Reduced dynamics and Lagrangian submanifolds of symplectic manifolds

In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincar\'e and Lagrange-Poincar\'e reduction. In fact, we show that solutions of the Hamilton-Poincar\'e equations and of the Lagrange-Poincar\'e equations are in one-to-one correspondence with distinguished curves in a Lagrangian submanifold of a symplectic manifold. For this purpose, we will combine the concept of a Tulczyjew triple with Marsden-Weinstein symplectic reduction.Comment: 26 page

Polyradical character and spin frustration in fullerene molecules: An ab initio non-collinear Hartree--Fock study

Most {\em ab initio} calculations on fullerene molecules have been carried out based on the paradigm of the H\"uckel model. This is consistent with the restricted nature of the independent-particle model underlying such calculations, even in single-reference-based correlated approaches. On the other hand, previous works on some of these molecules using model Hamiltonians have clearly indicated the importance of short-range inter-atomic spin-spin correlations. In this work, we consider {\em ab initio} non-collinear Hartree--Fock (HF) solutions for representative fullerene systems: the bowl, cage, ring, and pentagon isomers of C$_{20}$, and the larger C$_{30}$, C$_{36}$, C$_{60}$, C$_{70}$, and C$_{84}$ fullerene cages. In all cases but the ring we find that the HF minimum corresponds to a truly non-collinear solution with a torsional spin density wave. Optimized geometries at the generalized HF (GHF) level lead to fully symmetric structures, even in those cases where Jahn-Teller distortions have been previously considered. The nature of the GHF solutions is consistent with the $\pi$-electron space becoming polyradical in nature: each $p$-orbital remains effectively singly occupied. The spin frustration, induced by the pentagon rings in an otherwise anti-ferromagnetic background, is minimized at the HF level by aligning the spins in non-collinear arrangements. The long-range magnetic ordering observed is reminiscent of the character of broken symmetry HF solutions in polyacene systems.Comment: 16 figure

Multi-reference symmetry-projected variational approximation for the ground state of the doped one-dimensional Hubbard model

A multi-reference configuration mixing scheme is used to describe the ground state, characterized by well defined spin and space group symmetry quantum numbers as well as doping fractions $N_{e}/N_{sites}$, of one dimensional Hubbard lattices with nearest-neighbor hopping and periodic boundary conditions. Within this scheme, each ground state is expanded in a given number of nonorthogonal and variationally determined symmetry-projected configurations. The results obtained for the ground state and correlation energies of half-filled and doped lattices with 30, 34 and 50 sites, compare well with the exact Lieb-Wu solutions as well as with the ones obtained with other state-of-the-art approximations. The structure of the intrinsic symmetry-broken determinants resulting from the variational procedure is interpreted in terms of solitons whose translational and breathing motions can be regarded as basic units of quantum fluctuations. It is also shown that in the case of doped 1D lattices, a part of such fluctuations can also be interpreted in terms of polarons. In addition to momentum distributions, both spin-spin and density-density correlation functions are studied as functions of doping. The spectral functions and density of states, computed with an ansatz whose quality can be well-controlled by the number of symmetry-projected configurations used to approximate the $N_{e} \pm 1$ electron systems, display features beyond a simple quasiparticle distribution, as well as spin-charge separation trends.Comment: 16 pages, 11 figure

Theories of the evolution of cooperative behaviour: A critical survey plus some new results

Gratuitous cooperation (in favour of non-relatives and without repeated interaction) eludes traditional evolutionary explanations. In this paper we survey the various theories of cooperative behaviour, and we describe our own effort to integrate these theories into a self-contained framework. Our main conclusions are as follows. First: altruistic punishment, conformism and gratuitous cooperation co-evolve, and group selection is a necessary ingredient for the co-evolution to take place. Second: people do not cooperate by mistake, as most theories imply; on the contrary, people knowingly sacrifice themselves for others. Third: in cooperative dilemmas conformism is an expression of preference, not a learning rule. Fourth, group-mutations (e.g., the rare emergence of a charismatic leader that brings order to the group) are necessary to sustain cooperation in the long run.Cooperation; altruism; altruistic punishment; conformism; group-selection

Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection–diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank–Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the t2+hp+12 error estimates for the L2-norm under either the standard hyperbolic CFL condition, when piecewise affine (p=1) approximation is used, or in the case of finite element approximation of order p≥1, a stronger, so-called 4/3-CFL, i.e. t≤Ch4/3. The theory is illustrated with some numerical examples

Internal Kinematics of Luminous Compact Blue Galaxies

We describe the dynamical properties which may be inferred from HST/STIS spectroscopic observations of luminous compact blue galaxies (LCBGs) between 0.1<z<0.7. While the sample is homogeneous in blue rest-frame color, small size and line-width, and high surface-brightness, their detailed morphology is eclectic. Here we determine the amplitude of rotation versus random, or disturbed motions of the ionized gas. This information affirms the accuracy of dynamical mass and M/L estimates from Keck integrated line-widths, and hence also the predictions of the photometric fading of these unusual galaxies. The resolved kinematics indicates this small subset of LCBGs are dynamically hot, and unlikely to be embedded in disk systems.Comment: To appear in "Starbursts: from 30 Doradus to Lyman Break Galaxies" 2005, eds. R. de Grijs and R. M. Gonzalez Delgado (Kluwer