Solutions of systems of nonlinear functional-differential equations bounded in the entire real axis and their properties

For a system of nonlinear functional-differential equations with a linearly transformed argument, we establish the existence and uniqueness conditions for a solution bounded in the entire real axis and study the properties of this solution. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 737–747, June, 1994.     

Representations of relations of the form i[A,B]=f(A)+G(B)

It is proved that all nontrivial representations of quadratic relation i[A, B]=f(A)+g(B) with self-adjoint operators A, B are unbounded if f and g are nonnegative; for any f and g this relation does not have nontrivial finite-dimensional representations and factor-representations of type II₁, but can have infinite-dimensional irreducible representations with bounded operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 110–114, January, 1991.     

Averaging of Boundary-Value Problems with Parameters for Multifrequency Impulsive Systems

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillating systems with pulse influence at fixed times. We also obtain estimates for the deviation of solutions of the averaged problem from solutions of the original problem.     

The Poincaré-Mel'nikov geometric analysis of the transversal splitting of manifolds of slowly perturbed nonlinear dynamical systems. I

On the basis of the geometric ideas of Poincaré and Mel'nikov, we study sufficient criteria of the transversal splitting of heteroclinic separatrix manifolds of slowly perturbed nonlinear dynamical systems with a small parameter. An example of adiabatic invariance breakdown is considered for a system on a plane.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1668–1681, December, 1993.     

Error Estimates for the Averaging Method for Pulse Oscillation Systems

We prove new theorems on the justification of the averaging method on a segment and semiaxis in multifrequency oscillation systems with pulse action at fixed times.     

Reducibility of nonlinear almost periodic systems of difference equations on an infinite-dimensional torus

We establish sufficient conditions for the reducibility of a nonlinear system of difference equationsx(t+1)=x(t)++P(x(t),t+ to a system y(t+1)= y(t)+, wherex, , m and the infinite-dimensional vector function P(x(t),t) is 2p-periodic inx _i i=1,2,...) and almost periodic int with a frequency basis.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1216–1223, September, 1994.     

Hole burning in the absorption spectrum of chlorin in polymer films: stark effect and temperature dependence

Optical hole-burning experiments at 2 K with chlorin in polymethylmethacrylate (PMMA), polystyrene (PST) and polyvinylbutanal (PVB) gave hole widths of 1.2, 1.6 and 0.75 GHz. Preliminary temperature-dependent measurements in PVB suggest that the linewidth at 0 K approximates the lifetime-limited value. Due to the Stark effect on the randomly distributed molecules, a significant reversible broadening was observed when an electric field of 6000 V/cm was applied to the sample.     

Hetarylation of cyclopentadiene,indene, and azulene

The direct hetarylation of cyclopentadiene, indene, and azulene with several benzopyridines in the presence of acyl halides was studied. It is shown that the most electrophilic N-benzoyl-isoquinolinium salt is capable of hetarylating CH acids with pK_a<21.     

Effect of the nature of the active component and support on the activity of catalysts for the hydrolysis of sodium borohydride

The effect of the nature of an active component and a support on the rate of hydrolysis of aqueous sodium borohydride solutions was studied. It was found that the activity of supported catalysts, which were reduced in a reaction medium of sodium borohydride, decreased in the order Rh > Pt ≈ Ru ? Pd regardless of the nature of the support (γ-Al₂O₃, a Sibunit carbon material, or TiO₂). The catalysts based on TiO₂ exhibited the highest activity. As found by UV-vis diffuse reflectance spectroscopy, the composition and structure of the supported precursor of an active component depend on the nature of the support. It is likely that rhodium clusters with different reaction properties were formed on various supports under the action of a reaction medium.