The higher integrability for very weak solutions of $A$-harmonic form equations $d^{*}A(x,u,du)=B(x,u,du)$ has been proved.

Animating expressive facial animation is a very challenging topic within the graphics community. In this paper, we introduce a novel ERI (expression ratio image) driving framework based on SVR and MPEG-4 for automatic 3D facial expression animation. Through using the method of support vector regression (SVR), the framework can learn and forecast the regression relationship between the facial animation parameters (FAPs) and the parameters of expression ratio image. Firstly, we build a 3D face animation system driven by FAP. Secondly, through using the method of principle component analysis (PCA), we generate the parameter sets of eigen-ERI space, which will rebuild reasonable expression ratio image. Then we learn a model with the support vector regression mapping, and facial animation parameters can be synthesized quickly with the parameters of eigen-ERI. Finally, we implement our 3D face animation system driving by the result of FAP and it works effectively.

The Caccioppoli inequality of weakly A-harmonic tensors has been proved, which can be used to consider the weak reverse Hölder inequality, regularity property, and zeros of weakly A-harmonic tensors.

We introduce a hybrid iterative scheme for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the varitional inequality problem and the equilibrium problem in Hilbert space. Under suitable conditions, some strong convergence theorems are obtained. Our results improve and extend the corresponding results in (Chang et al. (2009), Min and Chang (2012), Plubtieng and Punpaeng (2007), S. Takahashi and W. Takahashi (2007), Tada and Takahashi (2007), Gang and Changsong (2009), Ying (2013), Y. Yao and J. C. Yao (2007), and Yong-Cho and Kang (2012)).

The purpose of this paper is to introduce a new relaxed extragradient algorithms for the split feasibility problem. Our relaxed extragradient algorithm is new and it generalized some results for solving the split feasibility problem.

The purpose of this paper is to investigate the strong convergence problem of a modified mixed Ishikawa iterative sequence with errors for approximating the fixed points of an asymptotically nonexpansive mapping in the intermediate sense and an asymptotically quasi-pseudo-contractive-type mapping in an arbitrary real Banach space. The results here improve and extend the corresponding results reported by some other authors recently.

We investigate the inverse problem in the nonhomogeneous heat equation involving the recovery of the initial temperature from measurements of the final temperature. This problem is known as the backward heat problem and is severely ill-posed. We show that this problem can be converted into the first Fredholm integral equation, and an algorithm of inversion is given using Tikhonov's regularization method. The genetic algorithm for obtaining the regularization parameter is presented. We also present numerical computations that verify the accuracy of our approximation.

We first introduce a new mixed equilibrium problem with a relaxed monotone mapping in Banach spaces and prove the existence of solutions of the equilibrium problem. Then we introduce a new iterative algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a quasi-$\varphi$-nonexpansive mapping and prove some strong convergence theorems of the iteration. Our results extend and improve the corresponding ones given by Wang et al., Takahashi and Zembayashi, and some others.