Understanding Heisenberg's 'Magical' Paper of July 1925: a New Look at the Calculational Details

In July 1925 Heisenberg published a paper [Z. Phys. 33, 879-893 (1925)] which ended the period of `the Old Quantum Theory' and ushered in the new era of Quantum Mechanics. This epoch-making paper is generally regarded as being difficult to follow, perhaps partly because Heisenberg provided few clues as to how he arrived at the results which he reported. Here we give details of calculations of the type which, we suggest, Heisenberg may have performed. We take as a specific example one of the anharmonic oscillator problems considered by Heisenberg, and use our reconstruction of his approach to solve it up to second order in perturbation theory. We emphasize that the results are precisely those obtained in standard quantum mechanics, and suggest that some discussion of the approach - based on the direct computation of transition amplitudes - could usefully be included in undergraduate courses in quantum mechanics.Comment: 24 pages, no figures, Latex, submitted to Am. J. Phy

Timbral attributes for objective quality assessment of the Irish Tin Whistle

In this paper we extract various timbral attributes for a variety of Irish tin whistles, and use these attributes to form an objective quality assessment of the instruments. This assessment is compared with the subjective experiences of a number of professional musicians. The timbral attributes are drawn from those developed in the Timbre Mode

Dynamic Time Warping for Tin Whistle Partial Envelope Analysis

This paper examines techniques for the analysis of the amplitude envelope of harmonic partials in a tin whistle sound. Due to the nature of the envelope, a robust technique is required to estimate envelope parameters. This is achieved using a dynamic time warping based method. Furthermore, a heuristic scheme for addressing difficulties in envelope modelling is also introduced. The results presented show performance improvements over more established methods

Comparing synthetic and real templates for dynamic time warping to locate partial envelope features

In this paper we compare the performance of a number of different templates for the purposes of split point identification of various clarinet envelopes. These templates were generated with Attack-Decay-Sustain-Release (ADSR) descriptions commonly used in musical synthesis, along with a set of real templates obtained using k-means clustering of manually prepared test data. The goodness of fit of the templates to the data was evaluated using the Dynamic Time Warping (DTW) cost function, and by evaluating the square of the distance of the identified split points to the manually identified split points in the test data. It was found that the best templates for split point identification were the synthetic templates followed by the real templates having a sharp attack and release characteristic, as is characteristic of the clarinet envelope

Partial Envelope Analysis with Nonnegative Matrix Factorization

This paper investigates the application of the technique of Nonnegative Matrix Factorisation (NMF) to the analysis of partial envelopes extracted from a variety of instrumental sounds. Firstly, the initialisation of the NMF method is shown to be important for obtaining useful results. This is followed by an experimental analysis of the bases produced by NMF that demonstrates how they represent prototypical signal features

An evaluation of warping techniques applied to partial envelope analysis

Traditional methods for partial envelope analysis run into difficulties in the presence of tremolo and noise. This paper investigates an alternative method of analysis that uses a variety of non-linear warping techniques to match a synthetic envelope template to the partial envelope. Generation of suitable templates is discussed and the variety of implementations for the warping algorithm is examined. A complete procedure for matching the template is then explained and justified. Application of this procedure to partial envelope analysis of flute sounds is described and the results presented demonstrate the best choice of warping implementation. This technique allows for better envelope segmentation thus enabling improved modelling in representations such as the Timbre Mode

Considerations for a Computational Estimation of the Complex FM Spectrum

Complex frequency modulation (FM) is a useful method for sound synthesis as it allows the creation of rich timbres using only a small number of waveforms. However, the mathematical expression for the frequency domain representation of a frequency modulated (FM) wave with a non-sinusoidal modulator is a complex equation with multiple summation terms. Its implementation in software requires nested loops, with the number of loops depending on the number of sinusoidal components required to describe the complex modulator wave. Although the spectral representation can be easily computed from the FFT of the time domain waveform, this is at the expense of the insight that can be gained by direct evaluation of the analytic expression for the frequency domain representation. This paper looks at creating a software implementation of the analytic expression that overcomes the need for many nested loops and can be generalized. It will also suggest how it could be applied in a distributed processing environment

Considerations for a Computational Estimation of the Complex FM Spectrum

Complex frequency modulation (FM) is a useful method for sound synthesis as it allows the creation of rich timbres using only a small number of waveforms. However, the mathematical expression for the frequency domain representation of a frequency modulated (FM) wave with a non-sinusoidal modulator is a complex equation with multiple summation terms. Its implementation in software requires nested loops, with the number of loops depending on the number of sinusoidal components required to describe the complex modulator wave. Although the spectral representation can be easily computed from the FFT of the time domain waveform, this is at the expense of the insight that can be gained by direct evaluation of the analytic expression for the frequency domain representation. This paper looks at creating a software implementation of the analytic expression that overcomes the need for many nested loops and can be generalized. It will also suggest how it could be applied in a distributed processing environment

Los rohingya del estado de Rakhine (Myanmar): la evolución de un proceso genocida

Tomando como base los aportes teóricos de Feierstein sobre el genocidio y el paradigma de crimen de Estado, el presente artículo analiza información empírica obtenida en el estado de Rakhine (Myanmar), con el fin de brindar una mayor comprensión sobre los riesgos que enfrenta la población rohingya en la actualidad. La investigación revela que allí el genocidio –entendido como un proceso continuo– se encuentra en plena ejecución y predice que el paso hacia una próxima etapa –el aniquilamiento– es una posibilidad cierta si el gobierno de Myanmar no detiene los procesos de deshumanización, hostigamiento, aislamiento y debilitamiento sistemático que viene llevando a cabo. El presente trabajo sostiene que, ante la ausencia de mecanismos formales y legales de prevención, es responsabilidad de la sociedad civil exponer el proceso genocida para evitar que se produzca una matanza a gran escala (Green y Ward, 2004)

Liver regeneration in dogs: Morphologic and chemical changes

Forty-four percent and 72% hepatectomy were carried out in dogs and the animals were sacrificed for biochemical and pathologic studies from 0.5 to 6 days later. Compensatory hypertrophy and hyperplasia ("regeneration") were evident within 1 day, reached a maximum in 3 days, and were almost complete by 6 days. Coincident with the histologic events of regeneration were decreases in responsiveness of receptor adenyl cyclase to glucagon stimulation, increases of cyclic AMP, inconsistent changes in plasma insulin, and increases in plasma glucagon. These studies have standardized hepatic resection in dogs and they have focused attention upon some possible mechanisms that will require further study. © 1978 Academic Press, Inc. All rights of reserved