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### Open Access Journals

In this paper, we analyze the positive definiteness of DKI. We first characterize the positive definiteness of DKI through the positive definiteness of a tensor constructed by diffusion tensor and diffusion kurtosis tensor. Then, a conic linear optimization method and its simplified version are proposed to handle the positive definiteness of DKI from the perspective of numerical computation. Some preliminary numerical tests on both synthetical and real data show that the method discussed in this paper is promising.

In this paper, an SDP relaxation algorithm is proposed to test the copositivity of higher order tensors. By solving finitely many SDP relaxations, the proposed algorithm can determine the copositivity of higher order tensors. Furthermore, for any copositive but not strictly copositive tensor, the algorithm can also check it exactly. Some numerical results are reported to show the efficiency of the proposed algorithm.

As a natural extension of the linear complementarity problem, the tensor complementarity problem has been studied recently; and many theoretical results have been obtained. In this paper, we investigate the global error bound for the tensor complementarity problem with a *P*-tensor. We give two global error bounds for this class of complementarity problems with the help of two positively homogeneous operators defined by a *P*-tensor. When the order of the involved tensor reduces to 2, the results obtained in this paper coincide exactly with the one for the linear complementarity problem.

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