We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotically almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ). While it is known that the almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2 [4, 9], the almost sure mixing time drops from N only to N (up to poly-logarithmic fact...

Numerous types of hydrophilic polymer nanoparticles (NPs) have recently become research hotspots because their ability to dissolve in water and can be adapted with respect physical, chemical, biological properties meet the requirements different applications. Synthetic polymeric NPs had successfully gained much attention unique physicochemical properties, such as low toxicity, biodegradability,...

hydrogels, due to their unique potentials such as high-water content and hydrophilicity are interest for the controlled release of drug molecules. the present study aims to create a controlled-release system through the preparation and characterization of hydrogels based on ph-sensitive polymers such as poly (acrylic acid). poly (acrylic acid), p(aa), hydrogel has been synthesized by radical po...