Equation of motion of small bodies in relativity

There is proven a theorem, to the effect that a material body in general relativity, in a certain limit of sufficiently small size and mass, moves along a geodesic.     

Methylene-azaphosphirane as a reactive intermediate

Reaction of the transient phosphinidene complexes R-P=W(CO)5 with N-substituted-diphenylketenimines leads unexpectedly to the novel 2-aminophosphindoles, as confirmed by an X-ray crystal structure determined for one of the derivatives. Experimental evidence for a methylene-azaphosphirane intermediate was found by using the iron-complexed phosphinidene iPr2N-P=Fe(CO)4, which affords the 2-aminophosphindole together with the novel methylene-2,3-dihydro-1H-benzo[1,3]azaphosphole. Analysis of the reaction pathways with DFT indicates that the initially formed methylene-azaphosphirane yields both phosphorus heterocycles by way of a [1,5]- or [1,3]-sigmatropic shift, respectively, followed by a H-shift. Strain underlies both rearrangements, which causes these remarkably selective conversions that can be tuned by changing the substituents.     

Contributions to the relativistic mechanics of continuous media

This is a translation from German of an article originally published inProceedings of the Mathematical-Natural Science Section of the Mainz Academy of Science and Literature, Nr. 11, 1961 (pp. 792–837) (printed by Franz Steiner and Co, Wiesbaden), which is Paper IV in the series Exact Solutions of the Field Equations of General Relativity Theory by Pascual Jordan, Jürgen Ehlers, Wolfgang Kundt and Rainer K. Sachs. The translation has been carried out by G. F. R. Ellis (Department of Applied Mathematics, University of Cape Town), assisted by P. K. S. Dunsby, so that this outstanding review paper can be readily accessible to workers in the field today. As far as possible, the translation has preserved both the spirit and the form of the original paper. Despite its age, it remains one of the best reviews available in this area.     

Modellkomplexe mit Hydrazonchelaten von (S)- und (R)-1-Amino-2-methoxymethylpyrrolidin („SAMP”︁ und „RAMP”︁) und Dimethylhydrazin - Struktur des Pyridinaldehyd-RAMP-hydrazon-tetracarbonyl-molybdäns

Model Complexes with Hydrazone Chelates of (S)- and (R)-1-Amino-2-methoxymethylpyrrolidine (?SAMP”? and ?RAMP”?) and Dimethylhydrazine — Structure of (Pyridine-2-aldehyde-RAMP-hydrazone)tetracarbonyl-molybdenum The hydrazones of glyoxal, biacetyl, pyridine-2-aldehyde and 2-acetylpyridine with N,N-dimethylhydrazine and (R)- and (S)-1-amino-2-methoxymethylpyrrolidine are synthesized and reacted with hexacarbonylmolybdenum to give the chelate hydrazone tetracarbonylmolybdenum complexes. In contrast to the pure electronic effect of methyl groups complexes derived from biacetyl (relativ to glyoxal) or from acetylpyridine (rel to the aldehyde) show a long-wavelength shift of the characteristic MLCT absorption band in their electric spectra, indicativ of a dominant steric effect. A single crystal x-ray diffraction study on (pyridine-2-aldehyde-RAMP-hydrazone)tetracarbonylmolybdenum 12 proves the conjugative interaction of the sp²-hybridized amino nitrogen with the chelate π-system. Orthorhombic 12 crystallizes in space group P2₁2₁2 with 4 molecules in the unit cell: a = 14.689(4), b = 15.699(1) and c = 7.866(6) Å, V = 1814 ų. Complete ¹H and ¹³C nmr parameter are given for the ligands and the complexes.     

Synthesis, structures, and strain energies of dispirophosphiranes. Comparisons with dispirocyclopropanes

Six novel dispirophosphirane complexes have been synthesized from the reaction of bicycloalkylidenes with the electrophilic phosphinidene complex PhPW(CO)(5). They contain a central phosphirane ring, which is spirofused on one side to a cyclopropane or cyclobutane ring and on the other side with a three-, four-, five-, or six-membered ring. Their crystal structures and MP2/6-31G-computed geometries for simplified parent systems suggest that spirofusion with small rings results in a tightening of the central three-membered phosphaheterocycle, while spirofusion with larger rings results in a relaxation of the phosphirane geometry. Similar theoretical predictions are made for the corresponding annulated hydrocarbons. Strain energies for both the hydrocarbon and phosphorus series of structures have been calculated at G3(MP2). Whereas the [3]triangulane hydrocarbon and phospha[3]triangulane have a significant excess strain of 8.1 and 5.2 kcal/mol per spiroatom, respectively, the excess strain for systems spirofused with larger rings are negligible for the hydrocarbons and even negative for the phosphorus-containing species because of hyperconjugative stabilization.     

Facile Phenylphosphinidene Transfer Reactions from Carbene–Phosphinidene Zinc Complexes

Phosphinidenes [R-P] are convenient P₁ building blocks for the synthesis of a plethora of organophosphorus compounds. Thus far, transition-metal-complexed phosphinidenes have been used for their singlet ground-state reactivity to promote selective addition and insertion reactions. One disadvantage of this approach is that after transfer of the P₁ moiety to the substrate, a challenging demetallation step is required to provide the free phosphine. We report a simple method that enables the Lewis acid promoted transfer of phenylphosphinidene, [PhP], from NHC=PPh adducts (NHC=N-heterocyclic carbene) to various substrates to produce directly uncoordinated phosphorus heterocycles that are difficult to obtain otherwise.     

Phosphinine-Based Ligands in Gold-Catalyzed Reactions

A series of substituted phosphinines, 1-phosphabarrelenes and 5-phosphasemibullvalenes were synthesized and evaluated for their potential application as ligands in homogeneous catalytic reactions. While their buried volume (%V_bur) was calculated to get insight into the steric properties, [LNi(CO)₃] complexes were prepared in order to determine the corresponding Tolman electronic parameter. ETS-NOCV (extended-transition-state natural orbital for chemical valence) calculations on [LAuCl] complexes further allowed an estimation of the σ- and π-contributions to the L− M interaction. Au^I coordination compounds of selected examples were prepared and characterized by single crystal X-ray diffraction. Finally, the three classes of P^III compounds were successfully used in the Au^I-catalyzed cycloisomerization of N-2-propyn-1-ylbenzamide, showing very good activities and selectivities, which are comparable with the reported data of cationic phosphorus-based gold catalysts.     

Application and modification of the POD method and the POD-DEIM for model reduction in porous-media simulations

Researchers with a continuum-mechanical background typically use a multi-phasic and multi-component modelling approach for materials with a saturated porous microstructure. Therefore, the mechanical behaviour is considered in a continuum-mechanical manner and solved using the finite-element method (FEM). The developed models need to be complex enough to capture the relevant properties of the considered materials, what often results in expensive simulations with a very large number of degrees of freedom (DOF). The aim of the present contribution is to reduce the computing time of these simulations through model-reduction methods, while the accuracy of the solution needs to be maintained. Therefore, the method of proper-orthogonal decomposition (POD) for linear problems and the discrete-empirical-interpolation method (DEIM) in combination with the POD method (POD-DEIM) for nonlinear problems are investigated. Using the POD method, a given data set is approximated with a low-dimensional subspace. To generate this data set, the vector of unknowns of the FE simulation is stored in a pre-computation in the full (unreduced) system in each time-step (so-called “snapshots” of the system). Dealing with porous-media problems, the primary variables are the solid displacement, the pore pressure and, depending on the particular problem, other primary variables. Following this, the primary variables have entries with very huge differences in their absolute values. As a result, non-negligible rounding errors may occur when applying the POD method. To overcome this problems, modifications of the classical POD method need to be performed for such problems. The present contribution discusses this issue and presents results for the reduced simulations of porous media. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)