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With α irrational the graph of [jα] against integer j, displays interesting patterns of self-matching. This is best seen by comparing the Bernoulli (characteristic Sturmian) or difference sequence 〈βj〉, term by term with the Bernoulli sequence displaced by k terms 〈βj−k〉, where βj=[(j+1)α]−[jα].It is shown that the fraction of such self-matching is the surprisingly simple M(k)=max(|1−2{α}|,|1−2{kα}|).Of particular interest is the graph of M(k) against k as it is seen to exhibit an unexpected Moiré pattern obtained simply by folding the lower half of the graph of {kα} over the upper half.

We give a detailed account of an $\it{ab}$ $\it{initio}$ spectral approach for the calculation of energy spectra of two active electron atoms in a system of hyperspherical coordinates. In this system of coordinates, the Hamiltonian has the same structure as the one of atomic hydrogen with the Coulomb potential expressed in terms of a hyperradius and the nuclear charge replaced by an angle dependent effective charge. The simplest spectral approach consists in expanding the hyperangular wave function in a basis of hyperspherical harmonics. This expansion however, is known to be very slowly converging. Instead, we introduce new hyperangular sturmian functions. These functions do not have an analytical expression but they treat the first term of the multipole expansion of the electron-electron interaction potential, namely the radial electron correlation, exactly. The properties of these new functions are discussed in detail. For the basis functions of the hyperradius, several choices are possible. In the present case, we use Coulomb sturmian functions of half integer angular momentum. We show that, in the case of H$^-$, the accuracy of the energy and the width of the resonance states obtained through a single diagonalization of the Hamiltonian, is comparable to the values given by state-of-the-art methods while using a much smaller basis set. In addition, we show that precise values of the electric-dipole oscillator strengths for $S\rightarrow P$ transitions in helium are obtained thereby confirming the accuracy of the bound state wave functions generated with the present method., Comment: 28 pages, 4 figures

The previous paper in this series introduced a class of infinite binary strings, called two-pattern strings, that constitute a significant generalization of, and include, the much-studied Sturmian strings. The class of two-pattern strings is a union of a sequence of increasing (with respect to inclusion) subclasses Tλ of two-pattern strings of scope λ, λ=1,2,…. Prefixes of two-pattern strings are interesting from the algorithmic point of view (their recognition, generation, and computation of repetitions and near-repetitions) and since they include prefixes of the Fibonacci and the Sturmian strings, they merit investigation of how many finite two-pattern strings of a given size there are among all binary strings of the same length. In this paper we first consider the frequency fλ(n) of occurrence of two-pattern strings of length n and scope λ among all strings of length n on {a,b}: we show that limn→∞fλ(n)=0, but that for strings of lengths n⩽2λ, two-pattern strings of scope λ constitute more than one-quarter of all strings. Since the class of Sturmian strings is a subset of two-pattern strings of scope 1, it was natural to focus the study of the substring complexity of two-pattern strings to those of scope 1. Though preserving the aperiodicity of the Sturmian strings, the generalization to two-pattern strings greatly relaxes the constrained substring complexity (the number of distinct substrings of the same length) of the Sturmian strings. We derive upper and lower bounds on C1(k) (the number of distinct substring of length k) of two-pattern strings of scope 1, and we show that it can be considerably greater than that of a Sturmian string. In fact, we describe circumstances in which limk→∞(C1(k)−k)=∞.

Three new characterisations of balanced words are presented. Each of these characterisations is based on the ordering of a shift orbit, either lexicographically or with respect to the norm |·|1 (which counts the number of occurrences of the symbol 1).