The Editors-in-Chief have retracted this article [1] because its results are invalid. It also shows considerable overlap with an article by Lu and Sun [2] that was simultaneously under consideration. Additionally, the article shows evidence of authorship manipulation. The authors have not responded to any correspondence regarding this retraction.

With the continuous development of graphic image processing technology, threedimensional data is being widely used in various fields. However, difficulty still exists in graphic and image processing in three-dimensional data field. This paper applied ray projection algorithm and interpolation algorithm for graphic processing and combined acceleration algorithm with texture mapping technology to draw real-time images. By drawing a semi-sphere with the computer, the effect of the algorithms was detected. Meanwhile, in order to reduce the influence of image noise in the three-dimensional data field, three-dimensional polishing algorithm was applied for optimization. It was found that the drawn images were clear and the drawing speed was fast, suggesting that the algorithms were helpful in drawing 3D graphics and worth being promoted for wide application.

With the development of the society, the increased amount of information has extensively appeared on the Internet. It includes almost all the content we need. But information overload makes people unable to correctly find the information they need. Collaborative filtering recommendation algorithm can recommend items for users according to their demands. But traditional recommendation algorithm which has defects such as data sparsity needs to be improved. In this study, the collaborative filtering recommendation algorithm was analyzed, an improved collaborative filtering recommendation algorithm based on the probability matrix decomposition was put forward, and the feasibility of the algorithm was verified. Moreover the traditional algorithms including user based collaborative filtering algorithm, item based collaborative filtering algorithm, singular value decomposition based collaborative filtering algorithm and basic matrix based collaborative filtering algorithm were tested. The test results demonstrated that the proposed algorithm had a higher accuracy compared to the traditional algorithms, and its mean absolute error and root-mean-square error were significantly smaller than those of the traditional algorithms. Therefore it can be applied in the daily life.

Using boundary behaviors of solutions for certain Laplace equation proved by Yan and Ychussie (Adv. Difference Equ. 2015:226, 2015) and applying a new method to dispose of the impulsive term with finite mass subject presented by Shi and Liao (J. Inequal. Appl. 2015:363, 2015) from another point of view, we prove that there exists a supra-open in \((X,\tau)\) for each \(V \in\sigma\) in which the modified equilibrium equation has normal families of solutions. Moreover, we establish a new expression of a harmonic multifunction for the above equation. As applications, we not only prove the existence of normal families of solutions for modified equilibrium equations but also obtain several characterizations and fundamental properties of these new classes of superharmonic multifunctions.

Nano-fluid is a stable and homogeneous novel heat transferring medium developed by mixing nano-metal or non-metal particles with traditional heat transferring media such as water, oil and alcohol in certain proportion. The heat transfer property of nano-fluid is more advantageous than that of traditional fluids. To investigate the natural convection heat transfer characteristics of nano-fluid and promote its application in fields of physics and chemistry, this study has established a relevant physical models for natural convection at different states using computer numerical simulation and verification. The experimental results demonstrated that the heat transfer of nano-fluid improved with the increase of volume share of nano-particles. The presented research has promising applications in the areas of engineering, energy and information technology.

The quintile regression is a new regression method. This paper describes the theory and the theoretical foundation of the quintile regression. Compared with the general least-squares regression, quintile regression is more robust. Moreover, the paper uses China's energy consumption as an example to explain the characteristics of the quintile regression. The analysis of the example shows that quintile regression results are significantly different from the ordinary least-squares regression results, in different quintiles, the effect of the per capita GDP and the level of urbanization on energy consumption has a significant difference. ? Springer-Verlag Berlin Heidelberg 2014.

The strong limit theorem is one of the central questions for studying in the international Probability theory. The purpose of this paper is to give a strong limit theorem for functions of two-ordered Markov chains indexed by a kind of nonhomogeneous tree.

Objective Voronoi diagrams are important in many fields in a series of sciences. Network Voronoi diagrams are useful to investigate dominance regions in a grid street system or a radial-circular street system. However, all generators may have different effect. To deal with a network Voronoi diagram with varied functions of generators, it must be worth formulating a power network Voronoi diagram. Method Adding weight value on generators, which is used to indicate factors related to are difficult to construct when the position relation of generators. Results A new concept of power network Voronoi diagram are proposed. In accordance with discrete construction method, achieved the construction of power network Voronoi diagram. Conclution The application example shows that the algorithm is both simple and useful, and it is of high potential value in practice. Power network Voronoi diagram both perfected the theory about Voronoi diagrams, and extended the range of applications of Voronoi diagrams.

This Paper gives an introduction of Random Forest. Random Forest is a new Machine Learning Algorithm and a new combination Algorithm. Random Forest is a combination of a series of tree structure classifiers. Random Forest has many good characters. Random Forest has been wildly used in classification and prediction, and used in regression too. Compared with the traditional algorithms Random Forest has many good virtues. Therefore the scope of application of Random Forest is very extensive.

The classical complete convergence theorem concerns the arithmetic means which is a regular method of summability. In this paper, to obtain the main results, a new large class of summability methods is introduced. The results for complete convergence for negatively associated random variable sequence are obtained. To investigate this results, by restricting the moment conditions and use a new method of summability. Then the result of the complete convergence for NA random variables sequences are obtained by applying the Kolmogorov-type inequality for NA random variable sequence.

The classical theorem concerns the arithmetic means which is a regular method of summability. In this paper, under a large class of summability methods, a result for strong law of large numbers for negatively associated random variables is obtained. To investigate this result, this paper establishes a moment inequality for negatively associated random variables. Then, by restricting the moment conditions and use the method of summability, the result is extended for negatively associated random variables, which is closely related to classical theorems. Namely, it links in some sense the strong law of large numbers of Kolmogorov and that of Marcinkiewicz.

Power Network Voronoi diagrams are difficult to construct when the position relation of road segments are complicated. In traditional algorithm, The distance between objects must be calculated by selecting the minimum distance to their shared borders and doubling this value. When road segments cross or coincide with each other, production process will be extremely complex because we have to consider separately these parts. In this paper, we use discrete construction of network Voronoi diagrams. The algorithm can get over all kinds of shortcomings that we have just mentioned. So it is more useful and effective than the traditional ones. We also construct model according the algorithm. And the application example shows that the algorithm is both simple and useful, and it is of high potential value in practice.

The power Voronoi diagrams are difficult to construct because of their complicated structures. In traditional algorithm, production process which is based on the Delaunay diagram was extremely complex. While dynamic algorithm is only concerned with positions of generators, so it is effective for constructing Voronoi diagrams with complicated shapes of Voronoi polygons. It can be applied to power Voronoi diagram with any generators, and can get over most shortcomings of traditional algorithm. So it is more useful and effective. Model is constructed with dynamic algorithm. And the application example shows that the algorithm is both simple and practicable and of high potential value in practice.