Computational Harmonic Analysis

"The Harmonic Math, Chromatic numbers and sounds" presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic orders. The concept revolves around the relationship between different numerical bases and their visual representations, specifically focusing on Base 7. The paper explores the mathematical properties of cyclic numbers, monotonic functions, and their implications for establishing a harmonic system that maps numbers to colors in a structured way, thereby creating a cohesive language for sounds, colors, and numbers.
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• Elaborar um Programa em LabVIEW para a Análise Harmónica de Tensão e Corrente de uma Carga Não-Linear (Lâmpada Fluorescente Compacta) ligada à Rede Eléctrica;
• Configurar Entradas Analógicas de uma Carta de Aquisição de Sinal;
• Processar os sinais adquiridos com vista à determinação do conteúdo Harmónico de Tensão e Corrente e Potências associadas em regime Não-Sinusoidal;
• Calcular o Factor de Potência, Factor de desfasamento de 1.ª Harmónica e valores de Distorção Harmónica Total;
• Verificar o enquadramento dos parâmetros medidos de acordo com a Normalização da Qualidade de Tensão (EN 50160) e da Compatibilidade Electromagnética (IEC 61000-3-2).

We address the issue of approximating the pseudoinverse of the coecient matrix A for dynamically building preconditioning strategies for the numerical solution of large dense linear least-squares problems. The new preconditioning strategies are embedded into simple and well-known iterative schemes that avoid the use of the, usually ill-conditioned, normal equations. We analyze a scheme to approximate the pseudoinverse, based on Schulz iterative method, and also dierent iterative schemes, based on extensions of Richardson's method, and the conjugate gradient method, that are suitable for preconditioning strategies. We present preliminary numerical results to illustrate the advantages of the proposed schemes.

Resumen: Se explican en forma general los principales efectos de las armónicas en los componentes de los sistemas eléctricos actuales. Se muestra la necesidad de conocer los mismos para llevar a cabo un diseño o adaptación de los sistemas a las condiciones cambiantes establecidas por los dispositivos no lineales. Palabras Clave: armónicas, cargas no lineales, pérdidas, resonancia.

Finite difference schemes for the 2-D wave equation operating on hexagonal grids and the accompanying numerical dispersion properties have received little attention in comparison to schemes operating on rectilinear grids. This paper considers the hexagonal tiling of the wavenumber plane in order to show that the hexagonal grid is a more natural choice to emulate the isotropy of the Laplacian operator and the wave equation. Performance of the 7-point scheme on a hexagonal grid is better than previously reported as long as the correct stability limit and tiling of the wavenumber plane are taken into account. Numerical dispersion is analysed as a function of temporal frequency to demonstrate directional cutoff frequencies. A comparison to 9-point compact explicit schemes on rectilinear grids is presented using metrics relevant to acoustical simulation. It is shown that the 7-point hexagonal scheme has better computational efficiency than parameterised 9-point compact explicit rectilinear schemes. A novel multiply-free 7-point hexagonal scheme is introduced
and the 4-point scheme on a honeycomb grid is discussed.

In this work we consider numerical discretization of one dimensional cubic non-linear Schrödinger equation by finite differences in space and Strang splitting in time. The steps with the linear part are integrated with the midpoint rule. The aim of the paper is to explain an empirical evidence of some long-term high frequency oscillations and instabilities of this numerical scheme, first observed by U. Ascher in . Linearizing around spatially constant solution, the main result is a sufficient condition for the linearized frequencies of the numerical scheme to be purely imaginary, and thus not producing growing components. The condition is formulated in the form of restrictions on the time-step in terms of the space-step, suggesting that choosing the time-step k be smaller than h 2 , the space-step squared, prevented oscillations from emerging. In this work we provide theoretical support for this evidence and derive it by using plain wave analysis.

A tide analysis in Santa Martha bay was performed. Tide measurements were effectuated at Santa Marta bay using
WLR7 Mareograph of Aandera family. The amplitude and phase of principal constituents of the tide were established from the time series using the harmonic analysis with least square technique. The tides are mixed mainly diurnal with 12 main constituents, six diurnal (K1, O1, P1, Q1, J1, OO1), fi ve semi-diurnal (M2, S2, K2, N2, Mu2) and the lunisolar synodic fortnightly tidal constituent (MSf)

We derive the weakly nonlocal limit of a one-population neuronal field model of the Wilson-Cowan type in one spatial dimension. By transforming this equation to an equation in the firing rate variable, it is shown that stationary periodic solutions exist by appealing to a pseudo-potential analysis. The solutions of the full nonlocal equation obey a uniform bound, and the stationary periodic solutions in the weakly nonlocal limit satisfying the same uniform bound are characterized by finite ranges of pseudo energy constants. The time dependent version of the model is reformulated as a Ginzburg-Landau-Khalatnikov type of equation in the firing rate variable where the maximum (minimum) points correspond stable (unstable) homogeneous solutions of the weakly nonlocal limit. Based on this formulation it is also conjectured that the stationary periodic solutions are unstable. We implement a numerical method for the weakly nonlocal limit of the Wilson-Cowan type of model based on the wavelet-Galerkin approach. We perform some numerical tests to illustrate the stability of homogeneous solutions and the evolution of the bumps.

Functional harmonic analysis is an important task in music composition, accompaniment, arrangement and others. However, the solutions are still not satisfactory. The proposed process is divided into two levels: the first one extends one of previous works in the domain to carry out a richer analysis of chord grids and is where the very analysis is performed, and the second one is devoted to correct some conceptual inconsistencies concerning enharmonic spelling of chords. Both levels use an engine to make inferences on some rule bases, which can be easily improved by addition of new rules.

We address the issue of approximating the pseudoinverse of the coefficient matrix for dynamically building preconditioning strategies for the numerical solution of large dense linear least-squares problems. The new preconditioning strategies are embedded into simple and well-known iterative schemes that avoid the use of the, usually ill-conditioned, normal equations. We analyze a scheme to approximate the pseudoinverse, based on Schulz iterative method, and also different iterative schemes, based on extensions of Richardson's method, and the conjugate gradient method, that are suitable for preconditioning strategies. We present preliminary numerical results to illustrate the advantages of the proposed schemes.

This report describes the development and evaluation of mathematical models for predicting human performance from discrete wavelet transforms (DWT) of event-related potentials (ERP) elicited by task-relevant stimuli. The DWT was compared to principal components analysis ( PCA) for representation of ERPs in linear regression and neural network models developed to predict a composite measure of human signal detection performance. Linear regression models based on coefficients of the decimated DWT predicted signal detection performance with half as many free parameters as comparable models based on PCA scores.

This report describes the development and evaluation of mathematical models for predicting human performance from discrete wavelet transforms (DWT) of event-related potentials (ERP) elicited by task-relevant stimuli. The DWT was compared to principal components analysis (PCA) for representation of ERPs in linear regression and neural network models developed to predict a composite measure of human signal detection performance. Linear regression models based on coefficients of the decimated DWT predicted signal detection performance with half as many free parameters as comparable models based on PCA scores. In addition, the DWT-based models were more resistant to model degradation due to over-fitting than PCA-based models. Feed-forward neural networks were trained using the backpropagation algorithm to predict signal detection performance based on raw ERPs, PCA scores, or high-power coefficients of the DWT. Neural networks based on high-power DWT coefficients trained with fewer iterations, generalized to new data better, and were more resistant to overfitting than networks based on raw ERPs. Networks based on PCA scores did not generalize to new data as well as either the DWT network or the raw ERP network. The results show that wavelet expansions represent the ERP efficiently and extract behaviorally important features for use in linear regression or neural network models of human performance. The efficiency of the DWT is discussed in terms of its decorrelation and energy compaction properties. In addition, the DWT models provided evidence that a pattern of low-frequency