On the genus of the semidirect product of ℤ9 by ℤ3

We show that the non-commutative semidirect product Γ of ?₉ by ?₃ has orientable genus 4. In other words, some Cayley graph of Γ embeds in an orientable surface of genus 4 (Euler characteristic ?6), but no Cayley graph of Γ embeds in an orientable surface of genus less than 4 (Euler characteristic greater than ?6). We also show that some Cayley graph of Γ embeds in a (non-orientable) surface of Euler characteristic ?3, but no Cayley graph of Γ embeds in a surface of Euler characteristic greater than ?3. Γ is the first known example of a group whose orientable Euler characteristic and non-orientable Euler characteristic differ by more than 1. Our results also complete the determination of the orientable genus of each group of order less than 32.     

Simulation of the operation of a laminar-flow diffusion chamber to study homogeneous nucleation: Part I

A new mathematical model of the operation of a laminar-flow diffusion chamber that includes the homogeneous nucleation (condensation) and growth of new-phase particles (droplets) is studied qualitatively. It is shown that numerical simulation must correctly take into account the concentration and temperature dependences of transport coefficients in a vapor-gas mixture. The domains of applicability of the previous mathematical models of a laminar-flow diffusion chamber are found.     

Negative temperature coefficient phenomenon in exchange reactions between alkene oxides and alkyl-β-hydroxyalkyl sulfides

Conclusions The temperature dependence of the initial rate of the exchange reaction in the propylene oxide-octyl--hydroxyethyl sulfide system shows a maximum and a minimum between 60 and 110C, the maximum at 90C. This behavior may be explained by the participation in the reaction of an associate that is formed from a molecule of propylene oxide and two molecules of hydroxyethyl sulfide. Computer calculations have been made for this scheme of the reaction over a wide temperature range.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 1, pp. 51–57, January, 1985.     

The topology of group extensions of C systems

The paper is concerned with the topological and metric properties of group extensions of C systems. The basic theorem describes the topologically transitive component, the ergodic component, and the K component of a group extension of a C system. It is shown that each of these components is a group sub-bundle of a principal bundle in which the group extension acts. The frame flow on a manifold of negative curvature is seen to be a special case of a group of extension of a C system. It is shown that the space of frames on a compact three-dimensional manifold with negative curvature does not have any group sub-bundles, so that the frame flow on manifolds of this class is topologically transitive, ergodic, and a K system.     

Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes

A Euclidean complex X is a simplicial complex whose simplices are (flat) Euclidean simplices. We construct a natural Brownian motion on X and show that if X has nonpositive curvature and satisfies Gromov's hyperbolicity condition, then, with probability one, Brownian motion tends to a random limit on the Gromov boundary. Applying a combination of geometric and probabilistic techniques we describe spaces of harmonic functions on X. Received November 18, 1999; in final form January 18, 2000 / Published online April 12, 2001     

On the spectrum of anosov diffeomorphisms

Any Anosov diffeomorphismf induces a linear operator in the Banach space of continuous vector fields, which spectrum is contained in two annuli both being disjoint with the unit circle. The main theorem states that, if these annuli are sufficiently “narrow”, thenf has a fixed point. It is also proved that, iff acts on an infranilmanifold (e.g., on a torus), then the spectrum annuli for the algebraic automorphism corresponding tof are always more “narrow” than forf.