A Pairing Between Super Lie-Rinehart and Periodic Cyclic Homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K⁰-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology. The author was partially supported by the KBN grant 1P03A 036 26.     

Discontinuous synchrony in an array of Van der Pol oscillators

We show the phenomenon of complete synchronization in an network of coupled oscillators. We confirm that non-diagonal coupling can lead to the appearance or disappearance of synchronous windows (ragged synchronizability phenomenon) in the coupling parameter space. We also show the appearance of clusters (synchronization in one or more group) between coupled systems. Our numerical studies are confirmed by an electronic experiment.     

The Galois Theory of Matrix C-rings

A theory of monoids in the category of bicomodules of a coalgebra C or C-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a C-ring (termed a matrix C -ring) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a Galois module is introduced and the structure theorem, generalising Schneider’s Theorem II [Schneider, Isr. J. Math., 72:167–195, 1990], is proven. This is then applied to the C-ring associated to a weak entwining structure and a structure theorem for a weak A-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed. Dedicated to Stef Caenepeel on the occasion of his 50th birthday.     

The nature of the rotational barriers in simple carbonyl compounds

The rotational barriers around the CO and CC bonds are investigated in formic acid, ethanedial and glycolaldedyde molecules on the basis of DFT-B3LYP/aug-cc-pVDZ calculations. Natural bond orbitals analysis is applied to enhance physical understanding of rotational barriers. In the case of attractive barriers in formic acid and Gc-glycolaldehyde, the barrier originates from the loss of hyperconjugation that determines the equilibrium structures while for the repulsive barriers in ethanedial and Go-glycolaldehyde, both Lewis and hyperconjugation terms contribute.     

Unidirectional transmission of electrons in a magnetic field gradient

This work presents an experimental demonstration of time-reversal asymmetry of electron states propagating along the boundary separating areas with opposite magnetic fields. For this purpose we have fabricated a hybrid ferromagnet– semiconductor device in the form of a Hall cross with two ferromagnets deposited on top. The magnets generated two narrow magnetic barriers of opposite polarity in the active Hall area. We have observed that if the signs of the barriers are reversed, the bend resistance changes its sign. Using the Landauer–Büttiker theory, we have demonstrated that this is a direct consequence of asymmetric transmission of the “snake” and the “cycloidal” trajectories formed at the boundary separating the regions with opposite magnetic field directions.     

Dimension of measures invariant with respect to the Wa?ewska partial differential equation

We study asymptotic properties of a nonlinear first-order partial differential equation which describes the reproduction of blood cells. This equation under conditions proposed by Wa?ewska generates a semigroup of transformations with highly chaotic behaviour of trajectories. We show that this semigroup has invariant measures with arbitrary large dimension.     

Local geometry of orbits for an ordinary classical lie supergroup

In this paper we identify a real reductive dual pair of Roger Howe with an Ordinary Classical Lie supergroup. In these terms we describe the semisimple orbits of the dual pair in the symplectic space, a slice through a semisimple element of the symplectic space, an analog of a Cartan subalgebra, the corresponding Weyl group and the corresponding Weyl integration formula.