On the number of conjugacy classes of a finite solvable group II

We prove that a finite solvable group G has at least (49p+1)/60 conjugacy classes whenever p is a prime such that p² divides the order of G. We also construct an infinite family of finite solvable groups, where this bound is attained.     

Formula 1 – A Mathematical Microworld with CAS: Analysis of Learning Opportunities and Experiences with Students

In this article we describe a mathematical microworld for investigating car motion on a racing course and its use with a group of grade 12 students. The microworld is concerned with the mathematical construction of courses and functions which describe car motion. It is implemented in the computer algebra system, Maple^®, which provides the means to represent courses and functions symbolically and graphically. We describe the learning opportunities offered by the microworld in relation to the research literature on functions. Various facets and layers of the function concept are addressed in the microworld, and we suggest how work in the microworld might help in overcoming some well-known misconceptions.This revised version was published online in September 2005 with corrections to the Cover Date.     

Optical spectra and laser-induced dissociation of supported Na particles

This paper reports new results on the optical spectra of Na particles and on laser-induced photodissociation of Na atoms from the surface of these particles. In continuation of our earlier studies we have performed experiments to elucidate the mechanism of thenonthermal dissociation process. Furthermore, theoretical calculations have been carried out with the goal to correlate the wavelength dependence of the photodissociation yield with the optical absorption spectra of the metal particles. In addition, manipulation of the size distribution of metal particles on supports is outlined as an application of the effect. This allows for the preparation of very special surfaces with novel physical and chemical properties.     

Stereoselective Michael Addition of Carbonyl Compounds to ( E )-β-Nitrostyrene Catalysed by N -Toluensulfonyl-l -proline Amide in Ionic Liquids

Summary. N-Toluensulfonyl-l-prolin amide was tested as catalyst in the enantioselective Michael addition of carbonyl compounds to (E)-β-nitrostyrene in nine ionic liquids under different reaction conditions. The reaction rates and enantioselectivities were strongly dependent on the ionic liquids. Change of enantioselectivity was observed too and it is attributed to both the cation and the anion of ionic liquid. The best yields (up to 98%) and enantioselectivity (70% ee) of product were obtained in a basic ionic liquid [bmim]BF₄ at room temperature.     

Fluorinated tellurium(IV) azides and their precursors

The perfluoroaryl tellurolates C₆F₅TeLi (1) and 4-CF₃C₆F₄TeLi (2) were prepared. These intermediates were identified by NMR spectroscopy and may form, depending on the reaction conditions, either the corresponding ditellanes C₆F₅TeTeC₆F₅ (3) and CF₃C₆F₄TeTeC₆F₄CF₃ (4) by subsequent oxidation, or in the case of 1, a telluranthrene (C₆F₄Te)₂ (5) by reaction with itself. The halogenation products of 5, ( C₆F₄Te)₂F₄ (6), (C₆F₄Te)₂Cl₄ (7), (C₆F₄Te)₂Br₄ (8), as well as the azidation product (C₆F₄Te)₂(N₃)₄ (9) were synthesized. Furthermore, in pursuit of our recent work on tellurium azides, the syntheses and properties of R₂Te(N₃)₂ (R=CF₃ (10), C₆F₂H₃ (11)) and RTe(N₃)₃ (R=CF₃ (12) and C₆F₅ (13)) are reported. The crystal structures of CF₃C₆F₄TeTeC₆F₄CF₃ (4), (C₆F₄Te)₂Br₄ (8), and (C₆F₂H₃)₂Te(N₃)₂ (11) were determined.