The photon two-point function in unitary approximation

The photon two-point function is calculated using the exponential representation for the time-evolution operator taking into account terms up to second order ine in the exponent. The limiting case of vanishing electron mass is examined and it is shown that mass singularities cancel by means of unitarity. Near mass shell the causal Green function exhibits in the limit of vanishing electron mass the expected cut behaviour, off mass shell we recover the result of the usual perturbation theory.     

Adaptive application of operators in standard representation

Recently adaptive wavelet methods have been developed which can be shown to exhibit an asymptotically optimal accuracy/work balance for a wide class of variational problems including classical elliptic boundary value problems, boundary integral equations as well as certain classes of noncoercive problems such as saddle point problems. A core ingredient of these schemes is the approximate application of the involved operators in standard wavelet representation. Optimal computational complexity could be shown under the assumption that the entries in properly compressed standard representations are known or computable in average at unit cost. In this paper we propose concrete computational strategies and show under which circumstances this assumption is justified in the context of elliptic boundary value problems. Dedicated to Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 41A25, 41A46, 65F99, 65N12, 65N55. This work has been supported in part by the Deutsche Forschungsgemeinschaft SFB 401, the first and third author are supported in part by the European Community's Human Potential Programme under contract HPRN-CT-202-00286 (BREAKING COMPLEXITY). The second author acknowledges the financial support provided through the European Union's Human Potential Programme, under contract HPRN-CT-2002-00285 (HASSIP) and through DFG grant DA 360/4–1.     

Algebraic properties of discrete box splines

The central objective of this paper is to discuss linear independence of translates of discrete box splines which we introduced earlier as a device for the fast computation of multivariate splines. The results obtained here allow us to draw conclusions about the structure of such discrete splines which have, in particular, applications to counting the number of nonnegative integer solutions of linear diophantine equations.     

Surface characterization of fluorinated polymers (PTFE, PVDF, PFA) for use in ultratrace analysis

Surface characterization of new unused PTFE, PFA and PVDF labware has been carried out by light microscopy, scanning electron microscopy, profilometry and atomic force microscopy. It has been found that in spite of higher micro-roughness, PFA exhibits the lowest nano-roughness and hence seems best suited as vessel material for relatively mild pressureless chemical operations for sample preparation of ultrapure substances, as container material for storage of ultrapure liquids, and for transport of such liquids, e.g., from the producer in the chemical to the end user in the microelectronics industry. This suitability refers only to the surface quality of the investigated materials. PTFE-surfaces, due to the sintering process of production, exhibit the most unfavourable surface quality of the investigated fluorinated polymers.     

Efficient synthesis of sulfonic,phosphoric, and phosphinic esters employing alkylating polymer-bound reagents

The efficient esterification of various sulfonic acids and sulfonates using polymer-bound triazenes based on the triazene T2 linker is described. Esterification of enantiopure alpha-substituted sodium sulfonates was performed in the presence of an alkylating resin without racemization. Racemization is a serious drawback in the esterification route via sulfonyl chlorides because of intermediate sulfene formation. To demonstrate the versatility of this protocol, phosphoric and phosphinic acids have been converted into the corresponding esters as well. All products were obtained in good yield and excellent purities without any further purification steps.     

Correction to: Preface

Foundations of Computational Mathematics -     

Electron microscopy observations on the role of twinning in the evolution of microstructures.

Twinning plays an important role in phase transformations and can have significant effects on microstructural evolution. Different roles of twinning in the development of microstructures during precipitation and phase transformations are reviewed and illustrated with examples from investigations by high-resolution electron microscopy, including the effect of multiple twinning on the development of Ge precipitates in Al-Ge and Ag-Ge alloys, the twin dissociation of grain boundaries in Au, the formation of hexagonal Si at twin intersections and the effect of twin boundaries on the equilibrium shape of Pb inclusions in Al.     

Triarylborane ammonia complexes as ideal precursors for arylzinc reagents in asymmetric catalysis

[reaction: see text] The value of arylboranes as precursors for arylzinc reagents in asymmetric catalysis is demonstrated. Kinetic studies on the transmetalation reaction with zinc reagents rationalize the observed differences of three classes of arylboranes in catalytic applications. By using the stable and easily accessible triarylborane ammonia complexes, an array of chiral diarylmethanols in high yield and enantioselectivity was synthesized.     

The Lewis basicity of the [Pt3(μ-CO)3(PR3)3] clusters and the X-ray crystal structure of [{ZnI2}{Pt3(μ-CO)3(PPh(i-Pr)2)3}]

The cluster compounds [Pt₃(μ-CO)₃(PR₃)₃] act as Lewis bases towards the metal halides of Group XI, MX, Group XII, MX₂, and Group XIII, MX₃, to form cluster compounds of the composition [{MX_n}{Pt₃(μ-CO)₃(PR₃)₃}]. The X-ray crystal structure, NMR and IR data are given for the compound [{ZnI}{Pt₃(μ-CO)₃(PPh(i-Pr)₂)₃}].