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 Riemann’s critique of Helmholtz’s work on hearing led Helmholtz to respond and study Riemann’s then-unpublished lecture on the foundations of geometry.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.     

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.     

Transition probabilities of the X1Σ+, A1Π, B1Δ, C1Σ+, and D1Π states of the AlO+ cation

ABSTRACT

This study computes the potential energy curves of the X¹Σ⁺, A¹Π, B¹Δ, C¹Σ⁺, and D¹Π states of AlO⁺ cation and the transition dipole moments between them. The orders of the rotationless radiative lifetimes are 10–100?μs for the A¹Π state, 1–1000?ms for the B¹Δ state, 10?ns for the first well and 100?ns for the second well of the C¹Σ⁺ state, and 1?μs for the D¹Π state. Emissions of the B¹Δ–A¹Π and D¹Π–C¹Σ⁺ systems are so weak that they are hardly measured via spectroscopy, the emissions of the C¹Σ⁺–X¹Σ⁺, C¹Σ⁺–A¹Π, and D¹Π–X¹Σ⁺ systems are so strong that they can be detected readily, and emissions of the A¹Π–X¹Σ⁺ and D¹Π–A¹Π systems can be observed through spectroscopy only by a significant effort. There is a strong great similarity between spontaneous emissions of the A¹Π–X¹Σ⁺ system of the AlO⁺ cation and the A²Π–X²Σ⁺ system of the AlO radical. The emissions of the A²Π–X²Σ⁺ system of the AlO radical have been measured in outer space Therefore, it is highly possible that the emissions of the A¹Π–X¹Σ⁺ system of the AlO⁺ cation can be detected in the astrophysical media.     

Geologic investigation and mapping of the Sinus Iridum quadrangle from Clementine,SELENE, and Chang’e-1 data

The objectives of lunar satellite remote sensing are to study lunar surface characteristics, inner structure, and its evolution history. The contents of TiO 2 and FeO are assessed from Clementine UV/VIS data for Sinus Iridum. The geologic stratigraphic units and crates are interpreted visually based on SELENE Terrain Camera (TC) images and the spatial resolution of which is up to 10 m. And the geologic ages of different stratigraphic units are calculated by the crater size-frequency distributions measurements. The gravity anomaly is generated from SELENE gravity model (SGM90d) to show its difference from Mare Imbrium. Furthermore, the thickness of lunar regolith is also derived from microwave radiometer data of Chang’e-1 satellite. Integrating these results, it shows that the Sinus Iridum is different from the Mare Imbrium in inner structure and surface sedimentation. And its history of subsidence, deposition, volcanism, and impact is described. It makes sense to the future soft-landing and sampling at potential Sinus Iridum by remote sensing geologic analysis.     

High-precision calculationsj of the 3,1 P 1 0 →1 S 0 E1 amplitudes for magnesium, calcium, and strontium

High-precision calculations of the ^3,1 P ₁ ⁰ (ns np→¹ S ₀(ns ²) E1 amplitudes were carried out for magnesium, calcium, and strontium (n=3, 4, and 5, respectively). The following results were obtained for the reduced matrix element 〈¹ P ₁ ⁰ ‖d‖ ¹ S ₀〉 of electric dipole moment operator: 4.03(2) au for Mg, 4.91(7) au for Ca, and 5.28(9) au for Sr. These matrix elements are necessary for calculating the van der Waals coefficients C ₆, which are used in evaluating the atomic scattering lengths. The latter determine the dynamics and stability of the Bose-Einstein condensate.     

Magnetic ordering and the Kondo effect in the alloys,Ce2Co1−xPdxSi3

The compound Ce₂CoSi₃, crystallizing in a AlB₂-derived hexagonal structure, has been recently identified as a Kondo lattice with a non-magnetic ground state. Here, we report the influence of gradual replacement of Co by Pd on the magnetic behaviour in the pseudo-ternary solid solution, Ce₂Co_1−xPd_xSi_3, by magnetization (2–300 K), electrical resistivity (2–300 K) and heat-capacity (0.7–30 K) measurements to bring out a transformation from non-magnetic to magnetic ordering. Distinct features attributable to the existence of a competition between the Kondo effect and magnetic ordering with varying x and temperature are observed in the electrical resistivity data. The results reveal that small substitutions of Pd (x=0.2–0.3) are sufficient to induce magnetic ordering of the Ce ions at low temperatures. The strength of the Kondo interaction as indicated by the paramagnetic Curie temperature decreases monotonically with increasing Pd content. A notable finding is that there are qualitative changes in the isothermal magnetization data in the magnetically ordered state, as if there are modifications in the magnetic structure with changes in the Co/Pd composition. The importance of electronic structure relative to unit-cell volume in deciding magnetic characteristics of this class of compounds is brought out taking into account the trends in the magnetic behaviour of isostructural Ce compounds.     

Peculiarities of the electronic structure of Yb, In, Ag, and Cu in the heavy-fermion system YbIn1−x AgxCu4

The method of x-ray spectral line displacement is used for studying the electronic structure, i.e., the population of the 4f shell of Yb, 5s shells of In and Ag, and 4s shell of Cu, in the YbIn_1?x Ag_xCu₄ heavy-fermion system (0≤x≤1, T=300 K; T=77, 300, and 1000 K for YbIn_0.7Ag_0.3Cu₄). It is shown that Yb is in a state with fractional valence whose value is independent of x (or on the type of the partner, i.e., In and Ag) in the entire range of compositions and is equal to $\bar m = 2.91 \pm 0.01$ at T=300 K. An increase in the population of the In s states of In, Ag, and Cu (as compared to metals) is observed: $\overline {\Delta n_s } (In) = 0.65 \pm 0.05 el/at$ , $\overline {\Delta n_s } (Ag) = 0.71 \pm 0.09 el/at$ , and $\overline {\Delta n_s } (Cu) = 0.08 \pm 0.02 el/at$ . A practically linear decrease in the valence of Yb to the value m(T=1000 K)=2.81±0.02 is observed in YbIn_0.7Ag_0.3Cu₄ upon an increase in temperature from T=77 to 1000 K.     

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.     

Dielectric, magnetoelectric, structure, and dissipative properties and the Mössbauer effect in PbFe1/2Nb1/2O3 ceramics in wide frequency and temperature ranges

High-quality fine-grained ceramic samples of classical multiferroics PbFe_1/2Nb_1/2O₃ (PFN) were synthesized. Their dielectric, magnetoelectric, and magnetic characteristics, including the Mössbauer effect, were measured over wide ranges of temperatures (10–1000 K) and field frequencies (from 25 Hz to 1 MHz). The temperature dependence of the dielectric loss exhibits a maximum between 150 and 170 K, likely due to magnetic ordering. The dependence of ? on the magnetic field displays an anomalous increase near the Curie temperature (370 K) that rises to 40% upon heating.     

Calculation of spectroscopic constants and radiative parameters for the B 1Π-X 1Σ+ electronic transitions of NaK,NaRb, and NaCs molecules

The vibrational, rotational, and centrifugal spectroscopic constants and the radiative parameters (the Einstein coefficients, oscillator strengths, Frank-Condon factors, r _v′v″ centroids, and wavenumbers of rotational lines) of electronic-vibrational-rotational transitions in the systems of bands B ¹Π -X ¹Σ⁺ of NaK (0 ≤ v′ ? 14, 0 ≤ v″ ≤ 52, j = 0, 30, 50, 70, 80, and 100), NaRb (0 ≤ v′ ≤ 12, 0 ≤ v″ ≤ 51, j = 0, 20, 30, 50, 70, 90, 100, and 120), and NaCs (0 ≤ v′ ≤ 10, 0 ≤ v″ ≤ 44, j = 0, 30, 50, 70, 90, 100, and 120) molecules, as well as the radiative lifetimes of excited electronic states, are calculated. Calculations are performed based on semiempirical potential curves constructed in this work. The calculated spectroscopic constants and the radiative lifetimes are compared with experimental values.     

Inflation, Extra Dimensions, and ΩΛ ≈ 1

An (N + 1)-dimensional quantum mechanical model for the origin of the universeresults in a 58e-fold inflation and a cosmological constant/vacuum energy densitywith 1.     

On the hopf structure of U p,q (gl(1∣1)) and the universal T-matrix of fun p,q (GL(1∣1))

The dually conjugate Hopf superalgebras Fun _p,q (GL(11)) and U _p,q (gl(11)) are studied using the Frønsdal-Galindo approach and the full Hopf structure of U _p,q (gl(11)) is extracted. A finite expression for the universal T-matrix, identified with the dual form and expressing the generalization of the exponential map of the classical groups, is obtained for Fun _p,q (GL(11)). In a representation with a colour index, the T-matrix assumes a form that satisfies a coloured graded Yang-Baxter equation.     

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.     

Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

We compute the phase diagram in the N→∞ limit for lattice RP^N−1, CP^N−1 and QP^N−1 σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.     

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.     

Dynamical Algebras in the 1+1 Dirac Oscillator and the Jaynes–Cummings Model

We study the algebraic structure of the one-dimensional Dirac oscillator by extending the concept of spin symmetry to a noncommutative case.An SO(4) algebra is found connecting the eigenstates of the Dirac oscillator,in which the two elements of Cartan subalgebra are conserved quantities.Similar results are obtained in the Jaynes-Cummings model.     

Analysis of the Vibrational-Rotational Structure of 2ν1, ν1 + ν3, and 2ν3 Bands of the D2Se Molecule

The absorption spectrum of the D₂Se molecule in the region of 2₁, ₁ + ₃, and 2₃ absorption bands is registered with a high-resolution Fourier spectrometer and is studied theoretically for a Hamiltonian model with allowance for resonant interactions among (200), (101), and (002) vibrational states.