Continuity of truncated Hardy-Littlewood maximal function

This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function.We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous.Then by investigating the behavior of the truncated Hardy-Littlewood maximal function when the truncated parameterγchanges,we obtain an equivalent condition of the continuity of the truncated Hardy-Littlewood maximal function.

Ensemble Learning Based Collaborative Filtering with Instance Selectionand Enhanced Clustering

Recommender system is a tool to suggest items to the users from the extensive history of the user’s feedback.Though,it is an emerging research area concerning academics and industries,where it suffers from sparsity,scalability,and cold start problems.This paper addresses sparsity,and scalability problems of model-based collaborative recommender system based on ensemble learning approach and enhanced clustering algorithm for movie recommendations.In this paper,an effective movie recommendation system is proposed by Classification and Regression Tree(CART)algorithm,enhanced Balanced Iterative Reducing and Clustering using Hierarchies(BIRCH)algorithm and truncation method.In this research paper,a new hyper parameters tuning is added in BIRCH algorithm to enhance the cluster formation process,where the proposed algorithm is named as enhanced BIRCH.The proposed model yields quality movie recommendation to the new user using Gradient boost classification with broad coverage.In this paper,the proposed model is tested on Movielens dataset,and the performance is evaluated by means of Mean Absolute Error(MAE),precision,recall and f-measure.The experimental results showed the superiority of proposed model in movie recommendation compared to the existing models.The proposed model obtained 0.52 and 0.57 MAE value on Movielens 100k and 1M datasets.Further,the proposed model obtained 0.83 of precision,0.86 of recall and 0.86 of f-measure on Movielens 100k dataset,which are effective compared to the existing models in movie recommendation.

Spatial correlation properties and phase singularity annihilation of Gaussian Schell-model beams in the focal region

By using the generalized Debye diffraction integral, this paper studies the spatial correlation properties and phase singularity annihilation of apertured Gaussian Schell-model （GSM） beams in the focal region. It is shown that the width of the spectral degree of coherence can be larger, less than or equal to the corresponding width of spectral density, which depends not only on the scalar coherence length of the beams, but also on the truncation parameter. With a gradual increase of the truncation parameter, a pair of phase singularities of the spectral degree of coherence in the focal plane approaches each other, resulting in subwavelength structures. Finally, the annihilation of pairs of phase singularities takes place at a certain value of the truncation parameter. With increasing scalar coherence length, the annihilation occurs at the larger truncation parameter. However, the creation process of phase singularities outside the focal plane is not found for GSM beams.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.