The maximal angle between 5×5 positive semidefinite and 5×5 nonnegative matrices

The paper is devoted to the study of the maximal angle between the $5\times 5$ semidefinite matrix cone and $5\times 5$ nonnegative matrix cone. A signomial geometric programming problem is formulated in the process to find the maximal angle. Instead of using an optimization problem solver to solve the problem numerically, the method of Lagrange Multipliers is used to solve the signomial geometric program, and therefore, to find the maximal angle between these two cones.

Weakly Coupled B-Type Kadomtsev-Petviashvili Equation: Lump and Rational Solutions

Through the method of Z N -KP hierarchy, we propose a new ( 3 + 1 )-dimensional weakly coupled B-KP equation. Based on the bilinear form, we obtain the lump and rational solutions to the dimensionally reduced cases by constructing a symmetric positive semidefinite matrix. Then, we do numerical analysis on the rational solutions and fit the trajectory equation of the crest. Furthermore, we verify the accuracy of the trajectory equation by numerical analysis. This method of solving the lump and rational solutions can also be applied to other nonlinear evolution equations.

Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

Monotone transformations on the cone of all positive semidefinite real matrices

Let $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) be the cone of all positive semidefinite (symmetric) n × n real matrices. Matrices from $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) play an important role in many areas of engineering, applied mathematics, and statistics, e.g. every variance-covariance matrix is known to be positive semidefinite and every real positive semidefinite matrix is a variance-covariance matrix of some multivariate distribution. Three of the best known partial orders that were mostly studied on various sets of matrices are the Löwner, the minus, and the star partial orders. Motivated by applications in statistics authors have recently investigated the form of maps on $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) that preserve either the Löwner or the minus partial order in both directions. In this paper we continue with the study of preservers of partial orders on $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ). We characterize surjective, additive maps on $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ), n ≥ 3, that preserve the star partial order in both directions. We also investigate the form of surjective maps on the set of all symmetric real n × n matrices that preserve the Löwner partial order in both directions.