Second quantization for systems with a constant number of particles in the Dirac notation

The relation between the completeness condition for an appropriate one-particle basis set and the occupation number representation (second quantization) is shown for the time-independent case. The explicit expressions for the basic symmetric operators are derived in the Dirac bra–ket notation. The physical meaning of these operators, the algebra as well as the connections with the one-electron density matrix and with the projector on the Fermi sea in the one-electron approximation, follow directly from these expressions. The generalization for a nonorthogonal basis and the algebra for corresponding basic operators are formulated. The connection with the notion of the molecular diagrams of different kinds for the nonorthogonal atomic orbitals is shown. The Mulliken populations and the Chirgwin–Coulson bond orders are equal to the diagonal and offdiagonal elements of the molecular diagram 1, respectively. The matrix elements of the projector on the Fermi sea in the one-electron approximation in the representation of nonorthogonal atomic orbitals are elements of the molecular diagram 2.     

Extension of the modified Sanderson method to ternary mixed oxides derived from hydrotalcite-like compounds and its correlation to CO2 (TPD-CO2) desorption peak maxima

A simple, intuitive and computationally non-intensive modified model, proposed first for the prediction of the catalyst basicity of binary mixed oxides, based simply on the chemical composition using the Sanderson Method was extended to ternary systems. The modification of the method allows the discrimination between the oxygen charge on different chemical environments (differences between local and global composition). TPD-CO₂ result was used to the validation between theoretical values and experimental results, letting us propose a correlation between the oxygen charge of the different clusters and the TPD-CO₂ peak maxima.     

Scaling laws in the vortex lattice model of turbulence

The lattice vortex model of the intertial range in turbulence theory is reviewed; the model consists of an array of vortex tubes whose axes coincide with the bonds on a regular lattice, subjected to random stretching and successive scaling, and constrained by conservation laws for energy, specific volume, circulation, helicity, and an energy/vorticity relation. The scaling laws for vorticity are examined in detail, a Hausdorff dimension for the active portion of the vortex array is calculated, the origin of intermittency is exhibited, and it is pointed out that the Kolmogorov — 5/3 power law already accounts for intermittency effects.Partially supported by the Office of Energy Research, U.S. Department of Energy, under contract DE-AC03-76SF0098, and in part by the Office of Naval Research under contract N00014-76-C-0316     

Odometer actions of the Heisenberg group

Let H₃(?) be the 3-dimensional real Heisenberg group. Given a family of lattices Γ₁ ? Γ₂ ? … ? H₃(?), let T be the associated uniquely ergodic H₃(?)-odometer, i.e., the inverse limit of the H₃(?)-actions by rotations on the homogeneous spaces H₃(?)/Γ_j, j ∈ ?. We describe explicitly the decomposition of the underlying Koopman unitary representation of H₃(?) into a countable direct sum of irreducible components and find the ergodic 2-fold self-joinings of T. We show that in general, the H₃(?)-odometers are neither isospectral nor spectrally determined.     

MIP models for production lot sizing problems with distribution costs and cargo arrangement

This study presents mixed integer programming (MIP) models for production lot sizing problems with distribution costs using unit load devices such as pallets and containers. Problems that integrate production lot sizing decisions and loading of the products in vehicles (bins) are also modelled, in which constraints such as weight limits, volume restrictions or the value of the cargo loaded in the bins are considered. In general, these problems involve a trade-off between production, inventory and distribution costs. Lot sizing decisions should take into account production capacity and product demand constraints. Distribution decisions are related to the loading and transport of products in unit load devices. The MIP models are solved by the branch-and-cut method of an optimization package and the results show that these approaches have the potential to address different practical situations.     

The formulation of dynamical contact problems with friction in the case of systems of rigid bodies and general discrete mechanical systems—Painlevé and Kane paradoxes revisited

The dynamics of mechanical systems with a finite number of degrees of freedom (discrete mechanical systems) is governed by the Lagrange equation which is a second-order differential equation on a Riemannian manifold (the configuration manifold). The handling of perfect (frictionless) unilateral constraints in this framework (that of Lagrange’s analytical dynamics) was undertaken by Schatzman and Moreau at the beginning of the 1980s. A mathematically sound and consistent evolution problem was obtained, paving the road for many subsequent theoretical investigations. In this general evolution problem, the only reaction force which is involved is a generalized reaction force, consistently with the virtual power philosophy of Lagrange. Surprisingly, such a general formulation was never derived in the case of frictional unilateral multibody dynamics. Instead, the paradigm of the Coulomb law applying to reaction forces in the real world is generally invoked. So far, this paradigm has only enabled to obtain a consistent evolution problem in only some very few specific examples and to suggest numerical algorithms to produce computational examples (numerical modeling). In particular, it is not clear what is the evolution problem underlying the computational examples. Moreover, some of the few specific cases in which this paradigm enables to write down a precise evolution problem are known to show paradoxes: the Painlevé paradox (indeterminacy) and the Kane paradox (increase in kinetic energy due to friction). In this paper, we follow Lagrange’s philosophy and formulate the frictional unilateral multibody dynamics in terms of the generalized reaction force and not in terms of the real-world reaction force. A general evolution problem that governs the dynamics is obtained for the first time. We prove that all the solutions are dissipative; that is, this new formulation is free of Kane paradox. We also prove that some indeterminacy of the Painlevé paradox is fixed in this formulation.     

Equation of State Grid with the Quark-Meson-Coupling Model

In this work, we present the preliminary results of an equation of state (EoS) grid for possible use in core-collapse supernova simulations. We treat uniform matter made of nucleons using the quark-meson coupling (QMC) model. We show a table with a variety of thermodynamic quantities, which covers the proton fraction range Y _p = 0?0.65 with the linear grid spacing ΔY _p = 0.01 (66 points) and the density range ρ _B = 10¹⁴ ? 10¹⁶g cm^?3 with the logarithmic grid spacing Δlog₁₀(ρ _B /[g cm^?3]) = 0.1 (21 points). This preliminary study is performed at zero temperature, and our results are compared with the widely used EoS already available in the literature.     

A version of Fabry's theorem for power series with regularly varying coefficients

For real power series whose non-zero coefficients satisfy , we prove a stronger version of Fabry's theorem relating the frequency of sign changes in the coefficients and analytic continuation of the sum of the power series.

     

Banach Spaces with Property (M) and their Szlenk Indices

We show that a separable Banach space with property (M*) has a Szlenk index equal to ω₀, and a norm with an optimal modulus of asymptotic uniform smoothness. From this we derive a condition on the Szlenk functions of the space and its dual which characterizes embeddability into c ₀ or an ℓ _p -sum of finite dimensional spaces. We also prove that two Lipschitz-isomorphic Orlicz sequence spaces contain the same ℓ _p -spaces.