Objetivou-se com este trabalho avaliar a utilização da metodologia no infravermelho com Transformada de Fourier (ITF) na determinação do pH e ponto de congelamento (PC) no leite bovino. No primeiro estudo foram avaliados os efeitos das diferentes condições de conservação em amostras de leite para posterior análise do pH e do PC. Para tanto, foram coletadas, do tanque de 57 propriedades, amostras de três litros de leite. Cada amostra foi subdividida e transferida para 45 frascos de 40mL e distribuídas de acordo com a temperatura de armazenamento (-30C° - Congelado; 7°C - Resfriado; 25°C - Ambiente), idade da amostra (0, 3, 6 e 9dias) e níveis de adição de água (0, 2, 4 e 6%). Foram adicionadas pastilhas de bronopol (Microtabs®) aos 44 frascos, sendo que um foi mantido sem adição do conservante para tratamento controle. Os resultados obtidos, considerando as diferentes condições de conservação das amostras, e ainda, o efeito da adição do bronopol, foram avaliados por comparação de médias. As metodologias (referência e alternativa) foram correlacionadas em função da idade da amostra para o PC por análise de regressão linear. A sensibilidade e a especificidade foram calculadas para avaliação do desempenho do equipamento MilkoScanTM FT+ na detecção de água. A adição do bronopol não alterou as médias do pH, porém reduziu significativamente o PC. Para eliminação desse efeito, foram calculados fatores de correção para os resultados do PC nas duas metodologias empregadas nesse estudo. O aumento da temperatura de armazenamento reduziu significativamente as médias do pH e do PC em amostras com seis e nove dias de coleta. Não houve efeito significativo nas médias do pH e do PC com o aumento na idade da amostra a -30° e 7°C. As correlações entre as metodologias em função da idade da amostra para o PC foram altas e significativas. A sensibilidade e especificidade do equipamento MilkoScanTM FT+ na detecção de água, em relação ao crioscópio eletrônico, foram de 90,9% e de 86,8%, respectivamente. No segundo estudo, objetivou-se caracterizar a atual situação do ponto de congelamento (PC) em rebanhos brasileiros, e avaliar o efeito da lactose e da contagem bacteriana total (CBT) sobre os resultados do PC. Foram analisados 137.443 dados por meio de estatística descritiva e de análise da variância, para caracterizar a atual situação do PC e para avaliação dos efeitos da lactose e da CBT sobre o PC. A média e o respectivo desviopadrão do PC foi de -0,522 (0,011)°C, e, constatou-se que o percentual de amostras em conformidade com a Instrução Normativa nº 51 foi de 92,48%, enquanto que para amostras em não conformidade, com indicativo de adição de água ou de soluto, foi de 7,10% e 0,41% respectivamente. A lactose e a CBT influenciaram significativamente nos níveis do PC. / The objective of this study was to evaluate the use of the methodology Fourier Transform Infrared (FTIR) for determining the pH and freezing point (FP) in bovine milk. The first study evaluated the effects of different storage conditions on samples of milk for subsequent analysis of pH and the PC. For both, were collected from the tank of 57 properties, samples of three liters of milk. Each sample was divided and transferred to 45 vials of 40mL and distributed according to storage temperature (-30C°- Frozen, 7°C - Cold, 25°C - Environment), age of the sample (0, 3, 6 and 9 days) and levels of added water (0, 2, 4 and 6%). Tablets to bronopol (Microtabs®) were added to 44 bottles, of which one was maintained without addition of preservative for control treatment. The results, considering the different conditions of storage of samples, and the effect of adding bronopol were evaluated by comparison of means. The methods (reference and alternative) were correlated with age of the sample to the FP for linear regression analysis. The sensitivity and specificity were calculated to evaluate the performance of MilkoScanTM FT+ equipment for detecting water. The addition of bronopol did not alter the average pH, but significantly reduced the FP. To combat this effect, were calculated correction factors for the results of the PC in the two methodologies employed in this study. Increasing the temperature of storage significantly reduced the average pH and the FP samples with six and nine days of collection. There was no significant effect on mean pH and the PC with the increasing age of the sample at -30°C and 7°C. The correlations between the methods depending on the age of the sample for the FP were high and significant. The sensitivity and specificity of the equipment MilkoScanTM FT+ in the detection of water on the thermistor cryoscope were 90.9% and 86.8% respectively. While in the second study aimed to characterize the current state of the freezing point (FP) in Brazilian herds and Evaluated the effect of lactose and the total bacterial count (TBC) on the results of the FP. 137,443 data were analyzed using descriptive statistics and analysis of variance, to characterize the current state of the FP and to evaluated the effects of lactose and TBC on the FP. The mean and standard deviation of the FP was -0.522 (0.011)°C, and it was found that the percentage of samples in accordance with Normative Instruction nº 51 was 92.48%, while for samples not in accordance with an indication of added water and solute was 7.10% and 0.41% respectively. The lactose and TBC significantly influenced the levels of PC.

A digital program has been developed to Fourier analyse the data obtained in experiments to study the de Haas-van Alphen effect. A physical account of the origin of this effect is given and the problem of analysing the data is explained. An account is given of the development of the program together with a consideration of the mathematical properties of the Fourier transform. General questions concerning the resolving power and accuracy of this method of analysis are discussed. The program was tested using artificial synthetic data of known analytic form and later applied to the analysis of data from a single crystal of mercury. / Thesis / Master of Science (MSc)

In this thesis, the main objective is to study the presence of Gibbs phenomenon and the Gibbs constant in Fourier-Legendre series. The occurrence of The Gibbs phenomenon is a well known consequence when approximating functions with Fourier series that have points of discontinuity. Consequently, the initial focus was to examine Fourier seriesand the occurrence of Gibbs phenomenon in this context. Next, we delve into Legendrepolynomials, showing their applicability to be expressed as a Fourier series due to theirorthogonality in [−1, 1]. We then continue to explore Gibbs phenomenon for Fourier-Legendre series. The findings proceeds to confirm the existence of the Gibbs phenomenon for Fourier-Legendre series, but most notebly, the values of the error seem to convergeto the same number as for Fourier series which is the Gibbs constant.

No description available.

Fourier analysis, Walsh-Fourier analysis, and wavelet analysis have often been used in time series analysis. Fourier analysis can be used to detect periodic components that have sinusoidal shape; however, it might be misleading when the periodic components are not sinusoidal. Walsh-Fourier analysis is suitable for revealing the rectangular trends of time series. The flaw of the Walsh-Fourier analysis is that Walsh functions are not periodic. The resulting Walsh-Fourier analysis is more difficult to interpret than classical Fourier analysis. Wavelet analysis is very useful in analyzing and describing time series with gradual frequency changes. Wavelet analysis also has a shortcoming by giving no exact meaning to the concept of frequency because wavelets are not periodic functions. In addition, all three analysis methods above require equally-spaced time series observations. In this dissertation, by using a sequence of periodic step functions, a new analysis method, adaptive Fourier analysis, and its theory are developed. These can be applied to time series data where patterns may take general periodic shapes that include sinusoids as special cases. Most importantly, the resulting adaptive Fourier analysis does not require equally-spaced time series observations. / Ph. D.

Master of Science / Department of Mathematics / Marianne Korten / Distribution theory is an important tool in studying partial differential equations. Distributions are linear functionals that act on a space of smooth test functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. There are different possible choices for the space of test functions, leading to different spaces of distributions. In this report, we take a look at some basic theory of distributions and their Fourier transforms. And we also solve some typical exercises at the end.

The adaptation of a general purpose laboratory minicomputer for nuclear medicine imaging is described. Electronic interfaces have been designed and constructed to link nucleonic equipment to a PDP 12 computer. A computer television display system has been developed to facilitate interactive processing of scintigraphic data. The main features of the television system are that it is relatively inexpensive and reliable. A domestic quality receiver has been adapted for use as a colour monitor. Any instrument that records data will produce a distorted or degraded version of the input signal. Generally, imaging equipment will produce a blurred image of the object, and in the case of scintigraphic imaging the blurs may be comparable to the size of the physiological structures being investigated. The process of refocussing the recorded data is called, in mathematical terms, deconvolution. In this study Fourier transform techniques have been developed as methods of implementing deconvolution. It is shown that the restoration of images in the presence of noise is likely to be a mathematically unstable process. Four methods of accommodating the problems associated with noise are described. Each method has built in optimisation of one form or another so that mathematically stable algorithms are used to implement deconvolution. This means that all the parameters used by the computer programs are determined automatically so that the computer operator is not required to select any parameters manually. A brief description of two dimensional digital filtering is given to enable comparison between filtering and deconvolution of scintigrams. A two dimensional lowpass filter is developed which automatically defines the passband frequency response appropriate to a particular scintigram. Finally, all the signal processing methods are tested on both simulated and clinical data. Results show that deconvolution offers advantages over digital filtering particularly for scintigrams obtained from morphic structures. Some of the problems of deconvolving certain types of scintigram are discussed.

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No description available.

L'objectif de la thèse est d'explorer les comportements mécaniques effectifs des matériaux poreux impliqués dans les problèmes de pores ou d'inclusion, avec pour objectif principal d'homogénéiser un matériau aussi hétérogène que possible, plus d'informations de géométrie physique à une échelle donnée. Nous examinons séparément ici quatre microstructures typiques de matériaux ressemblant à des roches couvrant différents constituants: nous avons cherché à déterminer comment la présence de pores et une inclusion de morphologies différentes influent sur les comportements macroscopiques élastiques, débit plastique et dépendant du temps. Un modèle d'homogénéisation par étapes pour les comportements élastoplastiques et viscoplastiques est proposé, basé sur une méthode d'homogénéisation par transformée de Fourier rapide, afin d'améliorer le comportement local depuis la transition à la micro-échelle vers la méso-échelle, puis vers la macroscopie. Une série de simulations est calculée et comparée. Nous portons une attention particulière au comportement anisotrope des matériaux à la fois pore et inclusion configurée à la même échelle et à différentes échelles Nous fournissons une solution de référence pour ce type de matériaux couvrant l’effet des géométries de pores et d’inclusion en raison de la limitation à l’obtention d’un critère analytique. En plus nous proposons un modèle d’endommagement plastique multi-échelle. les comparaisons entre les résultats numériques et les données expérimentales montrent que ce modèle peut bien caractériser son comportement en défaillance avec des micro-structures complexes. / The objective of this thesis aims to explore the effective mechanical behaviors of porous materials involved with pore or inclusion problem. The key point to this problem is to homogenize such a highly heterogeneous material incorporating more physical geometry information at a given scale. In this work, four typical microstructure of rock-like materials are respectively considered here covering different constituents. We sought to determine how the presence of pores and inclusion with different morphologies influence the macroscopic elastic, plastic flow and time-dependent behaviors. For this propose, a two-step homogenization model for elastoplastic and viscoplastic behaviors is proposed based on a Fast Fourier Transform homogenization method to upscale the local behavior from the micro-scale transition to meso-scale and then to macroscale. A series of simulations are computed and compared. Particular attentions are focused on the anisotropic behavior for the materials with both pore and inclusion configured at same and different scales. We provide a reference solution for this kinds of materials covering the effect of pore and inclusion geometries due to a limitation to obtain an analytical criterion. Moreover, we proposed a damage plastic model with multi-scale characters. Comparisons between numerical results and experimental data show that this model can well characterize its failure behavior with complex micro-structures.