Perturbed Markov Chains with Damping Component

The paper is devoted to studies of regularly and singularly perturbed Markov chains with damping component. In such models, a matrix of transition probabilities is regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter ε. We perform a detailed perturbation analysis for such Markov chains, particularly, give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, explicit coupling type upper bounds for the rate of convergence in ergodic theorems for n-step transition probabilities, as well as ergodic theorems in triangular array mode.

     

On a bifurcation governed by hysteresis nonlinearity

We consider autonomous systems with a nonlinear part depending on a parameter and study Hopf bifurcations at infinity. The nonlinear part consists of the nonlinear functional term and the Prandtl--Ishlinskii hysteresis term. The linear part of the system has a special form such that the close-loop system can be considered as a hysteresis perturbation of a quasilinear Hamiltonian system. The Hamiltonian system has a continuum of arbitrarily large cycles for each value of the parameter. We present sufficient conditions for the existence of bifurcation points for the non-Hamiltonian system with hysteresis. These bifurcation points are determined by simple characteristics of the hysteresis nonlinearity.     

Relation of orbital integrals on SO(5) and PGL(2)

We relate the “Fourier” orbital integrals of corresponding spherical functions on thep-adic groups SO(5) and PGL(2). The correspondence is defined by a “lifting” of representations of these groups. This is a local “fundamental lemma” needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.     

Metric tangent spaces to Euclidean spaces

We prove that the metric spaces pretangent to a finite-dimensional Euclidean or unitary space E are isometric to E. As a consequence of this result, we describe the metric pretangent spaces at the nonsingular points of smooth surfaces. It is also proved that there exist the spaces pretangent to the Hilbert space l ₂ , which are not isometric to it.     

Extended generalized octagons and the groupHe

Let be an extended generalized octagon such that the points of a triple {u, v, w} not on a block are pairwise adjacent if and only if the distance betweenv andw in the local generalized octagon _u equals 3 and there is a thick line through any point of _u . Then is one of the two examples related to the groups 2·L ₃(4).2² andHe. It is also shown that does not admit further extensions.     

Correction of systematic errors in the fabrication of fiber Bragg gratings

We suggest a novel method for significant improvement of fiber Bragg grating (FBG) spectral characteristics based on compensation of repeatable errors during the grating writing process. The method is based on an open loop scheme in which a test grating is written first and then a corrected design is obtained and implemented by writing another (compensated) grating. The results are presented for a FBG dispersion compensator.     

Modification of methylaluminoxane-activated ansa-zirconocene catalysts with triisobutylaluminum-transformations of reactive cations studied by NMR spectroscopy

When triisobutylaluminum (AliBu(3)) is added to solutions containing methylaluminoxane (MAO) and rac-[Me(2)Si(ind)(2)ZrCl(2)] (ind: indenyl) in C(6)D(6), NMR spectra show that methyl-bridged mixed-alkylaluminum dimers Al(mu-Me)(2)Me(4-x)iBu(x) predominate. These dimers react with MAO under partial transfer of isobutyl groups and induce a conversion of the initially prevailing cationic trimethylaluminum adduct rac-[Me(2)Si(ind)(2)Zr(mu-Me)(2)AlMe(2) (+)] to rac-[Me(2)Si(ind)(2)Zr(mu-Me)(2)AlMeiBu(+)] and rac-[Me(2)Si(ind)(2)Zr(mu-Me)(2)AliBu(2) (+)]. These species are unstable and release isobutene under formation of zirconocene hydrides.     

Activation of dimethyl zirconocene by methylaluminoxane (MAO)-size estimate for Me-MAO(-) anions by pulsed field-gradient NMR

In a study of the reaction system MAO/(C5H5)2ZrMe2, the size of the ion pair [(C5H5)2Zr(mu-Me)2AlMe2]+ [Me-MAO]- was determined by pulsed field-gradient NMR of its cationic moiety. A mean effective hydrodynamic radius of 12.2-12.5 A, determined from diffusion rates in benzene solution at different zirconocene and MAO concentrations, indicates that the ion pair remains associated even at the lowest concentrations studied. At elevated concentrations, aggregation to ion quadruples or higher aggregates is indicated by an apparent size increase and by shifts of the C5H5 and Me 1H NMR signals. The equilibrium constant for the reaction [(C5H5)2ZrMe+...Me-MAO-] + 1/2Al2Me6 right harpoon over left harpoon [(C5H5)2Zr(mu-Me)2AlMe2]+ [Me-MAO]- changes at different Al/Zr ratios; this indicates that MAO contains various species that produce Me-MAO- anions with different Lewis basicities. The volume of the Me-MAO- anion suggests that it contains 150-200 Al atoms.