Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.     

Dynamics of energy transfer in collisions of O(3P) atoms with a 1-decanethiol self-assembled monolayer surface

Chemical dynamics simulations are reported of energy transfer in collisions of O(3P) atoms with a 300 K 1-decanethiol self-assembled monolayer (H-SAM) surface. The simulations are performed with a nonreactive potential energy surface, developed from PMP2/aug-cc-pVTZ calculations of the O(3P) + H-SAM intermolecular potential, and the simulation results represent the energy transfer dynamics in the absence of O(3P) reaction. Collisions energies E(i) of 0.12, 2.30, 11.2, 75.0, and 120.5 kcal/mol and incident angles theta(i) of 15, 30, 45, 60, and 75 degrees were considered in the study (theta(i) = 0 degrees is the surface normal). The translational energy distribution of the scattered O(3P) atoms, P(E(f)), may be deconvoluted into Boltzmann and non-Boltzmann components, with the former fraction identified as f(B). The trajectories are also analyzed in terms of three types; that is, direct scattering from and physisorption on the top of the H-SAM and penetration of the H-SAM. There are three energy regimes in the scattering dynamics. For the low E(i) values of 0.12 and 2.30 kcal/mol, physisorption is important and both f(B) and the average final translational energy of the scattered O(3P) atom, E(f), are nearly independent of the incident angle. The dynamics is much different for hyperthermal energies of 75.0 and 120.5 kcal/mol, where penetration of the surface is important. For hyperthermal collisions, the penetration probability decreases as theta(i) is increased, with a significant transition between theta(i) of 60 and 75 degrees . Hyperthermal penetration occurs upon initial surface impact and is more probable if the impinging O(3P) atom may move down a channel between the chains. For E(i) = 120.5 kcal/mol, 90% of the trajectories penetrate at theta(i) = 15 degrees , while only 3% penetrate at theta(i) = 75 degrees. For the former theta(i), the energy transfer to the surface is efficient with E(f) = 4.04 kcal/mol, but for the latter theta(i), E(f) = 85.3 kcal/mol! Particularly interesting penetrating trajectories are those in which O(3P) is trapped in the H-SAM for times exceeding 60 ps, linger near the Au substrate, and strike the Au substrate and scatter directly. For E(i) = 11.2 kcal/mol, there is a transition between the scattering dynamics for the low and hyperthermal collision energies. Additional detail in the energy transfer dynamics is obtained from the final polar and azimuthal angles, the residence time on/in the H-SAM, the minimum height with respect to the Au substrate, and the number of inner turning points in the O-atom's velocity. Calculated values of E(f) vs the final polar angle, theta(f), are in qualitative agreement with experiment. The O(3P) + H-SAM nonreactive energy transfer dynamics, for E(i) of 11.2 kcal/mol and lower, are very similar to previously reported Ne + H-SAM simulations.     

Mixing cell for indirect measurements of excess volume

A mixing cell suitable for making liquid mixtures of accurately known composition is described and has been tested by measurements of the excess volume of benzene + cyclohexane.     

Test of a chemical timing method for measuring absolute vibrational relaxation rate constants for S1 p-difluorobenzene

A chemical timing (CT) method for measuring absolute rate constants for collisional vibrational relaxation has been tested for the 5(1) state of S(1) p-difluorobenzene (pDFB) where an alternative method exists to provide benchmark values. The CT method was originally developed to treat vibrational energy transfer (VET) in large molecules excited to high vibrational levels where the intramolecular vibrational redistribution (IVR) resulting from large vibrational state densities completely eliminates vibrational structure in the emission spectrum. Here we apply the same method to a low-lying state (5(1) with epsilon(vib) = 818 cm(-1)) located in the low-density region of the vibrational manifold where IVR plays no role. For high vibrational levels, the chemical timing method involves addition of high O(2) pressures (kTorr) to a low-pressure pDFB sample, introducing vibrational structure in the fluorescence spectrum. Response of this spectrum to vibrational relaxation by Ar is then examined. For levels such as 5(1), the fully structured fluorescence spectrum allows the rate constant for single-collision VET into the surrounding vibrational field to be measured directly without the presence of O(2). The measurements of 5(1) VET have been repeated with various O(2) pressures (kTorr) for comparison with the O(2)-free benchmark. In the presence of O(2), the rate constant for VET by Ar is (4.0 +/- 0.5) x 10(6) Torr(-1) s(-1) and independent of high O(2) pressure variations. The rate constant as found by the standard O(2)-free method is (3.6 +/- 0.4) x 10(6) Torr(-1) s(-1). This comparison suggests that the chemical timing method is capable of providing a reasonably accurate measure of the VET rate constant for high vibrational levels provided that details of the kinetics are known.     

Shock waves in diatomic chains—I. linear analysis

Solutions are presented for the response of semi-infinite diatomic chains to a step jump in velocity. Integral transform theory and contour integration are used to express the solutions as definite integrals for the acoustic and optical branch contributions to the response. The contribution of an end mode is indicated for the case of particles that are unequally spaced within a unit cell. Asymptotic approximations are obtained for the contribution of the optical and acoustic branches to the wave solution when the wave propagates far into the lattice. Asymptotic estimates are obtained, also, for the discontinuous speeds at which the head of the pulse travels in a general diatomic chain. Shock profiles for the special case of a simple diatomic chain and for a general diatomic chain are discussed, and upper bounds are obtained for the maximum shock responses that are possible in such chains with the given shock condition.     

Application of the “helium excitation standard” procedure to some hydrocarbon Hβ absolute emission cross section measurements

Absolute emission cross sections for H_β radiation are presented. The H_β emission is obtained by dissociative electron excitation of propane, propylene, n-butane and 1-butene under binary collision conditions. Optical excitation functions measured for these transitions are normalized by the “helium excitation standard” procedure, using the 4 ¹S-2 ¹P transition in He as a standard at an electron impact energy of 100 eV. The results obtained are determined with an accuracy of ± 15%. The investigated energy interval is 50–500 eV. The experimental results show that the emission cross sections for H_β radiation are not independent of the number of atoms in the parent molecule.     

The reaction of alkoxides with dicobalt octacarbonyl: Trapping of the Co(I) intermediate in the disproportionation (“base reaction”) with a hardLewis base

Dicobalt octacarbonyl reacts with alcoholates (RO^–) yielding alkoxycarbonylcobalt tetracarbonyls,ROC(O)Co(CO)₄.

---

Reaktion von Alkoxiden mit Dicobaltoctacarbonyl: Abfangen des Co(I) Zwischenproduktes in der Disproportionierung (Basen-Reaktion) mit einer hartenLewis Base (Kurze Mitteilung)

Zusammenfassung Dicobaltoctacarbonyl reagiert mit Alkoxiden (RO^–) zu Alkoxycarbonylcobalt tetracarbonylen,ROC(O)Co(CO)₄.

     

On the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P

Let I, H, S, P be the usual class operators on universal algebras. For a class K of universal algebras of the same type, let R({K}) be the class of all algebras isomorphic to a retract of a member of K and let R denote the corresponding class operator. In this paper the semigroup generated by class operators I, R, H, S, P and the corresponding partially ordered set are described. Also the standard semigroups of the above operators are determined for some varieties.