Surface Magnetization of Aperiodic Ising Quantum Chains

We study the surface magnetization of aperiodic Ising quantum chains. Using fermion techniques, exact results are obtained in the critical region for quasiperiodic sequences generated through an irrational number as well as for the automatic binary Thue-Morse sequence and its generalizations modulo p. The surface magnetization exponent keeps its Ising value, beta_s=1/2, for all the sequences studied. The critical amplitude of the surface magnetization depends on the strength of the modulation and also on the starting point of the chain along the aperiodic sequence.Comment: 11 pages, 6 eps-figures, Plain TeX, eps

Enumeration and Decidable Properties of Automatic Sequences

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems

Nilpotence order growth of recursion operators in characteristic p

We prove that the killing rate of certain degree-lowering "recursion operators" on a polynomial algebra over a finite field grows slower than linearly in the degree of the polynomial attacked. We also explain the motivating application: obtaining a lower bound for the Krull dimension of a local component of a big mod-p Hecke algebra in the genus-zero case. We sketch the application for p=2 and p=3 in level one. The case p=2 was first established in by Nicolas and Serre in 2012 using different methods

On formal inverse of the Prouhet-Thue-Morse sequence

Let $p$ be a prime number and consider a $p$-automatic sequence ${\bf u}=(u_{n})_{n\in\N}$ and its generating function $U(X)=\sum_{n=0}^{\infty}u_{n}X^{n}\in\mathbb{F}_{p}[[X]]$. Moreover, let us suppose that $u_{0}=0$ and $u_{1}\neq 0$ and consider the formal power series $V\in\mathbb{F}_{p}[[X]]$ which is a compositional inverse of $U(X)$, i.e., $U(V(X))=V(U(X))=X$. In this note we initiate the study of arithmetic properties of the sequence of coefficients of the power series $V(X)$. We are mainly interested in the case when $u_{n}=t_{n}$, where $t_{n}=s_{2}(n)\pmod{2}$ and ${\bf t}=(t_{n})_{n\in\N}$ is the Prouhet-Thue-Morse sequence defined on the two letter alphabet $\{0,1\}$. More precisely, we study the sequence ${\bf c}=(c_{n})_{n\in\N}$ which is the sequence of coefficients of the compositional inverse of the generating function of the sequence ${\bf t}$. This sequence is clearly 2-automatic. We describe the sequence ${\bf a}$ characterizing solutions of the equation $c_{n}=1$. In particular, we prove that the sequence ${\bf a}$ is 2-regular. We also prove that an increasing sequence characterizing solutions of the equation $c_{n}=0$ is not $k$-regular for any $k$. Moreover, we present a result concerning some density properties of a sequence related to ${\bf a}$.Comment: 16 pages; revised version will appear in Discrete Mathematic