Preservation of logarithmic concavity by the Mellin transform and applications to the Schrödinger equation for certain classes of potentials

We prove that the Mellin transform of a function log-concave (convex) is, after division by (+1), where is the argument of the transform, itself log-concave (convex) in . This theorem is first applied to the moments of the ground state wave function of the Schrödinger equation where the Laplacian of the central potential has a given sign, and generalized to other situations. This is used to derive inequalities linking thel ^th derivative of the ground state wave function at the origin for angular momentuml and the expectation value of the kinetic energy, and applied to quarkonium physics. A generalization to higher radial excitations is shown to be plausible by using the WKB approximation. Finally, new bounds on ground-state energies in power potentials are obtained.     

Generalized Lax pairs,the modified classical Yang-Baxter equation,and affine geometry of Lie groups

We generalize the usual Lax equationd/dt L=[M, L] byd/dt L=–(M)L, where is an arbitrary representation of a Lie algebra g (the values ofM) in a representation spaceV (the values ofL). The usual classicalr-matrix programme for Hamiltonian integrable systems is generalized tor-matrices taking values in gV. Ther-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie groupK (with Lie algebraV ^*). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness ofK whereas the Modified CYBE implies thatK is an affine locally symmetric space. An example is discussed.     

Yttrium aluminate crystals for slabs

The methods of growing YAG:Nd crystals with a flat interface and YAP:Nd crystals with a sharp conical interface are described. The form of the interface was controlled by the He and H₂O content in the reducing atmosphere composed mainly of Ar and H₂ as well as by axial temperature gradient above the melt level. The crystals of 120–180 mm in length were machined to the high-quality slabs.     

Nd3+ near UV and visible fluorescence in YAP:Nd

The present paper summarizes detailed investigations of Nd³⁺ fluorescence spectra in YAP:Nd laser crystal in the broad spectral range of 370–1100 nm at liquid nitrogen temperature. Especially, Nd³⁺ near UV and visible fluorescence spectra were studied for the first time in this crystal. The Nd³⁺ near UV and visible fluorescence spectra of this crystal consist of many narrow lines (up to 50). The Nd³⁺ near UV and visible fluorescence lines arise mainly from⁴D_3/2 and²P_3/2 Nd³⁺ terms (transitions from⁴D_3/2,²P_3/2 to lower lying Nd³⁺ levels are responsible for their appearance). At room temperature the fluorescence spectra consist of broad bands with narrow lines (mainly⁴D_3/2⁴F_3/2,⁴F_5/2 and²H_9/2 transitions).     

Non-steady state homogeneous nucleation in glass-forming systems: Dimensionless kinetic equation

The kinetic equation of the classical homogeneous nucleation theory is rewritten in dimensionless form and it is shown that the numerical computation of the dimensionless kinetic equation is approximately 20 times faster in comparison with the original kinetic equation. The results are compared with known analytical results and with previous numerical results in the case of the constant and variable temperature for the model Li₂O.2 SiO₂ melt. It is shown, e.g., that under non-isothermal conditions the nucleation rate is greater than the quasistationary nucleation rate and the cluster distribution function is greater than the corresponding quasiequilibrium cluster distribution function.     

XPS investigation of LPE garnet films

XPS spectra of YIG, YIG:Bi, YIG:Ca and YIG:Co(Ge, Ca) epitaxial garnet films were measured at room temperature. The positions of the main iron, yttrium, oxygen, bismuth, cobalt and lead spectral lines were determined and their changes with the surface treatments were investigated. It was found that divalent and trivalent states of iron and cobalt ions can be resolved, the chemical shifts being in oposite directions.The authors would like to thank Dr. J. imová for careful EPMA analysis of the samples.     

Krichever-Novikov algebras for more than two points

Krichever-Novikov algebras of meromorphic vector fields with more than two poles on higher genus Riemann surfaces are introduced. The structure of these algebras and their induced modules of forms of weight is studied.