Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.     

Small-x behavior of the structure function and its slope for “frozen” and analytic strong-coupling constants

Using the leading-twist approximation of the Wilson operator product expansion with “frozen” and analytic versions of the strong-coupling constant, we show that the Bessel-inspired behavior of the structure function F₂ and its slope ∂lnF₂/∂ln(1/x) at small values of x, obtained for a flat initial condition in the DGLAP evolution equations, leads to good agreement with experimental data of deep-inelastic scattering at DESY HERA.     

Spectral Determinants on Mandelstam Diagrams

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric \({|\omega|^2}\), where \({\omega}\) is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic form \({\omega}\). As an important intermediate result we prove a decomposition formula of the type of Burghelea–Friedlander–Kappeler for the determinant of the Laplacian for flat surfaces with cylindrical ends and conical singularities.     

Almost Invariant Half-Spaces of Operators on Banach Spaces

We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on an infinite-dimensional Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T(Y). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on ℓ_p (1 ≤ p < ∞) or c₀.     

Ligand‐site exchange in intramolecular complexes of silicon: substituent effects

The intramolecular complexes containing coordination bonds Si←N or Si←O are distinguished for their stereochemical nonrigidity resulting in interconversion between isomers, that is, ligand‐site exchange. The influence of the substituents bound to the silicon atom on the free energies of activation for ligand exchange ΔG^≠ of specific interest is poorly understood. In this work, the literature data on substituent influence on the energies ΔG^≠ for 13 series of the complexes have been considered, using the correlation analysis. On the basis of the obedience of the energies ΔG^≠ to the linear free energy relationship, it has been established for the first time that the ΔG^≠ values depend not only on the inductive and resonance effects but also on the polarizability and steric effects of substituents. The reason for the occurrence of the polarizability effect is the appearance of excess charges on Si and N (or O) atoms as a result of intramolecular coordination consisting in the charge transfer from the donor center (N or O atom) to the acceptor one (Si atom). In some series the contribution of the polarizability or steric effect to the overall change in ΔG^≠ because of the influence of substituents is a maximum. An understanding of these effects may give a better insight into the mechanism of nucleophilic substitution, involving organoelement compounds. Copyright © 2012 John Wiley & Sons, Ltd.     

Dichloromethane as solvent and reagent: a case study of photoinduced reactions in mixed phosphonium‐iodonium ylide

Several years ago the photoinduced reaction of mixed phosphonium‐iodonium ylides ( 1 ) with acetylenes ( 2 ) to give λ⁵‐phosphinolines ( 3 ) and substituted furans ( 4 ) was described. This reaction is one‐pot, metal‐free synthesis of heterocycles 3 and 4 with the yields of 40% to 80%. The reaction proceeds only in dichloromethane (DCM) at the high ylide concentrations (>0.01 mol/L). The product analysis by ³¹P NMR, electrospray ionization mass spectrometry, UV‐vis spectrophotometry, and the dynamic light scattering study of the self‐aggregation of the ylide in DCM showed a dual role of the solvent in the photoinduced reactions of mixed phosphonium‐iodonium ylide: (i) at the low ylide concentrations (<0.01 mol/L), the conjugated photoinitiation of the chain reaction in DCM results in the formation of chlorine‐containing products and (ii) at the high ylide concentrations (>0.01 mol/L), the photolysis mechanism is determined by self‐organization of the ylide molecules to give large stable aggregates in DCM, in which the target heterocycles are synthesized. Two important issues follow from the study. First, the annulation reaction between mixed phosphonium‐iodonium ylide and acetylenes occurs only when the reactive intermediates are in close proximity to one other, and, second, DCM is not inert reagent in reactions occurring with participation of radicals and in one form or another can participate in photoinduced radical reactions of various solutes.