English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $J_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\;x_{n-m+1}\cdots x_n,\; x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$be the $m$-path ideal of the cycle graph of length $n$, in the ring$S=K[x_1,\ldots,x_n]$.  Let $d=\gcd(n,m)$. We prove that $\operatorname{depth}(S/J_{n,m}^t)=d-1$ forall $t\geq n-1$. We show that$\operatorname{sdepth}(S/J_{n,n-1}^t)=\operatorname{depth}(S/J_{n,n-1}^t)=\max\{n-t-1,0\}$for all $t\geq 1$. Also, we give some bounds for$\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$,where $t\geq 1$.

Depth and Stanley depth of powers of the path ideal of a cycle graph