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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

A special class of Jordan algebras over a field $F$ of characteristic zero isconsidered. Such an algebra consists of an $r$-dimensional subspace of thevector space of all square matrices of a fixed order $n$ over $F$. It containsthe identity matrix, the all-one matrix; it is closed with respect to\correction{matrix transposition}, Schur-Hadamard (entrywise) multiplicationand the Jordan product $A*B=\frac 12 (AB+BA)$, where $AB$ is the usual matrixproduct.  The suggested axiomatics (with some natural additional requirements) impliesan equivalent reformulation in terms of symmetric binary relations on a vertexset of cardinality $n$. The appearing graph-theoretical structure is called aJordan scheme of order $n$ and rank $r$. A significant source of Jordan schemesstems from the symmetrization of association schemes. Each such structure iscalled a non-proper Jordan scheme. The question about the existence of properJordan schemes was posed a few times by Peter J. Cameron.  In the current text an affirmative answer to this question is given. Thefirst small examples presented here have orders $n=15,24,40$. Infinite classesof proper Jordan schemes of rank 5 and larger are introduced. A prolificconstruction for schemes of rank 5 and order $n=\binom{3^d+1}{2}$, $d\in{\mathbb N}$, is outlined.  The text is written in the style of an essay. The long exposition relies oninitial computer experiments, a large amount of diagrams, and finally issupported by a number of patterns of general theoretical reasonings. The essaycontains also a historical survey and an extensive bibliography.

Proper Jordan schemes exist. First examples, computer search, patterns of reasoning. An essay