Department of Electronics and Information Systems

Seminars Academic Year 2023-2024

Seminars organised in the academic year 2023-2024

Triangular decompositions of quaternionic non-selfadjoint operators (Uwe Kähler, University of Aveiro)

Abstract:

One of the principal problems in studying spectral theory for quaternionic or Clifford-algebra-valued operators lies in the fact that due to the noncommutativity many methods from classic spectral theory are not working anymore in this setting. For instance, even in the simplest case of finite rank operators there are different notions of a left and right spectrum. Hereby, the notion of a left spectrum has little practical use while the notion of a right spectrum is based on a nonlinear eigenvalue problem. In the present talk we will recall some basic facts about eigenvalues and linear algebra as well as the notion of S-spectrum  in a noncommutative setting and use it to study quaternionic non-selfadjoint operators. To this end  we will discuss quaternionic Volterra operators and triangular representation of quaternionic operators similar to the classic approaches by Gohberg, Krein, Livsic, Brodskii and de Branges. Hereby we introduce spectral integral representations with respect to quaternionic chains and discuss the concept of P-triangular operators in the quaternionic setting. This will allow us to study the localization of spectra of non-selfadjoint quaternionic operators.

Categories of Brauer type (Sigiswald Barbier, Ghent University)

Abstract:

In this talk we will introduce classes of monoidal (super)categories resembling the Brauer category. These classes encompass the Brauer category and its deformations as well as the periplectic Brauer category and its deformations in one framework but also include some new exotic categories. For all categories we can construct bases of the hom-spaces using Brauer diagrams.

Inverse Problems in Thermoelasticity (Frederick Maes, Ghent University)

Abstract:

Inverse problems arise in many applications as a process to determine the cause(s) of (a) desired or observed effect(s). A thermoelastic system describes the interaction between the changes in shape of an object and its fluctuation in temperature. Such a system is governed by a coupled set of equations, a hyperbolic equation dictating the displacement and a parabolic equation for the heat distribution. In this talk we will address some general aspects of inverse problems and discuss some questions for inverse source problems related to an unknown component in the load source of a thermoelastic system.

The Euler-Bernoulli equation with distributional coefficients and forces (Srdan Lazendic, Ghent University)

Abstract:

In this work we investigate a very weak solution to the initial-boundary value problem of an Euler-Bernoulli beam model. We allow for bending stiffness, axial- and transversal forces as well as for initial conditions to be irregular functions or distributions. We prove the wellposedness of this problem in the very weak sense. More precisely, we define the very weak solution to the problem and show its existence and uniqueness. For regular enough coefficients we show consistency with the weak solution. Numerical analysis shows that the very weak solution coincides with the weak solution, when the latter exists, but also offers more insights into the behavior of the very weak solution, when the weak solution does not exist.

Minimal representations of the metaplectic Lie supergroup and the super Segal-Bargmann transform (Sam Claerebout, Ghent University)

Abstract:

We construct Schrödinger models and Fock models of minimal representations of the metaplectic Lie supergroup Mp(2m | 2n, 2n). In the non-super case, the minimal representations of Mp(2m) give rise to the metaplectic (Segal-Shale-Weil) representation. Similarly, our Schrödinger model of the minimal representation leads to an already known Schrödinger model of the metaplectic representation of Mp(2m | 2n, 2n). Therefore, our Fock model of the minimal representation allows us to construct a Fock model of this metaplectic representation. We also construct an intertwining super Segal-Bargmann transform which extends the classical Segal-Bargmann transform.

Laplace transform method for hypercomplex Fourier transforms (Yang Ze, Ghent University)

Abstract:

In this talk we will introduce a modified Laplace transform method and its applications on some hypercomplex Fourier transforms. For the kernel of the radially deformed Fourier transform, we obtain the Laplace domain expressions of the kernel for the cases of m=2 and m > 2 for special values of the deformation parameter c. We will show that the expressions can be simplified using the Poisson kernel and the generating function of the Gegenbauer polynomials.  Then the inverse formulas are used to get the integral expressions of the kernel in terms of Mittag-Leffler functions. For the kernels of the Clifford-Fourier transform and the fractional Clifford-Fourier transform, some new integral expressions in terms of Bessel functions of the first kind and the Prabhakar functions will be given by adapting the method.

Paley-Wiener Type Theorems associated to Dirac Operators of Riesz-Feller type (Swanhild Bernstein, TU Bergakademie Freiberg)

Abstract:

Abstract: In my talk, I will present a recent paper with Nelson Faustino on Paley-Wiener-type theorems in the context of hypercomplex variables. We introduce and study the so-called \textit{generalized Bernstein spaces} $\mathcal{SB}^p_R(\D_\theta^\alpha)$ endowed by the fractional Dirac operator $\D_\theta^\alpha$ -- a space-fractional operator of order $\alpha$ and skewness $\theta$, encompassing the Dirac operator $\displaystyle \D$.

We will show that such family of function spaces seamlessly characterizes the interplay between Clifford-valued $L^p-$functions satisfying the support condition $\mbox{supp}\widehat{\f}\subseteq B(0,R)$, and the solutions of the Cauchy problems endowed by the space-time operator $\partial_{x_0}+\D_\theta^\alpha$ that are of exponential type $R^\alpha$.

Such construction allows us to generalize, in a meaningful way, the results obtained by Kou and Qian (2002) and Franklin, Hogan and Larkin (2017). Noteworthy, the exploitation of the well-known Kolmogorov-Stein inequalities to hypercomplex variables permits us to make the computation of the maximal radius $R$ for which $\mbox{supp}\widehat{\f}$ is compactly supported in $B(0,R)$ rather explicit.

On the relations of angular momentum algebras (Marcelo De Martino, Ghent University)

Abstract:

The angular momentum algebra (AMA) occurs as the centralizer algebra of the harmonic sl(2)-triple contaning the Laplace operator. Similarly for its square-root, the total angular momentum algebra (TAMA) occurs as the centralizer of the osp(1,2) algebra that contains the Dirac operator. For both these algebras there is a natural homomorphism from the universal enveloping algebra of the orthogonal Lie algebra onto a suitable subalgebra of the AMA or the TAMA. In this talk, I will fully describe the kernel of this homomorphism (following Feigin-Hakobyan), in the case of the AMA, and explain a conjectural description of the kernel in the case of the TAMA. This last part is an ongoing joint work with K. Calvert and R. Oste.