Finiteness conditions in the stable module category

We study groups whose cohomology functors commute with filtered colimits in high dimensions. We relate this condition to the existence of projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg–Mac Lane spaces with finitely many n-cells for all sufficiently large n. To that end, we determine the structure of completely finitary Gorenstein projective modules over group rings. The methods are inspired by representation theory and make use of the stable module category, in which morphisms are defined through complete cohomology. In order to carry out these methods, we need to restrict ourselves to certain classes of hierarchically decomposable groups.     

A general theoretical model for the magnetohydrodynamic flow of micropolar magnetic fluids. Application to Stokes flow

Many practical applications, which have an inherent interest of physical and mathematical nature, involve the hydrodynamic flow in the presence of a magnetic field. Magnetic fluids comprise a novel class of engineering materials, where the coexistence of liquid and magnetic properties provides us with the opportunity to solve problems with high mathematical and technical complexity. Here, our purpose is to examine the micropolar magnetohydrodynamic flow of magnetic fluids by considering a colloidal suspension of ferromagnetic material (usually non‐conductive) in a carrier magnetic liquid, which is in general electrically conductive. In this case, the ferromagnetic particles behave as rigid magnetic dipoles. Thus, the application of an external magnetic field, apart from the creation of an induced magnetic field of minor significance, will prevent the rotation of each particle, increasing the effective viscosity of the fluid and will cause the appearance of an additional magnetic pressure. Despite the fact that the general consideration consists of rigid particles of arbitrary shape, the assumption of spherical geometry is a very good approximation as a consequence of their small size. Our goal is to develop a general three‐dimensional theoretical model that conforms to physical reality and at the same time permits the analytical investigation of the partial differential equations, which govern the micropolar hydrodynamic flow in such magnetic liquids. Furthermore, in the aim of establishing the consistency of our proposed model with the principles of both ferrohydrodynamics and magnetohydrodynamics, we take into account both magnetization and electrical conductivity of the fluid, respectively. Under this consideration, we perform an analytical treatment of these equations in order to obtain the three‐dimensional effective viscosity and total pressure in terms of the velocity field, the total (applied and induced) magnetic field and the hydrodynamic and magnetic properties of the fluid, independently of the geometry of the flow. Moreover, we demonstrate the usefulness of our analytical approach by assuming a degenerate case of the aforementioned method, which is based on the reduction of the partial differential equations to a simpler shape that is similar to Stokes flow for the creeping motion of magnetic fluids. In view of this aim, we use the potential representation theory to construct a new complete and unique differential representation of magnetic Stokes flow, valid for non‐axisymmetric geometries, which provides the velocity and total pressure fields in terms of easy‐to‐find potentials, via an analytical fashion. Copyright © 2009 John Wiley & Sons, Ltd.     

On Periodic (Co)Homology of Groups

Here we show that a countable group G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H ^q (G, ?) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H ^q (G, ?). In [2 Asadollahi , J. , Hajizamani , A. , Salarian , Sh. Periodic flat resolutions and periodicity in group (co)homology. To appear in Forum Mathematicum.  [Google Scholar]] Asadollahi, Hajizamani, and Salarian showed that, if a group G is such that every flat ?G-module has finite projective dimension, then G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H ^q (G, ?) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H ^q (G, C), where C is the cotorsion envelope of the trivial ?G-module ?.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

Some aspects on the electromagnetic-impulse pendulum and Biot-Savart-Lorentz force

Summary The purpose of this paper is to cast light on various serious mistakes which have been involved during the analysis of two experiments, related to the correctness of the Biot-Savart-Lorentz force law. At first in the MIT experiment, carried out by Graneauet al., they claim that the calculated momentum, imparted to an electrodynamic-impulse pendulum, using the BSL force law, is 43% larger than the experimentally measured momentum of the pi-frame pendulum. We have found that this discrepancy, using their own data,is due to the use of a wrong value for the time constant parameter introduced in the current formula. In addition Pappas, like Graneau, although states that all the available energy is dissipated to Joule heating, considers that all the pendulum momentum is imparted to field momentum yielding an enormous amount of field-radiated energy, which is also completely wrong. To speed up pubblication, the authors of this paper have agreed to not receive the proofs for correction.