The Elastic-Plastic Torsion Problem: A Direct Boundary Approach∗

A new mechanics theorem pertaining to the elastic-plastic torsion problem is shown; i.e., the complementary energy evaluated on some admissible set of functions is minimized with respect to the transient (free) boundary between the elastic and plastic zones. The elastic-plastic torsion problem is then formulated as an unconstrained shape optimal design problem which treats the free boundary directly as a unknown. Unlike other formulations there is no inequality constraint involved in this formulation. The derivation of shape design sensitivity and numerical results for circular cross section and Sokolovsky's oval are presented.     

Reactions of [Y(BDI)(I)2(THF)] [BDI = {HC(CMeNAr)2}−, Ar = 2,6-diisopropylphenyl] with Na[M(Cp)(CO)3] (M = Cr,W): X-ray crystal structures of [{Y(BDI)[Cr(Cp)(CO)3]2(THF)}2] and [W(Cp)(CO)3][Na(THF)2]

Reaction of [Y(BDI)(I)₂(THF)] (1) with two equivalents of Na[Cr(Cp)(CO)₃] affords the dimeric complex [{Y(BDI)[Cr(Cp)(CO)₃]₂(THF)}₂] (2). Complex 2 contains two yttrium-BDI units which are each linked by two isocarbonyl bridging [Cr(Cp)(CO)₃]^? anions; a terminal, isocarbonyl bound [Cr(Cp)(CO)₃]^? anion and THF molecule completes the coordination sphere at each yttrium. This results in formation of a centrosymmetric, 12-membered C₄O₄Cr₂Y₂ ring. Forcing conditions were required to produce carbonyl metallate derivatives such as 2, as exemplified by the isolation of crystals of [W(Cp)(CO)₃][Na(THF)₂] (3) from the analogous reaction between 1 and two equivalents of Na[W(Cp)(CO)₃]. Complex 3 loses coordinated THF very easily and all isolated samples exhibit spectra consistent with the known, un-solvated form of Na[W(Cp)(CO)₃]. The crystal structure of 3 shows dimeric sodium units bridged by two THF molecules and one isocarbonyl group. Each sodium centre is further coordinated by one THF molecule and two isocarbonyl ligands. There are two crystallographically distinct [W(Cp)(CO)₃]^? units; one exhibits two bridging isocarbonyl groups and the other exhibits three bridging isocarbonyl groups to different sodium dimer units. This results in a 2-dimensional polymeric sheet network in the solid state. Complex 2 was characterised by single crystal X-ray diffraction, NMR spectroscopy, FTIR spectroscopy and CHN microanalysis; complex 3 was characterised by single crystal X-ray diffraction only.     

Centric-scan SPRITE magnetic resonance imaging with prepared magnetisation

The combination of contrast preparation with centric-scan SPRITE imaging readout is investigated. The main benefit of SPRITE, its ability to image objects with short T2, is retained. We demonstrate T1 and T2 mapping as examples of magnetisation preparation followed by magnetisation storage and spatially resolved encoding. A strategy for selection of the most advantageous imaging parameters for contrast mapping is presented.     

Zur elektrographischen Bestimmung von Blei in Legierungen

Zusammenfassung Die elektrographische Methode wurde zur Bestimmung von Pb in Pb-Sn-Legierungen verwendet. Die Fehler überschreiten nicht ±0,6%. Es wurde auch der Einfluß der Spannung, der Zeitdauer, der Konzentration des Elektrolyten sowie der Oberfläche der untersuchten Proben auf die entstandene Menge an Pb²⁺-Ionen untersucht.

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Summary The electrographic method has been applied to the determination of Pb in Pb-Sn alloys. The error is smaller than ±0,6%. The effects of voltage, time, electrolyte concentration and surface area of the sample on the amount of Pb²⁺ ions formed has been investigated.

     

Asymptotic Error Analysis of the Adaptive Verlet Method

The Adaptive Verlet method and variants are time-reversible schemes for treating Hamiltonian systems subject to a Sundman time transformation. These methods have been observed in computer experiments to exhibit superior numerical stability when implemented in a counterintuitive reciprocal formulation. Here we give a theoretical explanation of this behavior by examining the leading terms of the modified equation (backward error analysis) and those of the asymptotic error expansion. With this insight we are able to improve the algorithm by simply correcting the starting stepsize.     

Haar Measure and the Artin Conductor

Let be a connected reductive group, defined over a local, non-archimedean field . The group is locally compact and unimodular. In On the motive of a reductive group, Invent. Math. 130 (1997), by B. H. Gross, a Haar measure was defined on , using the theory of Bruhat and Tits. In this note, we give another construction of the measure , using the Artin conductor of the motive of over . The equivalence of the two constructions is deduced from a result of G. Prasad.