In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufficient condition for the positive definiteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efficiency of our results.

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Swarm intelligence (SI) is a new evolutionary computation technology, and its performance efficacy is usually affected by each individual behavior in the swarm. According to the genetic and sociological theory, the life evolution behavior process is influenced by the external and internal factors, so the mechanisms of external and internal environment change must be analyzed and explored. Therefore, in this paper, we used the thought of the famous American genetic biologist Morgan, “life = DNA + environment + interaction of environment + gene,” to propose the mutation and crossover operation of DNA fragments by the environmental change to improve the performance efficiency of intelligence algorithms. Additionally, PSO is a random swarm intelligence algorithm with the genetic and sociological property, so we embed the improved mutation and crossover operation to particle swarm optimization (PSO) and designed DNA-PSO algorithm to optimize single and multiobjective optimization problems. Simulation experiments in single and multiobjective optimization problems show that the proposed strategies can effectively improve the performance of swarm intelligence.

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.