Nonlinear rotations on a lattice

We consider a prototypical two-parameter family of invertible maps of $\mathbb{Z}^2$, representing rotations with decreasing rotation number. These maps describe the dynamics inside the island chains of a piecewise affine discrete twist map of the torus, in the limit of fine discretisation. We prove that there is a set of full density of points which, depending of the parameter values, are either periodic or escape to infinity. The proof is based on the analysis of an interval-exchange map over the integers, with infinitely many intervals.Comment: LaTeX, 34 pages with 4 figure

Approach to a rational rotation number in a piecewise isometric system

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps; this dynamical system features an orderly array of stable periodic orbits having a smooth parameter dependence, plus irregular contributions which become negligible in the limit.Comment: LaTeX, 57 pages with 13 figure

<b><i>Topoisomerase 1</i></b> Promoter Variants and Benefit from Irinotecan in Metastatic Colorectal Cancer Patients

Objective: Topoisomerase 1 (topo-1) is an important target for the treatment of metastatic colorectal cancer (CRC). The aim of the present study was to evaluate the correlation between topo-1 single-nucleotide polymorphisms (SNPs) and clinical outcome in metastatic CRC (mCRC) patients. Methods: With the use of specific software (PROMO 3.0), we performed an in silico analysis of topo-1 promoter SNPs; the rs6072249 and rs34282819 SNPs were included in the study. DNA was extracted from 105 mCRC patients treated with FOLFIRI ± bevacizumab in the first line. SNP genotyping was performed by real-time PCR. Genotypes were correlated with clinical parameters (objective response rate, progression-free survival, and overall survival). Results: No single genotype was significantly associated with clinical variables. The G allelic variant of rs6072249 topo-1 SNP is responsible for GC factor and X-box-binding protein transcription factor binding. The same allelic variant showed a nonsignificant trend toward a shorter progression-free survival (GG, 7.5 months; other genotypes, 9.3 months; HR 1.823, 95% CI 0.8904-3.734; p = 0.1). Conclusion: Further analyses are needed to confirm that the topo-1 SNP rs6072249 and transcription factor interaction could be a part of tools to predict clinical outcome in mCRC patients treated with irinotecan-based regimens

Renormalizable two-parameter piecewise isometries.

We exhibit two distinct renormalization scenarios for two-parameter piecewise isometries, based on 2π/5 rotations of a rhombus and parameter-dependent translations. Both scenarios rely on the recently established renormalizability of a one-parameter triangle map, which takes place if and only if the parameter belongs to the algebraic number field K=Q(5) associated with the rotation matrix. With two parameters, features emerge which have no counterpart in the single-parameter model. In the first scenario, we show that renormalizability is no longer rigid: whereas one of the two parameters is restricted to K, the second parameter can vary continuously over a real interval without destroying self-similarity. The mechanism involves neighbouring atoms which recombine after traversing distinct return paths. We show that this phenomenon also occurs in the simpler context of Rauzy-Veech renormalization of interval exchange transformations, here regarded as parametric piecewise isometries on a real interval. We explore this analogy in some detail. In the second scenario, which involves two-parameter deformations of a three-parameter rhombus map, we exhibit a weak form of rigidity. The phase space splits into several (non-convex) invariant components, on each of which the renormalization still has a free parameter. However, the foliations of the different components are transversal in parameter space; as a result, simultaneous self-similarity of the component maps requires that both of the original parameters belong to the field K

A data-driven method for the temporal estimation of soil water potential and its application for shallow landslides prediction

Soil water potential is a key factor to study water dynamics in soil and for estimating the occurrence of natural hazards, as landslides. This parameter can be measured in field or estimated through physically-based models, limited by the availability of effective input soil properties and pre-liminary calibrations. Data-driven models, based on machine learning techniques, could overcome these gaps. The aim of this paper is then to develop an innovative machine learning methodology to assess soil water potential trends and to implement them in models to predict shallow landslides. Monitoring data since 2012 from test-sites slopes in Oltrepò Pavese (northern Italy) were used to build the models. Within the tested techniques, Random Forest models allowed an outstanding recon-struction of measured soil water potential temporal trends. Each model is sensitive to meteorological and hydrological characteristics according to soil depths and features. Reliability of the proposed models was confirmed by correct estimation of days when shallow landslides were triggered in the study areas in December 2020, after implementing the modeled trends on a slope stability model, and by the correct choice of physically-based rainfall thresholds. These results confirm the potential application of the developed methodology to estimate hydrological scenarios that could be used for decision-making purposes