The rate of penetration (ROP) is one of the vital parameters which directly affects the drilling time and costs. There are various parameters that influence the drilling rate; they include weight on bit, rotational speed, mud weight, bit type, formation type, and bit hydraulic. Several approaches, including mathematical models and artificial intelligence have been proposed to predict the rate o...

Resolving a question of Banero, we show that for every integer K > 1, there exists a word with additive complexity identically K. This result is perhaps surprising in light of the rather strong restriction on the existence of words with constant abelian complexity, given in the work of Currie and Rampersad. To prove our result we generalize the notion of a sturmian word. We also pose some quest...

Cassaigne conjectured in 1994 that any pattern with m distinct variables of length at least 3(2m−1) is avoidable over a binary alphabet, and any pattern with m distinct variables of length at least 2 is avoidable over a ternary alphabet. Building upon the work of Rampersad and the power series techniques of Bell and Goh, we obtain both of these suggested strict bounds. Similar bounds are also o...

We answer a question of Harju: An infinite square-free ternary word with an n-stem factorization exists for any n ≥ 13. We show that there are uniform ternary morphisms of length k for every k ≥ 23. This resolves almost completely a problem of the author and Rampersad.

Answering a question of Richomme, Currie and Rampersad proved that 7/3 is the infimum of the real numbers α > 2 such that there exists an infinite binary word that avoids α-powers but is highly 2-repetitive, i.e., contains arbitrarily large squares beginning at every position. In this paper, we prove similar statements about βrepetitive words, for some other β’s, over the binary and the ternary...

The aim of this short note is to generalise the result of Rampersad--Shallit saying that an automatic sequence and a Sturmian sequence cannot have arbitrarily long common factors. We show that the same result holds if a Sturmian sequence is replaced by an arbitrary sequence whose terms are given by a generalised polynomial (i.e., an expression involving algebraic operations and the floor functi...

In previous work, Currie and Rampersad showed that the growth of the number of binary words avoiding the pattern xxx R was intermediate between polynomial and exponential. We now show that the same holds for the growth of the number of binary words avoiding the pattern xx R x. Curiously, the analysis for xx R x is much simpler than that for xxx R. We derive our results by giving a bijection bet...

Consider a sequence of letters or numbers. Does a pattern exist that is avoided by the sequence? This topic is a very popular area of research in mathematics for its promising utility in computer science and other branches of mathematics, the elegant proofs and solutions, and the many open problems that still remain. Section 1 of this paper provides definitions, notations, and some properties o...

We use results on Dyck words and lattice paths to derive a formula for the exact number of binary words of a given length with a given minimal abelian border length, tightening a bound on that number from Christodoulakis et al. (Discrete Applied Mathematics, 2014). We also extend to any number of distinct abelian borders a result of Rampersad et al. (Developments in Language Theory, 2013) on th...

In previous work, Currie and Rampersad showed that the growth of the number of binary words avoiding the pattern xxxR was intermediate between polynomial and exponential. We now show that the same result holds for the growth of the number of binary words avoiding the pattern xxRx. Curiously, the analysis for xxRx is much simpler than that for xxxR. We derive our results by giving a bijection be...