In this paper, we make use of the notion of the character of a transformation on a fixed set X , provided by Purisang and Rakbud in 2016, and the notion of a Δ -structure on X , provided by Magill Jr. and Subbiah in 1974, to define a sub-semigroup of the full-transformation semigroup T X . We also define a sub-semigroup of that semigroup. The regularity of those two semigroups is also studied.

In this paper, we study affine mappings on domains in R 3 that are unions of simplices. Let f i be an affine mapping of the form f i (x) = A i x, where A i is a 3×3 transformation matrix, on a simplex Q i . We establish the condition of these matrices A i in order to obtain a continuous piecewise affine mapping f on the domain Q that is the union of simplices Q i such that f |Q i = f i .

In this paper, we establish a necessary and sufficient condition for two algebraic integers in complex quadratic number fields to be consecutive terms of generalized Fibonacci numbers. We use this result to obtain all solutions of the Diophantine equation x 2 − a x y + b y 2 = c over Gaussian integers, where b and c are units in [i] and a ∈ [i] with |a| 2 10.

In this paper, we study the prescribable conformally equivariant dilatations for orientation preserving quasiconformal homeomorphisms. The complex dilatation is a prescribable conformally equivariant dilatation in R 2 . A Schottky set is a subset of the unit sphere S n whose complement is the union of at least three disjoint open balls. By using the result of Bonk, Kleiner, and Merenkov that there are rigid Schottky sets of positive measure in each dimension at least 3, we prove that it is not possible to have a prescribable conformally equivariant dilatation in R n , where n ≥ 3.