We generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by McKane and Tarlie to differential operators with multiple zero eigenvalues. We derive general formulas for such functional determinants of $r\times r$ matrix second order differential operators O with $0 < n \leqslant 2r$ linearly independent zero modes. We separately discuss the cases of the homogeneous Dirichlet boundary conditions, when the number of zero modes cannot exceed r, and the case of twisted boundary conditions, including the periodic and anti-periodic ones, when the number of zero modes is bounded above by 2r. In all cases the determinants with excluded zero eigenvalues can be expressed only in terms of the n zero modes and other $r-n$ or $2r-n$ (depending on the boundary conditions) solutions of the homogeneous equation $O h=0$ , in the spirit of Gel'fand–Yaglom approach. In instanton calculations, the contribution of the zero modes is taken into account by introducing the so-called collective coordinates. We show that there is a remarkable cancellation of a factor (involving scalar products of zero modes) between the Jacobian of the transformation to the collective coordinates and the functional fluctuation determinant with excluded zero eigenvalues. This cancellation drastically simplifies instanton calculations when one uses our formulas.
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Machine learning models have proven to be reliable methods in classification tasks. However, little research has been conducted on the classification of dwelling characteristics based on smart meter and weather data before. Gaining insights into dwelling characteristics, which comprise of the type of heating system used, the number of inhabitants, and the number of solar panels installed, can be helpful in creating or improving the policies to create new dwellings at nearly zero-energy standard. This paper compares different supervised machine learning algorithms, namely Logistic Regression, Support Vector Machine, K-Nearest Neighbor, and Long-short term memory, and methods used to correctly implement these algorithms. These methods include data pre-processing, model validation, and evaluation. Smart meter data, which was used to train several machine learning algorithms, was provided by Groene Mient. The models that were generated by the algorithms were compared on their performance. The results showed that the Long-short term memory performed the best with 96% accuracy. Cross Validation was used to validate the models, where 80% of the data was used for training purposes and 20% was used for testing purposes. Evaluation metrics were used to produce classification reports, which indicates that the Long-short term memory outperforms the compared models on the evaluation metrics for this specific problem.
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