We construct high accuracy trigonometric interpolants from equidistant evaluations of the Bessel functions \({{J}_{n}}(x)\) of the first kind and integer order. The trigonometric models are cosine or sine based depending on whether the Bessel function is even or odd. The main novelty lies in the fact that the frequencies in the trigonometric terms modelling \({{J}_{n}}(x)\) are also computed from the data in a Prony-type approach. Hence the interpolation problem is a nonlinear problem. Some existing compact trigonometric models for the Bessel functions \({{J}_{n}}(x)\) are hereby rediscovered and generalized.

Recently, various deep learning-based models have been applied in the study of recommender systems. Some researches have combined the classic collaborative filtering method with deep learning frameworks in order to obtain more accurate recommendations. However, these models either add additional features, but still recommend in the original linear manner, or only extract the global latent factors of the rating matrices in a non-linear way without considering some local special relationships. In this paper, we propose a deep learning framework for explicit recommender systems, named Attention Collaborative Autoencoder (ACAE). Based on the denoising autoencoder, our model can extract the global latent factors in a non-linear fashion from the sparse rating matrices. In ACAE, attention units are introduced during back propagation, enabling discovering potential relationships between users and items in the neighborhood, which makes the model obtain better results in the rating prediction tasks. In addition, we propose how to optimize the training process of the model by proposing a new loss function. Experiments on two public datasets demonstrate the effectiveness of ACAE and its outperformance of competitive baselines.

We present a method for determining the one-dimensional submodules of a Laurent-Ore module. The method is based on a correspondence between hyperexponential solutions of associated systems and one-dimensional submodules. The hyperexponential solutions are computed recursively by solving a sequence of first-order ordinary matrix equations. As the recursion proceeds, the matrix equations will have constant coefficients with respect to the operators that have been considered.