Euler e os números pentagonais

The present investigation includes a study of Leonhard Euler and the pentagonal numbers is his article Mirabilibus Proprietatibus Numerorum Pentagonalium - E524. After a brief review of the life and work of Euler, we analyze the mathematical concepts covered in that article as well as its historical context. For this purpose, we explain the concept of figurate numbers, showing its mode of generation, as well as its geometric and algebraic representations. Then, we present a brief history of the search for the Eulerian pentagonal number theorem, based on his correspondence on the subject with Daniel Bernoulli, Nikolaus Bernoulli, Christian Goldbach and Jean Le Rond d'Alembert. At first, Euler states the theorem, but admits that he doesn t know to prove it. Finally, in a letter to Goldbach in 1750, he presents a demonstration, which is published in E541, along with an alternative proof. The expansion of the concept of pentagonal number is then explained and justified by compare the geometric and algebraic representations of the new pentagonal numbers pentagonal numbers with those of traditional pentagonal numbers. Then we explain to the pentagonal number theorem, that is, the fact that the infinite product(1 x)(1 xx)(1 x3)(1 x4)(1 x5)(1 x6)(1 x7)... is equal to the infinite series 1 x1 x2+x5+x7 x12 x15+x22+x26 ..., where the exponents are given by the pentagonal numbers (expanded) and the sign is determined by whether as more or less as the exponent is pentagonal number (traditional or expanded). We also mention that Euler relates the pentagonal number theorem to other parts of mathematics, such as the concept of partitions, generating functions, the theory of infinite products and the sum of divisors. We end with an explanation of Euler s demonstration pentagonal number theorem

Comitês de Classificadores para o Reconhecimento Multibiométrico em Dados Biométricos Revogáveis

This work discusses the application of techniques of ensembles in multimodal recognition systems development in revocable biometrics. Biometric systems are the future identification techniques and user access control and a proof of this is the constant increases of such systems in current society. However, there is still much advancement to be developed, mainly with regard to the accuracy, security and processing time of such systems. In the search for developing more efficient techniques, the multimodal systems and the use of revocable biometrics are promising, and can model many of the problems involved in traditional biometric recognition. A multimodal system is characterized by combining different techniques of biometric security and overcome many limitations, how: failures in the extraction or processing the dataset. Among the various possibilities to develop a multimodal system, the use of ensembles is a subject quite promising, motivated by performance and flexibility that they are demonstrating over the years, in its many applications. Givin emphasis in relation to safety, one of the biggest problems found is that the biometrics is permanently related with the user and the fact of cannot be changed if compromised. However, this problem has been solved by techniques known as revocable biometrics, which consists of applying a transformation on the biometric data in order to protect the unique characteristics, making its cancellation and replacement. In order to contribute to this important subject, this work compares the performance of individual classifiers methods, as well as the set of classifiers, in the context of the original data and the biometric space transformed by different functions. Another factor to be highlighted is the use of Genetic Algorithms (GA) in different parts of the systems, seeking to further maximize their eficiency. One of the motivations of this development is to evaluate the gain that maximized ensembles systems by different GA can bring to the data in the transformed space. Another relevant factor is to generate revocable systems even more eficient by combining two or more functions of transformations, demonstrating that is possible to extract information of a similar standard through applying different transformation functions. With all this, it is clear the importance of revocable biometrics, ensembles and GA in the development of more eficient biometric systems, something that is increasingly important in the present day