Isometric embedding for homogeneous compact 3-manifolds

The anisotropic homogeneous ³ is isometrically globally embedded into a nine-dimensional Euclidean or pseudo-Euclidean space. In a special case the Euclidean space is six-dimensional. The space-sections of the anti-Mach metric appear as a submanifold of theE ⁹. We obtain also the isometric embedding ofSO(3) with its natural Killing metric.     

Untersuchungen an Hydriden im Bereich der Phase Ti2CO1—x Fe x (x=0–0,5)

The hydrogen absorption of the phase Ti₆₄Co_32–x Fe _x (x=0...16) and its influence on the magnetic properties have been investigated. Measurements of the vapour-pressure, wide-line NMR and magnetic susceptibility have been performed. Substitution of Co by Fe does not change the amount of absorbed hydrogen. However the reaction rate of hydriding process, the activation energy of diffusion and the magnetic quantities change upon this substitution.

     

Tin-rich Phases RE2Au3Sn6 with RE = La,Ce, Pr,Nd, Sm – Synthesis,Structure, Magnetic Properties,and 119Sn Mössbauer Spectra

The stannides RE₂Au₃Sn₆ (RE = La, Ce, Pr, Nd, Sm) were synthesized from the elements by arc-melting. Small single crystals were grown by annealing samples in sealed tantalum tubes in an induction furnace with a special annealing sequence. The polycrystalline phases were characterized through their X-ray powder diffraction pattern. The structures of Ce₂Au₃Sn₆, Pr₂Au₃Sn₆, and Nd₂Au₃Sn₆ were refined from single-crystal X-ray diffractometer data. The RE₂Au₃Sn₆ stannides crystallize with the orthorhombic La₂Zn₃Ge₆ type, space group Cmcm. The basic structural building units are Au1@Sn₄ tetrahedra and Au2@Sn₅ square pyramids. These units are condensed to layers and the structure can be described by a simple stacking of tetrahedral and pyramidal layers with the rare earth cations in between. Temperature dependent susceptibility studies indicate that all rare earth atoms are in the trivalent oxidation state, as their effective magnetic moments match the expected values of the free RE³⁺ ions. Pr₂Au₃Sn₆ and Nd₂Au₃Sn₆ exhibit antiferromagnetic ordering at T_N = 6.3(1) and 6.7(1) K. Investigations of the electrical resistivity of La₂Au₃Sn₆ and Ce₂Au₃Sn₆ confirmed that these compounds are metallic, for La₂Au₃Sn₆ a lower resistivity was observed, in line with the absence of screening unpaired electrons. ¹¹⁹Sn Mössbauer spectra for La₂Au₃Sn₆, Ce₂Au₃Sn₆, Pr₂Au₃Sn₆ and Nd₂Au₃Sn₆ show a complex superposition of three sub-spectra which can be differentiated through their distinctly different quadrupole splitting parameters. The isomer shifts (1.87 to 2.22 mm · s^–1) indicate significant s electron density at the tin nuclei.     

SrIrIn6 – An Intergrowth of SrIrIn4 with Indium Slabs

The indium-rich intermetallic compound SrIrIn₆ was synthesized from the elements in a sealed tantalum ampoule at 1173 K, followed by slow cooling for crystal growth. SrIrIn₆ crystallizes with a new structure type which was characterized by X-ray powder and single crystal diffraction: Pmma, a = 852.34(2), b = 434.54(5), c = 1059.18(6) pm, wR₂ = 0.0178, 884 F² values, and 32 variables. The SrIrIn₆ structure shows two basic building units: (i) Ir@In₉ tricapped trigonal prisms (261–292 pm Ir–In) and (ii) distorted bcc In@In₈ cubes (301 to 329 pm In–In). The strontium cations fill cages within the complex three-dimensional [IrIn₆] network and have coordination number 13 (Sr@In₁₃) in form of a tricapped pentagonal prism. The SrIrIn₆ structure can be described as a simple intergrowth variant of SrIrIn₄ (LaCoAl₄ type) with indium slabs. The crystal chemical similarities with the structures of SrIrIn₄, SrIr₂In₈ and Eu₃Ir₂In₁₅ are discussed.     

Ammonium chloride-promoted four-component synthesis of pyrrolo[3,4-b]pyridin-5-one

A novel multicomponent synthesis of 5-aminooxazole starting from simple and readily available inputs is described. Thus, simply heating a methanol solution of an aldehyde 3, an amine 4, and an alpha-isocyanoacetamide 5 provided the 5-aminooxazole (1) in good to excellent yield. The reaction of 1 with alpha,beta-unsaturated acyl chloride 13 lead to the formation of pyrrolo[3,4-b]pyridin-5-one (2) in a single operation. A triple domino sequence, acylation/IMDA/retro-Michael cycloreversion, is involved in this new scaffold-generating reaction. After the observation that ammonium chloride can significantly accelerate the oxazole formation in toluene, a one-pot four-component synthesis of 2 is developed.     

Cooperative and Non-Cooperative Differential Game Solutions to an Investment and Pricing Problem

A monopoly possesses a finite stock of a resource and wishes to determine an optimal pricing policy. The competitive fringe invests in production capacity and wishes to select an optimal investment rate. Demand towards the monopoly depends on price as well as on the sales rate of the competition. Modelling the situation as a differential game, non-cooperative (Nash and Stackelberg) and cooperative (Pareto) equilibria are determined. Owing to the special structure of the game, these solutions can be found in closed form.     

Das Dimerisierungsgleichgewicht 2 TiCl3,g = Ti2Cl6,g

The Dimerization Equilibrium 2 TiCl_3,g = Ti₂Cl_6,g The dimerization of gaseous titanium(III)chloride has been investigated. The measurement have been made in the presence of an excess of TiCl₄ by means of a static method using gold as a manometer liquid. The results for the equilibrium 2 TiCl_3,g = Ti₂Cl_6,g are ΔH°(298) = ?40.6 kcal; ΔS°(298) = ?36.4 cl; ΔCp = 4 cal/°, mole.     

Stochastic Integrals of Continuous Local Martingales,I

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITǒ integral of X. Using the method of random time change systematically, in the present paper the representation problem for continuous local martingales is treated. We describe a class of martingales Y that can be represented as stochastic integral of X by probabilistic conditions. This leads to sufficient conditions for the representation property of X being true. Besides, an interesting characterization of continuous processes with independent increments is obtained. In part II. we proceed with general examples, applications to the n-dimensional case, and, in particular, to the n-dimensional time change of continuous local martingales with orthogonal components.