Necessary and sufficient conditions for the operator equation I AXAX n*, to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X andAare given.

The present paper investigates the role of fear and predator dependent refuge in the prey-predator system. The system describes the interaction between prey and a stage structure of predator that incorporates Holling II functional response. The predator splits into two compartments immature (juvenile) and mature (adult). The mature predators can hunt and reproduce but this capability is not found in the immature predators, the immature depend on their parents. The growth rate of prey decreases due to the existence of mature predators. The existence, uniqueness, and boundedness of the solution of the system are investigated. Three equilibrium points of the system are determined. The local stability of the system is studied. The global stabil

In this paper, we study the growth of solutions of the second order linear complex differential equations  insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

Hazardous materials, heavy metals, and organic toxins released into the environment have caused considerable harm to microbes, plants, animals, and humans. Wastewater is one of the most contaminated ecosystems due to heavy metals emitted mostly by human activity. Bioremediation of wastewater is an ecologically acceptable and cost-effective method of removing heavy metals from sewage; the general purpose of this study is to analyse the dependability of anaerobic sludge biomass in removing sulfur compounds and heavy metals from waste water. The anaerobic sludge biomass evaluated in this work was taken from a wastewater treatment plant (WWTP) in Al-Rustumiya, Baghdad, and grown in the mineral medium for anaerobic growth. In serum bottl