Recently there has been some renewed interest in Laguerre differential geometry [1], [2], [3]. This geometry was to a large extent developed by Blaschke and his school and a large amount of material about it can be found in [4]. Consider the set of oriented spheres in Euclidean space. If we add to this set the space of oriented planes, i.e. the spheres of infinite radii, then we obtain the spac...

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σm = {0, ..., m−1}N that are invariant under multiplication by integers. The results apply to the sets {x ∈ Σm : ∀ k, xkx2k · · ·xnk = 0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.

Equivalences between conformal foliations on Euclidean 3-space, Hermitian structures on Euclidean 4-space, shear-free ray congruences on Minkowski 4-space, and holomorphic foliations on complex 4-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued harmonic morphism without critical points defined on an open subset of Minkowski spac...

The multiplicative and the functional renormalization group methods are applied for four-dimensional scalar theory in Minkowski space–time. It is argued that appropriate choice of subtraction point more important than Euclidean parameters cutoff theory, defined by a quasi-particle domain, complex due to mass-shell contributions flow becomes much involved its counterpart.