Global Operator Calculus on Spin Groups

Acknowledgements The work of P. Cerejeiras, M. Ferreira, and U. Kähler was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT– “Fundação para a Ciência e a Tecnologia”, within project UIDB/04106/2020 and UIDP/04106/2020. The present paper was supported by the project “Global operator calculi on compact and non-compact Lie groups”, Ações Integradas Luso-Alemãs – Acção No. A-42/16. Funding Open access funding provided by FCT|FCCN (b-on).n this paper, we use the representation theory of the group Spin(m) to develop aspects of the global symbolic calculus of pseudo-differential operators on Spin(3) and Spin(4) in the sense of Ruzhansky–Turunen–Wirth. A detailed study of Spin(3) and Spin(4)-representations is made including recurrence relations and natural differential operators acting on matrix coefficients. We establish the calculus of left-invariant differential operators and of difference operators on the group Spin(4) and apply this to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some that are locally neither invertible nor hypoelliptic. The paper presents a particular case study for higher dimensional spin groups.info:eu-repo/semantics/publishedVersio

An interpolation problem arising in a coupling of the finite element method with holomorphic basis functions

The purpose of this paper is to prove an interpolation theorem which arises in a method of coupling of a finite element and an analytical solution for boundary value problems with singularities

A wavelet based numerical method for nonlinear partial differential equations

The purpose of this paper is to present a wavelet–Galerkin scheme for solving nonlinear elliptic partial differential equations. We select as trial spaces a nested sequence of spaces from an appropriate biorthogonal multiscale analysis. This gives rise to a nonlinear discretized system. To overcome the problems of nonlinearity, we apply the machinery of interpolating wavelets to obtain knot oriented quadrature rules. Finally, Newton’s method is applied to approximate the solution in the given ansatz space. The results of some numerical experiments with different biorthogonal systems, confirming the applicability of our scheme, are presented.Instituto de Cooperação Científica e Tecnológica Internacional - Acções Integradas Luso-Alemãs (DAAD/ICCTI) - Projecto DAAD/ICCTI nº 01141

Analysis of BRAF and NRAS Mutation Status in Advanced Melanoma Patients Treated with Anti-CTLA-4 Antibodies: Association with Overall Survival?

Ipilimumab and tremelimumab are human monoclonal antibodies (Abs) against cytotoxic T-lymphocyte antigen-4 (CTLA-4). Ipilimumab was the first agent to show a statistically significant benefit in overall survival in advanced melanoma patients. Currently, there is no proven association between the BRAFV600 mutation and the disease control rate in response to ipilimumab. This analysis was carried out to assess if BRAFV600 and NRAS mutation status affects the clinical outcome of anti-CTLA-4-treated melanoma patients. This is a retrospective multi-center analysis of 101 patients, with confirmed BRAF and NRAS mutation status, treated with anti-CTLA-4 antibodies from December 2006 until August 2012. The median overall survival, defined from the treatment start date with the anti-CTLA-4. Abs-treatment to death or till last follow up, of BRAFV600 or NRAS mutant patients (n = 62) was 10.12 months (95% CI 6.78-13.2) compared to 8.26 months (95% CI 6.02-19.9) in BRAFV600/NRASwt subpopulation (n = 39) (p = 0.67). The median OS of NRAS mutated patients (n = 24) was 12.1 months and although was prolonged compared to the median OS of BRAF mutated patients (n = 38, mOS = 8.03 months) or BRAFV600/NRASwt patients (n = 39, mOS = 8.26 months) the difference didn't reach statistical significance (p = 0.56). 69 patients were able to complete 4 cycles of anti-CTLA-4 treatment. Of the 24 patients treated with selected BRAF- or MEK-inhibitors, 16 patients received anti-CTLA 4 Abs following either a BRAF or MEK inhibitor with only 8 of them being able to finish 4 cycles of treatment. Based on our results, there is no difference in the median OS in patients treated with anti-CTLA-4 Abs implying that the BRAF/NRAS mutation status alone is not sufficient to predict the outcome of patients treated with anti-CTLA-4 Abs

Differential influence of vemurafenib and dabrafenib on patients' lymphocytes despite similar clinical efficacy in melanoma

In this study, we demonstrate that vemurafenib but not dabrafenib reduces peripheral lymphocyte counts in melanoma patients while both agents show similar clinical efficacy. Within the lymphocyte compartment, vemurafenib selectively decreases circulating CD4+ T cells and changes their phenotype and function. This indicates that selective BRAFi need to be assessed individually for immunomodulatory effects, especially, when planning combinations with immunotherapie

A convenient coordinatization of Siegel-Jacobi domains

We determine the homogeneous K\"ahler diffeomorphism $FC$ which expresses the K\"ahler two-form on the Siegel-Jacobi ball \mc{D}^J_n=\C^n\times \mc{D}_n as the sum of the K\"ahler two-form on \C^n and the one on the Siegel ball \mc{D}_n. The classical motion and quantum evolution on \mc{D}^J_n determined by a hermitian linear Hamiltonian in the generators of the Jacobi group G^J_n=H_n\rtimes\text{Sp}(n,\R)_{\C} are described by a matrix Riccati equation on \mc{D}_n and a linear first order differential equation in z\in\C^n, with coefficients depending also on W\in\mc{D}_n. $H_n$ denotes the $(2n+1)$-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on \mc{D}_n. When the transform $FC:(\eta,W)\rightarrow (z,W)$ is applied, the first order differential equation in the variable \eta=(\un-W\bar{W})^{-1}(z+W\bar{z})\in\C^n becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel-Jacobi upper half plane \mc{X}^J_n=\C^n\times\mc{X}_n, where \mc{X}_n denotes the Siegel upper half plane.Comment: 32 pages, corrected typos, Latex, amsart, AMS font

Mental illness and COVID-19 vaccination: a multinational investigation of observational & register-based data

Individuals with mental illness are at higher risk of severe COVID-19 outcomes. However, previous studies on the uptake of COVID-19 vaccination in this population have reported conflicting results. Using data from seven cohort studies (N = 325,298) included in the multinational COVIDMENT consortium, and the Swedish registers (N = 8,080,234), this study investigates the association between mental illness (defined using self-report measures, clinical diagnosis and prescription data) and COVID-19 vaccination uptake. Results from the COVIDMENT cohort studies were pooled using meta-analyses, the majority of which showed no significant association between mental illness and vaccination uptake. In the Swedish register study population, we observed a very small reduction in the uptake of both the first and second dose of a COVID-19 vaccine among individuals with vs. without mental illness; the reduction was however greater among those not using psychiatric medication. Here we show that uptake of the COVID-19 vaccine is generally high among individuals both with and without mental illness, however the lower levels of vaccination uptake observed among subgroups of individuals with unmedicated mental illness warrants further attention