On three quantization methods for particle on hyperboloid

We compare the respective efficiencies of three quantization methods (group theoretical, coherent state and geometric) by quantizing the dynamics of a free massive particle in two-dimensional de Sitter space. For each case we consider the realization of the principal series representation of $SO_0(1,2)$ group and its two-fold covering SU(1,1). We demonstrate that standard technique for finding an irreducible representation within the geometric quantization scheme fails. For consistency we recall our earlier results concerning the other two methods, make some improvements and generalizations.Comment: 22 pages, no figures, revte

Statistical Study of the Galaxy Distribution

Large-scale structures in the Universe provide crucial information about formation of structures and can be used to test cosmological models. The good agreement between large-scale observations and the now-standard Lambda-Cold Dark Matter (ΛCDM) model gives hope for this model to be a lasting foundation. Going a step further, these observation

Topological data analysis of human vowels: Persistent homologies across representation spaces

Topological Data Analysis (TDA) has been successfully used for various tasks in signal/image processing, from visualization to supervised/unsupervised classification. Often, topological characteristics are obtained from persistent homology theory. The standard TDA pipeline starts from the raw signal data or a representation of it. Then, it consists in building a multiscale topological structure on the top of the data using a pre-specified filtration, and finally to compute the topological signature to be further exploited. The commonly used topological signature is a persistent diagram (or transformations of it). Current research discusses the consequences of the many ways to exploit topological signatures, much less often the choice of the filtration, but to the best of our knowledge, the choice of the representation of a signal has not been the subject of any study yet. This paper attempts to provide some answers on the latter problem. To this end, we collected real audio data and built a comparative study to assess the quality of the discriminant information of the topological signatures extracted from three different representation spaces. Each audio signal is represented as i) an embedding of observed data in a higher dimensional space using Taken's representation, ii) a spectrogram viewed as a surface in a 3D ambient space, iii) the set of spectrogram's zeroes. From vowel audio recordings, we use topological signature for three prediction problems: speaker gender, vowel type, and individual. We show that topologically-augmented random forest improves the Out-of-Bag Error (OOB) over solely based Mel-Frequency Cepstral Coefficients (MFCC) for the last two problems. Our results also suggest that the topological information extracted from different signal representations is complementary, and that spectrogram's zeros offers the best improvement for gender prediction