Wacker Oxidation of Methylenecyclobutanes: Scope and Selectivity in an Unusual Setting

Methylenecyclobutanes are found to undergo Wacker oxidation via a semi-pinacol-type rearrangement. Key to a successful process is the use of tert-butyl nitrite as oxidant, which not only enables efficient catalyst turn-over but also ensures high Markovnikov-selectivity under mild conditions. Thus, cyclopentanones (26 examples) can be accessed in an overall good yield and excellent selectivity (up to 97 % yield, generally >99 : 1 ketone:aldehyde ratio). Stereochemical analysis of the reaction sequence reveals migration aptitudes in line with related 1,2-shifts. By introducing a pyox ligand to palladium, prochiral methylenecyclobutanes can be desymmetrized, thus realizing the first enantioselective Wacker oxidation.     

On primeness of general skew inverse Laurent series ring

Ever since the introduction, skew inverse Laurent series rings have kept growing in importance, as researchers characterized their properties (such as Noetherianness, Armendarizness, McCoyness, etc.) in terms of intrinsic properties of the base ring and studied their relations with other fields of mathematics, as for example quantum mechanics theory. The goal of our paper is to study the primeness and semiprimeness of general skew inverse Laurent series rings R((x^?1;σ,δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ.     

On the dynamics of rings rotating with variable spin speed

This work investigates nonlinear dynamic response of circular rings rotating with spin speed which involves small fluctuations from a constant average value. First, Hamilton's principle is applied and the equations of motion are expressed in terms of a single time coordinate, representing the amplitude of an in-plane bending mode. For nonresonant excitation or for slowly rotating rings, a complete analysis is presented by employing phase plane methodologies. For rapidly rotating rings, periodic spin speed variations give rise to terms leading to parametric excitation. In this case, the vibrations that occur under principal parametric resonance are analyzed by applying the method of multiple scales. The resulting modulation equations possess combinations of trivial and nontrivial constant steady state solutions. The existence and stability properties of these motions are first analyzed in detail. Also, analysis of the undamped slow-flow equations provides a global picture for the possible motions of the ring. In all cases, the analytical predictions are verified and complemented by numerical results. In addition to periodic response, these results reveal the existence of unbounded as well as transient chaotic response of the rotating ring.     

Large amplitude free flexural vibration of rings using finite element approach

Here, the large amplitude free flexural vibrations of isotropic/laminated orthotropic rings are investigated, using a shear flexible curved beam element based on field consistency principle. A laminated refined beam theory is introduced for developing the element, which satisfies the interface transverse shear stress and displacement continuity, and has a vanishing shear stress on the inner and outer surfaces of the beam. The formulation includes in-plane and rotary inertia effects, and the non-linearity due to the finite deformation of the ring. The governing equations obtained using Lagrange's equations of motion are solved through the direct integration technique. Amplitude-frequency relationships evaluated from the dynamic response history are examined. Detailed numerical results are presented considering various parameters such as radius-to-thickness ratio, circumferential wave number and ovality for isotropic and laminated orthotropic rings. The nature and degree of the participation of various modes in non-linear asymmetric vibration of oval ring brought out through the present study are useful for accurate modelling of the closed non-circular structures.     

More on the averaged midpoint-trapezoid type rules

A generalization of the corrected averaged midpoint-trapezoid rule is derived. Various error bounds for this generalization are established.     

On a problem of Eidelheit from The Scottish Book concerning absolutely continuous functions

A negative solution of Problem 188 posed by Max Eidelheit in the Scottish Book concerning superpositions of separately absolutely continuous functions is presented. We discuss here this and some related problems which have also negative solutions. Finally, we give an explanation of such negative answers from the “embeddings of Banach spaces” point of view.     

Absolutely continuous spectrum of one random elliptic operator

We consider an elliptic random operator, which is the sum of the differential part and the potential. The potential considered in the paper is the same as the one in the Andersson model, however the differential part of the operator is different from the Laplace operator. We prove that such an operator has absolutely continuous spectrum on all of (0,∞).     

Schrödinger operators and associated hyperbolic pencils

For a large class of Schrödinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are significantly easier to study. Then, we give some applications to the original Schrödinger operators including one-dimensional Schrödinger operators with L²-operator-valued potentials, multidimensional Schrödinger operators with slowly decaying potentials.     

Local Rings of Rings of Quotients

The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a. Partially supported by the Ministerio de Educación y Ciencia and Fondos Feder, jointly, trough projects MTM2004-03845, MTM2007-61978 and MTM2004-06580-C02-02, MTM2007-60333, by the Junta de Andalucía, FQM-264, FQM336 and FQM02467 and by the Plan de Investigación del Principado de Asturias FICYT-IB05-017.