Analytic Functions in General Analysis

The theory of functions of a complex variable is distinguished from the theory of functions of a real variable by its simplicity - a simplicity is directly traceable to the complexity of the variable. Two of the remarkable simplicities of the theory are, first, that from the assumption that f(z) is differentiable throughout the neighborhood of a point z = z0 follows the existence of all higher derivatives and the convergence of the Taylor's series for f(z); and secondly, that we are able to classify in simple terms the possible singularities of an analytic function. It is the purpose of this work to generalize, insofar as is possible, the basic theorems of the classical theory, and to investigate in what measure the simplicities mentioned above are preserved when the arguments and function values lie in a Banach space. Of the three principally recognized points of view which are used in developing the theory of analytic functions we have used mainly the one due to Cauchy, which finds its natural extension in the ideas of Gateaux concerning differentials. Much of the work which we present was sketched in a memoir of Gateaux on functionals of continuous functions.* In addition we have developed the "Weierstrassian" properties of analytic functions, using as a foundation the notion of polynomial as set forth by R.S. Martin.** Finally, a brief section is devoted to a generalization of the Cauchy-Riemann equations. Nothing has been done with the implicitly suggested theory of pairs of conjugate harmonic functions, however. The study of differentials leads to an important result showing the relation of the Fréchet and Gateaux concepts of a differential. The classification of singular points is a most difficult problem. We have dealt completely with removable singularities, and showed to some extent the departures from classical theory which are caused by the generalization here undertaken. A more detailed investigation should be carried out in special cases. I freely express my admiration for the treaties of Professor W.F. Osgood, Lehrbuch der Funktionentheorie, to which I have had constant recourse in the writing of this thesis. Many of the proofs are directly carried over, with only the slight changes made necessary by the abstract nature of the quantities in hand. To professor A.D. Michal I am indebted for encouragement and advice at all times.</p

Cheating on the Edge

We present the results of an individual agent-based model of antibiotic resistance in bacteria. Our model examines antibiotic resistance when two strategies exist: “producers”–who secrete a substance that breaks down antibiotics–and nonproducers (“cheats”) who do not secrete, or carry the machinery associated with secretion. The model allows for populations of up to 10,000, in which bacteria are affected by their nearest neighbors, and we assume cheaters die when there are no producers in their neighborhood. Each of 10,000 slots on our grid (a torus) could be occupied by a producer or a nonproducer, or could (temporarily) be unoccupied. The most surprising and dramatic result we uncovered is that when producers and nonproducers coexist at equilibrium, nonproducers are almost always found on the edges of clusters of producers

The clinical effectiveness of individual behaviour change interventions to reduce risky sexual behaviour after a negative human immunodeficiency virus test in men who have sex with men: systematic and realist reviews and intervention development

Background: Men who have sex with men (MSM) experience significant inequalities in health and well-being. They are the group in the UK at the highest risk of acquiring a human immunodeficiency virus (HIV) infection. Guidance relating to both HIV infection prevention, in general, and individual-level behaviour change interventions, in particular, is very limited. Objectives: To conduct an evidence synthesis of the clinical effectiveness of behaviour change interventions to reduce risky sexual behaviour among MSM after a negative HIV infection test. To identify effective components within interventions in reducing HIV risk-related behaviours and develop a candidate intervention. To host expert events addressing the implementation and optimisation of a candidate intervention. Data sources: All major electronic databases (British Education Index, BioMed Central, Cumulative Index to Nursing and Allied Health Literature, EMBASE, Educational Resource Index and Abstracts, Health and Medical Complete, MEDLINE, PsycARTICLES, PsycINFO, PubMed and Social Science Citation Index) were searched between January 2000 and December 2014. Review methods: A systematic review of the clinical effectiveness of individual behaviour change interventions was conducted. Interventions were examined using the behaviour change technique (BCT) taxonomy, theory coding assessment, mode of delivery and proximity to HIV infection testing. Data were summarised in narrative review and, when appropriate, meta-analysis was carried out. Supplemental analyses for the development of the candidate intervention focused on post hoc realist review method, the assessment of the sequential delivery and content of intervention components, and the social and historical context of primary studies. Expert panels reviewed the candidate intervention for issues of implementation and optimisation. Results: Overall, trials included in this review (n = 10) demonstrated that individual-level behaviour change interventions are effective in reducing key HIV infection risk-related behaviours. However, there was considerable clinical and methodological heterogeneity among the trials. Exploratory meta-analysis showed a statistically significant reduction in behaviours associated with high risk of HIV transmission (risk ratio 0.75, 95% confidence interval 0.62 to 0.91). Additional stratified analyses suggested that effectiveness may be enhanced through face-to-face contact immediately after testing, and that theory-based content and BCTs drawn from ‘goals and planning’ and ‘identity’ groups are important. All evidence collated in the review was synthesised to develop a candidate intervention. Experts highlighted overall acceptability of the intervention and outlined key ways that the candidate intervention could be optimised to enhance UK implementation. Limitations: There was a limited number of primary studies. All were from outside the UK and were subject to considerable clinical, methodological and statistical heterogeneity. The findings of the meta-analysis must therefore be treated with caution. The lack of detailed intervention manuals limited the assessment of intervention content, delivery and fidelity. Conclusions: Evidence regarding the effectiveness of behaviour change interventions suggests that they are effective in changing behaviour associated with HIV transmission. Exploratory stratified meta-analyses suggested that interventions should be delivered face to face and immediately after testing. There are uncertainties around the generalisability of these findings to the UK setting. However, UK experts found the intervention acceptable and provided ways of optimising the candidate intervention. Future work: There is a need for well-designed, UK-based trials of individual behaviour change interventions that clearly articulate intervention content and demonstrate intervention fidelity

Cognitive behavioural therapy for adults with dissociative seizures (CODES): a pragmatic, multicentre, randomised controlled trial.

BACKGROUND: Dissociative seizures are paroxysmal events resembling epilepsy or syncope with characteristic features that allow them to be distinguished from other medical conditions. We aimed to compare the effectiveness of cognitive behavioural therapy (CBT) plus standardised medical care with standardised medical care alone for the reduction of dissociative seizure frequency. METHODS: In this pragmatic, parallel-arm, multicentre randomised controlled trial, we initially recruited participants at 27 neurology or epilepsy services in England, Scotland, and Wales. Adults (≥18 years) who had dissociative seizures in the previous 8 weeks and no epileptic seizures in the previous 12 months were subsequently randomly assigned (1:1) from 17 liaison or neuropsychiatry services following psychiatric assessment, to receive standardised medical care or CBT plus standardised medical care, using a web-based system. Randomisation was stratified by neuropsychiatry or liaison psychiatry recruitment site. The trial manager, chief investigator, all treating clinicians, and patients were aware of treatment allocation, but outcome data collectors and trial statisticians were unaware of treatment allocation. Patients were followed up 6 months and 12 months after randomisation. The primary outcome was monthly dissociative seizure frequency (ie, frequency in the previous 4 weeks) assessed at 12 months. Secondary outcomes assessed at 12 months were: seizure severity (intensity) and bothersomeness; longest period of seizure freedom in the previous 6 months; complete seizure freedom in the previous 3 months; a greater than 50% reduction in seizure frequency relative to baseline; changes in dissociative seizures (rated by others); health-related quality of life; psychosocial functioning; psychiatric symptoms, psychological distress, and somatic symptom burden; and clinical impression of improvement and satisfaction. p values and statistical significance for outcomes were reported without correction for multiple comparisons as per our protocol. Primary and secondary outcomes were assessed in the intention-to-treat population with multiple imputation for missing observations. This trial is registered with the International Standard Randomised Controlled Trial registry, ISRCTN05681227, and ClinicalTrials.gov, NCT02325544. FINDINGS: Between Jan 16, 2015, and May 31, 2017, we randomly assigned 368 patients to receive CBT plus standardised medical care (n=186) or standardised medical care alone (n=182); of whom 313 had primary outcome data at 12 months (156 [84%] of 186 patients in the CBT plus standardised medical care group and 157 [86%] of 182 patients in the standardised medical care group). At 12 months, no significant difference in monthly dissociative seizure frequency was identified between the groups (median 4 seizures [IQR 0-20] in the CBT plus standardised medical care group vs 7 seizures [1-35] in the standardised medical care group; estimated incidence rate ratio [IRR] 0·78 [95% CI 0·56-1·09]; p=0·144). Dissociative seizures were rated as less bothersome in the CBT plus standardised medical care group than the standardised medical care group (estimated mean difference -0·53 [95% CI -0·97 to -0·08]; p=0·020). The CBT plus standardised medical care group had a longer period of dissociative seizure freedom in the previous 6 months (estimated IRR 1·64 [95% CI 1·22 to 2·20]; p=0·001), reported better health-related quality of life on the EuroQoL-5 Dimensions-5 Level Health Today visual analogue scale (estimated mean difference 6·16 [95% CI 1·48 to 10·84]; p=0·010), less impairment in psychosocial functioning on the Work and Social Adjustment Scale (estimated mean difference -4·12 [95% CI -6·35 to -1·89]; p<0·001), less overall psychological distress than the standardised medical care group on the Clinical Outcomes in Routine Evaluation-10 scale (estimated mean difference -1·65 [95% CI -2·96 to -0·35]; p=0·013), and fewer somatic symptoms on the modified Patient Health Questionnaire-15 scale (estimated mean difference -1·67 [95% CI -2·90 to -0·44]; p=0·008). Clinical improvement at 12 months was greater in the CBT plus standardised medical care group than the standardised medical care alone group as reported by patients (estimated mean difference 0·66 [95% CI 0·26 to 1·04]; p=0·001) and by clinicians (estimated mean difference 0·47 [95% CI 0·21 to 0·73]; p<0·001), and the CBT plus standardised medical care group had greater satisfaction with treatment than did the standardised medical care group (estimated mean difference 0·90 [95% CI 0·48 to 1·31]; p<0·001). No significant differences in patient-reported seizure severity (estimated mean difference -0·11 [95% CI -0·50 to 0·29]; p=0·593) or seizure freedom in the last 3 months of the study (estimated odds ratio [OR] 1·77 [95% CI 0·93 to 3·37]; p=0·083) were identified between the groups. Furthermore, no significant differences were identified in the proportion of patients who had a more than 50% reduction in dissociative seizure frequency compared with baseline (OR 1·27 [95% CI 0·80 to 2·02]; p=0·313). Additionally, the 12-item Short Form survey-version 2 scores (estimated mean difference for the Physical Component Summary score 1·78 [95% CI -0·37 to 3·92]; p=0·105; estimated mean difference for the Mental Component Summary score 2·22 [95% CI -0·30 to 4·75]; p=0·084), the Generalised Anxiety Disorder-7 scale score (estimated mean difference -1·09 [95% CI -2·27 to 0·09]; p=0·069), and the Patient Health Questionnaire-9 scale depression score (estimated mean difference -1·10 [95% CI -2·41 to 0·21]; p=0·099) did not differ significantly between groups. Changes in dissociative seizures (rated by others) could not be assessed due to insufficient data. During the 12-month period, the number of adverse events was similar between the groups: 57 (31%) of 186 participants in the CBT plus standardised medical care group reported 97 adverse events and 53 (29%) of 182 participants in the standardised medical care group reported 79 adverse events. INTERPRETATION: CBT plus standardised medical care had no statistically significant advantage compared with standardised medical care alone for the reduction of monthly seizures. However, improvements were observed in a number of clinically relevant secondary outcomes following CBT plus standardised medical care when compared with standardised medical care alone. Thus, adults with dissociative seizures might benefit from the addition of dissociative seizure-specific CBT to specialist care from neurologists and psychiatrists. Future work is needed to identify patients who would benefit most from a dissociative seizure-specific CBT approach. FUNDING: National Institute for Health Research, Health Technology Assessment programme

Characteristics of 698 patients with dissociative seizures: A UK multicenter study

Objective We aimed to characterize the demographics of adults with dissociative (nonepileptic) seizures, placing emphasis on distribution of age at onset, male:female ratio, levels of deprivation, and dissociative seizure semiology. Methods We collected demographic and clinical data from 698 adults with dissociative seizures recruited to the screening phase of the CODES (Cognitive Behavioural Therapy vs Standardised Medical Care for Adults With Dissociative Non‐Epileptic Seizures) trial from 27 neurology/specialist epilepsy clinics in the UK. We described the cohort in terms of age, age at onset of dissociative seizures, duration of seizure disorder, level of socioeconomic deprivation, and other social and clinical demographic characteristics and their associations. Results In what is, to date, the largest study of adults with dissociative seizures, the overall modal age at dissociative seizure onset was 19 years; median age at onset was 28 years. Although 74% of the sample was female, importantly the male:female ratio varied with age at onset, with 77% of female but only 59% of male participants developing dissociative seizures by the age of 40 years. The frequency of self‐reported previous epilepsy was 27%; nearly half of these epilepsy diagnoses were retrospectively considered erroneous by clinicians. Patients with predominantly hyperkinetic dissociative seizures had a shorter disorder duration prior to diagnosis in this study than patients with hypokinetic seizures (P < .001); dissociative seizure type was not associated with gender. Predominantly hyperkinetic seizures were most commonly seen in patients with symptom onset in their late teens. Thirty percent of the sample reported taking antiepileptic drugs; this was more common in men. More than 50% of the sample lived in areas characterized by the highest levels of deprivation, and more than two‐thirds were unemployed. Significance Females with dissociative seizures were more common at all ages, whereas the proportion of males increased with age at onset. This disorder was associated with socioeconomic deprivation. Those with hypokinetic dissociative seizures may be at risk for delayed diagnosis and treatment

Prevalence and architecture of de novo mutations in developmental disorders.

The genomes of individuals with severe, undiagnosed developmental disorders are enriched in damaging de novo mutations (DNMs) in developmentally important genes. Here we have sequenced the exomes of 4,293 families containing individuals with developmental disorders, and meta-analysed these data with data from another 3,287 individuals with similar disorders. We show that the most important factors influencing the diagnostic yield of DNMs are the sex of the affected individual, the relatedness of their parents, whether close relatives are affected and the parental ages. We identified 94 genes enriched in damaging DNMs, including 14 that previously lacked compelling evidence of involvement in developmental disorders. We have also characterized the phenotypic diversity among these disorders. We estimate that 42% of our cohort carry pathogenic DNMs in coding sequences; approximately half of these DNMs disrupt gene function and the remainder result in altered protein function. We estimate that developmental disorders caused by DNMs have an average prevalence of 1 in 213 to 1 in 448 births, depending on parental age. Given current global demographics, this equates to almost 400,000 children born per year

Characteristics of 698 patients with dissociative seizures: A UK multicenter study

Objective We aimed to characterize the demographics of adults with dissociative (nonepileptic) seizures, placing emphasis on distribution of age at onset, male:female ratio, levels of deprivation, and dissociative seizure semiology. Methods We collected demographic and clinical data from 698 adults with dissociative seizures recruited to the screening phase of the CODES (Cognitive Behavioural Therapy vs Standardised Medical Care for Adults With Dissociative Non‐Epileptic Seizures) trial from 27 neurology/specialist epilepsy clinics in the UK. We described the cohort in terms of age, age at onset of dissociative seizures, duration of seizure disorder, level of socioeconomic deprivation, and other social and clinical demographic characteristics and their associations. Results In what is, to date, the largest study of adults with dissociative seizures, the overall modal age at dissociative seizure onset was 19 years; median age at onset was 28 years. Although 74% of the sample was female, importantly the male:female ratio varied with age at onset, with 77% of female but only 59% of male participants developing dissociative seizures by the age of 40 years. The frequency of self‐reported previous epilepsy was 27%; nearly half of these epilepsy diagnoses were retrospectively considered erroneous by clinicians. Patients with predominantly hyperkinetic dissociative seizures had a shorter disorder duration prior to diagnosis in this study than patients with hypokinetic seizures (P < .001); dissociative seizure type was not associated with gender. Predominantly hyperkinetic seizures were most commonly seen in patients with symptom onset in their late teens. Thirty percent of the sample reported taking antiepileptic drugs; this was more common in men. More than 50% of the sample lived in areas characterized by the highest levels of deprivation, and more than two‐thirds were unemployed. Significance Females with dissociative seizures were more common at all ages, whereas the proportion of males increased with age at onset. This disorder was associated with socioeconomic deprivation. Those with hypokinetic dissociative seizures may be at risk for delayed diagnosis and treatment

Multiple novel prostate cancer susceptibility signals identified by fine-mapping of known risk loci among Europeans

Genome-wide association studies (GWAS) have identified numerous common prostate cancer (PrCa) susceptibility loci. We have fine-mapped 64 GWAS regions known at the conclusion of the iCOGS study using large-scale genotyping and imputation in 25 723 PrCa cases and 26 274 controls of European ancestry. We detected evidence for multiple independent signals at 16 regions, 12 of which contained additional newly identified significant associations. A single signal comprising a spectrum of correlated variation was observed at 39 regions; 35 of which are now described by a novel more significantly associated lead SNP, while the originally reported variant remained as the lead SNP only in 4 regions. We also confirmed two association signals in Europeans that had been previously reported only in East-Asian GWAS. Based on statistical evidence and linkage disequilibrium (LD) structure, we have curated and narrowed down the list of the most likely candidate causal variants for each region. Functional annotation using data from ENCODE filtered for PrCa cell lines and eQTL analysis demonstrated significant enrichment for overlap with bio-features within this set. By incorporating the novel risk variants identified here alongside the refined data for existing association signals, we estimate that these loci now explain ∼38.9% of the familial relative risk of PrCa, an 8.9% improvement over the previously reported GWAS tag SNPs. This suggests that a significant fraction of the heritability of PrCa may have been hidden during the discovery phase of GWAS, in particular due to the presence of multiple independent signals within the same regio

Effect of Hydrocortisone on Mortality and Organ Support in Patients With Severe COVID-19: The REMAP-CAP COVID-19 Corticosteroid Domain Randomized Clinical Trial.

Importance: Evidence regarding corticosteroid use for severe coronavirus disease 2019 (COVID-19) is limited. Objective: To determine whether hydrocortisone improves outcome for patients with severe COVID-19. Design, Setting, and Participants: An ongoing adaptive platform trial testing multiple interventions within multiple therapeutic domains, for example, antiviral agents, corticosteroids, or immunoglobulin. Between March 9 and June 17, 2020, 614 adult patients with suspected or confirmed COVID-19 were enrolled and randomized within at least 1 domain following admission to an intensive care unit (ICU) for respiratory or cardiovascular organ support at 121 sites in 8 countries. Of these, 403 were randomized to open-label interventions within the corticosteroid domain. The domain was halted after results from another trial were released. Follow-up ended August 12, 2020. Interventions: The corticosteroid domain randomized participants to a fixed 7-day course of intravenous hydrocortisone (50 mg or 100 mg every 6 hours) (n = 143), a shock-dependent course (50 mg every 6 hours when shock was clinically evident) (n = 152), or no hydrocortisone (n = 108). Main Outcomes and Measures: The primary end point was organ support-free days (days alive and free of ICU-based respiratory or cardiovascular support) within 21 days, where patients who died were assigned -1 day. The primary analysis was a bayesian cumulative logistic model that included all patients enrolled with severe COVID-19, adjusting for age, sex, site, region, time, assignment to interventions within other domains, and domain and intervention eligibility. Superiority was defined as the posterior probability of an odds ratio greater than 1 (threshold for trial conclusion of superiority >99%). Results: After excluding 19 participants who withdrew consent, there were 384 patients (mean age, 60 years; 29% female) randomized to the fixed-dose (n = 137), shock-dependent (n = 146), and no (n = 101) hydrocortisone groups; 379 (99%) completed the study and were included in the analysis. The mean age for the 3 groups ranged between 59.5 and 60.4 years; most patients were male (range, 70.6%-71.5%); mean body mass index ranged between 29.7 and 30.9; and patients receiving mechanical ventilation ranged between 50.0% and 63.5%. For the fixed-dose, shock-dependent, and no hydrocortisone groups, respectively, the median organ support-free days were 0 (IQR, -1 to 15), 0 (IQR, -1 to 13), and 0 (-1 to 11) days (composed of 30%, 26%, and 33% mortality rates and 11.5, 9.5, and 6 median organ support-free days among survivors). The median adjusted odds ratio and bayesian probability of superiority were 1.43 (95% credible interval, 0.91-2.27) and 93% for fixed-dose hydrocortisone, respectively, and were 1.22 (95% credible interval, 0.76-1.94) and 80% for shock-dependent hydrocortisone compared with no hydrocortisone. Serious adverse events were reported in 4 (3%), 5 (3%), and 1 (1%) patients in the fixed-dose, shock-dependent, and no hydrocortisone groups, respectively. Conclusions and Relevance: Among patients with severe COVID-19, treatment with a 7-day fixed-dose course of hydrocortisone or shock-dependent dosing of hydrocortisone, compared with no hydrocortisone, resulted in 93% and 80% probabilities of superiority with regard to the odds of improvement in organ support-free days within 21 days. However, the trial was stopped early and no treatment strategy met prespecified criteria for statistical superiority, precluding definitive conclusions. Trial Registration: ClinicalTrials.gov Identifier: NCT02735707

Global patient outcomes after elective surgery: prospective cohort study in 27 low-, middle- and high-income countries.

BACKGROUND: As global initiatives increase patient access to surgical treatments, there remains a need to understand the adverse effects of surgery and define appropriate levels of perioperative care. METHODS: We designed a prospective international 7-day cohort study of outcomes following elective adult inpatient surgery in 27 countries. The primary outcome was in-hospital complications. Secondary outcomes were death following a complication (failure to rescue) and death in hospital. Process measures were admission to critical care immediately after surgery or to treat a complication and duration of hospital stay. A single definition of critical care was used for all countries. RESULTS: A total of 474 hospitals in 19 high-, 7 middle- and 1 low-income country were included in the primary analysis. Data included 44 814 patients with a median hospital stay of 4 (range 2-7) days. A total of 7508 patients (16.8%) developed one or more postoperative complication and 207 died (0.5%). The overall mortality among patients who developed complications was 2.8%. Mortality following complications ranged from 2.4% for pulmonary embolism to 43.9% for cardiac arrest. A total of 4360 (9.7%) patients were admitted to a critical care unit as routine immediately after surgery, of whom 2198 (50.4%) developed a complication, with 105 (2.4%) deaths. A total of 1233 patients (16.4%) were admitted to a critical care unit to treat complications, with 119 (9.7%) deaths. Despite lower baseline risk, outcomes were similar in low- and middle-income compared with high-income countries. CONCLUSIONS: Poor patient outcomes are common after inpatient surgery. Global initiatives to increase access to surgical treatments should also address the need for safe perioperative care. STUDY REGISTRATION: ISRCTN5181700