On Discrete Functions and Repetitive Arrangements with Algorithms to Construct All Discrete Functions and a Practical Problem
Nicolae Popoviciu · Mathematics Letters · 2021
The main idea of this work is based on the question: how can we control the electric circuits between a number of electric bulbs and a number of electric sources. This generates the correspondences between two discrete sets. The correspondence is based on the notion of discrete function and repetitive arrangements. The normal construction and notions of the work are introduced gradually and are detailed at every stage. Our constant endevour has been to ensure that every sentence in the work has a logical position. Here appears many questions: how to construct all discrete fuctions, which is the total number of these funtions, which is the relation between the number of bulbs and the number of sources, can we constract and control only a partial number of electric circuits (by direct access method) etc. The work answers all these questions by specialised algorithms: the construction algorithm and the decomposition algorithm. The algorithms use the rule from left to right to construct all possille discrete functions and, hence, all electric circuits. The decomposition algorithm supplies an access direct method. So we can control any part of the whole set of circuits. A lot of notions and specific notations are used to develop and illustrate the work. For combinations we have to show the constructon elements. A lot of examples explain this important notion. The work contains a lot of numerical examples and applications. The last section of the work deals with the bijective (and invertible) functions. Specialized notions and notations are used. Numerical examples and geometric designs illustrate the theory.