Abstract Arithmetic billiards show a nice interplay between arithmetics and geometry. The billiard table is rectangle with integer side lengths. A pointwise ball moves constant speed along segments making 45° angle the sides bounces on these. In classical setting, shooted from corner lands in corner. We allow to start at any point distances sides: either or trajectory periodic. length of path c...

A real-number to molecular structure mapping is a topological index. It graph invariant method for describing physico-chemical properties of structures specific substances. In that article, We examined pentacene’s chemical composition. The research on the subsequent indices reflected in our paper, we conducted an analysis several including general randic connectivity index, first zagreb sum-con...

Investigating banks’ status has always been a concern to its management and regulatory decisions, which can be carried out by means of income diversification and CAMEL composite. The importance of income diversification is to decline the operation risk. The soundness of banking is also another essential component in the banking industry that can be evaluated by CAMEL. This article’s two primary...

Similar sublattices of the root lattice A4 are possible [9] for each index that is the square of a non-zero integer of the form m + mn − n. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H(Q( √ 5)) and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given ...