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O número de Milnor de uma singularidade isolada
(Universidade Federal de São Carlos, 2011-11-24) Oréfice, Bruna; Tomazella, João Nivaldo; https://lattes.cnpq.br/0051564735964760; https://lattes.cnpq.br/6824383277098012
Given (X; 0) C (CN; 0) a weighted homogeneous germ of hypersurface with isolated singularity and f : (CN; 0) - C a germ of function finitely determined with respect to X, we show that UBR(f;X) = U(f) + U(X; f), where U(f) and U(X; f) denote the Milnor numbers of f and of the fiber X \ f��1(0), respectively, and UBR(f;X) is the Bruce-Roberts number of f with respect to X. We show that the logarithmic characteristic subvariety, LC(X), is Cohen-Macaulay and we get relations between the Bruce-Roberts number and the Euler obstruction. Given F : (CN; 0) ! Mm;n(C) a holomorphic function germ, let (X; 0) be the isolated determinantal singularity given by X = F-1(Ms m;n(C)) where Ms m;n(C) is the set of the complex matrices with rank less then s, with s an integer number between 0 and minfm; ng such that N < (m - s + 2)(n - s + 2), we will define the vanishing Euler characteristic of (X; 0) and the Milnor number of a holomorphic function germ with an isolated singularity at X, f : (X; 0) - C.
Ressonância magnética nuclear ultrarrápida: implementação, desenvolvimento e aplicações
(Universidade Federal de São Carlos, 2011-11-24) Queiroz Júnior, Luiz Henrique Keng; Ferreira, Antonio Gilberto; https://lattes.cnpq.br/3676462220401452; https://lattes.cnpq.br/5071272851467384
This work highlights the implementation, development and application of the Ultrafast Nuclear Magnetic Resonance (UFNMR) technique, proposed by Frydman and co-workers. Through this technique, it is possible to perform multidimensional experiments in a single scan, thus reducing the experiment time drastically, which can be a fraction of second. This study shows a practical protocol, which was optimized for the implementation of the first ultrafast 2D NMR COSY and HSQC experiments performed in Brazil and, to our knowledge, in the southern hemisphere. This process involves the setting of several parameters from the initial calibration of encoding gradients and pulses, to the setting of a specific acquisition scheme required by the technique. Furthermore, some important operational aspects are discussed in order to contribute, even more, to future implementations to be carried out safely and efficiently. Also, as part of this study, a software in C language was developed to obtain the real-time processing (1 second) of the UF-NMR data, and this was incorporated in the Bruker software for the acquisition and processing of NMR data - TopSpin® version 3.0. Finally, two studies were also performed using the UF-NMR experiments. The first one was the real-time identification of a mixture of flavonoids (epicatechin, naringenin and naringin), which underwent on-flow chromatographic separation, in a commercial HPLC-NMR system. UF-COSY experiments were performed during the chromatographic run, so that it was possible to obtain the 2D NMR spectra related to each flavonoid and to successfully follow their separation over the course of time. The second application was the monitoring of the hydrolysis reaction of the 2-(4-nitrophenyl)-1,3-dioxolane acetal. The hemiacetal intermediate of this reaction has a short lifetime, being difficult to identify by conventional NMR. Therefore, HSQC-UR experiments were performed during the reaction development, thus, it was possible to characterize the presence of the hemiacetal intermediate. This assignment was confirmed by quantum calculation of 1H and 13C NMR chemical shifts and of the NBOs (Natural Bonding Orbital).