The rr-Bell numbers

The notion of generalized Bell numbers has appeared in several works but there is no systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic properties of these numbers which generalize the similar properties of the Bell numbers. Most of these results seem to be new. It turns out that in a paper of Whitehead these numbers appeared in a very different context. In addition, we introduce the so-called $r$-Bell polynomials

Unitary transformations for testing Bell inequalities

It is shown that optical experimental tests of Bell inequality violations can be described by SU(1,1) transformations of the vacuum state, followed by photon coincidence detections. The set of all possible tests are described by various SU(1,1) subgroups of Sp(8,$\Bbb R$). In addition to establishing a common formalism for physically distinct Bell inequality tests, the similarities and differences of post--selected tests of Bell inequality violations are also made clear. A consequence of this analysis is that Bell inequality tests are performed on a very general version of SU(1,1) coherent states, and the theoretical violation of the Bell inequality by coincidence detection is calculated and discussed. This group theoretical approach to Bell states is relevant to Bell state measurements, which are performed, for example, in quantum teleportation.Comment: 3 figure

r−r-Bell polynomials in combinatorial Hopf algebras

We introduce partial $r$-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula.Comment: 7 page

Non-linear fe modelling of seismic pounding and damped-mitigating interconnection between a r/c tower and a masonry church

The finite element analysis of pounding represents one of the most critical issues for the assessment of the seismic performance of R/C structures built at poor distance from adjacent buildings. The effects of pounding can be particularly severe in slender R/C heritage structures, including civic or bell towers. An emblematic case study falling in this class of structures, i.e. a monumental R/C bell tower constructed in the early 1960s in Florence, is analyzed in this paper. Pounding collisions are simulated with a multi-link viscoelastic contact model originally implemented in this study. The results of the non-linear dynamic enquiry carried out with this model show that pounding affects the seismic response of the bell tower and the adjacent church as early as an input seismic action scaled at the amplitude of the normative basic design earthquake level. A retrofit hypothesis to prevent pounding is then proposed, which consists in linking the two structures by means of a pair of fluid-viscous dissipaters. Thanks to the supplemental damping action produced by these devices, the impacts are totally annulled, bringing the structural members of the tower to safe levels