The $${\bar\mu}$$-invariant of Seifert fibered homology spheres and the Dirac operator

We derive a formula for the [`(m)]{\bar\mu}-invariant of a Seifert fibered homology sphere in terms of the η-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

Ionic-charge dependence of the intermolecular coulombic decay time scale for aqueous ions probed by the core-hole clock

Auger electron spectroscopy combined with theoretical calculations has been applied to investigate the decay of the Ca 2p core hole of aqueous Ca(2+). Beyond the localized two-hole final states on the calcium ion, originating from a normal Auger process, we have further identified the final states delocalized between the calcium ion and its water surroundings and produced by core level intermolecular Coulombic decay (ICD) processes. By applying the core-hole clock method, the time scale of the core level ICD was determined to be 33 ± 1 fs for the 2p core hole of the aqueous Ca(2+). The comparison of this time constant to those associated with the aqueous K(+), Na(+), Mg(2+), and Al(3+) ions reveals differences of 1 and up to 2 orders of magnitude. Such large variations in the characteristic time scales of the core level ICD processes is qualitatively explained by different internal decay mechanisms in different ions as well as by different ion-solvent distances and interactions.     

Anomeric effect in N-azidomethylpyrrolidine: gas-phase electron diffraction and theoretical study

Gas-phase electron-diffraction data and high-level quantum chemical calculations have been used to study the conformational behaviour of N-azidomethylpyrrolidine. The two most stable conformers with a relative abundance of about 80% at 298 K possess gauche orientation of the azidomethyl group around the C-N(pyr) bond (C-N(azido)gauche with respect to the endocyclic N(pyr)-C bond). This orientation is a strong manifestation of an anomeric effect. The influence of the anomeric effect is also reflected in shortening of the C-N(pyr) bond and lengthening of the C-N(azido) bond as compared to such bonds in other compounds.     

Nash equilibrium in compact-continuous games with a potential

If the preferences of the players in a strategic game satisfy certain continuity conditions, then the acyclicity of individual improvements implies the existence of a (pure strategy) Nash equilibrium. Moreover, starting from any strategy profile, an arbitrary neighborhood of the set of Nash equilibria can be reached after a finite number of individual improvements.     

Shallow Donors and Deep-Level Color Centers in Bulk AlN Crystals: EPR, ENDOR, ODMR and Optical Studies

The results of studies of shallow donors and deep-level color centers in bulk AlN crystals are presented. Two shallow donors (presumably oxygen located on the nitrogen site and carbon located on the aluminum site) are suggested to exhibit the DX-relaxation. Third shallow donor (presumably silicon on the Al site) shows the shallow donor behavior up to the room temperature and can be observed without light excitation at temperatures above 200 K. The values of the Bohr radius of the shallow donors are estimated. The structure of deep-level color centers (neutral nitrogen vacancy V _N) in bulk AlN crystals is determined and analyzed by electron paramagnetic resonance, electron-nuclear double resonance, optical absorption and thermoluminescence induced by X-ray irradiation. Spin-dependent recombination processes in AlN crystals are studied by means of optically detected magnetic resonance.     

Towards relativistic ECP / DFT description of chemical bonding in E112 compounds: spin-orbit and correlation effects in E112X versus HgX (X=H, Au)

The relativistic effective core potential (RECP) approach combined with the spin-orbit DFT electron correlation treatment was applied to the study of the bonding of eka-mercury (E112) and mercury with hydrogen and gold atoms. Highly accurate small-core shape-consistent RECPs derived from Hartree-Fock-Dirac-Breit atomic calculations with Fermi nuclear model were employed. The accuracy of the DFT correlation treatment was checked by comparing the results in the scalar-relativistic (spin-orbit-free) limit with those of high level scalar-relativistic correlation calculations within the same RECP model. E112H was predicted to be slightly more stable than its lighter homologue (HgH). The E112-Au bond energy is expected to be ca. 25–30 % weaker than that of Hg-Au. The role of correlations and magnetic (spin-dependent) interactions in E112-X and Hg-X (X=H, Au) bonding is discussed. The present computational procedure can be readily applied to much larger systems and seems to be a promising tool for simulating E112 adsorption on metal surfaces.     

Unbiased estimates for gradients of stochastic network performance measures

Three classes of stochastic networks and their performance measures are considered. These performance measures are defined as the expected value of some random variables and cannot normally be obtained analytically as functions of network parameters in a closed form. We give similar representations for the random variables to provide a useful way of analytical study of these functions and their gradients. The representations are used to obtain sufficient conditions for the gradient estimates to be unbiased. The conditions are rather general and usually met in simulation study of the stochastic networks. Applications of the results are discussed and some practical algorithms of calculating unbiased estimates of the gradients are also presented.     

Spectroscopically determined potential energy surfaces of the H2O, H2O, and H2O isotopologues of water

Adiabatic potential energy surfaces (PESs) for three major isotopologues of water, H₂¹⁶O, H₂¹⁷O, and H₂¹⁸O, are constructed by fitting to observed vibration-rotation energy levels of the system using the nuclear motion program DVR3D employing an exact kinetic energy operator. Extensive tests show that the mass-dependent ab initio surfaces due to Polyansky et al. [O.L. Polyansky, A.G. Császár, S.V. Shirin, N.F. Zobov, P. Barletta, J. Tennyson, D.W. Schwenke, P.J. Knowles, Science 299 (2003) 539-542.] provide an excellent starting point for the fits. The refinements are performed using a mass-independent morphing function, which smoothly distorts the original adiabatic ab initio PESs. The best overall fit is based on 1788 experimental energy levels with the rotational quantum number J = 0, 2, and 5. It reproduces these levels with a standard deviation of 0.079 cm⁻¹ and gives, when explicit allowance is made for nonadiabatic rotational effects, excellent predictions for levels up to J = 40. Theoretical linelists for all three isotopologues of water involved in the PES construction were calculated up to 26 000 cm⁻¹ with energy levels up to J = 10. These linelists should make an excellent starting point for spectroscopic modelling and analysis.