### A Generalized Hecke Identity.

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The smoothing-type algorithm is a powerful tool for solving the second-order cone programming (SOCP), which is in general designed based on a monotone line search. In this paper, we propose a smoothing-type algorithm for solving the SOCP with a non-monotone line search. By using the theory of Euclidean Jordan algebras, we prove that the proposed algorithm is globally and locally quadratically convergent under suitable assumptions. The preliminary numerical results are also reported which indicate...

Binary operations on commutative Jordan algebras are used to carry out the ANOVA of a two layer model. The treatments in the first layer nests those in the second layer, that being a sub-model for each treatment in the first layer. We present an application with data retried from agricultural experiments.

First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.

Let $\U0001d510$ be a Jordan-Banach algebra with identity 1, whose norm satisfies:(i) $\parallel ab\parallel \le \parallel a\parallel \phantom{\rule{4pt}{0ex}}\parallel b\parallel $, $a,b\in \U0001d510$(ii) $\parallel {a}^{2}\parallel =\parallel a{\parallel}^{2}$(iii) $\parallel {a}^{2}\parallel \le \parallel {a}^{2}+{b}^{2}\parallel $.$\U0001d510$ is called a JB algebra (E.M. Alfsen, F.W. Shultz and E. Stormer, Oslo preprint (1976)). The set ${\U0001d510}^{+}$ of squares in $\U0001d510$ is a closed convex cone. $(\U0001d510,{\U0001d510}^{+},\mathbf{1})$ is a complete ordered vector space with $\mathbf{1}$ as a order unit. In addition, we assume $\U0001d510$ to be monotone complete (i.e. $\U0001d510$ coincides with the bidual ${\U0001d510}^{**}$), and that there exists a finite normal faithful trace $\phi $ on $\U0001d510$.Then the completion $\{{\U0001d510}^{+}{\}}_{\phi}$ of ${\U0001d510}^{+}$ with respect to the Hilbert structure...

Commutative Jordan algebras are used to drive an highly tractable framework for balanced factorial designs with a prime number p of levels for their factors. Both fixed effects and random effects models are treated. Sufficient complete statistics are obtained and used to derive UMVUE for the relevant parameters. Confidence regions are obtained and it is shown how to use duality for hypothesis testing.

A full Nesterov-Todd step infeasible interior-point algorithm is proposed for solving linear programming problems over symmetric cones by using the Euclidean Jordan algebra. Using a new approach, we also provide a search direction and show that the iteration bound coincides with the best known bound for infeasible interior-point methods.