Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

Closed Formula for Options with Discrete Dividends and Its Derivatives

Abstract

We present a closed pricing formula for European options under the Black–Scholes model as well as formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and a proposition that relates expectations of partial derivatives with partial derivatives themselves. The closed formulas are attained assuming the dividends are paid in any state of the world. The results are readily extensible to time-dependent volatility models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis, covering calls and puts, together with results on their partial derivatives. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods.     

CsC2H: Synthese,Kristallstruktur und spektroskopische Eigenschaften

CsC₂H: Synthesis, Crystal Structure, and Spectroscopic Properties CsC₂H was synthesised by the reaction of caesium solved in liquid ammonia with acetylene. The crystal structure could be solved and refined from X‐ray and neutron powder diffraction data (space group: R3c, Z = 18). The structure is characterised by C₂H^– trimers which are surrounded by caesium ions. Spectroscopic investigations (IR and Raman) of the stable monoalkalimetal acetylides mainly confirm the data given in the literature and show that the alkalimetal cation has a marked influence on the vibrational properties of the C₂H^– anion.     

Bromo­di­methyl(N‐methyl­pyrrolidin‐2‐one‐O)­tin(IV)‐di‐μ‐bromo‐bromo­di­methyl­tin(IV)

The preparation and X‐ray analysis of the title compound, [Sn₂Br₄(CH₃)₄(C₅H₉NO)], are described. The compound contains two Sn atoms in the asymmetric unit, that complexed by N‐methyl­pyrrolidin‐2‐one being hexacoordinated (a), the other exhibiting pentacoordination (b). The most important features are three different Sn—Br bond lengths at both Sn atoms with the following values: (a) 2.5060 (9), 2.7152 (10) and 3.7118 (10) Å; (b) 2.5084 (10), 2.5279 (9) and 3.5841 (10) Å.     

New open-chain tetrapyrroles as chromophores in the plant photoreceptor phytochrome

A series of six open-chain tetrapyrroles has been synthesized and used as chromophores for the plant photoreceptor protein phytochrome. The novel chromophores vary in the size of substituents 17 and 18 at ring D. This ring undergoes maximal conformational change upon light excitation ( Z --> E photoisomerization of the 15,16-double bond). Instead of methyl and vinyl substituents (positions 17, 18) as present in the native chromophore phytochromobilin, dimethyl, methyl and isopropyl, methyl and tert-butyl, ethyl and methyl, vinyl and methyl, and isopropyl and methyl substituents have been generated. All novel chromophores assemble with the apoprotein. The obtained chromoproteins show hypsochromic shifts of the absorbance maxima by 10 nm maximally, compared to the native pigment, except for the 17-isopropyl-18-methyl-substituted compound which showed a 100 nm hypsochromic shift of selectively the P r form. The assembly kinetics were slowed down in correlation to the increasing size of the substituents, with stronger effects for modified substituents at position 17. The thermal stability of the photoinduced P fr form for the 18-isopropyl and the 18- tert butyl substituents was even greater than that of the native pigments. Those chromophores carrying substituents at position 17 larger than the methyl group (ethyl and isopropyl) showed a very low stability of the respective P fr forms. Time-resolved detection of the P r to P fr conversion (laser-induced flash photolysis) revealed a slower formation of the P fr form for those chromophores carrying larger substituents at position 18, whereas the rise and decay kinetics of the early intermediates are only moderately changed. Introduction of larger substituents at position 17 (ethyl, vinyl, and isopropyl) causes drastic changes in the kinetics; in particular the formation of the first thermally stable intermediate, I 700, is significantly slowed, making a detection of its rise possible.     

Influence of Process Parameters on the Reaction Kinetics of the Chromium‐Catalyzed Trimerization of Ethylene

In this paper we report the results of an extensive experimental kinetic study carried out on the novel ethylene trimerization catalyst system, comprising the chromium source [CrCl₃(thf)₃] (thf=tetrahydrofuran), a Ph₂P‐N(iPr)‐P(Ph)‐N(iPr)H (PNPNH) ligand (Ph=phenyl, iPr=isopropyl), and triethylaluminum (AlEt₃) as activator. It could be shown that the initial activity shows a first‐order dependency on the ethylene concentration. Also, a first‐order dependency was found for the catalyst concentration. The initial activity follows a typical Arrhenius behavior with an experimentally determined activation energy of 52.6 kJ mol^?1. At elevated temperatures (ca. 80 °C), a significant deactivation was observed, which can be tentatively traced back to a ligand rearrangement in the presence of AlEt₃. After a fast initial phase, a pronounced ‘kink’ in the ethylene‐uptake curve is observed, followed by a slow, almost linear, further increase of the total ethylene consumption. The catalyst composition, in particular the ligand/chromium and the cocatalyst/chromium molar ratio, has a strong impact on the catalytic performance of the trimerization of ethylene.