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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.

Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular proper S-adic subshifts. They are generated by iterating sequences of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter. This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux-Rauzy subshifts and natural coding of interval exchange transformations. We compute their dimension group and investigate the relation between the triviality of the infinitesimal subgroup and rational independence of letter measures. We also introduce the notion of balanced functions and provide a topological characterization of bal-ancedness for primitive unimodular proper S-adic subshifts.

Summary Background Chimeric antigen receptor (CAR) T-cell therapy can induce side-effects such as cytokine release syndrome and immune effector cell-associated neurotoxicity syndrome (ICANS), which often require intensive care unit admission. The aim of this study was to describe management of critically ill CAR T-cell recipients in intensive care. Methods This international, multicentre, observational cohort study was done in 21 intensive care units in France, Spain, the USA, the UK, Russia, Canada, Germany, and Austria. Eligible patients were aged 18 years or older; had received CAR T-cell therapy in the past 30 days; and had been admitted to intensive care for any reason. Investigators retrospectively included patients admitted between Feb 1, 2018, and Feb 1, 2019, and prospectively included patients admitted between March 1, 2019, and Feb 1, 2020. Demographic, clinical, laboratory, treatment, and outcome data were extracted from medical records. The primary endpoint was 90-day mortality. Factors associated with mortality were identified using a Cox proportional hazard model. Findings 942 patients received CAR T-cell therapy, of whom 258 (27%) required admission to intensive care and 241 (26%) were included in the analysis. Admission to intensive care was needed within median 4·5 days (IQR 2·0–7·0) of CAR T-cell infusion. 90-day mortality was 22·4% (95% CI 17·1–27·7; 54 deaths). At initial evaluation on admission, isolated cytokine release syndrome was identified in 101 patients (42%), cytokine release syndrome and ICANS in 93 (39%), and isolated ICANS in seven (3%) patients. Grade 3–4 cytokine release syndrome within 1 day of admission to intensive care was found in 50 (25%) of 200 patients and grade 3–4 ICANS in 38 (35%) of 108 patients. Bacterial infection developed in 30 (12%) patients. Life-saving treatments were used in 75 (31%) patients within 24 h of admission to intensive care, primarily vasoactive drugs in 65 (27%) patients. Factors independently associated with 90-day mortality by multivariable analysis were frailty (hazard ratio 2·51 [95% CI 1·37–4·57]), bacterial infection (2·12 [1·11–4·08]), and lifesaving therapy within 24 h of admission (1·80 [1·05–3·10]). Interpretation Critical care management is an integral part of CAR T-cell therapy and should be standardised. Studies to improve infection prevention and treatment in these high-risk patients are warranted. Funding Groupe de Recherche Respiratoire en Reanimation Onco-Hematologique.

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $\mu$-almost every $\mathcal{C}$-adic sequence is balanced, where $\mu$ is any shift-invariant ergodic Borel probability measure on $\{1,2\}^\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $\mu$ is negative., Comment: 42 pages, 9 figures. Extended and augmented version of arXiv:1707.02741

International audience; Received (Day Month Year) Accepted (Day Month Year) Communicated by (xxxxxxxxxx) Dendric words are infinite words that are defined in terms of extension graphs. These are bipartite graphs that describe the left and right extensions of factors. Dendric words are such that all their extension graphs are trees. They are also called tree words. This class of words includes classical families of words such as Sturmian words, codings of interval exchanges, or else, Arnoux-Rauzy words. We investigate here the properties of substitutive dendric words and prove some rigidity properties, that is, algebraic properties on the set of substitutions that fix a dendric word. We also prove that aperiodic minimal dendric subshifts (generated by dendric words) cannot have rational topological eigenvalues, and thus, cannot be generated by constant length substitutions.

This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\ge 0}$ counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for $(S(n))_{n\ge 0}$. This leads to a connection with the $2$-regular Stern-Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on the Fibonacci sequence. Again our tree structure permits us to obtain recurrence relations for and the F-regularity of the corresponding sequence., Comment: 28 pages, 10 figures

Two finite words are k-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most k with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The k-binomial complexity of an infinite word \(\mathbf {x}\) maps the integer n to the number of classes in the quotient, by this k-binomial equivalence relation, of the set of factors of length n occurring in \(\mathbf {x}\). This complexity measure has not been investigated very much. In this paper, we characterize the k-binomial complexity of the Thue–Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue–Morse word is aperiodic, its k-binomial complexity eventually takes only two values. In this paper, we first express the number of occurrences of subwords appearing in iterates of the form \(\varPsi ^\ell (w)\) for an arbitrary morphism \(\varPsi \). We also thoroughly describe the factors of the Thue–Morse word by introducing a relevant new equivalence relation.