We study the measurable dynamical properties of the interval map generated by the model-case erasing substitution $\rho $, defined by

$$ \begin{align*} \rho(00)=\text{empty word},\quad \rho(01)=1,\quad \rho(10)=0,\quad \rho(11)=01. \end{align*} $$

We investigate several questions related to the notion of recognizable morphism. The main result is a new proof of Mossé’s theorem and actually of a generalization to a more general class of morphisms due to Berthé et al [Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys. 39(11) (2019), 2896–2931]. We actually prove the result of Berthé et al for the most general class of morphisms, including ones with erasable letters. Our result is derived from a result concerning elementary morphisms for which we also provide a new proof. We also prove some new results which allow us to formulate the property of recognizability in terms of finite automata. We use this characterization to show that for an injective morphism, the property of being recognizable on the full shift for aperiodic points is decidable.

Many dietary guidelines recommend restricting the consumption of processed red meat (PRM) in favour of healthier foods such as fish, to reduce the risk of chronic conditions such as hypertension and diabetes. The objective of this study was to estimate the potential effect of replacing PRM for fatty fish, lean fish, red meat, eggs, pulses, or vegetables, on the risk of incident hypertension and diabetes. This was a prospective study of women in the E3N cohort study. Cases of diabetes and hypertension were based on self-report, specific questionnaires, and drug reimbursements. In the main analysis, information on regular dietary intake was assessed with a single food history questionaire, and food substitutions were modelled using cox proportional hazard models. 95 % confidence intervals were generated via bootstrapping. 71 081 women free of diabetes and 45 771 women free of hypertension were followed for an average of 18·7 and 18·3 years, respectively. 2681 incident cases of diabetes and 12 327 incident cases of hypertension were identified. Relative to PRM, fatty fish was associated with a 15 % lower risk of diabetes (HR = 0·85, 95 CI (0·73, 0·97)) and hypertension (HR = 0 85 (0·79, 0·91)). Between 3 and 10 % lower risk of hypertension or diabetes was also observed when comparing PRM with vegetables, unprocessed red meat or pulses. Relative to PRM, alternative protein sources such as fatty fish, unprocessed red meat, vegetables or pulses was associated with a lower risk of hypertension and diabetes.

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux–Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal {S}$ -adic representation where the morphisms in $\mathcal {S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper, we investigate those $\mathcal {S}$ -adic representations, heading towards an $\mathcal {S}$ -adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with two vertices.

Tourmaline from the Solnechnoe hydrothermal granitoid-related tin deposit in the Khabarovsk Krai, Russian Far East has been studied with electron microprobe, infrared and Mössbauer spectroscopy. Tourmaline formed in three distinct stages with different types of chemical substitution. Tourmaline from the first unmineralised stage is classified as dravite or schorl, which could be enriched locally in Ca, the X-site vacancy and F. This tourmaline is characterised by the Fe ↔ Mg and X vacancy + Al ↔ Na + Fe substitutions. The second, molybdenum-stage tourmaline, is schorl–dravite and fluor-schorl–fluor-dravite enriched in Ca, and a few compositions belong to the calcic group. The predominant substitution is Ca + Mg ↔ Na + Al. The third, tin-stage tourmaline, is classified as schorl–dravite with some tourmalines being fluor-schorl, oxy-schorl, foitite and magnesio-foitite. The tin-stage tourmaline is characterised by the substitutions Fe2+ ↔ Mg, Altot + O2– ↔ Fe2+ + OH–, and Fe3+ ↔ Altot. An increase of the Fe3+/Fetot value from 3–9% in the molybdenum stage to 12–16% in the tin-stage tourmalines indicates an increase in oxidation potential, which possibly contributed to cassiterite deposition. Comparison of tourmalines from greisen, porphyry and intrusion-related tin deposits worldwide shows they differ in primary chemical substitutions so can be characterised by this mechanism. The Fe3+/Fetot value in tourmaline also appears to be one of the indications for the tin deposit type. The Fe3+/Fetot value increases from <10% in greisen tourmaline through 15% in tourmaline from intrusion-related deposits to 20% in tourmaline from porphyry deposits.

Euhedral crystals of complexly zoned niobian titanite (up to 0.3 mm) are enclosed in hedenbergite (Hd53—81Di15—43Jh3-5) and quartz from a hedenbergite vein skarn at Kamenne doly near Pisek, Czech Republic. They are associated with minor clinozoisite-epidote (Ps3—22), calcite, plagioclase (An95). scapolite (Me80—82), scheelite, pyrrhotite, fluorapatite, arsenopyrite, native bismuth and Bi,Te-minerals. The following textural and compositional subtypes were recognized: (I) Nb-rich titanite, (II) Nb-moderate titanite in the central zone, (III) Nb-poor, Sn-enriched titanite and (IV) Nb-poor, Al,F-rich titanite in the outer zone. The substitution Al(Nb,Ta)Ti—2 is dominant in subtypes I and II, the titanite subtype I being characterized by elevated contents of Al ≤ 0.257 atoms per formula unit (a.p.f.u.), Nb (≤ 0.161 a.p.f.u.) and Ta (≤ 0.037 a.p.f.u.). Amounts of Al, Nb and Ta in subtype II are smaller and more variable. The minor substitution SnTi—1 occurs chiefly in titanite subtype III with a content of Sn ≤ 0.039 a.p.f.u.. The substitution Al(F,OH)(TiO)_i is typical for titanite subtype IV exhibiting elevated contents of Al (s£ 0.221 a.p.f.u.), F (≤ 0.196 a.p.f.u.) and Fe (≤ 0.039 a.p.f.u.).

The negative relationship of substitutions Al(F,OH)(TiO)_i vs. SnTi—1 and Al(Nb,Ta)Ti—1 is constrained chiefly by crystal structure rather than by the composition of parent medium alone. Textural relations suggest that the Nb-moderate titanite in the core zone and entire outer zone are products of fluids-induced dissolution-reprecipitation processes. The studied niobian titanite represents a new F-enriched type from a medium-grade, calc-silicate rock.

Laminations are classic sets of disjoint and non-self-crossing curves on surfaces. Lamination languages are languages of two-way infinite words which code laminations by using associated labeled embedded graphs, and which are subshifts. Here, we characterize the possible exact affine factor complexities of these languages through bouquets of circles, i.e. graphs made of one vertex, as representative coding graphs. We also show how to build families of laminations together with corresponding lamination languages covering all the possible exact affine complexities.

Une substitution est un morphisme de monoïdes libres : chaque lettre a pour image un mot, et l'image d'un mot est la concaténation des images de ses lettres. Cet article introduit une généralisation de la notion de substitution, où l'image d'une lettre n'est plus un mot mais un motif, c'est-à-dire un “mot à trous”, l'image d'un mot étant obtenue en raccordant les motifs correspondant à chacune de ses lettres à l'aide de règles locales. On caractérise complètement les substitutions par des motifs qui sont définies sur toute suite biinfinie, et on explique comment les construire. On montre que toute suite biinfinie qui est point fixe d'une substitution par des motifs est substitutive, c'est-à-dire est l'image, par un morphisme lettre à lettre, d'un point fixe de substitution (au sens usuel).