A conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials

A synthesis of evidence for policy from behavioural science during COVID-19

Scientific evidence regularly guides policy decisions1, with behavioural science increasingly part of this process2. In April 2020, an influential paper3 proposed 19 policy recommendations (‘claims’) detailing how evidence from behavioural science could contribute to efforts to reduce impacts and end the COVID-19 pandemic. Here we assess 747 pandemic-related research articles that empirically investigated those claims. We report the scale of evidence and whether evidence supports them to indicate applicability for policymaking. Two independent teams, involving 72 reviewers, found evidence for 18 of 19 claims, with both teams finding evidence supporting 16 (89%) of those 18 claims. The strongest evidence supported claims that anticipated culture, polarization and misinformation would be associated with policy effectiveness. Claims suggesting trusted leaders and positive social norms increased adherence to behavioural interventions also had strong empirical support, as did appealing to social consensus or bipartisan agreement. Targeted language in messaging yielded mixed effects and there were no effects for highlighting individual benefits or protecting others. No available evidence existed to assess any distinct differences in effects between using the terms ‘physical distancing’ and ‘social distancing’. Analysis of 463 papers containing data showed generally large samples; 418 involved human participants with a mean of 16,848 (median of 1,699). That statistical power underscored improved suitability of behavioural science research for informing policy decisions. Furthermore, by implementing a standardized approach to evidence selection and synthesis, we amplify broader implications for advancing scientific evidence in policy formulation and prioritization

Chronic exposure to haloperidol and olanzapine leads to common and divergent shape changes in the rat hippocampus in the absence of grey-matter volume loss

One of the most consistently reported brain abnormalities in schizophrenia (SCZ) is decreased volume and shape deformation of the hippocampus. However, the potential contribution of chronic antipsychotic medication exposure to these phenomena remains unclear. We examined the effect of chronic exposure (8 weeks) to clinically relevant doses of either haloperidol (HAL) or olanzapine (OLZ) on adult rat hippocampal volume and shape using ex vivo structural MRI with the brain retained inside the cranium to prevent distortions due to dissection, followed by tensor-based morphometry (TBM) and elastic surface-based shape deformation analysis. The volume of the hippocampus was also measured post-mortem from brain tissue sections in each group. Chronic exposure to either HAL or OLZ had no effect on the volume of the hippocampus, even at exploratory thresholds, which was confirmed post-mortem. In contrast, shape deformation analysis revealed that chronic HAL and OLZ exposure lead to both common and divergent shape deformations (q = 0.05, FDR-corrected) in the rat hippocampus. In particular, in the dorsal hippocampus, HAL exposure led to inward shape deformation, whereas OLZ exposure led to outward shape deformation. Interestingly, outward shape deformations that were common to both drugs occurred in the ventral hippocampus. These effects remained significant after controlling for hippocampal volume suggesting true shape changes. Chronic exposure to either HAL or OLZ leads to both common and divergent effects on rat hippocampal shape in the absence of volume change. The implications of these findings for the clinic are discussed