Akshunna Shaurya Dogra

arXiv2018

Optimal Presentations of Mathematical Objects

We discuss the optimal presentations of mathematical objects under well defined symbol libraries. We shall examine what light our chosen symbol libraries and syntax shed upon the objects they represent. A major part of this work will focus on discrete sets, particularly the natural numbers, with results that describe the presentation of the natural numbers under specific symbol libraries and what those presentations may reveal about the properties of the natural numbers themselves. We shall present bounds and constraints on the length and shape of presentations, connect already existing problems in other fields of mathematics to questions relevant to these presentations and otherwise illuminate why such a study can produce exciting results.

math.HO↗

arXiv2018

Minimal Representations of Natural Numbers Under a Set of Operators

This paper studies the minimal length representation of the natural numbers. Let O be a fixed set of integer-valued functions (primarily hyperoperations). For each n, what is the shortest way of expressing n as a combinations of functions in O to the constant 1? For example, if O contains the two functions Sx (successor of x) and *xy (x times y) then the shortest representation of 12 is *SSS1SS1, with 8 symbols. This is taken to mean that 8 is complexity of 12 under O. We make a study of such minimal representations and complexities in this paper, proving and/or rightly predicting bounds on complexities, discussing some relevant patterns in the complexities and minimal representations of the natural numbers and listing the results gleaned from computational analysis. Computationally, the first 4.5 million natural numbers were probed to verify our mathematically obtained results. Due to the finiteness of the problem, we used the method of exhaustion of possibilities to state some other results as well.

math.HO↗