Efficient Continuous Wave-laser emission of Pr3+-doped fluorides at room temperature

We report, to our knowledge for the first time, Continuous Wave (CW) laser emission at room temperature of Pr:LiYF₄ (Pr.YLF) at six wavelengths: 522, 545, 607, 639.5, 720 and 907.4 nm. The pump source was an argon-ion laser tuned to a wavelength of = 457.9 nm. The maximum output powers at 522 nm (³P_i ³H₅) and 639.5 nm (³P₀ ³F₂) were 144 and 266 mW, respectively. We also observed CW laser action of Pr:GdLiF₄ (Pr: GLF) at = 639 nm and of Pr:KYF₄ (Pr: KYF) at = 642.5 nm.     

TrigonometricR matrices related to ‘dilute’ Birman-Wenzl-Murakami algebra

Explicit expressions for three series ofR matrices which are related to a dilute generalisation of the Birman-Wenzl-Murakami algebra are presented. Of those, one series is equivalent to the quantumR matrices of theD _n+1 ⁽²⁾ generalised Toda systems, whereas the remaining two series appear to be new.     

The quantum de Rham complexes associated with SL h (2)

Quantum de Rham complexes on the quantum plane and the quantum group itself are constructed for the nonstandard deformation of Fun(SL(2)). It is shown that in contrast to the standardq-deformation of SL(2), the above complexes are unique for SL _h (2). Also, as a byproduct, a new deformation of the two-dimensional Heisenberg algebra is obtained which can be used to construct models ofh-deformed quantum mechanics.     

Extended non-abelian gauge symmetries in the classical WZNW model

Higher spin extensions of the non-Abelian gauge symmetries for the classical WZNW model are considered. Both linear and nonlinear realizations of the extended affine Kac-Moody algebra are obtained. It is a characteristic property of the WZNW model that it admits a higher spin linear realization of the extended affine Kac-Moody algebra which is equivalent to the corresponding higher spin nonlinear realization of the same algebra. However, in both cases, the higher spin Noether currents do not span an invariant space with respect to their generating transformations. Here, the current space is extended to an invariant space which allows us to gauge the symmetry.Supported by Bulgarian Foundation on Fundamental Research under contract Ph-318/93-95.     

Cardinal hermite interpolation with box splines II

Summary In the present work we extent the results in [RS] on CHIP, i.e. Cardinal Hermite Interpolation by the span of translates of directional derivatives of a box spline. These directional derivatives are that ones which define the type of the Hermite Interpolation. We admit here several (linearly independent) directions with multiplicities instead of one direction as in [RS]. Under the same assumptions on the smoothness of the box spline and its defining matrixT we can prove as in [RS]: CHIP has a system of fundamental solutions which are inL L ² together with its directional derivatives mentioned above. Moreover, for data sequences inl ^p ( ^d ), 1p2, there is a spline function inL ^p, 1/p+1/p=1, which solves CHIP.Research supported in part by NSERC Canada under Grant # A7687. This research was completed while this author was supported by a grant from the Deutscher Akademischer Austauschdienst     

Equiconvergence of some complex interpolatory polynomials

Summary Walsh showed the close relation between the Lagrange interpolant in then ^th roots of unity and the corresponding Taylor expansion for functions belonging to a certain class of analytic functions. Recent extensions of this phenomena to Hermite interpolation and other linear processes of interpolation have been surveyed in [3, 5]. Following a recent idea of L. Yuanren [7], we show how new relations between other linear operators can be derived which exhibit Walsh equiconvergence.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThese authors were supported by NSERC A3094     

Suboptimal Kronrod extension formulae for numerical quadrature

Summary We consider cases where the Stieltjes polynomial associated with a Gaussian quadrature formula has complex zeros. In such cases a Kronrod extension of the Gaussian rule does not exist. A method is described for modifying the Stieltjes polynomial so that the resulting polynomial has no complex zeros. The modification is performed in such a way that the Kronrod-type extension rule resulting from the addition of then+1 zeros of the modified Stieltjes polynomial to the original knots of the Gaussian rule has only slightly lower degree of precision than normally achieved when the Kronrod extension rule exists. As examples of the use of the method, we present some new formulae extending the classical Gauss-Hermite quadrature rules. We comment on the limited success of the method in extending Gauss-Laguerre rules and show that several modified extensions of the Gauss Gegenbauer formulae exist in cases where the standard Kronrod extension does not.     

A shape-optimization technique for the capillary surface problem

Summary The problem of the construction of an equilibrium surface taking the surface tension into account leads to Laplace-Young equation which is a nonlinear elliptic free-boundary problem. In contrast to Orr et al. where an iterative technique is used for direct solution of the equation for problems with simple geometry, we propose here an alternative approach based on shape optimization techniques. The shape of the domain of the liquid is varied to attain the optimality condition. Using optimal control theory to derive expressions for the gradient, a numerical scheme is proposed and simple model problems are solved to validate the scheme.     

Tracking poles and representing Hankel operators directly from data

Summary We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL ² fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL ² which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH , analytic, approximation toG relative to theL norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.Partially supported by grants from the AFOSR and NSF     

Partitioning multi-edge graphs

We introduce a new kind of graph called multi-edge graph which arises in many applications such as finite state Markov chains and broadcasting communication networks. A cluster in such a graph is a set of nodes such that for any ordered pair of nodes, there is a path of multi-edges from one to the other such that these edges remain within the same set. We give two algorithms to partition a multi-edge graph into maximal clusters. Both these algorithms are based on the depth-first search algorithm to find strongly connected components of the directed graph. We also discuss some applications of clustering in multi-edge graphs.