Related papers: More Than 1700 Years of Word Equations
Most natural languages have a predominant or fixed word order. For example in English the word order is usually Subject-Verb-Object. This work attempts to explain this phenomenon as well as other typological findings regarding word order…
Introduction to the special issue of Phil. Trans. R. Soc. A 376, 2018, `Hilbert's Sixth Problem'. The essence of the Sixth Problem is discussed and the content of this issue is introduced. In 1900, David Hilbert presented 23 problems for…
A new formulation of the subject equation is presented. Several parametric and semi-parametric solutions are derived. The parametric solution for a=-1 was originally presented in 1972, but never published. A computer-generated version was…
Nobody has discovered any perfect cuboid and there is no formula to deliver all possible Euler bricks. During investigations of famous open problems regarding the perfect cuboid and Euler brick; I have found new important conjectures on…
We study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves…
We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Tribonacci-automatic". This class includes, for example, the famous Tribonacci word T =…
We implement four algorithms for solving linear Diophantine equations in the naturals: a lexicographic enumeration algorithm, a completion procedure, a graph-based algorithm, and the Slopes algorithm. As already known, the lexicographic…
Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as…
Words are fundamental linguistic units that connect thoughts and things through meaning. However, words do not appear independently in a text sequence. The existence of syntactic rules induces correlations among neighboring words. Using an…
By the theory of elliptic curves, we investigate the nontrivial rational parametric solutions of the Diophantine equation $f(x)f(y)=f(z)^n$, where $n=1,2$ and $f(X)$ are some simple Laurent polynomials.
The term complexity derives etymologically from the Latin plexus, which means interwoven. Intuitively, this implies that something complex is composed by elements that are difficult to separate. This difficulty arises from the relevant…
Word order is an important concept in natural language, and in this work, we study how word order affects the induction of world knowledge from raw text using language models. We use word analogies to probe for such knowledge. Specifically,…
We notice that one of the Diophantine equations, $knm=2kn+2km+2nm$, arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In…
We investigate systems of equations and the first-order theory of one-relator monoids. We describe a family $\mathcal{F}$ of one-relator monoids of the form $\langle A\mid w=1\rangle$ where for each monoid $M$ in $\mathcal{F}$, the…
In the fall 2011 issue of the Journal'Mathematics and Computer Education', author Unal Hasan, in the one page article "Proof without Words", gives a purely geometric proof of the equality, arctan(1/3)+ arctan(1/7) = arctan(1/2) (1) (See…
The heights of iterates of the discrete Painleve equations over number fields appear to grow no faster than polynomials while the heights of generic solutions of non-integrable discrete equations grow exponentially. This gives rise to a…
A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. Such cuboids are not yet discovered and their non-existence is also not proved. Perfect Euler cuboids…
We investigate the equational theory for Kleene algebra terms with variable complements and constant complements -- (language) complement where it applies only to variables or constants -- w.r.t. languages. While the equational theory…
In this article, we present a fresh perspective on language, combining ideas from various sources, but mixed in a new synthesis. As in the minimalist program, the question is whether we can formulate an elegant formalism, a universal…
By following the same construction pattern which Martin Davis proposed in a 1968 paper of his, we have obtained six quaternary quartic Diophantine equations that candidate as `rule-them-all' equations: proving that one of them has only a…