On universal central extensions of Hom-Leibniz algebras

In the category of Hom-Leibniz algebras we introduce the notion of Hom-corepresentation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of α-central extension, universal α-central extension and α-perfect Hom-Leibniz algebra due to the fact that the composition of two central extensions of Hom-Leibniz algebras is not central. We also provide the recognition criteria for these kind of universal central extensions. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them. In case α = Id we recover the corresponding results on universal central extensions of Leibniz algebras.

PROGRAMA PARA A JUSTIÇA PORTUGUESA

Há a verdade, há a mentira e há as estatísticas. Muitas outras medidas poderiam ser alvitradas, mas estas são algumas das ideias fundamentais para um programa de Governo que visa uma Justiça humanista, na qual o Ser Humano, organizado ou individual, é o centro da questão. A Justiça portuguesa também não pode ser uma competição entre a cor-de-laranja e o rosa, vermelho, azul ou qualquer outra cor, como amarelo, cinzento, castanho, entre outras. A Justiça portuguesa deve fazer-se sentir e respeitar. Não é mentira se dissermos que temos entre os melhores Magistrados do mundo, entre os melhores Advogados e Solicitadores do mundo, entre os melhores Funcionários Judiciais, Conservadores, Notários e outros Auxiliares de Justiça, entre os melhores polícias e investigadores do mundo. Bem hajas, Portugal, pois és capaz de muito mais e melhor. § There is the fact there are lies and there are statistics. Many other measures could be tell, but these are some of the key ideas for a government program aimed at a humanistic Justice, in which the human being organized or individual, is the heart of the matter. The Portuguese Justice can not be a competition between the pink-orange and pink, red, blue or any other color, such as yellow, gray, brown, and others. The Portuguese Justice must be felt and respect. It's not a lie if we say we have among the best Magistrates of the world, among the best Lawyers and Solicitors of the world, among the best Judicial Officers, Conservatives, notaries and other auxiliaries of Justice, police and among the best researchers in the world. Well wilt, Portugal, for you are capable of much more and better.

Computer-aided recognition of dental implants in X-ray images

Dental implant recognition in patients without available records is a time-consuming and not straightforward task. The traditional method is a complete user-dependent process, where the expert compares a 2D X-ray image of the dental implant with a generic database. Due to the high number of implants available and the similarity between them, automatic/semi-automatic frameworks to aide implant model detection are essential. In this study, a novel computer-aided framework for dental implant recognition is suggested. The proposed method relies on image processing concepts, namely: (i) a segmentation strategy for semi-automatic implant delineation; and (ii) a machine learning approach for implant model recognition. Although the segmentation technique is the main focus of the current study, preliminary details of the machine learning approach are also reported. Two different scenarios are used to validate the framework: (1) comparison of the semi-automatic contours against implant’s manual contours of 125 X-ray images; and (2) classification of 11 known implants using a large reference database of 601 implants. Regarding experiment 1, 0.97±0.01, 2.24±0.85 pixels and 11.12±6 pixels of dice metric, mean absolute distance and Hausdorff distance were obtained, respectively. In experiment 2, 91% of the implants were successfully recognized while reducing the reference database to 5% of its original size. Overall, the segmentation technique achieved accurate implant contours. Although the preliminary classification results prove the concept of the current work, more features and an extended database should be used in a future work.