Mixed models will be considered using the Commutative Jordan Algebra of Symmetric matrices approach. Prime basis factorial models will now be considered in the framework provided by Commutative Jordan Algebra of Symmetric matrices. This will enable to obtain fractional replicates when the number of levels is neither a prime or a power of a prime. We present an application to the effect of lidocaine, at an enzymatic level, on the heart muscle of beagle dogs

This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $𝔽S_n$ of the symmetric group $S_n$ over a field $𝔽$ of characteristic 0 (or $p>n$). The goal is to obtain a constructive version of the isomorphism $\psi :{⨁}_{\lambda }{M}_{d_{\lambda }}\left(𝔽\right)⟶𝔽S_n$ where $\lambda$ is a partition of $n$ and $d_{\lambda }$ counts the standard tableaux of shape $\lambda$....

We show that any sequence $\left({\phi }_n\right)$ of mutually orthogonal pure states on a JB algebra A such that $\left({\phi }_n\right)$ forms an almost discrete sequence in the relative topology induced by the primitive ideal space of A admits a sequence $\left(a_n\right)$ consisting of positive, norm one, elements of A with pairwise orthogonal supports which is supporting for $\left({\phi }_n\right)$ in the sense of ${\phi }_n\left(a_n\right)=1$ for all n. Moreover, if A is separable then $\left(a_n\right)$ can be taken such that $\left({\phi }_n\right)$ is uniquely determined by the biorthogonality condition ${\phi }_n\left(a_n\right)=1$. Consequences of this result improving...

We obtain conditions on a JB*-algebra X so that the canonical embedding of X into its associated quasi-invertible manifold has dense range. We prove that if a JB* has this density property then the quasi-invertible manifold is homogeneous for biholomorphic mappings. Explicit formulae for the biholomorphic mappings are also given.

We characterize an important class of generalized projective geometries $(X,X^{\text{'}})$ by the following essentially equivalent properties: (1) $(X,X^{\text{'}})$ admits a central null-system; (2) $(X,X^{\text{'}})$ admits inner polarities: (3) $(X,X^{\text{'}})$ is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s...

Let $𝒩=N_n\left(R\right)$ be the algebra of all $n\times n$ strictly upper triangular matrices over a unital commutative ring $R$. A map $\varphi$ on $𝒩$ is called preserving commutativity in both directions if $xy=yx\iff \varphi \left(x\right)\varphi \left(y\right)=\varphi \left(y\right)\varphi \left(x\right)$. In this paper, we prove that each invertible linear map on $𝒩$ preserving commutativity in both directions is exactly a quasi-automorphism of $𝒩$, and a quasi-automorphism of $𝒩$ can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.

Any bounded sequence in an L¹-space admits a subsequence which can be written as the sum of a sequence of pairwise disjoint elements and a sequence which forms a uniformly integrable or equiintegrable (equivalently, a relatively weakly compact) set. This is known as the Kadec-Pełczyński-Rosenthal subsequence splitting lemma and has been generalized to preduals of von Neuman algebras and of JBW*-algebras. In this note we generalize it to JBW*-triple preduals.