Global analytic hypoellipticity for a class of left-invariant operators on T1 x S3

Orientador: Prof Dr. Alexandre KirilovTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Matemática. Defesa : Curitiba, 27/02/2020Inclui referências: p. 72-74Resumo: Apresentamos uma caracterização completa da hipoeliticidade global analítica de uma classe de operadores de primeira ordem definidos em alguns produtos de grupos de Lie compactos, principalmente T1 × S3. No caso de coeficientes com valores reais, provamos que o operador 'e conjugado a um operador com coeficientes constantes e que tal conjugação preserva a hipoeliticidade global anal'?tica. No caso em que a parte imaginária não 'e identicamente nula, nós mostramos que o operador 'e globalmente analítico hipoelítico se a condição (P) de Nirenberg-Treves vale em conjunto com uma condição Diofantina. Também estendemos parte de nossos resultados para uma classe de operadores definidos em produtos da forma T1 × S3 × ·· ·×S3. Palavras-chaves: Hipoeliticidade Global Analítica. Séries de Fourier em grupos de Lie compactos. Condições Diofantinas analíticas. Condição (P) de Nirenberg-TrevesAbstract: We present a complete characterization to the global analytic hypoellipticity of a class of first-order operators defined on some products of compact Lie groups, mainly T1 × S3. In the case of real-valued coefficients, we prove that the operator is conjugated to a constant coefficient operator and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is not identically zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition (P) holds in addition to a Diophantine condition. We also extend part of our results for a class of operators defined on products of the type T1 × S3 × ·· ·×S3. Keywords: Global analytic hypoellipticity. Fourier Series on compact Lie groups. analytic Diophantine conditions. Nirenberg-Treves condition (P)

Global analytic hypoellipticity for a class of evolution operators on T1×S3\mathbb{T}^1\times\mathbb{S}^3

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on $\mathbb{T}^1 \times \mathbb{S}^3$. In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is non-zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition ($\mathcal{P}$) holds, in addition to a Diophantine condition.Comment: 24 page

Simetrias em geografia e física.

Orientador : Prof. Dr. Eduardo Outeiral Correa HoefelDissertaçao (mestrado) - Universidade Federal do Paraná.Setor de Ciencias Exatas. Programa de Pós-Graduação em Matemática Aplicada. Defesa: Curitiba, 15/03/2011Bibliografia: fls. 99-100Resumo: O objetivo desta dissertação é fazer um estudo mostrando algumas ligações entre geometria e física utilizando conexões em fibrados principais. Mais especificamente, descreveremos as geometrias clássicas como exemplos de espaços simétricos e, além disso, mostraremos exemplos de field strenghts como expressões de conexões em certos fibrados principais. Começamos fazendo uma apresentação sucinta dos conceitos: fibrado principal, conexão e curvatura. Em seguida, dividimos o trabalho em duas direções. Na primeira, usamos a linguagem de fibrados principais e conexões para definir o conceito de espaço simétrico e escrevemos, nestes termos, as geometrias clássicas (simplesmente conexas) de curvatura constante: a geometria Esférica, a geometria Hiperbólica e a geometria Euclidiana. No segundo, apresentamos uma relação entre as fibrações de Hopf e alguns temas oriundos da física teórica: o eletromagnetismo e a teoria de Yang-Mills.Abstract: The aim of this work is to make a study showing some relations between geometry and physics using connections on principal fiber bundles. More specifically,we will describe the classical geometries as examples of symmetric spaces and, besides,we will showexamples of field strenghts as expressions of connections in some principal fiber bundles. We start doing a quick presentation of the concepts: principal fiber bundle, connection and curvature. In the following, we divide the work in two directions. First, we use the language of principal fiber bundles and connections to define the concept of symmetric space and we write, in these terms, the classical geometries (simple connected) of constant curvature: the Spherical geometry, the Hyperbolic geometry and the Euclidean geometry. Next, we present a relation between the Hopf’s fibrations and some topics arising from theoretical physics: the eletromagnetism and the Yang-Mills’ theory

30-Day Morbidity and Mortality of Bariatric Surgery During the COVID-19 Pandemic: a Multinational Cohort Study of 7704 Patients from 42 Countries.

BACKGROUND There are data on the safety of cancer surgery and the efficacy of preventive strategies on the prevention of postoperative symptomatic COVID-19 in these patients. But there is little such data for any elective surgery. The main objectives of this study were to examine the safety of bariatric surgery (BS) during the coronavirus disease 2019 (COVID-19) pandemic and to determine the efficacy of perioperative COVID-19 protective strategies on postoperative symptomatic COVID-19 rates. METHODS We conducted an international cohort study to determine all-cause and COVID-19-specific 30-day morbidity and mortality of BS performed between 01/05/2020 and 31/10/2020. RESULTS Four hundred ninety-nine surgeons from 185 centres in 42 countries provided data on 7704 patients. Elective primary BS (n = 7084) was associated with a 30-day morbidity of 6.76% (n = 479) and a 30-day mortality of 0.14% (n = 10). Emergency BS, revisional BS, insulin-treated type 2 diabetes, and untreated obstructive sleep apnoea were associated with increased complications on multivariable analysis. Forty-three patients developed symptomatic COVID-19 postoperatively, with a higher risk in non-whites. Preoperative self-isolation, preoperative testing for SARS-CoV-2, and surgery in institutions not concurrently treating COVID-19 patients did not reduce the incidence of postoperative COVID-19. Postoperative symptomatic COVID-19 was more likely if the surgery was performed during a COVID-19 peak in that country. CONCLUSIONS BS can be performed safely during the COVID-19 pandemic with appropriate perioperative protocols. There was no relationship between preoperative testing for COVID-19 and self-isolation with symptomatic postoperative COVID-19. The risk of postoperative COVID-19 risk was greater in non-whites or if BS was performed during a local peak

Safety of Bariatric Surgery in ≥ 65-Year-Old Patients During the COVID-19 Pandemic

Background Age >= 65 years is regarded as a relative contraindication for bariatric surgery. Advanced age is also a recognised risk factor for adverse outcomes with Coronavirus Disease-2019 (COVID-19) which continues to wreak havoc on global populations. This study aimed to assess the safety of bariatric surgery (BS) in this particular age group during the COVID-19 pandemic in comparison with the younger cohort.Methods We conducted a prospective international study of patients who underwent BS between 1/05/2020 and 31/10/2020. Patients were divided into two groups - patients >= 65-years-old (Group I) and patients < 65-years-old (Group II). The two groups were compared for 30-day morbidity and mortality.Results There were 149 patients in Group 1 and 6923 patients in Group II. The mean age, preoperative weight, and BMI were 67.6 +/- 2.5 years, 119.5 +/- 24.5 kg, and 43 +/- 7 in Group I and 39.8 +/- 11.3 years, 117.7 +/- 20.4 kg, and 43.7 +/- 7 in Group II, respectively. Approximately, 95% of patients in Group 1 had at least one co-morbidity compared to 68% of patients in Group 2 (p = < 0.001). The 30-day morbidity was significantly higher in Group I ( 11.4%) compared to Group II (6.6%) (p = 0.022). However, the 30-day mortality and COVID-19 infection rates were not significantly different between the two groups.Conclusions Bariatric surgery during the COVID-19 pandemic is associated with a higher complication rate in those >= 65 years of age compared to those < 65 years old. However, the mortality and postoperative COVID-19 infection rates are not significantly different between the two groups

Effect of BMI on safety of bariatric surgery during the COVID-19 pandemic, procedure choice, and safety protocols - An analysis from the GENEVA Study

Background: It has been suggested that patients with a Body Mass Index (BMI) of > 60 kg/m2 should be offered expedited Bariatric Surgery (BS) during the Coronavirus Disease-2019 (COVID-19) pandemic. The main objective of this study was to assess the safety of this approach. Methods: We conducted a global study of patients who underwent BS between 1/05/2020 and 31/10/2020. Patients were divided into three groups according to their preoperative BMI -Group I (BMI < 50 kg/m2), Group II (BMI 50-60 kg/m2), and Group III (BMI > 60 kg/m2). The effect of preoperative BMI on 30-day morbidity and mortality, procedure choice, COVID-19 specific safety protocols, and comorbidities was assessed. Results: This study included 7084 patients (5197;73.4 % females). The mean preoperative weight and BMI were 119.49 & PLUSMN; 24.4 Kgs and 43.03 & PLUSMN; 6.9 Kg/m2, respectively. Group I included 6024 (85 %) patients, whereas Groups II and III included 905 (13 %) and 155 (2 %) patients, respectively.The 30-day mortality rate was higher in Group III (p = 0.001). The complication rate and COVID-19 infection were not different. Comorbidities were significantly more likely in Group III (p = < 0.001). A significantly higher proportion of patients in group III received Sleeve Gastrectomy or One Anastomosis Gastric Bypass compared to other groups. Patients with a BMI of > 70 kg/m2 had a 30-day mortality of 7.7 % (2/26). None of these patients underwent a Roux-en-Y Gastric Bypass. Conclusion: The 30-day mortality rate was significantly higher in patients with BMI > 60 kg/m2. There was, however, no significant difference in complications rates in different BMI groups, probably due to differences in procedure selection

30-Day morbidity and mortality of bariatric metabolic surgery in adolescence during the COVID-19 pandemic – The GENEVA study

Background: Metabolic and bariatric surgery (MBS) is an effective treatment for adolescents with severe obesity. Objectives: This study examined the safety of MBS in adolescents during the coronavirus disease 2019 (COVID-19) pandemic. Methods: This was a global, multicentre and observational cohort study of MBS performed between May 01, 2020, and October 10,2020, in 68 centres from 24 countries. Data collection included in-hospital and 30-day COVID-19 and surgery-specific morbidity/mortality. Results: One hundred and seventy adolescent patients (mean age: 17.75 ± 1.30 years), mostly females (n = 122, 71.8%), underwent MBS during the study period. The mean pre-operative weight and body mass index were 122.16 ± 15.92 kg and 43.7 ± 7.11 kg/m2, respectively. Although majority of patients had pre-operative testing for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) (n = 146; 85.9%), only 42.4% (n = 72) of the patients were asked to self-isolate pre-operatively. Two patients developed symptomatic SARS-CoV-2 infection post-operatively (1.2%). The overall complication rate was 5.3% (n = 9). There was no mortality in this cohort. Conclusions: MBS in adolescents with obesity is safe during the COVID-19 pandemic when performed within the context of local precautionary procedures (such as pre-operative testing). The 30-day morbidity rates were similar to those reported pre-pandemic. These data will help facilitate the safe re-introduction of MBS services for this group of patients