A note on the Holst action, the time gauge, and the Barbero–Immirzi parameter

In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero–Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero–Immirzi parameter as a potential way to investigate its fate and relevance in the quantum theory.     

Helmholtz, Riemann, and the Sirens: Sound, Color, and the “Problem of Space”

Emerging from music and the visual arts, questions about hearing and seeing deeply affected Hermann Helmholtz’s and Bernhard Riemann’s contributions to what became called the “problem of space [Raumproblem],” which in turn influenced Albert Einstein’s approach to general relativity. Helmholtz’s physiological investigations measured the time dependence of nerve conduction and mapped the three-dimensional manifold of color sensation. His concurrent studies on hearing illuminated musical evidence through experiments with mechanical sirens that connect audible with visible phenomena, especially how the concept of frequency unifies motion, velocity, and pitch. Riemann’s critique of Helmholtz’s work on hearing led Helmholtz to respond and study Riemann’s then-unpublished lecture on the foundations of geometry. During 1862–1870, Helmholtz applied his findings on the manifolds of hearing and seeing to the Raumproblem by supporting the quadratic distance relation Riemann had assumed as his fundamental hypothesis about geometrical space. Helmholtz also drew a “close analogy … in all essential relations between the musical scale and space.” These intersecting studies of hearing and seeing thus led to reconsideration and generalization of the very concept of “space,” which Einstein shaped into the general manifold of relativistic space-time.     

The Schrödinger Equation,the Zero-Point Electromagnetic Radiation,and the Photoelectric Effect

A Schrödinger type equation for a mathematical probability amplitude Ψ(x,t) is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M and charge e, moving in a potential V(x). The particle phase space probability density is denoted Q(x,p,t), and the entire system is immersed in the “vacuum” zero-point electromagnetic radiation. We show, in the first part of the paper, that the generalized Liouville equation is reduced to a simpler Liouville equation in the equilibrium limit where the small radiative corrections cancel each other approximately. This leads us to a simpler Liouville equation that will facilitate the calculations in the second part of the paper. Within this second part, we address ourselves to the following task: Since the Schrödinger equation depends on \(\hbar \), and the zero-point electromagnetic spectral distribution, given by \(\rho _{0}{(\omega )} = \hbar \omega ^{3}/2 \pi ^{2} c^{3}\), also depends on \(\hbar \), it is interesting to verify the possible dynamical connection between ρ₀(ω) and the Schrödinger equation. We shall prove that the Planck’s constant, present in the momentum operator of the Schrödinger equation, is deeply related with the ubiquitous zero-point electromagnetic radiation with spectral distribution ρ₀(ω). For simplicity, we do not use the hypothesis of the existence of the L. de Broglie matter-waves. The implications of our study for the standard interpretation of the photoelectric effect are discussed by considering the main characteristics of the phenomenon. We also mention, briefly, the effects of the zero-point radiation in the tunneling phenomenon and the Compton’s effect.     

MIDI,the 10 μm interferometer of the VLT

We report in this paper on the current status of the mid-infrared beam combiner of the VLTI: MIDI. We explain the interest of VLTI for this range of wavelentghs and give an overview of the main characteristics of MIDI and present its scientific targets.     

On the Convergence Rate of the Euler-α, an Inviscid Second-Grade Complex Fluid, Model to the Euler Equations

We study the convergence rate of the solutions of the incompressible Euler-α, an inviscid second-grade complex fluid, equations to the corresponding solutions of the Euler equations, as the regularization parameter α approaches zero. First we show the convergence in H ^s , s>n/2+1, in the whole space, and that the smooth Euler-α solutions exist at least as long as the corresponding solution of the Euler equations. Next we estimate the convergence rate for two-dimensional vortex patch with smooth boundaries.     

Analysis of the common characteristics of the Hartmann,Ronchi, and Shack–Hartmann tests

An overview of the settings of the planes for the filters and observed patterns in the Hartmann and Ronchi tests is presented. Also a new set of filters for both test were developed. In a similar way, it is easy to extend this analysis to the Shack–Hartmann test, and to propose a new Null Shack–Hartmann filter.     

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p_x} if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore, we show that the usual proof of the Lovász local lemma – which provides a sufficient condition for this to occur – corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer⁽⁹⁸⁾ and explicitly by Dobrushin.^(37,38) We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for soft dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.     

Effects of the 3d transition metal doping on the structural,electronic, and magnetic properties of BeO nanotubesEffects of the 3d transition metal doping on the structural,electronic, and magnetic properties of BeO nanotubesEffects of the 3d transition metal doping on the structural,electronic, and magnetic properties of BeO nanotubes

First-principles calculations have been performed on the structural, electronic, and magnetic properties of seven 3d transition-metal （TM） impurities （V, Cr, Mn, Fe, Co, Ni, and Cu） doped armchair （5,5） and zigzag （8,0） beryllium oxide nanotubes （BeONTs）. The results show that there exists a structural distortion around the 3d TM impurities with respect to the pristine BeONTs. The magnetic moment increases for V- and Cr-doped BeONTs and reaches a maximum for Mn-doped BeONT, and then decreases for Fe-, Co-, Ni-, and Cu-doped BeONTs successively, consistent with the predicted trend of Hund＇s rule to maximize the magnetic moments of the doped TM ions. However, the values of the magnetic moments are smaller than the predicted values of Hund＇s rule due to the strong hybridization between the 2p orbitals of the near O and Be ions of BeONTs and the 3d orbitals of the TM ions. Furthermore, the V-, Co-, and Ni-doped （5,5） and （8,0） BeONTs with half-metal ferromagnetism and thus 100% spin polarization character are good candidates for spintronic applications.     

Existence,regularity, and asymptotic behavior of the solutions to the Ginzburg-Landau equations on ℝ3

This paper studies the solutions of the Ginzburg-Landau equations on ³ in the presence of an arbitrarily distributed external magnetic field. The existence and regularity of the solutions at the lowest energy level are established. The solutions found are in the Coulomb gauge. If the external field is sufficiently regular, the solutions are shown to have nice asymptotic decay properties at infinity.     

The ζ-Determinant and the Additivity of the η-Invariant on the Smooth, Self-Adjoint Grassmannian

In this paper we discuss the existence of the ‘-determinant of a Dirac operator \Dd\Dd on a compact manifold with boundary. We show that the determinant is well defined in the case of the operator \Dd\Dd with a domain determined by a boundary condition from the smooth, self-adjoint Grassmannian \Grass_￥^*(\Dd)\Grass_{\infty}^*(\Dd) discussed in the papers [5, 13, 29]. We prove a generalization of a pasting formula for the m-invariant (see [34]). The results of the paper are used in the recent proof of the projective equality of the ‘-determinant and Quillen determinant on \Grass_￥^*(\Dd)\Grass_{\infty}^*(\Dd) (see [30, 31]).     

Ions,bubbles, and vortices at the superfluid λ-transition

A vortex-ring theory of the superfluid -transition is reviewed, and new results are presented for the vortex density and for the superfluid density of helium in confined geometries. Possible experiments using ion probes to detect the thermally excited vortices are discussed. It appears that the usual helium ions are not suitable for this purpose, and that larger ion complexes such as multielectron bubbles will need to be employed. An experiment to stably trap multielectron bubbles with acoustic standing waves is outlined.     

Charge,Geometry, and Effective Mass in the Kerr-Newman Solution to the Einstein Field Equations

It has been shown that for the Reissner-Nordström solution to the vacuum Einstein field equations charge, like mass, has a unique space-time signature (Marsh, Found. Phys. 38:293–300, 2008). The presence of charge results in a negative curvature. This work, which includes a discussion of effective mass, is extended here to the Kerr-Newman solution.     

Quantitative estimates of the content of the metastable Si-XII,Si-III,and α-Si phases of silicon in the areas of indenter impressions

A quantitative estimate is presented of the specific and total volumes of the metastable Si-XII, Si-III, and α-Si phases of silicon in a locally strained (Berkovich pyramid) area. Calculations are performed using experimental data obtained via Raman spectroscopy and in situ registration of the Si-I → Si-II phase transitions of silicon under the indenter.     

Configuration interaction study including the effects of spin–orbit coupling for the electronic states of the LiX molecules (X = C,Si, Ge,Sn)

Ab initio multireference configuration interaction calculations including spin–orbit coupling effects have been carried out for four LiX molecules (X?=?C, Si, Ge and Sn). Potential energy curves of the ground and low-lying excited states have been obtained in each case as well as the corresponding spectroscopic constants. Transition moments have also been computed in order to give estimates of the radiative lifetimes of the excited states for each system. Trends in a variety of quantities such as T _e values, spin–orbit splittings, equilibrium bond lengths and vibrational frequencies for this series of molecules are discussed in detail and comparison with the corresponding data reported earlier for the PbLi system is also made.     

First-principles calculations of the structural,electronic, optical and thermal properties of the BNxAs1–x alloys

The present paper aims to study the structural, electronic, optical and thermal properties of the boron nitride (BN) and BAs bulk materials as well as the BN_xAs_1–x ternary alloys by employing the full-potential-linearised augmented plane wave method within the density functional theory. The structural properties are determined using the Wu–Cohen generalised gradient approximation that is based on the optimisation of the total energy. For band structure calculations, both the Wu–Cohen generalised gradient approximation and the modified Becke–Johnson of the exchange-correlation energy and potential, respectively, are used. We investigated the effect of composition on the lattice constants, bulk modulus and band gap. Deviations of the lattice constants and the bulk modulus from the Vegard’s law and the linear concentration dependence, respectively, were observed for the alloys where this result allows us to explain some specific behaviours in the electronic properties of the alloys. For the optical properties, the calculated refractive indices and the optical dielectric constants were found to vary nonlinearly with the N composition. Finally, the thermal effect on some of the macroscopic properties was predicted using the quasi-harmonic Debye model in which the lattice vibrations are taken into account.