Seminars from until

Biological, physical, and ecological systems offer a lot of complexity that should be well understood before valuable interventions can be made. We will address both complex and extreme measurements from these systems. There is a necessity to classify appropriate learning mechanisms and define transfer functions and statistics. A natural question may arise: how to address extreme parts of data? How to define boundaries of the datasets and what can be the effects on statistical properties of estimated structures (e.g. uniqueness of copulas? Can we provide efficient estimators of extremes? For closed physical systems, all can be well integrated into both natural and technical sciences, which gives us an optimal instrument for the decomposition of data into stochastic, deterministic, and chaotic part. In particular, we will introduce SPOCU transfer function and provide some of its unique properties for processing of complex data, statistical learning will be discussed, and tuning of parameters of SPOCU-based neural networks will be explained. During the talk, I will acknowledge the contributions of the Portuguese Extreme group and outline some relations to t-Hill-based estimators. The t-Hill approach will be introduced from a robustness perspective, mentioning and interconnecting with articles, among others. Attractive applications to biological systems, such as mass balance in the ecosystem of glaciers in Patagonia, or methane emissions from wetlands will be addressed.

Quantum Signal Processing (QSP) is an algorithmic process by which one represents a function $f$ on the unit interval as the upper left entry of a product of $SU(2)$ matrices parametrized by the input variable $x ∈ [0,1]$ and some “phase factors” $\{ψ_k\}_{k ≥ 0}$ depending on $f$. We show that, after a change of variables, QSP is actually the $SU(2)$-valued nonlinear Fourier transform, and the phase factors correspond to the nonlinear Fourier coefficients. By exploiting a nonlinear Plancherel identity and using some basic spectral theory, we extend QSP to represent any function $f$ satisfying a mild $\log$ integrability condition.

In this talk the derivation and dynamics of some 1D models for the propagation of internal waves are reviewed. From the starting point of the corresponding Euler equations and under certain physical hypotheses, Boussinesq-type systems are derived. Then a numerical analysis of the models, based on the approximation with spectral methods and efficient time integrators, is developed. This will be finally used to study, by computational means, some issues of their dynamics, mainly focused on the solitary wave solutions.

We consider a generalization of the classical perimeter, called nonlocal bi-axial discrete perimeter, where not only the external boundary of a polyomino $\mathcal{P}$ contributes to the perimeter, but all internal and external components of $\mathcal{P}$.

Formally, the nonlocal perimeter $Per_{\lambda}(\mathcal{P})$ of the polyomino $\mathcal{P}$ with parameter $\lambda>1$ is defined as:

$$ Per_{\lambda}(\mathcal{P}):=\sum_{x \in \mathbb{Z}^2 \cap \mathcal{P}, \, y \in \mathbb{Z}^2 \cap \mathcal{P}^c} \frac{1}{d^{\lambda}(x,y)} $$

where $d^{\lambda}(x,y)$ is the fractional bi-axial function defined by the relation:

$$ \frac{1}{d^{\lambda}(x,y)} := \frac{1}{|x_2-y_2|^\lambda}\textbf{1}_{\{ x_1=y_1, \, x_2 \neq y_2\}} + \frac{1}{|x_1-y_1|^{\lambda}} \textbf{1}_{\{ x_2=y_2, \, x_1 \neq y_1\}} $$

with $x=(x_1,x_2)$, $y=(y_1,y_2)$ and $\mathcal{P}^c=\mathbb{R}^2 \setminus \mathcal{P}$.

We tackle the nonlocal discrete isoperimetric problem analyzing and characterizing the minimizers within the class of polyominoes with a fixed area $n$.

The solution of this isoperimetric problem provides a foundation for rigorously investigating the metastable behavior of the long-range bi-axial Ising model.