Venue

The workshop will take place at Department of Mathematics of University of Aveiro at room Sousa Pinto (second floor).

Schedule

Mouse over the titles shows the abstracts.  

14:00 - 14:15  Opening

14:15 - 15:00  Gerhard Opfer

Zeros of unilateral quaternionic and coquaternionic polynomials

Let ${\cal A}$ be a finite dimensional algebra over the reals. For ${\cal A}$ we will consider $\mathbb{H}$ (quaternions), $\mathbb{H}_{\rm coq}$ (coquaternions), $\mathbb{H}_{\rm nec}$ (nectarines), and $\mathbb{H}_{\rm con}$ (conectarines), and study the possibility of finding the zeros of unilateral polynomials over these algebras, which are the four noncommutative algebras in $\mathbb{R}^4$. A polynomial $p$ will be defined by $$p(z):=\sum_{j=0}^n a_jz^j,\quad a_j,z\in {\cal A},$$ and for finding the zeros we use of the so-called companion polynomial, which has real coefficients, and is defined by $$q(z):=\sum_{j,k=0}^n \overline{a_j}a_kz^{j+k}=\sum_{\ell=0}^{2n}b_\ell z^\ell \Rightarrow b_\ell\in\mathbb{R}.$$ See D. Janovská and G. O.: SIAM J. Numer. Anal. 48 (2010), 244-256, for quaternionic polynomials and ETNA 41 (2014), 133-158 for coquaternionic polynomials. The real or complex roots of the companion polynomial $q$ will provide information on similarity classes which contain zeros of $p$. It will be shown, that the companion polynomial $q$ has more capacity than formerly described in our papers, valid in all noncommutative algebras of $\mathbb{R}^4$. There will be numerical examples.

15:05 - 15:25  Friederike Luther

On zeros of monogenic functions

In this talk we study zeros of monogenic functions in $L_2(\mathbb{B}_3;\mathbb{H})\cap \ker \overline \partial$, which are generated by an orthogonal Appell basis. We prove the basis functions have no other zeros than one non isolated, furthermore we show for a certain class of monogenic functions all zeros are either isolated or circle lines. Finally we give an algorithm to compute the zeros of an arbitrary monogenic function under a certain condition.

15:30 - 15:50  Maria Joana Soares

Quaternionic Weierstrass method

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas. As a consequence, the design of efficient methods for numerically approximating the zeros of quaternionic polynomials has attracted the attention of several authors. In fact, one can find in the literature important contributions to this subject, but numerical methods based on quaternion arithmetic remain scarce. In this talk we propose a Weierstrass-like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.
Joint work with Maria Irene Falcão, Fernando Miranda and Ricardo Severino.

15:55 - 16:15  Fernando Miranda

QuaternionAnalysis: a Mathematica package

In this talk we present a Mathematica add-on application, QuaternionAnalysis package, for numeric and symbolic manipulation of quaternion valued functions.
This package adds functionalities to the standard Quaternions package (implementing Hamilton's quaternion algebra) and provides tools to handle regular quaternion valued functions. Some of the added new features include the possibility of performing operations on special polynomials as well as on functions defined in $\mathbb{R}^{n+1}$, $n\geq2$.
Joint work with Maria Irene Falcão.

16:45 - 17:05  Ricardo Severino

Iteration of quaternion maps

In 1995, Bedding and Briggs proved that the dynamics obtained from the iteration of the quaternion quadratic polynomials $P(x)=x^2+q$, are, in some sense, equivalent to the dynamics of quadratic complex polynomials. Does this mean that the dynamics of quadratic polynomials are insensible to the noncommutative nature of the phase space?

17:10 - 17:30  Cecília Costa

A note on quaternion block-tridiagonal and block quasi-tridiagonal systems

We propose a direct method for solving a system of linear equations having matrices with quaternionic entries as coefficients and independent terms. For the calculation of the needed inverse matrix we use the second kind Chebyshev polynomials, generalizing the method proposed by Kershaw (1969) and Rózsna and Romani (1991). This method is applicable when the system matrix is a block-tridiagonal matrix or a block quasi-tridiagonal matrix, with blocks that allow the use of Chebyshev polynomials with quaternionic matrix argument.
Joint work with R. Serôdio (University of Trás-os-Montes e Alto Douro) and José Vitória (University of Beira Interior).

17:35- 17:55  Paula Catarino

Some special sequences of quaternions and octonions

A recurrence relation is a "mathematical technique" which allows us to define sequences, sets, even operations or algorithms, from particular cases to general cases. When we use this "technique", is of fundamental importance to have attention, firstly, at the initial condition(s) -- which must be known -- and, on the other hand, the "recurrence equation" -- which is not more than the rule that will calculate the next terms in the light of predecessors. In this workshop, we introduce some special sequences of quaternions and octonions defined by a second-order recurrence, presenting some of its properties, generating function and generating matrix.

18:00 - 18:45  Drahoslava Janovská

Matrices without eigenvalues

We are focused on matrices over nondivision algebras and show by an example from an $\mathbb{R}^4$ algebra that these matrices do not necessarily have eigenvalues, even if they are invertible. The standard condition for eigenvectors to be nonzero is replaced by the condition that $x$ contains at least one invertible component.
Our example raises several key questions:
What qualifies a matrix over a nondivision algebra to have eigenvalues?
Are these matrices diagonalizable or triangulizable? Do they allow a Schur decomposition?

Information available soon.

09:30 - 10:15  Susanna Pumplün

Factoring skew polynomials over Hamilton's quaternion algebra and the complex numbers

We show that all polynomials in a skew-polynomial ring $C[t; s, d]$ over the complex numbers decompose into a product of linear and quadratic irreducible factors. We also give a new conceptual proof for the Fundamental Theorem of Algebra for left polynomials over Hamilton's quaternion algebra. Our proofs use nonassociative algebras constructed out of skew-polynomial rings as introduced by Petit.

10:20 - 10:40  Ricardo Pereira

Quaternions and linear systems theory - I

Within the behavioral framework, a linear dynamical system is characterized by its behavior, which is the solution space of some linear differential or difference equation with constant coefficients. In the real or complex case, this equation is represented by a matrix polynomial operator and the classical properties of control system (as, for instance, controllability and stability) are related to corresponding properties of the operator. The extension of this theory to quaternionic coefficients is not straightforward, making it necessary to introduce new tools and techniques.
In this talk, starting with a suitable definition of quaternionic polynomials, these algebraic tools (polynomial determinant, quaternionic Smith form, etc) will be presented, along with their properties, and applied to the analysis of some aspects of linear dynamical systems.

10:40 - 11:00  Paolo Vettori

Quaternions and linear systems theory - II

Within the behavioral framework, a linear dynamical system is characterized by its behavior, which is the solution space of some linear differential or difference equation with constant coefficients. In the real or complex case, this equation is represented by a matrix polynomial operator and the classical properties of control system (as, for instance, controllability and stability) are related to corresponding properties of the operator. The extension of this theory to quaternionic coefficients is not straightforward, making it necessary to introduce new tools and techniques.
In this talk, starting with a suitable definition of quaternionic polynomials, these algebraic tools (polynomial determinant, quaternionic Smith form, etc) will be presented, along with their properties, and applied to the analysis of some aspects of linear dynamical systems.

11:30 - 11:50  Patrícia Beites

Standard composition algebras of type II associated to quaternion and octonion algebras

Over a field of characteristic different from two, standard composition algebras of type II associated to quaternion and octonion algebras are considered. Some identities satisfied by the former algebras are studied. Furthermore, a characterization of the type II is obtained through one of those identities.

11:55 - 12:15  José María Sanchez Delgado

Quasicrossed products and systems on graded quasialgebras

G-graded quasialgebras were introduced by H. Albuquerque and S. Majid about a decade ago. This class of algebras includes several types of algebras. Indeed, the class of associative algebras fits into this concept, as well as some notable nonassociative algebras such as deformed group algebras (for example, Cayley algebras and Clifford algebras).
Inspired by the theory of graded rings and graded algebras, we show an extension of the notion of crossed product to the setting of graded quasialgebras.
By collecting basic definitions and properties related to graded quasialgebras we introduce the notion of quasicrossed product, including some examples and the relationship with quasicrossed system. We show that the quasicrossed system corresponding to the deformed group algebra obtained from the Cayley-Dickson process applied to a deformed group algebra is related to the quasicrossed system corresponding to the initial algebra. Finally we obtain results about simple quasicrossed products.

14:30 - 14:50  Paulo Saraiva

Best pair of two skew lines over the octonions

This talk concerns an application of octonions to Analytic Geometry. In the octonionic context, the orthogonal projection of a point onto a straight line is presented. Further, the best approximation pair of points of two skew lines over the octonions is studied.

14:55 - 15:15  Sebastian Bock

Constructive orthogonal series expansions in dimensions 2, 3 and 4: properties, representation formulae, applications

In the talk we shall present an unified approach of orthogonal series expansions in dimensions 2, 3 and 4 using the framework of hypercomplex function theory. The essential tools are recently developed orthogonal Appell bases of inner and outer solid spherical monogenics that have special properties with regard to the (hyper-)complex derivation and primitivation. In this context, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented which yield novel definitions of the canonical series expansions in dimensions 3 and 4. The analogy to the complex theory is emphasized. Finally, the presented series expansions are applied to solve some classical problems in linear elasticity theory in $\mathbb{R}^3$.

15:20 - 15:40  Dmitrii Legatiuk

On pseudo-complex polynomials and their applications

In this talk we plan to present recent results on a special type of monogenic polynomials so-called Pseudo-Complex Polynomials (PCP). In the end of the talk we will discuss an application of PCP to a problem of interpolation in $\mathbb{R}^{3}$ and $\mathbb{R}^{4}$.

15:45 - 16:30  Klaus Gürlebeck

Interpolation and approximation with quaternion-valued monogenic functions

Based on abstract theory of interpolation we discuss relations between interpolation and best approximation. If the result of interpolation and best approximation coincide then we speak about optimal interpolation. Starting with this concept we study different known and some new ideas of the interpolation of monogenic functions by using Fueter polynomials, rational functions and Appell systems.