Cosmic age, Statefinder and OmOm diagnostics in the decaying vacuum cosmology

As an extension of $\Lambda$CDM, the decaying vacuum model (DV) describes the dark energy as a varying vacuum whose energy density decays linearly with the Hubble parameter in the late-times, $\rho_\Lambda(t) \propto H(t)$, and produces the matter component. We examine the high-$z$ cosmic age problem in the DV model, and compare it with $\Lambda$CDM and the Yang-Mills condensate (YMC) dark energy model. Without employing a dynamical scalar field for dark energy, these three models share a similar behavior of late-time evolution. It is found that the DV model, like YMC, can accommodate the high-$z$ quasar APM 08279+5255, thus greatly alleviates the high-$z$ cosmic age problem. We also calculate the Statefinder $(r,s)$ and the {\it Om} diagnostics in the model. It is found that the evolutionary trajectories of $r(z)$ and $s(z)$ in the DV model are similar to those in the kinessence model, but are distinguished from those in $\Lambda$CDM and YMC. The ${\it Om}(z)$ in DV has a negative slope and its height depends on the matter fraction, while YMC has a rather flat ${\it Om}(z)$, whose magnitude depends sensitively on the coupling.Comment: 12 pages, 4 figures, with some correction

Berry phase in a composite system

The Berry phase in a composite system with only one subsystem being driven has been studied in this Letter. We choose two spin-$\frac 1 2$ systems with spin-spin couplings as the composite system, one of the subsystems is driven by a time-dependent magnetic field. We show how the Berry phases depend on the coupling between the two subsystems, and what is the relation between these Berry phases of the whole system and those of the subsystems.Comment: 4 pages, 6 figure

Geometric, Variational Integrators for Computer Animation

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physics-based animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically preserves important invariants, such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the implementation of the method. These properties are achieved using a discrete form of a general variational principle called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate the applicability of our integrators to the simulation of non-linear elasticity with implementation details

Discrete Lie Advection of Differential Forms

In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite volume methods for scalar hyperbolic conservation laws, we first discretize the interior product (also called contraction) through integrals over Eulerian approximations of extrusions. This, along with Cartan's homotopy formula and a discrete exterior derivative, can then be used to derive a discrete Lie derivative. The usefulness of this operator is demonstrated through the numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC