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We obtain expansions of ergodic integrals for $\Z^d$-covers of compact self-similar translation flows, and as a consequence we obtain a form of weak rational ergodicity with optimal rates. As examples, we consider the so-called self-similar $(s,1)$-staircase flows ($\Z$-extensions of self-similar translations flows of genus-$2$ surfaces), and particular cases of the Ehrenfest wind-tree model., Comment: This research was the result of the 2023 Research-in-Teams project 0223 "Limit Theorems for Parabolic Dynamical Systems" at the Erwin Schr\"odinger Institute, Vienna. This updated version contains some improved versions of proofs

Finite element analysis of soft tissues is a well-developed method that allows estimation of mechanical quantities (e.g. stresses, strains). A constitutive law has to be used to characterise the individual tissues. This is complex as biological tissues are generally visco-hyperelastic, anisotropic, and heterogenous. A specific characteristic, their nearly incompressibility, was well reported in the literature, but very little effort has been made to compare volume variations computed by the simulations with in vivo measurements. In the present study, volume changes of the fat pad during controlled indentations of the human heel region were estimated from segmented medical images using digital volume correlation. Indentations were repeated with high and mild intensity normal and shear loads. The experiment was reproduced using finite element modelling with several values of Poisson's ratio for the fat pad, extracted from literature values (from 0.4500 to 0.4999). Estimated fat pad volume changes were compared to the measured ones to assess the best value of Poisson's ratio in each indentation case. The impact of the Poisson's ratio on the Jacobian of the deformation gradient and the volumetric strains was also computed. A single value of Poisson's ratio could not fit all the indentation cases. Estimated volume changes were between 0.9 % - 11.7 % with a Poisson's ratio from 0.4500 to 0.4999. The best fit was obtained with a 0.4900 Poisson's ratio except for the high normal load where a value of 0.4999 resulted in less error. In conclusion, special care should be taken when setting the Poisson's ratio as the resulting estimated deformations may become unrealistic when the value is far from incompressible materials.

Diabetic foot ulcers are triggered by mechanical loadings applied to the surface of the plantar skin. Strain is considered to play a crucial role in relation to ulcer etiology and can be assessed by Finite Element (FE) modelling. A difficulty in the generation of these models is the choice of the soft tissue material properties. In the literature, many studies attempt to model the behavior of the heel soft tissues by implementing constitutive laws that can differ significantly in terms of mechanical response. Moreover, current FE models lack of proper evaluation techniques that could estimate their ability to simulate realistic strains. In this article, we propose and evaluate a FE model of the human heel for diabetic foot ulcer prevention. Soft tissue constitutive laws are defined through the fitting of experimental stretch-stress curves published in the literature. The model is then evaluated through Digital Volume Correlation (DVC) based on non-rigid 3D Magnetic Resonance Image Registration. The results from FE analysis and DVC show similar strain locations in the fat pad and strain intensities according to the type of applied loads. For additional comparisons, different sets of constitutive models published in the literature are applied into the proposed FE mesh and simulated with the same boundary conditions. In this case, the results in terms of strains show great diversity in locations and intensities, suggesting that more research should be developed to gain insight into the mechanical properties of these tissues.

Recently, a new bi-layer dressing was proposed by Urgo RID to reduce the healing time of pressure ulcers (PU). This dressing was numerically evaluated in previously published work. In the current work, the influence on the maximal shear strains of modelling parameters such as the dressing local geometry, the pressure applied by the gauze inside the wound, the wound deepness, and the mattress stiffness, was assessed. A sensitivity analysis was performed on these four parameters. Among all experiments, the mean maximal Green--Lagrange shear strain was 0.29. The gauze pressure explained 60% of the model response in terms of the volume of tissues under strains of 0.3, while the wound deepness explained 28%. The mattress had a significant, but low impact, whereas the dressing local geometry had no significant impact. As expected, the wound deepness was one of the most influential parameters. The gauze turned out to be more significant than expected. This may be explained by the large range of values chosen for this study. The results should be extended to more subjects, but still suggest that the gauze is a parameter that might not be neglected. Care should also be taken in clinical practice when using gauze that could have either a positive or negative impact on the soft tissues' strains. This may also depend on the wound deepness.

We exhibit an infinite family of discrete subgroups of ${Sp}_4(\mathbb R)$ which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters $\left(\tfrac{N-3}{2N},\tfrac{N-1}{2N}, \tfrac{N+1}{2N}, \tfrac{N+3}{2N}\right)$ at infinity and maximal unipotent monodromy at zero, for any integer $N\geq 4$. Additionally, we relate the cones used for ping-pong in $\mathbb R^4$ with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian., Comment: 53 pages, 12 figures

We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3. Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic., Comment: 17 pages, 8 figures. Transactions of the Moscow Mathematical Society, American Mathematical Society, In press

Effects of Single Event Upsets (SEU) and Single Event Transients (SET) are studied in the FE-I4B chip of the innermost layer of the ATLAS pixel system. SEU/SET affect the FE-I4B Global Registers as well as the settings for the individual pixels, causing, among other things, occupancy losses, drops in the low voltage currents, noisy pixels, and silent pixels. Quantitative data analysis and simulations indicate that SET dominate over SEU on the load line of the memory. Operational issues and mitigation techniques are presented., Comment: 30 pages with 32 figures

We propose a new point of view on multidimensional continued fraction algorithms inspired by Rauzy induction. The generic behaviour of such an algorithm is described here as a random walk on a graph that we call simplicial system. These systems provide a family of examples for random walks with memory recorded by a finite dimensional vector. We introduce a general criterion on these graphs that induces ergodicity together with a bundle of many other dynamical properties. In particular, after computing the representation of Brun, Selmer and Arnoux-Rauzy-Poincare algorithm in this formalism, it provides a unified proof of ergodicity for these classical examples as well as new results such as uniqueness of the measure of maximal entropy on a canonical suspension. These objects also bring a new perspective to some fractal sets such as Rauzy gaskets. We show general explicit upper bound on Hausdorff dimensions of fractals described in this formalism as well as a construction of their measure of maximal entropy. This implies in particular that the Rauzy gasket in all dimensions has Hausdorff dimension strictly smaller than its ambient space, as well as sharper bounds on the dimension and an asymptotic result.

In the current paper we prove simplicity for the spectrum of Lyapunov exponents for triangle sequence and Selmer algorithm in dimension 3. We introduce a strategy that can be applied for a wide class of Markovian MCF.

Consider a windtree model with several parallel arbitrary right-angled obstacles placed periodically on the plane. We show that its diffusion rate is the largest Lyapunov exponent of some stratum of quadratic differentials and exhibit a new general strategy to compute the generic diffusion rate of such models. This result enables us to compute numerically the diffusion rates of a large family of models, and to observe its asymptotic behaviour according to the shape of the obstacles.

During the shutdown of the CERN Large Hadron Collider in 2013-2014, an additional pixel layer was installed between the existing Pixel detector of the ATLAS experiment and a new, smaller radius beam pipe. The motivation for this new pixel layer, the Insertable B-Layer (IBL), was to maintain or improve the robustness and performance of the ATLAS tracking system, given the higher instantaneous and integrated luminosities realised following the shutdown. Because of the extreme radiation and collision rate environment, several new radiation-tolerant sensor and electronic technologies were utilised for this layer. This paper reports on the IBL construction and integration prior to its operation in the ATLAS detector., Comment: 90 pages in total. Author list: ATLAS IBL Collaboration, starting page 2. 69 figures, 20 tables. Published in Journal of Instrumentation. All figures available at: https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PLOTS/PIX-2018-001

Consider the flat bundle on $\mathrm{CP}^1 - \{0,1,\infty \}$ corresponding to solutions of the hypergeometric differential equation $ \prod_{i=1}^h (\mathrm D - \alpha_i) - z \prod_{j=1}^h (\mathrm D - \beta_j) = 0$ where $\mathrm D = z \frac {d}{dz}$. For $\alpha_i$ and $\beta_j$ distinct real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on $\mathrm{CP}^1 - \{0,1,\infty \}$, we associate $n$ Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge structure and study the dependence of the Lyapunov exponents in terms of these degrees by means of numerical simulations.

We describe in this article the dynamics of a $1$-parameter family of affine interval exchange transformations. It amounts to studying the directional foliations of a particular affine surface, the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial, but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. s study is achieved through the analysis the dynamics of the Veech group of this surface combined a modified version of Rauzy induction in the context of affine interval exchange transformations.

We show an upper bound for the sum of positive Lyapunov exponents of any Teichm\"uller curve in strata of quadratic differentials with at least one zero of large multiplicity. As a corollary, it holds for any $SL(2,\mathbb R)$-invariant subspaces defined over $\mathbb Q$ in these strata. This proves Grivaux-Hubert's conjecture about the asymptotics of Lyapunov exponents for strata with a large number of poles in the situation when at least one zero has large multiplicity.

We introduce a class of objects which we call 'affine surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalise the notion of Veech group to affine surfaces, and we prove a structure result about these Veech groups., Comment: 24 pages, 22 figures