# 2021 – 22

Speaker: **Joan Torregrosa**

Affiliation: Universitat Autònoma de Barcelona

Title: Oscil·lacions de la funció de període per a centres al pla

Date: 24th of May at 12.00pm

Place: Aula 6 Anselm Turmeda

Summary: Per a camps polinomials definits al pla, la finitud del nombre d’oscil·lacions en funció del grau és un problema obert. Assumint la finitud, i en el cas de famílies de codimensió més alta (centres reversibles i hamiltonians), té sentit plantejar-se quin podria ser aquest nombre. Donarem, per a camps vectorials definits per polinomis de grau petit, quins són els resultats coneguts i els recents avenços que s’han fet, tant per a grau petit, com per a qualsevol grau.

Speaker: **María Jesús Álvarez**

Affiliation: Universitat de les Illes Balears

Title: Uniqueness of limit cycles ofr complex differential equations with two monomials.

Date: 8th of April at 12.30pm

Place: Zoom Webminar

Summary: We prove that the family of complex differential equations

with two monomials, ż = az^k z̄^l + bz^m z̄^n, with k, l, m, n non negative

integers and a, b ∈ C, has at most one limit cycle. Moreover, we char-

acterize when it exists and prove that it is hyperbolic. For this family

we also solve the center-focus problem.

Speaker: **Antoni Buades**

Affiliation: Universitat de les Illes Balears

Title: **Joint denoising and HDR for RAW sequences**

Date: 15th of March at 12.30pm

Place: Zoom Webminar

Summary: We propose a patch-based method for the simultaneous denoising and fusion of a sequence of RAW multi-exposed images. A spatio-temporal criterion is used to select similar patches along the sequence, and a weighted principal component analysis permits to both denoise and fuse the multi exposed data simultaneously. The overall strategy permits to denoise and fuse the set of images without the need of recovering each denoised image in the multi-exposure set, leading to a very efficient procedure. Several experiments show that the proposed method permits to obtain state-of-the-art fusion results with real RAW data.

Speaker:** Onofre Martorell**

Affiliation: Universitat de les Illes Balears

Title: ** Variational Temporal Optical Flow for Multi-exposure Video**

Date: Wednesday 12th January from 11:30 to 12:30.

Place: Zoom Webminar

Summary: High Dynamic Range (HDR) reconstruction for multi-exposure video sequences is a very challenging task. One of its main steps is registration of input frames. We propose a novel variational model for optical flow estimation for multi-exposure video sequences. We introduce data terms for consecutive and non consecutive frames, the latter term comparing frames with the same exposure. We also compute forward and backward flow terms for the current frame, naturally introducing temporal regularization. We present a particular formulation with sequences with two exposures, that can be extended to larger number of exposures. We compare the proposed method with state of the art variational models.

Speaker:** Jordi Penalva**

Affiliation: Universitat de les Illes Balears

Title: ** Slow passage through a Hopf-like bifurcation in piecewise linear systems**

Date: Wednesday 22nd December from 11:30 to 12:30.

Place: Zoom Webminar

Speaker: **Cristina Giossi**

Affiliation: Universitat de les Illes Balears

Title: **The Pause-then-Stop Neural Model. How do we make decisions and interrupt ongoing actions**

Date: Wednesday 17th November from 11:30 to 12:30.

Place: Zoom Webminar

# 2020 – 21

Speaker: ** Antonio E Teruel **

Affiliation: Universitat de les Illes Balears

Title: ** Transition and maduration of canard cycles**

Date: Wednesday 10th Feruary from 15:00 to 16:00.

Place: Zoom Webminar

Summary: In smooth slow-fast systems of Van der Pol type $\dot{x}=y-f(x),\, \dot{y}=\varepsilon(x-a)$, with $f(x)$ a cubic shaped curve, it is stablished the existence of a one parameter family of limit cycles starting at a supercritical Hopf bifurcation and ending at a relaxation oscillation. Moreover, along this family the limit cycles are organized in Hopf type limit cycles, canard limit cycles (both headless and with head) and relaxation oscillations, being the range of the parameter where the canard cycles do exist very narrow, what is called the canard explosion. The existence of this family is reported in many works, both theoretical and applied ones, nevertheless, up to our knowledge, the analysis of the transition between the different limit cycles is not reported anywhere, even when the transition between headless canard cycles and canard cycles with head is important in applications since it defines the excitability threshold of the system.

By using a PWL caricature of the Van der Pol system where previously it has been proved the existence of such a one-parameter-family of limit cycles, in this

work we address the analysis of both the transition from headless canard cycles to canard cycles with head and from canard cycles with head to relaxation

oscillations. In the first transition, we prove that the canard cycle acting as a boundary between headless canards and canards with head is different from the

maximal canard cycle and from the canard cycle with maximal period.

Speaker: Alberto Pérez-Cervera

Affiliation: Higher School of Economics – Moscow – Russia.

Title: **Phase Amplitude reduction of Oscillatory dynamics.**

Date: Wednesday 16th December from 15:00 to 16:00.

Place: BBCollaborate webminar.

Summary: In this talk we aim to introduce a methodology providing an accurate description of oscillatory dynamics. From the mathematical perspective, oscillations correspond to attracting limit cycles in the phase space whose dynamics can be described by a single variable: the phase. For attracting limit cycles, the orbit of each point on the basin of attraction approaches asymptotically the orbit of a point on the limit cycle. Therefore, we can assign to each point on the basin of attraction of the cycle an asymptotic phase corresponding to the phase of the point on the cycle whose orbit approaches to. By doing this, we can define the isochrons of a cycle as the sets of points having the same asymptotic phase. As we will show during the talk, the isochron distribution allows for a geometrical understanding of the phase shifts caused by external perturbations. This understanding is completed by means of the Phase Response Curves.

As we will describe during the talk, although the usage of the phase variable is quite extended, it has some limitations. To overcome these limitations, an alternative strategy which has gained a lot of interest in the recent years, consists in introducing extra variables, denoted as amplitude variables, which describe the dynamics along the directions transversal to the limit cycle. The addition of these variables allows to describe the effects of the perturbations acting during the transient states in which trajectories have not relaxed yet to the limit cycle.

In this talk we will introduce these phase-amplitude tools providing an accurate geometrical and dynamical description of oscillatory dynamics.

This is a joint work with my PhD advisors: Gemma Huguet and Tere M.Seara from Universitat Politècnnica de Catalunya.

Speaker: **Iván Pereira **

Affiliation; Universitat de les Illes Balears

Title: ** Compleció de vídeo a partir de la distribució espaciotemporal de patches. **

Date: Monday 26th October, from 12:30h to 13:30h.

Place: BBCollaborate webminar.

Summary: En aquest treball es pretén tractar el problema de compleció d’imatges i vídeo. La compleció d’imatges i vídeo és un problema que pretén reconstruir una regió desconeguda d’una imatge o vídeo, amb la intenció que quedi visualment plausible. Aquest problema esdevé tant de voler restaurar imatges que s’han degradat com d’editar objectes que s’estenen al llarg d’un vídeo. Exposarem l’extensió a vídeo que hem desenvolupat a partir d’un mètode per a la compleció d’imatges. A partir de la distribució dels \textit{patches} dels fotogrames d’un vídeo extraurem un conjunt de desplaçaments. Els desplaçaments faran que els píxels de la regió desconeguda adoptin la intensitat d’altres píxels del vídeo. Llavors, definirem una funció d’etiquetatge que assignarà a cada píxel el seu desplaçament. Finalment, construirem una energia de manera que el mínim s’assolirà en la funció d’etiquetatge que ens permetrà reconstruir la imatge.

# 2019 – 20

Speaker: **M.J. Álvarez **

Affiliation; Universitat de les Illes Balears

Title: ** Alien limit cycles in Abel equations **

Date: Thursday 12th March, from 13:30h to 14:30h.

Place: Room 1B, Antoni Maria Alcover Sureda.

Summary: The aim of this paper is to study the existence of limit cycles for a family of generalized Abel equations $x’=A(t)x^m+B(t)x^n$, $m,n\ge2$.

Under certain assumptions, it is proved that there exists a non-trivial limit cycle. This limit cycle has the characteristic that it arises from neither a Hopf bifurcation nor a perturbation of periodic orbits in a period annulus around the center at the origin.

Speaker: **Julia Navarro**

Affiliation: Universitat de les Illes Balears.

Title: ** Multi-view imaging: depth estimation and enhancement**

Date: Monday 18th November, from 14:30h to 15:30h

Place:Room 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: Multi-view imaging is the process of using multiple cameras to capture several pictures from the scene. In this thesis, we have studied the problems of depth estimation and spatio-angular super-resolution given several images from the same scene. First, we study the combination of local and global approaches to deal with the two-view stereo problem. This approach is then extended to the multi-view case. In particular, for depth recovery given a light field (LF) image. We exploit the LF configuration and design a method to robustly combine two-view estimations. Then, we study the super-resolution problem for the multi-view setting in both spatial and angular dimensions. The spatial super-resolution is applied to videos, LF and depth videos. The proposed approach comprises inter-frame registration, upsampling and deconvolution. Finally, a learning-based method is tailored for light field angular super-resolution. We propose three sequential CNNs for feature extraction, geometry estimation and view selection. The method, which is initially designed for plenoptic LF, is also adapted and tested on wide-baseline LF.

# 2018 – 19

Speaker: **Rafel Prohens**

Affiliation: Universitat de les Illes Balears.

Title: **Cyclicity of 2-saddle conection in a discontinuous piecewise linear differential systems**

Date: Monday 25th March 2019, from 15:00h to 16:00h

Place:Room 3B, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: This work is concerned with the study of the number of limit cycles perturbing from two-polycycles in discontinuous piecewise linear differential systems with three regions of linearity (DPWL). In particular, we consider DPWL systems having a piecewise algebraic first integral. In this family we select a system exhibiting a connection between two saddles and we perturb it inside the same family. The analysis of the cyclicity is performed via the Poincar\'{e} return map, which is accessible through the first integral. From breaking the connection, at first order of perturbation in ε on the 3-zones center, can bifurcate either 5 limit cycles or 3 algebraic limit cycles, crossing the 3-zones. This is a joint work with A.E. Teruel (UIB) and J. Torregrosa (UAB).

Speaker: **Julia Navarro**

Affiliation: Universitat de les Illes Balears.

Title: **Deep new view synthesis for light field images**

Date: Wednesday 16th January 2019, from 14:00h to 15:00h

Place:Room 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: In this work we present a novel learning-based approach to synthesize new views of a light field image. In particular, given the four corner views of a light field, the presented method estimates the center image. We use three sequential convolutional neural networks for feature extraction, scene geometry and view selection. Compared to state-of-the-art approaches, in order to handle occlusions we propose to estimate a different disparity map per view, leading to proper reconstructions near object boundaries. Ablation studies and comparison against the state of the art on Lytro light fields show the superior performance of the proposed method. Furthermore, the method is tested on light fields with wide baselines acquired with a camera array and, in spite of having to deal with large occluded areas, the proposed approach yields very promising results.

Speaker: **Onofre Martorell**

Affiliation: Universitat de les Illes Balears.

Title: **DCT based multi exposure image fusion**

Date: Wednesday 13th February 2019, from 14:00h to 15:00h

Place:Room 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: We propose a novel algorithm for multi-exposure fusion (MEF). This algorithm decomposes image patches with the DCT transform. Coefficients from patches with different exposure are combined. The luminance and chrominance of the different images are fused separately. Details of the fused image are finally enhanced as a post-processing. Experiments with several data sets show that the proposed algorithm performs better than state-of-the-art.

Speaker: **Antonio E. Teruel**

Affiliation: Universitat de les Illes Balears.

Title: **Canard phenomena in slow-fast piecewise linear systems **

Date: Wednesday 12th December, 14:00h — 15:00h

Place:Room 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: Canard phenomenon consists in a sudden growing of the size of a periodic orbit which occurs in slow-fast systems. Even when this phenomena was observed in piecewise linear (PWL) slow-fast systems at the beginning of the 1990s, it was not analyzed from the singular perturbation theory point of view. By using a singular perturbation approach, in this talk we analyze the existence and stability of canard solutions in a class of planar PWL slow-fast systems with a N-shaped critical manifold. Moreover, we show that two canard orbits can coexist, and annihilate one each other in a saddle-node bifurcation.

Speaker: **Jamila Mifdal**

Affiliation: Université de Bretagne Sud.

Title: **A variational formulation for hyperspectral and multispectral image fusion **

Date: Wednesday 7th November, 14:00h — 15:00h

Place:Room 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: Due to technical constraints, satellite sensors cannot provide images with high spectral and spatial resolution at the same time but they can acquire either a hyperspectral image (HS) or a multispectral one (MS). In this work we present a variational model for HS and MS image fusion. The novelty is the non-local regularization term that makes the fusion robust to additive noise. Moreover, we introduce a geometry constraint that forces the fused and the MS images to share high modulated frequencies.

# 2017 – 18

Speaker: **Jordi Penalva**

Affiliation: Universitat de les Illes Balears (UIB), Math. Dept.

Title: **Takens’ Embedding Theorem **

Date: Thursday, June 28th, 2018, 10:00 – 11:00

Place:Room 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: Takens’ Embedding Theorem is a very extended result into time-series analyses and it is commonly used into a lot of branches, such as psychology or economies, where it is natural the usage of time-series. It allows to recover the attractor from the dynamical system. From the attractor, we can find some basic properties from the system, such as the dimension or the fractal dimension.

In this talk, I will explain the meaning of the Theorem, next I will give a sketch of the proof and finally we will see how to obtain the dynamical system from the signals. We will use some classic examples: the circumference, the torus and the Lorenz system.

Speaker: **José Luis Lisani**

Affiliation: Universitat de les Illes Balears (UIB), Math. Dept.

Title: **Adaptive local image enhancement based on logarithmic mappings**

Date: Thursday, June 14th, 2018, 11:30 – 12:30

Place:Room 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: In this talk, a new local image enhancement technique is presented. It is based on a logarithmic mapping function that is adapted to the luminance characteristics of the neighborhood of each pixel. The method permits to simultaneously increase the luminance in dark regions and decrease it in bright regions. The result is an image with an overall improved contrast.

Speaker: **Adrian C. Murza**

Affiliation: University of Texas

Title: **Integrability of dynamical systems: overview of main techniques with emphasis on the Darboux Theory of Integrability
**

Date: Wednesday, June 6th, 2018, 12:45

Place: Room ATA10, Math Dep. Edifici Anselm Turmeda at UIB

Summary: Proving the existence (or nonexistence) of first integrals is of great interest in the study of dynamical systems. Their existence simplifies notably the analysis of the system. However, their detection and derivation of explicit expression is often difficult, in spite of the different existing techniques. The first part of the talk is a survey on the main techniques of integrability for polynomial vector fields on $R^n$ and a discussion about the interplay between symmetry and integrability. Then we provide new results about application of the Darboux Theory of Integrability on systems of type Lotka-Volterra and on polynomial vector fields defined on $S^n$.

Speaker: **Antonio E Teruel**

Affiliation: Universitat de les Illes Balears (UIB), Math. Dept.

Title: **Saddle-node of limit cycles in PWL systems and applications**

Date: Thursday, May 24th, 2018, 11:30 – 12:30

Place:Room 3A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: In this talk, we prove the existence of a saddle-node bifurcation of limit cycles in continuous piecewise linear systems with three zones. This bifurcation, sometimes called Hopf-like-saddle-node, arises from the perturbation of a non-generic situation, where there exists a linear center in the middle zone. We also obtain an approximation of the relation between the parameters of the system, such that the saddle-node bifurcation takes place, as well as of the period and amplitude of the non-hyperbolic limit cycle that bifurcates. We apply the obtained results to a version of the McKean model, which is a piecewise linear version of the FitzHugh-Nagumo neuron model of spike generation.

Speaker: **Rafel J. Prohens
**

Affiliation: Universitat de les Illes Balears (UIB), Math. Dept.

Title: **New lower bounds for the Hilbert numbers using reversible centers**

Date: Monday, January 15th, 2017, 14:00 – 15:00

Place: Aula 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: In this seminar we present the work (preprint), of the same title, of the authors R. Prohens and J. Torregrosa (Universitat Autònoma de Barcelona). In this work we provide the best lower bounds, that are known up to now, for the Hilbert number of a polynomial vector field of degree N, H(N), for small values of N. These limit cycles appear bifurcating from new symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric about it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles writes in the form 2n+m, for some values of n and m. The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, H(4)≥28, H(5)≥37, H(6)≥53, H(7)≥74, H(8)≥96, H(9)≥120, and H(10)≥142..

# 2016 – 17

Speaker: **Peter De Maesschalck**

Affiliation: Hasselt University, Campus Diepenbeek, Agoralaan, gebouw D, 3590 Diepenbeek, BELGICA

Title: **Normal forms of saddle connections in the plane**

Date: Wednesday, June 21th, 2017, 11:30 – 12:30

Place: Anselm Turmeda, Sala de Reunions

Summary: When studying limit cycles in the plane near singular points the local behaviour around the singular point becomes important, and for degenerate singular points a lot of open questions remain. In order to deal with the questions we propose to find the most simple local expressions at those singular points by blowing up first, identifying saddle connections close to the blow up locus and extending normal form theory to a nonlocal theory near these saddle connections.

Speaker: ** Antonio E Teruel **

Affiliation: University of Balearic Islands

Title: ** Global phase portrait of a 3D integrable flow **

Date: Wednesday, May 24th, 2017, 15:00 – 16:00

Place: Anselm Turmeda, A128

Summary: In many cases, the knowledge about the integrablility of a flow is synonymous with knowing its behavior. However, there is still an long way to travel for going from integrability to the global description of the phase portrait. Although the necessary tools are elementary, this trip is not always easy to do. In this work, and starting from a system of three-dimensional and integrable differential equations, a description of its overall phase portrait is presented.

Speaker: ** Joan Duran **

Affiliation: University of Balearic Islands

Title: ** Nonlocal Regularizing Constraints in Variational Optical Flow **

Date: Wednesday, March 22th, 2017, 15:00 – 16:00

Place: Anselm Turmeda, A128

Summary: Optical flow methods try to estimate a dense correspondence field describing the motion of the objects between an image pair. We introduce novel nonlocal regularizing constraints for variational optical flow computation. While the use of similarity weights has been restricted to the regularization term so far, the proposed data terms permit to implicitly use the image geometry in order to regularize the flow and better locate motion discontinuities. We show some preliminar results comparing the classical brightness constancy assumption with the proposed data constraints.

Speaker: ** Julia Navarro **

Affiliation: University of Balearic Islands

Title: ** Cost adaptive window for local stereo matching **

Date: Tuesday, February 7th, 2017, 14:00 to 15:00

Place: Mateu Orfila, Room A16

Summary: The goal of stereovision is to estimate the depth of the scene from at least two images taken from different viewpoints. For an epipolar rectified image pair, the problem is reduced to estimate the displacement (disparity) between both views. Local stereo methods estimate disparity by means of a block-matching approach. These methods assume that disparity is constant within the matching window. We present a novel stereo block-matching algorithm which uses adaptive windows. The shape of the window is selected to minimize the matching cost. Such a window might be the less distorted by the disparity function and thus the optimal one for matching. Moreover, we use a coarse-to-fine strategy to limit the number of ambiguous matches and reduce the computational cost. The proposed approach performs as state of the art local matching methods.

Speaker: ** Antonio E. Teruel **

Affiliation: University of Balearic Islands

Title: ** Electrophysiology of Neurons: the Hodgkin-Huxley model **

Date: Tuesday, January 10th, 2017, 14:00 to 15:00

Place: Mateu Orfila, Room A16

Summary: In this introductory talk we will introduce step by step the well known neural model of the giant squid due to Hodgkin and Huxley. The action potential exhibited by the model will be simulated by using numerical tools.

Speaker: ** Bartomeu Coll **

Affiliation: University of Balearic Islands

Title: ** An introduction to optimal transport applied to image fusion **

Date: Tuesday, December 13th, 2016, 14:00 to 15:00.

Place: Mateu Orfila, Room A16

Summary: In this talk, we shall give an introduction to the optimal transport theory which defines a natural and useful geometry to compare measures supported on metric probability spaces as the Wasserstein distance. Based on the optimal transport theory, we propose an image fusion application for the hyperspectral and multispectral images by merging the spectral information of the hyperspectral image with the relevant spatial one contained in the multispectral image. The diffusion process consists in minimizing the sum of the weighted Wasserstein distances.

Speaker: ** Catalina Vich Llompart.**

Affiliation: University of Balearic Islands.

Title: **Inverse methods to estimate synaptic conductances – Two compartment neurons**

Date: Tuesday, November 15th, 2016, 14:00 to 15:00.

Place: Seminari del grup de Matemàtica Aplicada.

Summary: Un dels problemes de neuro-ciència en que s’està treballant avui en dia és el problema d’estimació de conductàncies, important per a entendre la connectivitat cerebral. Les tècniques d’estimació han anat millorant els darrers anys per a fer estimacions en zones de poca activitat neuronal. Ara bé, tots aquests mètodes suposen que la neurona és un únic punt enlloc de tota una estructura morfològica. En aquest seminari es vol donar a conèixer un problema obert d’estimació de conductàncies, el qual involucre models neuronals que combinen equacions diferencials ordinàries i parcials, a més de possible renou.

# 2015-16

Speaker: **Dr. Ana Belén Petro Balaguer.**

Affiliation: University of Balearic Islands.

Title: **The online Retinex benchmark for contrast improvement of Earth observation images.**

Date: Thursday, May 12th, 2016, 13:00 to 14:00.

Place: 3A at the ground floor of Antoni Maria Alcover i Sureda building.

Summary: We present the results of the ranking demo for the evaluation of 13 contrast enhancements of three sorts (retinex, tone mapping, gradient based methods) that were implemented, analyzed and compared on line on Pleiades images by experts at CNES and two universities.This led to the selection of four algorithms, all performing a local enhancement, three of them belonging to the multiscale retinex family. In a second inquiry, we reviewed 9 algorithms that have been proposed in the recent rush toward High dynamic Range. After a careful analysis of these algorithms and of their implementations, we have decided to include five of them in a new online benchmark.

Our conclusion is that a new benchmark must be immediately performed to classify our four preceding winners and match them to our five new ones. We organize this benchmark following the considerations: 1) to make all users rate the same pieces of images; 2) to divide the evaluation into two marks, as the experts will be invited to rate separately the overall contrast, but also to pay a special attention to the loss of contrast in bright areas, resulting in an apparent saturation; 3) to completely divide the evaluation of the grey level process from the evaluation of the color process.

Speaker: **Cati Vich.**

Affiliation: University of Balearic Islands.

Title:** A strategy for non-linear estimation of synaptic conductances**

Date: Thursday, Match 3th, 2016, 13:00 to 14:00.

Place: 3A at the ground floor of Antoni Maria Alcover i Sureda building.

Summary Estimating the synaptic conductances impinging on a single neuron directly from a single trace of its membrane potential is an important problem in order to understand the flow of information in the brain. Despite the existence of some methods, they all present the inconvenience that the estimation is done in subthreshold activity regime assuming the absence of subthreshold ionic currents.

Recently we have proved how these methods cannot estimated correctly the conductances under the presence of subthreshold activated currents; presenting a deterministic model, which provides significant improvements when results are compared with the ones obtained by the usual linear model.

However, experimental recordings are noisy, and stochastic models are called for. In this work we extend the procedure to estimate synaptic conductances based on a quadratization of a stochastic model: a stochastic version of the Quadratic Integrate-and-Fire model. Given the membrane potential trace, we estimate the time course of the model coefficients by approximate maximum likelihood, which allows to estimate the time-varying excitatory and inhibitory conductances. This new procedure has been applied both to simulated data from computational models and using real data, obtaining good estimates in the first case, and reasonable results in the real data.

Speaker: **Dr. María Jesús Álvarez Torres.**

Affiliation: University of Balearic Islands.

Title: **Centers and limit cycles for a family of Abel equations**.

Date: Thursday, February 25th, 2016, 13:00 to 14:00.

Place: 3A at the ground floor of Antoni Maria Alcover i Sureda building.

Summary: We consider the class of generalized Abel equations of the form $x’=(a_1A_1(t)+a_2A_2(t))x^m+(a_3A_3(t)+a_4A_4(t))x^n$, where $n,m>1$ are natural numbers, $a_1,a_2,a_3,a_4\in\mathbb{R}$ and $A_1,A_2,A_3,A_4$ are trigonometric monomials. In this talk we give a brief summary of the known results concerning the existence and non-existence of limit cycles of the previous differential equation, paying special attention to the new theorems, that apply to the case for which $A_1,A_3$ have the same signs as $\sin t$, and $A_2,A_4$ the same signs as $\cos t$. More concretely, these new results prove that the center variety for the previous equation (i.e. the set of values $a_1,a_2,a_3,a_4$ such that the Abel equation has a center) has codimension one or two. Moreover, it is one if and only if $A_1=A_3$ and $A_2=A_4$ and it is two if and only if the family has non-trivial limit cycles (different from $x(t)\equiv 0$) for some values of the parameters.

Speaker: **Dr. Catalina Sbert Juan**

Affiliation: University of Balearic Islands

Title: **A white-patch model for color cast detection in natural images.
**

Date: Friday, November 20th, 2015, 12:30 to 13:30

Place: 2B at the ground floor of Antoni Maria Alcover i Sureda building

Summary: We introduce the image formation model and discuss in a mathematical framework the classical hypotheses for reflectance and illuminant estimation. We show the different role that reflectance and illuminance play for distinguish between dominant color images and color cast images. Then we discuss some methods for color cast detection in natural images.

Speaker: **Dr. Antoni Buades Capó**

Affiliation: University of Balearic Islands

Title: **Video denoising and applications to satellite multi-image restoration**

Date: Friday, October 30th, 2015, 12:30 to 13:30

Place: 2B at the ground floor of Antoni Maria Alcover i Sureda building

Summary: A novel image sequence denoising algorithm is presented. The proposed approach takes advantage of self similarity and redundancy of adjacent frames. The algorithm is inspired by fusion algorithms and, as the number of frames increases, it tends to a pure temporal average. The use of motion compensation by regularized optical flow methods permits robust patch comparison in a spatiotemporal volume. The use of Principal Component Analysis ensures the correct preservation of fine texture and details. An extensive comparison with state of the art methods illustrates the superior performance of the proposed approach, with improved texture and detail reconstruction. The proposed algorithm is applied for the restoration of series of satellite images acquired nearly simultaneously, including compression and aliasing artefact removal.

# 2014-15

Speaker: **Michael Möller.**

Affiliation: Technical University of Munich

Title: **Nonlinear multiscale methods for image and signal analysis**

Date: Thursday, July 16, 2015

Place: AT-D108

Summary: Classical strategies for image and signal analysis apply an orthonormal linear transform, filter the resulting coefficients, and apply the inverse transform. If the filtering itself is also linear, one typically interprets the whole process with the help of the singular value decomposition of the linear operator. This talk discusses the possibility to introduce similar concepts for nonlinear transformations arising from the solution of variational, scale space, or inverse scale space methods. We introduce the notion of generalized frequency and wavelength representations as well as filterings of the corresponding coefficients. First numerical results demonstrate interesting properties of the resulting image decomposition.

Speaker: **Antoni Manel Ferragut**

Affiliation: Universitat Jaume I.

Títol: **Aproximació algebraica de separatrius de sella d’un sistema de Lotka-Volterra**

Date: Tuesday, July 14, 2015

Place: AT-D108

Summary: Given a differential system of Lotka-Volterra type with three positive parameters, consider the families that have a saddle in the first quadrant. Once characterized these families, we focus on the cases where the separatrix joining the origin with the saddle is algebraic, and characterize these subfamilies.

For the remainder subfamilies (where the separatrix not algebraic) we give lower and upper bounds of the separatrix by algebraic functions. From them we can calculate the probability that, as an initial condition in the first quadrant, extinguished species or the other.

Speaker:** Edoardo Provenzi**

Title: **Second order stationarity and tensor product spatiochromatic features of natural images**

Date: Thursday, Juny 4, 2015

Place: AT-D108

Summary: In the seminar, it will be shown that two simple assumptions on the covariance matrices ofcolor images yield eigenvectors made by the Kronecker product of Fourier features times the triadgiven by luminance plus color opponent channels. The first of these assumptions is second orderstationarity while the second one is commutativity between color correlation matrices. The validity ofthese assumptions and the predicted shape of the PCA components of color images are experimentally observed on two large image databases. As a by-product of this experimental study, we will provide novel data to support an exponential decay law of the spatiochromatic covariancebetween pairs of pixels as a function of their spatial distance.

Speaker:** Joan Torregrosa**

Affiliation: Department of Mathematics, Universitat de les Illes Balears

Title: **Center, weak-focus and cyclicity problems for planar systems**

Date: Thursday, May 14, 2015

Place: AT-D108

Summary: The center-focus problem consists in distinguishing whether a monodromic singular point is a center or a focus. For singular points with imaginary eigen-

values, usually called nondegenerate singular points, this problem was already

solved by Poincar ́ and Lyapunov, see [3]. The solution consists in computing several quantities called commonly the Poincar ́–Lyapunov constants, and study whether they are zero or not.

Despite the existence of many methods, the solution of the center-focus problem for simple families, like for instance the complete cubic systems or the quartic systems with homogeneous nonlinearities, has resisted all the attempts. For this reason, we propose to push on this question in another direction. We study this problem for a natural family of differential systems with few free parameters but arbitrary degree. We consider planar systems with a linear center at the origin that in complex coordinates the nonlinearity terms are formed by the sum of few monomials. For some families in this class, we study the center problem, the maximum order of a weak-focus and the cyclicity problem. Several

centers inside this family are done. The list includes a new class of Darboux centers that are also persistent centers. We study if the given list is exhaustive or not.

For small degrees we provide explicit systems with weak foci or high-order centers that, after perturbation, give new lower bounds for the number of limit cycles surrounding a single critical point. These lower bounds are higher than the corresponding Hilbert number known until now for these degrees. The talk will be a review of the results [1,2].

*References*

1. A. Gasull, J. Gin ́, and J. Torregrosa. Center problem for systems with two monomial nonlinearities. Preprint. 2014.

2. H. Liang and J. Torregrosa. Some new results on the order and the cyclicity of weak focus of planar polynomial system. Preprint. 2015.

3. A. M. Lyapunov. The general problem of the stability of motion. Taylor & Francis, Ltd., London, 1992. Translated from Edouard Davaux’s French translation (1907) of the 1892 Russian original and edited by A. T. Fuller. Reprint of Internat. J. Control 55 (1992), no. 3, 521–790.

Speaker: **Antonio E. Teruel**

**Folded singularities in piecewise linear systems**

Speaker: **M.J. Álvarez**

Title: **Period Function of degenerate systems**

Date: Thursday, April 16, 2015

Place: AT-D108

Summary: In this work we study the period function of a system that is the sum of two quasi-homogeneous hamiltonian vector fields, none of them being linear. We prove that, when the period annulus is global, the period function is monotone.

We also give a result in the homogeneous case when the period annulus is nonglobal and we get that,

for an unbounded region of the parameters (quasi-degrees of the vector fields) the maximum number of critical periods is one, and this bound is sharp. We also pay a particular attention to the case of a quasi-homogeneous vector field.

Speaker: **Toni Buades **

Title**: Self-similarity and spectral correlation adaptive algorithm for color demosaicking**

Date: Thursday, March 12, 2015

Place: AT-D108

Summary:** **Most common cameras use a CCD sensor device measuring a single color per pixel. The other two color values of each pixel must be interpolated from the neighboring pixels in the so-called demosaicking process. State-of-the-art demosaicking algorithms take advantage of inter-channel correlation locally selecting the best interpolation direction. These methods give impressive results except when local geometry cannot be inferred from neighboring pixels or channel correlation is low. In these cases, they create interpolation artifacts.

We introduce a new algorithm involving non-local image self-similarity in order to reduce interpolation artifacts when local geometry is ambiguous. The proposed algorithm introduces a clear and intuitive manner of balancing how much channel-correlation must be taken advantage of.

Comparison shows that the proposed algorithm gives state-of-the-art methods in several image bases.

Speaker: **Rafel Prohens**

Title: **Asymptotic behaviours in difference equations having equilibrium parabolic points**

Date: Thursday, February 26, 2015

Place: AT-D108

Summary: Some classes of non-linear difference equations present a parabolic equilibrium point and a natural question is to find the asymptotic behaviour of the solutions tending to it. Roughly speaking, we are mainly interested in to face some questions concerning known asymptotic behaviours appearing in the literature such as the problem to bound (upper and lower) the solutions tending to the equilibrium point. More concretely, we explain why these bounds are guessed. We prove why these guesses are used as natural bounds for the solutions, by proving that they are not only bounds (upper and lower) but the natural bounds of the difference equation itself. These bounds are obtained when looking for the asymptotic behaviour of the solutions in the invariant manifold of the parabolic point. Even more, for some families of difference equations we prove that this invariant manifold is, actually, the graph of a Lipschitz function, i.e. a one dimensional variety in R^n. Since this is a work in progress, we aim to address the more interesting question that is to prove that the solutions bounding the referred invariant manifold are, not only bounds but that they (asymptotically) are the solution itself.

Speaker: **Joan Duran Grimalt**

Title:** A Novel Framework for Vectorial Total Variation and Applications to Image De-noising**

**
**Date: Monday, January 19, 2015

Place: AT-D108Summary: Even after over twenty years of research, the total variation (Tv) introduced by Rudin, Osher and Fatemi remains one of the most popular regularization criteria for image restoration. It has proved to be quite efficient for regularizing images without over-smoothing the boundaries of the objects. By observing that the discrete gradient of a colour image can be considered as a 3D matrix with dimensions corresponding to the pixels, the (local or non-local) derivatives and the colour channels, we propose several vector-valued TV regularizations that penalize the “mixed matrix norms” of the data structure, that is, $\ell^{p,q,r}$ or $(S^p,\ell^q)$ norms.

In this talk, we will discuss the implementation of different local and non-local colour TV regularizations by using the primal-dual hybrid gradient method. We will first analyse the de-noising properties of each mixed matrix norm for suppressing colour artefacts while preserving image features and aligning edges. We will describe then in detail how to compute the proximity operators – a generalization of the projection mapping – of the proposed norms. Finally, a performance comparison between several mixed matrix TV methods on image de-noising will be provided.

Speaker: **Marko Miladinovic**

Title:

Date: Monday, November 3, 2014

Place: AT-D108

Summary: Two topics are going to be covered by the following talk: **Numerical optimization and Computations of generalized inverses of matrices.**

abstract: In optimization gradient methods and quasi-Newton methods play very important role. We will see how original gradient method can be improved by introducing some acceleration parameters. Also two point step size methods will be consider as quasi-Newton methods. The improvement of the original BB method given in terms of scalar correction will be present.

Generalized inverses of matrices are very important tool in mathematics and can be applied on different fields such as: linear statistical models, optimization, singular linear difference equations, probability theory and statistics, financial and insurance mathematics. Successive Matrix Squaring (SMS) algorithm for computing generalized inverses of Toeplitz matrices will be explained. Also, some interesting representations of {2,4} and {2,3}-inverses will be present. Additionally, some of the gradient methods will be apply in order to determine some generalized inverses or to solve the system of linear equations.

Speaker: **Antonio E. Teruel **

Title:** MMOs induced by PWL canards.**

Date: Monday, October 27, 2014

Place: AT-D124

Summary: Mixed mode oscillations (MMOs) are periodic behaviours which are characterized by a sequence of small amplitude oscillations followed by large amplitude oscillations. In the literature, slow-fast differential system-based models are proposed. In this talk we present minimal models based on piecewise linear slow-fast systems. We will show that these systems reproduce qualitative and quantitatively the MMOs exhibited by the differential systems.

# 2013-14

- Speaker:
**Xavier Jarque**

Affiliation: Department of Mathematics, Universitat de Barcelona

Title: **Sobre la connectivitat del conjut de Julia per funcions polinomials, racionals i transcendents**

Date: Thursday, March 27, 2014

Place: Room ATA09

Time: 11:00

Summary : The talk will be about the connectivity of the Julia set, that is the region of the complex plane in which the family of iterates of a holomorphic function is not a normal family. I will present first the polynomial case (the dichotomy of the quadratic family) and some other results for hyperbolic rational maps. Another interesting approach follows Shishikura’s Theorem which state that if the Julia set of a rational map is disconnected then the map has at least two weakly repelling fixed points. In the last part of the seminar I will present some work done by the Barcelona’s research group in holomorphic dynamics about the connectivity of the Julia set for transcendental meromorphic maps. In particular I will state the extension of Shishikura’s Theorem for this case which is a joint work with K. Baranski, N. Fagella and B. Karpinska involving the solution of an old conjecture on the existence of absorbing domains for Baker domains.

- Speaker:
**Rafel Prohens**

Affiliation: Department of Mathematics, Universitat de les Illes Balears

Title: **Alien limit cycles in Liénard equations**

Date: Tuesday, January 21, 2014

Place: Room ATA09

Time: 14:00

Summary : In this work we prove the existence of alien limit cycles in the polynomial Liénard differential equations by constructing an example of a family of polynomial differential equations Liénard exhibiting a limit cycle limit cycle of type alien. This limit cycle is perturbed from a 2-saddle cycle in the boundary of an annulus of

periodic orbits given by a Hamiltonian vector field. The Hamiltonian represents a truncated pendulum of degree $4$. In comparison to a former polynomial example, not only the equations are simpler but a lot of tedious calculations can be avoided, making the example also interesting with respect to simplicity in treatment.

- Speaker:
**Antoni Guillamon**

Affiliation: Department of Applied Mathematics, University (Catalonia)

Title: **A survey on estimation of synaptic conductance**

Date: Wednesday, October 23, 2013

Place: Room ATA09

Time: 13:30

Summary : Brain connectivity is Currently one of the hottest topics in neuroscience . Among all sub Emerging from it , the estimation of conductance from intracellular recordings , Intended to Obtain information about the local synaptic inputs to neuron That receiver , you have a special relevance . Despite of the existing literature , no satisfactory methods have yet BEEN GIVEN , the main obstacle BEING That it is an inverse problem analysis and sound data it requires to have a good Supporting behind in order to estimate model parameters . Associated to Different computational modelling strategies have BEEN proposed alternatives , but none of Them have reached a consensus . However , They constitute , altogether , a nice compendium of mathematical resources to handle a specific practical problem . In this informal talk we will present and discuss some Attempt That BOTH use statistical methods , tools from dynamical systems , master equations for processes or stochastic filtering methods . We will try to shed light on the drawback of EACH approach ( sometimes of practical nature , sometimes of mathematical nature ) Hoping to Encourage discussion to stimulate fresh opinions on the problem .

- Speaker:
**Paolo Massobrio**

Affiliation: Neuroengineering and Bio-nanoTechnology Lab (NBT), Department of Informatics, Bioengineering, Robotics and System Engineering (DIBRIS), University of Genova (Italy)

Title: **Playing with living and in silico neuronal assemblies: the neuroengineering approach**

Date: Wednesday, October 16, 2013

Place: Room ATA04B

Time: 12:30

Summary : In this seminar, I will first introduce the basic concepts of neuronal networks both from a physiological and computational point of view. Issues like how to describe the interactions among neurons (i.e., synaptic transmission), how to estimate, infer and model network connectivity (i.e., functional, effective, morphological) in large-scale networks will be described and discussed.

The second part of the seminar is devoted to present some examples of engineered neuronal networks recreated in vitro over Micro-Electrode Arrays (MEAs) by “playing” on parameters like connectivity, cell density and composition, network organization. By exploiting the MEA technology it is possible to record extracellularly the activity of neuronal assemblies from several electrodes, creating a neuro-electronic interface. Experimental and in silico results regarding the interplay between network connectivity and dynamics, as well as the interactions between heterogeneous neuronal populations (cortical-thalamic circuits) will be presented and discussed. Finally, possible perspectives for the development of hybrid networks where living neuronal assemblies are dynamically interfaced with in silico ones will be discussed as possible proof of concepts for the design of innovative neuroprostheses.

Keywords: Neuronal networks modelling, in vitro neuronal networks, network dynamics, interacting assemblies, neuroengineering.