# scope

**The Third Workshop New Trends in Quaternions and Octonions** aims to present recent advances in the research on quaternions and octonions gathering scientists working in pure as well as applied mathematics, scientific computation and applications in physics, engineering and other applied sciences.

This workshop represents an opportunity to discuss recent developments in the field of quaternions and octonions, may they be theoretical, computational or real-world applications.

The workshop will take place at University of Coimbra, February 15-16, and will count with several invited lectures and short communications.

# Invited speakers

Mouse pointer over speaker name shows title talk and abstract.

## Alberto Elduque

University of Zaragoza, Spain
## Alexandre Correia

University of Coimbra, Portugal
## Rogério Serôdio

University of Beira Interior, Portugal
## Sebastian Bock

Bauhaus-Universitat Weimar, Germany

# Organization

The meeting is jointly organized by the Centre for Earth and Space Research (CITEUC) and the Centre for Mathematics (CMUC) of the University of Coimbra,
the Center for Research & Development in Mathematics and Applications (CIDMA) of University of Aveiro,
the Centre of Mathematics and Applications (CMA) of the University of Beira Interior and the Centre of Mathematics (CMAT) of University of Minho.

## Organizers

# Previous editions

CITEUC is funded by National Funds through FCT - Foundation for Science
and Technology (project: UID/Multi/00611/2013) and FEDER - European
Regional Development Fund through
COMPETE 2020 - Operational Programme Competitiveness and
Internationalization (project: POCI-01-0145-FEDER-006922)

The Centre for Mathematics of the University of Coimbra (CMUC) is
funded by the Portuguese Government through FCT (UID/MAT/00324/2013)
and co-funded by the European Regional Development Fund through
the Partnership Agreement PT2020.

This event is supported by Portuguese funds through the CMA - Research Centre of Mathematics and Applications of the University of Beira Interior - and the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), within project UID/MAT/00212/2019.

This event is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications - and the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), within project UID/MAT/04106/2013.

This event is supported by Portuguese funds through the CMAT - Research Centre of Mathematics of the University of Minho - and the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), within project UID/MAT/00013/2013.

## Octonions in low characteristics

Some special features of Cayley algebras, and their Lie algebras of derivations, over fields of low characteristics will be presented. As an example, over fields of characteristic two, the isomorphism class of the Lie algebra of derivations of a Cayley algebra does not depend on the Cayley algebra itself.

## Chaotic dynamics in gravitational systems using quaternions

Beyond the orbit of Neptune resides a system of three small bodies, known by the Lempo system. The dynamics of this system is very rich, but depends on many parameters that are presently unknown. We derive a full 3D N-body model that takes into account the orbital and spin evolution of all bodies, which are assumed triaxial ellipsoids. We show that, for reasonable values of the shapes and rotational periods, the present orbital solution is chaotic and unstable in short time-scales. Since the spin axes can assume any orientation, they are modelled using quaternions, so we can avoid singularities in the equations of motion.

## Bounds for the zeros of unilateral octonionic polynomials

It is proved that the zeros of an unilateral octonionic polynomial are the latent roots of an appropriate lambda-matrix. This allows the use of matricial norms, matrix norms in particular, to obtain (upper and lower) bounds for the zeros of unilateral octonionic polynomials. Unexpected results arise due to the nonassociative setting, improving known algorithms.

## Special classes of monogenic functions and applications in linear elastic fracture mechanics

Recently, the classical orthogonal function systems of inner and outer monogenic Appell functions were used to find new classes of monogenic functions with (logarithmic) line singularities, which extend the known function classes in a natural way.
In the lecture we will discuss some essential properties (orthogonality, Appell property, three-term recurrence relation) of the classical and the special classes
of monogenic functions as well as their relations to each other.
Furthermore, on the basis of the generalized Kolosov-Muskhelishvili formulas, it will be shown how these special functions can be used to construct analytical near-field solutions of spatial crack fronts in linear elastic fracture mechanics.