Determination of the spacetime from local time measurements

We consider an inverse problem for a Lorentzian spacetime (M, g), and show that time measurements, that is, the knowledge of the Lorentzian time separation function on a submanifold \(\Sigma \subset M\) determine the \(C^\infty \)-jet of the metric in the Fermi coordinates associated to \(\Sigma \). We use this result to study the global determination of the spacetime (M, g) when it has a real-analytic structure or is stationary and satisfies the Einstein-scalar field equations. In addition to this, we require that (M, g) is geodesically complete modulo scalar curvature singularities. The results are Lorentzian counterparts of extensively studied inverse problems in Riemannian geometry—the determination of the jet of the metric and the boundary rigidity problem. We give also counterexamples in cases when the assumptions are not valid, and discuss inverse problems in general relativity.     

Pathwise Integrals and Itô–Tanaka Formula for Gaussian Processes

We prove an Itô–Tanaka formula and existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue–Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of the Itô–Tanaka formula for pricing and hedging of financial derivatives.     

Determination of aluminium in AlCl3 and Al2O3 modified silica catalyst supports

X-ray fluorescence spectrometry (XRF) was applied to determine aluminium in AlCl₃- and Al₂O₃-modified silica catalyst supports that were prepared by gas-solid reactions in an atomic layer epitaxy (ALE) process using aluminium chloride or aluminium chloride and water as adsorbates and silica as support. INAA and AAS were used as reference methods to determine the aluminium content of the supports. The calibration of XRF results was done by plotting the Al/Si intensity ratios against the aluminium content as determined by atomic absorption spectrometry (AAS) and verified by instrumental neutron activation analysis (INAA). Correlation factors for the calibration graphs were 0.984 for AlCl₃/SiO₂ and 0.995 for Al₂O₃/ SiO₂ samples in the aluminium content range 0–2.6 g Al per 100 g of sample. Received: 19 October 1998 / Accepted: 14 December 1998     

Comparison between the classical theory predictions and molecular simulation results for heterogeneous nucleation of argon

We have performed Monte Carlo simulations of homogeneous and heterogeneous nucleations of Lennard-Jones argon clusters. The simulation results were interpreted using the major concept posing a difference between the homogeneous and heterogeneous classical nucleation theories-the contact parameter. Our results show that the multiplication concept of the classical heterogeneous nucleation theory describes the cluster-substrate interaction surprisingly well even for small molecular clusters. However, in the case of argon nucleating on a rigid monolayer of fcc(111) substrate at T=60 K, the argon-substrate atom interaction being approximately one-third as strong as the argon-argon interaction, the use of the classical theory concept results in an underestimation of the heterogeneous nucleation rate by two to three orders of magnitude even for large clusters. The main contribution to this discrepancy is induced by the failure of the classical theory of homogeneous nucleation to predict the energy involved in bringing one molecule from the vapor to the cluster for clusters containing less than approximately 15 molecules.     

Abnormal inversion splitting in NH2D: rotational analysis of the nu5 bending vibrational band system

A high-resolution Fourier transform infrared spectrum of the nu(5) bending vibrational band system region of the partially deuterated ammonia molecule NH(2)D has been measured and rotationally analyzed. The spectrum consists of strong a-type transitions between the states of same vibrational symmetry and weaker c-type transitions between the states of different vibrational symmetry. The Hamiltonian model used includes interaction terms between the rotational states of both upper and lower inversion doublets. The vibrational term values for the symmetric and the antisymmetric component of the upper-inversion doublet are 1,605.637 965(620) cm(-1) and 1,590.993 82(100) cm(-1), respectively, where the numbers in parentheses are one-standard deviations in the least significant digit. These figures are close to the corresponding values 1,605.62 cm(-1) and 1,590.72 cm(-1) obtained recently from results based on high-level ab initio calculations. The order of the vibrational term values is abnormal in the ammonia family, as typically the symmetric state is lower in wavenumber than the antisymmetric one.     

Substituent effects on the basicity of 3,7-diazabicyclo[3.3.1]nonanes

Basicity constants for a series of 3,7-diazabicyclo[3.3.1]nonane derivatives in acetonitrile with a variation over 13 orders of magnitude have been determined using a spectrophotometric titration technique. An excellent correlation between basicity and calculated proton affinities obtained at PCM-B3LYP/6-31+G(d)//B3LYP/6-31G(d) level was found. The results are discussed in terms of substituent effects and compared to (15)N NMR chemical shifts.     

A nine-dimensional calculation of the vibrational OH stretching and HOH bending spectrum of the water trimer

We have studied the vibrational high-frequency spectrum of the water trimer computationally. We expand an earlier study [J. Chem. Phys. A 2009, 113, 9124-9132] where we approximated the water trimer as three individually vibrating water monomer units. Some intramolecular potential energy coupling terms are now included in the previous model. The six OH bond lengths and the three HOH bending angles are used as the internal coordinates. The kinetic energy operator is a sum of the kinetic energy operators of the monomer units. We use the coupled cluster method with single, double, and perturbative triple excitations method [CCSD(T)] with augmented correlation consistent polarized valence triple-ζ (aug-cc-pVTZ) basis set to calculate the potential energy surface (PES). The counterpoise correction is included in the one-dimensional part of the PES. We calculate the vibrational energy eigenvalues using the variational method. The corresponding eigenfunctions are used to obtain the absorption intensities.     

Ab initio structural and vibrational investigation of sulfuric acid monohydrate

We employ ab initio methods to find stable geometries and to calculate potential energy surfaces and vibrational wavenumbers for sulfuric acid monohydrate. Geometry optimizations are carried out with the explicitly correlated coupled-cluster approach that includes single, double, and perturbative triple excitations (CCSD(T)-F12a) with a valence double-ζ basis set (VDZ-F12). Four different stable geometries are found, and the two lowest are within 0.41 kJ mol(-1) (or 34 cm(-1)) of each other. Vibrational harmonic wavenumbers are calculated at both the density-fitted local spin component scaled second-order M?ller-Plesset perturbation theory (DF-SCS-LMP2) with the aug-cc-pV(T+d)Z basis set and the CCSD-F12/VDZ-F12 level. Water O-H stretching vibrations and two highly anharmonic large-amplitude motions connecting the three lowest potential energy minima are considered by limiting the dimensionality of the corresponding potential energy surfaces to small two- or three-dimensional subspaces that contain only strongly coupled vibrational degrees of freedom. In these anharmonic domains, the vibrational problem is solved variationally using potential energy surfaces calculated at the CCSD(T)-F12a/VDZ-F12 level.     

Kinetics of the reactions of chlorinated methyl radicals (CH2Cl, CHCl2, and CCl3) with NO2 in the temperature range 220-360 K

The kinetics of the reactions of chlorinated methyl radicals (CH2Cl, CHCl2, and CCl3) with NO2 have been studied in direct measurements at temperatures between 220 and 360 K using a tubular flow reactor coupled to a photoionization mass spectrometer. The radicals have been homogeneously generated at 193 or 248 nm by pulsed laser photolysis of appropriate precursors. Decays of radical concentrations have been monitored in time-resolved measurements to obtain the reaction rate coefficients under pseudo-first-order conditions with the amount of NO2 being in large excess over radical concentrations. The bimolecular rate coefficients of all three reactions are independent of the bath gas (He or N2) and pressure within the experimental range (1-6 Torr) and are found to depend on temperature as follows: k(CH2Cl + NO2) = (2.16 +/- 0.08) x 10(-11) (T/300 K)(-1.12+/-0.24) cm3 molecule(-1) s(-1) (220-363 K), k(CHCl2 + NO2) = (8.90 +/- 0.16) x 10(-12) (T/300 K)(-1.48+/-0.13) cm3 molecule(-1) s(-1) (220-363 K), and k(CCl3 + NO2) = (3.35 +/- 0.10) x 10(-12) (T/300 K)(-2.2+/-0.4) cm3 molecule(-1) s(-1) (298-363 K), with the uncertainties given as one-standard deviations. Estimated overall uncertainties in the measured bimolecular reaction rate coefficients are about +/-25%. In the reactions CH2Cl + NO2, CHCl2 + NO2, and CCl3 + NO2, the products observed are formaldehyde, CHClO, and phosgene (CCl2O), respectively. In addition, a weak signal for the HCl formation has been detected for the CHCl2 + NO2 reaction.