deadlines
Registration with submission of a talk:
November 25
Registration:
December 14
sponsors
Venue
The workshop will take place at University of Minho, School of Economics and Management (Escola de Economia e Gestão) Room 1.01, Campus de Gualtar, Braga.
Schedule
Mouse over the titles shows the abstracts.
Friday, Dec 16
14:15 - 14:30 Opening
14:30 - 15:30 Swanhild Bernstein
Fractional Riesz-Hilbert transforms and fractional monogenic signals
The analytic signal was proposed by Gabor [2] as a complex signal corresponding to a given real signal. The Hilbert transform is the key component in Gabor's analytic signal construction. The analytic signal has been generalized the monogenic signal by Felsberg and Sommer [1] based on Clifford analysis.
Fractional transformations are widely used in optics and based on applied needs as well as a mathematical constructions based on eigenvalue decompositions. We will explain how fractional transform can be constructed.
We will generalize the fractional Hilbert transform to the quaternionic fractional Riesz-Hi [3] and Seelamantula [4].
[1] M. Felsberg, G. Sommer, The monogenic signal. IEEE Trans. Signal Proc., 49 (12) (2001), 3136--3144.
[2] D. Gabor, Theory of communication. J. of the Institution of Electrical Engineers - Part III: Radio and Communication Engineering, 93 (26) (1946), 429--457.
[3] A.W. Lohmann, D. Mendlovic, Z. Zalevsky, Fractional Hilbert transform. Optics Letters, 21 (1996), 281--283.
[4] A. Venkitaraman, C.S. Seelamantula, Fractional Hilbert transform extensions and associated analytic signal construction. Signal Processing, 94 (2014), 359--372.
[5] B. Heise et al, Fourier Plane Filterung revisited -- Analogies in Optics and Mathematics, Sampling theory in Signal and Image Processing, 13 (3), (2014), 231--248.
[1] M. Felsberg, G. Sommer, The monogenic signal. IEEE Trans. Signal Proc., 49 (12) (2001), 3136--3144.
[2] D. Gabor, Theory of communication. J. of the Institution of Electrical Engineers - Part III: Radio and Communication Engineering, 93 (26) (1946), 429--457.
[3] A.W. Lohmann, D. Mendlovic, Z. Zalevsky, Fractional Hilbert transform. Optics Letters, 21 (1996), 281--283.
[4] A. Venkitaraman, C.S. Seelamantula, Fractional Hilbert transform extensions and associated analytic signal construction. Signal Processing, 94 (2014), 359--372.
[5] B. Heise et al, Fourier Plane Filterung revisited -- Analogies in Optics and Mathematics, Sampling theory in Signal and Image Processing, 13 (3), (2014), 231--248.
15:30 - 16:00 Paula Catarino
The dual version of $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell quaternions and octonions
Clifford, in [1], extended the real numbers and introduced the dual number $A$ defined by $A=a+\epsilon a^*$, where $a,~a^* \in \mathbb{R}$ and $\epsilon$ is called by the dual unit and it has the
following properties:
\begin{equation*}
\epsilon \not = 0,~0 \epsilon = \epsilon 0 = 0,~1 \epsilon = \epsilon 1 = \epsilon,~\epsilon^2 = 0.
\end{equation*}
The set of dual numbers shall be denoted by $\mathbb{D}$ and forms a commutative ring (not field) having $\epsilon a^*$ as divisors of zero.
A dual quaternion $\widehat{q}$ can be defined in a similar way to the dual numbers as $\widehat{q} = q + \epsilon q^*$, where $q,~q^* \in \mathbb{H}$. Let us denote by $\mathbb{D}_q$ the set of dual quaternions and any $\widehat{q} \in \mathbb{D}_q$ can be written as $$\widehat{q} = A_0e_0 + A_1e_1 + A_2e_2 + A_3e_3,$$
where $A_i \in \mathbb{D}, A_i = a_i + \epsilon a_i^*, a_i,~a_i^* \in \mathbb{R}$, $i=0, 1, 2, 3$. Note that the set of dual quaternions $\mathbb{D}_q$ is an algebra over the real numbers which is noncommutative but associative. In a similar way, we define a dual octonion.
In this presentation, we state some fundamental algebraic properties of the dual $k$-Pell, the dual $k$-Pell-Lucas, and the dual Modified $k$-Pell quaternions and octonions, as well as we give their Binet's formulae and generating functions.
[1] W. K. Clifford, Preliminary Sketch of Biquaternions. Proc. London Math. Soc. 4 (1) (1873), 381--395.
[1] W. K. Clifford, Preliminary Sketch of Biquaternions. Proc. London Math. Soc. 4 (1) (1873), 381--395.
16:00 - 16:30 Helmuth Malonek
Quaternions and much more: On the 1940 Fueter Festschrift
From time to time it seems to be worthwhile to look back to the beginnings of Quaternionic Analysis in the work of R. Fueter.
The dedicated to him "Festschrift RUDOLF FUETER zur Vollendung seines sechzigsten Altersjahres 30.VI. 1940" published as "Beiblatt zur Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 1940, Nº. 32 Jahrg. 85" (not so well known supplement to the quarterly of the Naturalist Society in Zurich founded in 1746) reads like "Who's Who in the Mathematical World 1940".
The names of OYSTEIN ORE, HENRI LEBESGUE, M. PLANCHEREL, N. TSCHEBOTARÖW, PAUL MONTEL, L. J. MORDELL, FRANCESCO SEVERI, T. CARLEMAN, E. HECKE, H. BRANDT, C. CARATHEODORY, LUDWIG BIEBERBACH, PAUL FINSLER, HEINZ HOPF, H. BEHNKE, K. STEIN, ELIE CARTAN, and ANDREAS SPEISER (among others of 27 authors) remind us easily that Fueter was a world wide highly praised mathematician who worked on the cross road of Number Theory, Algebra, Geometry, and Analysis.
Very briefly, the talk tries to use the Festschrift as pretext for remembering some of Fueter's steps to apply quaternions for problems in Analytic Number Theory.
The dedicated to him "Festschrift RUDOLF FUETER zur Vollendung seines sechzigsten Altersjahres 30.VI. 1940" published as "Beiblatt zur Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 1940, Nº. 32 Jahrg. 85" (not so well known supplement to the quarterly of the Naturalist Society in Zurich founded in 1746) reads like "Who's Who in the Mathematical World 1940".
The names of OYSTEIN ORE, HENRI LEBESGUE, M. PLANCHEREL, N. TSCHEBOTARÖW, PAUL MONTEL, L. J. MORDELL, FRANCESCO SEVERI, T. CARLEMAN, E. HECKE, H. BRANDT, C. CARATHEODORY, LUDWIG BIEBERBACH, PAUL FINSLER, HEINZ HOPF, H. BEHNKE, K. STEIN, ELIE CARTAN, and ANDREAS SPEISER (among others of 27 authors) remind us easily that Fueter was a world wide highly praised mathematician who worked on the cross road of Number Theory, Algebra, Geometry, and Analysis.
Very briefly, the talk tries to use the Festschrift as pretext for remembering some of Fueter's steps to apply quaternions for problems in Analytic Number Theory.
Coffee break
17:00 - 18:00 Lidia Aceto
Matrix representation of algebra-valued Appell polynomials
Appell polynomials are any polynomial sequence $\{p_n(x)\}_{n\geq 0}$
whose elements verify the conditions
\begin{equation}
\frac{d}{d x}p_n(x) = n \, p_{n-1}(x), \qquad n=1,2,3,\dots,
\end{equation}
with
$p_0(x)=c_0,\ c_0 \in \mathbb{R} \setminus \{0 \}.$
In the last years the Appell polynomials, named after Paul Émile Appell which introduced them in 1880 [1], have gained renewed interest and has been studied by several authors. In particular, some new characterizations of Appell polynomials themselves through new approaches have been considered. To quote some of them we mention, for instance, the novel approach developed in [5], which makes use of the generalized Pascal functional matrices introduced in [6], and also a new characterization based on a determinantal definition proposed in [3] and recently applied in [4] to Sheffer sequences too. Both of them have allowed to derive some properties of Appell polynomials by employing only linear algebra tools and to generalize some classical Appell polynomials.
In this talk we present a unified approach to matrix representations of different types of real Appell polynomials. Besides the most common examples like, Euler, Bernoulli, Hermite and Laguerre polynomials, also Legendre and Chebychev polynomials (both of the first and second kind) are referred; in the latter cases, however, the Appell polynomial nature can be disclosed by a substitution suggested by Carlson [2]. In addition, we introduce an extention of this approach for the case of monogenic polynomials in arbitrary dimensions and in a noncommutative hypercomplex setting, i.e. for Clifford algebra-valued polynomials in the kernel of a generalized Cauchy-Riemann operator.
The approach we propose is based on the creation matrix - a special matrix which has only the natural numbers as entries and is closely related to the well known Pascal matrix. It also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations. Furthermore, the effectiveness of the unified matrix representation is confirmed by the fact that some new types of recently introduced Appell polynomials can immediately be deduced.
Work in collaboration with Helmuth R. Malonek (University of Aveiro, Portugal) and Graça Tomaz (Polytechnic Institute of Guarda, Portugal).
[1] Appell P. Sur une classe de polynômes. Ann Sci École Norm. Supér. 1880;9:119-144.
[2] B. C. Carlson, Polynomials satisfying a binomial theorem, J. Math. Anal. Appl. 32 (1970) 543-558.
[3] Costabile F, Longo E. A determinantal approach to Appell polynomials. J Comp Appl Math. 2010;234:1528-1542.
[4] Costabile F, Longo E. An algebraic approach to Sheffer polynomial sequences. Integral transforms and special functions. 2014;25:295-311.
[5] Yang Y, Youn H. Appell polynomials sequences: a linear algebra approach. JP Journal of Algebra, Number Theory and Applications. 2009;13:65-98.
[6] Yang Y, Micek C. Generalized Pascal functional matrix and its applications. Linear Algebra and its Applications. 2007;423:230-245.
In the last years the Appell polynomials, named after Paul Émile Appell which introduced them in 1880 [1], have gained renewed interest and has been studied by several authors. In particular, some new characterizations of Appell polynomials themselves through new approaches have been considered. To quote some of them we mention, for instance, the novel approach developed in [5], which makes use of the generalized Pascal functional matrices introduced in [6], and also a new characterization based on a determinantal definition proposed in [3] and recently applied in [4] to Sheffer sequences too. Both of them have allowed to derive some properties of Appell polynomials by employing only linear algebra tools and to generalize some classical Appell polynomials.
In this talk we present a unified approach to matrix representations of different types of real Appell polynomials. Besides the most common examples like, Euler, Bernoulli, Hermite and Laguerre polynomials, also Legendre and Chebychev polynomials (both of the first and second kind) are referred; in the latter cases, however, the Appell polynomial nature can be disclosed by a substitution suggested by Carlson [2]. In addition, we introduce an extention of this approach for the case of monogenic polynomials in arbitrary dimensions and in a noncommutative hypercomplex setting, i.e. for Clifford algebra-valued polynomials in the kernel of a generalized Cauchy-Riemann operator.
The approach we propose is based on the creation matrix - a special matrix which has only the natural numbers as entries and is closely related to the well known Pascal matrix. It also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations. Furthermore, the effectiveness of the unified matrix representation is confirmed by the fact that some new types of recently introduced Appell polynomials can immediately be deduced.
Work in collaboration with Helmuth R. Malonek (University of Aveiro, Portugal) and Graça Tomaz (Polytechnic Institute of Guarda, Portugal).
[1] Appell P. Sur une classe de polynômes. Ann Sci École Norm. Supér. 1880;9:119-144.
[2] B. C. Carlson, Polynomials satisfying a binomial theorem, J. Math. Anal. Appl. 32 (1970) 543-558.
[3] Costabile F, Longo E. A determinantal approach to Appell polynomials. J Comp Appl Math. 2010;234:1528-1542.
[4] Costabile F, Longo E. An algebraic approach to Sheffer polynomial sequences. Integral transforms and special functions. 2014;25:295-311.
[5] Yang Y, Youn H. Appell polynomials sequences: a linear algebra approach. JP Journal of Algebra, Number Theory and Applications. 2009;13:65-98.
[6] Yang Y, Micek C. Generalized Pascal functional matrix and its applications. Linear Algebra and its Applications. 2007;423:230-245.
18:00 - 18:30 Graça Tomaz
A matrix approach to general monogenic orthogonal polynomial systems
The matrix representation of real and hypercomplex Appell sequences uses the Pascal matrix
and its creation matrix. In this talk we are looking for the matrix representation of general
homogeneous orthogonal Appell polynomials obtained through the so called shifted Pascal
matrix.
(In collaboration with I. Cação and H. Malonek)
(In collaboration with I. Cação and H. Malonek)
Workshop dinner
Saturday, Dec 17
09:30 - 10:30 João Morais
Prolate spheroidal wave functions associated with the quaternionic Fourier transform
One of the fundamental problems in communications is related to finding the energy distribution of signals in time and frequency domains. It should therefore be of great interest to find the quaternionic signal whose time-frequency energy distribution is most concentrated in a given time-frequency domain.
In this talk, we find a new kind of quaternionic signals whose energy concentration is maximal in both time and frequency under the quaternionic Fourier transform. The new signals are a generalization of the Prolate Spheroidal Wave Functions (also known as Slepian functions) to a quaternionic space and they are called the Prolate Spheroidal Quaternion Wave Signals (PSQWSs). We show that the PSQWSs are orthogonal and complete over two different intervals: the space of square integrable functions over a finite interval and the three-dimensional Paley-Wiener space of bandlimited functions. To progress in this direction, we compute the PSQWSs restricted in frequency to the unit sphere. The representation of these functions in terms of generalized spherical harmonics is explicitly given, from which several fundamental properties can be derived. In the applications part of this talk, we show that the PSQWSs can reach the extreme case in the energy concentration problem both from a theoretical and experimental point of view.
In this talk, we find a new kind of quaternionic signals whose energy concentration is maximal in both time and frequency under the quaternionic Fourier transform. The new signals are a generalization of the Prolate Spheroidal Wave Functions (also known as Slepian functions) to a quaternionic space and they are called the Prolate Spheroidal Quaternion Wave Signals (PSQWSs). We show that the PSQWSs are orthogonal and complete over two different intervals: the space of square integrable functions over a finite interval and the three-dimensional Paley-Wiener space of bandlimited functions. To progress in this direction, we compute the PSQWSs restricted in frequency to the unit sphere. The representation of these functions in terms of generalized spherical harmonics is explicitly given, from which several fundamental properties can be derived. In the applications part of this talk, we show that the PSQWSs can reach the extreme case in the energy concentration problem both from a theoretical and experimental point of view.
10:30 - 11:00 Marco Pérez de la Rosa
On some boundary value properties of oblate spheroidal quaternionic wave functions
This work introduces the Oblate Spheroidal Quaternionic Wave Functions (OSQWFs), which extend the oblate spheroidal wave functions introduced in the late 1950s by C. Flammer. We show that the theory of the OSQWFs is determined by a Moisil-Teodorescu type operator with quaternionic variable coefficients. Moreover, we present the connections between the solutions of the radial and angular equations and of the Chebyshev equation, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions on the other. We further establish an analogue of the Cauchy's integral formula as well as some analogues of the boundary value properties for this version of the quaternionic function theory.
Coffee break
11:30 - 12:00 Klaus Gürlebeck
Characterization of monogenic functions by their Taylor coefficients
We consider Taylor series of monogenic functions with respect to complete orthogonal Appell systems. The coefficients of these series can be used to characterize whether the function belongs to a certain function space or not. Examples of such spaces are the Dirichlet space, the scale of weighted Dirichlet spaces and the scale of $Q_p$ spaces.
12:00 - 12:30 Sebastian Bock
On a hypercomplex version of the Kelvin solution in linear elasticity
In the talk we shall present an overview about recently developed generalized Kolosov-Muskhelishvili formulae in $\mathbb{R}^{3}$ using the framework of hypercomplex function theory. Based on these results, a hypercomplex representation of the fundamental solution of three dimensional linear elasticity, also known as Kelvin solution, is constructed. For this purpose a new class of monogenic functions with line singularities is studied and an associated two step recurrence formula is proved.
Lunch
14:30 - 15:00 João Fernandes
Quaternions applications: from the Earth to the Universe
As it's very well known, the quaternions are used in several applications to represent rotations (eg. navigation or robotics). In this talk we will discuss how quaternions can be used in two different realities: to stablish relations between different geodetical coordinate systems (datum), named Bursa-Wolf transformations, something crucial for the local and global positioning at Earth; and how to move from geocentric to heliocentric coordinate systems in the framework of the observations of millions of stars made by space missions, as the case of the newest GAIA (European Space Agency).
15:00 - 15:30 Celino Miguel
The commuting graph of a full matrix algebra over the quaternions
Let $M_n(\mathbb H)$ be the ring of $n\times n$ matrices over the quaternions. The commuting graph $\Gamma(M_n(\mathbb H))$ is a graph whose vertices are all noncentral matrices and two distinct vertices $A$ and $B$ are adjacent if and only if $AB=BA$.
In this talk we will study the connectedness and the diameter of $\Gamma(M_n(\mathbb H))$.
15:30 - 16:00 Isabel Cação
The hypercomplex approach to generalized Vietoris number sequences
The construction of orthogonal bases of polynomials in the hypercomplex context, made by R. Laviscka in 2012, leads to polynomial sequences with a special structure, where the building blocks satisfy several properties, including recurrence relations. The restriction of the building blocks to the hyperplane $x_0=0$ results on polynomial sequences with coefficients that are generalizations of the Vietoris number sequence. In this talk we study in detail those number sequences: origins, properties and representations in terms of the Clifford algebra elements.
16:00 - 17:00 Ricardo Severino
Iteration of quadratic maps on coquaternions
Quaternions are usually considered as the natural generalization of complex numbers. Hence, it is not surprising that the first attempts of extending the well-established iteration theory of quadratic maps in the complex plane to higher dimensions involved quaternionic maps.
It turns out that this generalization proved not to be very fruitful, with the new results being very closely related to the corresponding ones in the complex case.
However, quaternions are not the only four dimensional hypercomplex real algebra generalizing complex numbers; for example, one may also consider coquaternions, introduced by Sir James Cockle at about the same time as Sir William Hamilton discovered the quaternions.
In this talk we focus on the the iteration of the quadratic coquaternionic map $f_{c}(q)=q^2+c$, where $c$ is a fixed coquaternionic parameter, with particular interest in the determination and subsequent stability analysis of its fixed points and periodic points of period two. Some computational results for points of period higher than two will also be presented.
This is joint work with: M.I. Falcão, F. Miranda and M.J. Soares
It turns out that this generalization proved not to be very fruitful, with the new results being very closely related to the corresponding ones in the complex case.
However, quaternions are not the only four dimensional hypercomplex real algebra generalizing complex numbers; for example, one may also consider coquaternions, introduced by Sir James Cockle at about the same time as Sir William Hamilton discovered the quaternions.
In this talk we focus on the the iteration of the quadratic coquaternionic map $f_{c}(q)=q^2+c$, where $c$ is a fixed coquaternionic parameter, with particular interest in the determination and subsequent stability analysis of its fixed points and periodic points of period two. Some computational results for points of period higher than two will also be presented.
This is joint work with: M.I. Falcão, F. Miranda and M.J. Soares