Displaying similar documents to “Binary equality words with two b ’s”

On ☆ - associated comonotone functions

Ondrej Hutník, Jozef Pócs (2018)

Kybernetika

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We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper [1]. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to + -associatedness of functions (as proved by Boczek and Kaluszka), but also to their ☆ -associatedness with ☆ being an arbitrary...

On some properties of doubly-periodic words

Claudio Baiocchi (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We study the functional equation: 1 ⁢ A ⁢ B ⁢ C = C ⁢ D ⁢ A where A , B , C and D are words over an alphabet A . In particular we prove a «structure result» for the inner factors B , D : for suitably chosen words X , Y , Z one has: 2 ⁢ B = X ⁢ Y ⁢ Z , D = Z ⁢ Y ⁢ X 2 ⁢ B = X ⁢ Y ⁢ Z , D = Z ⁢ Y ⁢ X 2 ⁢ B = X ⁢ Y ⁢ Z , D = Z ⁢ Y ⁢ X 2 ⁢ B = X ⁢ Y ⁢ Z , D = Z ⁢ Y ⁢ X . It is a generalization of the Lyndon-Schützenberger's Theorem (see [7]): if in (1) A or C is empty, formula (2) holds true with one among X , Y , Z which can be chosen empty.

Average Value of the Euler Function on Binary Palindromes

William D. Banks, Igor E. Shparlinski (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if ℬ 2 ℓ denotes the set of binary palindromes with precisely 2ℓ binary digits, we derive an asymptotic formula for the average value of the Euler function on ℬ 2 ℓ .

Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial

Florian Luca, Augustine O. Munagi (2014)

Colloquium Mathematicae

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We note that every positive integer N has a representation as a sum of distinct members of the sequence d ( n ! ) n ≥ 1 , where d(m) is the number of divisors of m. When N is a member of a binary recurrence u = u ₙ n ≥ 1 satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂. ...

Pattern avoidance in partial words over a ternary alphabet

Adam Gągol (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Blanched-Sadri and Woodhouse in 2013 have proven the conjecture of Cassaigne, stating that any pattern with m distinct variables and of length at least 2 m is avoidable over a ternary alphabet and if the length is at least 3 · 2 m - 1 it is avoidable over a binary alphabet. They conjectured that similar theorems are true for partial words – sequences, in which some characters are left “blank”. Using method of entropy compression, we obtain the partial words version of the theorem for ternary words. ...

ℱ -hypercyclic and disjoint ℱ -hypercyclic properties of binary relations over topological spaces

Marko Kostić (2020)

Mathematica Bohemica

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We examine various types of ℱ -hypercyclic ( ℱ -topologically transitive) and disjoint ℱ -hypercyclic (disjoint ℱ -topologically transitive) properties of binary relations over topological spaces. We pay special attention to finite structures like simple graphs, digraphs and tournaments, providing a great number of illustrative examples.

A note on univoque self-sturmian numbers

Jean-Paul Allouche (2008)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number β in ( 1 , 2 ) is univoque and self-sturmian if and only if the β -expansion of 1 is of the form 1 v , where v is a characteristic...

Every braid admits a short sigma-definite expression

Jean Fromentin (2011)

Journal of the European Mathematical Society

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A result by Dehornoy (1992) says that every nontrivial braid admits a σ -definite expression, defined as a braid word in which the generator σ i with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a σ -definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid...

TeX in a Nutshell

Petr Olšák (2021)

Zpravodaj Československého sdružení uživatelů TeXu

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Nowadays, many users discover TeX through high-level formats that hide the complexity of typesetting behind a facade of a friendly markup language. However, all except the simplest of typesetting tasks require that the user can understand what happens under the hood and knows how they can influence the algorithms of TeX when needed. In this article, the author introduces the foundations of most high-level TeX formats, which will help the readers with their day-to-day work with TeX...

From binary cube triangulations to acute binary simplices

Brandts, Jan, van den Hooff, Jelle, Kuiper, Carlo, Steenkamp, Rik

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Cottle’s proof that the minimal number of 0 / 1 -simplices needed to triangulate the unit 4 -cube equals 16 uses a modest amount of computer generated results. In this paper we remove the need for computer aid, using some lemmas that may be useful also in a broader context. One of the 0 / 1 -simplices involved, the so-called antipodal simplex, has acute dihedral angles. We continue with the study of such acute binary simplices and point out their possible relation to the Hadamard determinant problem. ...

Regular elements and Green's relations in Menger algebras of terms

Klaus Denecke, Prakit Jampachon (2006)

Discussiones Mathematicae - General Algebra and Applications

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Defining an (n+1)-ary superposition operation S n on the set W τ ( X n ) of all n-ary terms of type τ, one obtains an algebra n - c l o n e τ : = ( W τ ( X n ) ; S n , x 1 , . . . , x n ) of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation S n there are different possibilities to define binary associative operations on the set W τ ( X n ) and on the cartesian power W τ ( X n ) n . In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative...

Multi-island finite automata and their even computation

Dušan Kolář, Alexander Meduna, Martin Tomko (2021)

Kybernetika

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This paper discusses n -island finite automata whose transition graphs can be expressed as n -member sequences of islands i 1 , i 2 , ⋯ , i n , where there is a bridge leaving i j and entering i j + 1 for each 1 ≤ j ≤ n - 1 . It concentrates its attention on even computation defined as any sequence of moves during which these automata make the same number of moves in each of the islands. Under the assumption that these automata work only in an evenly computational way, the paper proves its main result stating that n -island finite...

Mobile TeX: Porting TeX to the iPad

Arthur Reutenauer (2011)

Zpravodaj Československého sdružení uživatelů TeXu

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The paper presents the achievement of Richard Koch, amongst others author of TeXShop and MacTeX developer, who has successfully compiled and used TeX on Apple's iPad.