COGARCH as a continuous-time limit of GARCH(1,1)

COGARCH is an extension of the GARCH time series concept to continuous time, which has been suggested by Klüppelberg, Lindner and Maller [C. Klüppelberg, A. Lindner, R. Maller, A continuous-time GARCH process driven by a Lévy process: Stationarity and second order behaviour, Journal of Applied Probability 41 (2004) 601–622]. We show that any COGARCH process can be represented as the limit in law of a sequence of GARCH(1,1) processes. As a by-product we derive the infinitesimal generator of the bivariate Markov process representation of COGARCH. Moreover, we argue heuristically that COGARCH and the classical bivariate diffusion limit of Nelson [D. Nelson, ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38] are probably the only continuous-time limits of GARCH.     

Set-valued processes induced by parameterised stochastic differential equations with an application to Tuned Mass Dampers

In this paper ordinary stochastic differential equations whose coefficients depend on uncertain parameters are considered. An approach is presented how to combine both types of uncertainty (stochastic excitation and parameter uncertainty) leading to set-valued stochastic processes. The latter serve as a robust representation of solutions of the underlying stochastic differential equations. The mathematical concept is applied to a problem from earthquake engineering, where it is shown how the efficiency of Tuned Mass Dampers can be realistically assessed in the presence of uncertainty. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trichloroberyllat‐Komplexe von Dimethylcyanamid,Morpholin und 4,4′‐Bipyridin

Trichloroberyllate Complexes of Dimethyl Cyanamide, Morpholine, and 4,4′‐Bipyridine The trichloroberyllate complexes (Ph₄P)[BeCl₃(NCNMe₂)] ( 1 ), (Ph₄P)[BeCl₃{HN(CH₂)₄O}] ( 2 ), and (Ph₄P)₂[(BeCl₃)₂(4,4′‐bipy)] ( 3 ) were prepared by reactions of (Ph₄P)₂[Be₂Cl₆] with dimethyl cyanamide, trimethylsilylmorpholinate, and 4,4′‐bipyridine, respectively, in dichloromethane solutions. 1 ‐ 3 were characterized by X‐ray crystallography and by IR‐spectroscopy. 1 ·CH₂Cl₂: Space group P1, Z = 1, lattice dimensions at 173 K: a = 714.2(1), b = 919.5(2), c = 1233.4(2) pm, α = 94.97(1)°, β = 90.86(1)°, γ = 111.90(1)°, R₁ = 0.0310. In the complex anion [BeCl₃(NCNMe₂)]^? the dimethyl cyanamide ligand is coordinated via a linear Be–N≡C‐NMe₂ arrangement, the CH₂Cl₂ molecules forming linear chains by hydrogen bridges ···HCH···Cl··· with the chlorine atoms of the {BeCl₃^?} groups. 2 : Space group , Z = 2, lattice dimensions at 173 K: a = 1050.9(1), b = 1099.7(1), c = 1308.3(2) pm, α = 87.57(1)°, β = 70.97(1)°, γ = 74.58(1)°, R₁ = 0.0397. The complex anions are dimerized by centrosymmetric puckered eight‐membered [Be–N–H···Cl]₂ rings. 3 ·2CH₂Cl₂: Space group , Z = 2, lattice dimensions at 173 K: a = 1095.4(1), b = 1559.6(2), c = 1869.8(3) pm, α = 79.12(1)°, β = 73.83(1)°, γ = 78.76(1)°, R₁ = 0.0548. The complex contains dianions [Cl₃Be(μ‐bipy)BeCl₃]^2? with Be–N distances of 177.0(6) and 178.5(6) pm. Both {BeCl₃}^? groups form C–H···Cl hydrogen bridges with the dichloromethane molecules.     

Photoinduced bimolecular electron transfer investigated by femtosecond time-resolved infrared spectroscopy

Ultrafast infrared transient absorption spectroscopy is used to study the photoinduced bimolecular electron transfer reaction between perylene in the first singlet excited state and 1,4-dicyanobenzene in acetonitrile and dichloromethane. Following vibrational marker modes on both donor and acceptor sides in real time provides direct insight into the structural dynamics during the reaction. A band narrowing on a time scale of a few tens of picoseconds observed on the antisymmetric CN stretching vibration of the dicyanobenzene radical anion indicates that a substantial part of the excess energy is channeled into vibrational modes of the product, despite the fact that the reaction is weakly exergonic. An additional narrowing of the same band on a time scale of several hundreds of picoseconds observed in acetonitrile only is interpreted as a signature of the dissociation of the geminate ion pairs into free ions.     

Barbituric acid as a substituent at aryl methylium ions

Activation of different benzophenone derivatives with triflic anhydride for electrophilic aromatic substitution of 5-phenylbarbituric acids leads to regioselective formation of the ortho-substituted product. The resulting triphenylmethylium salt can be isolated when the Michlers ketone is used. More electrophilic cations form cyclic enol ethers such as 1-n-butyl-9,9-diaryl-1,9-dihydro-10-oxa-1,3-diazaphenanthrene-2,4-diones. Alternatively, supramolecular complex formation with 2,6-diacetamido pyridine as well as carbenium ion generation have been studied. Although in dilute acid only protonation of one of the carbonyl oxygens occurs, ring opening of the cyclic enol ether toward the carbenium ion is observed in 96% sulfuric acid.     

Vitrification of polymer solutions as a function of solvent quality, analyzed via vapor pressures

Vapor pressures (headspace sampling in combination with gas chromatography) and glass transition temperatures [differential scanning calorimetry (DSC)] have been measured for solutions of polystyrene (PS) in either toluene (TL) (10-70 degrees C) or cyclohexane (CH) (32-60 degrees C) from moderately concentrated solutions up to the pure polymer. As long as the mixtures are liquid, the vapor pressure of TL (good solvent) is considerably lower than that of CH (theta solvent) under other identical conditions. These differences vanish upon the vitrification of the solutions. For TL the isothermal liquid-solid transition induced by an increase of polymer concentration takes place within a finite composition interval at constant vapor pressure; with CH this phenomenon is either absent or too insignificant to be detected. For PS solutions in TL the DSC traces look as usual, whereas these curves may become bimodal for solutions in CH. The implications of the vitrification of the polymer solutions for the determination of Flory-Huggins interaction parameters from vapor pressure data are discussed. A comparison of the results for TL/PS with recently published data on the same system demonstrates that the experimental method employed for the determination of vapor pressures plays an important role at high polymer concentrations and low temperatures.     

Pyrazine as a building block for molecular architectures with PtII

A series of pyrazine (pz) complexes containing cis-(NH(3))(2)Pt(II), (tmeda)Pt(II) (tmeda = N,N,N',N'-tetramethylethylenediamine), and trans-(NH(3))(2)Pt(II) entities have been prepared and characterized by X-ray crystallography and/or 1H NMR spectroscopy. In these compounds, the pz ligands act as monodentate (1-3) or bidentate bridging ligands (4-7). Three variants of the latter case are described: a dinuclear complex [Pt(II)]2 (4b), a cyclic tetranuclear [Pt(II)](4) complex (5), and a trinuclear mixed-metal complex [Pt2Ag] (7). Mono- and bidentate binding modes are readily differentiated by 1H NMR spectroscopy, and the assignment of pz protons in the case of monodentate coordination is aided by the observation of (195)Pt satellites. Formation of the open molecular box cis-[{(NH3)2Pt(pz)}4](NO3)8.3.67H2O (5) from cis-(NH3)2Pt(II) and pz follows expectations of the "molecular library approach" for the generation of a cyclic tetramer.     

Experimental and theoretical study of the coordination of 1,2,4-triazolato, tetrazolato, and pentazolato ligands to the [K(18-crown-6)]+ fragment

Treatment of 3,5-diisopropyltriazole, 3,5-diphenyltriazole, 3,5-di-3-pyridyltriazole, phenyltetrazole, pyrrolidinyltetrazole, or tert-butyltetrazole with equimolar quantities of potassium hydride and 18-crown-6 in tetrahydrofuran at ambient temperature led to slow hydrogen evolution and formation of (3,5-diisopropyl-1,2,4-triazolato)(18-crown-6)potassium (88%), (3,5-diphenyl-1,2,4-triazolato)(tetrahydrofuran)(18-crown-6)potassium (87%), (3,5-di-3-pyridyl-1,2,4-triazolato)(18-crown-6)potassium (81%), (phenyltetrazolato)(18-crown-6)potassium (94%), (pyrrolidinyltetrazolato)(18-crown-6)potassium (90%), and (tert-butyltetrazolato)(18-crown-6)potassium (94%) as colorless crystalline solids. (1,2,4-Triazolato)(18-crown-6)potassium was isolated as a hemi-hydrate in 81% yield upon treatment of 1,2,4-triazole with potassium metal in tetrahydrofuran. The X-ray crystal structures of these new complexes were determined, and the solid-state structures consist of the nitrogen heterocycles bonded to the (18-crown-6)potassium cationic fragments with eta2-bonding interactions. In addition, (3,5-diphenyl-1,2,4-triazolato)(tetrahydrofuran)(18-crown-6)potassium has one coordinated tetrahydrofuran ligand on the same face as the 3,5-diphenyl-1,2,4-triazolato ligand, while (3,5-di-3-pyridyl-1,2,4-triazolato)(18-crown-6)potassium forms a polymeric solid through coordination of the distal 3-pyridyl nitrogen atoms to the potassium ion on the face opposite the 1,2,4-triazolato ligand. The solid-state structures of the new complexes show variable asymmetry in the potassium-nitrogen distances within the eta2-interactions and also show variable bending of the heterocyclic C2N3 and CN4 cores toward the best plane of the 18-crown-6 ligand oxygen atoms. Molecular orbital and natural bond order calculations were carried out at the B3LYP/6-311G(d,p) level of theory on the model complex, (phenyltetrazolato)(18-crown-6)potassium, and demonstrate that the asymmetric potassium-nitrogen distances and bending of the CN4 core toward the 18-crown-6 ligand are due to hydrogen bond-like interactions between filled nitrogen-based orbitals and carbon-hydrogen sigma orbitals on the 18-crown-6 ligands. Calculations carried out on the model pentazolato complex (pentazolato)(18-crown-6)potassium predict a structure in which the pentazolato ligand N5 core is bent by 45 degrees toward the best plane of the 18-crown-6 oxygen atoms. Such bending is induced by the formation of intramolecular nitrogen-hydrogen-carbon hydrogen bonds. Examination of the solid-state structures of the new complexes reveals many intramolecular and intermolecular nitrogen-hydrogen distances of < or =3.0 A which support the presence of nitrogen-hydrogen-carbon hydrogen bonds.     

Nodes of large degree in random trees and forests

We study the asymptotic behavior of the number N_k,n of nodes of given degree k in unlabeled random trees, when the tree size n and the node degree k both tend to infinity. It is shown that N_k,n is asymptotically normal if and asymptotically Poisson distributed if . If , then the distribution degenerates. The same holds for rooted, unlabeled trees and forests. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006