On the cone eigenvalue complementarity problem for higher-order tensors

In this paper, we consider the tensor generalized eigenvalue complementarity problem (TGEiCP), which is an interesting generalization of matrix eigenvalue complementarity problem (EiCP). First, we given an affirmative result showing that TGEiCP is solvable and has at least one solution under some reasonable assumptions. Then, we introduce two optimization reformulations of TGEiCP, thereby beneficially establishing an upper bound of cone eigenvalues of tensors. Moreover, some new results concerning the bounds of number of eigenvalues of TGEiCP further enrich the theory of TGEiCP. Last but not least, an implementable projection algorithm for solving TGEiCP is also developed for the problem under consideration. As an illustration of our theoretical results, preliminary computational results are reported.Comment: 26 pages, 2 figures, 3 table

von Neumann type trace inequality for dual quaternion matrices

As a powerful tool to represent rigid body motion in 3D spaces, dual quaternions have been successfully applied to robotics, 3D motion modelling and control, and computer graphics. Due to the important applications in multi-agent formation control, this paper addresses the concept of spectral norm of dual quaternion matrices. We introduce a von Neumann type trace inequality and a Hoffman-Wielandt type inequality for general dual quaternion matrices, where the latter characterizes a simultaneous perturbation bound on all singular values of a dual quaternion matrix. In particular, we also present two variants of the above two inequalities expressed by eigenvalues of dual quaternion Hermitian matrices. Our results are helpful for the further study of dual quaternion matrix theory, algorithmic design, and applications

A regularization-patching dual quaternion optimization method for solving the hand-eye calibration problem

The hand-eye calibration problem is an important application problem in robot research. Based on the 2-norm of dual quaternion vectors, we propose a new dual quaternion optimization method for the hand-eye calibration problem. The dual quaternion optimization problem is decomposed to two quaternion optimization subproblems. The first quaternion optimization subproblem governs the rotation of the robot hand. It can be solved efficiently by the eigenvalue decomposition or singular value decomposition. If the optimal value of the first quaternion optimization subproblem is zero, then the system is rotationwise noiseless, i.e., there exists a ``perfect'' robot hand motion which meets all the testing poses rotationwise exactly. In this case, we apply the regularization technique for solving the second subproblem to minimize the distance of the translation. Otherwise we apply the patching technique to solve the second quaternion optimization subproblem. Then solving the second quaternion optimization subproblem turns out to be solving a quadratically constrained quadratic program. In this way, we give a complete description for the solution set of hand-eye calibration problems. This is new in the hand-eye calibration literature. The numerical results are also presented to show the efficiency of the proposed method

The Social Income Inequality, Social Integration and Health Status of Internal Migrants in China

Background: To examine the interaction between social income inequality, social integration, and health status among internal migrants (IMs) who migrate between regions in China. Methods: We used the data from the 2014 Internal Migrant Dynamic Monitoring Survey in China, which sampled 15,999 IMs in eight cities in China. The Gini coefficient at the city level was calculated to measure social income inequality and was categorized into low (0.2 \u3c Gini \u3c= 0.3), medium (0.3 \u3c Gini \u3c= 0.4), high (0.4 \u3c x \u3c = 0.5), and very high (Gini \u3e 0.5). Health status was measured based upon self-reported health, subjective well-being, and perceptions of stress and mental health. Social integration was measured from four perspectives (acculturation and integration willingness, social insurance, economy, social communication). Linear mixed models were used to examine the interaction effects between health statuses, social integration, and the Gini coefficient. Results: Factors of social integration, such as economic integration and acculturation and integration willingness, were significantly related to health. Social income inequality had a negative relationship with the health status of IMs. For example, IMs in one city, Qingdao, with a medium income inequality level (Gini = 0.329), had the best health statuses and better social integration. On the other hand, IMs in another city, Shenzhen, who had a large income inequality (Gini = 0.447) were worst in health statues and had worse social integration. Conclusion: Policies or programs targeting IMs should support integration willingness, promote a sense of belonging, and improve economic equality. In the meantime, social activities to facilitate employment and create social trust should also be promoted. At the societal level, structural and policy changes are necessary to promote income equity to promote IMs\u27 general health status