On a theorem due to Murray

In this paper, we introduce the notions of $\alpha$-quasicomplemented and totally $\alpha$-quasicomplemented subspaces and we established some results under these contexts. We show, for example, that if $X$ is a separable or reflexive Banach space and $Y$ is a closed infinite codimensional subspace of $X$, then $Y$ is totally$\mathit{\ }\alpha$-quasicomplemented if, and only if, $\alpha<\aleph_{0}$ $\left( \text{this is an analogue of the theorem of Murray-Mackey and Lindenstrauss}\right)$. We also show that if $H$ is a Hilbert space and $Y,W\subset H$ are closed subspaces of $H$ such that $W$ is orthogonal to $Y$ and $\operatorname{codim}\left( Y+W\right) =\infty$, then $Y$ has a quasicomplement $Z$ containing $W$ with $\dim Z/W=\infty$. Other results in the different contexts are also included. Such results establish a connection between the theory of quasicomplemented subspaces and $\left( \alpha,\beta\right)$-spaceability

A case of equine multicentric lymphoma: Clinical, microscopical, and molecular findings

Background: Although relatively uncommon, lymphoma is the most prevalent haematopoietic neoplasia in horses, and multicentric lymphoma remains the most common presentation of the disease. The pathogenesis of equine lymphoma is still poorly understood and the diagnosis is usually confirmed at an advanced stage of the disease, compromising the prognosis. This study investigated the clinical, pathological, and molecular features of a case of equine multicentric lymphoma. Case description: An apparently healthy 5-year-old crossbreed mare hospitalized at the Centre of Animal Reproduction of Vairão, Portugal, suddenly presented clinical signs of supraorbital oedema and mandibular lymph node enlargement, developing fever, facial oedema, and generalized lymphadenopathy. The mare ended up dying twenty-four days after the first clinical signs due to multisystem organ failure. Haematological and biochemical analyses, necropsy, and microscopic and molecular evaluation of affected tissues were performed. At necropsy, the main findings were multiple multinodular lesions, distributed along the serous surface of oropharynx, trachea, pericardium, gastrointestinal tract, and mesentery. Microscopically, these consisted of solid proliferations of neoplastic round cells that exhibited immunopositivity for CD3 (T cells). Based on these findings, a medium-grade multicentric T-cell lymphoma was diagnosed. Conclusion: There is still very little research regarding the molecular characterization of lymphoma in horses. As an entity itself is quite heterogeneous, it is important to describe the interspecies particularities to understand its development and behaviour.GFS (2022.14765.BD) acknowledges the Portuguese Foundation for Science and Technology (FCT), for financial support. IPATIMUP integrates the i3S Research Unit, which is partially supported by FCT

From collinear to vortex magnetic structures in Mn corrals on Pt(111)

We study the magnetic properties of small Mn ring shaped clusters on a Pt(111) surface in the framework of density functional theory. We find that the Mn atoms possess large magnetic moments, of the order of 4uB/atom, and have dominating antiferromagnetic interatomic exchange interactions. A quantum confinement effect within the ring like clusters was found, indicating that even very small clusters can be seen as quantum corrals. The antiferromagnetic exchange couplings lead to collinear magnetic arrangements in simple corrals, as well as complex non-collinear ordering, as vortex-like structures, for the case of corrals with particular geometry where antiferromagnetism becomes frustrated.Comment: Accepted for publication as a Regular Article in Physical Review B (2010

Análise crítica da reconstrução da raiz da aorta com a preservação da valva aórtica: 11 anos de seguimento

INTRODUCTION: The composite mechanical valve conduit replacement is the standardized operation for aneurysms of the aortic root. The objective of this study is to evaluate the long-term surgical results of aortic valve-preserving procedures to the root reconstruction. METHODS: From 1996 to 2008, 54 consecutive patients underwent two different techniques of valve-sparing aortic root operation (40 Yacoub operations and 14 David operations). Mean age was 48 ± 14 years (range 17 to 74). 36 patients (66.7%) were male and 16 (29.6%) experienced Marfan's syndrome. The mean Euroscore was 4 ± 1.25. The mean follow up time was 4.1 years (from 49 days to 10.9 years). Clinical and echocardiographic parameters were analysed. T-Student paired test, the McNemar Non Parametric test and the Kaplan-Meyer Outcome Curves have been used. RESULTS: The hospital mortality was 5.6% and the average hospitalization time was 9±4 days. One non related late death (2%) was reported. The actuarial survival and freedom from reoperation were respectively 94.4% and 96% within 11 years of follow-up. There were benefits in reduction of functional class (P=0.002; 78% CF I), in reduction of aortic regurgitation (P<0.001; 78% with or without discrete reflux), in reduction of systolic and diastolic diameters, end-sytolic and end-diastolic volumes of left ventricle (respectively P=0.004; P<0.0001; P=0.036 and P<0.001). Two (3.9%) patients required aortic valve replacement due to severe aortic regurgitation during this same period. No thromboembolic, endocarditis or bleeding events were reported during the follow-up. CONCLUSION: The valve-sparing operation for aortic root aneurysms is an effective alternative to the use of a mechanical valve conduit replacement.INTRODUÇÃO: A utilização do tubo valvulado é a operação clássica para a reconstrução da raiz da aorta. O objetivo deste trabalho é avaliar a reconstrução da aorta ascendente com a preservação da valva aórtica. MÉTODOS: Entre 1996 e 2008, 54 pacientes consecutivos (66,7% do sexo masculino), com idade média de 48 ± 14 anos, foram submetidos à reconstrução da aorta ascendente e preservação da valva aórtica (40 remodelamentos e 14 reimplantes). O Euroscore médio foi de 4 ± 1,25 e 29,6% eram portadores de síndrome de Marfan. O tempo médio de seguimento foi de 4,1 anos (49 dias até 10,9 anos). Foram avaliados por parâmetros clínicos e ecocardiográficos. Para a análise dos dados foram utilizados os testes t de Student pareado, o não-paramétrico de McNemar e a curva de sobrevida de Kaplan-Meyer. RESULTADOS: A mortalidade hospitalar foi de 5,6%. O tempo médio de internação foi de 9 ± 4 dias. Houve um óbito tardio não relacionado (2%). A sobrevida e sobrevida livre de reoperação nos 11 anos de seguimento foram respectivamente de 94,4% e 96%. Houve melhora da classe funcional (P=0,002) (78% CF I), redução da insuficiência aórtica (P<0,001) (78% sem ou com refluxo discreto), redução dos diâmetros sistólico e diastólico, dos volumes sistólico final e diastólico final do ventrículo esquerdo, respectivamente P=0,004; P<0,001; P=0,036 e P<0,001. Dois pacientes foram submetidos à troca de valva aórtica (3,9%) com 4 e 10 anos da operação. Não foram observados fenômenos tromboembólicos, hemorrágicos ou endocardite durante o seguimento. CONCLUSÃO: A reconstrução da raiz da aorta com a preservação da valva aórtica é uma alternativa eficaz ao uso do tubo valvulado