Venue

The workshop will take place at the School of Technology and Management of the Polytechnic Institute of Guarda, Av. Dr. Francisco Sá Carneiro nº50, Guarda.

Schedule

Quaternions are hyper-complex numbers applied in 3D space. They find applications in several knowledge fields, namely physics, geometry, image processing, computational biology, computational chemistry, computer graphics, and computer games. In this talk, we overview the theory of quaternions, as well as a few applications in computational science and engineering. Specifically, we take advantage of quaternions to show how quaternions solve the Gimbal Lock problem, consume less memory space, and speed up geometric operations in computer graphics and games.

Recently, we presented a complete characterization of the zero set of left unilateral polynomials over coquaternions. In this talk we consider other approaches to obtain the nth roots of a coquaternion and explicit expressions for the zeros of quadratic polynomials.
This is a joint work with F. Miranda, R. Severino and M.J. Soares

The Mandelbrot set for the one-parameter family of complex quadratics maps shows that, in the parameter plane, the dichotomy existence/non-existence of an attractor for the dynamics has a topological and geometric complexity far beyond what we would expect. In this work, we address the same question for the coquaternionic family of quadratic maps $q^2+bq$, with b a complex parameter, and construct its Mandelbrot set.

This note deals with the spectral radius and the spread of an octonionic polynomial.Tools include matricial norms and matrix norms of matrices of blocks.

Using an appropriate multiplication, we can introduce a left matrix representation of an octonion ${a}$, which we will represent by $\omega({a})$. Given two octonions ${a}$ and ${b}$, the spectrum of $\omega({ab})$ is equal to the spectrum of $\omega({a})\omega({b})$. However, if we add an extra octonion ${c}$ the spectrum of $\omega({ab}+{c})$ and $\omega({a})\omega({b})+\omega({c})$ may differ. When this occurs, the conjugacy class of spectrum of $\omega({ab}+{c})$ splits into two conjugacy classes, one bigger and one smaller in absolute value. This property allows the improvement of the bounds for zeros of unilateral octonionic polynomials, namely the bound proposed by J. Vitória and generalized to octonionic polynomials.

It is well known that octonions (both real and complex) are very involved in the structure of the exceptional Lie algebras. We will explore several aspects of this relationship: how octonions provide models of the exceptional Lie algebras, coordinatizating them by means of structurable algebras. This plays an important role in the description of their inner ideals, and, in turn, these appear naturally in certain point-line geometries. In general, these constructions are related to gradings over the integers. We will use also the octonions and related structures to approach gradings on exceptional Lie algebras but over finite groups. Contrary to the situation above, these gradings are related to semisimple elements, and the nice symmetry behind the constructions is reflected in the corresponding Lie groups and in Particle Physics.

It is well-known that quaternions are a real associative noncommutative algebra, as well as the hypercomplex numbers. There are other noncommutative structures of interest in mathematics. In this talk we will concentrate on gyrogroups, which are a generalization of groups. Gyrogroups appeared first in the study of the parametrization of the Lorentz transformation group. They form a non-associative structure, where the associative law is replaced by a left gyroassociative law, governed by the presence of gyrations, which belong to the automorphism group of the gyrogroup. We concentrate on the study of real inner product gyrogroups and we show an orthogonal decomposition theorem, together with the construction of let and right coset spaces, quotient spaces, fiber bundles, and sections. The general theory is exemplified for some well-known gyrogroups such as Einstein, Möbius, Proper Velocity, and Chen's gyrogroups. In the end we show the importance of such gyrogroups in the construction of integral transforms of Fourier or wavelet type in non-Euclidean manifolds, such as the sphere or the ball.

The asymptotic growth behavior of functions satisfying higher dimensional Cauchy-Riemann type equations in terms of growth orders will be studied. Generalized growth orders and growth types in the sense of Shah and Seremeta in the context of polymonogenic functions will be introduced, as well as related inequalities.
(Joint work with R.S. Kraußhar)

Special integers sequences have been the center of attention for many researchers, as well as the sequences of quaternions where its components are the elements of these sequences.
Motivated by a rational sequence, we consider the quaternions with components Vietoris' numbers and investigate some of its properties.
(Joint work with R. de Almeida)

The aim of the talk is to relate a sequence of odd integers with a combinatorial identity over the generators of the quaternions.
The approach is based on the consideration of a quaternionic simplex obtained via the generalization of Faà di Bruno's theorem and some combinatorial relations.

We obtain a sufficient condition for the convexity of quaternionic numerical range for complex matrices in terms of its complex numerical range. It is also shown that the Bild coincides with complex numerical range for real matrices. From this result we derive that all real matrices have convex quaternionic numerical range. As an example we fully characterize the quaternionic numerical range of $2\times2$ real matrices.

We prove that the quaternionic numerical range is always star-shaped and its star-center is given by the equivalence classes of the star-center of the bild. We determine the star-center of the bild, and consequently of the numerical range, by showing that the geometrical shape of the upper part of the center is defined by two lines, tangents to the lower bild.

It is argued that the 3-dimensional unit sphere plays an important role in the description reality, though in a very different manner than current beliefs based on the standard models of physics. A methodological analysis of the history of physics reveals that significant progress has always been accompanied by the reduction of the number of fundamental constants. Consequently, a rational description of reality must do without any constants of nature, which are, after all, arbitrary postulates. Therefore, there have to be purely mathematical reasons for the existence of fundamental quantities such as the speed of light c and the quantum of action h. Since these two constants are intimately related to the fact that reality is commonly described within the paradigm of space and time, one must question whether these concepts are appropriate for a fundamental description of reality. It is argued that c and h may indeed be a consequence of the peculiar mathematical properties of the 3-dimensional unit sphere. Quaternions are therefore an excellent candidate for a deeper understanding of the laws of nature, as already suggested by Rowan William Hamilton.

In the preface of the book "generatingfunctionology" [2] following ideas of [1] we can find the sentence
Generating functions are a bridge between discrete mathematics, on the one hand, and continuous analysis (particularly complex variable theory) on the other.
The aim of the lecture is to show that hypercomplex analysis can also benefit from such a philosophy of the interaction of two sides of mathematics.
Therefore we consider a Sturm-Liouville equation of the form $$(x^ny')'+x^{-1+n}(1+x)y=0$$ and link in two different ways its solutions to sequences of rational numbers for obtaining old and new results in continuous (function theoretic) and discrete (enumerative combinatorial) hypercomplex analysis in $\mathbb{R}^{n+1}$. Specifying the results to quaternions ($n=3$) and octonions ($n=7$) we arrive to some devisability properties of special sequences of integers.
The underlying research carried out together with Isabel Cação, Irene Falcão and Graça Tomaz was very much inspired by our recent results connecting real, complex and hypercomplex analysis.

[1] Graham, R. L., Knuth, D. E., Patashnik, O., Concrete Mathematics - A foundation for computer science. Addison Wesley; (2nd ed.) (1994)
[2] Wilf, H., generatingfunctionology, CRC Press; (3rd ed.) (2005)