Gaussian Behavior of Quadratic Irrationals

We study the probabilistic behaviour of the continued fraction expansion of a quadratic irrational number, when weighted by some "additive" cost. We prove asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal with the underlying dynamical system associated with the Gauss map, and its weighted periodic trajectories. We work with analytic combinatorics methods, and mainly with bivariate Dirichlet generating functions; we use various tools, from number theory (the Landau Theorem), from probability (the Quasi-Powers Theorem), or from dynamical systems: our main object of study is the (weighted) transfer operator, that we relate with the generating functions of interest. The present paper exhibits a strong parallelism with the methods which have been previously introduced by Baladi and Vall\'ee in the study of rational trajectories. However, the present study is more involved and uses a deeper functional analysis framework.Comment: 39 pages In this second version, we have added an annex that provides a detailed study of the trace of the weighted transfer operator. We have also corrected an error that appeared in the computation of the norm of the operator when acting in the Banach space of analytic functions defined in the paper. Also, we give a simpler proof for Theorem

Surjective H-Colouring over reflexive digraphs

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen (2014) proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL; otherwise, it is NP-complete. By combining this result with some known and new results, we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3

An Average-case Analysis of the Gaussian Algorithm for Lattice Reduction

The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, the probability distribution decays geometrically, and the dynamics is characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Analysis in the discrete model and detailed numerical data are also presented

An Analysis of the gaussian algorithm for lattice reduction

Résumé disponible dans le fichier PD

Exome sequencing identifies germline variants in DIS3 in familial multiple myeloma

[Excerpt] Multiple myeloma (MM) is the third most common hematological malignancy, after Non-Hodgkin Lymphoma and Leukemia. MM is generally preceded by Monoclonal Gammopathy of Undetermined Significance (MGUS) [1], and epidemiological studies have identified older age, male gender, family history, and MGUS as risk factors for developing MM [2]. The somatic mutational landscape of sporadic MM has been increasingly investigated, aiming to identify recurrent genetic events involved in myelomagenesis. Whole exome and whole genome sequencing studies have shown that MM is a genetically heterogeneous disease that evolves through accumulation of both clonal and subclonal driver mutations [3] and identified recurrently somatically mutated genes, including KRAS, NRAS, FAM46C, TP53, DIS3, BRAF, TRAF3, CYLD, RB1 and PRDM1 [3,4,5]. Despite the fact that family-based studies have provided data consistent with an inherited genetic susceptibility to MM compatible with Mendelian transmission [6], the molecular basis of inherited MM predisposition is only partly understood. Genome-Wide Association (GWAS) studies have identified and validated 23 loci significantly associated with an increased risk of developing MM that explain ~16% of heritability [7] and only a subset of familial cases are thought to have a polygenic background [8]. Recent studies have identified rare germline variants predisposing to MM in KDM1A [9], ARID1A and USP45 [10], and the implementation of next-generation sequencing technology will allow the characterization of more such rare variants. [...]French National Cancer Institute (INCA) and the Fondation Française pour la Recherche contre le Myélome et les Gammapathies (FFMRG), the Intergroupe Francophone du Myélome (IFM), NCI R01 NCI CA167824 and a generous donation from Matthew Bell. This work was supported in part through the computational resources and staff expertise provided by Scientific Computing at the Icahn School of Medicine at Mount Sinai. Research reported in this paper was supported by the Office of Research Infrastructure of the National Institutes of Health under award number S10OD018522. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. The authors thank the Association des Malades du Myélome Multiple (AF3M) for their continued support and participation. Where authors are identified as personnel of the International Agency for Research on Cancer / World Health Organization, the authors alone are responsible for the views expressed in this article and they do not necessarily represent the decisions, policy or views of the International Agency for Research on Cancer / World Health Organizatio

Hematopoietic stem cell transplantation for adolescents and adults with inborn errors of immunity: an EBMT IEWP study.

peer reviewedAllogeneic hematopoietic stem cell transplantation (HSCT) is the gold standard curative therapy for infants and children with many inborn errors of immunity (IEI), but adolescents and adults with IEI are rarely referred for transplant. Lack of published HSCT outcome data outside small, single-center studies and perceived high risk of transplant-related mortality have delayed the adoption of HSCT for IEI patients presenting or developing significant organ damage later in life. This large retrospective, multicenter HSCT outcome study reports on 329 IEI patients (age range, 15-62.5 years at HSCT). Patients underwent first HSCT between 2000 and 2019. Primary endpoints were overall survival (OS) and event-free survival (EFS). We also evaluated the influence of IEI-subgroup and IEI-specific risk factors at HSCT, including infections, bronchiectasis, colitis, malignancy, inflammatory lung disease, splenectomy, hepatic dysfunction, and systemic immunosuppression. At a median follow-up of 44.3 months, the estimated OS at 1 and 5 years post-HSCT for all patients was 78% and 71%, and EFS was 65% and 62%, respectively, with low rates of severe acute (8%) or extensive chronic (7%) graft-versus-host disease. On univariate analysis, OS and EFS were inferior in patients with primary antibody deficiency, bronchiectasis, prior splenectomy, hepatic comorbidity, and higher hematopoietic cell transplant comorbidity index scores. On multivariable analysis, EFS was inferior in those with a higher number of IEI-associated complications. Neither age nor donor had a significant effect on OS or EFS. We have identified age-independent risk factors for adverse outcome, providing much needed evidence to identify which patients are most likely to benefit from HSCT

Genealogy of lattice reduction : algorithmic description and dynamical analyses (Natural extension of arithmetic algorithms and S-adic system)

"Natural extension of arithmetic algorithms and S-adic system". July 20～24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The present study describes the main algorithms devoted to solving the lattice reduction problem. This is a central algorithmic problem, due to its intrinsic theoretical interest, together to its multiple possible applications, located at many various areas in the interface between mathematics and computer science : computational number theory, integer programming but also complexity theory and cryptology. We first describe the algorithms themselves, inside their genealogy, and explain how the main ideas of small dimensions are used in higher dimensions. We are mainly interested in their probabilistic analysis, and wish to describe in a probabilistic way the main properties of their execution or the geometry of their outputs. Finally, the methodology that conducts these analyses is itself a main subject of interest, as it involves an original mixing between probabilistic modelling of the inputs, analytic combinatorics, and also tools that come from dynamical systems. This method, called dynamical analysis, is completely fruitful in small dimensions, and well explains the transition between the two smaller dimensions. For higher dimensions, such a direct approach is no longer possible, but it can be adapted via the introduction of simplified models

The Euclid algorithm is “totally ” gaussian

We consider Euclid’s gcd algorithm for two integers (p, q) with 1 ≤ p ≤ q ≤ N, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function d on the set of possible digits, asymptotically for N → ∞. For any additive cost of moderate growth d, Baladi and Vallée obtained a central limit theorem, and –in the case when the cost d is lattice – a local limit theorem. In both cases, the optimal speed was attained. When the cost is non lattice, the problem was later considered by Baladi and Hachemi, who obtained a local limit theorem under an intertwined diophantine condition which involves the cost d together with the “canonical ” cost c of the underlying dynamical system. The speed depends on the irrationality exponent that intervenes in the diophantine condition. We show here how to replace this diophantine condition by another diophantine condition, much more natural, which already intervenes in simpler problems of the same vein, and only involves the cost d. This “replacement ” is made possible by using the additivity of cost d, together with a strong property satisfied by the Euclidean Dynamical System, which states that the cost c is both “strongly ” non additive and diophantine in a precise sense. We thus obtain a local limit theorem, whose speed is related to the irrationality exponent which intervenes in the new diophantine condition. We mainly use the previous proof of Baladi and Hachemi, and “just ” explain how their diophantine condition may be replaced by our condition. Our result also provides a precise comparison between the rational trajectories of the Euclid dynamical system and the real trajectories