Related papers: On Brlek-Reutenauer conjecture

We will be defining a type of perverse equivalence that always corresponds to a derived equivalence with two-term tilting complexes. We are going to show that the tilting considered by Okuyama and Yoshii for the proof of Brou\'e's…

Given a countable set X (usually taken to be N or Z), an infinite permutation $\pi$ of X is a linear ordering $<_\pi$ of X. This paper investigates the combinatorial complexity of infinite permutations on N associated with the image of…

It was conjectured that the augmentation ideal of a dihedral quandle of even order $n>2$ satisfies $|\Delta^k(\text{R}_n)/\Delta^{k+1}(\text{R}_{n})|=n$ for all $k\geq 2$. In this article we provide a counterexample against this conjecture.

The Morse-Hedlund Theorem states that a bi-infinite sequence $\eta$ in a finite alphabet is periodic if and only if there exists $n\in\N$ such that the block complexity function $P_\eta(n)$ satisfies $P_\eta(n)\leq n$. In dimension two,…

Generalized pseudostandard words have been introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated…

Let $p$ be a prime and ${\mathfrak P}_p$ the set of positive integers which are prime to $p$. Recently, Wang and Cai proved that for every positive integer $r$ and prime $p>2$ $$ \sum_{\substack{i+j+k=p^r\\ i,j,k\in{\mathfrak P}_p}}…

In 1938, Morse and Hedlund proved that the subword complexity function of an infinite word is either bounded or at least linearly growing. In 1982, Ehrenfeucht and Rozenberg proved that this gap property holds for the subword complexity…

In intuitionistic mathematics, the Brouwer Continuity Theorem states that all total real functions are (uniformly) continuous on the unit interval. We study this theorem and related principles from the point of view of Reverse Mathematics…

We prove the universality of the regular realizability problems for several classes of filters. The filters are encodings of finite relations on the set of non-negative integers in the format proposed by P. Wolf and H. Fernau. The…

The relation between the trace and R-current anomalies in supersymmetric theories implies that the U$(1)_RF^2$, U$(1)_R$ and U$(1)_R^3$ anomalies which are matched in studies of N=1 Seiberg duality satisfy positivity constraints. Some…

G{\"o}del's second incompleteness theorem forbids to prove, in a given theory U, the consistency of many theories-in particular, of the theory U itself-as well as it forbids to prove the normalization property for these theories, since this…

Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an…

We study word structures of the form $(D,<,P)$ where $D$ is either $\mathbb{N}$ or $\mathbb{Z}$, $<$ is the natural linear ordering on $D$ and $P\subseteq D$ is a predicate on $D$. In particular we show: (a) The set of recursive…

A special inverse monoid is one defined by a presentation where all the defining relations have the form $r = 1$. By a result of Ivanov Margolis and Meakin the word problem for such an inverse monoid can often be reduced to the word problem…

We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…

A two-dimensional ($2$D) word is a $2$D palindrome if it is equal to its reverse and it is an HV-palindrome if all its columns and rows are $1$D palindromes. We study some combinatorial and structural properties of HV-palindromes and its…

We show that the uniform Littlewood Conjecture (ULC) recently introduced by Bandi, Fregoli and Kleinbock is false. More precisely the counterexamples form a residual set, the method further suggests positive Hausdorff dimension. For a…

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a…

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many n, every length-n factor is a product of two palindromes. We show that every Sturmian word has this property,…

Let $m$, $r$ and $n$ be positive integers. We denote by ${\bf k}\vdash n$ any tuple of odd positive integers ${\bf k}=(k_1,\dots,k_t)$ such that $k_1+\dots+k_t=n$ and $k_j\ge 3$ for all $j$. In this paper we prove that for every…