Undecidability in number theory

These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers are universally definable in the rationals.Comment: 48 pages. arXiv admin note: text overlap with arXiv:1011.342

Association of Apolipoprotein E ɛ4 allele with clinical and multimodal biomarker changes of Alzheimer Disease in adults with Down syndrome

Importance: Alzheimer disease (AD) is the leading cause of death in individuals with Down syndrome (DS). Previous studies have suggested that the APOE ɛ4 allele plays a role in the risk and age at onset of dementia in DS; however, data on in vivo biomarkers remain scarce. Objective: To investigate the association of the APOE ɛ4 allele with clinical and multimodal biomarkers of AD in adults with DS. Design, setting, and participants: This dual-center cohort study recruited adults with DS in Barcelona, Spain, and in Cambridge, UK, between June 1, 2009, and February 28, 2020. Included individuals had been genotyped for APOE and had at least 1 clinical or AD biomarker measurement; 2 individuals were excluded because of the absence of trisomy 21. Participants were either APOE ɛ4 allele carriers or noncarriers. Main outcomes and measures: Participants underwent a neurological and neuropsychological assessment. A subset of participants had biomarker measurements: Aβ1-42, Aβ1-40, phosphorylated tau 181 (pTau181) and neurofilament light chain (NfL) in cerebrospinal fluid (CSF), pTau181, and NfL in plasma; amyloid positron emission tomography (PET); fluorine 18-labeled-fluorodeoxyglucose PET; and/or magnetic resonance imaging. Age at symptom onset was compared between APOE ɛ4 allele carriers and noncarriers, and within-group local regression models were used to compare the association of biomarkers with age. Voxelwise analyses were performed to assess topographical differences in gray matter metabolism and volume. Results: Of the 464 adults with DS included in the study, 97 (20.9%) were APOE ɛ4 allele carriers and 367 (79.1%) were noncarriers. No differences between the 2 groups were found by age (median [interquartile range], 45.9 [36.4-50.2] years vs 43.7 [34.9-50.2] years; P = .56) or sex (51 male carriers [52.6%] vs 199 male noncarriers [54.2%]). APOE ɛ4 allele carriers compared with noncarriers presented with AD symptoms at a younger age (mean [SD] age, 50.7 [4.4] years vs 52.7 [5.8] years; P = .02) and showed earlier cognitive decline. Locally estimated scatterplot smoothing curves further showed between-group differences in biomarker trajectories with age as reflected by nonoverlapping CIs. Specifically, carriers showed lower levels of the CSF Aβ1-42 to Aβ1-40 ratio until age 40 years, earlier increases in amyloid PET and plasma pTau181, and earlier loss of cortical metabolism and hippocampal volume. No differences were found in NfL biomarkers or CSF total tau and pTau181. Voxelwise analyses showed lower metabolism in subcortical and parieto-occipital structures and lower medial temporal volume in APOE ɛ4 allele carriers. Conclusions and relevance: In this study, the APOE ɛ4 allele was associated with earlier clinical and biomarker changes of AD in DS. These results provide insights into the mechanisms by which APOE increases the risk of AD, emphasizing the importance of APOE genotype for future clinical trials in DS.This study was funded in part by the Fondo de Investigaciones Sanitario; Instituto de Salud Carlos III (grants PI14/01126 and PI17/01019 to Dr Fortea, grants PI13/01532 and PI16/01825 to Dr Blesa, grant PI18/00335 to Dr Carmona-Iragui, grants PI18/00435 and INT19/00016 to Dr Alcolea, grant PI18/00327 to Dr Belbin, and grants PI14/1561 and PI17/01896 to Dr Lleó); Center of Biomedical Investigation Network for Neurodegenerative Diseases (CIBERNED) Program 1, Alzheimer Disease (Dr Lleó) and SIGNAL study, which was partly jointly funded by Fondo Europeo de Desarrollo Regional, Unión Europea, and Una Manera de Hacer Europa; National Institutes of Health (NIH; grants 1R01AG056850-01A1, R21AG056974, and R01AG061566 to Dr Fortea); Departament de Salut de la Generalitat de Catalunya, Pla Estratègic de Recerca i Innovació en Salut (grant SLT002/16/00408 to Dr Lleó); Fundació La Marató de TV3 (grant 20141210 to Dr Fortea and grant 044412 to Dr Blesa); Fundació Catalana Síndrome de Down and Fundació Víctor Grífols i Lucas (Dr Fortea); Generalitat de Catalunya (grant SLT006/17/00119 to Dr Fortea, grant SLT006/17/95 to Dr Vilaplana, and grant SLT006/17/00125 to Dr Alcolea); Fundació Bancaria La Caixa to Dr Blesa; NIHR Cambridge Biomedical Research Centre; NIHR Collaborations in Leadership for Applied Health Research and Care (CLAHRC) for the East of England; NIHR Cambridge Dementia Biomedical Research Unit; Down Syndrome Association; and the Health Foundation. Dr Bejanin was supported by a Juan de la Cierva Incorporación grant (IJCI-2017-32609) from the Spanish Ministry of Economy and Competitiveness and by a Miguel Servet I grant (CP20/00038) from the Carlos III Health Institute. Dr Iulita was supported by the Jérôme Lejeune Foundation and Sisley D’Ornano Foundations. Dr Estellés was supported by grant CM19/00017 from Pío del Río Hortega. Dr Padilla was supported as a Sara Borrell Postdoctoral Fellow (CD20/00133) at the Carlos III Health Institute. Dr Illán-Gala was supported by a Juan Rodés contract (JR21-00018) and the Pilot Award for Global Brain Health Leaders (GBHI ALZ UK-21-720973). Dr Belbin was supported by a Miguel Servet II grant (CP18/00011). Dr Osorio was supported by grants R01AG056031, R01AG056531, R01AG066870, and R21AG067549 from the NIH and grant 98480 from the Medical Research Council. Dr Lehmann was supported by grants from the Programme Hospitalier de Recherche Clinique National program, grant NeuroMET2 #18HLT09 from the European Union (EU) European Metrology Programme for Innovation and Research, and the Marie Skłodowska-Curie grant agreement 860197 from the EU Horizon 2020 research and innovation programme. Dr Holland was supported by CLAHRC for the East of England at Cambridgeshire and Peterborough NHS Foundation Trust. Dr Zetterberg was supported by grant 2018-02532 from the Swedish Research Council; grant 681712 from the European Research Council; grant ALFGBG-720931 from the Swedish State Support for Clinical Research; grant 201809-2016862 from the Alzheimer Drug Discovery Foundation (ADDF); grants ADSF-21-831376-C, ADSF-21-831381-C, and ADSF-21-831377-C from the Alzheimer’s Strategic Fund and the Alzheimer's Association; Olav Thon Foundation; Erling-Persson Family Foundation; grant FO2019-0228 from the Stiftelsen för Gamla Tjänarinnor, Hjärnfonden, Sweden; the Marie Skłodowska-Curie grant agreement 860197 from the EU Horizon 2020 research and innovation programme; and the UK Dementia Research Institute at UCL. Dr Blennow was supported by grant 2017-00915 from the Swedish Research Council; grant RDAPB-201809-2016615 from the ADDF; grant AF-742881 from the Swedish Alzheimer Foundation; grant FO2017-0243 from Hjärnfonden, Sweden, the Swedish state under the agreement between the Swedish government and the County Councils; grant ALFGBG-715986 from the ALF agreement; and grant JPND2019-466-236 from the EU Joint Program for Neurodegenerative Disorders. Dr Zaman was supported by CLAHRC for the East of England at Cambridgeshire and Peterborough NHS Foundation Trust and grant 98480 from the Medical Research Council. Dr Fortea was supported by grants from the Jérome Lejeune Foundation

Predicting mortality in hospitalized patients with 2009 H1N1 influenza pneumonia

BACKGROUND: Community-acquired pneumonia (CAP) severity scores can identify patients at low risk for mortality who may be suitable for ambulatory care. Here, we follow the clinical course of hospitalized patients with CAP due to 2009 H1N1 influenza. OBJECTIVE: To evaluate the role of CAP severity scores as predictors of mortality. METHODS: This was a secondary data analysis of patients hospitalized with CAP due to 2009 H1N1 influenza confirmed by reverse transcriptase polymerase chain reaction enrolled in the CAPO (Community-Acquired Pneumonia Organization) international cohort study. CAP severity scores PSI (Pneumonia Severity Index), CURB-65 (confusion, urea, respiratory rate, blood pressure, age 65 65 years) and CRB-65 (confusion, respiratory rate, blood pressure, age 65 65 years) were calculated. Actual and predicted mortality rates were compared. A total of 37 predictor variables were evaluated to define those associated with mortality. RESULTS: Data from 250 patients with CAP due to 2009 H1N1 influenza were analyzed. Patients with low predicted mortality rates (0-1.5%) had actual mortality rates ranging from 2.6% to 17.5%. Obesity and wheezing were the only novel variables associated with mortality. CONCLUSIONS: The decision to hospitalize a patient with CAP due to 2009 H1N1 influenza should not be based on current CAP severity scores, as they underestimate mortality rates in a significant number of patients. Patients with obesity or wheezing should be considered at an increased risk for mortality

Reconstruction Of Inclined Air Showers Detected With The Pierre Auger Observatory

20148NSF; National Science Foundatio