Proof of a conjecture on irredundance perfect graphs

Let ir(G) and γ(G) be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H)=γ(H), for every induced subgraph H of G. In this article we present a result which immediately implies three known conjectures on irredundance perfect graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 292–306, 2002     

Geometrical Molecular Structure Design Concept in Approach to Homo- and Heterometallic Precursors of Advanced Materials in Sol–Gel Technology

The article describes the principles of the Single Source Precursor approach to inorganic materials and introduces the Geometrical Molecular Structure Design Concept (MSDC) based on the choice of a proper molecular structure type for the desired precursor and completing it with ligands providing both the necessary number of donor atoms and the sterical protection of the chosen core. Application of MSDC is illustrated with examples taken from development of new approaches in the synthesis of oxide and sulfide catalysts and ferroelectric oxide materials.     

Theory of A ϕ-periodic one-dimensional bundle of dielectric waveguides

Mogil' Division, Institute of Physics, Belorussian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 35, No. 2, pp. 200–204, February, 1992.     

The Poisson boundary of covering Markov operators

Covering Markov operators are a measure theoretical generalization of both random walks on groups and the Brownian motion on covering manifolds. In this general setup we obtain several results on ergodic properties of their Poisson boundaries, in particular, that the Poisson boundary is always infinite if the deck group is non-amenable, and that the deck group action on the Poisson boundary is amenable. For corecurrent operators we show that the Radon-Nikodym cocycles of two quotients of the Poisson boundary are cohomologous iff these quotients coincide. It implies that the Poisson boundary is either purely non-atomic or trivial, and that the action of any normal subgroup of the deck group on the Poisson boundary is conservative. We show that the Poisson boundary is trivial for any corecurrent covering operator with a nilpotent (or, more generally, hypercentral) deck group. Other applications and examples are discussed. Supported by a British SERC Advanced Fellowship. A part of this work was done during my stay at MSRI, Berkeley supported by NSF Grant DMS 8505550.     

Equivariant quantization on quotients of simple Lie groups by reductive subgroups of maximal rank

We consider a class of homogeneous manifolds including all semisimple coadjoint orbits. We describe manifolds of that class admitting deformation quantizations equivariant under the action ofG and the corresponding quantum group. We also classify Poisson brackets relating to such quantizations. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.