On the correlation of the Thue–Morse sequence

The Ramanujan Journal - Let $$s_{q}$$ denote the sum of digits function in base q. The aim of this work is to estimate the exponential sums involving the sum of digits of shifted integers, namely...     

Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences,II

We prove that the Thue–Morse sequence t along subsequences indexed by ?n ^c ? is normal, where 1 < c < 3/2. That is, for c in this range and for each ω ∈ {0, 1} ^L , where L ≥ 1, the set of occurrences of ω as a factor (contiguous finite subsequence) of the sequence \(n \mapsto {t_{\left\lfloor {{n^c}} \right\rfloor }}\) has asymptotic density 2^?L . This is an improvement over a recent result by the second author, which handles the case 1 < c < 4/3.In particular, this result shows that for 1 < c < 3/2 the sequence \(n \mapsto {t_{\left\lfloor {{n^c}} \right\rfloor }}\) attains both of its values with asymptotic density 1/2, which improves on the bound c < 1.4 obtained by Mauduit and Rivat (who obtained this bound in the more general setting of q-multiplicative functions, however) and on the bound c ≤ 1.42 obtained by the second author.In the course of proving the main theorem, we show that 2/3 is an admissible level of distribution for the Thue–Morse sequence, that is, it satisfies a Bombieri–Vinogradov type theorem for each exponent η < 2/3. This improves on a result by Fouvry and Mauduit, who obtained the exponent 0.5924. Moreover, the underlying theorem implies that every finite word ω ∈ {0, 1} ^L is contained as an arithmetic subsequence of t.     

Transcendence of Thue–Morse p-Adic Continued Fractions

It has been proved that real numbers defined as a limit of continued fractions with a Thue–Morse sequence as the sequence of positive integer partial quotients are transcendental. The same holds for more general sequences of partial quotients. Moreover, transcendence results have been proved for p-adic numbers defined by Hensel development. Here, we tackle the transcendence question in case of p-adic numbers defined as a limit of continued fractions by proving a similar result as the first one.     

More infinite products: Thue–Morse and the gamma function

The Ramanujan Journal - Letting $$(t_n)$$ denote the Thue–Morse sequence with values 0,&nbsp;1, we note that the Woods–Robbins product $$\begin{aligned} \prod _{n \ge 0}\left(...     

Moments of a Thue–Morse generating function

We study the moments of even order of the generating function

$$\prod\limits_{0 \leqslant r \leqslant n} {(1 - e({2^r}x))} $$

of the Thue–Morse sequence and present several conjectures related to these moments.

     

Maxwell’s equations,the Euler index,and Morse theory

We show that the singularities of the Fresnel surface for Maxwell’s equation on an anisotrpic material can be accounted from purely topological considerations. The importance of these singularities is that they explain the phenomenon of conical refraction predicted by Hamilton. We show how to desingularise the Fresnel surface, which will allow us to use Morse theory to find lower bounds for the number of critical wave velocities inside the material under consideration. Finally, we propose a program to generalise the results obtained to the general case of hyperbolic differential operators on differentiable bundles.     

Subword complexity of the Fibonacci–Thue–Morse sequence: The proof of Dekking’s conjecture

Recently F. M. Dekking conjectured the form of the subword complexity function for the Fibonacci–Thue–Morse sequence. In this note we prove his conjecture by purely computational means, using the free software Walnut.     

The Prouhet–Tarry–Escott problem,indecomposability of polynomials and Diophantine equations

The Ramanujan Journal - In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of $$A subseteq {1, dots , n}$$ in k-sets such...     

Quasi-Neutral Limit of the Euler–Poisson and Euler–Monge–Ampère Systems

ABSTRACT

This paper studies the pressureless Euler–Poisson system and its fully nonlinear counterpart, the Euler–Monge–Ampère system, where the fully nonlinear Monge–Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.     

Mineyev–Dicks proof of the HN-conjecture and the Euler–Poincaré characteristic

A proof of the Hanna Neumann conjecture (HN-conjecture) based on the ideas of Mineyev andDicks is presented. The new ingredients are theQuillen formula for the Euler–Poincaré characteristic and the “abstract HN-conjecture.”     

Summation formulae of products of the Apostol–Bernoulli and Apostol–Euler polynomials

In this paper, a further investigation for the Apostol–Bernoulli and Apostol–Euler polynomials is performed, and some summation formulae of products of the Apostol–Bernoulli and Apostol–Euler polynomials are established by applying some summation transform techniques. Some illustrative special cases as well as immediate consequences of the main results are also considered.     

Multidimensional extension of the Morse–Hedlund theorem

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence x$x$ over a finite alphabet is ultimately periodic if and only if, for some n$n$, the number of different factors of length n$n$ appearing in x$x$ is less than n+1$n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let d≥2$d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of Z^d${\mathbb{Z}}^d$ definable by a first order formula in the Presburger arithmetic 〈Z;<,+〉$〈\mathbb{Z};<,+〉$. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse–Hedlund theorem to an arbitrary dimension d$d$ and characterize sets of Z^d${\mathbb{Z}}^d$ definable in 〈Z;<,+〉$〈\mathbb{Z};<,+〉$ in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often.     

On the theory of Besov–Herz spaces and Euler equations

Inspired by Kalton and Wood’s work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier–Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism T between the associated Fourier algebras (resp. Fourier–Stieltjes algebras) with completely bounded norm \({\left\| T \right\|_{cb}} < \sqrt {3/2} \left( {{\text{resp}}{\text{.}}{{\left\| T \right\|}_{cb}} < \sqrt {5/2} } \right)\). We show similar results involving the norm distortion ‖T‖‖T ^?1‖ with universal but non-explicit bound. Our results subsume Walter’s well-known structural theorems and also Lau’s theorem on the second conjugate of Fourier algebras.     

La suite de Thue–Morse et la catégorie Rec

We define the category $\mathrm{Rec}(\mathbf{K})$ of recurrence matrices over a field K and use it to calculate determinants of Hankel matrices related to the Thue–Morse sequence. To cite this article: R. Bacher, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Hydrodynamic Coherence and Vortex Solutions of the Euler–Helmholtz Equation

The form of the general solution of the steady-state Euler–Helmholtz equation (reducible to the Joyce–Montgomery one) in arbitrary domains on the plane is considered. This equation describes the dynamics of vortex hydrodynamic structures.