Formalizing the Ring of Witt Vectors

The ring of Witt vectors $\mathbb{W} R$ over a base ring $R$ is an important tool in algebraic number theory and lies at the foundations of modern $p$-adic Hodge theory. $\mathbb{W} R$ has the interesting property that it constructs a ring of characteristic $0$ out of a ring of characteristic $p > 1$, and it can be used more specifically to construct from a finite field containing $\mathbb{Z}/p\mathbb{Z}$ the corresponding unramified field extension of the $p$-adic numbers $\mathbb{Q}_p$ (which is unique up to isomorphism). We formalize the notion of a Witt vector in the Lean proof assistant, along with the corresponding ring operations and other algebraic structure. We prove in Lean that, for prime $p$, the ring of Witt vectors over $\mathbb{Z}/p\mathbb{Z}$ is isomorphic to the ring of $p$-adic integers $\mathbb{Z}_p$. In the process we develop idioms to cleanly handle calculations of identities between operations on the ring of Witt vectors. These calculations are intractable with a naive approach, and require a proof technique that is usually skimmed over in the informal literature. Our proofs resemble the informal arguments while being fully rigorous

Novelty-Related fMRI Responses of Precuneus and Medial Temporal Regions in Individuals at Risk for Alzheimer Disease

BACKGROUND AND OBJECTIVES: We assessed whether novelty-related fMRI activity in medial temporal lobe regions and the precuneus follows an inverted U-shaped pattern across the clinical spectrum of increased Alzheimer disease (AD) risk as previously suggested. Specifically, we tested for potentially increased activity in individuals with a higher AD risk due to subjective cognitive decline (SCD) or mild cognitive impairment (MCI). We further tested whether activity differences related to diagnostic groups were accounted for by CSF markers of AD or brain atrophy. METHODS: We studied 499 participants aged 60-88 years from the German Center for Neurodegenerative Diseases Longitudinal Cognitive Impairment and Dementia Study (DELCODE) who underwent task-fMRI. Participants included 163 cognitively normal (healthy control, HC) individuals, 222 SCD, 82 MCI, and 32 patients with clinical diagnosis of mild AD. CSF levels of β-amyloid 42/40 ratio and phosphorylated-tau181 were available from 232 participants. We used region-based analyses to assess novelty-related activity (novel > highly familiar scenes) in entorhinal cortex, hippocampus, and precuneus as well as whole-brain voxel-wise analyses. First, general linear models tested differences in fMRI activity between participant groups. Complementary regression models tested quadratic relationships between memory impairment and activity. Second, relationships of activity with AD CSF biomarkers and brain volume were analyzed. Analyses were controlled for age, sex, study site, and education. RESULTS: In the precuneus, we observed an inverted U-shaped pattern of novelty-related activity across groups, with higher activity in SCD and MCI compared with HC, but not in patients with AD who showed relatively lower activity than MCI. This nonlinear pattern was confirmed by a quadratic relationship between memory impairment and precuneus activity. Precuneus activity was not related to AD biomarkers or brain volume. In contrast to the precuneus, hippocampal activity was reduced in AD dementia compared with all other groups and related to AD biomarkers. DISCUSSION: Novelty-related activity in the precuneus follows a nonlinear pattern across the clinical spectrum of increased AD risk. Although the underlying mechanism remains unclear, increased precuneus activity might represent an early signature of memory impairment. Our results highlight the nonlinearity of activity alterations that should be considered in clinical trials using functional outcome measures or targeting hyperactivity

Long-term modification of cortical synapses improves sensory perception

Synapses and receptive fields of the cerebral cortex are plastic. However, changes to specific inputs must be coordinated within neural networks to ensure that excitability and feature selectivity are appropriately configured for perception of the sensory environment. Long-lasting enhancements and decrements to rat primary auditory cortical excitatory synaptic strength were induced by pairing acoustic stimuli with activation of the nucleus basalis neuromodulatory system. Here we report that these synaptic modifications were approximately balanced across individual receptive fields, conserving mean excitation while reducing overall response variability. Decreased response variability should increase detection and recognition of near-threshold or previously imperceptible stimuli, as we found in behaving animals. Thus, modification of cortical inputs leads to wide-scale synaptic changes, which are related to improved sensory perception and enhanced behavioral performance

BHPR research: qualitative1. Complex reasoning determines patients' perception of outcome following foot surgery in rheumatoid arhtritis

Background: Foot surgery is common in patients with RA but research into surgical outcomes is limited and conceptually flawed as current outcome measures lack face validity: to date no one has asked patients what is important to them. This study aimed to determine which factors are important to patients when evaluating the success of foot surgery in RA Methods: Semi structured interviews of RA patients who had undergone foot surgery were conducted and transcribed verbatim. Thematic analysis of interviews was conducted to explore issues that were important to patients. Results: 11 RA patients (9 ♂, mean age 59, dis dur = 22yrs, mean of 3 yrs post op) with mixed experiences of foot surgery were interviewed. Patients interpreted outcome in respect to a multitude of factors, frequently positive change in one aspect contrasted with negative opinions about another. Overall, four major themes emerged. Function: Functional ability & participation in valued activities were very important to patients. Walking ability was a key concern but patients interpreted levels of activity in light of other aspects of their disease, reflecting on change in functional ability more than overall level. Positive feelings of improved mobility were often moderated by negative self perception ("I mean, I still walk like a waddling duck”). Appearance: Appearance was important to almost all patients but perhaps the most complex theme of all. Physical appearance, foot shape, and footwear were closely interlinked, yet patients saw these as distinct separate concepts. Patients need to legitimize these feelings was clear and they frequently entered into a defensive repertoire ("it's not cosmetic surgery; it's something that's more important than that, you know?”). Clinician opinion: Surgeons' post operative evaluation of the procedure was very influential. The impact of this appraisal continued to affect patients' lasting impression irrespective of how the outcome compared to their initial goals ("when he'd done it ... he said that hasn't worked as good as he'd wanted to ... but the pain has gone”). Pain: Whilst pain was important to almost all patients, it appeared to be less important than the other themes. Pain was predominately raised when it influenced other themes, such as function; many still felt the need to legitimize their foot pain in order for health professionals to take it seriously ("in the end I went to my GP because it had happened a few times and I went to an orthopaedic surgeon who was quite dismissive of it, it was like what are you complaining about”). Conclusions: Patients interpret the outcome of foot surgery using a multitude of interrelated factors, particularly functional ability, appearance and surgeons' appraisal of the procedure. While pain was often noted, this appeared less important than other factors in the overall outcome of the surgery. Future research into foot surgery should incorporate the complexity of how patients determine their outcome Disclosure statement: All authors have declared no conflicts of interes

Dissecting the Shared Genetic Architecture of Suicide Attempt, Psychiatric Disorders, and Known Risk Factors

Background Suicide is a leading cause of death worldwide, and nonfatal suicide attempts, which occur far more frequently, are a major source of disability and social and economic burden. Both have substantial genetic etiology, which is partially shared and partially distinct from that of related psychiatric disorders. Methods We conducted a genome-wide association study (GWAS) of 29,782 suicide attempt (SA) cases and 519,961 controls in the International Suicide Genetics Consortium (ISGC). The GWAS of SA was conditioned on psychiatric disorders using GWAS summary statistics via multitrait-based conditional and joint analysis, to remove genetic effects on SA mediated by psychiatric disorders. We investigated the shared and divergent genetic architectures of SA, psychiatric disorders, and other known risk factors. Results Two loci reached genome-wide significance for SA: the major histocompatibility complex and an intergenic locus on chromosome 7, the latter of which remained associated with SA after conditioning on psychiatric disorders and replicated in an independent cohort from the Million Veteran Program. This locus has been implicated in risk-taking behavior, smoking, and insomnia. SA showed strong genetic correlation with psychiatric disorders, particularly major depression, and also with smoking, pain, risk-taking behavior, sleep disturbances, lower educational attainment, reproductive traits, lower socioeconomic status, and poorer general health. After conditioning on psychiatric disorders, the genetic correlations between SA and psychiatric disorders decreased, whereas those with nonpsychiatric traits remained largely unchanged. Conclusions Our results identify a risk locus that contributes more strongly to SA than other phenotypes and suggest a shared underlying biology between SA and known risk factors that is not mediated by psychiatric disorders.Peer reviewe

Shared genetic risk between eating disorder- and substance-use-related phenotypes:Evidence from genome-wide association studies

First published: 16 February 202

Laser Shock Peening on Zr-based Bulk Metallic Glass and Its Effect on Plasticity: Experiment and Modeling

The Zr-based bulk metallic glasses (BMGs) are a new family of attractive materials with good glass-forming ability and excellent mechanical properties, such as high strength and good wear resistance, which make them candidates for structural and biomedical materials. Although the mechanical behavior of BMGs has been widely investigated, their deformation mechanisms are still poorly understood. In particular, their poor ductility significantly impedes their industrial application. In the present work, we show that the ductility of Zr-based BMGs with nearly zero plasticity is improved by a laser shock peening technique. Moreover, we map the distribution of laser-induced residual stresses via the micro-slot cutting method, and then predict them using a three-dimensional finite-element method coupled with a confined plasma model. Reasonable agreement is achieved between the experimental and modeling results. The analyses of serrated flows reveal plentiful and useful information of the underlying deformation process. Our work provides an easy and effective way to extend the ductility of intrinsically-brittle BMGs, opening up wider applications of these materials

Results of the IEA Bioenergy Round Robin on the Analysis of Heteroatoms in Biomass Liquefaction Oils

A round robin study evaluating the analysis of biomass liquefaction oils (BLOs) from fast pyrolysis and hydrothermal liquefaction (HTL) was performed, involving fifteen laboratories in seven countries in order to assess the current status of analytical techniques for the determination of nitrogen, sulfur, and chlorine content in BLOs and evaluate potential differences in origin (i.e. fast pyrolysis vs HTL). The BLOs were produced from a range of feedstocks including pine, mixed softwoods, forest residues, micro-algae, miscanthus, and wheat straw to cover a variety in nitrogen, sulfur, and chlorine content and speciation. Nine samples were distributed, comprised of eight separate BLOs and one blind duplicate produced by five producers. The samples were analyzed for water, carbon, hydrogen, nitrogen, sulfur, and chlorine content. No analytical test method was mandated; laboratories were encouraged to utilize whichever method they determined would be most applicable, relying on the existing body of BLO literature as a guide. The results of this round robin study are presented in this paper. The results of the carbon, hydrogen, and water measurements as reference analyses had relative standard deviations (2.9%, 3.5, and 5.6%, respectively) that were comparable to those found in past round robin studies on fast pyrolysis bio-oil. The analysis of nitrogen, sulfur, and chlorine showed higher levels of variability. Laboratories mostly chose the same method for water, carbon, hydrogen and nitrogen determination whereas there were a variety of methods chosen for sulfur and chlorine determination. The results suggest that specific analytical methods for the determination of nitrogen, sulfur, and chlorine should be further refined to ensure reproducible and accurate results for BLO analysis due to their importance on emissions, material selection, and catalyst activity