Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos corrected; improvements on the statements and comments suggested by a referee. Keywords: singular flows, singular-hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm la

Recurrence and algorithmic information

In this paper we initiate a somewhat detailed investigation of the relationships between quantitative recurrence indicators and algorithmic complexity of orbits in weakly chaotic dynamical systems. We mainly focus on examples.Comment: 26 pages, no figure

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.Comment: 19 page

Statistical properties of Lorenz like flows, recent developments and perspectives

We comment on mathematical results about the statistical behavior of Lorenz equations an its attractor, and more generally to the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statisitcal behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated to the existence of physical measures: \emph{in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure}. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors.Comment: 40 pages; 10 figures; Keywords: sensitive dependence on initial conditions, physical measure, singular-hyperbolicity, expansiveness, robust attractor, robust chaotic flow, positive Lyapunov exponent, large deviations, hitting and recurrence times. Minor typos corrected and precise acknowledgments of financial support added. To appear in Int J of Bif and Chaos in App Sciences and Engineerin

Integrative organelle-based functional proteomics: in silico prediction of impaired functional annotations in SACS KO cell model

Autosomal recessive spastic ataxia of Charlevoix-Saguenay (ARSACS) is an inherited neurodegenerative disease characterized by early-onset spasticity in the lower limbs, axonal-demyelinating sensorimotor peripheral neuropathy, and cerebellar ataxia. Our understanding of ARSACS (genetic basis, protein function, and disease mechanisms) remains partial. The integrative use of organelle-based quantitative proteomics and whole-genome analysis proposed in the present study allowed identifying the affected disease-specific pathways, upstream regulators, and biological functions related to ARSACS, which exemplify a rationale for the development of improved early diagnostic strategies and alternative treatment options in this rare condition that currently lacks a cure. Our integrated results strengthen the evidence for disease-specific defects related to bioenergetics and protein quality control systems and reinforce the role of dysregulated cytoskeletal organization in the pathogenesis of ARSACS

Robust exponential decay of correlations for singular-flows

We construct open sets of Ck (k bigger or equal to 2) vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure. In particular we prove that the geometric Lorenz attractor has exponential decay of correlations with respect to the unique physical measure.Comment: Final version accepted for publication with added corrections (not in official published version) after O. Butterley pointed out to the authors that the last estimate in the argument in Subsection 4.2.3 of the previous version is not enough to guarantee the uniform non-integrability condition claimed. We have modified the argument and present it here in the same Subsection. 3 figures, 34 page

Application of deep learning model in the sonographic diagnosis of uterine adenomyosis

Background: This study aims to evaluate the diagnostic performance of Deep Learning (DL) machine for the detection of adenomyosis on uterine ultrasonographic images and compare it to intermediate ultrasound skilled trainees. Methods: Prospective observational study were conducted between 1 and 30 April 2022. Transvaginal ultrasound (TVUS) diagnosis of adenomyosis was investigated by an experienced sonographer on 100 fertile-age patients. Videoclips of the uterine corpus were recorded and sequential ultrasound images were extracted. Intermediate ultrasound-skilled trainees and DL machine were asked to make a diagnosis reviewing uterine images. We evaluated and compared the accuracy, sensitivity, positive predictive value, F1-score, specificity and negative predictive value of the DL model and the trainees for adenomyosis diagnosis. Results: Accuracy of DL and intermediate ultrasound-skilled trainees for the diagnosis of adenomyosis were 0.51 (95% CI, 0.48–0.54) and 0.70 (95% CI, 0.60–0.79), respectively. Sensitivity, specificity and F1-score of DL were 0.43 (95% CI, 0.38–0.48), 0.82 (95% CI, 0.79–0.85) and 0.46 (0.42–0.50), respectively, whereas intermediate ultrasound-skilled trainees had sensitivity of 0.72 (95% CI, 0.52–0.86), specificity of 0.69 (95% CI, 0.58–0.79) and F1-score of 0.55 (95% CI, 0.43–0.66). Conclusions: In this preliminary study DL model showed a lower accuracy but a higher specificity in diagnosing adenomyosis on ultrasonographic images compared to intermediate-skilled trainees

(Non)Invariance of dynamical quantities for orbit equivalent flows

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions

Infinite ergodic theory and Non-extensive entropies

We bring into account a series of result in the infinite ergodic theory that we believe that they are relevant to the theory of non-extensive entropie