Related papers: Strings in abstract root systems

We consider $\Phi(x)=x^{-\frac{1}{4}}\left[1-2\sqrt{x}\Sigma e^{-p^2\pi x}\ln p\right]$ on $x>0$, where the sum is over all primes $p$. If $\Phi$ is bounded on $x>0$, then the Riemann hypothesis is true or there are infinitely many zeros…

If $\alpha \in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $\alpha$ is $\Phi(\alpha) = \{(i, j) \, | \, 1 \leq i < j \leq n, \alpha(i) > \alpha(j)\}$. We describe all $r$-tuples $\alpha_1, \alpha_2, \ldots, \alpha_r…

The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension…

Firstly studied by Kempa and Prezza in 2018 as the cement of text compression algorithms, string attractors have become a compelling object of theoretical research within the community of combinatorics on words. In this context, they have…

The theory of finite automata concerns itself with words in a free monoid together with concatenation and without further structure. There are, however, important applications which use alphabets which are structured in some sense. We…

A result of Kaufmann shows that if $L_\alpha$ is countable, admissible and satisfies $\Pi_n\textsf{-Collection}$, then $\langle L_\alpha, \in \rangle$ has a proper $\Sigma_{n+1}$-elementary end extension. This paper investigates to what…

A notion of one-dimensional formal ring is presented. It consists of a triple $(A,\Phi,\Psi)$ where $A$ is a unital ring and $\Phi$ and $\Psi$ are two formal power series in $2$ variables ${\Phi(x,y),\Psi(x,y)\in A\llbracket…

Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the…

We study the palindromic complexity of infinite words $u_\beta$, the fixed points of the substitution over a binary alphabet, $\phi(0)=0^a1$, $\phi(1)=0^b1$, with $a-1\geq b\geq 1$, which are canonically associated with quadratic non-simple…

Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type…

We present a class of static, spherically symmetric, non-singular solutions of the tree-level string effective action, truncated to first order in $\alpha'$. In the string frame the solutions approach asymptotically (at $r\to 0$ and $r\to…

In this article we consider several probabilistic processes defining random grapha. One of these processes appeared recently in connection with a factorization problem in the symmetric group. For each of the probabilistic processes, we…

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property $\Phi$. What happens if this question is modified in a way that we get a possibly infinite family of graphs…

We consider, for infinite cardinals kappa and alpha <= kappa^+, the group Pi(kappa,< alpha) of sequences of integers, of length kappa, with non-zero entries in fewer than alpha positions. Our main result tells when Pi(kappa,< alpha) can be…

We introduce the concept of a $\lambda$-Hopf algebra as a Hopf algebra obtained as the partial smash product algebra of a Hopf algebra and its base field, and show that every Hopf algebra is a $\lambda$-Hopf algebra. Moreover, a method to…

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…

A fundamental theme in automata theory is regular languages of words and trees, and their many equivalent definitions. Salvati has proposed a generalization to regular languages of simply typed $\lambda$-terms, defined using denotational…

To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much…

Integer partitions express the different ways that a positive integer may be written as a sum of positive integers. Here we explore the analytic properties of a new polynomial $f_\lambda(x)$ that we call the partition polynomial for the…

The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root…