The Sch\"{o}nberg-Chandrasekhar limit in post main sequence evolution for stars of masses the range $1.4\lesssim M/M_{\odot}\lesssim 6$ gives maximum pressure that stellar core can withstand, once central hydrogen is exhausted. It usually expressed as a quadratic function $1/\alpha$, with $\alpha$ being ratio mean molecular weight to envelope. Here, we revisit this scenarios where balance equat...

Abstract In this article, we introduce some special several variables mappings which are quadratic in each variable and show that such can be defined as a single equation is the generalized multi-quadratic functional equation. We also apply fixed point theorem to establish Hyers–Ulam stability for equations. Furthermore, present an example few corollaries corresponding known results.

This paper constructs the penalty cost functional for optimizing the two-dimensional control operator of the energized wave equation. In some multiplier methods such as the Lagrange multipliers and Pontrygean maximum principle, the cost of merging the constraint equation to the integral quadratic objective functional to obtain an unconstraint equation is normally guessed or obtained from the fi...