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Register a new userProper Jordan schemes exist. First examples, computer search, patterns of reasoning. An essay
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A special class of Jordan algebras over a field $F$ of characteristic zero is considered. Such an algebra consists of an $r$-dimensional subspace of the vector space of all square matrices of a fixed order $n$ over $F$. It contains the identity matrix, the all-one matrix; it is closed with respect to correction{matrix transposition}, Schur-Hadamard (entrywise) multiplication and the Jordan product $A*B=frac 12 (AB+BA)$, where $AB$ is the usual matrix product. The suggested axiomatics (with some natural additional requirements) implies an equivalent reformulation in terms of symmetric binary relations on a vertex set of cardinality $n$. The appearing graph-theoretical structure is called a Jordan scheme of order $n$ and rank $r$. A significant source of Jordan schemes stems from the symmetrization of association schemes. Each such structure is called a non-proper Jordan scheme. The question about the existence of proper Jordan schemes was posed a few times by Peter J. Cameron. In the current text an affirmative answer to this question is given. The first small examples presented here have orders $n=15,24,40$. Infinite classes of proper Jordan schemes of rank 5 and larger are introduced. A prolific construction for schemes of rank 5 and order $n=binom{3^d+1}{2}$, $din {mathbb N}$, is outlined. The text is written in the style of an essay. The long exposition relies on initial computer experiments, a large amount of diagrams, and finally is supported by a number of patterns of general theoretical reasonings. The essay contains also a historical survey and an extensive bibliography.
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In 2003 Peter Cameron introduced the concept of a Jordan scheme and asked whether there exist Jordan schemes which are not symmetrisations of coherent configurations (proper Jordan schemes). The question was answered affirmatively by the authors last year and some of the examples were presented in an essay uploaded to the arXiv. In this paper we describe several infinite series of proper Jordan schemes and present first developments in the theory of Jordan schemes - a new class of algebraic-combinatorial objects.
The refined inertia of a square real matrix $A$ is the ordered $4$-tuple $(n_+, n_-, n_z, 2n_p)$, where $n_+$ (resp., $n_-$) is the number of eigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is the number of zero eigenvalues of $A$, and $2n_p$ is the number of nonzero pure imaginary eigenvalues of $A$. For $n geq 3$, the set of refined inertias $mathbb{H}_n={(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)}$ is important for the onset of Hopf bifurcation in dynamical systems. We say that an $ntimes n$ sign pattern ${cal A}$ requires $mathbb{H}_n$ if $mathbb{H}_n={text{ri}(B) | B in Q({cal A})}$. Bodine et al. conjectured that no $ntimes n$ irreducible sign pattern that requires $mathbb{H}_n$ exists for $n$ sufficiently large, possibly $nge 8$. However, for each $n geq 4$, we identify three $ntimes n$ irreducible sign patterns that require $mathbb{H}_n$, which resolves this conjecture.
We construct new symmetric Hadamard matrices of orders $92,116$, and $172$. While the existence of those of order $92$ was known since 1978, the orders $116$ and $172$ are new. Our construction is based on a recent new combinatorial array discovered by N. A. Balonin and J. Seberry. For order $116$ we used an adaptation of an algorithm for parallel collision search. The adaptation pertains to the modification of some aspects of the algorithm to make it suitable to solve a 3-way matching problem. We also point out that a new infinite series of symmetric Hadamard matrices arises by plugging into the GP array the matrices constructed by Xia, Xia, Seberry, and Wu in 2005.
In this paper we aim to characterize association schemes all of whose symmetric fusion schemes have only integral eigenvalues, and classify those obtained from a regular action of a finite group by taking its orbitals.
We construct twelve infinite families of pseudocyclic and non-amorphic association schemes, in which each nontrivial relation is a strongly regular graph. Three of the twelve families generalize the counterexamples to A. V. Ivanovs conjecture by Ikuta and Munemasa [15].
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