Non-European roots of mathematics

We express invariants of Finsler manifolds in a geometrical way by means of using moving planes and their associated Jacobi curves, which are curves in a fixed homogeneous Grassmannian manifold. We also use this language to review some applications.

We establish asymptotic stability results and provide explicit decay rate for linear symmetric hyperbolic systems in terms of the dissipative structure. In particular, we are concerned with the model systems which have a non-symmetric relaxation term and therefore their decay estimates and energy estimates are of regularity-loss type. We present a kind of S & K mixed criterion which is applicable also to weakly dissipative models including the Timoshenko–Cattaneo system, for which any already known criterion did not work.

There are studied algebraic properties of quadratic Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Their relations both with derivations of symmetric tensor algebras and Yang–Baxter structures on the adjacent Lie algebras are demonstrated. Special attention is paid to quadratic Poisson brackets of Lie–Poisson type, examples of Balinsky–Novikov and Leibniz algebras are discussed. The non-associative structures of commutative algebras related with Balinsky–Novikov, Leibniz, Lie, and Zinbiel algebras are studied in detail.

This report summarizes some representative talks presented at the 12th Maghreb Colloquium on the History of Arab Mathematics (COMHISMA12), held on May 26-28, 2016, at Cadi Ayyad University in Marrakesh, Morocco. This rich and diverse colloquium series, the longest-running and still leading international conference on the overall history of Arab mathematics, is part of the effort to uncover and disseminate the mathematical contributions of Arab and other Islamic cultures.