X-ray spectroscopic study of chemical bonding in NiSe and NiSe2

The K absorption spectra of nickel and selenium have been recorded photographically in the pure metals and the compounds NiSe and NiSe₂. It has been observed that the Ni discontinuity shifts toward the high energy side with respect to that in the pure metal by about 5.0 eV for NiSe and about 6.7 eV for NiSe₂. The Se edge in both these compounds is found to shift by about 2.6 eV toward the low energy side with respect to that in pure selenium. From these results it is possible to obtain the bonding pictures for the compounds. The compound NiSe appears to have resonating p³ type of bonding. For NiSe₂ the orbitals involved in the bonding are d²sp³ for the metal atom and sp³ for the metalloid atom. The electrical behaviour of the compounds has been explained on the basis of these bonding pictures.     

X-ray spectroscopic study of chemical bonding in MnSe2 and CoSe2

X-ray absorption study of two intermetallic compounds MnSe₂ and CoSe₂ has been carried out using a Cauchois type bent crystal spectrograph. The metal K absorption edges in both the compounds are found to shift toward the high energy side with respect to the discontinuities in the pure metals. On the other hand, the Se K absorption edge in both these compounds is found to shift toward the lower energies. Emission study of the compound MnSe₂ has shown that the Mn Kβ ₅ band in this compound is shifted toward the high energy side with respect to that in the pure metal. From the magnetic data and the results obtained in this work it is possible to obtain the chemical bonding pictures in these compounds. For MnSe₂ the bondings sp ³ d ² for the metal atom andsp ³ for the metalloid atom have been suggested. For CoSe₂ the bondings appear to bed ² sp ³ for the metal atom andsp ³ for the metalloid atom. These bondings are compatible with the pyrite type structure of these compounds. It is possible to explain the electrical behaviour of the compounds on the basis of these bonding pictures.     

Piecewise constant positive potentials with practically the same fixed energy phase shifts

It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level k² > 0. However, numerical experiments show that two quite different potentials can produce almost identical phase shifts. It has been guessed by physicists that such examples are possible only for “less physical” oscillating and changing sign potentials. In this note it is shown that the above guess is incorrect: we give examples of four positive spherically symmetric compactly supported quite different potentials having practically identical phase shifts. The note also describes a hybrid stochastic deterministic method for global minimization for the construction of these potentials.     

A family of simple C1-algebras related to weighted shift operators

In this paper we study a family of C¹-algebras which occurs naturally in the study of C¹-algebras generated by weighted shifts. We show that these algebras are simple modulo the compacts, and while they share many of the properties of uniformly hyperfinite C¹-algebras, they are not approximately finite dimensional.     

On the (U+K)-orbits of certain weighted shifts

D. A. Herrero has defined the (U+K)-orbit of an operatorT acting on an Hilbert space as (U+K)(T)={R ^–1 TR: R invertible of the form unitary plus compact}. In this paper we consider the (U+K)-orbit and the closure thereof for bilateral and unilateral weighted shifts. In particular, we determine which shifts are in the (U+K)-orbits of injective weighted shifts and which shifts are in the closure of the (U+K)-orbit of periodic injective shifts. Also, the closure of the (U+K)-orbit of injective essentially normal shifts is determined.Research supported in part by an NSERC operating grant.     

X-Ray spectroscopic evidence for chemical bonding in manganese selenide (MnSe)

K absorption spectra of manganese and selenium have been recorded photographically in the pure metals and in the intermetallic compound manganese selenide (MnSe), employing a bent crystal X-ray spectrograph. It has been observed that the Mn K absorption discontinuity in MnSe shifts by 5·6 eV toward the high energy side with respect to the discontinuity in the pure metal. On the other hand, the Se K edge in MnSe is found to shift by 3·4 eV toward the low energy side with respect to the discontinuity in pure selenium. Also, emission work has revealed that the Mn K_β5 band in the compound shifts by about 5 eV toward the high energy side with respect to that in the pure metal. It has been shown that these results confirm the resonatingp ³ type of chemical bonding in the compound, as suggested theoretically by Pearson.     

Varying the time-frequency lattice of Gabor frames

A Gabor or Weyl-Heisenberg frame for is generated by time-frequency shifts of a square-integrable function, the Gabor atom, along a time-frequency lattice. The dual frame is again a Gabor frame, generated by the dual atom. In general, Gabor frames are not stable under a perturbation of the lattice constants; that is, even for arbitrarily small changes of the parameters the frame property can be lost.

In contrast, as a main result we show that this kind of stability does hold for Gabor frames generated by a Gabor atom from the modulation space , which is a dense subspace of . Moreover, in this case the dual atom depends continuously on the lattice constants. In fact, we prove these results for more general weighted modulation spaces. As a consequence, we obtain for Gabor atoms from the Schwartz class that the continuous dependence of the dual atom holds even in the Schwartz topology. Also, we complement these main results by corresponding statements for Gabor Riesz sequences and their biorthogonal system.

     

Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions

We characterize finite codimensional linear isometries on two spaces, C ⁽ⁿ⁾[0; 1] and Lip [0; 1], where C ⁽ⁿ⁾[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C ⁽ⁿ⁾[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.     

Substitution and solvent effects on C13-F19 coupling constants

C¹³-F¹⁹ coupling constants have been measured inp-substituted fluorobenzenes. It has been shown that they vary linearly with Hammett’s constant (σ _v ). Further the change in the ionic character of the C-F bond in fluorobenzenes calculated on the basis of the change in the coupling constants agrees well with that calculated from F¹⁹ chemical shifts. The coupling constant in trifluoroacetic acid has also been measured in various solvents at several concentrations. A remarkable solvent effect on these couplings has been discovered; this appears-to be related to the dipole moment of the solvent.     

Sobolev Error Estimates and a Bernstein Inequality for Scattered Data Interpolation via Radial Basis Functions

Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes of functions still were not "large" enough to be applicable to a number of practical situations. In this paper we obtain Sobolev-type error estimates on compact regions of Rⁿ when the RBFs have Fourier transforms that decay algebraically. In addition, we derive a Bernstein inequality for spaces of finite shifts of an RBF in terms of the minimal separation parameter.     

Theβ- andγ-band systems of14NS and15NS

The spectra of¹⁴NS and¹⁵NS molecules are obtained in a 2450 mc./s. microwave oscillator discharge. Vibrational assignments of theβ- andγ-systems are studied in more detail. Several hitherto unrecorded bands are satisfactorily explained as belonging to higherv′,v″ levels in the Deslandres schemes of bothβ- andγ-systems. The observed isotope shifts (¹⁴NS-¹⁵NS) provide confirmatory evidence for the proposed vibrational analysis. In the case of theγ-system, thev′=2 level of the upper state appears to be strongly perturbed showing a shift of about 38 cm.^?1 from the expected position. In theβ-system, the isotopic shifts in the band-heads involving thev′=0 and 1 levels of the²Δ_5/2 of the upper state show small deviations from expected values. The reality of these small deviations is established beyond doubt by the occurrence of the effect on the 0–0 sub-band which exhibits the isotopic head in a wavelength direction opposite to the expected one.     

Hotelling’s T charts with variable sample size and control limit

The ideas of variable sampling interval (VSI), variable sample size (VSS), variable sample size and sampling interval (VSSI), and variable parameters (VP) in the univariate case have been successfully applied to the multivariate case to improve the efficiency of Hotelling’s T² chart with fixed sampling rate (FSR) in detecting small process shifts. However, the main disadvantage in using most of these control schemes is an increasing in the complexity due to the adaptive changes in sampling intervals. In this paper, retaining the lengths of sampling intervals constant, a variable sample size and control limit (VSSC) T² chart is proposed and described. The statistical efficiency of the VSSC T² chart in terms of the average time to signal a shift in process mean vector is compared with that of the VP, VSSI, VSS, VSI, and FSR T² charts. From the results of comparison, it shows that the VSSC T² chart for a (very) small shift in the process mean vector gives a better performance than the VSSI, VSS, VSI, and FSR T² charts; meanwhile, it presents a similar performance to the VP T² chart. Furthermore, from the viewpoint of practicability, it is more convenient for administrating the control chart than the VSI, VSSI, and VP T² chart. Thus, it may provide a good option for quick response to small shifts in a multivariate process.     

Multi-Box Splines

We construct local generators, comprising r functions, for refinable spaces of bivariate C^n-1 spline functions of degree n on meshes comprising all lines through points of the integer lattice in the directions of n + r + 1 pairwise linearly independent vectors with integer components. The generators are characterised by their Fourier transforms. Their shifts are shown to form a Riesz basis if and only if at most r lines in the mesh intersect other than in the integer lattice, which can occur for n ≤ 2r - 1. The symmetry of these generators is studied and examples are given.     

Invertible composition operators on Hp

The weakly closed algebras generated by certain sets of composition operators are shown to be reflexive. A structure theorem for invertible composition operators on H² is obtained and used to show that such operators are reflexive. The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts. Another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal. Thus the general invariant subspace problem for Hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators.     

Approximate dual Gabor atoms via the adjoint lattice method

Regular Gabor frames for \({\boldsymbol {L}{^{2}}(\mathbb {R}^d)}\) are obtained by applying time-frequency shifts from a lattice in \(\boldsymbol {\Lambda } \vartriangleleft {\mathbb {R}^{d} \times \mathbb {\widehat {R}}}\) to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some \(g \in {\boldsymbol {S}_{0}(\mathbb {R}^{d})}\) . There is always a canonical dual frame, generated by the dual Gabor atom \({\widetilde g}\) . The paper promotes a numerical approach for the efficient calculation of good approximations to the dual Gabor atom for general lattices, including the non-separable ones (different from \({a\mathbb {Z}^{d}\,{\times }\,b\mathbb {Z}^{d}}\) ). The theoretical foundation for the approach is the well-known Wexler-Raz biorthogonality relation and the more recent theory of localized frames. The combination of these principles guarantees that the dual Gabor atom can be approximated by a linear combination of a few time-frequency shifted atoms from the adjoint lattice \(\boldsymbol {\Lambda }\circ\) . The effectiveness of this approach is justified by a new theoretical argument and demonstrated by numerical examples.     

Subsystem entropy for ? sofic shifts

We prove that any ?^d shift of finite type with positive topological entropy has a family of subsystems of finite type whose entropies are dense in the interval from zero to the entropy of the original shift. We show a similar result for ?^d sofic shifts, and also show every ?^d sofic shift can be covered by a ?^d shift of finite type arbitrarily close in entropy.     

Semiclassical quantization of Bohr orbits in the helium atom

We use the complex WKB-Maslov method to construct the semiclassical spectral series corresponding to the resonance Bohr orbits in the helium atom. The semiclassical energy levels represented as the Rydberg tetra series correspond to the doubly symmetrically excited states of helium-like atoms. This level series contains the Rydberg triple series reported by Richter and Wintgen in 1991, which corresponds to the Z²⁺e⁻e⁻ configuration of electrons observed by Eichmann and his collaborators in experiments on the laser excitation of the barium atom in 1992. The lower-level extrapolation of the formula obtained for the semiclassical spectrum gives the value of the ground state energy, which differs by 6% from the experimental value obtained by Bergeson and his collaborators in 1998. We also calculate the fine structure of the semiclassical spectrum due to the spin-orbit and spin-spin interactions of electrons. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 261–286, May, 2007.     

Deriving hydrodynamic equations for lattice systems

We study the dynamics of lattice systems in ℤ^d, d ≥ 1. We assume that the initial data are random functions. We introduce the family of initial measures {μ₀^ɛ, ɛ > 0}. The measures μ₀^ɛ are assumed to be locally homogeneous or “slowly changing” under spatial shifts of the order o(ɛ⁻¹ ) and inhomogeneous under shifts of the order ɛ⁻¹ . Moreover, correlations of the measures μ₀^ɛ decrease uniformly in ɛ at large distances. For all τ ∈ ℝ \ 0, r ∈ ℝ^d, and κ > 0, we consider distributions of a random solution at the instants t = τ/ɛ^κ at points close to [r/ɛ] ∈ ℤ^d. Our main goal is to study the asymptotic behavior of these distributions as ɛ → 0 and to derive the limit hydrodynamic equations of the Euler and Navier-Stokes type.     

Orthogonal wavelets on direct products of cyclic groups

We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group G which is the weak direct product of a countable set of cyclic groups of pth order. For all integers p, n ≥ 2, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with p ⁿ numerical coefficients generate multiresolution analyses in L ²(G). It is noted that the coefficients of these scaling equations can be calculated from the given values of p ⁿ parameters using the discrete Vilenkin-Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in L ²(G) is stable and has a linearly independent system of “integer” shifts. We present several examples illustrating these results.     

On a Bernoulli property of some piecewiseC 2-diffeomorphisms in ℝ d

Some piecewiseC ²-diffeomorphisms which expand distances uniformly (in all directions) are considered. It is shown that they have the so-called Bernoulli property, i.e. their natural extensions are isomorphic to Bernoulli shifts.