Numerical instability of the Akhmediev breather and a finite-gap model of it

In this paper we study the numerical instabilities of the NLS Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be written in terms of different elementary functions in different time regions and, ultimately, it shows an exact recurrence of rogue waves described, at each appearance, by the Akhmediev breather. We discover a remarkable empirical formula connecting the recurrence time with the number of time steps used in the SSFM and, via our recent theoretical findings, we establish that the SSFM opens up a vertical unstable gap whose length can be computed with high accuracy, and is proportional to the inverse of the square of the number of time steps used in the SSFM. This neat picture essentially changes when the round-off error is sufficiently large. Indeed experiments in standard double precision show serious instabilities in both the periods and phases of the recurrence. In contrast with it, as predicted by the theory, replacing the exact Akhmediev Cauchy datum by its first harmonic approximation, we only slightly modify the numerical output. Let us also remark, that the first rogue wave appearance is completely stable in all experiments and is in perfect agreement with the Akhmediev formula and with the theoretical prediction in terms of the Cauchy data.Comment: 27 pages, 8 figures, Formula (30) at page 11 was corrected, arXiv admin note: text overlap with arXiv:1707.0565

What Is the Negative Predictive Value of Multiparametric Magnetic Resonance Imaging in Excluding Prostate Cancer at Biopsy? A Systematic Review and Meta-analysis from the European Association of Urology Prostate Cancer Guidelines Panel

Context: It remains unclear whether patients with a suspicion of prostate cancer (PCa) and negative multiparametric magnetic resonance imaging (mpMRI) can safely obviate prostate biopsy. Objective: To systematically review the literature assessing the negative predictive value (NPV) of mpMRI in patients with a suspicion of PCa. Evidence acquisition: The Embase, Medline, and Cochrane databases were searched up to February 2016. Studies reporting prebiopsy mpMRI results using transrectal or transperineal biopsy as a reference standard were included. We further selected for meta-analysis studies with at least 10-core biopsies as the reference standard, mpMRI comprising at least T2-weighted and diffusion-weighted imaging, positive mpMRI defined as a Prostate Imaging Reporting Data System/Likert score of ≥3/5 or ≥4/5, and results reported at patient level for the detection of overall PCa or clinically significant PCa (csPCa) defined as Gleason ≥7 cancer. Evidence synthesis: A total of 48 studies (9613 patients) were eligible for inclusion. At patient level, the median prevalence was 50.4% (interquartile range [IQR], 36.4–57.7%) for overall cancer and 32.9% (IQR, 28.1–37.2%) for csPCa. The median mpMRI NPV was 82.4% (IQR, 69.0–92.4%) for overall cancer and 88.1% (IQR, 85.7–92.3) for csPCa. NPV significantly decreased when cancer prevalence increased, for overall cancer (r = –0.64, p < 0.0001) and csPCa (r = –0.75, p = 0.032). Eight studies fulfilled the inclusion criteria for meta-analysis. Seven reported results for overall PCa. When the overall PCa prevalence increased from 30% to 60%, the combined NPV estimates decreased from 88% (95% confidence interval [95% CI], 77–99%) to 67% (95% CI, 56–79%) for a cut-off score of 3/5. Only one study selected for meta-analysis reported results for Gleason ≥7 cancers, with a positive biopsy rate of 29.3%. The corresponding NPV for a cut-off score of ≥3/5 was 87.9%. Conclusions: The NPV of mpMRI varied greatly depending on study design, cancer prevalence, and definitions of positive mpMRI and csPCa. As cancer prevalence was highly variable among series, risk stratification of patients should be the initial step before considering prebiopsy mpMRI and defining those in whom biopsy may be omitted when the mpMRI is negative. Patient summary This systematic review examined if multiparametric magnetic resonance imaging (MRI) scan can be used to reliably predict the absence of prostate cancer in patients suspected of having prostate cancer, thereby avoiding a prostate biopsy. The results suggest that whilst it is a promising tool, it is not accurate enough to replace prostate biopsy in such patients, mainly because its accuracy is variable and influenced by the prostate cancer risk. However, its performance can be enhanced if there were more accurate ways of determining the risk of having prostate cancer. When such tools are available, it should be possible to use an MRI scan to avoid biopsy in patients at a low risk of prostate cancer

Nonlinear Supersymmetry as a Hidden Symmetry

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A new algorithm for computing the Geronimus transformations for large shifts

A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. The basic Geronimus transformation with shift ? transforms the monic Jacobi matrix associated with a measure d? into the monic Jacobi matrix associated with d?/(x????)?+?C?(x????), for some constant C. In this paper we examine the algorithms available to compute this transformation and we propose a more accurate algorithm, estimate its forward errors, and prove that it is forward stable. In particular, we show that for C?=?0 the problem is very ill-conditioned, and we present a new algorithm that uses extended precision

The benefits and harms of different extents of lymph node dissection during radical prostatectomy for prostate cancer: a systematic review

Context: There is controversy regarding the therapeutic role of pelvic lymph node dissection (PLND) in patients undergoing radical prostatectomy for prostate cancer (PCa). Objective: To systematically review the relevant literature assessing the relative benefits and harms of PLND for oncological and non-oncological outcomes in patients undergoing radical prostatectomy for PCa. Evidence acquisition: MEDLINE, MEDLINE In-Process, Embase, and the Cochrane Central Register of Controlled Trials were searched up to December 2015. Comparative studies evaluating no PLND, limited, standard, and (super)-extended PLND that reported oncological and non-oncological outcomes were included. Risk-of-bias and confounding assessments were performed. A narrative synthesis was undertaken. Evidence synthesis: Overall, 66 studies recruiting a total of 275,269 patients were included (44 full-text articles and 22 conference abstracts). Oncological outcomes were addressed by 29 studies, one of which was a randomized clinical trial (RCT). Non-oncological outcomes were addressed by 43 studies, three of which were RCTs. There were high risks of bias and confounding in most studies. Conflicting results emerged when comparing biochemical and clinical recurrence, while no significant differences were observed among groups for survival. Conversely, the majority of studies showed that the more extensive the PLND, the greater the adverse outcomes in terms of operating time, blood loss, length of stay, and postoperative complications. No significant differences were observed in terms of urinary continence and erectile function recovery. Conclusions: Although representing the most accurate staging procedure, PLND and its extension are associated with worse intraoperative and perioperative outcomes, whereas a direct therapeutic effect is still not evident from the current literature. The current poor quality of evidence indicates the need for robust and adequately powered clinical trials. Patient summary: Based on a comprehensive review of the literature, this article summarizes the benefits and harms of removing lymph nodes during surgery to remove the prostate because of PCa. Although the quality of the data from the studies was poor, the review suggests that lymph node removal may not have any direct benefit on cancer outcomes and may instead result in more complications. Nevertheless, the procedure remains justified because it enables accurate assessment of cancer spread

Bidirectional Coupling between Astrocytes and Neurons Mediates Learning and Dynamic Coordination in the Brain: A Multiple Modeling Approach

In recent years research suggests that astrocyte networks, in addition to nutrient and waste processing functions, regulate both structural and synaptic plasticity. To understand the biological mechanisms that underpin such plasticity requires the development of cell level models that capture the mutual interaction between astrocytes and neurons. This paper presents a detailed model of bidirectional signaling between astrocytes and neurons (the astrocyte-neuron model or AN model) which yields new insights into the computational role of astrocyte-neuronal coupling. From a set of modeling studies we demonstrate two significant findings. Firstly, that spatial signaling via astrocytes can relay a “learning signal” to remote synaptic sites. Results show that slow inward currents cause synchronized postsynaptic activity in remote neurons and subsequently allow Spike-Timing-Dependent Plasticity based learning to occur at the associated synapses. Secondly, that bidirectional communication between neurons and astrocytes underpins dynamic coordination between neuron clusters. Although our composite AN model is presently applied to simplified neural structures and limited to coordination between localized neurons, the principle (which embodies structural, functional and dynamic complexity), and the modeling strategy may be extended to coordination among remote neuron clusters