Every Binary Pattern of Length Greater Than 14 Is Abelian-2-Avoidable

We show that every binary pattern of length greater than 14 is abelian-2-avoidable. The best known upper bound on the length of abelian-2-unavoidable binary pattern was 118, and the best known lower bound is 7. We designed an algorithm to decide, under some reasonable assumptions, if a morphic word avoids a pattern in the abelian sense. This algorithm is then used to show that some binary patterns are abelian-2-avoidable. We finally use this list of abelian-2-avoidable pattern to show our result. We also discuss the avoidability of binary patterns on 3 and 4 letters

Ten Conferences WORDS: Open Problems and Conjectures

In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

Reflections on Restoring Integrity to the United States Immigration System: A Personal Vision

In the debate on US immigration reform, a number of legislative proposals have been introduced.To be effective, reforms must take into account the lessons learned from implementing the Immigration Reform and Control Act of 1986 (IRCA).This policy brief summarizes these lessons learned as follows: * The robust and growing demand for work and family reunification visas must be incorporated into new policies;* Legalization should not be done halfway;* Reducing incentives for fraud should be a top policy goal; and* Migration is a complex phenomenon that cannot be managed unilaterally; it requires cooperation among neighboring countries.The policy brief argues that there are three "E"s required to achieve stable reform:* Enforcement that devises smarter border and interior controls that are consistent with America's values, temperament, and philosophy of government powers;* Expanded numbers of visas that address the continuing demand, especially from citizens of the countries on America's borders; and* Earned regularization that offers a realistic and fair opportunity to the unauthorized resident population in the United States to earn lawful permanent status

Avoiding Patterns in the Abelian Sense

We classify all 3 letter patterns that are avoidable in the abelian sense. A short list of four letter patterns for which abelian avoidance is undecided is given. Using a generalization of Zimin words we deduce some properties of ω-words avoiding these patterns.Research of both authors supported by NSERC Operating Grants.https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/avoiding-patterns-in-the-abelian-sense/42148B0781A38A6618A537AAD7D39B8

A characterization of power-free morphisms

AbstractA word is called kth power-free if it does not contain any non-empty factor uk. A morphism is kth power-free if it preserves kth power-free words. A morphism is power-free if it is kth power-free for every K > 1.We show that it is decidable whether a morphism is power-free; more precisely, we prove that a morphism h is power-free iff: h is a square-free morphism and, for each letter a, the image h(a2) is cube-free