Fourier series for quaternions and the square of the error theorem

In this paper we introduce a type of Hypercomplex Fourier Series based on Quaternions, and discuss on a Hypercomplex version of the Square of the Error Theorem. Since their discovery by Hamilton (Sinegre [1]), quaternions have provided beautifully insights either on the structure of different areas of Mathematics or in the connections of Mathematics with other fields. For instance: I) Pauli spin matrices used in Physics can be easily explained through quaternions analysis (Lan [2]); II) Fundamental theorem of Algebra (Eilenberg [3]), which asserts that the polynomial analysis in quaternions maps into itself the four dimensional sphere of all real quaternions, with the point infinity added, and the degree of this map is n. Motivated on earlier works by two of us on Power Series (Pendeza et al. [4]), and in a recent paper on Liouville’s Theorem (Borges and Mar˜o [5]), we obtain an Hypercomplex version of the Fourier Series, which hopefully can be used for the treatment of hypergeometric partial differential equations such as the dumped harmonic oscillation.

Square of the Error Octonionic Theorem and Hypercomplex Fourier Series

The focus of this paper is to address some classical results for a class of hypercomplex numbers. More specifically we present an extension of the Square of the Error Theorem and a Bessel inequality for octonions.

Transformada de Fourier: teoria como base para aplicações em mecânica celeste

Pós-graduação em Matemática Universitária - IGCE

Rotor Failure Diagnosis of Induction Motors by Wavelet Transform and Fourier Transform in Function of the Load

This paper presents two diagnostic methods for the online detection of broken bars in induction motors with squirrel-cage type rotors. The wavelet representation of a function is a new technique. Wavelet transform of a function is the improved version of Fourier transform. Fourier transform is a powerful tool for analyzing the components of a stationary signal. But it is failed for analyzing the non-stationary signal whereas wavelet transform allows the components of a non-stationary signal to be analyzed. In this paper, our main goal is to find out the advantages of wavelet transform compared to Fourier transform in rotor failure diagnosis of induction motors.

Development and validation of the quantitative analysis of ampicillin sodium in powder for injection by Fourier-transform infrared spectroscopy (FT-IR)

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Oscilador harmônico unidimensional via transformada unilateral de Fourier e mapeamento do potencial de morse unidimensional nos potenciais pseudo-harmônico e de kratzer tridimensionais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Oscilador harmônico unidimensional via transformada unilateral de Fourier e mapeamento do potencial de morse unidimensional nos potenciais pseudo-harmônico e de kratzer tridimensionais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Fractal descriptors based on Fourier spectrum applied to texture analysis

This work proposes the development and study of a novel technique lot the generation of fractal descriptors used in texture analysis. The novel descriptors are obtained from a multiscale transform applied to the Fourier technique of fractal dimension calculus. The power spectrum of the Fourier transform of the image is plotted against the frequency in a log-log scale and a multiscale transform is applied to this curve. The obtained values are taken as the fractal descriptors of the image. The validation of the proposal is performed by the use of the descriptors for the classification of a dataset of texture images whose real classes are previously known. The classification precision is compared to other fractal descriptors known in the literature. The results confirm the efficiency of the proposed method. (C) 2012 Elsevier B.V. All rights reserved.

Hybrid Water Demand Forecasting Model Associating Artificial Neural Network with Fourier Series

This paper addressed the problem of water-demand forecasting for real-time operation of water supply systems. The present study was conducted to identify the best fit model using hourly consumption data from the water supply system of Araraquara, Sa approximate to o Paulo, Brazil. Artificial neural networks (ANNs) were used in view of their enhanced capability to match or even improve on the regression model forecasts. The ANNs used were the multilayer perceptron with the back-propagation algorithm (MLP-BP), the dynamic neural network (DAN2), and two hybrid ANNs. The hybrid models used the error produced by the Fourier series forecasting as input to the MLP-BP and DAN2, called ANN-H and DAN2-H, respectively. The tested inputs for the neural network were selected literature and correlation analysis. The results from the hybrid models were promising, DAN2 performing better than the tested MLP-BP models. DAN2-H, identified as the best model, produced a mean absolute error (MAE) of 3.3 L/s and 2.8 L/s for training and test set, respectively, for the prediction of the next hour, which represented about 12% of the average consumption. The best forecasting model for the next 24 hours was again DAN2-H, which outperformed other compared models, and produced a MAE of 3.1 L/s and 3.0 L/s for training and test set respectively, which represented about 12% of average consumption. DOI: 10.1061/(ASCE)WR.1943-5452.0000177. (C) 2012 American Society of Civil Engineers.

Quantification of Retinal Neural Loss in Patients with Neuromyelitis Optica and Multiple Sclerosis with or without Optic Neuritis Using Fourier-Domain Optical Coherence Tomography

PURPOSE. We compared retinal nerve fiber layer (RNFL) and macular thickness measurements in patients with multiple sclerosis (MS) and neuromyelitis optica (NMO) with or without a history of optic neuritis, and in controls using Fourier-domain (FD) optical coherence tomography (OCT). METHODS. Patients with MS (n = 60), NMO (n = 33), longitudinal extensive transverse myelitis (LETM, n = 28) and healthy controls (n = 41) underwent ophthalmic examination, including automated perimetry, and FD-OCT RNFL and macular thickness measurements. Five groups of eyes were compared: MS with or without previous optic neuritis, NMO, LETM, and controls. Correlation between OCT and visual field (VF) findings was investigated. RESULTS. With regard to most parameters, RNFL and macular thickness measurements were significantly smaller in eyes of each group of patients compared to controls. MS eyes with optic neuritis did not differ significantly from MS eyes without optic neuritis, but measurements were smaller in NMO eyes than in all other groups. RNFL (but not macular thickness) measurements were significantly smaller in LETM eyes than in controls. While OCT abnormalities were correlated significantly with VF loss in NMO/LETM and MS, the correlation was much stronger in the former. CONCLUSIONS. Although FD-OCT RNFL and macular thickness measurements can reveal subclinical or optic neuritis-related abnormalities in NMO-spectrum and MS patients, abnormalities are predominant in the macula of MS patients and in RFNL measurements in NMO patients. The correlation between OCT and VF abnormalities was stronger in NMO than in MS, suggesting the two conditions differ regarding structural and functional damage. (ClinicalTrials.gov number, NCT01024985.) Invest Ophthalmol Vis Sci. 2012;53:3959-3966) DOI:10.1167/iovs.11-9324

Effect of dental tissue conditioners and matrix metalloproteinase inhibitors on type I collagen microstructure analyzed by Fourier transform infrared spectroscopy

This study aimed to evaluate the chemical interaction of collagen with some substances usually applied in dental treatments to increase the durability of adhesive restorations to dentin. Initially, the similarity between human dentin collagen and type I collagen obtained from commercial bovine membranes of Achilles deep tendon was compared by the Attenuated Total Reflectance technique of Fourier Transform Infrared (ATR-FTIR) spectroscopy. Finally, the effects of application of 35% phosphoric acid, 0.1M ethylenediaminetetraacetic acid (EDTA), 2% chlorhexidine, and 6.5% proanthocyanidin solution on microstructure of collagen and in the integrity of its triple helix were also evaluated by ATR-FTIR. It was observed that the commercial type I collagen can be used as an efficient substitute for demineralized human dentin in studies that use spectroscopy analysis. The 35% phosphoric acid significantly altered the organic content of amides, proline and hydroxyproline of type I collagen. The surface treatment with 0.1M EDTA, 2% chlorhexidine, or 6.5% proanthocyanidin did not promote deleterious structural changes to the collagen triple helix. The application of 6.5% proanthocyanidin on collagen promoted hydrogen bond formation. (c) 2012 Wiley Periodicals, Inc. J Biomed Mater Res Part B: Appl Biomater, 2012.