The article describes Euler’s group Γ (z), formed by the invertible elements of the ring of the residues of the integer complex numbers modulo numbers divisible by z. The answers are proved for the primary cases z = (1+ i)m and z = (p+ iq)m, p2 +q2 = 4k+3 being prime.

That is, W0 is the set of all finite products of the generators, allowing repetitions. Let W signify the larger multiplicative semigroup generated by W0 together with {1/2}. We call W the wild semigroup and W0 the Wooley semigroup. The question we consider is: Which integers belong to these semigroups? The sets of integer elements W(Z) := W ∩ Z and W0(Z) := W0 ∩ Z themselves form multiplicative...

We prove that an infinite class of Copeland-Erdős numbers is not strongly normal and provide the analogous result for Bugeaud’s Mahler-inspired extension of the Copeland-Erdős numbers. After the presentation of our results, we o↵er several open questions concerning normality and strong normality.

The projective complex plane and the "twisted" S3 bundle over Sl are proved to be the unique closed prime connected (smooth or PL) 4-manifolds of genus two. Then the classification of the nonorientable 4-manifolds of genus 4 is given. Finally the genus of a manifold M is shown to be related with the 2nd Betti number of M and some applications are proved in the general (resp. simply-connected) c...

We show that very weak topological assumptions are enough to ensure the existence of a Hellytype theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If F is a finite family of subsets of R such that β̃i ( ⋂G) ≤ b for any G ( F and every 0 ≤ i ≤ dd/2e−1 then F has Helly number at most h(b, d). Here β̃i denotes t...