A central limit theorem for time-dependent dynamical systems

The work [8] established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random, i. e. specific examples are also found, which are not small perturbations of a given map. Our approach uses martingale approximation technique in the form of [9]

Deep Treatment Response Assessment and Prediction of Colorectal Cancer Liver Metastases

Evaluating treatment response is essential in patients who develop colorectal liver metastases to decide the necessity for second-line treatment or the admissibility for surgery. Currently, RECIST1.1 is the most widely used criteria in this context. However, it involves time-consuming, precise manual delineation and size measurement of main liver metastases from Computed Tomography (CT) images. Moreover, an early prediction of the treatment response given a specific chemotherapy regimen and the initial CT scan would be of tremendous use to clinicians. To overcome these challenges, this paper proposes a deep learning-based treatment response assessment pipeline and its extension for prediction purposes. Based on a newly designed 3D Siamese classification network, our method assigns a response group to patients given CT scans from two consecutive follow-ups during the treatment period. Further, we extended the network to predict the treatment response given only the image acquired at first time point. The pipelines are trained on the PRODIGE20 dataset collected from a phase-II multi-center clinical trial in colorectal cancer with liver metastases and exploit an in-house dataset to integrate metastases delineations derived from a U-Net inspired network as additional information. Our approach achieves overall accuracies of 94.94% and 86.86% for treatment response assessment and early prediction respectively, suggesting that both treatment response assessment and prediction issues can be effectively solved with deep learning

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Abstract. Sufficient conditions have been given for the convergence in norm and a.e. of the ergodic Hilbert transform ([11], [5], [6]). Here we apply these conditions to the rotated ergodic Hilbert transform ∑∞ n=1 λn n T nf, where λ is a complex number of modulus 1. When T is a contraction in a Hilbert space, we show that the logarithmic Hausdorff dimension of the set of λ’s for which this series does not converge is at most 2 and give examples where this bound is attained