For all positive integers n, we prove the following divisibility properties: (2n + 3) ( 2n n ) ∣∣∣∣3(6n 3n )( 3n n ) and (10n + 3) ( 3n n ) ∣∣∣∣21(15n 5n )( 5n n ) . This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers m and n, the product am ( am+bm−1 am )( an+bn an ) is divisible by m + n. In fact,...

? 10n is effected by crawling on the balls of the eet and on the hands. Lathyrism was first recognised in Europe over yo. hundred and fifty years ago, and it is pvious that even in those early and unscientific ^lrnes it was ascribed to the ingestion of the ^eeds of various species of the genus Lathyrus, Gcause as early as 1671 the Duke of Wurtum^GlS issued an edict forbidding the use of these ^...

Abstract In this paper, we study the Koszul property of homogeneous coordinate ring a generic collection lines in $\mathbb {P}^n$ and general linear position {P}^n.$ We show that if $\mathcal {M}$ is m with $2m \leq n+1$ R {M},$ then Koszul. Furthermore, even $m +1\leq n$ or odd +2\leq n,$ Lastly, such $$ \begin{align*} m> \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\end{ali...

The hand exoskeleton has been widely studied in the fields of rehabilitation and grasping assistance tasks. Current exoskeletons face challenges combining a user-friendly design with lightweight structure accurate modeling motion. In this study, we developed finger rolling contact involute joint. Specific implementation methods were investigated, including an analysis mechanical characteristics...

The Kukis practise the traditional institution called Chieftainship, which i~ stil.l in exis~ence today. Every village is governed under such an headed by a chief Haosa. along wtth hts counctl~en, ~ho are elected. on ter'ilire basis, works for welfare of villagers. In return villagers have obligation pay~ng tnbu~es to t~e ~ht~f, purely voluntary nature. However, there continuing debates among y...