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

A statistical mechanical analysis on the possibility of achieving fair cylindrical dice

Many have dedicated their time trying to determine the ideal conditions for a cylinder to have equal probabilities of falling with one of its faces facing upwards or on its side. However, to this day, there is no concrete analysis of what these conditions should be. In order to determine such circumstances, a theoretical analysis was conducted, considering approaches from Rigid Body Dynamics and Statistical Mechanics. An experimental system was also built to improve control over the launches, and a comparative analysis was performed between the results obtained experimentally and the theory. It was concluded that the environment and other launching conditions have a significant influence; nevertheless, it is possible, under controlled conditions, to determine, within certain limits, the expected probabilities.Comment: 27 pages, 25 figure

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