Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Let $$\Gamma $$ Γ be a torsion-free lattice of $$PU (p,1)$$ P U ( p , 1 ) with $$p \ge 2$$ p ≥ 2 and let $$(X,\mu _X)$$ ( X , μ X ) be an ergodic standard Borel probability $$\Gamma $$ Γ -space. We prove that any maximal Zariski dense measurable cocycle $$\sigma : \Gamma \times X \longrightarrow SU (m,n)$$ σ : Γ × X ⟶ S U ( m , n ) is cohomologous to a cocycle associated to a representation of $$PU (p,1)$$ P U ( p , 1 ) into $$SU (m,n)$$ S U ( m , n ) , with $$1 \le m \le n$$ 1 ≤ m ≤ n . The proof follows the line of Zimmer’ Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, there cannot exist maximal measurable cocycles with the above properties when $$1< m < n$$ 1 < m < n .

Conceptual Study and Manufacturing of a Configurable and Weld-Free Lattice Base for Automatic Food Machines

The study is aimed at developing a modular lattice base for automatic food machines, starting with a solution already patented by some of the authors. In this case, welded carpentry modules were interlocked with a system of profiles and metal inserts, also in welded carpentry, and the union was stabilized by structural adhesive bonding. Since welding involves long processing times and thermal distortions to be restored later, the driver of this study is to limit the use of welding as much as possible while increasing the modularity of the construction. For this purpose, various solution concepts have been generated where a common feature is the presence of rods of the same geometry and section to be joined together in configurable structural nodes. The concepts are qualitatively evaluated in light of the requirements, and the selected concept is digitally and physically prototyped. The prototype has been in service from over 5 years without showing any problems whatsoever.

Gradient of Residual Stress and Lattice Parameter in Mechanically Polished Tungsten Measured Using Classical X-rays and Synchrotron Radiation

In this work, the stress gradient in mechanically polished tungsten sample was studied using X-ray diffraction methods. To determine in-depth stress evolution in the very shallow subsurface region (up to 10 μm), special methods based on reflection geometry were applied. The subsurface stresses (depth up to 1 μm) were measured using the multiple-reflection grazing incidence X-ray diffraction method with classical characteristic X-rays, while the deeper volumes (depth up to 10 μm) were investigated using energy-dispersive diffraction with white high energy synchrotron beam. Both complementary methods allowed for determining in-depth stress profile and the evolution of stress-free lattice parameter. It was confirmed that the crystals of tungsten are elastically isotropic, which simplifies the stress analysis and makes tungsten a suitable material for testing stress measurement methods. Furthermore, it was found that an important compressive stress of about − 1000 MPa was generated on the surface of the mechanically polished sample, and this stress decreases to zero value at the depth of about 9 μm. On the other hand, the strain-free lattice parameter does not change significantly in the examined subsurface region.

Lattices do not distribute over powerset

We show that there is no distributive law of the free lattice monad over the powerset monad. The proof presented here also works for other classes of lattices such as (bounded) distributive/modular lattices and also for some variants of the powerset monad such as the (nonempty) finite powerset monad.