Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

We study the interlacing property of zeros of Laguerre polynomials of adjacent degree, where the free parameters differ by an integer, and of the same degree, where the free parameter is shifted continuously. Similar interlacing results are proven for the positive zeros of Gegenbauer polynomials. Research by Kathy Driver is supported by the National Research Foundation of South Africa under grant number 2053730. Research by Kerstin Jordaan is partially supported by the National Research Foundation of South Africa under grant number 2054423.     

Products of hypergeometric functions and the zeros of 4F3 polynomials

We recall a known result (cf. [1]) expressing certain ₄ F ₃ hypergeometric functions as products of ₂ F ₁ hypergeometric functions. We consider the polynomial case and show how recent results (cf. [2]) concerning the zero distribution of Gauss hypergeometric polynomials can be used to obtain information about the location of the zeros of three types of ₄ F ₃ hypergeometric polynomials. Numerical and graphical evidence of the zeros is provided with the help of Mathematica.     

Quantum hysteresis loops in microscopic system: The loop area as a dynamical parameter

The area enclosed by hysteresis loops in a periodically forced bistable microscopic system at zero-temperature is examined by using the time dependent Hellmann-Feynman theorem and the Fourier grid Hamiltonian recipe for solving time-dependent Schrödinger equation. Effects of non-zero temperatures are explored with reference to a symmetric double well potential. The barrier crossing or, relaxation rates are shown to correlate systematically with the area of the loop. The possible use of hysteresis loop area in designing field parameters for optimal control is suggested.     

Convergence of theU(1)4 lattice gauge theory to its continuum limit

It is shown that in four space-time dimensions the compactU(1) lattice gauge theory with general energy function converges to a renormalized free electromagnetic field on the current sector as the lattice spacing approaches zero, provided the coupling constant is sufficiently large. For the Wilson energy function, it is possible, by judicious choice of the Gibbs state, to get convergence for arbitrary coupling strengths. Furthermore, for all but a countable number of values of the coupling constant, the limit exists and is independent of the particular state chosen to define the lattice model.This research was supported in part by an Alfred P. Sloan Foundation Doctoral Dissertation Fellowship and in part by N.S.F. Grant Nos. DMS 84-01997 and DMS-8610730(1)     

Stabilizing CO on Au with NO2: electronegative species as promoters on coinage metals?

CO adsorption on NO(2)-predosed Au[111] reveals an unexpected attractive coadsorbate interaction, associated with an unprecedented blueshift of the CO stretch frequency, a sizeable attenuation of the infrared NO(2) symmetric stretch band, and a (sq.rt(7) x sq.rt(7))R19 degrees structure characterized by scanning tunneling microscopy and low energy electron diffraction. Density functional calculations allow us to rationalize these observations, and point towards a general pattern of behavior for electronegative coadsorbates on coinage metals, with important implications for catalytic promotion.     

Trajectories of the zeros of hypergeometric polynomials F(−n, b; 2b; z) for b < − 1/2

In a previous paper [2] we studied the zeros of hypergeometric polynomials F(−n, b; 2b; z), where b is a real parameter. Making connections with ultraspherical polynomials, we showed that for b > − 1/2 all zeros of F(−n, b; 2b; z) lie on the circle |z − 1| = 1, while for b < 1 − n all zeros are real and greater than 1. Our purpose now is to describe the trajectories of the zeros as b descends below the critical value − 1/2 to 1 − n. The results have counterparts for ultraspherical polynomials and may be said to “explain” the classical formulas of Hilbert and Klein for the number of zeros of Jacobi polynomials in various intervals of the real axis. These applications and others are discussed in a further paper [3].     

YM2:Continuum expectations,lattice convergence,and lassos

The two dimensional Yang-Mills theory (YM₂) is analyzed in both the continuum and the lattice. In the complete axial gauge the continuum theory may be defined in terms of a Lie algebra valued white noise, and parallel translation may be defined by stochastic differential equations. This machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesC. The expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group. The time parameters of the heat kernels are determined by the areas enclosed by the collectionC, and the arguments are determined by the crossing topologies of the curves inC. The expectations for the Wilson lattice models have a similar structure, and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations. It is also shown that the lasso variables advocated by L. Gross [36] exist and are sufficient to generate all the measurable functions on the YM₂-measure space.     

Synthesis,growth, structure,spectral, thermal and first-order molecular hyperpolarizability of benzophenone-2-furoyl hydrazone single crystal

Single crystals of benzophenone-2-furoyl hydrazone are grown by slow evaporation solution growth technique from ethanol at room temperature. It belongs to monoclinic system with the space group P2₁/c and the cell parameters are, a = 6.1631(3) Å, b = 13.1397(8) Å, c = 18.0030(11) Å and V = 1457.72(14) Å³. NMR spectral studies reveal the structure and powder XRD indicates the crystallinity of the specimen. The characteristic functional groups present in the molecule are confirmed by Fourier transform infrared spectroscopy. The crystals are transparent in the visible region having a lower optical cut-off at ∼406 nm and the band gap energies are estimated by the application of Kubelka–Munk algorithm. Thermal analysis by TG/DTA indicates the stability of the material. The scanning electron microscopy studies reveal the surface morphology of the as-grown crystal. Theoretical calculations were performed using Hartree–Fock method with 6-31G(d,p) as the basis set for to derive the optimized geometry, dipole moment and first-order molecular hyperpolarizability (β) values.