The three space problem in topological groups

We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p=c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups.     

X-ray metrology for advanced silicon processes

X-ray reflectivity (XRR), X-ray fluorescence (XRF) and small angle X-ray scattering (SAXS) techniques are used to the monitoring of Cu/porous low κ processes, which are developed for the next generation (≤65 nm) integrated circuits. Sensitivity of XRR and XRF is sufficient to detect drifts of the copper barrier layer, copper seed layer and Cu CMP (chemical-mechanical polishing) processes. Their metrology key parameters comply with production requirements. SAXS allows determining the pore structure of low κ films: average pore size and pore size distribution.     

Hopf monads

We introduce and study Hopf monads on autonomous categories (i.e., monoidal categories with duals). Hopf monads generalize Hopf algebras to a non-braided (and non-linear) setting. In particular, any monoidal adjunction between autonomous categories gives rise to a Hopf monad. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence of integrals, Maschke's criterium of semisimplicity, etc.) to Hopf monads. We also introduce and study quasitriangular and ribbon Hopf monads (again defined in a non-braided setting).     

Locally compact spaces of countable core and Alexandroff compactification

We introduce a new cardinal invariant, core of a space, defined for any locally compact Hausdorff space X and denoted by cor(X). Locally compact spaces of countable core generalize locally compact σ-compact spaces in a way that is slightly exotic, but still quite natural. We show in Section 1 that under a broad range of conditions locally compact spaces of countable core must be σ-compact. In particular, normal locally compact spaces of countable core and realcompact locally compact spaces of countable core are σ-compact. Perfect mappings preserve the class of spaces of countable core in both directions (Section 2). The Alexandroff compactification aX is weakly first countable at the Alexandroff point a if and only if cor(X)=ω (Section 3). Two examples of non-σ-compact locally compact spaces of countable core are discussed in Section 3. We also extend the well-known theorem of Alexandroff and Urysohn on the cardinality of perfectly normal compacta to compacta satisfying a weak version of perfect normality. Several open problems are formulated.     

Synthesis and magnetic properties of multilayer Ni/Cu and NiFe/Cu nanowires

Highly ordered composite nanowires with multilayer Ni/Cu and NiFe/Cu have been fabricated by pulsed electrodeposition into nanoporous alumina membrane. The diameter of wires can be easily varied by pore size of alumina, ranging from 30 to 100 nm. The applied potential and the duration of each potential square pulse determine the thickness of the metal layers. The nanowires have been characterized by transmission electron microscopy (TEM), magnetic force microscopy (MFM), and vibrating sample magnetometer (VSM) measurements. The MFM images indicate that every ferromagnetic layer separated by Cu layer was present as single isolated domain-like magnet. This technique has potential use in the measurement and application of magnetic nanodevices.     

On Genuine Bernstein–Durrmeyer Operators

We continue the studies on the so–called genuine Bernstein–Durrmeyer operators U _n by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by U _n . Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of U _n , compute the moments of T(t) and establish asymptotic formulas. Received: January 26, 2007.     

Another independent self-dual basis for the trivial variety

We give a new independent self-dual 3-basis for the trivial variety with two binary operations. Received October 24, 2006; accepted in final form January 25, 2007.     

Monotone versions of countable paracompactness

One possible natural monotone version of countable paracompactness, MCP, turns out to have some interesting properties. We investigate various other possible monotone versions of countable paracompactness and how they are related.