Finite-Band Potentials with Trigonal Curves

We construct Dubrovin equations and trace formulas for finite-band operators with trigonal curves and present some examples.     

Parametric oscillation at 2.1 μm in periodically poled RbTiOAsO4

An Nd:YAG laser-pumped periodically poled RbTiOAsO₄ (PPRTA) optical parametric oscillator (OPO) near degeneracy is reported. At a pump power of 3.7 W, a net conversion efficiency of 43% was obtained. These crystals were found to be phase-matchable for the higher order quasi-phase-matched second-harmonic generation (SHG) in the visible range. Detailed analyses of the phase-matching properties for these processes were made by using the published Sellmeier and thermo-optic dispersion formulas.     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Relations among eigenvalues of Sturm-Liouville problems with different types of leading coefficient functions

For any Sturm-Liouville problem with a separable boundary condition and whose leading coefficient function changes sign (exactly once), we first give a geometric characterization of its eigenvalues λn using the eigenvalues of some corresponding problems with a definite leading coefficient function. Consequences of this characterization include simple proofs of the existence of the λn's, their Prüfer angle characterization, and a way for determining their indices from the zeros of their eigenfunctions. Then, interlacing relations among the λn's and the eigenvalues of the corresponding problems are obtained. Using these relations, a simple proof of asymptotic formulas for the λn's is given.     

A general converse theorem for mean‐value theorems in linear elasticity

We prove a rather general mean‐value formula in the theory of elasticity, which expresses the value of the displacement at the centre of a sphere in terms of certain combinations of integral averages over the sphere itself of the traction and the displacement. We also establish the corresponding converse to this mean‐value formula under minimal smoothness assumptions on the displacement. Copyright © 2006 John Wiley & Sons, Ltd.     

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

Convergent Star Product Algebras on ‘ax+b’

We define nontempered (exponential growth) function spaces on the Lie group ax+b which are stable under some left-invariant (convergent) star product. The techniques used to achieved the latter come from symmetric spaces geometry and star representation theory.     

Modified Kramers formulas for the decay rate in agreement with dynamical modeling

Accuracy of the Kramers approximate formulas for the thermal decay rate of the metastable state is studied for the anharmonic shapes of the potential pocket and the barrier. This is done by the comparison with the quasistationary rate resulting from the dynamical modeling. Disagreement between the Kramers rate and the dynamical one is shown to reach 15% in the cases when much better agreement is expected. Corrections to the Kramers formulas accounting for the higher derivatives of the potential are obtained. The small parameters are the ratios of the thermal energy to the stiffnesses at the extremes of the potential. The distance between the potential barrier and the absorptive border is accounted for as well. This corrected Kramers rate is demonstrated to agree with the dynamical rate typically within 2%. Probably the most interesting result is that despite the corrections are derived in the case of the overdamped Brownian motion, the above 2% agreement holds even in the case of medium friction.     

Capacitance and potential coefficients at large distances

The design of capacitive sensors and devices for new and emerging applications would benefit from simple and reliable methods to estimate the capacitance between conductors in terms of the capacitance of the isolated bodies and of the distance between them. The coefficients of potential and capacitance of a pair of conductor are approximated with the first terms of an expansion formula in the inverse of their distance. The form given applies to conductors of generic shapes and position in space. A comparison with the exact value for two spheres shows agreement even for rather small distances.