# Schedule

Mouse over the titles shows the abstracts.

## Friday, Feb 15

09:30 - 10:15 **Registration**

10:15 - 10:30 **Opening**

10:30 - 11:20 **Alberto Elduque**

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.

11:30 - 11:55 **Margarida Camarinha**

Kustaanheimo-Stiefel regularization

In this talk we review the regularization theory of the spatial Kepler problem. The aim of this theory is to transform singular differential equations into regular ones.
The most efficient regularization method of the spatial two-body equation was proposed by Kustaanheimo and Stiefel and is based on the so-called KS-transformation. We explain how the regularizing transformation can be described by means of the quaternion language and give an overview of the geometry behind it.

12:00 - 12:25 **Fernando Miranda**

Computational aspects of quaternionic polynomials

In the first edition of the workshop New Trends in Quaternions and Octonions we have presented a Mathematica package, QuaternionAnalysis, to perform numerical and symbolic operations on quaternions valued functions.
Since then new Mathematica functions were written to deal, in particular, with polynomial computations.

In this talk we describe a collection of Mathematica functions for solving classical polynomial problems and present several examples illustrating its use in practice.

Join work with M. I. Falcão, R. Severino, and M. J. Soares.

##### Lunch

14:00 - 14:50 **Alexandre Correia**

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 modeled using quaternions, so we can avoid singularities in the equations of motion.

15:00 - 15:25 **Paula Catarino**

Bicomplex Generalized Tribonacci Quaternions and their applications in matrices

In this talk, we introduce the Bicomplex Generalized Tribonacci Quaternions, study some properties of this sequence of quaternions, give the respective generating functions, Binet's formula and summation formula. Also, we show that by the use of a determinant of a special matrix we can obtain the $n$th term of this sequence of quaternions.

15:30 - 15:55 **Graça Tomaz**

Combinatorial identities via hypercomplex Appel polynomials

Combinatorial identities are not only of importance in abstract combinatorial problems but often are byproducts in other mathematical fields giving significance to their inner relationship or indicating the quantitative connection of qualitative similar problems. More than a dozen years ago, the development of quaternion valued and, more general, Clifford algebra valued Appell polynomials, was motivated by our curiosity of a better understanding of the structure of monogenic polynomials and dealing with them in a suggestive way. Since then hypercomplex Appell polynomials gained interest of many authors, particularly also in different applications. Surprisingly, meanwhile the coefficients themselves of those polynomials (now called Vietoris numbers) became objects of own interest, partially due to their re-detection in positivity problems of real analysis which have been studied half a century ago.

The aim of our talk is to show how some combinatorial identities are emerging from the different representations of a special hypercomplex Appell polynomial sequence. A deep look inside two different structures of those polynomials permit to establish combinatorial identities that follow general combinatorial pattern.

Joint work with: Helmuth Malonek, Isabel Cação, and Irene Falcão

##### Coffee break

16:30 - 16:55 **Sérgio Mendes**

On the convexity and circularity of the numerical range over quaternionic matrices

Let $A\in\mathcal{M}_n(\mathbb{H})$ be a $n\times n$ matrix over the quaternions $\mathbb{H}$. The quaternionic numerical range of $A$ is the subset $W_{\mathbb{H}}(A)\subset\mathbb{H}$ defined by
$$W_{\mathbb{H}}(A)=\{x^*Ax:x\in\mathbb{D}_{\mathbb{H}^n}(0,1)\}$$
where $\mathbb{D}_{\mathbb{H}^n}$ denotes the unit ball with centre in the origin of $\mathbb{H}^n$. Contrary to the case of complex matrices where the numerical range is always convex (Toeplitz-Hausdorff Theorem), convexity is no longer a property of every quaternionic numerical range. We study the convexity of the numerical range over quaternionic matrices. Quite specific, we prove that a certain class of quaternionic matrices always has convex numerical range and we give necessary and sufficient conditions for a $3\times 3$ nilpotent quaternionic matrix to have convex numerical range.
Another property that has been studied for complex and quaternionic matrices is the circularity of the numerical range. We establish the circularity of the numerical range for a class of quaternionic matrices. Moreover, we give necessary and sufficient conditions for a $3\times 3$ nilpotent quaternionic matrix to have circular numerical range.
Joint work with Luís Carvalho and Cristina Diogo from ISCTE-IUL.

17:00 - 17:25 **Helena Albuquerque**

Quaternions and octonions: algebras and coalgebras

In this talk we present quaternions and octonions as commutative and associative algebras and as coassociative coalgebras in the monoidal category of G-graded vector spaces, studying the compatibility of these two structures.

17:30 - 17:55 **Helmuth Malonek**

Generalized Function Theory via Harmonic Analysis - approaches and their limits

About 50 years ago, E. M. Stein and G. Weiss proved in their seminal paper [1], the *"correspondence of irreducible representations of several rotation groups to first order constant coefficient partial differential equations generalizing the Cauchy-Riemann equations."* They showed how certain properties of complex one-dimensional function theory extend to solutions of those systems, in particular the fact of being harmonic solutions. In their list of systems one can find a *generalized Riesz system*, the *Moisil-Theodoresco system*, *spinor systems* as $n$-dimensional generalization of *Diracs equations*, *Hodge - de Rham equations * and special cases of them. Indeed, the aim of proving that correspondence between representation groups and partial differential equations were merely of qualitative nature and deeply connected with properties of harmonic functions in several real variables.
According to the statistics of the Zentralblatt, between the first citation from 1969 and the until now last citation in January 2019 it has been cited during 50 years 74 times in papers published in 35 different journals by 78 different authors. This seems to be more than sufficient for being a seminal paper. But what is very striking is the fact that an overwhelming number of those publications uses their methodical connection via representation theory with [1] as automatical justification for their relation to complex one-dimensional function theory *without* noticing that this is only true so far it concerns very special qualitative questions. Naturally, like in [1] itself they do not go deeper into complex one-dimensional function theoretic topics as the properties of irreducible representations of several rotation groups allow. In our talk we will discuss E. M. Stein's and G. Weiss' as well as its followers approaches to the generalization of certain qualitative properties of complex one-dimensional function theory and their limits.

**References**

[1] Stein, E. M., Weiss, G.:Generalization of the Cauchy-Riemann Equations and Representations of the Rotation Group.
American Journal of Mathematics **90** (1), 163-196 (1968).

##### 20:00 Workshop dinner

## Saturday, Feb 16

10:00 - 10:50 **Rogério Serôdio**

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.

##### Coffee break

11:30 - 11:55 **Joana Soares**

On the zero-sets of quaternionic and coquaternionic polynomials

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is vast and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra of coquaternions have received much less attention from researchers.

The main purpose of this talk is to present some recent results on the structure of the zero-sets of coquaternionic polynomials.
The differences from the quaternionic case, which appear mainly due to the fact that coquaternions, contrary to what happens with quaternions, do not constitute a division algebra, are emphasized.

This is joint work with: M.I. Falcão, F. Miranda, and R. Severino

12:00 - 12:25 **Ricardo Severino**

Coquaternionic dynamics

Building on our previous study of the fixed points and 2-cycles of the family of coquaternionic polynomials $f(q)=q^2+c$,
we consider the following two questions: (1) is it reasonable to expect the existence of new types of sets of fixed points, for other families of quadratic coquaternionic polynomials? (2) what can be said about non-periodic dynamics?

In this talk, we give a positive answer to the first question, by describing the fixed points of the one-parameter family of quadratic coquaternionic polynomials $f(q)=q^2+a\,q$.

Computational examples with the two families of quadratic coquaternionic polynomials referred above will also be presented, giving a first answer to the second question.

Joint work with M. I. Falcão, F. Miranda, and M. J. Soares.

##### Lunch

15:30 - 15:55 **Regina De Almeida**

Wiman-Valiron theory for higher dimensional polynomial Cauchy-Riemann equations

Some basic properties of growth orders and growth types for entire solutions to higher dimensional polynomial Cauchy-Riemann equations, will be presented. Furthermore, a generalization of the famous Lindelöf-Pringsheim theorem linking these growth orders and growth types with the Taylor series coefficients in the context of this function class is introduced.

(Joint work with R.S. Kraußhar)

16:00 - 16:25 **Isabel Cação**

From the Minimal Exponent Integer Sequence to a n-parameter generalization of the Vietoris' Sequence: a hypercomplex journey

The classical definition of Appell polynomials (P. Appell, 1880), intensively studied in the real (and complex case), was extended to the quaternionic setting by H. Malonek et al. in 2006. One year later the same authors generalized their approach to arbitrary dimensions via hypercomplex function theory, and ever since hypercomplex Appell polynomials have been widely studied by several authors from different points of view and with different purposes.

In this talk, we travel around sequences of numbers via the hypercomplex Appell polynomials road. As a starting point we study the Minimal Exponent Integer Sequence (MEIS) that arises from the coefficients of quaternion-valued Appell polynomials and its connection to the sequence of rational numbers S that appears in a celebrated theorem of L. Vietoris in 1958 about the positivity of certain trigonometric sums (Vietoris' Sequence). We continue our journey with the generalization of S to a n-parameter sequence S(n), for which S=S(2), through n-dimensional hypercomplex Appell polynomials.

16:30 - 17:20 **Sebastian Bock**

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.

##### Coffee and goodbye