Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2,R)

The restriction principle is used to implement a realization of the holomorphic representations of SL(2,R) on L ² (R ⁺,t dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character e^–i(2n++1); under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.     

Holomorphic H-spherical distribution vectors in principal series representations

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L²(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown . In this article we construct a minimal G-invariant subdomain _H of with G/H as Shilov boundary. Let be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of , which admit a holomorphic extension to _H, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L²(G/H)_mc in a space of holomorphic functions.     

A Paley-Wiener theorem for the spherical Laplace transform on causal symmetric spaces of rank 1

We formulate and prove a topological Paley-Wiener theorem for the normalized spherical Laplace transform defined on the rank 1 causal symmetric spaces , for .

     

Support properties and Holmgren's uniqueness theorem for differential operators with hyperplane singularities

Let W be a finite Coxeter group acting linearly on Rn. In this article we study the support properties of a W-invariant partial differential operator D on Rn with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(suppDf)=conv(suppf) holds for all compactly supported smooth functions f so that conv(suppf) is W-invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.     

On the length of simplex paths: The assignment case

This paper reports on an experimental study of the distribution of the length of simplex paths for the Optimal Assignment Problem. We study the distribution of the pivot counts for a version of the simplex method that with essentially equal probabilities introduces any variable with negative reduced cost into the basis. In this situation the distribution of the pivot counts turns out to be normally distributed and independent of the actual cost coefficients, provided these are sufficiently spread out. Further, the mean and standard deviation grow only moderately with the size of the problem, namely asd ^1.8, andd ^1.5 respectively for ad×d problem, implying in particular that the pivot counts concentrate around the mean with growingd. The usual simplex method on the other hand gives a growth ofd ^1.6. Hence a large part of the favourable polynomial growth experienced on practical problems may be attributed to the fact that the simplex paths are rather short on the average, at least for assignment problems.     

<Superscript>57</Superscript>Fe emission Mössbauer spectroscopy following dilute implantation of <Superscript>57</Superscript>Mn into In <Subscript>2</Subscript>O<Subscript>3</Subscript>

Emission Mössbauer spectroscopy has been utilised to characterize dilute ⁵⁷Fe impurities in In ₂O₃ following implantation of ⁵⁷Mn (T _1/2 = 1.5 min.) at the ISOLDE facility at CERN. From stoichiometry considerations, one would expect Fe to adopt the valence state 3 + , substituting In ³⁺, however the spectra are dominated by spectral lines due to paramagnetic Fe²⁺. Using first principle calculations in the framework of density functional theory (DFT), the density of states of dilute Fe and the hyperfine parameters have been determined. The hybridization between the 3d-band of Fe and the 2p band of oxygen induces a spin-polarized hole on the O site close to the Fe site, which is found to be the cause of the Fe²⁺ state in In ₂O₃. Comparison of experimental data to calculated hyperfine parameters suggests that Fe predominantly enters the 8b site rather than the 24d site of the cation site in the Bixbyite structure of In ₂O₃. A gradual transition from an amorphous to a crystalline state is observed with increasing implantation/annealing temperature.     

Emission Mössbauer spectroscopy study of fluence dependence of paramagnetic relaxation in Mn/Fe implanted ZnO

Emission Mössbauer Spectroscopy following the implantation of radioactive precursor isotope ⁵⁷Mn⁺ (T _1/2= 1.5 min) into ZnO single crystals at ISOLDE/CERN shows that a large fraction of ⁵⁷Fe atoms produced in the ⁵⁷Mn beta decay is created as paramagnetic Fe³⁺ with relatively long spin-lattice relaxation times. Here we report on ZnO pre-implanted with ⁵⁶Fe to fluences of 2×10¹³, 5×10 ¹³ and 8 × 10¹³ ions/cm² in order to investigate the dependence of the paramagnetic relaxation rate of Fe³⁺ on fluence. The spectra are dominated by magnetic features displaying paramagnetic relaxation effects. The extracted spin-lattice relaxation rates show a slight increase with increasing ion fluence at corresponding temperatures and the area fraction of Fe³⁺ at room temperature reaches a maximum contribution of 80(3)% in the studied fluence range.     

Three-Way Tiling Sets in Two Dimensions

In this article we show that there exist measurable sets W⊂ℝ² with finite measure that tile ℝ² in a measurable way under the action of a expansive matrix A, an affine Weyl group [(W)\tilde]\widetilde{W} , and a full rank lattice [\varGamma\tilde] ì \mathbbR²\widetilde{\varGamma}\subset\mathbb{R}^{2} . This note is follow-up research to the earlier article “Coxeter groups and wavelet sets” by the first and second authors, and is also relevant to the earlier article “Coxeter groups, wavelets, multiresolution and sampling” by M. Dobrescu and the third author. After writing these two articles, the three authors participated in a workshop at the Banff Center on “Operator methods in fractal analysis, wavelets and dynamical systems,” December 2–7, 2006, organized by O. Bratteli, P. Jorgensen, D. Kribs, G. ólafsson, and S. Silvestrov, and discussed the interrelationships and differences between the articles, and worked on two open problems posed in the Larson-Massopust article. We solved part of Problem 2, including a surprising positive solution to a conjecture that was raised, and we present our results in this article.     

An optimization approach to partitional data clustering

Scalability of clustering algorithms is a critical issue facing the data mining community. One method to handle this issue is to use only a subset of all instances. This paper develops an optimization-based approach to the partitional clustering problem using an algorithm specifically designed for noisy performance, which is a problem that arises when using a subset of instances. Numerical results show that computation time can be dramatically reduced by using a partial set of instances without sacrificing solution quality. In addition, these results are more persuasive as the size of the problem is larger.     

Fourier Series on Compact Symmetric Spaces: K-Finite Functions of Small Support

The Fourier coefficients of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter μ, which determines the representation, and they can be represented by elements [^(f)](m)\hat{f}(\mu) in a common Hilbert space ℋ.