Homogeneous structures on three-dimensional Lorentzian manifolds

We prove that any non-symmetric three-dimensional homogeneous Lorentzian manifold is isometric to a Lie group equipped with a left-invariant Lorentzian metric. We then classify all three-dimensional homogeneous Lorentzian manifolds.     

Lorentz manifolds modelled on a Lorentz symmetric space

We give examples of Lorentz manifolds modelled on an indecomposable Lorentz symmetric space which are geodesically complete and not locally homogeneous.     

Letter: On the Global Hyperbolicity of Lorentz 2-Step Nilpotent Lie Groups

This work is concerned with the existence of Lorentz 2-step nilpotent Lie groups having a timelike center and which are not globally hyperbolic. Namely, we prove that any left invariant Lorentz metric with a timelike center on the Heisenberg group H ²ⁿ⁺¹ is not globally hyperbolic.     

Hamiltonian circle actions on symplectic manifolds and the signature

Let M be a symplectic manifold with a Hamiltonian circle action with isolated fixed points. We prove that σ (M) = b₀(M) − b₂(M) + b₄(M) − b₆(M) + … where σ (M) is the signature of M and b_i(M) is the ith Betti number of M.     

A Unified Approach to Poisson Reduction

Given any Poisson action G×PP of a Poisson–Lie group G we construct an object =T ^*G*T^* P which has both a Lie groupoid structure and a Lie algebroid structure and which is a half-integrated form of the matched pair of Lie algebroids which J.-H. Lu associated to a Poisson action in her development of Drinfeld's classification of Poisson homogeneous spaces. We use to give a general reduction procedure for Poisson group actions, which applies in cases where a moment map in the usual sense does not exist. The same method may be applied to actions of symplectic groupoids and, most generally, to actions of Poisson groupoids.     

High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy

In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier–Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the machine accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.     

One-parameter Lie groups admitted by time-dependent Schrödinger equation: Atoms,molecules and nucleons in harmonic oscillator field

Lie’s method of differential equation is used to obtain the one-parameter Lie groups admitted by the time-dependent Schrödinger equations for atoms, molecules and nucleons in harmonic oscillator field. This group for atoms and molecules is isomorphic to 10-parameter inhomogeneous orthogonal group in 4 dimensions, irrespective of the numbers of nuclei and electrons. For Z protons andN neutrons in a harmonic oscillator field, both isotropic and anisotropic, the r-parameter Lie groups are seraidirect products of an invariant subgroup and a factor group. In the case of isotropic oscillator field r is 1/2[3Z(3Z-1) +3N (3N -1)+2], while for the anisotropic oscillator field r is 1/2[3Z (Z+1)+3N(JV+1)+2].     

Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations

This is the fourth in a series of papers on developing a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group to include transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. In previous work, we have shown that there exist representations of the Galilean line group that uphold the non-relativistic equivalence principle as well as representations that violate the equivalence principle. In these previous studies, the focus was on linear accelerations. In this paper, we undertake an extension of the formulation to include rotational accelerations. We show that the incorporation of rotational accelerations requires a class of loop prolongations of the Galilean line group and their unitary cocycle representations. We recover the centrifugal and Coriolis force effects from these loop representations. Loops are more general than groups in that their multiplication law need not be associative. Hence, our broad theoretical claim is that a Galilean quantum theory that holds in arbitrary non-inertial reference frames requires going beyond groups and group representations, the well-established framework for implementing symmetry transformations in quantum mechanics.     

Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor

The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field B_ij (x,x′,t) and isometries of the correlation space K³ equipped with a (pseudo-) Riemannian metrics ds²(t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds²(t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (M^t, ds²(t)), M^t ? K³. Here the metric ds²(t), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds²(t) in K³ and present symmetries (or isometric motions in K³) of the metric ds²(t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds²(t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale λ_g is a second-order differential invariant and show how λ_g can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.     

Time evolution of the classical and quantum mechanical versions of diffusive anharmonic oscillator: an example of Lie algebraic techniques

We present the general solutions for the classical and quantum dynamics of the anharmonic oscillator coupled to a purely diffusive environment. In both cases, these solutions are obtained by the application of the Baker-Campbell-Hausdorff (BCH) formulas to expand the evolution operator in an ordered product of exponentials. Moreover, we obtain an expression for the Wigner function in the quantum version of the problem. We observe that the role played by diffusion is to reduce or to attenuate the the characteristic quantum effects yielded by the nonlinearity, as the appearance of coherent superpositions of quantum states (Schr?dinger cat states) and revivals.