Strachan (2+1) dimensional nonlinear Schrödinger equations, respectively. © 1998 American Institute of Physics.

Notes: A proof of the Poincaré–Birkhoff–Witt theorem is given for a class of generalized Lie algebras closely related to the Gurevich S-Lie algebras. As concrete examples, we construct the positive (negative) parts of the quantized universal enveloping algebras of type An and Mp,q,ε(n,K), which is a nonstandard quantum deformation of GL(n). In particular, we get, for both algebras, a unified proof of the Poincaré–Birkhoff–Witt theorem, and we show that they are genuine universal enveloping algebras of certain generalized Lie algebras. © 1998 American Institute of Physics.

Notes: In physical geodesy one considers several (exterior) Robin boundary value problems for the Laplace equation in three dimensions. The ellipsoidal Stokes problem, which occurs in context of gravimetric determination of the geoid, belongs to this class. Up to now, this and related problems have been treated with high order series expansions of spherical and spheroidal harmonics. In this article we investigate the nullfield method for this class of boundary value problems. An integral equation formulation is achieved, and existence and uniqueness conditions are attained in view of the Fredholm alternative. For the case that the underlying surface is a triaxial ellipsoid or an oblate spheroid, explicit expressions for the eigenvalues and eigenfunctions for the boundary integral operator are provided. © 1998 American Institute of Physics.

Notes: The interaction energy between vortices of the generalized Abelian Higgs vortices is studied numerically. These are the solitons of the p-th member of the hierarchy of MaxwellHiggs systems, the p=1 member of which is the usual Abelian Higgs model. It is found that for all p, these vortices have both attractive and repulsive phases, the strengths of these interactions decreasing successively for increasing p. A detailed comparison of the p=1,2 and p=3 cases is made quantitatively, and an extended model, composed of the p=1 and p=2 members, is studied in some detail and shown to exhibit attractive and repulsive phases. © 1998 American Institute of Physics.

Notes: We analyze constrained quantum systems where the dynamics do not preserve the constraints. This is done, in particular, for the restriction of a quantum particle in Rn to a curved submanifold, and we propose a method of constraining and dynamics adjustment that produces the right Hamiltonian on the submanifold when tested on known examples. This method will be the germ of a "Dirac algorithm for quantum constraints." We generalize it to the situation where the constraint is a general self-adjoint operator with some additional structures. © 1998 American Institute of Physics.

Notes: Simple cosmological models are used to show that gravitation can be quantized as an ordinary gauge system if the Hamilton–Jacobi equation for the model under consideration is separable. In this situation, a canonical transformation can be performed such that in terms of the new variables the model has a linear and homogeneous constraint, and therefore canonical gauges are admissible in the path integral. This has the additional practical advantage that gauge conditions that do not generate Gribov copies are then easy to choose. © 1998 American Institute of Physics.

Notes: Separability of bi-Hamiltonian finite-dimensional chains with two degenerated Poisson tensors, which have Pfaffian quasi-bi-Hamiltonian representation, is proved. © 1998 American Institute of Physics.

Notes: For slowly rotating fluids, we establish the existence of a critical point similar to the one found for nonrotating systems. As the fluid approaches the critical point, the effective inertial mass of any fluid element decreases, vanishing at that point and changing sign beyond it. This result implies that the first-order perturbative method is not always reliable to study dissipative processes occurring before relaxation. Physical consequences that might follow from this effect are commented on. © 1998 American Institute of Physics.

Notes: The spin-torsion sector of a new gauge-theoretic formulation of gravity is analyzed and the relationship to the Einstein–Cartan–Kibble–Sciama theory of gravity is discussed. The symmetries of the Riemann tensor and the conservation laws of the theory are derived. This formalism is applied to the problem of a Dirac field coupled self-consistently to gravity. The equations derived from a minimally coupled gauge-invariant Lagrangian naturally give the gauge-theoretic analogs of the Einstein–Cartan–Dirac equations. Finally, a semiclassical model for a spinning point-particle moving in a gravitational background with torsion is considered. © 1998 American Institute of Physics.

Notes: As a basis for electronic structure calculations, Gaussians are inconvenient because they show unsuitable behavior at larger distances, while Hankel functions are singular at the origin. This paper discusses a new set of special functions which combine many of the advantageous features of both families. At large distances from the origin, these "smoothed Hankel functions" resemble the standard Hankel functions and therefore show behavior similar to that of an electronic wave function. Near the origin, the functions are smooth and analytical. Analytical expressions are derived for two-center integrals for the overlap, the kinetic energy, and the electrostatic energy between two such functions. We also show how to expand such a function around some point in space and discuss how to evaluate the potential matrix elements efficiently by numerical integration. This supplies the elements needed for a practical application in an electronic structure calculation. © 1998 American Institute of Physics.

Notes: We consider the double sinh-Gordon potential which is a quasi-exactly solvable problem and show that in this case one has two sets of Bender–Dunne orthogonal polynomials. We study in some detail the various properties of these polynomials and the corresponding quotient polynomials. In particular, we show that the weight functions for these polynomials are not always positive. We also study the orthogonal polynomials of the double sine-Gordon potential which is related to the double sinh-Gordon case by an anti-isospectral transformation. Finally we discover a new quasi-exactly solvable problem by making use of the anti-isospectral transformation. © 1998 American Institute of Physics.

Notes: The complete classification of Lie symmetries for the Kramers equation is described. Group invariance under infinitesimal transformations is used to generate a wide class of exact solutions. Some of the known solutions are obtained as particular cases. The general results can also be used to solve some types of moving-boundary problems. We researched some Q-conditional symmetry properties of this equation. © 1998 American Institute of Physics.

Notes: We describe gauge fixing at the level of virtual paths in the path integral as a nonsymplectic BRST-type of flow on the path phase space. As a consequence, a gauge-fixed, nonlocal symplectic structure arises. Restoring of locality is discussed. A pertinent anti-Lie-bracket and an infinite dimensional group of gauge fermions are introduced. Generalizations to Sp(2)-symmetric BLT theories are made. © 1998 American Institute of Physics.

Notes: We propose a path integral formulation of noncommutative generalizations of space–time manifold in even dimensions, characterized by a length scale λP. The commutative case is obtaind in the limit λP=0. © 1998 American Institute of Physics.

Notes: We consider a paraboliclike system of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds. We find that in the two-dimensional case there is a simple gauge theoretic flow for a connection built from a Riemannian structure, and that the convergence of the flow to the fixed points is consistent with the Poincare Uniformization Theorem. We construct a similar system for the three-dimensional case. Here the connection is built from a Riemannian geometry, an SO(3) connection and two other 1-form fields which take their values in the SO(3) algebra. The flat connections include the eight homogeneous geometries relevant to the three-dimensional uniformization theorem conjectured by Thurston. The fixed points of the flow include, besides the flat connections (and their local deformations), nonflat solutions of the Yang–Mills equations. These latter "instanton" configurations may be relevant to the fact that generic 3-manifolds do not admit one of the homogeneous geometries, but may be decomposed into "simple 3-manifolds" which do. © 1998 American Institute of Physics.

Notes: The product of two Heisenberg–Weil algebras contains the Jordan–Schwinger representation of su(2). This algebra is quotiented by the square-root of the Casimir to produce a nonassociative algebra denoted by Ψ. This algebra may be viewed as the right-module over one of its associative subalgebras which corresponds to the algebra of scalar fields on the noncommutative sphere. It is now possible to interpret other subspaces as the space of spinor or vector fields on the noncommutative sphere. A natural basis of Ψ is given which may be interpreted as the deformed entries in the rotation matrices of SU(2). © 1998 American Institute of Physics.

Notes: It is well known, and important for applications, that Ricci-flat Riemannian manifolds of nongeneric holonomy always admit a parallel (covariant constant) spinor if they are simply connected. The nonsimply connected case is much more subtle, however. We show that a parallel spinor can still be found in this case provided that the (real) dimension is not a multiple of four, and provided that the spin structure is carefully chosen. © 1998 American Institute of Physics.

Notes: Ginzburg–Landau theory has provided an effective method for understanding the onset of superconductivity in the presence of an external magnetic field. In this paper we examine the instability of the normal state to superconductivity with decreasing magnetic field for a closed smooth cylindrical region of arbitrary cross-section subject to a vertical magnetic field. We examine the problem asymptotically in the boundary layer limit (i.e., when the Ginzburg–Landau parameter, k, is large). We demonstrate that instability first occurs in a region exponentially localized near the point of maximum curvature on the boundary. The transition occurs at a value of the magnetic field associated with the half-plane at leading order, with a small positive correction due to the curvature (which agrees with the transition problem for the disc), and a smaller correction due to the second derivative of the curvature at the maximum. © 1998 American Institute of Physics.

Notes: The theory of adiabatic asymptotics is adapted to systems with discrete time evolution. The corresponding theorems about the approximation of physical time evolution by the adiabatic time evolution are shown to hold true in a discrete setting. © 1998 American Institute of Physics.

Notes: We propose stochastic hidden variables model in which hidden variables have a p-adic probability distribution ρ(λ) and at the same time conditional probabilistic distributions P(U,λ), U=A,A′,B,B′, are ordinary probabilities defined on the basis of the Kolmogorov measure-theoretical axiomatics. A frequency definition of p-adic probability is quite similar to the ordinary frequency definition of probability. p-adic frequency probability is defined as the limit of relative frequencies νn but in the p-adic metric. We study a model with p-adic stochastics on the level of the hidden variables description. But, of course, responses of macroapparatuses have to be described by ordinary stochastics. Thus our model describes a mixture of p-adic stochastics of the microworld and ordinary stochastics of macroapparatuses. In this model probabilities for physical observables are the ordinary probabilities. At the same time Bell's inequality is violated. © 1998 American Institute of Physics.

Notes: The relationship between topology and quantum mechanics is considered in two different ways. We suggest regarding quantum mechanics as a first-order approximation to topology theory on a fixed set. In this spirit, the approach of quantizing the lattice τ(X) of topologies on a set X [C. J. Isham, "Quantum topology and quantization on the lattice of topologies," Class. Quantum. Grav. 6, 1509–1534 (1989)] is shown to be easily applicable to a quantum mechanical lattice. In the main part of the paper we deal with the question of the quantization of the whole category of topological spaces and continuous injective maps. Ideas of the modern categorical approach to quantization, as known from topological quantum field theory, are invoked in this part. The final paragraphs are devoted to the problem of extending quantization, even to the level of the underlying set theory. © 1998 American Institute of Physics.