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/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

If $R$ is a commutative ring, $I$ an ideal of $R$ and $v, w \in Um_{2n}(R,I)$ then we show that $v, w$ are in the same orbit of elementary action if andonly if they are in the same orbit of elementary symplectic action. We alsoshow that if $A$ is a non-singular affine algebra of dimension $d$ over analgebraically closed field $k$ such that $d! A = A$, $d \equiv 2 \pmod 4$ and$I$ an ideal of $A$, then $Um_d(A, I) = e_1{Sp}_d(A, I)$. As a consequence itis proved that if $A$ is a non-singular affine algebra of dimension $d$ over analgebraically closed field $k$ such that $(d + 1)!A = A$, $d \equiv 1 \pmod 4$and $I$ a principal ideal then $Sp_{d-1}(A, I) \cap {ESp}_{d+1}(A, I) ={ESp}_{d -1}(A, I)$. We give an example to show that the above result does nothold true for an affine algebra over a $C_2$ field and also show by an examplethat the above stability estimate is optimal.

Optimal injective stability for the symplectic $K_1Sp$ group