Theq-harmonic oscillator and the Al-Salam and Carlitz polynomials

One more model of aq-harmonic oscillator based on theq-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form ofq-creation andq-annihilation operators,q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.     

Comprehensive DFT and MO studies on glyoxilic acid oxime and related ions in gas phase and solution: Conformations,basicities and acidities

Density functional (BLYP, B3LYP and BHLYP) and highly correlated MP2 and CCSD(T) calculations have been performed to investigate conformers, energy barriers, intramolecular H-bond strength, gas-phase basicity and deprotonation energies of glyoxilic acid oxime (gao) and related ions in gas phase and in aqueous solution (SCRF-PCM method). BHLYP/6-311G(d,p) and B3LYP/6-31++G(d) predictions for the global minimum conformer of gao were consistent with experiment. BLYP level overestimated the H-bond and stabilized incorrectly the H-bonded conformer. The calculations in solution indicated destabilization of H-bonded conformers due to the small polarizability and weaken of the H-bond. The same global minimum structures in gas phase and aqueous solution were found for gao-neutral (ectt) and gao-dianion (e⁻²), whereas they were different for gao-anion because of the strong decrease of the conformational energies in solution. The global minimum structures of the neutral, anion and dianion of gao, obtained in solution, are in agreement with experiment. The gas-phase basicity (GB) and molecular electrostatic potential (MEP) calculations revealed the same sites for electrophilic attack, supported by the nature of HOMO: the carbonylic oxygen for the neutral, the carboxylic oxygen for the anion and the oxime nitrogen for the dianion. MEP results in gas phase and in solution suggested a region between the two atoms, but not on one atom in accordance with bidentate binding of gao ions to a metal. The BHLYP/6-31++G(d,p) molecular properties of gao were in best consistent with CCSD(T) results. The thermodynamical properties (GB and bond deprotonation energy) of gao were better estimated at B3LYP level.     

On a pair of functional equations of combinatorial interest

The equations of the title appear in the author's paper Chromatic Sums for Rooted Planar Triangulations, V: Special Equations. (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy ₂ of two variablesy ₁ andz. A parameter is involved. The main result is that for = 2 cos (2/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.The present work is basically an expository paper on the theory given in Chromatic Sums, V, but it includes some new results and many simplifications.     

Characterization of the normal distribution by constant regression of quadratic statistics on linear statistics

This paper contains applications of theorems of [1] for quadratic statistics which have constant regression on linear statistics. Two theorems are proved. The first is a sufficient condition which assumes that the characteristic function of a sample is an entire function. The second gives a new characterization of the normal distribution.     

On the divergence of certain Hermite-Fejér interpolation

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the ebyev roots) not describing the derivative values at exceptional nodes {_n} _n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever –1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [–1, 1].     

Natural spline functions,their associated eigenvalue problem

Summary We consider the problem of studying the behaviour of the eigenvalues associated with spline functions with equally spaced knots. We show that they are wherem is the order of the spline andn, the number of knots.This result is of particular interest to prove optimality properties of the Generalized Cross-Validation Method and had been conjectured by Craven and Wahba in a recent paper.     

KT-nearfields of rank 2

Summary Nearfields of rank 2 over their kernel which are not Dickson nearfields have recently been constructed by H. Zassenhaus. Here it is shown that KT-nearfields of rank 2 over their kernels and of characteristic 2 are necessarily Dickson nearfields which are coupled to quadratic field extensions and their Dickson group is isomorphic to ₂. Using results of Gröger one obtains all KT-nearfields of rank 2 over the field of rational numbers.     

Some identities for products and degree raising of splines

In this paper it is shown how the algebraic product of two spline functions, each represented in terms of B-splines, can again be represented as a linear combination of suitable B-splines. As a corollary to this result we obtain an explicit representation of a given B-spline function in terms of B-splines of some arbitrary higher degree. This generalizes some known results for raising the degree by one. Recurrence relations for both products and degree raising are established that may be useful for computation.Communicated by Larry L. Schumaker.     

Problems with the ζ-function method

The -function method for evaluating effective actions is applied to an operator containing a potential localized on two parallel planes. This operator is characterized by a continuous spectrum and broken translation invariance. In this case, the -method leads to a divergent volume factor independent of the physical parameters. In a suitable regularization scheme, only the next to leading order term reproduces a physically interesting result.