In this work, we try to build a theory for random double tensor integrals (DTI). We begin with the definition of DTI and discuss how randomness structure is built upon DTI. Then, the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means, e.g., arithmetic mean, geometric mean, harmonic mean, and general mean. By associating DTI with perturbation formula, i.e., a formula to relate the tensor-valued function difference with respect the difference of the function input tensors, the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case, respectively. We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean, and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.

In this study, we develop a set of ordinary differential equations that represents the dynamics of an ecosystem with two predators and one prey, but only the first predator population is affected by an infectious disease. The Lotka-Volterra predator-prey system’s model stability have been examined using the Takagi-Sugeno (T-S) impulsive control model and the Fuzzy impulsive control model. Following the formulation of the model, the global stabilities and the Fuzzy solution are carried out through numerical simulations and graphical representations with appropriate discussion for better understanding the dynamics of our proposed model.

In this paper, we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition. A three-level average technique is utilized, thereby leading to a linearized difference scheme. The main work lies in the pointwise error estimate in $H^2$-norm. The optimal fourth-order convergence order is proved in combination of induction, the energy method and the embedded inequality. Moreover, we establish the stability of the difference scheme with respect to the initial value under very mild condition, however, does not require any step ratio restriction. Extensive numerical examples with/without exact solutions under diverse cases are implemented to validate the theoretical results.

In this paper, we study some inclusion sets of $US$-eigenvalues and $U$-eigenvalues based on quantum information. We give three inclusion sets theorems of $US$-eigenvalues and two inclusion sets theorems of $U$-eigenvalues. And we obtain the relationships among these inclusion sets. Some numerical examples are shown to illustrate the conclusions.