Efficient laser operation of heavily doped Cr4+:forsterite

Pulsed laser operation of a new high Cr4+ concentration forsterite crystal is reported. Slope efficiencies of 13% and 5% have been obtained for 1064 and 532nm pumping. Results are compared with the performance of three other forsterite crystals. A threefold enhancement in operating efficiency is obtained over standard low dopant-level material. Tuning ranges have also been investigated. This revised version was published online in November 2006 with corrections to the Cover Date.     

Construction of Jacobi forms

Partially supported KOSEF Research Grant 91-08-00-07 and KOSEF 921-0100-018-2     

Dirichlet series of Rankin-Cohen brackets

Given modular forms f and g of weights k and ?, respectively, their Rankin-Cohen bracket corresponding to a nonnegative integer n is a modular form of weight k+?+2n, and it is given as a linear combination of the products of the form f(r)g(n−r) for 0?r?n. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin-Cohen brackets.     

Chemo-biological preparation of the chiral building block (R)-4-acetoxy-2-methyl-1-butanol using Pseudomonas putida

In order to develop new methyl substituted chiral building blocks which are useful for the synthesis of methyl branched natural products, the enantioselective bioreduction of an exo-methylene to a methyl group was investigated. 4-Acetoxy-2-methylene-1-butanol 3 was prepared from itaconic acid over four steps and converted to the chiral alcohol (R)-4-acetoxy-2-methyl-1-butanol 4, by growing cells of Pseudomonas putida. The bioconversion achieved a high enantioselectivity (92% ee) and a high chemical yield (65%) within a relatively short reaction time (18–20 h).     

Quasiconformal deformations of complex hyperbolic <Emphasis Type="Italic">p</Emphasis>-Schottky groups

In this paper, we generalize a Schottky group construction to complex hyperbolic space. In particular, we construct a fundamental domain whose sides consist of disjoint non-asymptotic packs for the action of the Schottky group acting on complex hyperbolic space. Then we prove that a smooth deformation of such a Schottky group is quasiconformally stable.     

Construction of Jacobi forms associated to indefinite quadratic forms

Vignéras constructs non-holomorphic theta functions according to indefinite quadratic forms with arbitrary signature. We use Vignéras’ theta functions to create examples of non-holomorphic Jacobi forms associated to indefinite theta series by two different methods.     

Geometrically infinite surfaces with discrete length spectra

It is well-known that the length spectrum of a geometrically finite hyperbolic manifold is discrete. In this paper, we begin a study of the length spectrum for geometrically infinite hyperbolic surfaces. In this generality, it is possible that the spectrum is not discrete and the main focus of this work is to find necessary and sufficient conditions for a geometrically infinite surface to have a discrete spectrum. After deriving a number of properties of the length spectrum, we show that every topological surface of infinite type admits both an infinite dimensional family of quasiconformally distinct hyperbolic structures having a discrete length spectrum, and an infinite dimensional family of quasiconformally distinct structures with a nondiscrete spectrum. Moreover, there exists such an infinite dimensional subspace arbitrarily close to (in the Fenchel-Nielsen topology) any hyperbolic structure.      

Values of <Emphasis Type="Italic">L</Emphasis>-functions at integers outside the critical strip

We incorporate the non-critical values of L-functions of cusp forms into a cohomological set-up analogous to the one of Eichler, Manin and Shimura. We use the 1-cocycles we associate in this way to non-critical values to prove an expression for such values which is similar in structure to Manin’s formula for the critical value of the L-function of a weight 2 cusp form. YoungJu Choie is partially supported by KOSEF R01-2003-00011596-0 and by ITRC Research Fund. N. Diamantis is partially supported by EPSRC grant EP/D032350/1.