Erst Computeralgebra nutzen, dann technologiefrei Umformen lernen?

Inwieweit bauen Lernende konzeptuelles Wissen in der Algebra auf und inwieweit begreifen sie den Wert von Algebra, ohne dass sie technologiefrei Terme umformen können? Im Beitrag wird die Konzeption einer Studie zur Beantwortung dieser Frage vorgestellt. Bei dem eingesetzten Material, das auf der Basis zweier explorativer Forschungszyklen entwickelt wurde, wird durch CAS ermöglicht, dass gleich zu Beginn des Lernprozesses Problemlösen und Modellieren mit Variablen im Vordergrund steht

The interface of gravity and quantum mechanics illuminated by Wigner phase space

We provide an introduction into the formulation of non-relativistic quantum mechanics using the Wigner phase-space distribution function and apply this concept to two physical situations at the interface of quantum theory and general relativity: (i) the motion of an ensemble of cold atoms relevant to tests of the weak equivalence principle, and (ii) the Kasevich-Chu interferometer. In order to lay the foundations for this analysis we first present a representation-free description of the Kasevich-Chu interferometer based on unitary operators.Comment: 69 pages, 6 figures, minor changes to match the published version. The original publication is available at http://en.sif.it/books/series/proceedings_fermi or http://ebooks.iospress.nl/volumearticle/3809

Palladium(II) complexes of a bridging amine bis(phenolate) ligand featuring κ² and κ³ coordination modes

Bidentate and tridentate coordination of a 2,4-di-tert-butyl-substituted bridging amine bis­(phenolate) ligand to a palladium(II) center are observed within the same crystal structure, namely di­chlorido­({6,6′-[(ethane-1,2-diylbis(methyl­aza­nedi­yl)]bis­(methyl­ene)}bis­(2,4-di-tert-butyl­phenol))palladium(II) chlorido­(2,4-di-tert-butyl-6-{[(2-{[(3,5-di-tert-butyl-2-hy­droxy­phen­yl)meth­yl](meth­yl)amino}­eth­yl)(meth­yl)amino]­meth­yl}phenolato)palladium(II) methanol 1.685-solvate 0.315-hydrate, [PdCl2(C34H56N2O2)][PdCl(C34H55N2O2)]·1.685CH3OH·0.315H2O. Both complexes exhibit a square-planar geometry, with unbound phenol moieties participating in inter­molecular hydrogen bonding with co-crystallized water and methanol. The presence of both κ2 and κ3 coordination modes arising from the same solution suggest a dynamic process in which phenol donors may coordinate or dissociate from the metal center, and offers insight into catalyst speciation throughout Pd-mediated processes. The unit cell contains di­chlorido­({6,6′-[(ethane-1,2-diylbis(methyl­aza­nedi­yl)]bis­(methyl­ene)}bis­(2,4-di-tert-butyl­phenol))palladium(II), {(L2)PdCl2}, and chlorido­(2,4-di-tert-butyl-6-{[(2-{[(3,5-di-tert-butyl-2-hy­droxy­phen­yl)meth­yl](methyl)amino}eth­yl)(meth­yl)amino]­meth­yl}phenolato)palladium(II), {(L2X)PdCl}, mol­ecules as well as fractional water and methanol solvent mol­ecules

Synthesis of an Ortho-Triazacyclophane: N,N\u27,N\u27\u27-Trimethyltribenzo-1,4,7-triazacyclononatriene

N,N\u27,N\u27\u27-Trimethyltribenzo-1,4,7-triazacyclononatriene has been synthesized via sequential palladium-catalyzed Buchwald-Hartwig N-arylation reactions affording the 9-membered triaza o-cyclophane in 35% overall yield. An X-ray crystal structure shows the new cyclophane to have a C(2)-symmetric saddle conformation, as compared to the crown conformation exhibited by the related carbocyclic cyclotriveratrylene (CTV)

Di-μ-acetato-bis­(dimethyl­formamide)­penta­kis­(μ-N,2-dioxidobenzene-1-car­boximidato)tetra­kis­(1-ethyl­imidazole)­penta­manganese(III)­manganese(II)–diethyl ether–dimethyl­foramide–methanol–water (1/1/1/1/0.12)

The title compound [Mn6(C7H4NO3)5(CH3CO2)2(C5H8N2)4(C3H7NO)2]·(C2H5)2O·C3H7NO·CH3OH·0.12H2O, abbreviated as MnII(OAc)2[15-MCMnIII(N)shi-5](EtIm)4(DMF)2·diethyl ether·DMF·MeOH·0.12H2O (where −OAc is acetate, MC is metallacrown, shi3− is salicylhydroximate, EtIM is n-ethylimidazole, DMF is N,N-dimethylformamide, and MeOH is methanol) contains five MnIII ions as members of the metallacrown ring and an MnII ion bound in the central cavity. The central MnII ion is seven-coordinate with a distorted face-capped trigonal–prismatic geometry. The five MnIII ions of the metallacrown ring are six-coordinate with distorted octa­hedral geometries. The configuration of the MnIII ions about the metallacrown ring follow a ΔΛΔPP pattern, with P representing planar. The four 1-ethyl­imidazole ligands are bound to four different MnIII ions. A diethyl ether solvent mol­ecule was found to be disordered over two mutually exclusive sites with an occupancy ratio of 0.568 (7):0.432 (7). A methanol solvent mol­ecule was found to be disordered over two mutually exclusive sites by being hydrogen bonded either to a dimethyl­formamide solvent mol­ecule (major occupancy component) or to an O atom of the main mol­ecule (minor occupancy component). The occupancy ratio refined to 0.678 (11):0.322 (11). Associated with the minor component is a partially occupied water mol­ecule [total occupancy 0.124 (15)]

Bis[μ-pentane-2,4-dionato(1−)]bis­{aqua­[1,1,1,5,5,5-hexa­fluoro­pentane-2,4-dionato(1−)]cobalt(II)}

The title complex, [Co2(C5HF6O2)2(C5H7O2)2(H2O)2], is centrosymmetric with a crystallographic inversion center in the middle of the mol­ecule. The octa­hedrally coordinated CoII atoms are bridged by two chelating acetyl­acetonate (acac) ligands and two more electron-poor 1,1,1,5,5,5-hexa­fluoro­pentane-2,4-dionato (hfac) ligands are bonded terminally in a solely chelating manner. The coordinated water mol­ecules form inter­molecular O—H⋯O hydrogen bonds with electron-rich acac O atoms of neighboring mol­ecules, leading to strings of mol­ecules along the a axis

Beckmann Rearrangement of Cyclotriveratrylene (CTV) Oxime: Tandem Beckmann-Electrophilic Aromatic Addition

The Beckmann rearrangement has been performed on the oxime of cyclotriveratrylene (CTV) with thionyl chloride affording the ring-expanded 10-membered ring amide exclusively in high yield. Modified conditions afford a helical pentacycle derived from an unusual tandem Beckmann rearrangement and electrophilic aromatic addition followed by demethylation and tautomerization