Modular invariance and uniqueness of conformal characters

We show that the conformal characters of various rational models ofW-algebras can be already uniquely determined if one merely knows the central charge and the conformal dimensions. As a side result we develop several tools for studying representations of SL(2,) on spaces of modular functions. These methods, applied here only to certain rational conformal field theories, may be useful for the analysis of many others.     

Lower Bounds for the Lp-Norm in Terms of the Mellin Transform

Given a measurable function f on (0, ) with Mellin transformF(s), let |f|_p denote the L^p-norm of f with respect to the measuredx/x. We prove that under certain assumptions, for instanceif f is real and non-negative and F() converges for in an openinterval and F() 0, then wherec_p (2e)^–1. We derive similar inequalities for complex-valuedf, for the L^p-norm of the derivative of f, and for the supremumof real-valued f and of its derivative. The lower bounds areeminently applicable when f is a convolution product.     

Picosecond laser ablation of SiO2 layers on silicon substrates

In this work, we report on laser ablation of thermally grown SiO₂ layers from silicon wafer substrates, employing an 8–9 ps laser, at 1064 (IR), 532 (VIS) and 355 nm (UV) wavelengths. High-intensity short-pulse laser radiation allows direct absorption in materials with bandgaps higher than the photon energy. However, our experiments show that in the intensity range of our laser pulses (peak intensities of <2×10¹² W/cm²) the removal of the SiO₂ layer from silicon wafers does not occur by direct absorption in the SiO₂ layer. Instead, we find that the layer is removed by a “lift off” mechanism, actuated by the melting and vaporisation of the absorbing silicon substrate. Furthermore, we find that exceeding the Si melting threshold is not sufficient to remove the SiO₂ layer. A second threshold exists for breaking of the layer caused by sufficient vapour pressure. For SiO₂ layer ablation, we determine layer thickness dependent minimum fluences of 0.7–1.2 J/cm² for IR, 0.1–0.35 J/cm² for VIS and 0.2–0.4 J/cm² for UV wavelength. After correcting the fluences by the reflected laser power, we show that, in contrast to the melting threshold, the threshold for breaking the layer depends on the SiO₂ thickness.     

Explicit formulas for Hecke Gauss sums in quadratic number fields

We derive an explicit formula for Hecke Gauss sums of quadratic number fields. As an immediate consequence we obtain a quadratic reciprocity law in quadratic number fields which generalizes the classical one given by Hecke. The proofs use, apart from the well-known formulas for ordinary Gauss sums, only elementary algebraic manipulations.     

A Poisson Summation Formula for Extensions of Number Fields

A formula of Poisson summation type is established for sumsinvolving relative norms in an extension of number fields.     

Analytische Anwendung substituierter Selenoniumsalze

Zusammenfassung Die extraktive Isolierung von Quecksilbermengen ab 10 g als [(C₆H₅)₃Se]-[HgCl₄] ermöglicht seine photometrische Bestimmung als kolloidales HgS in Methylenchlorid.

---

Summary Extractive isolation of mercury as [(C₆H₅)₃Se][HgCl₄] permits the determination of 10 g of Hg²⁺ as colloidal HgS in non-aqueous phase (CH₂Cl₂).

---

---

Herr Wolfgang Renner führte die erforderlichen präparativen Arbeiten aus. Der Deutschen Forschungsgemeinschaft danken wir für die Förderung der Arbeit durch Sachbeihilfen.