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

Two sesquilinear forms $\Phi:\mathbb C^m\times\mathbb C^m\to \mathbb C$ and$\Psi:\mathbb C^n\times\mathbb C^n\to \mathbb C$ are called topologicallyequivalent if there exists a homeomorphism $\varphi :\mathbb C^m\to \mathbbC^n$ (i.e., a continuous bijection whose inverse is also a continuousbijection) such that $\Phi(x,y)=\Psi(\varphi (x),\varphi (y))$ for all $x,y\in\mathbb C^m$. R.A.Horn and V.V.Sergeichuk in 2006 constructed a regularizingdecomposition of a square complex matrix $A$; that is, a direct sum$SAS^*=R\oplus J_{n_1}\oplus\dots\oplus J_{n_p}$, in which $S$ and $R$ arenonsingular and each $J_{n_i}$ is the $n_i$-by-$n_i$ singular Jordan block. Inthis paper, we prove that $\Phi$ and $\Psi$ are topologically equivalent if andonly if the regularizing decompositions of their matrices coincide up topermutation of the singular summands $J_{n_i}$ and replacement of $R\in\mathbbC^{r\times r}$ by a nonsingular matrix $R'\in\mathbb C^{r\times r}$ such that$R$ and $R'$ are the matrices of topologically equivalent forms. Analogousresults for real and complex bilinear forms are also obtained.

Topological classification of sesquilinear forms: reduction to the nonsingular case