In this note, we construct families of functionals of the type of F-functional and W-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman's no… (More)
In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem 2.1.
Environmental factors play an important role in the development of rheumatoid arthritis (RA). Among these factors, smoking is generally considered to be an established risk factor for RA. Data regarding the impact of diet on risk of RA development is limited. This study assessed the impact of dietary patterns on RA susceptibility in Chinese populations.… (More)