We give in this paper some closed-form formulas for the maximal and minimal values of the rank and inertia of the Hermitian matrix expression A − BX ± (BX)∗ with respect to a variable matrix X. As applications, we derive the extremal values of the ranks/inertias of the matrices X and X ± X∗, where X is a (Hermitian) solution to the matrix equation AXB = C, respectively, and give necessary and s...

We explore a new general-purpose heuristic for finding high-quality solutions to hard discrete optimization problems. The method, called extremal optimization, is inspired by self-organized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively updates extremely undesirable variables of a single suboptimal solution, assigning th...

We explore a new general-purpose heuristic for finding high-quality solutions to hard discrete optimization problems. The method, called extremal optimization, is inspired by self-organized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively updates extremely undesirable variables of a single suboptimal solution, assigning th...

We prove that a non-degenerate extremal 3 point Pick problem on the bidisk always has a unique solution.

In the present paper we prove a multiplicity theorem for a quasilinear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach relies on extremal solutions for an auxiliary pa...

We first obtain Liouville type results for stable entire solutions of the biharmonic equation −∆2u = u−p in R for p > 1 and 3 ≤ N ≤ 12. Then we consider the Navier boundary value problem for the corresponding equation and improve the known results on the regularity of the extremal solution for 3 ≤ N ≤ 12. As a consequence, in the case of p = 2, we show that the extremal solution u∗ is regular w...

the present note is devoted to establish some extremal results for the zeroth-order general randi'{c} index of cacti, characterize the extremal polyomino chains with respect to the aforementioned index, and hence to generalize two already reported results.

The present note is devoted to establish some extremal results for the zeroth-order general Randi'{c} index of cacti, characterize the extremal polyomino chains with respect to the aforementioned index, and hence to generalize two already reported results.

<p style='text-indent:20px;'>We study the quasilinear Dirichlet boundary problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $\end{document} </tex-mat...