Geometric Approach to Pontryagin's Maximum Principle

Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page

A robust statistical procedure to discover expression biomarkers using microarray genomic expression data

Microarray has become increasingly popular biotechnology in biological and medical researches, and has been widely applied in classification of treatment subtypes using expression patterns of biomarkers. We developed a statistical procedure to identify expression biomarkers for treatment subtype classification by constructing an F-statistic based on Henderson method III. Monte Carlo simulations were conducted to examine the robustness and efficiency of the proposed method. Simulation results showed that our method could provide satisfying power of identifying differentially expressed genes (DEGs) with false discovery rate (FDR) lower than the given type I error rate. In addition, we analyzed a leukemia dataset collected from 38 leukemia patients with 27 samples diagnosed as acute lymphoblastic leukemia (ALL) and 11 samples as acute myeloid leukemia (AML). We compared our results with those from the methods of significance analysis of microarray (SAM) and microarray analysis of variance (MAANOVA). Among these three methods, only expression biomarkers identified by our method can precisely identify the three human acute leukemia subtypes

Tipos de miosinas de linhagens de frangos de corte criados em sistemas de confinamento e semiconfinamento Types of myosin of chickens of different strains reared in confinement and semi-confinement systems

O objetivo neste trabalho foi avaliar o peso vivo, o peso de pernas, os aspectos morfológicos das fibras musculares do músculo flexor longo do hálux e o perfil eletroforético das miosinas de cadeia pesada de quatro linhagens de frangos de corte criados nos sistemas de confinamento e semiconfinamento. Foram utilizados 1.440 pintos distribuídos em delineamento inteiramente casualisado em esquema fatorial 4 &#215; 2, composto de quatro linhagens (Ross 305, Máster Gris, Label Rouge e Vermelhão Pesado) e dois sistemas de criação (confinamento e semiconfinamento), cada combinação avaliada com quatro repetições. Aos 28 e 84 dias de idade, foram abatidas quatro aves por tratamento, totalizando 64 aves. A eletroforese identificou a presença das três isoformas de miosinas, tipo MyHC-I, MyHC-IIa e MyHC-IIb, no músculo flexor longo do hálux dos frangos de corte. Com aumento da idade, a isoforma de miosina MyHC-II aumenta, enquanto a MyHC-I diminui. Somente aos 84 dias de idade, a expressão das isoformas de miosina do tipo MyHC-II foram influenciadas pela linhagem, confirmando o reflexo da seleção na linhagem Ross no músculo mais glicolítico.A linhagem Ross apresenta maior peso vivo, peso de perna, peso e área do músculo flexor longo do hálux em comparação às linhagens tipo caipira.<br>The objective was to evaluate body weight, leg weight and morphologic aspects of the muscle fibers of the flexor hallucis longus muscle and electrophoretic profile of myosin heavy chain of four strains of broilers, reared in confinement and semi-confinement systems. 1440 chicks were randomly assigned in a 4 &#215; 2 factorial arrangement: four strains (Ross 305, Master Gris, Label Rouge and Vermelhao Pesado), two production systems (confinement and semi-confinement), with four replicates for each treatment. Four birds were sacrificed for each treatment, at 28 and 84 d, totaling 64 animals. Electrophoresis technique identified the presence of three myosin heavy chain types: MyHC-I, MyHC-IIa and MyHC-IIb, in the flexor hallucis longus muscle of the broilers. There was increase in MyHC-II and decrease in MyHC-I, as age passed. Only at 84 days-old was the MyHC-II expression influenced by strain, confirming a response of selection in Ross lineage on a more glycolytic muscle. The Ross lineage has higher live weight, leg weight, weight and area of the flexor hallucis longus, compared to different chicken strains