Global stability of equilibrium manifolds,and “peaking” behavior in quadratic differential systems related to viscoelastic models

We study a model inspired by the Oldroyd-B equations for viscoelastic fluids. The objective is to better understand the nonlinear coupling between the stress and velocity fields in viscoelastic flows, and thus gain insight into the reasons that cause the loss of accuracy of numerical computations at high Weissenberg number. We derive a model system by discarding the stress-advection and stress-relaxation terms in the Oldroyd-B model. The reduced (unphysical) model, which bears some resemblance to a viscoelastic solid, only retains the stretching of the stress due to velocity gradients and the induction of velocity by the stress field. Our conjecture is that such a system always evolves toward an equilibrium in which the stress builds up such to cancel the external forces. This conjecture is supported by numerous simulations. We then turn our attention to a finite dimensional model (i.e., a set of ordinary differential equations) that has the same algebraic structure as our model system. Numerical simulations indicate that the finite-dimensional analog has a globally attracting equilibrium manifold. In particular, it is found that subsets of the equilibrium manifold may be unstable, leading to a “peaking” behavior, where trajectories are repelled from the equilibrium manifold at one point, and are eventually attracted to a stable equilibrium point on the same manifold. Generalizations and implications to solutions of the Oldroyd-B model are discussed.     

Nonlocal Wigner-like correlation energy density functional: parametrization and tests on two-electron systems

Reparametrization of Wigner's correlation energy density functional yields a very close fit to the correlation energies of the helium isoelectronic sequence. However, a quite different reparametrization is required to obtain an equally close fit to the isoelectronic sequence of Hooke's atom. In an attempt to avoid having to reparametrize the functional for different choices of the one-body potential, we propose a parametrization that depends on global characteristics of the ground-state electron density as quantified by scale-invariant combinations of expectation values of local one-body operators. This should be viewed as an alternative to the density-gradient paradigm, allowing one to introduce the nonlocal dependence of the density functional on the density in a possibly more effective way. Encouraging results are obtained for two-electron systems with one-body potentials of the form r(zeta) with zeta=-12,+12,1, which span the range between the Coulomb potential (zeta=-1) and the Hooke potential (zeta=2).     

Shape restricted wavefunctions: Ellipsoidal limits for ground and excited σg states of H2

Ellipsoidal as well as spherical limits for the energies of the 1sσ_g, 2sσ_g and 3sσ_g states of H₂⁺ have been obtained at various internuclear distances. The improvement obtained for the ground-state energy in the ellipsoidal limit is about 40% of the difference between the spherical limit and the exact energy.     

Coupling of radial and angular correlations in two-electron atoms

The nature of interelectronic correlation in the ground and low lying excited states of the helium sequence is investigated. It is demonstrated that as a consequence of the coupling between radial and angular correlation the electrons are actually pulled together rather than pushed apart, in the low nuclear charge end of the isoelectronic sequence. A uniform treatment of correlation effects in both the ground and the excited states is thus shown to be feasible.     

Anthropic arguments and the cosmological constant, with and without the assumption of typicality

We reexamine claims that anthropic arguments provide an explanation for the observed smallness of the cosmological constant, and we argue that correlations between the cosmological constant value and the existence of life could at best be demonstrated only under restrictive assumptions. Causal effects are more subtle to uncover. The assumption of our typicality is crucial to such arguments.     

Many‐particle Dirac identities for arbitrary elementary spins

The eigenvalues of arbitrary conjugacy class‐sums of the symmetric group, within subspaces that contain irreducible representations with at most k rows, are considered. Explicit expressions for these eigenvalues in terms of the eigenvalues of single‐cycle class‐sums with cycle lengths up to k are obtained. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 407–411, 2000     

Nonlinear computer-generated holograms

We propose a novel technique for arbitrary wavefront shaping in quadratic nonlinear crystals by introducing the concept of computer-generated holograms (CGHs) into the nonlinear optical regime. We demonstrate the method experimentally showing a conversion of a fundamental Gaussian beam pump light into the first three Hermite-Gaussian beams at the second harmonic in a stoichiometric lithium tantalate nonlinear crystal, and we characterize its efficiency dependence on the fundamental power and the crystal temperature. Nonlinear CGHs open new possibilities in the fields of nonlinear beam shaping, mode conversion, and beam steering.     

Why is there a molecular relativistic effect?

The almost exclusive association of the molecular geometry dependence of the relativistic correction with the valence orbital contribution to the mass-velocity and Darwin terms is investigated using SCF and MCSCF wavefunctions. The requirement of orthogonality of the valence orbitals to the core orbitals is confirmed to be the mechanism responsible for the increase in (the absolute value of) relativistic energy upon decrease of the internuclear distance. Certain “fingerprint”-type features of the valence relativistic correction, revealing the identity of the particular core orbital giving rise to it, are identified.     

Chirality of silver nanoparticles synthesized on DNA

Silver nanocrystals grown on a poly(dG)-poly(dC) double stranded DNA scaffold displayed circular dichroism at their surface plasmon excitation band. This chiral plasmon signature was not observed in a control experiment where silver nanocrystals of similar size were produced without the DNA template and adsorbed to the DNA. It is concluded that the DNA templated Ag nanocrystals had a preferred structural handedness.