We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

Applying the Faber polynomial expansions, we obtain the general coefficient bounds for the class of biunivalent functions with bounded boundary rotations.

A finite group G is called splitting or splittable if it is a union of some collections of its proper subgroups intersecting pairwise at the identity. A special kind of splitting is known to be normal splitting. Also, a group G is said to have the basis property if, for each subgroup H≤G, H has a basis (minimal generating set), and any two bases have the same cardinality. In this work, I discuss a relation between classes of finite groups that possess both normal splitting and the basis property. This paper shows mainly that any non-p-group with basis property is normal splitting. However, the converse is not true in general. A counterexample is given. It is well known that any p-group has basis property. I demonstrate some types of p-groups which are splitting as well.

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Boundary value problems arise while modeling various physical and engineering reality. In this communication we investigate windowed Fourier frames focusing two-point BVPs. We approximate BVPs using windowed Fourier frames. We present some numerical results to demonstrate the efficiency of such frame functions to approximate BVPs.