On Lagrangian tangent sweeps and Lagrangian outer billiards

Given a Lagrangian submanifold in linear symplectic space, its tangent sweep is the union of its (affine) tangent spaces, and its tangent cluster is the result of parallel translating these spaces so that the foot point of each tangent space becomes the origin. This defines a multivalued map from the tangent sweep to the tangent cluster, and we show that this map is a local symplectomorphism (a well known fact, in dimension two). We define and study the outer billiard correspondence associated with a Lagrangian submanifold. Two points are in this correspondence if they belong to the same tangent space and are symmetric with respect to its foot pointe. We show that this outer billiard correspondence is symplectic and establish the existence of its periodic orbits. This generalizes the well studied outer billiard map in dimension two.Comment: revision as requested by the refere

Polymer-Mode-Coupling Theory of Finite-Size-Fluctuation Effects in Entangled Solutions, Melts and Gels. I. General Formulation and Predictions

The transport coefficients of dense polymeric fluids are approximately calculated from the microscopic intermolecular forces. The following finite molecular weight effects are discussed within the Polymer-Mode-Coupling theory (PMC) and compared to the corresponding reptation/ tube ideas: constraint release mechanism, spatial inhomogeneity of the entanglement constraints, and tracer polymer shape fluctuations. The entanglement corrections to the single polymer Rouse dynamics are shown to depend on molecular weight via the ratio N/N_e, where the entanglement degree of polymerization, N_e, can be measured from the plateau shear modulus. Two microscopically defined non-universal parameters, an entanglement strength 1/alpha and a length scale ratio, delta= xi_rho/b, where xi_rho and b are the density screening and entanglement length respectively, are shown to determine the reduction of the entanglement effects relative to the reptation- -like asymptotes of PMC theory. Large finite size effects are predicted for reduced degrees of polymerization up to N/N_e\le10^3. Effective power law variations for intermediate N/N_e of the viscosity, eta\sim N^x, and the diffusion constant, D\sim N^{-y}, can be explained with exponents significantly exceeding the asymptotic, reptation-like values, x\ge 3 and y\ge2, respectively. Extensions of the theory to treat tracer dielectric relaxation, and polymer transport in gels and other amorphous systems, are also presented.Comment: Latex, figures and styles files included; Macromolecules, in press (1997

Polymer-Mode-Coupling Theory of Finite-Size-Fluctuation Effects in Entangled Solutions, Melts and Gels. II. Comparison with Experiment

The predictions of the polymer mode coupling theory for the finite size corrections to the transport coefficients of entangled polymeric systems are tested in comparisons with various experimental data. It is found that quantitative descriptions of the viscosities, eta, dielectric relaxation time, tau_e, and diffusion coefficients, D, of polymer melts can be achieved with two microscopic structural fit parameters whose values are in the range expected from independent theoretical or experimental information. An explanation for the (apparent) power law behaviors of eta, taue, and D in (chemically distinct) melts for intermediate molecular weights as arising from finite size corrections, mainly the self-consistent constraint release mechanism, is given. The variation of tracer dielectric relaxation times from Rouse to reptation-like behavior upon changes of the matrix molecular weight is analyzed. Self and tracer diffusion constants of entangled polymer solutions can be explained by the theory as well, if one further parameter of the theory is adjusted. The anomalous scaling of the tracer diffusion coefficients in semidilute and concentrated polystyrene solutions, D\sim N^{-2.5}, is predicted to arise due to the spatial correlations of the entanglement constraints, termed ``constraint porosity''. Extensions of the theory to polymer tracer diffusion through polyvinylmethylether and polyacrylamide gels provide an explanation of the observation of anomalously high molecular weight scaling exponents in a range where the size of the tracer, R_g, already considerably exceeds the gel pore size, xi_g.Comment: Latex, figures and styles files included; Macromolecules, in press (1997

Recent developments in effective field theory

We will give a short introduction to the one-nucleon sector of chiral perturbation theory and will address the issue of a consistent power counting and renormalization. We will discuss the infrared regularization and the extended on-mass-shell scheme. Both allow for the inclusion of further degrees of freedom beyond pions and nucleons and the application to higher-loop calculations. As applications we consider the chiral expansion of the nucleon mass to order O(q^6) and the inclusion of vector and axial-vector mesons in the calculation of nucleon form factors.Comment: 8 pages, 6 figures, invited talk given at International School of Nuclear Physics, 29th Course "Quarks in Hadrons and Nuclei", Erice, Sicily, 16 - 24 September 200

Analysis and interpretation of new low-energy Pi-Pi scattering data

The recently published E865 data on charged K_e4 decays and Pi-Pi phases are reanalyzed to extract values of the two S-wave scattering lengths, of the subthreshold parameters alpha and beta, of the low-energy constants l3-bar and l4-bar as well as of the main two-flavour order parameters: and F_pi in the limit m_u = m_d = 0 taken at the physical value of the strange quark mass. Our analysis is exclusively based on direct experimental information on Pi-Pi phases below 800 MeV and on the new solutions of the Roy equations by Ananthanarayan et al. The result is compared with the theoretical prediction relating 2 a_0^0 - 5 a_0^2 and the scalar radius of the pion, which was obtained in two-loop Chiral Perturbation Theory. A discrepancy at the 1-sigma level is found and commented upon.Comment: Published version, to appear in Eur. Phys. J.