Abstract: At the same time that Unmanned Aerial Vehicles and Inertial Measurement Units have become more and more popular, attitude estimation for the solid body has attracted increasing attention. Among the many mathematical tools in this area, since its inception in 1960, Kalman filtering has become the algorithm par excellence. In this talk we review some well-known approaches to attitude estimation with quaternion-based Kalman filters. 

Abstract: This work aims to initiate the study of aperiodic attractors in the context of dynamic systems in coquaternions. Therefore, this initial study will focus on the two families of quadratics x2 + c  and  x2 + bx, with both  b  and  c  coquaternionic parameters.

Abstract: In this talk, a new elliptic biquaternion sequence, which is constructed using Vietoris' numbers as its components, is introduced. Some properties and several identities are studied, and its generating function is presented. Moreover, a connection between this elliptic biquaternion sequence and a particular class of tridiagonal matrices is shown. Specifically, by computing the determinant of this type of tridiagonal matrices, this sequence is obtained. This is a joint work with Paula Catarino. 

Abstract:  Some results on arithmetic for octonionic closed balls are presented: inversion, power, square roots. Equations with octonionic balls are dealt with as well. 

Joint work with Patrícia Beites and Rogério Serôdio.

Abstract:  Monogenic hypercomplex variables and polynomials named after R. Fueter (1880-1950) are important stones in the foundation of the theory of hypercomplex functions. They were objects of their own interest in the beginning 70ies in the work of R. Delanghe about totally analytic variables. But only later with an increasing number of papers and books published in the field of hypercomplex analysis they received their name.

Nowadays, one can notice a revival of the interest in Fueter variables. This motivated us to remember some moments from their past life and add some almost unknown details to complete the “folklore” around them.  

Abstract: Robot learning and control often requires us to deal with orientations and rotational motions in Cartesian space. As there exists no singularity-free minimal representation of orientation, quaternions are often used instead. In my talk I will review some quaternion-based motion representations that are commonly used in robotics as well as the methodologies used for their learning. 

Abstract: Analytical methods for the computation of roots of quaternion-valued equations, namely polynomial equations, are a recent topic in the literature. Nevertheless, a quite impressive number of papers devoted to this subject have been produced in the last 15 years. In this work, we provide a brief historical background for the Quaternionic's Newton Method that takes us to some exciting new ideas on iteration functions. Several numerical examples are presented. 

Abstract: In this talk, I will introduce non-associative generalizations of skew Laurent polynomial rings and some related rings, such as skew polynomial rings, skew power series rings and skew Laurent series rings. In particular, I will show how classical Cayley-Dickson algebras such as quaternions and octonions naturally give rise to such rings.

The focus will mainly be on non-associative skew Laurent polynomial rings and results concerning their ideals, such as when they are simple and generalizations of the famous Hilbert’s basis theorem. For non-associative skew Laurent polynomial rings, I will show that both a left and a right version of Hilbert’s basis theorem hold. For non-associative skew polynomial rings, I will show that a right version holds, but will give a counterexample to a left version; a difference that does not appear in the associative setting. The talk is based on joint work with J. Richter.

Abstract:  Relation graph is a useful tool to visualize an arbitrary binary algebraic relation R on an algebraic structure S. Its vertices represent elements or their equivalence classes in S, and there is an edge from x to y if and only if xRy. The most popular relation graphs of various algebras are commutativity, orthogonality, and zero divisor graphs.

The talk is devoted to the commutativity graph of the real sedenion algebra. We show that any element, whose imaginary part is not a zero divisor, corresponds to an isolated vertex of this graph. All other elements belong to the same connected component, and its diameter equals 3. Our proof is based on the complete description of zero divisors and their annihilators in the sedenion algebra.

Abstract: In this talk, we introduce the Linear Canonical Transforms associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann-Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish a quaternionic version of the Heisenberg uncertainty principle for the right-sided QHLCT. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations. 

Joint work with Milton Ferreira.

Abstract: This is a joint work with Alexander Zubkov from United Arab Emirates University. Over an algebraically closed field, we described a minimal set of representatives for G2-orbits on the set of pairs of octonions. As an application, the case of traceless octonions is considered. The corresponding preprint can be found here: https://arxiv.org/abs/2208.08122.