Combinatorial species provide a unified framework to study families of combinatorial objects. If the family of combinatorial objects has natural operations to merge and break structures, the corresponding species becomes a Hopf monoid. In this talk, we present a novel definition of type B Hopf monoids. In the same spirit as the work of Bergeron and Choquette, we consider structures over finite sets with a fixed-point free involution. However, instead of defining a monoidal structure on the category of type B species, our construction involves an action of the monoidal category of (standard) species on the category of Type B species. We present the basic definitions and some examples of Type B Hopf monoids constructed from (type B) set compositions, (symplectic) matroids, and (type B) generalized permutahedra. This is work in progress with M. Aguiar.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: José Bastidas (LACIM/Université du Québec à Montréal, Canadá) - titulo: Chromatic symmetric functions for Dyck paths and \(q\)-rook theory abstract: | - Given a graph and a set of colors, a coloring is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to \(\mathbb{Z}^+\). In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as \(q\)-analogues. - In the particular case of Dyck paths, Stanley and Stembridge described the connection between chromatic symmetric functions of abelian Dyck paths and square hit numbers, and Guay-Paquet described their relation to rectangular hit numbers. Recently, Abreu-Nigro generalized the former connection for the Shareshian-Wachs \(q\)-analogue, and in unpublished work, Guay-Paquet generalized the latter. - In this talk, I want to give an overview of the framework and present another proof of Guay-Paquet's identity using \(q\)-rook theory. Along the way, we will also discuss \(q\)-hit numbers, two variants of their statistic, and some deletion-contraction relations. This is recent work with Alejandro H. Morales and Greta Panova. +Given a graph and a set of colors, a coloring is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to \(\mathbb{Z}^+\). In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as \(q\)-analogues.
+In the particular case of Dyck paths, Stanley and Stembridge described the connection between chromatic symmetric functions of abelian Dyck paths and square hit numbers, and Guay-Paquet described their relation to rectangular hit numbers. Recently, Abreu-Nigro generalized the former connection for the Shareshian-Wachs \(q\)-analogue, and in unpublished work, Guay-Paquet generalized the latter.
+In this talk, I want to give an overview of the framework and present another proof of Guay-Paquet's identity using \(q\)-rook theory. Along the way, we will also discuss \(q\)-hit numbers, two variants of their statistic, and some deletion-contraction relations. This is recent work with Alejandro H. Morales and Greta Panova.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Laura Colmenarejo (North Carolina State University, Estados Unidos) - titulo: Peak algebra for combinatorial Hopf algebras - abstract: The peak algebra is originally introduced by Stembridge using enriched \(P\)-partitions. Using the character theory by Aguiar-Bergeron-Sottile, the peak algebra is also the image of \(\Theta\), the universal morphism between certain combinatorial Hopf algebras. We introduce a shuffle basis of quasi-symmetric functions that has a shuffle-like Hopf structure and is also the eigenfunctions of \(\Theta\). Using this new basis, we extend the notion of peak algebras and theta maps to shuffle, tensor and symmetric algebras. As examples, we study the peak algebras of symmetric functions in non-commuting variables and the graded associated Hopf algebra on permutations. + abstract:The peak algebra is originally introduced by Stembridge using enriched \(P\)-partitions. Using the character theory by Aguiar-Bergeron-Sottile, the peak algebra is also the image of \(\Theta\), the universal morphism between certain combinatorial Hopf algebras. We introduce a shuffle basis of quasi-symmetric functions that has a shuffle-like Hopf structure and is also the eigenfunctions of \(\Theta\). Using this new basis, we extend the notion of peak algebras and theta maps to shuffle, tensor and symmetric algebras. As examples, we study the peak algebras of symmetric functions in non-commuting variables and the graded associated Hopf algebra on permutations.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Shu Xiao Li (Dalian University of Technology, China) - titulo: A multiset partition algebra - abstract: Classical Howe duality provides a representation theoretic framework for classical invariant theory. In the classical Howe duality the general linear group \(GL_n\) is dual to \(GL_k\) when acting on the polynomial ring on variables \(x_{ij}\) where \(1\leq i\leq n\) and \(1\leq j \leq k\). In this talk we restrict the action of \(GL_n\) to the group of permutation matrices and show that the Howe dual is an algebra whose basis is indexed by multiset partition algebra. + abstract:Classical Howe duality provides a representation theoretic framework for classical invariant theory. In the classical Howe duality the general linear group \(GL_n\) is dual to \(GL_k\) when acting on the polynomial ring on variables \(x_{ij}\) where \(1\leq i\leq n\) and \(1\leq j \leq k\). In this talk we restrict the action of \(GL_n\) to the group of permutation matrices and show that the Howe dual is an algebra whose basis is indexed by multiset partition algebra.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Rosa C. Orellana (Dartmouth College, Estados Unidos) - titulo: Valuations and the Hopf Monoid of Generalized Permutahedra - abstract: Many combinatorial objects, such as matroids, graphs, and posets, can be realized as generalized permutahedra - a beautiful family of polytopes. This realization respects the natural multiplication of these objects as well as natural "breaking" operations. Surprisingly many of the important invariants of these objects, when viewed as functions on polytopes, satisfy an inclusion-exclusion formula with respect to subdivisions. Functions that satisfy this formula are known as valuations. In this talk, I will discuss recent work with Federico Ardila that completely describes the relationship between the algebraic structure on generalized permutahedra and valuations. Our main contribution is a new easy-to-apply method that converts simple valuations into more complicated ones. We also describe an universality property of the Hopf monoid of indicator functions of generalized permutahedra that extends the relationship between Combinatorial Hopf algebras and QSYM. + abstract:Many combinatorial objects, such as matroids, graphs, and posets, can be realized as generalized permutahedra - a beautiful family of polytopes. This realization respects the natural multiplication of these objects as well as natural "breaking" operations. Surprisingly many of the important invariants of these objects, when viewed as functions on polytopes, satisfy an inclusion-exclusion formula with respect to subdivisions. Functions that satisfy this formula are known as valuations. In this talk, I will discuss recent work with Federico Ardila that completely describes the relationship between the algebraic structure on generalized permutahedra and valuations. Our main contribution is a new easy-to-apply method that converts simple valuations into more complicated ones. We also describe an universality property of the Hopf monoid of indicator functions of generalized permutahedra that extends the relationship between Combinatorial Hopf algebras and QSYM.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Mario Sanchez (UC Berkeley, Estados Unidos) - titulo: Computation of Kronecker coefficients abstract: | - A Kronecker coefficient is a non-negative integer that depends on three partitions of a natural number n. It is the multiplicity of an irreducible representation of the symmetric group of degree n in the tensor (or Kronecker) product of two other irreducible representations of the same group. - The study of ways of computing Kronecker coefficients is an important topic on algebraic combinatorics. It was initiated by Francis Murnaghan more than eighty years ago. Several tools have been used to try to understand them, notably from representation theory, symmetric functions theory and Borel-Weil theory. These numbers generalize the well-known Littlewood-Richardson coefficients, but are still very far to be fully grasped. - It is known that each Kronecker coefficient can be described as an alternating sum of numbers of integer points in convex polytopes. - In this talk we present a new family of polytopes that permit very fast computations on Kronecker coefficients associated to partitions with few parts. This family provides, in particular, insight into some properties of Kronecker coefficients as well as into Murnaghan stability. +A Kronecker coefficient is a non-negative integer that depends on three partitions of a natural number n. It is the multiplicity of an irreducible representation of the symmetric group of degree n in the tensor (or Kronecker) product of two other irreducible representations of the same group.
+The study of ways of computing Kronecker coefficients is an important topic on algebraic combinatorics. It was initiated by Francis Murnaghan more than eighty years ago. Several tools have been used to try to understand them, notably from representation theory, symmetric functions theory and Borel-Weil theory. These numbers generalize the well-known Littlewood-Richardson coefficients, but are still very far to be fully grasped.
+It is known that each Kronecker coefficient can be described as an alternating sum of numbers of integer points in convex polytopes.
+In this talk we present a new family of polytopes that permit very fast computations on Kronecker coefficients associated to partitions with few parts. This family provides, in particular, insight into some properties of Kronecker coefficients as well as into Murnaghan stability.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Ernesto Vallejo (Universidad Nacional Autónoma de México, México) - titulo: The Castelnuovo-Mumford Regularity of Matrix Schubert Varieties - abstract: The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchogt, Ren, Robichaux, and St. Dizier, we noted that the CM-regularity of matrix Schubert varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for these degrees for symmetric Grothendieck polynomials. In this talk, I will present a general degree formula for Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer. + abstract:The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchogt, Ren, Robichaux, and St. Dizier, we noted that the CM-regularity of matrix Schubert varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for these degrees for symmetric Grothendieck polynomials. In this talk, I will present a general degree formula for Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Anna E. Weigand (Massachusetts Institute of Technology, Estados Unidos) - titulo: 'Chromatic symmetric homology for graphs: some new developments' abstract: | - In his study of the four colour problem, Birkhoff showed that the number of ways to colour a graph with k colours is a polynomial chi(k), which he called the chromatic polynomial. Later, Stanley defined the chromatic symmetric function X(x_1, x_2, ... ), which is a multivariable lift of the chromatic polynomial so that when the first k variables are set to 1, it recovers chi(k). This can be further lifted to a homological setting; we can construct a chain complex of graded S_n-modules whose homology has a bigraded Frobenius characteristic that recovers X upon setting q=t=1. - In this talk, we will explain the construction of the homology, discuss some new results regarding the strength of the homology as a graph invariant, and state some surprising conjectures regarding integral symmetric homology for graphs. This is based on joint work with Chandler, Sazdanovic, and Stella. +In his study of the four colour problem, Birkhoff showed that the number of ways to colour a graph with k colours is a polynomial chi(k), which he called the chromatic polynomial. Later, Stanley defined the chromatic symmetric function X(x_1, x_2, ... ), which is a multivariable lift of the chromatic polynomial so that when the first k variables are set to 1, it recovers chi(k). This can be further lifted to a homological setting; we can construct a chain complex of graded S_n-modules whose homology has a bigraded Frobenius characteristic that recovers X upon setting q=t=1.
+In this talk, we will explain the construction of the homology, discuss some new results regarding the strength of the homology as a graph invariant, and state some surprising conjectures regarding integral symmetric homology for graphs. This is based on joint work with Chandler, Sazdanovic, and Stella.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Martha Yip (University of Kentucky, Estados Unidos) @@ -305,44 +305,44 @@ mail: drodrig@dm.uba.ar charlas: - titulo: Differentially private inference via noisy optimization - abstract: We propose a general optimization-based framework for computing differentially private M-estimators and a new method for the construction of differentially private confidence regions. Firstly, we show that robust statistics can be used in conjunction with noisy gradient descent and noisy Newton methods in order to obtain optimal private estimators with global linear or quadratic convergence, respectively. We establish global convergence guarantees, under both local strong convexity and self-concordance, showing that our private estimators converge with high probability to a neighborhood of the non-private M-estimators. The radius of this neighborhood is nearly optimal in the sense it corresponds to the statistical minimax cost of differential privacy up to a logarithmic term. Secondly, we tackle the problem of parametric inference by constructing differentially private estimators of the asymptotic variance of our private M-estimators. This naturally leads to the use of approximate pivotal statistics for the construction of confidence regions and hypothesis testing. We demonstrate the effectiveness of a bias correction that leads to enhanced small-sample empirical performance in simulations. + abstract:We propose a general optimization-based framework for computing differentially private M-estimators and a new method for the construction of differentially private confidence regions. Firstly, we show that robust statistics can be used in conjunction with noisy gradient descent and noisy Newton methods in order to obtain optimal private estimators with global linear or quadratic convergence, respectively. We establish global convergence guarantees, under both local strong convexity and self-concordance, showing that our private estimators converge with high probability to a neighborhood of the non-private M-estimators. The radius of this neighborhood is nearly optimal in the sense it corresponds to the statistical minimax cost of differential privacy up to a logarithmic term. Secondly, we tackle the problem of parametric inference by constructing differentially private estimators of the asymptotic variance of our private M-estimators. This naturally leads to the use of approximate pivotal statistics for the construction of confidence regions and hypothesis testing. We demonstrate the effectiveness of a bias correction that leads to enhanced small-sample empirical performance in simulations.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Marco Avella Medina (Columbia University, Estados Unidos) - titulo: Adaptive regression with Brownian path covariate - abstract: In this talk, we will study how to obtain optimal estimators in problems of non-parametric estimation. More specifically, we will present the Goldenshluger-Lepski (2011) method that allows one to obtain estimators that adapt to the smoothness of the function to be estimated. We will show how to extend this statistical procedure in regression with functional data when the regressor variable is a Wiener process \(W\). Using the Wiener-Ito decomposition of m(W), where \(m\) is the regression function, we will define a family of estimators that satisfy an oracle inequality, are proved to be adaptive and converge at polynomial rates over specific classes of functions. + abstract:In this talk, we will study how to obtain optimal estimators in problems of non-parametric estimation. More specifically, we will present the Goldenshluger-Lepski (2011) method that allows one to obtain estimators that adapt to the smoothness of the function to be estimated. We will show how to extend this statistical procedure in regression with functional data when the regressor variable is a Wiener process \(W\). Using the Wiener-Ito decomposition of m(W), where \(m\) is the regression function, we will define a family of estimators that satisfy an oracle inequality, are proved to be adaptive and converge at polynomial rates over specific classes of functions.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Karine Bertin (Universidad de Valparaíso, Chile) - titulo: 'Adjusting ROC curves for covariates: a robust approach' abstract: | - ROC curves are a popular tool to describe the discriminating power of a binary classifier based on a continuous marker as the threshold is varied. They become an interesting strategy to evaluate how well an assignment rule based on a diagnostic test distinguishes one population from the other. Under certain circumstances the marker's discriminatory ability may be affected by certain covariates. In this situation, it seems sensible to include this information in the ROC analysis. This task can be accomplished either by the induced or the direct method. - In this talk we will focus on ROC curves in presence of covariates. We will show the impact of outliers on the conditional ROC curves and we will introduce a robust proposal. We follow a semiparametric approach where we combine robust parametric estimators with weighted empirical distribution estimators based on an adaptive procedure that downweights outliers. - We will discuss some aspects concerning consistency and through a Monte Carlo study we will compare the performance of the proposed estimators with the classical ones both, in clean and contaminated samples. +ROC curves are a popular tool to describe the discriminating power of a binary classifier based on a continuous marker as the threshold is varied. They become an interesting strategy to evaluate how well an assignment rule based on a diagnostic test distinguishes one population from the other. Under certain circumstances the marker's discriminatory ability may be affected by certain covariates. In this situation, it seems sensible to include this information in the ROC analysis. This task can be accomplished either by the induced or the direct method.
+In this talk we will focus on ROC curves in presence of covariates. We will show the impact of outliers on the conditional ROC curves and we will introduce a robust proposal. We follow a semiparametric approach where we combine robust parametric estimators with weighted empirical distribution estimators based on an adaptive procedure that downweights outliers.
+We will discuss some aspects concerning consistency and through a Monte Carlo study we will compare the performance of the proposed estimators with the classical ones both, in clean and contaminated samples.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Ana M. Bianco (Universidad de Buenos Aires, Argentina) - titulo: Least trimmed squares estimators for functional principal component analysis - abstract: 'Classical functional principal component analysis can yield erroneous approximations in presence of outliers. To reduce the influence of atypical data we propose two methods based on trimming: a multivariate least trimmed squares (LTS) estimator and its coordinatewise variant. The multivariate LTS minimizes the multivariate scale corresponding to \(h-\)subsets of curves while the coordinatewise version uses univariate LTS scale estimators. Consider a general setup in which observations are realizations of a random element on a separable Hilbert space \(\mathcal{H}\). For a fixed dimension \(q\), we aim to robustly estimate the \(q\) dimensional linear space in \(\mathcal{H}\) that gives the best approximation to the functional data. Our estimators use smoothing to first represent irregularly spaced curves in a high-dimensional space and then calculate the LTS solution on these multivariate data. The solution of the multivariate data is subsequently mapped back onto \(\mathcal{H}\). Poorly fitted observations can therefore be flagged as outliers. Simulations and real data applications show that our estimators yield competitive results when compared to existing methods when a minority of observations is contaminated. When a majority of the curves is contaminated at some positions along its trajectory coordinatewise methods like Coordinatewise LTS are preferred over multivariate LTS and other multivariate methods since they break down in this case.' + abstract: 'Classical functional principal component analysis can yield erroneous approximations in presence of outliers. To reduce the influence of atypical data we propose two methods based on trimming: a multivariate least trimmed squares (LTS) estimator and its coordinatewise variant. The multivariate LTS minimizes the multivariate scale corresponding to \(h-\)subsets of curves while the coordinatewise version uses univariate LTS scale estimators. Consider a general setup in which observations are realizations of a random element on a separable Hilbert space \(\mathcal{H}\). For a fixed dimension \(q\), we aim to robustly estimate the \(q\) dimensional linear space in \(\mathcal{H}\) that gives the best approximation to the functional data. Our estimators use smoothing to first represent irregularly spaced curves in a high-dimensional space and then calculate the LTS solution on these multivariate data. The solution of the multivariate data is subsequently mapped back onto \(\mathcal{H}\). Poorly fitted observations can therefore be flagged as outliers. Simulations and real data applications show that our estimators yield competitive results when compared to existing methods when a minority of observations is contaminated. When a majority of the curves is contaminated at some positions along its trajectory coordinatewise methods like Coordinatewise LTS are preferred over multivariate LTS and other multivariate methods since they break down in this case.
' start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Holger Cevallos-Valdiviezo (Escuela Superior Politécnica del Litoral, Ecuador y Ghent University, Bélgica) - titulo: A non-asymptotic analysis of certain high-dimensional estimators for the mean - abstract: Recent work in Statistics and Computer Science has considered the following problem. Given a distribution \(P\) over \({\bf R}^d\) and a fixed sample size \(n\), how well can one estimate the mean \(\mu = \bf E_{X\sim P} X\) from a sample \(X_1,\dots,X_n\stackrel{i.i.d.}{\sim}P\) while only requiring finite second moments and allowing for sample contamination? It turns out that the best estimators are not related to the sample mean. In this talk we present a new analysis of certain approaches to this problem, and reproduce or improve previous results by several authors. + abstract:Recent work in Statistics and Computer Science has considered the following problem. Given a distribution \(P\) over \({\bf R}^d\) and a fixed sample size \(n\), how well can one estimate the mean \(\mu = \bf E_{X\sim P} X\) from a sample \(X_1,\dots,X_n\stackrel{i.i.d.}{\sim}P\) while only requiring finite second moments and allowing for sample contamination? It turns out that the best estimators are not related to the sample mean. In this talk we present a new analysis of certain approaches to this problem, and reproduce or improve previous results by several authors.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Roberto Imbuzeiro Oliveira (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: Stick-breaking priors via dependent length variables - abstract: In this talk, we present new classes of Bayesian nonparametric prior distributions. By allowing length random variables, in stick-breaking constructions, to be exchangeable or Markovian, appealing models for discrete random probability measures appear. As a result, by tuning the stochastic dependence in such length variables allows to recover extreme families of random probability measures, i.e. Dirichlet and Geometric processes. As a byproduct, the ordering of the weights, in the species sampling representation, can be controlled and thus tuned for efficient MCMC implementations in density estimation or unsupervised classification problems. Various theoretical properties and illustrations will be presented. + abstract:In this talk, we present new classes of Bayesian nonparametric prior distributions. By allowing length random variables, in stick-breaking constructions, to be exchangeable or Markovian, appealing models for discrete random probability measures appear. As a result, by tuning the stochastic dependence in such length variables allows to recover extreme families of random probability measures, i.e. Dirichlet and Geometric processes. As a byproduct, the ordering of the weights, in the species sampling representation, can be controlled and thus tuned for efficient MCMC implementations in density estimation or unsupervised classification problems. Various theoretical properties and illustrations will be presented.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Ramsés Mena Chávez (Universidad Nacional Autónoma de México, México) - titulo: 'Modelling in pandemic times: using smart watch data for early detection of COVID-19' abstract: | - The COVID-19 pandemic brought many challengers to statisticians and modelers across all quantitative disciplines. From accelerated clinical trials to modelling of epidemiological interventions at a feverish pace, to the study of the impact of the pandemic in mental health outcomes and racial disparities. In this talk, we would like to share our experience using classical statistical modelling and machine learning to use data obtained from wearable devices as digital biomarkers of COVID-19 infection. - Early in the pandemic, health care workers in the Mount Sinai Health System (New York city) were prospectively followed in an observational study using the custom Warrior Watch Study app, to collect weekly information about stress, symptoms and COVID-19 infection. Participants wore an Apple Watch for the duration of the study, measuring heart rate variability (HRV), a digital biomarker previously associated with infection in other settings, throughout the follow-up period. - The HRV data collected through the Apple Watch was characterized by a circadian pattern, with sparse sampling over a 24-hour period, and non-uniform timing across days and participants. These characteristics preclude us from using easily derived features (ie mean, maximum, CV etc) with Machine learning methods to develop a diagnostic tool. As such, suitable modelling of the non-uniform, sparsely sampled circadian rhythm data derived from wearable devices are an important step to advance the use of integrated wearable data for prediction of health outcomes. - To circumvent such limitations, we introduced the mixed-effects COSINOR model, where the daily circadian rhythm is express as a non-linear function with three rhythm characteristics: the rhythm-adjusted mean (MESOR), half the extent of variation within a cycle (amplitude), and an angle relating to the time at which peak values recur in each cycle (acrophase). The longitudinal changes in the circadian patterns can then be evaluated extending the COSINOR model to a mixed-effect model framework, allowing for random effects and interaction between COSINOR parameters and time-varying covariates. In this talk, we will discuss our model framework, boostrapped-based hypothesis testing and prediction approaches, as well as our evaluation of HRV measures as early biomarkers of COVID-19 diagnosis. To facilitate the future use of the mixed-effect COSINOR model, we implemented in an R package cosinoRmixedeffects. +The COVID-19 pandemic brought many challengers to statisticians and modelers across all quantitative disciplines. From accelerated clinical trials to modelling of epidemiological interventions at a feverish pace, to the study of the impact of the pandemic in mental health outcomes and racial disparities. In this talk, we would like to share our experience using classical statistical modelling and machine learning to use data obtained from wearable devices as digital biomarkers of COVID-19 infection.
+Early in the pandemic, health care workers in the Mount Sinai Health System (New York city) were prospectively followed in an observational study using the custom Warrior Watch Study app, to collect weekly information about stress, symptoms and COVID-19 infection. Participants wore an Apple Watch for the duration of the study, measuring heart rate variability (HRV), a digital biomarker previously associated with infection in other settings, throughout the follow-up period.
+The HRV data collected through the Apple Watch was characterized by a circadian pattern, with sparse sampling over a 24-hour period, and non-uniform timing across days and participants. These characteristics preclude us from using easily derived features (ie mean, maximum, CV etc) with Machine learning methods to develop a diagnostic tool. As such, suitable modelling of the non-uniform, sparsely sampled circadian rhythm data derived from wearable devices are an important step to advance the use of integrated wearable data for prediction of health outcomes.
+To circumvent such limitations, we introduced the mixed-effects COSINOR model, where the daily circadian rhythm is express as a non-linear function with three rhythm characteristics: the rhythm-adjusted mean (MESOR), half the extent of variation within a cycle (amplitude), and an angle relating to the time at which peak values recur in each cycle (acrophase). The longitudinal changes in the circadian patterns can then be evaluated extending the COSINOR model to a mixed-effect model framework, allowing for random effects and interaction between COSINOR parameters and time-varying covariates. In this talk, we will discuss our model framework, boostrapped-based hypothesis testing and prediction approaches, as well as our evaluation of HRV measures as early biomarkers of COVID-19 diagnosis. To facilitate the future use of the mixed-effect COSINOR model, we implemented in an R package cosinoRmixedeffects.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Mayte Suarez-Farinas (Icahn School of Medicine at Mount Sinai, Estados Unidos) @@ -410,51 +410,51 @@ charlas: - titulo: Regularity criteria for weak solutions to the three-dimensional MHD system abstract: | - In this talk we will first review various known regularity criteria and partial regularity theory for 3D incompressible Navier-Stokes equations. I will then present two generalizations of partial regularity theory of Caffarelli, Kohn and Nirenberg to the weak solutions of MHD equations. The first one is based on the framework of parabolic Morrey spaces. We will show parabolic Hölder regularity for the "suitable weak solutions" to the MHD system in small neighborhoods. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory and makes it possible to weaken the hypotheses on the pressure. The second one is a regularity result relying on the notion of "dissipative solutions". By making use of the first result, we will show the regularity of the dissipative solutions to the MHD system with a weaker hypothesis on the pressure. - This is a joint work with Diego Chamorro (Université Paris-Saclay, site Evry). +In this talk we will first review various known regularity criteria and partial regularity theory for 3D incompressible Navier-Stokes equations. I will then present two generalizations of partial regularity theory of Caffarelli, Kohn and Nirenberg to the weak solutions of MHD equations. The first one is based on the framework of parabolic Morrey spaces. We will show parabolic Hölder regularity for the "suitable weak solutions" to the MHD system in small neighborhoods. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory and makes it possible to weaken the hypotheses on the pressure. The second one is a regularity result relying on the notion of "dissipative solutions". By making use of the first result, we will show the regularity of the dissipative solutions to the MHD system with a weaker hypothesis on the pressure.
+This is a joint work with Diego Chamorro (Université Paris-Saclay, site Evry).
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Jiao He (Université Paris-Saclay, Francia) - titulo: On the infinite energy weak solutions for the MHD equations abstract: | - In this talk, we will consider the magneto-hydrodynamics (MHD) equations placed on the whole three-dimensional space. These equations write down as a nonlinear coupled system of two Navier-Stokes type equations, where the unknowns are the velocity field, the magnetic field and the pressure term. In the first part of this talk, within the framework of a kind of weighted L^2 spaces, we expose some new energy controls which allow us to prove the existence of global in time weak solutions. These solutions are also called the infinite energy weak solutions, in contrast with the classical theory of finite energy weak solutions in the L^2 space. The uniqueness of both finite energy and infinite energy weak solutions remains a very challenging open question. Thus, in the second part of this talk, we study a a priori condition, known as weak-strong uniqueness criterion, to ensure the uniqueness of the infinite energy weak solutions. This result is given in a fairly general multipliers type space, which contains some well-known functional spaces previously used to prove some weak-strong uniqueness criteria. - This is a joint work with Pedro Fernandez-Dalgo (Université Paris-Saclay). +In this talk, we will consider the magneto-hydrodynamics (MHD) equations placed on the whole three-dimensional space. These equations write down as a nonlinear coupled system of two Navier-Stokes type equations, where the unknowns are the velocity field, the magnetic field and the pressure term. In the first part of this talk, within the framework of a kind of weighted L^2 spaces, we expose some new energy controls which allow us to prove the existence of global in time weak solutions. These solutions are also called the infinite energy weak solutions, in contrast with the classical theory of finite energy weak solutions in the L^2 space. The uniqueness of both finite energy and infinite energy weak solutions remains a very challenging open question. Thus, in the second part of this talk, we study a a priori condition, known as weak-strong uniqueness criterion, to ensure the uniqueness of the infinite energy weak solutions. This result is given in a fairly general multipliers type space, which contains some well-known functional spaces previously used to prove some weak-strong uniqueness criteria.
+This is a joint work with Pedro Fernandez-Dalgo (Université Paris-Saclay).
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Oscar Jarrín (Universidad de las Américas, Ecuador) - titulo: Long-time asymptotics for a damped Navier-Stokes-Bardina model - abstract: We consider finite energy solutions for a damped Navier-Stokes-Bardina’s model and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension. Furthermore, we examine the long-term behavior of solutions of the damped Navier-Stokes-Bardina’s equation in the energy space. + abstract:We consider finite energy solutions for a damped Navier-Stokes-Bardina’s model and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension. Furthermore, we examine the long-term behavior of solutions of the damped Navier-Stokes-Bardina’s equation in the energy space.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Fernando Cortez (Escuela Politécnica Nacional, Ecuador) - titulo: Compact embeddings of p-Sobolev-like cones of nuclear operators - abstract: We prove that a cone of nuclear operators, whose eigenfunctions belong to a p-Sobolev space and that have finite total energy, is compactly embedded in the trace norm. This result is analogous to the classical Sobolev embeddings but at operators level. In the path we prove regularity properties for the density function of any operator living in the cone. Also, departing from Lieb-Thirring type conditions, we obtain some Gagliardo-Nirenberg inequalities. By using the compactness property, several free-energy functionals for operators are shown to have a minimizer. The entropy term of these free-energy functionals is generated by a Casimir-class function related to the eigenvalue problem of the Schrödinger operator. + abstract:We prove that a cone of nuclear operators, whose eigenfunctions belong to a p-Sobolev space and that have finite total energy, is compactly embedded in the trace norm. This result is analogous to the classical Sobolev embeddings but at operators level. In the path we prove regularity properties for the density function of any operator living in the cone. Also, departing from Lieb-Thirring type conditions, we obtain some Gagliardo-Nirenberg inequalities. By using the compactness property, several free-energy functionals for operators are shown to have a minimizer. The entropy term of these free-energy functionals is generated by a Casimir-class function related to the eigenvalue problem of the Schrödinger operator.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Juan Mayorga (Yachay Tech, Ecuador) - titulo: Scattering for quadratic-type Schrödinger systems in dimension five without mass-resonance - abstract: In this talk we will discuss the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our approach is based on the recent ideas introduced by Dodson and Murphy, which relies on an interaction Morawetz estimate. + abstract:In this talk we will discuss the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our approach is based on the recent ideas introduced by Dodson and Murphy, which relies on an interaction Morawetz estimate.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Ademir Pastor (Universidade Estadual de Campinas, Brasil) - titulo: Stability of smooth periodic traveling waves in the Camassa-Holm equation abstract: | - Smooth periodic travelling waves in the Camassa–Holm (CH) equation are revisited in this talk. We show that these periodic waves can be characterized in two different ways by using two different Hamiltonian structures. The standard formulation, common to the Korteweg–de Vries (KdV) equation, has several disadvantages, e.g., the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We explore the nonstandard formulation common to evolution equations of CH type and prove that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region of three parameters where the smooth periodic waves exist. - This is a joint work with Dmitry E. Pelinovsky (McMaster University), AnnaGeyer (Delft University of Technology) and Renan H. Martins (State University of Maringa). +Smooth periodic travelling waves in the Camassa–Holm (CH) equation are revisited in this talk. We show that these periodic waves can be characterized in two different ways by using two different Hamiltonian structures. The standard formulation, common to the Korteweg–de Vries (KdV) equation, has several disadvantages, e.g., the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We explore the nonstandard formulation common to evolution equations of CH type and prove that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region of three parameters where the smooth periodic waves exist.
+This is a joint work with Dmitry E. Pelinovsky (McMaster University), AnnaGeyer (Delft University of Technology) and Renan H. Martins (State University of Maringa).
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Fábio Natali (Universidade Estadual de Maringá, Brasil) - titulo: Spectral stability of monotone traveling fronts for reaction diffusion-degenerate Nagumo equations - abstract: This talk addresses the spectral stability of monotone traveling front solutions for reaction diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusivities which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). Spectral stability is understood as the property that the spectrum of the linearized operator around the wave, acting on an exponentially weighted space, is contained in the complex half plane with non-positive real part. The degenerate fronts studied in this paper travel with positive speed above a threshold value and connect the(diffusion-degenerate) zero state with the unstable equilibrium point of the reaction function. In this case, the degeneracy of the diffusion coefficient is responsible of the loss of hyperbolicity of the asymptotic coefficient matrices of the spectral problem at one of the end points, precluding the application of standard techniques to locate the essential spectrum. This difficulty is overcome with a suitable partition of the spectrum, a generalized convergence of operators technique, the analysis of singular (or Weyl) sequences and the use of energy estimates. The monotonicity of the fronts, as well as detailed descriptions of the decay structure of eigenfunctions on a case by case basis, are key ingredients to show that all traveling fronts under consideration are spectrally stable in a suitably chosen exponentially weighted L2 energy space. + abstract:This talk addresses the spectral stability of monotone traveling front solutions for reaction diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusivities which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). Spectral stability is understood as the property that the spectrum of the linearized operator around the wave, acting on an exponentially weighted space, is contained in the complex half plane with non-positive real part. The degenerate fronts studied in this paper travel with positive speed above a threshold value and connect the(diffusion-degenerate) zero state with the unstable equilibrium point of the reaction function. In this case, the degeneracy of the diffusion coefficient is responsible of the loss of hyperbolicity of the asymptotic coefficient matrices of the spectral problem at one of the end points, precluding the application of standard techniques to locate the essential spectrum. This difficulty is overcome with a suitable partition of the spectrum, a generalized convergence of operators technique, the analysis of singular (or Weyl) sequences and the use of energy estimates. The monotonicity of the fronts, as well as detailed descriptions of the decay structure of eigenfunctions on a case by case basis, are key ingredients to show that all traveling fronts under consideration are spectrally stable in a suitably chosen exponentially weighted L2 energy space.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Ramón G. Plaza (Universidad Nacional Autónoma de México, México) - titulo: Blow-up solutions of the intercritical inhomogeneous NLS equation abstract: | - We consider the inhomogeneous nonlinear Schrödinger (INLS) equation +We consider the inhomogeneous nonlinear Schrödinger (INLS) equation $$i u_t +\Delta u+|x|^{-b}|u|^{2\sigma} u = 0, \,\,\, x\in \mathbb{R}^N,$$ - with \(N\geq 3\) and \(0 \lt b \lt \min\{\frac{N}{2},2\}\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma}\) satisfies \(0 \lt s_c \lt 1\). In this talk, for initial data in \(\dot H^{s_c}\cap \dot H^1\), we discuss the existence of blow-up solutions and also a lower bound for the blow-up rate in the radial and non-radial settings. - This is a joint work with Mykael Cardoso (Universidade Federal do Piauí, Brasil). + with \(N\geq 3\) and \(0 \lt b \lt \min\{\frac{N}{2},2\}\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma}\) satisfies \(0 \lt s_c \lt 1\). In this talk, for initial data in \(\dot H^{s_c}\cap \dot H^1\), we discuss the existence of blow-up solutions and also a lower bound for the blow-up rate in the radial and non-radial settings.
+This is a joint work with Mykael Cardoso (Universidade Federal do Piauí, Brasil).
start: 2021-09-17T16:45 end: 2021-09-16T17:30 speaker: Luiz Gustavo Farah (Universidade Federal de Minas Gerais, Brasil). @@ -466,14 +466,14 @@ end: 2021-09-17T16:30 speaker: Michal Kowalczyk (Universidad de Chile, Chile) - titulo: Korteweg de-Vries limit for the Fermi-Pasta-Ulam System - abstract: In this talk, we are going to discuss dispersive properties for the Fermi-Pasta-Ulam (FPU) system with infinitely many oscillators. Precisely, we see that FPU systems are reformulated and their solutions satisfies Strichartz, local smoothing, and maximal function estimates in comparison with linear Korteweg–de Vries (KdV) flows. With these properties, we finally show that the infinite FPU system can be approximated by counter-propagating waves governed by the KdV equation as the lattice spacing approaches zero. + abstract:In this talk, we are going to discuss dispersive properties for the Fermi-Pasta-Ulam (FPU) system with infinitely many oscillators. Precisely, we see that FPU systems are reformulated and their solutions satisfies Strichartz, local smoothing, and maximal function estimates in comparison with linear Korteweg–de Vries (KdV) flows. With these properties, we finally show that the infinite FPU system can be approximated by counter-propagating waves governed by the KdV equation as the lattice spacing approaches zero.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Chulkwang Kwak (Ehwa Womans University, Corea del Sur) - titulo: Spectral stability in the nonlinear Dirac equation with Soler type nonlinearity. abstract: | - This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies w in (0,m), where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction. - The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems. +This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies w in (0,m), where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction.
+The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Hanne Van Den Bosch (Universidad de Chile, Chile) @@ -488,10 +488,10 @@ charlas: - titulo: Complete colorings on circulant graphs and digraphs abstract: | - A complete \(k\)-vertex-coloring of a graph \(G\) is a vertex-coloring of \(G\) using \(k\) colors such that for every pair of colors there is at least two incident vertices in \(G\) colored with this pair of colors. The \emph{chromatic} \(\chi(G)\) and \emph{achromatic} \(\alpha(G)\) numbers of \(G\) are the smallest and the largest number of colors in a complete proper \(k\)-vertex-coloring of \(G\), therefore \(\chi(G)\leq\alpha(G)\). - The dichromatic number and the diachromatic number, generalice the concepts of chromatic number and achromatic number. An acyclic \(k\)-vertex-coloring of a digraph \(D\) is vertex coloring using \(k\) colors such that \(D\) has no monochromatic cycles and a \emph{complete} \(k\)-vertex-coloring of a digraph \(D\) is a vertex coloring using \(k\) colors such that for every ordered pair \((i,j)\) of different colors, there is at least one arc \((u,v)\) such that \(u\) has color \(i\) and \(v\) has color \(j\). The dichromatic number \(dc(D)\) and diachromatic number \(dac(D)\) of \(D\) are the smallest and the largest number of colors in a complete proper \(k\)-vertex-coloring of \(D\), therefore \(dc(D)\leq dac(D)\). - We determine the achromatic and diachromatic numbers of some specific circulant graphs and digraphs and give general bounds for these two parameters on these graphs and digraphs. Also, we determine the achromatic index for circulant graphs of order \(q^2+q+1\) using projective planes. - Joint work with: Juan José Montellano-Ballesteros, Mika Olsen and Christian Rubio-Montiel. +A complete \(k\)-vertex-coloring of a graph \(G\) is a vertex-coloring of \(G\) using \(k\) colors such that for every pair of colors there is at least two incident vertices in \(G\) colored with this pair of colors. The \emph{chromatic} \(\chi(G)\) and \emph{achromatic} \(\alpha(G)\) numbers of \(G\) are the smallest and the largest number of colors in a complete proper \(k\)-vertex-coloring of \(G\), therefore \(\chi(G)\leq\alpha(G)\).
+The dichromatic number and the diachromatic number, generalice the concepts of chromatic number and achromatic number. An acyclic \(k\)-vertex-coloring of a digraph \(D\) is vertex coloring using \(k\) colors such that \(D\) has no monochromatic cycles and a \emph{complete} \(k\)-vertex-coloring of a digraph \(D\) is a vertex coloring using \(k\) colors such that for every ordered pair \((i,j)\) of different colors, there is at least one arc \((u,v)\) such that \(u\) has color \(i\) and \(v\) has color \(j\). The dichromatic number \(dc(D)\) and diachromatic number \(dac(D)\) of \(D\) are the smallest and the largest number of colors in a complete proper \(k\)-vertex-coloring of \(D\), therefore \(dc(D)\leq dac(D)\).
+We determine the achromatic and diachromatic numbers of some specific circulant graphs and digraphs and give general bounds for these two parameters on these graphs and digraphs. Also, we determine the achromatic index for circulant graphs of order \(q^2+q+1\) using projective planes.
+Joint work with: Juan José Montellano-Ballesteros, Mika Olsen and Christian Rubio-Montiel.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Gabriela Araujo-Pardo (Universidad Nacional Autónoma de México, México) @@ -505,71 +505,71 @@ speaker: Charles J. Colbourn (Arizona State University, Estados Unidos) - titulo: On the \(\Delta\)-interval and \(\Delta\)-convexity numbers of graphs and graph products abstract: | - Given a graph \(G\) and a set \(S \subseteq V(G)\), the \(\Delta\)-interval of \(S\), \([S]_{\Delta}\), is the set formed by the vertices of \(S\) and every \(w \in V(G)\) forming a triangle with two vertices of \(S\). If \([S]_\Delta = S\), then \(S\) is \(\Delta\)-convex of \(G\); if \([S]_{\Delta} = V(G)\), then \(S\) is a \(\Delta\)-interval set of \(G\). The \(\Delta\)-interval number of \(G\) is the minimum cardinality of a \(\Delta\)-interval set and the \(\Delta\)-convexity number of \(G\) is the maximum cardinality of a proper \(\Delta\)-convex subset of \(V(G)\). In this work, we show that the problem of computing the \(\Delta\)-convexity number is {\sf W}[1]-hard and {\sf NP}-hard to approximate within a factor \(O(n^{1-\varepsilon})\) for any constant \(\varepsilon > 0\) even for graphs with diameter \(2\) and that the problem of computing the \(\Delta\)-interval number is {\sf NP}-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the \(\Delta\)-convexity number of chordal graphs and for computing the \(\Delta\)-interval number of block graphs. We also present results on the \(\Delta\)-hull, \(\Delta\)-interval and \(\Delta\)-convexity numbers concerning the three standard graph products, namely, the Cartesian, strong and lexicographic products, in function of these and well-studied parameters of the operands. - Joint work with: Bijo S. Anand, Prasanth G. Narasimha-Shenoi, and Sabeer S. Ramla. +Given a graph \(G\) and a set \(S \subseteq V(G)\), the \(\Delta\)-interval of \(S\), \([S]_{\Delta}\), is the set formed by the vertices of \(S\) and every \(w \in V(G)\) forming a triangle with two vertices of \(S\). If \([S]_\Delta = S\), then \(S\) is \(\Delta\)-convex of \(G\); if \([S]_{\Delta} = V(G)\), then \(S\) is a \(\Delta\)-interval set of \(G\). The \(\Delta\)-interval number of \(G\) is the minimum cardinality of a \(\Delta\)-interval set and the \(\Delta\)-convexity number of \(G\) is the maximum cardinality of a proper \(\Delta\)-convex subset of \(V(G)\). In this work, we show that the problem of computing the \(\Delta\)-convexity number is {\sf W}[1]-hard and {\sf NP}-hard to approximate within a factor \(O(n^{1-\varepsilon})\) for any constant \(\varepsilon > 0\) even for graphs with diameter \(2\) and that the problem of computing the \(\Delta\)-interval number is {\sf NP}-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the \(\Delta\)-convexity number of chordal graphs and for computing the \(\Delta\)-interval number of block graphs. We also present results on the \(\Delta\)-hull, \(\Delta\)-interval and \(\Delta\)-convexity numbers concerning the three standard graph products, namely, the Cartesian, strong and lexicographic products, in function of these and well-studied parameters of the operands.
+Joint work with: Bijo S. Anand, Prasanth G. Narasimha-Shenoi, and Sabeer S. Ramla.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Mitre Dourado (Universidade Federal do Rio de Janeiro, Brasil) - titulo: Unavoidable patterns - abstract: Ramsey's theorem states that, for any integer \(t\), if \(n\) is a sufficiently large integer, then every \(2\)-edge-coloring of a complete graph \(K_n\) contains a monochromatic \(K_t\). On the other hand, Turán's theorem says that, if a graph has sufficiently many edges, then it contains a \(K_t\) as a subgraph. In relation to these two types of problems, we study which \(2\)-colored patterns are forced to appear in differently saturated \(2\)-edge-colorings of the complete graph \(K_n\) provided \(n\) is large enough. This is a joint work with Yair Caro and Amanda Montejano. + abstract:Ramsey's theorem states that, for any integer \(t\), if \(n\) is a sufficiently large integer, then every \(2\)-edge-coloring of a complete graph \(K_n\) contains a monochromatic \(K_t\). On the other hand, Turán's theorem says that, if a graph has sufficiently many edges, then it contains a \(K_t\) as a subgraph. In relation to these two types of problems, we study which \(2\)-colored patterns are forced to appear in differently saturated \(2\)-edge-colorings of the complete graph \(K_n\) provided \(n\) is large enough. This is a joint work with Yair Caro and Amanda Montejano.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Adriana Hansberg (Universidad Nacional Autónoma de México, México) - titulo: Approximating graph eigenvalues using tree decompositions abstract: | - Spectral graph theory aims to extract structural information about graphs from spectra of different types of matrices associated with them, such as the adjacency matrix, Laplacian matrices, and many others. A basic step in any such application consists of computing or approximating the spectrum or some prescribed subset of eigenvalues. One strategy for this is to compute diagonal matrices that are congruent to the original matrices, which can be done quite efficiently if we have a graph decompositions such as the tree decomposition with bounded width. To be precise, let \(M=(m_{ij})\) be a symmetric matrix of order \(n\) whose elements lie in an arbitrary field \(\mathbb{F}\), and let \(G\) be the graph with vertex set \(\{1,\ldots,n\}\) such that distinct vertices \(i\) and \(j\) are adjacent if and only if \(m_{ij} \neq 0\). We present an algorithm that finds a diagonal matrix that is congruent to \(M\) in time \(O(k^2 n)\), provided that we are given a suitable tree decomposition of \(G\) with width \(k\). In addition to the above applications, this allows one to compute the determinant and to find the rank of a symmetric matrix in time \(O(k^2 n)\). - This is joint work with M. FŸrer (Pennsylvannia State University) and Vilmar Trevisan (Universidade Federal do Rio Grande do Sul). +Spectral graph theory aims to extract structural information about graphs from spectra of different types of matrices associated with them, such as the adjacency matrix, Laplacian matrices, and many others. A basic step in any such application consists of computing or approximating the spectrum or some prescribed subset of eigenvalues. One strategy for this is to compute diagonal matrices that are congruent to the original matrices, which can be done quite efficiently if we have a graph decompositions such as the tree decomposition with bounded width. To be precise, let \(M=(m_{ij})\) be a symmetric matrix of order \(n\) whose elements lie in an arbitrary field \(\mathbb{F}\), and let \(G\) be the graph with vertex set \(\{1,\ldots,n\}\) such that distinct vertices \(i\) and \(j\) are adjacent if and only if \(m_{ij} \neq 0\). We present an algorithm that finds a diagonal matrix that is congruent to \(M\) in time \(O(k^2 n)\), provided that we are given a suitable tree decomposition of \(G\) with width \(k\). In addition to the above applications, this allows one to compute the determinant and to find the rank of a symmetric matrix in time \(O(k^2 n)\).
+This is joint work with M. FŸrer (Pennsylvannia State University) and Vilmar Trevisan (Universidade Federal do Rio Grande do Sul).
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Carlos Hoppen (Universidade Federal do Rio Grande do Sul, Brasil) - titulo: Fault tolerance in cryptography using cover-free families abstract: | - Cover-free families (CFFs) have been investigated under different names and as a solution to many problems related to combinatorial group testing. A \(d\)-cover-free family \(d\)-CFF\((t, n)\) is a set system with n subsets of a \(t\)-set, where the union of any d subsets does not contain any other. A \(d\)-CFF\((t, n)\) allows for the identification of up to \(d\) defective elements in a set of \(n\) elements by performing only \(t\) tests (typically \(t \ll n\)). - We explore different aspects of cover-free families in order to achieve fault tolerance in cryptography. For instance, while CFFs are used as a solution to many problems in this area, we note that some of those problems require CFFs with increasing \(n\). In this context, we investigate new infinite families and their constructions in order to better approach fault tolerance in cryptography. This is joint work with Lucia Moura. - [1] IDALINO, T. B.; MOURA, L., Nested cover-free families for unbounded fault-tolerant aggregate signatures. Theoretical Computer Science, 854 (2021), 116--130. - [2] IDALINO, T. B.; MOURA, L., Embedding cover-free families and cryptographical applications. Advances in Mathematics of Communications, 13 (2019), 629--643. +Cover-free families (CFFs) have been investigated under different names and as a solution to many problems related to combinatorial group testing. A \(d\)-cover-free family \(d\)-CFF\((t, n)\) is a set system with n subsets of a \(t\)-set, where the union of any d subsets does not contain any other. A \(d\)-CFF\((t, n)\) allows for the identification of up to \(d\) defective elements in a set of \(n\) elements by performing only \(t\) tests (typically \(t \ll n\)).
+We explore different aspects of cover-free families in order to achieve fault tolerance in cryptography. For instance, while CFFs are used as a solution to many problems in this area, we note that some of those problems require CFFs with increasing \(n\). In this context, we investigate new infinite families and their constructions in order to better approach fault tolerance in cryptography. This is joint work with Lucia Moura.
+[1] IDALINO, T. B.; MOURA, L., Nested cover-free families for unbounded fault-tolerant aggregate signatures. Theoretical Computer Science, 854 (2021), 116--130.
+ [2] IDALINO, T. B.; MOURA, L., Embedding cover-free families and cryptographical applications. Advances in Mathematics of Communications, 13 (2019), 629--643.
An \((n,k,\lambda)\) perfect sequence covering array is a subset of the \(n!\) permutations of the sequence \((1, 2, \dots, n)\) whose elements collectively contain each ordered \(k\)-subsequence exactly \(\lambda\) times. The central question is: for given \(n\) and \(k\), what is the smallest value of \(\lambda\) (denoted \(g(n,k)\)) for which such a configuration exists? We interpret the sequences of a perfect sequence covering array as elements of the symmetric group \(S_n\), and constrain its structure to be a union of cosets of a prescribed subgroup of \(S_n\). By adapting a search algorithm due to Mathon and van Trung for finding spreads, we obtain highly structured examples of perfect sequence covering arrays. In particular, we determine that \(g(6,3) = g(7,3) = g(7,4) = 2\) and that \(g(8,3)\) is either 2 or 3.
+This is joint work with Jingzhou Na.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Jonathan Jedwab (Simon Fraser University, Canada) - titulo: The status of the \((r, l)\) well-covered graph sandwich problem abstract: | - An \((r, l)\)-partition of a graph \(G\) is a partition of its vertex set into \(r\) independent sets and \(l\) cliques. A graph is \((r,l)\) if it admits an \((r, l)\)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is \((r,l)\)-well-covered if it is both \((r,l)\) and well-covered. We have proved the full classification of the decision problem \((r,l)\)-Well-Covered Graph problem, where we are given a graph \(G\), and the question is whether \(G\) is an \((r, l)\)-well-covered graph. We have shown that this problem is polynomial only in the cases \((0, 1)\), \((0, 2)\), \((1, 0)\), \((1, 1)\), \((1, 2)\), and \((2, 0)\) and hard otherwise. The Sandwich Problem for the Property \(\pi\) has as an instance a pair of graphs \(G1 = (V, E_1)\) and \(G2 = (V, E_2)\) with \(E_1 \subset E_2\), plus the question whether there is a graph \(G=(V,E)\) with \(E_1 \subset E\subset E_2\), such that \(G\) has the property \(\pi\). When a recognition problem for a property \(\pi\) is hard, we can consider the sets \(E_1 = E_2\) to obtain that the sandwich problem for the property \(\pi\) reduces to the recognition and so also hard. Hence, the only cases where the \((r, l)\)-well-covered sandwich problem can be no longer hard are in the \(6\) polynomial cases. In this talk we prove that \((r, l)\)-well-covered sandwich problem is polynomial in the cases that \((r, l) = (0, 1), (1, 0), (1, 1)\) or \((0, 2)\), and NP-complete if we consider the property of being \((1, 2)\)-well-covered. - This work was done with the collaboration of Sancrey Rodrigues Alves (FAETEC), Fernanda Couto (UFRRJ), Luerbio Faria (UERJ), Sylvain Gravier (CNRS), and Uéverton S. Souza (UFF). +An \((r, l)\)-partition of a graph \(G\) is a partition of its vertex set into \(r\) independent sets and \(l\) cliques. A graph is \((r,l)\) if it admits an \((r, l)\)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is \((r,l)\)-well-covered if it is both \((r,l)\) and well-covered. We have proved the full classification of the decision problem \((r,l)\)-Well-Covered Graph problem, where we are given a graph \(G\), and the question is whether \(G\) is an \((r, l)\)-well-covered graph. We have shown that this problem is polynomial only in the cases \((0, 1)\), \((0, 2)\), \((1, 0)\), \((1, 1)\), \((1, 2)\), and \((2, 0)\) and hard otherwise. The Sandwich Problem for the Property \(\pi\) has as an instance a pair of graphs \(G1 = (V, E_1)\) and \(G2 = (V, E_2)\) with \(E_1 \subset E_2\), plus the question whether there is a graph \(G=(V,E)\) with \(E_1 \subset E\subset E_2\), such that \(G\) has the property \(\pi\). When a recognition problem for a property \(\pi\) is hard, we can consider the sets \(E_1 = E_2\) to obtain that the sandwich problem for the property \(\pi\) reduces to the recognition and so also hard. Hence, the only cases where the \((r, l)\)-well-covered sandwich problem can be no longer hard are in the \(6\) polynomial cases. In this talk we prove that \((r, l)\)-well-covered sandwich problem is polynomial in the cases that \((r, l) = (0, 1), (1, 0), (1, 1)\) or \((0, 2)\), and NP-complete if we consider the property of being \((1, 2)\)-well-covered.
+This work was done with the collaboration of Sancrey Rodrigues Alves (FAETEC), Fernanda Couto (UFRRJ), Luerbio Faria (UERJ), Sylvain Gravier (CNRS), and Uéverton S. Souza (UFF).
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Sulamita Klein (Universidade Federal do Rio de Janeiro, Brasil) - titulo: Counting Segment, Rays, Lines in Quasimetric spaces - abstract: In the Euclidean plane, it is a well-known result that a set of n points defines exactly \(n(n-1)/2\) distinct segments, at least \(2(n-1)-1\) different rays and, when the points are not collinear, at least \(n\) different lines. Segments, rays and lines can be define in any quasimetric space by using its quasimetric. In this talk we survey recent results concerning the generalization of the above mentioned properties of the Euclidean plane, specially those of lines, to finite quasimetric spaces, mainly those defined by connected graphs and strongly connected digraphs. + abstract:In the Euclidean plane, it is a well-known result that a set of n points defines exactly \(n(n-1)/2\) distinct segments, at least \(2(n-1)-1\) different rays and, when the points are not collinear, at least \(n\) different lines. Segments, rays and lines can be define in any quasimetric space by using its quasimetric. In this talk we survey recent results concerning the generalization of the above mentioned properties of the Euclidean plane, specially those of lines, to finite quasimetric spaces, mainly those defined by connected graphs and strongly connected digraphs.
start: 2021-09-14T17:30 end: 2021-09-13T18:15 speaker: Martín Matamala (Universidad de Chile, Chile) - titulo: Uniformly Optimally-Reliable Graphs abstract: | - The reliability polynomial of a simple graph G represents the probability that G will remain connected given a fixed probability \(1-p\) of each edge failing. A graph \(G\) is uniformly most reliable if its reliability polynomial is greater than or equal to the reliability polynomial of all other graphs with the same number of nodes and edges for all \(p \in [0, 1]\). - Which is the most-reliable graph with n nodes and m edges? - This problem has several variants, according to the notion of optimality (in a local or uniform sense), failure type (either nodes or edges), or reliability model (all-terminal connectedness, source-terminal or general multi-terminal setting). - This presentation shows a summary of the multiple proposals to attack the problem, together with recent trends and important conjectures proposed decades ago which require further research. - Particularly, we will present recently proved conjectures regarding UMGR (Uniformly Most-Reliable Graphs) as well as new evaluation techniques for this property. - Keywords: uniformly most-reliable graph, uniformly least-reliable graph, all-terminal reliability, source-terminal reliability, failure-type, graph theory. +The reliability polynomial of a simple graph G represents the probability that G will remain connected given a fixed probability \(1-p\) of each edge failing. A graph \(G\) is uniformly most reliable if its reliability polynomial is greater than or equal to the reliability polynomial of all other graphs with the same number of nodes and edges for all \(p \in [0, 1]\).
+Which is the most-reliable graph with n nodes and m edges?
+This problem has several variants, according to the notion of optimality (in a local or uniform sense), failure type (either nodes or edges), or reliability model (all-terminal connectedness, source-terminal or general multi-terminal setting).
+This presentation shows a summary of the multiple proposals to attack the problem, together with recent trends and important conjectures proposed decades ago which require further research.
+Particularly, we will present recently proved conjectures regarding UMGR (Uniformly Most-Reliable Graphs) as well as new evaluation techniques for this property.
+Keywords: uniformly most-reliable graph, uniformly least-reliable graph, all-terminal reliability, source-terminal reliability, failure-type, graph theory.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Franco Robledo (Universidad de la Republica, Uruguay) - titulo: Circularly compatible ones, \(D\)-circularity, and proper circular-arc bigraphs - abstract: In 1969, Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results led to the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as remarked there, this work did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). We solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related \(D\)-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu et al. [J. Graph Theory, 73 (2013), pp. 361--376]. Our findings generalize some known results about \(D\)-interval hypergraphs and proper interval bigraphs. + abstract:In 1969, Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results led to the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as remarked there, this work did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). We solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related \(D\)-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu et al. [J. Graph Theory, 73 (2013), pp. 361--376]. Our findings generalize some known results about \(D\)-interval hypergraphs and proper interval bigraphs.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Martín Safe, (Universidad Nacional del Sur, Argentina) - titulo: Active clustering abstract: | - Given a set (known to us), and a partition of this set, unknown to us, consider the following task. We have to determine the partition by making successive queries about pairs of elements, each time querying whether the pair belong to the same class or not. Later queries are allowed to depend on the outcome of earlier queries. There is a natural way to assign a graph to each step of the querying process, by starting with an edgeless graph where each vertex represents an element of the set, and after queries identifying vertices for a yes-answer and adding edges for no-answers. We prove the at first sight surprising result that the partitioning algorithms reaching the minimal average number of queries are exactly those that keep these graphs chordal. (The average is taken over all possible partitions of the base set.) The optimal algorithms even share the same distribution on the number of queries, which we characterize. We also analyse a related model where the number of partition classes is fixed, and each element of the base set chooses its partition class randomly. - This is joint work with Élie de Panafieu, Quentin Lutz and Alex Scott. +Given a set (known to us), and a partition of this set, unknown to us, consider the following task. We have to determine the partition by making successive queries about pairs of elements, each time querying whether the pair belong to the same class or not. Later queries are allowed to depend on the outcome of earlier queries. There is a natural way to assign a graph to each step of the querying process, by starting with an edgeless graph where each vertex represents an element of the set, and after queries identifying vertices for a yes-answer and adding edges for no-answers. We prove the at first sight surprising result that the partitioning algorithms reaching the minimal average number of queries are exactly those that keep these graphs chordal. (The average is taken over all possible partitions of the base set.) The optimal algorithms even share the same distribution on the number of queries, which we characterize. We also analyse a related model where the number of partition classes is fixed, and each element of the base set chooses its partition class randomly.
+This is joint work with Élie de Panafieu, Quentin Lutz and Alex Scott.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Maya Stein (Universidad de Chile, Chile) @@ -584,54 +584,54 @@ charlas: - titulo: Aproximación unificada del problema acoplado de Stokes-Darcy abstract: | - En esta charla nos abocaremos al análisis y la resolución numérica, por elementos finitos mixtos, del problema acoplado de Stokes-Darcy en el plano. Introduciremos una formulación modificada del problema con el propósito de permitir el uso de la misma familia de elementos, en el sector del dominio gobernado por la ecuación de Stokes y en la porción del dominio gobernada por la ecuación de Darcy. - En primer lugar consideraremos el caso de que el dominio sea poligonal, donde presentaremos resultados tanto teóricos como numéricos de la resolución del problema utilizando MINI-elements, los cuales son uno de los más sencillos de implementar. También, aprovechando las ventajas de nuestra formulación, mostraremos la buena performance de la aplicación de otros elementos como los P2-P1 o P1-P1 estabilizados. - Luego trataremos el problema en dominios curvos, incluso con la posibilidad de tener interfase curva, haciendo uso de triángulos curvos. Con este enfoque obtendremos también estimaciones de error de orden óptimo, extendiendo así los resultados obtenidos para el caso poligonal. Por último presentaremos experimentos numéricos que confirman la buena performance del método propuesto. - Trabajo en colaboración con María Lorena Stockdale. Departamento de Matematica, FCEyN, Universidad de Buenos Aires. +En esta charla nos abocaremos al análisis y la resolución numérica, por elementos finitos mixtos, del problema acoplado de Stokes-Darcy en el plano. Introduciremos una formulación modificada del problema con el propósito de permitir el uso de la misma familia de elementos, en el sector del dominio gobernado por la ecuación de Stokes y en la porción del dominio gobernada por la ecuación de Darcy.
+En primer lugar consideraremos el caso de que el dominio sea poligonal, donde presentaremos resultados tanto teóricos como numéricos de la resolución del problema utilizando MINI-elements, los cuales son uno de los más sencillos de implementar. También, aprovechando las ventajas de nuestra formulación, mostraremos la buena performance de la aplicación de otros elementos como los P2-P1 o P1-P1 estabilizados.
+Luego trataremos el problema en dominios curvos, incluso con la posibilidad de tener interfase curva, haciendo uso de triángulos curvos. Con este enfoque obtendremos también estimaciones de error de orden óptimo, extendiendo así los resultados obtenidos para el caso poligonal. Por último presentaremos experimentos numéricos que confirman la buena performance del método propuesto.
+Trabajo en colaboración con María Lorena Stockdale. Departamento de Matematica, FCEyN, Universidad de Buenos Aires.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Gabriela Armentano (Universidad de Buenos Aires, Argentina) - titulo: 'Transversally-Enriched Pipe Element Method: an efficient approach for computational hemodynamics' - abstract: The development of novel numerical and computational strategies for the simulation of blood flow in complex patient-specific vasculatures has gained strong attention in the scientific community aiming both theoretical and applied research. The need for fast, yet accurate methods is at the core of any translational research enterprise. In this talk, we will introduce a middle fidelity numerical approach to tackle the demanding blood-flow simulation in deformable arteries. This strategy can be placed in-between oversimplified (low-fidelity) reduced 1D models and expensive (high-fidelity) full 3D models. The main feature of the method is the discretization of the pipe-like domain in which the fluid flow problem is addressed. This pipe-oriented discretization is exploited by placing a differentiated polynomial approximation for cross-sectional and longitudinal directions. We will show the basic idea of the method, a numerical study of its convergence properties and applications for patient-specific blood flow modeling in large branching arterial geometries. + abstract:The development of novel numerical and computational strategies for the simulation of blood flow in complex patient-specific vasculatures has gained strong attention in the scientific community aiming both theoretical and applied research. The need for fast, yet accurate methods is at the core of any translational research enterprise. In this talk, we will introduce a middle fidelity numerical approach to tackle the demanding blood-flow simulation in deformable arteries. This strategy can be placed in-between oversimplified (low-fidelity) reduced 1D models and expensive (high-fidelity) full 3D models. The main feature of the method is the discretization of the pipe-like domain in which the fluid flow problem is addressed. This pipe-oriented discretization is exploited by placing a differentiated polynomial approximation for cross-sectional and longitudinal directions. We will show the basic idea of the method, a numerical study of its convergence properties and applications for patient-specific blood flow modeling in large branching arterial geometries.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Pablo Blanco (Laboratório Nacional de Computação Científica, Brasil) joint with Alonso M. Alvarez, Raúl A. Feijóo - titulo: Local error estimates for nonlocal problems abstract: | - The integral fractional Laplacian of order \(s \in (0,1)\) is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions. - In this talk we shall discuss regularity of solutions on bounded Lipschitz domains. For finite element discretizations, we derive local error estimates in the \(H^s\)-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples. - The talk is based on joint work with Dmitriy Leykekhman (University of Connecticut) and Ricardo Nochetto (University of Maryland). +The integral fractional Laplacian of order \(s \in (0,1)\) is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions.
+In this talk we shall discuss regularity of solutions on bounded Lipschitz domains. For finite element discretizations, we derive local error estimates in the \(H^s\)-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.
+The talk is based on joint work with Dmitriy Leykekhman (University of Connecticut) and Ricardo Nochetto (University of Maryland).
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Juan Pablo Borthagaray (Universidad de la República, Uruguay) - titulo: A priori and a posteriori error analysis of a momentum conservative mixed-FEM for the stationary Navier-Stokes problem abstract: | - In this talk we present a new conforming mixed finite element method for the Navier-Stokes problem posed on non-standard Banach spaces, where a pseudostress tensor and the velocity are the main unknowns of the system. The associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the pseudostress and discontinuous piecewise polynomials of degree k for the velocity. Next, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed scheme. - Joint work with S. Caucao (Universidad Católica de la Santísima Concepción, Chile), C. Garcia (Pontificia Universidad Católica de Chile), R. Oyarzúa (Universidad del Bío Bío, Chile) and S. Villa (Universidad del Bío Bío, Chile). +In this talk we present a new conforming mixed finite element method for the Navier-Stokes problem posed on non-standard Banach spaces, where a pseudostress tensor and the velocity are the main unknowns of the system. The associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the pseudostress and discontinuous piecewise polynomials of degree k for the velocity. Next, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed scheme.
+Joint work with S. Caucao (Universidad Católica de la Santísima Concepción, Chile), C. Garcia (Pontificia Universidad Católica de Chile), R. Oyarzúa (Universidad del Bío Bío, Chile) and S. Villa (Universidad del Bío Bío, Chile).
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Jessika Camaño (Universidad Católica de la Santísima Concepción, Concepción, Chile) - titulo: A fully discrete adaptive scheme for parabolic equations abstract: | - We present an adaptive algorithm for solving linear parabolic equations using hierarchical B-splines and the implicit Euler method for the spatial and time discretizations, respectively. - Our development improves upon one from 2018 by Gaspoz and collaborators, where fully discrete adaptive schemes have been analyzed within the framework of classical finite elements. Our approach is based on an a posteriori error estimation that essentially consists of four indicators: a time and a consistency error indicator that dictate the time-step size adaptation, and coarsening and a space error indicator that are used to obtain suitably adapted hierarchical meshes (at different time steps). Even though we use hierarchical B-splines for the space discretization, a straightforward generalization to other methods, such as FEM, is possible. The algorithm is guaranteed to reach the final time within a finite number of operations, and keep the space-time error below a prescribed tolerance. Some numerical tests document the practical performance of the proposed adaptive algorithm. +We present an adaptive algorithm for solving linear parabolic equations using hierarchical B-splines and the implicit Euler method for the spatial and time discretizations, respectively.
+Our development improves upon one from 2018 by Gaspoz and collaborators, where fully discrete adaptive schemes have been analyzed within the framework of classical finite elements. Our approach is based on an a posteriori error estimation that essentially consists of four indicators: a time and a consistency error indicator that dictate the time-step size adaptation, and coarsening and a space error indicator that are used to obtain suitably adapted hierarchical meshes (at different time steps). Even though we use hierarchical B-splines for the space discretization, a straightforward generalization to other methods, such as FEM, is possible. The algorithm is guaranteed to reach the final time within a finite number of operations, and keep the space-time error below a prescribed tolerance. Some numerical tests document the practical performance of the proposed adaptive algorithm.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Eduardo Garau (Universidad Nacional del Litoral, Santa Fe, Argentina) - titulo: LSD for multiscale problems abstract: | - The Localized Spectral Decomposition finite element method is based on a hybrid formulation of elliptic partial differential equations, that is then transformed via a FETI-like space decomposition. Such decomposition make the formulation embarrassingly parallel and efficient, in particular in the presence of multiscale coefficients. It differs from most of the methods out there since it requires solution's minimum regularity. The major novelty is that we base our decomposition in local spectral problems, resulting in methods that are robust with respect to high contrast coefficients. - This is a joint work with Marcus Sarkis, of WPI (Worcester Polytechnic Institute). +The Localized Spectral Decomposition finite element method is based on a hybrid formulation of elliptic partial differential equations, that is then transformed via a FETI-like space decomposition. Such decomposition make the formulation embarrassingly parallel and efficient, in particular in the presence of multiscale coefficients. It differs from most of the methods out there since it requires solution's minimum regularity. The major novelty is that we base our decomposition in local spectral problems, resulting in methods that are robust with respect to high contrast coefficients.
+This is a joint work with Marcus Sarkis, of WPI (Worcester Polytechnic Institute).
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Alexandre Madureira (Laboratório Nacional de Computação Científica, Brasil) - titulo: Virtual Element Spectral Analysis for the Transmission Eigenvalue Problem. - abstract: The aim of this talk is to analyze a C1 Virtual Element Method (VEM) on polygonal meshes for solving a quadratic and non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained. Numerical experiments will be provided to verify the theoretical error estimates. + abstract:The aim of this talk is to analyze a C1 Virtual Element Method (VEM) on polygonal meshes for solving a quadratic and non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained. Numerical experiments will be provided to verify the theoretical error estimates.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: David Mora (Universidad del Bio-Bio, Concepción, Chile) - titulo: Analysis and approximation of fluids under singular forcing - abstract: We present a well-posedness theory for Newtonian and some non-Newtonian fluids under singular forcing in Lipschitz domains and in convex polytopes. The main idea, that allows us to deal with such forces, is that we study the fluid problems in suitably weighted Sobolev spaces. We develop an a priori approximation theory, which requires the development of the stability of the Stokes projection over weighted spaces. We conclude by presenting existence results for a singular Bousinessq system and, in the case that the forcing is a linear combination of Dirac deltas, we also present an a posteriori error estimator. + abstract:We present a well-posedness theory for Newtonian and some non-Newtonian fluids under singular forcing in Lipschitz domains and in convex polytopes. The main idea, that allows us to deal with such forces, is that we study the fluid problems in suitably weighted Sobolev spaces. We develop an a priori approximation theory, which requires the development of the stability of the Stokes projection over weighted spaces. We conclude by presenting existence results for a singular Bousinessq system and, in the case that the forcing is a linear combination of Dirac deltas, we also present an a posteriori error estimator.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Enrique Otárola (Universidad Técnica Federico Santamaría, Valparaíso, Chile) @@ -702,7 +702,7 @@ speaker: María José Benac (Universidad Nacional de Santiago del Estero, Argentina) - titulo: From closures of unitary group actions to groupoid actions abstract: | - The set of all normal operators on a Hilbert space is naturally acted on both by the group of unitary operators and by the groupoid of partial isometries. We show that the norm closures of these group orbits are related to the above groupoid orbits, and this relation allows one to study the differentiable structures of these orbit closures. Some of the earlier results on closed unitary orbits are recovered in the natural degree of generality in our framework. This presentation is based on joint work with Gabriel Larotonda. +The set of all normal operators on a Hilbert space is naturally acted on both by the group of unitary operators and by the groupoid of partial isometries. We show that the norm closures of these group orbits are related to the above groupoid orbits, and this relation allows one to study the differentiable structures of these orbit closures. Some of the earlier results on closed unitary orbits are recovered in the natural degree of generality in our framework. This presentation is based on joint work with Gabriel Larotonda.
start: 2021-09-15T15:00 end: 2021-09-16T15:45 speaker: Daniel Beltita (Institute of Mathematics of the Romanian Academy, Rumania) @@ -862,73 +862,73 @@ charlas: - titulo: Geometría Lipschitz para espacios de operadores abstract: | - La teoría no lineal de espacios de Banach ha sido un área muy activa en las últimas décadas, pero su contraparte no conmutativa (es decir, la teoría no lineal de espacios de operadores) apenas está siendo explorada. - En un trabajo previo con Bruno M. Braga hemos mostrado que la noción más obvia de transformación Lipschitz entre espacios de operadores desafortunadamente da lugar a una teoría trivial, en el sentido de que las únicas transformaciones que satisfacen la definición son lineales. En este trabajo, conjunto con Bruno M. Braga y Thomas Sinclair, introducimos una noción más sutil de encaje Lipschitz entre espacios de operadores: estrictamente más débil que la noción lineal, pero suficientemente rígida para imponer restricciones en la estructura lineal de los espacios de operadores. Con este propósito introducimos una noción de espacios libres Lipschitz para espacios de operadores, y probamos algunas de sus propiedades al estilo del trabajo clásico de G. Godefroy y N. Kalton. +La teoría no lineal de espacios de Banach ha sido un área muy activa en las últimas décadas, pero su contraparte no conmutativa (es decir, la teoría no lineal de espacios de operadores) apenas está siendo explorada.
+En un trabajo previo con Bruno M. Braga hemos mostrado que la noción más obvia de transformación Lipschitz entre espacios de operadores desafortunadamente da lugar a una teoría trivial, en el sentido de que las únicas transformaciones que satisfacen la definición son lineales. En este trabajo, conjunto con Bruno M. Braga y Thomas Sinclair, introducimos una noción más sutil de encaje Lipschitz entre espacios de operadores: estrictamente más débil que la noción lineal, pero suficientemente rígida para imponer restricciones en la estructura lineal de los espacios de operadores. Con este propósito introducimos una noción de espacios libres Lipschitz para espacios de operadores, y probamos algunas de sus propiedades al estilo del trabajo clásico de G. Godefroy y N. Kalton.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Alejandro Chávez Domínguez (University of Oklahoma, Estados Unidos) - titulo: Singular perturbations of symplectic isomorphism of Banach spaces abstract: | - Given a real reflexive Banach space \(X\), an onto isomorphism \(\alpha: X \to X^*\) is said to be \emph{symplectic} if \(\alpha^*= - \alpha\) with the canonical identification. We study some properties of strictly singular perturbations of symplectic isomorphism in the direction of the relations obtained by V. Ferenczi and E. Galego (2007) on complex structures of a Banach and its hyperplanes with the elements of square -1 of the algebra \(\mathcal L(X)/S(X)\) quotient of the algebra \(\mathcal L(X)\) of linear and continuous operators on \(X\) by the ideal of strictly singular operators. +Given a real reflexive Banach space \(X\), an onto isomorphism \(\alpha: X \to X^*\) is said to be \emph{symplectic} if \(\alpha^*= - \alpha\) with the canonical identification. We study some properties of strictly singular perturbations of symplectic isomorphism in the direction of the relations obtained by V. Ferenczi and E. Galego (2007) on complex structures of a Banach and its hyperplanes with the elements of square -1 of the algebra \(\mathcal L(X)/S(X)\) quotient of the algebra \(\mathcal L(X)\) of linear and continuous operators on \(X\) by the ideal of strictly singular operators.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Wilson Cuellar Carrera (Universidade de Sâo Paulo, Brasil) - titulo: Normas tensoriales en operator spaces abstract: | - En esta charla presentamos el inicio de una teoría sistemática de normas tensoriales en operator spaces y su interacción con ideales de operadores lineales asociados. Basados en la teoría de productos tensoriales en espacios de Banach, proponemos las definiciones naturales correspondientes a este nuevo ámbito y obtenemos algunos resultados análogos. Sin embargo, hay también notables discrepancias con la teoría clásica que requieren nuevos conocimientos, técnicas, ideas o hipótesis. En particular, mostramos varias diferencias sustanciales y muchas preguntas abiertas en la lista de las denominadas normas naturales (en el sentido de Grothendieck). - Trabajo en colaboración con Alejandro Chávez-Domínguez y Daniel Galicer. +En esta charla presentamos el inicio de una teoría sistemática de normas tensoriales en operator spaces y su interacción con ideales de operadores lineales asociados. Basados en la teoría de productos tensoriales en espacios de Banach, proponemos las definiciones naturales correspondientes a este nuevo ámbito y obtenemos algunos resultados análogos. Sin embargo, hay también notables discrepancias con la teoría clásica que requieren nuevos conocimientos, técnicas, ideas o hipótesis. En particular, mostramos varias diferencias sustanciales y muchas preguntas abiertas en la lista de las denominadas normas naturales (en el sentido de Grothendieck).
+Trabajo en colaboración con Alejandro Chávez-Domínguez y Daniel Galicer.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Verónica Dimant (Universidad de San Andrés, Argentina) - titulo: Linear dynamics of operators on non-metrizable topological vector spaces abstract: | - In this talk we will explore the following notions of chaos for operators on non-metrizable topological vector spaces: mixing, hypercyclicity, cyclicity, \(n\)-supercyclicity, Li-Yorke chaos and Devaney chaos. We will discuss the main differences between these notions for operators on Fr\'echet and on non-metrizable topological vector spaces. - To illustrate some of these differences, we will explore results about the linear dynamics of convolution operators on spaces of entire functions of finitely and infinitely many complex variables. A classical result due to Godefroy and Shapiro states that every nontrivial convolution operator on the Fréchet space \(\mathcal{H}(\mathbb{C}^n)\) of all entire functions of \(n\) complex variables is hypercyclic. In sharp contrast to this result, in a joint work with J. Mujica it was showed that no translation operator on the space \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) (which is a complete non-metrizable locally convex space) of entire functions of infinitely many complex variables is hypercyclic. Recently, in a joint work with B. Caraballo it was showed that no convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is cyclic or \(n\)-supercyclic for any positive integer \(n\). In the opposite direction, it was proved that every nontrivial convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is mixing. Exploring the concept of Li-Yorke chaos on non-metrizable topological vector spaces, it was also proved that nontrivial convolution operators on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) are Li-Yorke chaotic. +In this talk we will explore the following notions of chaos for operators on non-metrizable topological vector spaces: mixing, hypercyclicity, cyclicity, \(n\)-supercyclicity, Li-Yorke chaos and Devaney chaos. We will discuss the main differences between these notions for operators on Fr\'echet and on non-metrizable topological vector spaces.
+To illustrate some of these differences, we will explore results about the linear dynamics of convolution operators on spaces of entire functions of finitely and infinitely many complex variables. A classical result due to Godefroy and Shapiro states that every nontrivial convolution operator on the Fréchet space \(\mathcal{H}(\mathbb{C}^n)\) of all entire functions of \(n\) complex variables is hypercyclic. In sharp contrast to this result, in a joint work with J. Mujica it was showed that no translation operator on the space \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) (which is a complete non-metrizable locally convex space) of entire functions of infinitely many complex variables is hypercyclic. Recently, in a joint work with B. Caraballo it was showed that no convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is cyclic or \(n\)-supercyclic for any positive integer \(n\). In the opposite direction, it was proved that every nontrivial convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is mixing. Exploring the concept of Li-Yorke chaos on non-metrizable topological vector spaces, it was also proved that nontrivial convolution operators on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) are Li-Yorke chaotic.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Vinícius V. Fávaro (Universidade Federal de Uberlândia, Brasil) - titulo: Constante de proyección de espacios de polinomios abstract: | - Un resultado elemental en la teoría de espacios normados asegura que todo espacio de dimensión finita está complementado en cualquier superespacio que lo contenga. Luego, resulta natural querer cuantificar la norma de una determinada proyección sobre un espacio dado. - Una cantidad clásica relacionada con este propósito es la denominada \emph{constante de proyección} del espacio \( X \), definida como la mejor constante \(c>0\) que asegura para \emph{cualquier} superespacio \( Y \supset X\), la existencia de una proyección de \( Y \) sobre \( X \) de norma menor o igual a \( c \). - En esta charla discutiremos la constante de proyección de \(\mathcal P_m(X_n)\) (el espacio de polinomios \(m\)-homogéneos sobre un espacio \(n\)-dimensional \(X_n\)) y la conexión existente con algunos invariantes geométricos relativos a \(X_n\). Nos focalizamos en la dependencia respecto de ambos parámetros \(m\) y \(n\), i.e., el grado de homogeneidad y la dimensión del espacio, respectivamente. También trataremos la constante de proyección para otras clases de polinomios (e.g., polinomios de Dirichlet, polinomios sobre el cubo Booleano, trigonométricos analíticos, polinomios de grado menor o igual a \(m\) y tetraedrales homogéneos, entre otros). Mencionaremos cómo la constante de proyección de espacios de polinomios se conecta con el estudio de incondicionalidad en espacios de polinomios y el radio de Bohr. - Trabajo en conjunto con A. Defant, M. Mansilla, M. Mastyło y S. Muro. +Un resultado elemental en la teoría de espacios normados asegura que todo espacio de dimensión finita está complementado en cualquier superespacio que lo contenga. Luego, resulta natural querer cuantificar la norma de una determinada proyección sobre un espacio dado.
+Una cantidad clásica relacionada con este propósito es la denominada \emph{constante de proyección} del espacio \( X \), definida como la mejor constante \(c>0\) que asegura para \emph{cualquier} superespacio \( Y \supset X\), la existencia de una proyección de \( Y \) sobre \( X \) de norma menor o igual a \( c \).
+En esta charla discutiremos la constante de proyección de \(\mathcal P_m(X_n)\) (el espacio de polinomios \(m\)-homogéneos sobre un espacio \(n\)-dimensional \(X_n\)) y la conexión existente con algunos invariantes geométricos relativos a \(X_n\). Nos focalizamos en la dependencia respecto de ambos parámetros \(m\) y \(n\), i.e., el grado de homogeneidad y la dimensión del espacio, respectivamente. También trataremos la constante de proyección para otras clases de polinomios (e.g., polinomios de Dirichlet, polinomios sobre el cubo Booleano, trigonométricos analíticos, polinomios de grado menor o igual a \(m\) y tetraedrales homogéneos, entre otros). Mencionaremos cómo la constante de proyección de espacios de polinomios se conecta con el estudio de incondicionalidad en espacios de polinomios y el radio de Bohr.
+Trabajo en conjunto con A. Defant, M. Mansilla, M. Mastyło y S. Muro.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Daniel Galicer (Universidad de Buenos Aires, Argentina) - titulo: Ideals of multilinear operators from a geometric approach abstract: | - A recent approach for studying collections of multilinear operators, based on the geometry of tensor products, allows to generalize collections of linear operators to the multilinear setting. Besides defining new collections, this approach allows to characterize them in terms of summing properties, factorizations and duality relations with tensor products. The defined collections to this day include compact, summing, dominated and even factorable operators. These collections of multilinear operators enjoy an ideal behavior, that is, they are closed under compositions that preserves the geometry of the multilinear domain. This ideal property, together with the geometry of tensor products, has inspired the study of the so called \(\Sigma\)-ideals. - This talk is dedicated to present the foundations of \(\Sigma\)-ideals and their classification in terms of tensor norms. Moreover we will explore the \(\Sigma\)-ideal of \((p,q)\)-dominated multilinear operators and its classification in terms of tensor spaces as well as the factorization through \(L_p\) spaces that they admit. +A recent approach for studying collections of multilinear operators, based on the geometry of tensor products, allows to generalize collections of linear operators to the multilinear setting. Besides defining new collections, this approach allows to characterize them in terms of summing properties, factorizations and duality relations with tensor products. The defined collections to this day include compact, summing, dominated and even factorable operators. These collections of multilinear operators enjoy an ideal behavior, that is, they are closed under compositions that preserves the geometry of the multilinear domain. This ideal property, together with the geometry of tensor products, has inspired the study of the so called \(\Sigma\)-ideals.
+This talk is dedicated to present the foundations of \(\Sigma\)-ideals and their classification in terms of tensor norms. Moreover we will explore the \(\Sigma\)-ideal of \((p,q)\)-dominated multilinear operators and its classification in terms of tensor spaces as well as the factorization through \(L_p\) spaces that they admit.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Samuel García Hernández (Universidad Tecnológica de México, México) - titulo: Geometría de los espacios de sucesiones de Marcinkiewicz abstract: | - Los espacios de sucesiones de Marcinkiewicz \(m_{\Psi}^0\) y sus duales \((m_\Psi^0)'\) y \(m_\Psi\), son espacios invariantes por reordernamientos, determinados por un símbolo \(\Psi\) (una sucesión no decreciente de números reales positivos). Para símbolos apropiados \(\Psi\), estos espacios devienen en espacios de Lorentz (sus preduales y duales) \(d_*(w,1), d(w,1)\) y \(d^*(w,1)\). - El objetivo de esta charla es entender, para un símbolo general \(\Psi\), la geometría de la bola unidad de \(m_{\Psi}^0,(m_\Psi^0)'\) y \(m_\Psi\) a partir de la caracterización de sus puntos extremales (reales y complejos) y de sus puntos expuestos. - Veremos que los puntos extremales complejos de la bola unidad de \(m_\Psi^0\) están determinados por la geometría de los subespacios finito-dimensionales \(m_\Psi^n\). Mientras que la geometría de la bola unidad de \(m_\Psi\) depende fuertemente del comportamiento asintótico de los valores de \(\Psi\). También caracterizaremos los puntos extremales y expuestos de la bola unidad de \((m^0_\Psi)'\). Como consecuencia, extendemos resultados de Kamińska, Lee y Lewicki (2009) y de Ciesielski y Lewicki (2019). Trabajo en conjunto con Chris Boyd (UCD). +Los espacios de sucesiones de Marcinkiewicz \(m_{\Psi}^0\) y sus duales \((m_\Psi^0)'\) y \(m_\Psi\), son espacios invariantes por reordernamientos, determinados por un símbolo \(\Psi\) (una sucesión no decreciente de números reales positivos). Para símbolos apropiados \(\Psi\), estos espacios devienen en espacios de Lorentz (sus preduales y duales) \(d_*(w,1), d(w,1)\) y \(d^*(w,1)\).
+El objetivo de esta charla es entender, para un símbolo general \(\Psi\), la geometría de la bola unidad de \(m_{\Psi}^0,(m_\Psi^0)'\) y \(m_\Psi\) a partir de la caracterización de sus puntos extremales (reales y complejos) y de sus puntos expuestos.
+Veremos que los puntos extremales complejos de la bola unidad de \(m_\Psi^0\) están determinados por la geometría de los subespacios finito-dimensionales \(m_\Psi^n\). Mientras que la geometría de la bola unidad de \(m_\Psi\) depende fuertemente del comportamiento asintótico de los valores de \(\Psi\). También caracterizaremos los puntos extremales y expuestos de la bola unidad de \((m^0_\Psi)'\). Como consecuencia, extendemos resultados de Kamińska, Lee y Lewicki (2009) y de Ciesielski y Lewicki (2019). Trabajo en conjunto con Chris Boyd (UCD).
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Silvia Lassalle (Universidad de San Andrés, Argentina) - titulo: Cluster value problems in Banach spaces abstract: | - In this talk I will discuss a couple of problems about the Banach algebra \(H^{\infty}(B)\) of bounded holomorphic functions on the ball \(B\) of a Banach space \(X\). First I will talk about the Corona problem, which asks if the evaluations on each point of \(B\) of functions in \(H^{\infty}(B)\) is a dense subset of \(M_{H^{\infty}(B)}\), the nonzero algebra homomorphisms from \(H^{\infty}(B)\) into \(\mathbb{C}\). An easier problem deals with a comparison of the limit behavior of functions in \(H^{\infty}(B)\) towards each point \(x^{**}\) of the closed ball \(\overline{B}^{**}\) of \(X^{**}\) with the elements of \(M_{H^{\infty}(B)}\) that in a sense correspond to \(x^{**}\). I will discuss why the last problem, the cluster value problem, is indeed easier than the Corona problem, and I will talk about some of the essential ideas behind the cluster value theorems known up to now. +In this talk I will discuss a couple of problems about the Banach algebra \(H^{\infty}(B)\) of bounded holomorphic functions on the ball \(B\) of a Banach space \(X\). First I will talk about the Corona problem, which asks if the evaluations on each point of \(B\) of functions in \(H^{\infty}(B)\) is a dense subset of \(M_{H^{\infty}(B)}\), the nonzero algebra homomorphisms from \(H^{\infty}(B)\) into \(\mathbb{C}\). An easier problem deals with a comparison of the limit behavior of functions in \(H^{\infty}(B)\) towards each point \(x^{**}\) of the closed ball \(\overline{B}^{**}\) of \(X^{**}\) with the elements of \(M_{H^{\infty}(B)}\) that in a sense correspond to \(x^{**}\). I will discuss why the last problem, the cluster value problem, is indeed easier than the Corona problem, and I will talk about some of the essential ideas behind the cluster value theorems known up to now.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Sofía Ortega Castillo (Universidad de Guadalajara, México) - titulo: Spaceability and Residuality in some sets of analytic functions on the open unit disk abstract: | - In this talk we will present some results concerning algebraic and topological structure inside the following sets of special functions: the set \(\mathcal{F}\) of all Bloch functions defined on the open unit disk that are unbounded, and the set \(\mathcal{W}^p\) of all Bergman functions defined on the open unit disk that are not Bloch functions. We show that \(\mathcal{F}\) and \(\mathcal{W}^p\) are spaceable and residual, that is, \(\mathcal{F}\cup\{0\}\) (and \(\mathcal{W}^p\cup\{0\}\)) contains a closed infinite dimensional vector space. Residuality of the set \(\mathcal{F}\) (and \(\mathcal{W}^p\)) means that their complements are of first category. We also investigate the set of all holomorphic functions of bounded type defined on a Banach algebra \(E\) into \(E\), which are not Lorch-analytic. We show that this set is spaceable but not residual. - Joint work with M. Lilian Lourenço, IME-USP, São Paulo, Brazil. +In this talk we will present some results concerning algebraic and topological structure inside the following sets of special functions: the set \(\mathcal{F}\) of all Bloch functions defined on the open unit disk that are unbounded, and the set \(\mathcal{W}^p\) of all Bergman functions defined on the open unit disk that are not Bloch functions. We show that \(\mathcal{F}\) and \(\mathcal{W}^p\) are spaceable and residual, that is, \(\mathcal{F}\cup\{0\}\) (and \(\mathcal{W}^p\cup\{0\}\)) contains a closed infinite dimensional vector space. Residuality of the set \(\mathcal{F}\) (and \(\mathcal{W}^p\)) means that their complements are of first category. We also investigate the set of all holomorphic functions of bounded type defined on a Banach algebra \(E\) into \(E\), which are not Lorch-analytic. We show that this set is spaceable but not residual.
+Joint work with M. Lilian Lourenço, IME-USP, São Paulo, Brazil.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Daniela M. Vieira ( Universidade de São Paulo, Brasil) - titulo: Extremos de polinomios - un enfoque probabilístico abstract: | - Consideremos un polinomio \(k\)-homogéneo \(P:\mathbb R^n \longrightarrow \mathbb R\). ¿Cuál es la probabilidad de que \(P\) alcance un máximo relativo en algún vértice de la bola-1 (i.e., la bola unidad de la norma \(\Vert \cdot \Vert_1\))? ¿Y en un v\'ertice de la bola-\(\infty\)? Se sabe que si \(k \gt 2\) la probabilidad de alcanzar un máximo relativo en algún vértice de la bola-1 tiende a uno a medida que la dimensión \(n\) crece. Esto es falso para \(k=2\), y es un problema abierto para la bola-\(\infty\). - En esta charla veremos algunas de las herramientas utilizadas para encarar estas cuestiones, y algunas de las dificultades que se presentan. - Veremos también un resultado reciente, obtenido en conjunto con Damián Pinasco y Ezequiel Smucler, para polinomios sobre un simple: si \(k>4\), la probabilidad de que un polinomio \(k\)-homogéneo alcance un máximo relativo en algún vértice del simple \(n\)-dimensional tiende a uno al crecer la dimensión \(n\). Esto requiere un aporte a un viejo problema estadístico: el de las probabilidades ortantes. +Consideremos un polinomio \(k\)-homogéneo \(P:\mathbb R^n \longrightarrow \mathbb R\). ¿Cuál es la probabilidad de que \(P\) alcance un máximo relativo en algún vértice de la bola-1 (i.e., la bola unidad de la norma \(\Vert \cdot \Vert_1\))? ¿Y en un v\'ertice de la bola-\(\infty\)? Se sabe que si \(k \gt 2\) la probabilidad de alcanzar un máximo relativo en algún vértice de la bola-1 tiende a uno a medida que la dimensión \(n\) crece. Esto es falso para \(k=2\), y es un problema abierto para la bola-\(\infty\).
+En esta charla veremos algunas de las herramientas utilizadas para encarar estas cuestiones, y algunas de las dificultades que se presentan.
+Veremos también un resultado reciente, obtenido en conjunto con Damián Pinasco y Ezequiel Smucler, para polinomios sobre un simple: si \(k>4\), la probabilidad de que un polinomio \(k\)-homogéneo alcance un máximo relativo en algún vértice del simple \(n\)-dimensional tiende a uno al crecer la dimensión \(n\). Esto requiere un aporte a un viejo problema estadístico: el de las probabilidades ortantes.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Ignacio Zalduendo (Universidad Torcuato Di Tella, Argentina) @@ -945,73 +945,73 @@ charlas: - titulo: On the averaging theory for computing periodic orbits abstract: | - This talk deals with the averaging theory, which is a classical tool allowing us to study periodic solutions of the nonlinear differential systems. - We present some applications of averaging theory. Also, we shall show a system where the averaging method fails to detect periodic orbits. +This talk deals with the averaging theory, which is a classical tool allowing us to study periodic solutions of the nonlinear differential systems.
+We present some applications of averaging theory. Also, we shall show a system where the averaging method fails to detect periodic orbits.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Martha Alvarez Ramírez (Universidad Autónoma Metropolitana, México) - titulo: Integrable models close to slow-fast Hamiltonian systems abstract: | - Slow-fast Hamiltonian systems are characterized by a separation of the phase space into slow and fast parts typically identified by a small (or slow) parameter. This kind of Hamiltonian system is not integrable, in general; even though in one degree of freedom. In this lecture, we study an integrable model associated with a slow-fast Hamiltonian system of two degrees of freedom. Thinking of the slow parameter as a perturbative one, making suitable symmetry assumptions, and using normal form theory, we show that a slow-fast Hamiltonian system in two degrees of freedom is close to an integrable Hamiltonian model. What we gain with this model is the possibility to associate a family of Lagrangian 2-tori which is almost invariant with respect to the original slow-fast Hamiltonian system. As an important application, this family of almost invariant Lagrangian 2-tori can be used to compute approximations to the spectrum of the quantum model associated with the slow-fast Hamiltonian systems. +Slow-fast Hamiltonian systems are characterized by a separation of the phase space into slow and fast parts typically identified by a small (or slow) parameter. This kind of Hamiltonian system is not integrable, in general; even though in one degree of freedom. In this lecture, we study an integrable model associated with a slow-fast Hamiltonian system of two degrees of freedom. Thinking of the slow parameter as a perturbative one, making suitable symmetry assumptions, and using normal form theory, we show that a slow-fast Hamiltonian system in two degrees of freedom is close to an integrable Hamiltonian model. What we gain with this model is the possibility to associate a family of Lagrangian 2-tori which is almost invariant with respect to the original slow-fast Hamiltonian system. As an important application, this family of almost invariant Lagrangian 2-tori can be used to compute approximations to the spectrum of the quantum model associated with the slow-fast Hamiltonian systems.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Misael Avendaño-Camacho (Universidad de Sonora, México) - titulo: The Maximum Principle and solutions to nonlinear elliptic problems abstract: | - In this talk, we will use a form of the maximum principle and an iteration method to show existence of solutions to the Dirichlet problem: - \[\left\{\begin{array}{l}-\Delta u = \lambda\left(x\right)f\left(u\right)+h\left(x\right)\quad \mbox{in}\quad \Omega,\\u=0 \quad \mbox{on}\quad \partial \Omega,\end{array}\right.\]in bounded and thin unbounded domains. Also, we will show how these methods can be applied to show existence of nontrivial solutions to the Lane-Emden system\[\left\{\begin{array}{l}-\Delta u=v^p,\quad -\Delta v=u^p, \quad \mbox{in}\quad \Omega\\u=v=0, \quad\mbox{on}\quad \partial \Omega. \end{array}\right.\] - This is joint work with Jonatán Torres-Orozco. +In this talk, we will use a form of the maximum principle and an iteration method to show existence of solutions to the Dirichlet problem: + \[\left\{\begin{array}{l}-\Delta u = \lambda\left(x\right)f\left(u\right)+h\left(x\right)\quad \mbox{in}\quad \Omega,\\u=0 \quad \mbox{on}\quad \partial \Omega,\end{array}\right.\]in bounded and thin unbounded domains. Also, we will show how these methods can be applied to show existence of nontrivial solutions to the Lane-Emden system\[\left\{\begin{array}{l}-\Delta u=v^p,\quad -\Delta v=u^p, \quad \mbox{in}\quad \Omega\\u=v=0, \quad\mbox{on}\quad \partial \Omega. \end{array}\right.\]
+This is joint work with Jonatán Torres-Orozco.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Jean C. Cortissoz (Universidad de los Andes, Colombia) - titulo: Singularities of the Focal Set of a Line Congruence abstract: | - A line congruence is a \(2\)-dimensional arrangement of lines in \(3\)-space. It is a classical topic of projective differential geometry and has many relations with binary differential equations. In this talk we give a geometric description of the generic singularities of the focal surface of a line congruence. We show that these singularities occur along the ridge and double eigenvalue curves. A basic tool is the support function associated with an eqüiaffine vector field transversal to a surface in \(\mathbb{R}^3\). - This is a joint work with Ronaldo Garcia (UFG, Brazil). +A line congruence is a \(2\)-dimensional arrangement of lines in \(3\)-space. It is a classical topic of projective differential geometry and has many relations with binary differential equations. In this talk we give a geometric description of the generic singularities of the focal surface of a line congruence. We show that these singularities occur along the ridge and double eigenvalue curves. A basic tool is the support function associated with an eqüiaffine vector field transversal to a surface in \(\mathbb{R}^3\).
+This is a joint work with Ronaldo Garcia (UFG, Brazil).
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Marcos Craizer (Pontifícia Universidade Católica do Rio de Janeiro, Brasil) - titulo: Line congruence in singular surface in \(\mathbb R^3\) abstract: | - A line congruence in the Euclidean space of dimension \(3\) is a \(2\) parameter family of lines in \(\mathbb R^{3}\). - The first record about line congruences appeared in "Mémoire sur la Théorie des Déblais et des Remblais" (1776,1784) where Gaspard Monge seeks to solve a minimizing cost problem of transporting an amount of land from one place to another, preserving the volume. - After Monge, Ernst Eduard Kummer in ''Allgemeine Theorie der geradlinigen Strahien systeme", was the first to deal exclusively with the general theory of line congruences. Mainly due to optical applications, line congruences started to gain importance, increasing even more with the development of technology. In this lecture, we will deal with line congruence when the parameters vary in a surfaces with singularities. We will build the theory of congruence lines when the parameter space is a Frontal. As application, we will consider line congruence of cuspidal edges. We will show relations between singularities of the congruence lines and geometric properties of initial cuspidal edges. - This a joint work with Tito Medina(ICMC/USP), Igor Chagas Santos(ICMC/USP) and Maria Aparecida S. Ruas(ICMC/USP). +A line congruence in the Euclidean space of dimension \(3\) is a \(2\) parameter family of lines in \(\mathbb R^{3}\).
+The first record about line congruences appeared in "Mémoire sur la Théorie des Déblais et des Remblais" (1776,1784) where Gaspard Monge seeks to solve a minimizing cost problem of transporting an amount of land from one place to another, preserving the volume.
+After Monge, Ernst Eduard Kummer in ''Allgemeine Theorie der geradlinigen Strahien systeme", was the first to deal exclusively with the general theory of line congruences. Mainly due to optical applications, line congruences started to gain importance, increasing even more with the development of technology. In this lecture, we will deal with line congruence when the parameters vary in a surfaces with singularities. We will build the theory of congruence lines when the parameter space is a Frontal. As application, we will consider line congruence of cuspidal edges. We will show relations between singularities of the congruence lines and geometric properties of initial cuspidal edges.
+This a joint work with Tito Medina(ICMC/USP), Igor Chagas Santos(ICMC/USP) and Maria Aparecida S. Ruas(ICMC/USP).
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Débora Lopes da Silva (Universidade Federal de Sergipe, Brasil) - titulo: Frobenius manifolds in the context of \(\mathbb{A}\)-manifolds abstract: | - The theory of Frobenius manifolds has important relevance in mathematics since the application to enumerative geometry with the counting of the number of zero genus curves of degree \(d\) passing through \(3d-1\) points in the projective space \(\mathbb{C} P^2\) (M. Kontsevich). - Let \(\mathbb{A}\) be a finite-dimensional commutative associative algebra with unit over \(\mathbb{R}\). A map \(F: \mathbb{A}^n \to \mathbb{A}^n\) is called \(\mathbb{A}\)-differentiable if \(dF\) is \(\mathbb{A}\)-lineal. Let \(\Gamma_\mathbb{A}\) be the pseudogroup of local \(\mathbb{A}\)-diffeomorphisms of \(\mathbb{A}^n\). An \(\mathbb{A}\)-manifold is a manifold endowed with \(\Gamma_\mathbb{A}\)-atlas. An important example of \(\mathbb{A}\)-manifold is the total space of A. Weil's bundle of \(\mathbb{A}\)-closed points. - The theory of \(\mathbb{A}\)-manifolds (A.P. Shirokov, V.V. Vishnevskii, G.I. Kruchkovich, V.V. Shurygin) has strong relations to the foliation theory and to geometry of jet bundles and the theory of natural operations in differential geometry (I. Kolář, J. Slovák, P. W. Michor). - In our talk we will show that Frobenius manifolds in the sense of B. Dubrovin and N.J. Hitchin have the structure of an \(\mathbb{A}\)-manifold, where \(\mathbb{A}\) is a Frobenius algebra. We will introduce Weil coalgebras, the \(G\)-equivariant graduated versions of Weil algebras, and the corresponding bundle structures. +The theory of Frobenius manifolds has important relevance in mathematics since the application to enumerative geometry with the counting of the number of zero genus curves of degree \(d\) passing through \(3d-1\) points in the projective space \(\mathbb{C} P^2\) (M. Kontsevich).
+Let \(\mathbb{A}\) be a finite-dimensional commutative associative algebra with unit over \(\mathbb{R}\). A map \(F: \mathbb{A}^n \to \mathbb{A}^n\) is called \(\mathbb{A}\)-differentiable if \(dF\) is \(\mathbb{A}\)-lineal. Let \(\Gamma_\mathbb{A}\) be the pseudogroup of local \(\mathbb{A}\)-diffeomorphisms of \(\mathbb{A}^n\). An \(\mathbb{A}\)-manifold is a manifold endowed with \(\Gamma_\mathbb{A}\)-atlas. An important example of \(\mathbb{A}\)-manifold is the total space of A. Weil's bundle of \(\mathbb{A}\)-closed points.
+The theory of \(\mathbb{A}\)-manifolds (A.P. Shirokov, V.V. Vishnevskii, G.I. Kruchkovich, V.V. Shurygin) has strong relations to the foliation theory and to geometry of jet bundles and the theory of natural operations in differential geometry (I. Kolář, J. Slovák, P. W. Michor).
+In our talk we will show that Frobenius manifolds in the sense of B. Dubrovin and N.J. Hitchin have the structure of an \(\mathbb{A}\)-manifold, where \(\mathbb{A}\) is a Frobenius algebra. We will introduce Weil coalgebras, the \(G\)-equivariant graduated versions of Weil algebras, and the corresponding bundle structures.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Mikhail Malakhaltsev (Universidad de los Andes, Colombia) y Carlos Segovia (Universidad Nacional Autónoma de México, México) - titulo: Classification of complex polynomials in \(\mathbb{C}^2\) and its applications abstract: | - The huge group of polynomial diffeomorphisms of \(\mathbb{C}^2\) acts on the space of polynomials \(\{ f: \mathbb{C}^2 \longrightarrow \mathbb{C} \}\) by coordinate changes. The classification of polynomials under this action is a challenging open problem. We obtain a very concrete result for degree three polynomials. An application to perturbation of singular Hamiltonian foliations on \(\mathbb{C}^2\), having \(\mathbb{C} \backslash \{ k \, \hbox{points}\}\), as generic leafs, is provided. - Joint work with John Alexander Arredondo (Colombia) and Salomón Rebollo (Chile). +The huge group of polynomial diffeomorphisms of \(\mathbb{C}^2\) acts on the space of polynomials \(\{ f: \mathbb{C}^2 \longrightarrow \mathbb{C} \}\) by coordinate changes. The classification of polynomials under this action is a challenging open problem. We obtain a very concrete result for degree three polynomials. An application to perturbation of singular Hamiltonian foliations on \(\mathbb{C}^2\), having \(\mathbb{C} \backslash \{ k \, \hbox{points}\}\), as generic leafs, is provided.
+Joint work with John Alexander Arredondo (Colombia) and Salomón Rebollo (Chile).
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Jesús R. Muciño-Raymundo (Universidad Nacional Autónoma de México, México) - titulo: Umbilic Points at Infinity of Certain Algebraic Surfaces abstract: | - In this lecture, we study the global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function \(f\) on the plane. We will prove that every umbilic point at infinity of the projective extension of these direction fields has a Poincar\'e-Hopf index equal to 1/2 and the topological type of a Lemon or a Monstar. As a consequence, we will provide a Poincar\'e-Hopf type formula for the graph of \(f\) pointing out that, if all umbilics are isolated, the sum of all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of \(f\). - This is a joint work with B. Guilfoyle. +In this lecture, we study the global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function \(f\) on the plane. We will prove that every umbilic point at infinity of the projective extension of these direction fields has a Poincar\'e-Hopf index equal to 1/2 and the topological type of a Lemon or a Monstar. As a consequence, we will provide a Poincar\'e-Hopf type formula for the graph of \(f\) pointing out that, if all umbilics are isolated, the sum of all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of \(f\).
+This is a joint work with B. Guilfoyle.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Adriana Ortiz-Rodríguez (Universidad Nacional Autónoma de México, México) - titulo: Perturbations of Hamiltonian foliations abstract: | - At the International Congress of Mathematics held in Paris in 1900, D. Hilbert presented a list of 23 problems for the century. After more than one hundred years some of those problems are still unsolved. Among them is the 16th problem. This problem, in its second part, asks for the number and position of limit cycles (isolated periodic solutions) of planar polynomial differential equations. In the 70's, Y. Ilyashenko and V. I. Arnold proposed, in this context, to investigate what happens for systems infinitely close to Hamiltonians. This approach is known as the infinitesimal Hilbert 16th problem. More precisely, let \(F\in\mathbb{R}[x,y]\) a polynomial and \(\omega\) a polynomial 1-form. We consider the perturbation \(dF+\epsilon\omega=0\) of the Hamiltonian foliation \(dF=0\), where \(\epsilon\) is a small parameter. Let \(\gamma(z)\subset F^{-1}(z)\) a continuous family of regular periodic orbits of \(dF=0\). The displacement map with respect to this family \(\gamma(z)\) and with respect to the foliation \(dF+\epsilon\omega=0\) is an analytic function \(\Delta(z,\epsilon)=\epsilon^\mu M_\mu(z)+\cdots\). The first function \(M_\mu\) which does not vanish identically keeps information about limit cycles born from this perturbation. By passing to the complexification of \(F\), we can consider the orbit under monodromy of \(\gamma(z)\), which is a normal subgroup \(\mathcal{O}\) of the first homotopy group of the regular fiber \(F^{-1}(z)\). To understand the orbit under monodromy one must focus on the critical values of the function \(F\). To each critical value corresponds a vanishing cycle; the monodromy related to such cycle detects the traces of change in the topology due to the presence of the singular fiber. In non generic cases, not every cycle can be reached by the orbit under monodromy of the family of regular periodic orbits \(\gamma(z)\). Thus, to detect the cycles that are not reached one considers the quotient \(\frac{\mathcal{O}}{[\mathcal{O},\pi_1(F^{-1}(z),p_0)]}\) and defines a constant \(\kappa\), called orbit depth. It turns out that the function \(M_\mu\) is an iterated integral of length at most \(\kappa\). We stress that \(\kappa\) by its definition depends only on the topology of the regular fiber of \(F\), and on the orbit \(\mathcal{O}\) of \(\gamma(z)\), while the function \(M_\mu\) depends also on the perturbation given by \(\omega\). We will talk about this result and discuss some conjectures around this bound for non-generic polynomials. +At the International Congress of Mathematics held in Paris in 1900, D. Hilbert presented a list of 23 problems for the century. After more than one hundred years some of those problems are still unsolved. Among them is the 16th problem. This problem, in its second part, asks for the number and position of limit cycles (isolated periodic solutions) of planar polynomial differential equations. In the 70's, Y. Ilyashenko and V. I. Arnold proposed, in this context, to investigate what happens for systems infinitely close to Hamiltonians. This approach is known as the infinitesimal Hilbert 16th problem. More precisely, let \(F\in\mathbb{R}[x,y]\) a polynomial and \(\omega\) a polynomial 1-form. We consider the perturbation \(dF+\epsilon\omega=0\) of the Hamiltonian foliation \(dF=0\), where \(\epsilon\) is a small parameter. Let \(\gamma(z)\subset F^{-1}(z)\) a continuous family of regular periodic orbits of \(dF=0\). The displacement map with respect to this family \(\gamma(z)\) and with respect to the foliation \(dF+\epsilon\omega=0\) is an analytic function \(\Delta(z,\epsilon)=\epsilon^\mu M_\mu(z)+\cdots\). The first function \(M_\mu\) which does not vanish identically keeps information about limit cycles born from this perturbation. By passing to the complexification of \(F\), we can consider the orbit under monodromy of \(\gamma(z)\), which is a normal subgroup \(\mathcal{O}\) of the first homotopy group of the regular fiber \(F^{-1}(z)\). To understand the orbit under monodromy one must focus on the critical values of the function \(F\). To each critical value corresponds a vanishing cycle; the monodromy related to such cycle detects the traces of change in the topology due to the presence of the singular fiber. In non generic cases, not every cycle can be reached by the orbit under monodromy of the family of regular periodic orbits \(\gamma(z)\). Thus, to detect the cycles that are not reached one considers the quotient \(\frac{\mathcal{O}}{[\mathcal{O},\pi_1(F^{-1}(z),p_0)]}\) and defines a constant \(\kappa\), called orbit depth. It turns out that the function \(M_\mu\) is an iterated integral of length at most \(\kappa\). We stress that \(\kappa\) by its definition depends only on the topology of the regular fiber of \(F\), and on the orbit \(\mathcal{O}\) of \(\gamma(z)\), while the function \(M_\mu\) depends also on the perturbation given by \(\omega\). We will talk about this result and discuss some conjectures around this bound for non-generic polynomials.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Jessie Pontigo Herrera (Universidad Nacional Autónoma de México, México) - titulo: Solutions of algebraic linear ordinary differential equations abstract: | - A classical result of F. Klein states that, given a finite primitive group \(G\subseteq SL_2(\mathbb{C})\), there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to \(G\) is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group \(G\subseteq SL_n(\mathbb{C})\), there exist a positive integer \(d=d(G)\) and a standard equation such that any LODE whose differential Galois group is isomorphic to \(G\) is gauge equivalent, over a field extension \(F\) of degree \(d\), to an equation projectively equivalent to the pullback by a map in \(F\) of this standard equation. For \(n=3\), these standard equations can be chosen to be hypergeometric. +A classical result of F. Klein states that, given a finite primitive group \(G\subseteq SL_2(\mathbb{C})\), there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to \(G\) is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group \(G\subseteq SL_n(\mathbb{C})\), there exist a positive integer \(d=d(G)\) and a standard equation such that any LODE whose differential Galois group is isomorphic to \(G\) is gauge equivalent, over a field extension \(F\) of degree \(d\), to an equation projectively equivalent to the pullback by a map in \(F\) of this standard equation. For \(n=3\), these standard equations can be chosen to be hypergeometric.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Camilo Sanabria (Universidad de Los Andes, Colombia) @@ -1150,58 +1150,58 @@ charlas: - titulo: Asymptotic analysis of matrix orthogonal polynomials abstract: | - In this talk we consider matrix orthogonal polynomials (MVOPs) on a finite interval of the real line, with Jacobi-type weights. We are particularly interested in the asymptotic behavior as the degree of the polynomials tends to infinity, that we study with the method of steepest descent applied to the corresponding Riemann-Hilbert problem, as described by Grünbaum, de la Iglesia and Martínez-Finkelshtein. We include examples motivated by group theory. - This is joint work with Arno Kuijlaars (KU Leuven, Belgium) and Pablo Román (Universidad de Córdoba, Argentina). +In this talk we consider matrix orthogonal polynomials (MVOPs) on a finite interval of the real line, with Jacobi-type weights. We are particularly interested in the asymptotic behavior as the degree of the polynomials tends to infinity, that we study with the method of steepest descent applied to the corresponding Riemann-Hilbert problem, as described by Grünbaum, de la Iglesia and Martínez-Finkelshtein. We include examples motivated by group theory.
+This is joint work with Arno Kuijlaars (KU Leuven, Belgium) and Pablo Román (Universidad de Córdoba, Argentina).
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Alfredo Deaño (Universidad Carlos III de Madrid, España) - titulo: Sobolev spaces on graded groups abstract: | - Fractional Calculus is one of the areas where Special Functions naturally arise. We will characterize fractional Sobolev spaces of potential type on graded groups. The approach is based on the Littlewood-Paley g-function. This is a joint work with Pablo de Nápoli. +Fractional Calculus is one of the areas where Special Functions naturally arise. We will characterize fractional Sobolev spaces of potential type on graded groups. The approach is based on the Littlewood-Paley g-function. This is a joint work with Pablo de Nápoli.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Rocío Diaz Martín (Universidad Nacional de Córdoba, Argentina) - titulo: Elliptic Kac-Sylvester matrix from difference Lamé equation abstract: | - Through a finite-dimensional reduction of the difference Lamé equation, an elliptic analog of the Kac-Sylvester tridiagonal matrix is found. We solve the corresponding finite discrete Lamé equation by constructing an orthogonal basis of eigenvectors for this novel elliptic Kac-Sylvester matrix. (Based on work in collaboration with Tamás Görbe.) +Through a finite-dimensional reduction of the difference Lamé equation, an elliptic analog of the Kac-Sylvester tridiagonal matrix is found. We solve the corresponding finite discrete Lamé equation by constructing an orthogonal basis of eigenvectors for this novel elliptic Kac-Sylvester matrix. (Based on work in collaboration with Tamás Görbe.)
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Jan Felipe van Diejen (Universidad de Talca, Chile) - titulo: Sobolev Type orthogonal polynomials on the cone of revolution abstract: | - We present the Sobolev Type orthogonal polynomials of several variables defined on the cone and on the surface of the cone as a particular case of the polynomials on a quadratic surface of revolution. Using the results from [1], [2] and [3], we show that this kind of polynomials satisfy some properties. - [1] L. Fernández, T. E Pérez, M. Piñar and Y. Xu. Krall-type orthogonal polynomials in several variables. Comp. Appl. Math. Vol 81. Pags 1519-1524. (2001). - [2] Y. Xu, Sobolev orthogonal polynomials defined via gradient on the unit ball, J. Approx. Theory. 152 (2008), 52--65. (2006). - [3] Y. Xu, Fourier series in orthogonal polynomials on a cone of revolution, arXiv:1905.07587 2019 +We present the Sobolev Type orthogonal polynomials of several variables defined on the cone and on the surface of the cone as a particular case of the polynomials on a quadratic surface of revolution. Using the results from [1], [2] and [3], we show that this kind of polynomials satisfy some properties.
+[1] L. Fernández, T. E Pérez, M. Piñar and Y. Xu. Krall-type orthogonal polynomials in several variables. Comp. Appl. Math. Vol 81. Pags 1519-1524. (2001).
+ [2] Y. Xu, Sobolev orthogonal polynomials defined via gradient on the unit ball, J. Approx. Theory. 152 (2008), 52--65. (2006).
+ [3] Y. Xu, Fourier series in orthogonal polynomials on a cone of revolution, arXiv:1905.07587 2019
In this contribution, we present some algebraic and analytic properties related to spectral transformations of orthogonality matrix measures on the unit circle, that constitute generalizations from well-known results on the scalar case. We focus our attention on the so-called Christoffel, Geronimus and Uvarov transformations, and deal with connection formulas, factorizations of block Hessenberg and CVM matrices, and relative asymptotics for the associated orthogonal matrix polynomials. This is a joint work with Edinson Fuentes.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Luis E. Garza Gaona (Universidad de Colima, México) - titulo: Toda lattice, special functions and their matrix analogues abstract: | - The classical Toda lattice is a model for a one-dimensional crystal. After a transformation in Flaschka coordinates there exists a Lax pair, for which the operator acts as a three-term recurrence operator. This gives a link to orthogonal polynomials, special functions and Lie algebra representations. In the case of orthogonal polynomials, the time dependence in the Toda lattice corresponds to deformation of the orthogonality measure by an exponential. The nonabelian Toda lattice is a generalisation of the Toda lattice for which matrix valued orthogonal polynomials play a similar role. We discuss matrix polynomials, and we discuss an explicit example of such a nonabelian Toda lattice. +The classical Toda lattice is a model for a one-dimensional crystal. After a transformation in Flaschka coordinates there exists a Lax pair, for which the operator acts as a three-term recurrence operator. This gives a link to orthogonal polynomials, special functions and Lie algebra representations. In the case of orthogonal polynomials, the time dependence in the Toda lattice corresponds to deformation of the orthogonality measure by an exponential. The nonabelian Toda lattice is a generalisation of the Toda lattice for which matrix valued orthogonal polynomials play a similar role. We discuss matrix polynomials, and we discuss an explicit example of such a nonabelian Toda lattice.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Erik Koelink (Radboud Universiteit, Países Bajos) - titulo: Multiple orthogonal polynomials with respect to hypergeometric functions abstract: | - In this talk I will discuss on two new sets of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Confluent and Gauss' hypergeometric functions. The former on unbounded support and the latter with bounded support on the real line. It was recently shown in [3] that the latter are indeed random walk polynomials. In both cases, this type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total-positivity problems in enumerative combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. - References: - [1] H. Lima and A. Loureiro, Multiple orthogonal polynomials with respect to Gauss’ hypergeometric function, to appear in Stud. Appl. Math. arXiv:2001.06820. - [2] H. Lima and A. Loureiro, Multiple orthogonal polynomials associated with confluent hypergeometric functions, J. Approx. Theory, 260 (2020), pp 105484. - [3] A. Branquinho, J. E. Fernández-Díaz, A. Foulquié-Moreno and M. Mañas, Hypergeometric Multiple Orthogonal Polynomials and Random Walks, (2021) arXiv:2107.00770v2 +In this talk I will discuss on two new sets of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Confluent and Gauss' hypergeometric functions. The former on unbounded support and the latter with bounded support on the real line. It was recently shown in [3] that the latter are indeed random walk polynomials. In both cases, this type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total-positivity problems in enumerative combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property.
+References:
+ [1] H. Lima and A. Loureiro, Multiple orthogonal polynomials with respect to Gauss’ hypergeometric function, to appear in Stud. Appl. Math. arXiv:2001.06820.
+ [2] H. Lima and A. Loureiro, Multiple orthogonal polynomials associated with confluent hypergeometric functions, J. Approx. Theory, 260 (2020), pp 105484.
+ [3] A. Branquinho, J. E. Fernández-Díaz, A. Foulquié-Moreno and M. Mañas, Hypergeometric Multiple Orthogonal Polynomials and Random Walks, (2021) arXiv:2107.00770v2
We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szegö polynomials. These results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions.
+This is a joint work with Markus Hunziker, Taylor Poe, and Brian Simanek, all at Baylor University.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Andrei Martínez Finkelshtein (Baylor University, Estados Unidos, y Universidad de Almería, España) @@ -1463,49 +1463,49 @@ charlas: - titulo: Orbit equivalence and dynamical properties of minimal group actions on the Cantor set abstract: | - Two topological dynamical systems \((X, T, G)\) and \((Y,S,\Gamma)\) are said to be (topologically) orbit equivalent if there exists a homeomorphism \(h:X\to Y\) sending \(T\)-orbits onto \(S\)-orbits. A general question in the framework of orbit equivalence is how the dynamical properties of a given system are related to its orbit equivalence class. In this talk we will discuss about what are the natural restrictions that arise from being orbit equivalent, in the case of minimal shift actions on the Cantor set given by countable amenable groups. We will also discuss about the problem of realization of Choquet simplices as sets of invariant measures of topological dynamical systems and see how this is connected to the problem of classification up to orbit equivalence. +Two topological dynamical systems \((X, T, G)\) and \((Y,S,\Gamma)\) are said to be (topologically) orbit equivalent if there exists a homeomorphism \(h:X\to Y\) sending \(T\)-orbits onto \(S\)-orbits. A general question in the framework of orbit equivalence is how the dynamical properties of a given system are related to its orbit equivalence class. In this talk we will discuss about what are the natural restrictions that arise from being orbit equivalent, in the case of minimal shift actions on the Cantor set given by countable amenable groups. We will also discuss about the problem of realization of Choquet simplices as sets of invariant measures of topological dynamical systems and see how this is connected to the problem of classification up to orbit equivalence.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Paulina Cecchi (Universidad de Chile, Chile) - titulo: Random Walks and CAT(0) Cube Complexes abstract: | - Let G be a group acting on a finite dimensional CAT(0) cube complex X. By studying equivariant maps from the Furstenberg-Poisson boundary to the Roller boundary, we deduce a variety of phenomena concerning the push-forward of the random walk from G to an orbit in X. Under mild and natural assumptions, we deduce positivity of the drift, sublinear tracking, and a central limit theorem. Along the way we prove that regular elements are plentiful and establish a homeomorphism between the boundary of the contact graph of X with a special subset of the Roller boundary called the regular points. This is joint work with Jean Lécureux and Frédéric Mathéus. +Let G be a group acting on a finite dimensional CAT(0) cube complex X. By studying equivariant maps from the Furstenberg-Poisson boundary to the Roller boundary, we deduce a variety of phenomena concerning the push-forward of the random walk from G to an orbit in X. Under mild and natural assumptions, we deduce positivity of the drift, sublinear tracking, and a central limit theorem. Along the way we prove that regular elements are plentiful and establish a homeomorphism between the boundary of the contact graph of X with a special subset of the Roller boundary called the regular points. This is joint work with Jean Lécureux and Frédéric Mathéus.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Talia Fernós (The University of North Carolina at Greensboro, Estados Unidos) - titulo: Optimal regularity of mapping class group actions on the circle abstract: | - We prove that for each finite index subgroup \(H\) of the mapping class group of a closed hyperbolic surface, and for each real number \(r>0\) there does not exist a faithful \(C^{1+r}\)--action of \(H\) on a circle. (Joint with Thomas Koberda and Cristobal Rivas) +We prove that for each finite index subgroup \(H\) of the mapping class group of a closed hyperbolic surface, and for each real number \(r>0\) there does not exist a faithful \(C^{1+r}\)--action of \(H\) on a circle. (Joint with Thomas Koberda and Cristobal Rivas).
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Sang-hyun Kim (Korea Institute for Advanced Study, Corea del Sur) - titulo: Hyperbolic groups acting on their boundaries abstract: | - A hyperbolic group acts on its Gromov boundary by homeomorphisms. In recent joint work with Jason Manning, we show these actions are topologically stable whenever the boundary is a sphere: any small perturbation of the action is semi-conjugate to the original action. This is also true for free groups, with cantor set boundary, and in ongoing work we are investigating the general case. In my talk, I will explain some of the strategy of the proof and motivation for the problem. +A hyperbolic group acts on its Gromov boundary by homeomorphisms. In recent joint work with Jason Manning, we show these actions are topologically stable whenever the boundary is a sphere: any small perturbation of the action is semi-conjugate to the original action. This is also true for free groups, with cantor set boundary, and in ongoing work we are investigating the general case. In my talk, I will explain some of the strategy of the proof and motivation for the problem.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Kathryn Mann (Cornell University, Estados Unidos) - titulo: 'Locally moving groups acting on the real line I: C^1 actions' abstract: | - Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F. - In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action. - In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it. - The talks are based on joint work of the speakers with J. Brum and C. Rivas. +Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F.
+In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action.
+In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it.
+The talks are based on joint work of the speakers with J. Brum and C. Rivas.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Michele Triestino (Université de Bourgogne, Francia) - titulo: 'Locally moving groups acting on the real line II: exotic actions' abstract: | - Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F. - In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action. - In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it. - The talks are based on joint work of the speakers with J. Brum and C. Rivas. +Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F.
+In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action.
+In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it.
+The talks are based on joint work of the speakers with J. Brum and C. Rivas.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Nicolás Matte Bon (Université de Lyon, Francia) - titulo: Projective manifolds, hyperbolic manifolds and the Hessian of Hausdorff dimension abstract: | - Let \(\Gamma\) be the fundamental group of a closed (real) hyperbolic \(n\)-manifold \(M.\) We study the second variation of the Hausdorff dimension of the limit set of convex co-compact morphisms acting on the complex-hyperbolic space \(\rho:\Gamma\to Isom(\mathbb H^n_\mathbb C)\), obtained by deforming a discrete and faithful representation of \(\Gamma\) that preserves a totally geodesic (and totally real) copy of the real-hyperbolic space \(\mathbb H^n_\mathbb R\subset\mathbb H^n_\mathbb C\). This computation is based on the study of the space of convex projective structures on \(M\) and a natural metric on it induced by the Pressure form. This is joint work with M. Bridgeman, B. Pozzetti and A. Wienhard. +Let \(\Gamma\) be the fundamental group of a closed (real) hyperbolic \(n\)-manifold \(M.\) We study the second variation of the Hausdorff dimension of the limit set of convex co-compact morphisms acting on the complex-hyperbolic space \(\rho:\Gamma\to Isom(\mathbb H^n_\mathbb C)\), obtained by deforming a discrete and faithful representation of \(\Gamma\) that preserves a totally geodesic (and totally real) copy of the real-hyperbolic space \(\mathbb H^n_\mathbb R\subset\mathbb H^n_\mathbb C\). This computation is based on the study of the space of convex projective structures on \(M\) and a natural metric on it induced by the Pressure form. This is joint work with M. Bridgeman, B. Pozzetti and A. Wienhard.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Andrés Sambarino (Sorbonne Université, Francia) @@ -1518,68 +1518,68 @@ charlas: - titulo: Coupled reaction-diffusion and difference system with nonlocal dispersal term and implicit time delay abstract: | - We consider a class of biological models represented by a coupled system between reaction-diffusion and difference equations (renewal equation) with nonlocal dispersal term and implicit time delay. The difference equation generally arises, using the characteristic method, from an age-structured partial differential system. We start by studying the existence, uniqueness, positivity, and boundedness of solutions. We then investigate the stability analysis and obtain a threshold condition for the global asymptotic stability of the trivial steady state by using a Lyapunov functional. We also obtain a sufficient condition for the existence and uniqueness of a positive steady-state by using the method of lower and upper solutions. In the case of an unbounded domain, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior. +We consider a class of biological models represented by a coupled system between reaction-diffusion and difference equations (renewal equation) with nonlocal dispersal term and implicit time delay. The difference equation generally arises, using the characteristic method, from an age-structured partial differential system. We start by studying the existence, uniqueness, positivity, and boundedness of solutions. We then investigate the stability analysis and obtain a threshold condition for the global asymptotic stability of the trivial steady state by using a Lyapunov functional. We also obtain a sufficient condition for the existence and uniqueness of a positive steady-state by using the method of lower and upper solutions. In the case of an unbounded domain, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Mostafa Adimy (Institut de la Recherche en Informatique et Automatique, Francia) - titulo: Periodic positive solutions of superlinear delay equations via topological degree abstract: | - We present a joint work with Pablo Amster (Universidad de Buenos Aires, Argentina) and Julián Haddad (Universidade Federal de Minas Gerais, Brazil), which deals with an extension to delay problems of some recent results by G. Feltrin and F. Zanolin. They obtained existence and multiplicity results for positive periodic solutions to nonlinear differential equations of the form +We present a joint work with Pablo Amster (Universidad de Buenos Aires, Argentina) and Julián Haddad (Universidade Federal de Minas Gerais, Brazil), which deals with an extension to delay problems of some recent results by G. Feltrin and F. Zanolin. They obtained existence and multiplicity results for positive periodic solutions to nonlinear differential equations of the form \[u''(t)=f(t,u(t),u'(t))\] with periodic or Neumann boundary conditions and special assumptions on $f$. We obtain analogous results for a delay problem of the type \[u''(t)=f(t,u(t),u(t-\tau),u'(t))\] - Our approach, as well as that of Feltrin and Zanolin, is topological and based on the coincidence degree introduced by J. Mawhin. + Our approach, as well as that of Feltrin and Zanolin, is topological and based on the coincidence degree introduced by J. Mawhin.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Pierluigi Benevieri (Universidade de Sao Paulo, Brasil) - titulo: A Krasnoselskii relatedness principle for delay differential equations abstract: | - In this talk, we shall present an \textbf{Abstract} formulation of a duality principle established by Krasnoselskii. Under appropriate conditions, it shall be shown that, if the solutions of a nonlinear functional equation can be obtained by finding fixed points of certain operators in possibly different Banach spaces, then these operators have the same topological index. An application to delay differential equations shall be given. This is a joint work with P. Amster. +In this talk, we shall present an \textbf{Abstract} formulation of a duality principle established by Krasnoselskii. Under appropriate conditions, it shall be shown that, if the solutions of a nonlinear functional equation can be obtained by finding fixed points of certain operators in possibly different Banach spaces, then these operators have the same topological index. An application to delay differential equations shall be given. This is a joint work with P. Amster.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Julian Epstein (Universidad de Buenos Aires, Argentina) - titulo: Stability and boundedness of solutions of retarded equations abstract: | - Generalized ODEs were introduced by J. Kurzweil in 1957 and are known to encompass several other types of equations. In this talk, we explore converse Lyapunov theorems for this type of equations. In particular, we derive results for certain integral forms of measure functional differential equations. We also relate Lyapunov stability to the boundedness of solutions of these equations which have non--absolute integrable right--hand sides. +Generalized ODEs were introduced by J. Kurzweil in 1957 and are known to encompass several other types of equations. In this talk, we explore converse Lyapunov theorems for this type of equations. In particular, we derive results for certain integral forms of measure functional differential equations. We also relate Lyapunov stability to the boundedness of solutions of these equations which have non--absolute integrable right--hand sides.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Márcia Federson (Universidade de Sao Paulo, Brasil) - titulo: Global bifurcation-like Theorems in presence of non-vanishing Spectral Flow abstract: | - Given a one-parameter family of functions \(f_t:H \to \mathbb R\) where \(f_t (0) = 0, \nabla f_t (0) = 0\) and \(H\) is a real Hilbert space, the Spectral Flow of the Hessian of \(f\) was related recently to local bifurcation results by Fitzpatrick, Pejsachowicz and Waterstraat. While this invariant is finer than the topological index, the existent results are of local nature, in contrast to the global bifurcation theorem of Krasnoselski and Rabinowitz. We prove a global bifurcation theorem for “target values” of \(f\) under Spectral Flow hypothesis. +Given a one-parameter family of functions \(f_t:H \to \mathbb R\) where \(f_t (0) = 0, \nabla f_t (0) = 0\) and \(H\) is a real Hilbert space, the Spectral Flow of the Hessian of \(f\) was related recently to local bifurcation results by Fitzpatrick, Pejsachowicz and Waterstraat. While this invariant is finer than the topological index, the existent results are of local nature, in contrast to the global bifurcation theorem of Krasnoselski and Rabinowitz. We prove a global bifurcation theorem for “target values” of \(f\) under Spectral Flow hypothesis.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Julian Haddad (Universidade Federal de Minas Gerais, Brasil) - titulo: On the Stability and the Existence and Non-existence of \(T-\)Periodic Solutions for Nonlinear Delayed Differential Equations with Friction and \(\varphi\)-Laplacian abstract: | - Let us consider the following problem +Let us consider the following problem \[(\varphi(x'(t) ))' + h(x(t),x'(t))x'(t)+ g(x(t-r)) = p(t), \ \ \ t\in [0,\infty),\] - where \(\varphi:\mathbb{R} \rightarrow \mathbb{R}\) is an increasing homeomorphism such that \(\varphi(0)=0\), \(r\) is a positive constant, \(g:\mathbb{R} \rightarrow \mathbb{R}\) is a continuous differentiable function, and \(h:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}\) and \(p:[0,\infty) \rightarrow \mathbb{R}\) are continuous functions such that \(p\) is \(T\)-periodic in \(t\) for \(T\) a positive constant. - Using Lyapunov-Krasovskii functional and under appropriate assumptions we obtain new results on the global stability, boundedness of solutions, existence and non-existence of \(T-\)periodic solutions. This is a joint work with P. Amster and D.P. Santos. + where \(\varphi:\mathbb{R} \rightarrow \mathbb{R}\) is an increasing homeomorphism such that \(\varphi(0)=0\), \(r\) is a positive constant, \(g:\mathbb{R} \rightarrow \mathbb{R}\) is a continuous differentiable function, and \(h:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}\) and \(p:[0,\infty) \rightarrow \mathbb{R}\) are continuous functions such that \(p\) is \(T\)-periodic in \(t\) for \(T\) a positive constant.
+Using Lyapunov-Krasovskii functional and under appropriate assumptions we obtain new results on the global stability, boundedness of solutions, existence and non-existence of \(T-\)periodic solutions. This is a joint work with P. Amster and D.P. Santos.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Mariel Paula Kuna (Universidad de Buenos Aires, Argentina) - titulo: Linearized instability for neutral FDEs abstract: | - In this talk, we will give a brief overview of a class of equations called neutral functional differential equations with state-dependent delays, describing some important applications. After this, we will show some recents results in the area, and we will present a principle of linearized instability for these equations.This is a joint work with Professor Bernhard Lani-Wayda. +In this talk, we will give a brief overview of a class of equations called neutral functional differential equations with state-dependent delays, describing some important applications. After this, we will show some recents results in the area, and we will present a principle of linearized instability for these equations.This is a joint work with Professor Bernhard Lani-Wayda.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Jaqueline Godoy Mesquita (Universidade de Brasilia, Brasil) - titulo: Existence of bifurcation point for impulsive differential equations via generalized ordinary differential equations abstract: | - This is a joint work with Professors M. Federson and J. Mawhin. In this work, we establish conditions for the existence of a bifurcation point with respect to the trivial solution of a generalized ordinary differential equation, whose integral form displays the nonabsolute Kurzweil integral. The main tools employed here are the coincidence degree theory and an Arzela-Ascoli-type theorem for regulated functions. We also present applications to impulsive differential equations. +This is a joint work with Professors M. Federson and J. Mawhin. In this work, we establish conditions for the existence of a bifurcation point with respect to the trivial solution of a generalized ordinary differential equation, whose integral form displays the nonabsolute Kurzweil integral. The main tools employed here are the coincidence degree theory and an Arzela-Ascoli-type theorem for regulated functions. We also present applications to impulsive differential equations.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Maria Carolina Mesquita (Universidade de Sao Paulo, Brasil) - titulo: A Lazer-Leach type result for functional-differential equations at resonance abstract: | - In this talk we will discuss some recent results regarding Lazer-Leach type conditions for the existence of periodic solutions to systems of functional-differential equations at resonance. We consider even-dimensional kernels and general delays using Coincidence Degree Theory. +In this talk we will discuss some recent results regarding Lazer-Leach type conditions for the existence of periodic solutions to systems of functional-differential equations at resonance. We consider even-dimensional kernels and general delays using Coincidence Degree Theory.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Arturo Sanjuán (Universidad Distrital Francisco José de Caldas, Colombia) - titulo: On the nonlinearly determined wavefronts for the Mackey-Glass type diffusive equations abstract: | - We study the Mackey-Glass type monostable delayed reaction-diffusion equation with a unimodal birth function \(g(u)\). This model, designed to describe evolution of single species populations, is considered in the presence of the weak Allee effect (\(g(u_0)>g'(0)u_0\) for some \(u_0>0\)). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. By considering the particular case of Nicholson's diffusive equation, we also discuss other situation leading to the appearance of nonlinearly determined wavefronts. This is a joint work with Karel Hasík, Jana Kopfová and Petra Nábělková, Mathematical Institute, Silesian University, Czech Republic. +We study the Mackey-Glass type monostable delayed reaction-diffusion equation with a unimodal birth function \(g(u)\). This model, designed to describe evolution of single species populations, is considered in the presence of the weak Allee effect (\(g(u_0)>g'(0)u_0\) for some \(u_0>0\)). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. By considering the particular case of Nicholson's diffusive equation, we also discuss other situation leading to the appearance of nonlinearly determined wavefronts. This is a joint work with Karel Hasík, Jana Kopfová and Petra Nábělková, Mathematical Institute, Silesian University, Czech Republic.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Sergei Trofimchuk (Universidad de Talca, Chile) @@ -1847,91 +1847,91 @@ charlas: - titulo: Medición de la actividad eléctrica cardiaca utilizando algoritmos basados en la transformada wavelet abstract: | - El electrocardiograma (ECG) es una señal que representa la actividad eléctrica del corazón. A partir de la digitalización del ECG se han implementado una diversidad de estudios que requieren algoritmos de medición automática. Algunos estudios son los registros Holter para monitoreo ambulatorio, el análisis del ECG de alta resolución, las pruebas de esfuerzo, monitoreo continuo en unidades coronarias o inclusive el monitoreo telemétrico en humanos y/o animales. - Como primera etapa se requiere la detección de las posiciones de inicios, máximos y fines de las ondas que constituyen la señal electrocardiográfica. Posteriormente, utilizando las posiciones previamente detectadas, se calculan duraciones, amplitudes e intervalos del ECG, utilizados para el monitoreo y/o diagnóstico cardiovascular. - El estado del arte presenta una gran diversidad de técnicas para detección, monitoreo y diagnóstico de registros electrocardiográficos digitales. Una de ellas se basa en el uso de la Transformada Wavelet (TW), siendo hasta el momento una de las herramientas más poderosas y con mayor rendimiento en el delineado del registro de ECG. Se presentaran las bases del algoritmo de delineado del ECG basado en la TW y resultados obtenidos de estudios en registros de humanos y animales. +El electrocardiograma (ECG) es una señal que representa la actividad eléctrica del corazón. A partir de la digitalización del ECG se han implementado una diversidad de estudios que requieren algoritmos de medición automática. Algunos estudios son los registros Holter para monitoreo ambulatorio, el análisis del ECG de alta resolución, las pruebas de esfuerzo, monitoreo continuo en unidades coronarias o inclusive el monitoreo telemétrico en humanos y/o animales.
+Como primera etapa se requiere la detección de las posiciones de inicios, máximos y fines de las ondas que constituyen la señal electrocardiográfica. Posteriormente, utilizando las posiciones previamente detectadas, se calculan duraciones, amplitudes e intervalos del ECG, utilizados para el monitoreo y/o diagnóstico cardiovascular.
+El estado del arte presenta una gran diversidad de técnicas para detección, monitoreo y diagnóstico de registros electrocardiográficos digitales. Una de ellas se basa en el uso de la Transformada Wavelet (TW), siendo hasta el momento una de las herramientas más poderosas y con mayor rendimiento en el delineado del registro de ECG. Se presentaran las bases del algoritmo de delineado del ECG basado en la TW y resultados obtenidos de estudios en registros de humanos y animales.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Pedro D. Arini (Universidad de Buenos Aires, Argentina), joint with Javier G. Mendieta, and Santiago F. Caracciolo - titulo: Comportamiento de series de criptomonedas al inicio de la pandemia Covid-19 abstract: | - Estudiamos el comportamiento de la memoria a largo plazo de rendimiento y volatilidad de siete de las criptomonedas más relevantes, durante un período que abarca antes y después del inicio de la pandemia. Se calcula el exponente de Hurst por un método wavelet que hemos modificado inspiradas en la bibliografía. Comparamos los resultados obtenidos en diferentes frecuencias, llegando a altas frecuencias, de muestreo y discutimos el efecto de la pandemia de Covid-19 sobre el rendimiento y la volatilidad. +Estudiamos el comportamiento de la memoria a largo plazo de rendimiento y volatilidad de siete de las criptomonedas más relevantes, durante un período que abarca antes y después del inicio de la pandemia. Se calcula el exponente de Hurst por un método wavelet que hemos modificado inspiradas en la bibliografía. Comparamos los resultados obtenidos en diferentes frecuencias, llegando a altas frecuencias, de muestreo y discutimos el efecto de la pandemia de Covid-19 sobre el rendimiento y la volatilidad.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Ma. Belén Arouxet (Universidad Nacional de La Plata, Argentina), joint with Verónica E. Pastor - titulo: Análisis de coherencia wavelet en finanzas con datos alternativos abstract: | - Los mercados financieros son muy ricos en datos y permiten el estudio y modelización de procesos estocásticos de una gran importancia económica. Tradicionalmente, los estudios se realizaban con datos generados dentro de los mercados (precios, volúmenes negociados, etc.). Sin embargo, en los últimos años han surgido datos "alternativos". Entendemos por "alternativos" a aquellos datos que, si bien no surgen del propio mercado, pueden explicar el comportamiento de series temporales financieras: Twitts, consultas de Wikipedia, geolocalización de consumidores, etc. - Nosotros estudiamos la interacción de los precios y volatilidades de criptomonedas con la atención del público en dicho mercado medido por medio de Google Trends. Comparamos resultados utilizando diferentes conjuntos de palabras clave contenidas en Google Trends, y en distintos momentos del mercado (mercado bajista, alcista y plano). +Los mercados financieros son muy ricos en datos y permiten el estudio y modelización de procesos estocásticos de una gran importancia económica. Tradicionalmente, los estudios se realizaban con datos generados dentro de los mercados (precios, volúmenes negociados, etc.). Sin embargo, en los últimos años han surgido datos "alternativos". Entendemos por "alternativos" a aquellos datos que, si bien no surgen del propio mercado, pueden explicar el comportamiento de series temporales financieras: Twitts, consultas de Wikipedia, geolocalización de consumidores, etc.
+Nosotros estudiamos la interacción de los precios y volatilidades de criptomonedas con la atención del público en dicho mercado medido por medio de Google Trends. Comparamos resultados utilizando diferentes conjuntos de palabras clave contenidas en Google Trends, y en distintos momentos del mercado (mercado bajista, alcista y plano).
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Aurelio F. Bariviera, (Universitat Rovira i Virgili, España) - titulo: Bases B-spline wavelet ortogonales en el intervalo en la solución de problemas de valores de contorno abstract: | - Wavelets y análisis multirresolución constituyen una herramienta atractiva para la resolución numérica de ecuaciones diferenciales. La utilización de bases wavelet en los métodos wavelet-Galerkin permiten obtener esquemas numéricos eficientes y de elevada precisión. - En el marco de un Análisis Multirresolución (AMR) en el intervalo, se propone la construcción de una base B-spline wavelet con un requerimiento de ortogonalidad sobre sus derivadas entre distintas escalas de aproximación. La base está formada por wavelets interiores (que se obtienen de traslaciones y dilaciones una wavelet madre) cuyo soporte está contenido en el intervalo y wavelets de borde especialmente diseñadas. Al aplicar estas bases en la discretización de ecuaciones diferenciales de segundo orden, mediante esquemas del tipo wavelet-Galerkin, conducen a la resolución de sistemas lineales. Debido al soporte compacto y la condición de ortogonalidad requerida, las matrices asociadas tienen buenas propiedades, son ralas o esparcidas (diagonales por bloques y cada bloque es una matriz banda) con número de condición uniformemente acotado, lo que permite eficiencia en los cálculos y buenos resultados de convergencia a la solución con bajo costo computacional. +Wavelets y análisis multirresolución constituyen una herramienta atractiva para la resolución numérica de ecuaciones diferenciales. La utilización de bases wavelet en los métodos wavelet-Galerkin permiten obtener esquemas numéricos eficientes y de elevada precisión.
+En el marco de un Análisis Multirresolución (AMR) en el intervalo, se propone la construcción de una base B-spline wavelet con un requerimiento de ortogonalidad sobre sus derivadas entre distintas escalas de aproximación. La base está formada por wavelets interiores (que se obtienen de traslaciones y dilaciones una wavelet madre) cuyo soporte está contenido en el intervalo y wavelets de borde especialmente diseñadas. Al aplicar estas bases en la discretización de ecuaciones diferenciales de segundo orden, mediante esquemas del tipo wavelet-Galerkin, conducen a la resolución de sistemas lineales. Debido al soporte compacto y la condición de ortogonalidad requerida, las matrices asociadas tienen buenas propiedades, son ralas o esparcidas (diagonales por bloques y cada bloque es una matriz banda) con número de condición uniformemente acotado, lo que permite eficiencia en los cálculos y buenos resultados de convergencia a la solución con bajo costo computacional.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Lucila D. Calderón, (Universidad Nacional de La Plata, Argentina), joint with María T. Martín - titulo: 'Algoritmo para recuperar señales espacio-temporales con descomposición rala utilizando diccionarios wavelet: aplicación al mapeo electrocardiográfico' abstract: | - La construcción de imágenes a partir del electrocardiograma de superficie es una técnica de diagnóstico médico utilizada para estimar, no invasivamente, el potencial eléctrico en la superficie del corazón (epicardio) a partir del potencial eléctrico en el torso del paciente. - Para llevar a cabo dicha técnica, se requiere 1) una imagen médica con la cual se puede crear un modelo del torso y del músculo cardíaco, 2) la adquisición no invasiva de una gran cantidad de canales electrocardiográficos. A partir de lo mencionado, puede estimarse el potencial eléctrico sobre el epicardio, a través de la solución a un problema inverso mal condicionado el cual debe ser regularizado. - En el presente trabajo se desarrolla un algoritmo de optimización rala (del inglés, sparse) diseñado para modelos lineales con estructura de Kronecker el cual fue penalizado con una mezcla entre la norma 21 (LASSO de grupo) y la norma 2 al cuadrado (Tikhonov). Nuestra hipótesis se basó en que el potencial epicárdico puede descomponerse de manera rala en diccionarios creados a partir de funciones y/o escalas wavelet. +La construcción de imágenes a partir del electrocardiograma de superficie es una técnica de diagnóstico médico utilizada para estimar, no invasivamente, el potencial eléctrico en la superficie del corazón (epicardio) a partir del potencial eléctrico en el torso del paciente.
+Para llevar a cabo dicha técnica, se requiere 1) una imagen médica con la cual se puede crear un modelo del torso y del músculo cardíaco, 2) la adquisición no invasiva de una gran cantidad de canales electrocardiográficos. A partir de lo mencionado, puede estimarse el potencial eléctrico sobre el epicardio, a través de la solución a un problema inverso mal condicionado el cual debe ser regularizado.
+En el presente trabajo se desarrolla un algoritmo de optimización rala (del inglés, sparse) diseñado para modelos lineales con estructura de Kronecker el cual fue penalizado con una mezcla entre la norma 21 (LASSO de grupo) y la norma 2 al cuadrado (Tikhonov). Nuestra hipótesis se basó en que el potencial epicárdico puede descomponerse de manera rala en diccionarios creados a partir de funciones y/o escalas wavelet.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Santiago F. Caracciolo (Universidad de Buenos Aires, Argentina) joint with César F. Caiafa, Francisco D. Martínez Pería and Pedro D. Arini - titulo: Detection and analysis of micro saccadic movements during reading using the Continuous Wavelet Transform abstract: | - Reading requires the integration of several central cognitive subsystems from attention and oculomotor control to word identification and language comprehension. When reading, the eyes alternate between long movements and relative stillness, that are called saccadic movements and fixations, respectively. The average fixation lasts for 150 to 250 ms and it is composed by three movements called microsaccades (or microsaccadic movements), tremor and drift. Drift and tremor are slow movements with small amplitude; microsaccades represent a ballistic component of fixational eye movements. Then, microsaccades are characterized as roughly linear movement epochs with durations up to 30ms and a frequency of one to two per second in fixations not related with reading. They are considered as binocular movements with the standard definition of binocularity used in the literature. There are just a few works analyzing microsaccades while subjects are processing complex information and fewer when doing predictions about upcoming events. In all of them there is evidence that microsaccades are sensitive to changes of perceptual inputs as well as modulations of cognitive states. Changes in perceptual inputs are related to the type of sentences (low/high predictability, proverbs) and the characteristics of the words in the sentence (frequency, predictability, length, etc.). For this reason we think it is important to detect and characterise microsaccadics during the reading process. - It is well known that the Continuous Wavelet Transform (CWT) is an efficient method for displaying and analyzing characteristics of nonstationary signals that are dependent on time and scale and then on frequency. Taking this into account, it is possible to say that it provides a very useful tool for detecting and identifying particular spectral features of the analyzed signal, transient information content and the nonstationary properties, among others. In the context of eye movements, it has been used to characterize and extract microsaccades. Microsaccades can be modeled as smoothed singularities within a time series. These local singularities can be identified using, for example, the CWT and the results are not sensitive to the choice of the mother wavelet. - In time-frequency representation, a singularity at time \(t_0\) appears as a cone-like structure in the modulus of the wavelet transform. At lower frequencies, the wavelet is broader and the width of the cone scales with the width of the wavelet function at each voice or scale a. The wavelet focuses the whole variance at time points of singularities into the cone and its maximum at each voice corresponds to the position of the singularity in the time series. Therefore, the singularities can be detected using the modulus of the wavelet transform. - The method of maximum modulus lines is a numerical method for detecting singularities in time series using the CWT. A maximum modulus line is a line on the time-scale plane on which the modulus of the wavelet transform has a local maximum with respect to small variation of the translation parameter \(b_0\), that is - $$ |W_{f,\psi}(a,b_0)|>|W_{f,\psi}(a,b_0\pm\epsilon)|.$$ - As usual, \(W_{f,\psi}\) stands for the CWT of the time series of the signal \(f\) using the mother wavelet \(\psi\) and \(a, b_0\) the dilation and translation parameters, respectively. This points of maximum modulus are then connected giving as resulta the maximum modulus lines. It can be shown that if a signal has a singularity at a given point, then there is a maximum modulus line that converges towards the location of the singularity at small scales. In smoothed singularities the maximum modulus line may end at a scale which is approximately the smoothing scale of the singularity. For this reason, the maximum modulus lines, which go from a fixed highest frequency/smallest scale to a smallest frequency/largest scale were considered and the estimated position of the singularity is simply the small-scale end of the corresponding maximum modulus line. - In our work we have considered as mother wavelet the following function, +Reading requires the integration of several central cognitive subsystems from attention and oculomotor control to word identification and language comprehension. When reading, the eyes alternate between long movements and relative stillness, that are called saccadic movements and fixations, respectively. The average fixation lasts for 150 to 250 ms and it is composed by three movements called microsaccades (or microsaccadic movements), tremor and drift. Drift and tremor are slow movements with small amplitude; microsaccades represent a ballistic component of fixational eye movements. Then, microsaccades are characterized as roughly linear movement epochs with durations up to 30ms and a frequency of one to two per second in fixations not related with reading. They are considered as binocular movements with the standard definition of binocularity used in the literature. There are just a few works analyzing microsaccades while subjects are processing complex information and fewer when doing predictions about upcoming events. In all of them there is evidence that microsaccades are sensitive to changes of perceptual inputs as well as modulations of cognitive states. Changes in perceptual inputs are related to the type of sentences (low/high predictability, proverbs) and the characteristics of the words in the sentence (frequency, predictability, length, etc.). For this reason we think it is important to detect and characterise microsaccadics during the reading process.
+It is well known that the Continuous Wavelet Transform (CWT) is an efficient method for displaying and analyzing characteristics of nonstationary signals that are dependent on time and scale and then on frequency. Taking this into account, it is possible to say that it provides a very useful tool for detecting and identifying particular spectral features of the analyzed signal, transient information content and the nonstationary properties, among others. In the context of eye movements, it has been used to characterize and extract microsaccades. Microsaccades can be modeled as smoothed singularities within a time series. These local singularities can be identified using, for example, the CWT and the results are not sensitive to the choice of the mother wavelet.
+In time-frequency representation, a singularity at time \(t_0\) appears as a cone-like structure in the modulus of the wavelet transform. At lower frequencies, the wavelet is broader and the width of the cone scales with the width of the wavelet function at each voice or scale a. The wavelet focuses the whole variance at time points of singularities into the cone and its maximum at each voice corresponds to the position of the singularity in the time series. Therefore, the singularities can be detected using the modulus of the wavelet transform.
+The method of maximum modulus lines is a numerical method for detecting singularities in time series using the CWT. A maximum modulus line is a line on the time-scale plane on which the modulus of the wavelet transform has a local maximum with respect to small variation of the translation parameter \(b_0\), that is + $$ |W_{f,\psi}(a,b_0)|>|W_{f,\psi}(a,b_0\pm\epsilon)|.$$
+As usual, \(W_{f,\psi}\) stands for the CWT of the time series of the signal \(f\) using the mother wavelet \(\psi\) and \(a, b_0\) the dilation and translation parameters, respectively. This points of maximum modulus are then connected giving as resulta the maximum modulus lines. It can be shown that if a signal has a singularity at a given point, then there is a maximum modulus line that converges towards the location of the singularity at small scales. In smoothed singularities the maximum modulus line may end at a scale which is approximately the smoothing scale of the singularity. For this reason, the maximum modulus lines, which go from a fixed highest frequency/smallest scale to a smallest frequency/largest scale were considered and the estimated position of the singularity is simply the small-scale end of the corresponding maximum modulus line.
+In our work we have considered as mother wavelet the following function, $$\psi(t)=-\theta'(t), \qquad \theta(t)=\text{e}^{-t^2}.$$ - This mother wavelet has a null moment and the wavelet transform of the signal has a similar behaviour to the derivative of the signal. In this way, the CWT allows to detect the abrupt changes of the position, i.e. the singularities of its velocity. - Another method for detecting microsaccades is to consider them as outliers of the velocity since, as we have said before, they can be thought as ballistic movements. For the algorithm used in this method of detection, it is necessary to determine the value of a parameter \(\lambda\) and a minimum duration for the movement to be considered as a microsaccadic. Different works use different values for these parameters since it depends on the application and the determined value so that the detection is coherent. For our work we determined that \(\lambda=1.5\) and a minimum duration of 6 samples (i.e. 5ms duration) turn out to be a mean rate according to different works related to reading experiments. - Comparing the method of maximum modulus lines and the velocity algorithm using these parameters we have precisely determined the microsaccadic movements during the reading process. - For the experiment, we have considered two different groups of healthy people with similar education: one group of young adults and other group of elderly people. The first group consisted of 40 adults with mean age 28 (SD=4.2 years) and mean education 18.2 (in years). The second one involved 40 adults with mean age 71 (SD=6.1 years), mean education 15.1 (in years). We only took into account data of 19 subjects from the first group and 18 from the second one. To perform the experiment, we used a Spanish sentence corpus composed of 76 regular sentences that represent a large variety of grammatical structures, also called Low Predictability Sentences, and 64 Proverbs that are common our Argentinian culture. Each sentence was displayed in the centerline of a 20-inch LCD Monitor (1024\(\times\)768 pixels resolution; font: regular New Courier, 18 point, vertical size of one character: 0.2 in height). The participants were seated in front of the monitor at a distance of 60 cm. Head movements were minimized using a chin rest. The participants eye movements were recorded with an EyeLink 1000 Desktop Mount (SR Research) eyetracker, with a sampling rate of 1000 Hz and an eye position resolution of 20-s arc. All recordings and calibration were binocular. The participants gaze was calibrated with a standard 13-point grid for both eyes. After validating the calibration, a fixation point appeared at the position where the first letter of the sentence was to be presented. As soon as both eyes were detected within a one grade radius relative to the fixation point, the sentence was presented. After reading it, participants had to move their eyes to a dot in the lower right corner of the screen to end the trial. - The sentences ranged from 5 to 14 words in length; mean length being 7.3 (SD=1.9) words. The words ranged from 1 to 14 letters in length; the mean word length was 4.0 (SD=2) letters. All the participants read the whole set of sentences but the data with registration or reading errors were discarded. So the group of young participants read 3385 sentences and the group of adults read 2569 sentences. We have first detected the fixations on the words and then extracted the microsaccadic movements. For the first group were counted 27594 fixations with mean duration of 206.83ms (SD=105.58) while for the second one were counted 22571 fixations with a mean duration of 203.80ms (SD=100.30). - Now let us recall the definition of maxjumpword (MJW): it is the word with the largest difference between the close predictability of two consecutive words and can be determined according to the following equation + This mother wavelet has a null moment and the wavelet transform of the signal has a similar behaviour to the derivative of the signal. In this way, the CWT allows to detect the abrupt changes of the position, i.e. the singularities of its velocity.
+Another method for detecting microsaccades is to consider them as outliers of the velocity since, as we have said before, they can be thought as ballistic movements. For the algorithm used in this method of detection, it is necessary to determine the value of a parameter \(\lambda\) and a minimum duration for the movement to be considered as a microsaccadic. Different works use different values for these parameters since it depends on the application and the determined value so that the detection is coherent. For our work we determined that \(\lambda=1.5\) and a minimum duration of 6 samples (i.e. 5ms duration) turn out to be a mean rate according to different works related to reading experiments.
+Comparing the method of maximum modulus lines and the velocity algorithm using these parameters we have precisely determined the microsaccadic movements during the reading process.
+For the experiment, we have considered two different groups of healthy people with similar education: one group of young adults and other group of elderly people. The first group consisted of 40 adults with mean age 28 (SD=4.2 years) and mean education 18.2 (in years). The second one involved 40 adults with mean age 71 (SD=6.1 years), mean education 15.1 (in years). We only took into account data of 19 subjects from the first group and 18 from the second one. To perform the experiment, we used a Spanish sentence corpus composed of 76 regular sentences that represent a large variety of grammatical structures, also called Low Predictability Sentences, and 64 Proverbs that are common our Argentinian culture. Each sentence was displayed in the centerline of a 20-inch LCD Monitor (1024\(\times\)768 pixels resolution; font: regular New Courier, 18 point, vertical size of one character: 0.2 in height). The participants were seated in front of the monitor at a distance of 60 cm. Head movements were minimized using a chin rest. The participants eye movements were recorded with an EyeLink 1000 Desktop Mount (SR Research) eyetracker, with a sampling rate of 1000 Hz and an eye position resolution of 20-s arc. All recordings and calibration were binocular. The participants gaze was calibrated with a standard 13-point grid for both eyes. After validating the calibration, a fixation point appeared at the position where the first letter of the sentence was to be presented. As soon as both eyes were detected within a one grade radius relative to the fixation point, the sentence was presented. After reading it, participants had to move their eyes to a dot in the lower right corner of the screen to end the trial.
+The sentences ranged from 5 to 14 words in length; mean length being 7.3 (SD=1.9) words. The words ranged from 1 to 14 letters in length; the mean word length was 4.0 (SD=2) letters. All the participants read the whole set of sentences but the data with registration or reading errors were discarded. So the group of young participants read 3385 sentences and the group of adults read 2569 sentences. We have first detected the fixations on the words and then extracted the microsaccadic movements. For the first group were counted 27594 fixations with mean duration of 206.83ms (SD=105.58) while for the second one were counted 22571 fixations with a mean duration of 203.80ms (SD=100.30).
+Now let us recall the definition of maxjumpword (MJW): it is the word with the largest difference between the close predictability of two consecutive words and can be determined according to the following equation< \[ maxjumpword=\text{max}[\text{logit}(pred_{N+1}-\text{logit}(pred_{N}], \] - being \(\text{logit}\big(pred_{N}\big)=0.5\hspace{2mm}\text{ln} \Big(\frac{pred_{N}}{1-pred_{N}}\Big)\) and \(pred_{N}\) the predictability of the read word \(N\). - In this work we extract the microsaccadics and perform a first analysis of changes in some of its characteristics respect to the relative position of the MJW. The considered characteristics are: length of the movement, vertical and horizontal variations, maximum velocity and duration. + being \(\text{logit}\big(pred_{N}\big)=0.5\hspace{2mm}\text{ln} \Big(\frac{pred_{N}}{1-pred_{N}}\Big)\) and \(pred_{N}\) the predictability of the read word \(N\).
+In this work we extract the microsaccadics and perform a first analysis of changes in some of its characteristics respect to the relative position of the MJW. The considered characteristics are: length of the movement, vertical and horizontal variations, maximum velocity and duration.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Liliana R. Castro (Universidad Nacional del Sur, Argentina), joint with Juan M. Arriola and Marcela P. Álvarez - titulo: Entropía y Complejidad Wavelet para identificar la inestabilidad eléctrica cardíaca en pacientes con infarto de miocardio abstract: | - Se ha demostrado que aquellos individuos que han sufrido un infarto de miocardio (IM) tienen una alta probabilidad de desarrollar arritmias ventriculares malignas y/o muerte súbita cardíaca. Las anomalías de la conducción eléctrica que aparecen en la región infartada del miocardio se reflejan en el electrocardiograma (ECG) como fragmentaciones del complejo QRS (fQRS). Dichas fragmentaciones no siempre son posibles de detectar visualmente. Si bien puede suceder que los pacientes no desarrollen taquicardia y/o fibrilación venticular (VT/VF), son potencialmente riesgosos porque pueden desarrollarla inesperadamente y sin síntomas previos. - Hay pocas técnicas no invasivas para capturar dichas inestabilidades eléctricas, como la técnica de electrocardiograma de señal promediada, y la varianza espectral del complejo QRS, entre otras. - Por ello, hemos evaluado la Entropía y Complejidad Wavelet del complejo QRS del ECG, utilizando la Transformada Wavelet Continua, como un método eficaz para cuantificar estas alteraciones anormales en la actividad eléctrica cardíaca en pacientes post IM que no presentan VT/VF. +Se ha demostrado que aquellos individuos que han sufrido un infarto de miocardio (IM) tienen una alta probabilidad de desarrollar arritmias ventriculares malignas y/o muerte súbita cardíaca. Las anomalías de la conducción eléctrica que aparecen en la región infartada del miocardio se reflejan en el electrocardiograma (ECG) como fragmentaciones del complejo QRS (fQRS). Dichas fragmentaciones no siempre son posibles de detectar visualmente. Si bien puede suceder que los pacientes no desarrollen taquicardia y/o fibrilación venticular (VT/VF), son potencialmente riesgosos porque pueden desarrollarla inesperadamente y sin síntomas previos.
+Hay pocas técnicas no invasivas para capturar dichas inestabilidades eléctricas, como la técnica de electrocardiograma de señal promediada, y la varianza espectral del complejo QRS, entre otras.
+Por ello, hemos evaluado la Entropía y Complejidad Wavelet del complejo QRS del ECG, utilizando la Transformada Wavelet Continua, como un método eficaz para cuantificar estas alteraciones anormales en la actividad eléctrica cardíaca en pacientes post IM que no presentan VT/VF.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Gisela Clemente, (Universidad Nacional de La Plata, Argentina), joint with Esteban Valverde - titulo: 'Wavelet Analysis in Space Research at the Applied Mathematics and Geophysics group at INPE: advances and overview' abstract: | - This presentation aims at a comprehensive view of ongoing research efforts in multiscale analysis applied to challenges in the context of space environment and its electrodynamical effects on the Earth and nearby regions. Based on the developments by the INPE group LANCE and collaborators, the content explored takes an overview and the next steps on the multiscale research. We discussed some results obtained in the wavelet analysis of space and geophysical dataset and their implication on understanding the physics processes. Moreover, we present the first successful results obtained in the wavelet-based multiresolution regularity detection methodology with the generic block-structured mesh adaptation in evolutionary partial differential equations. In particular, the solvers that will permit fast and reliable space weather magneto-hydrodynamic simulations and their needs in the multiscale real time-space data modelling in future. +This presentation aims at a comprehensive view of ongoing research efforts in multiscale analysis applied to challenges in the context of space environment and its electrodynamical effects on the Earth and nearby regions. Based on the developments by the INPE group LANCE and collaborators, the content explored takes an overview and the next steps on the multiscale research. We discussed some results obtained in the wavelet analysis of space and geophysical dataset and their implication on understanding the physics processes. Moreover, we present the first successful results obtained in the wavelet-based multiresolution regularity detection methodology with the generic block-structured mesh adaptation in evolutionary partial differential equations. In particular, the solvers that will permit fast and reliable space weather magneto-hydrodynamic simulations and their needs in the multiscale real time-space data modelling in future.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Margarete Oliveira Domingues (Instituto Nacional de Pesquisas Espaciais, Brasil), joint with Odim Mendes and Muller Moreira Lopes - titulo: 'Cryopreservation Process: analyzed through Wavelet Transforms' abstract: | - Rapid and ultra rapid cooling methods, in which liquid nitrogen (\(LN_2\)) is used as cooling agent, require special equipment and methods in order to record and analyze the thermal history of the sample to be cryopreserved. The temperature profile during the process of cooling is an important parameter in the design and description of any protocol for cryopreservation. We present a home made device, used to record the temperature at high rate during the cooling in \(LN_2\). System noise at room temperature is about \(\pm ~ 2.9 ^{\circ}\)C and at \(LN_2\) is about \(\pm ~ 6.9 ^{\circ}\)C, when in the liquid nitrogen it is (\(-196^{\circ}\)C). This noise introduced when measuring the temperature may, in some cases, difficult the analysis and interpretation of the signal. A related use of Wavelet transform is for smoothing and or de noising data based on wavelet coefficients, by means of removing the undesired frequency components. With the intention of eliminate high frequencies noise, we apply a filter in such a way that in the level ``\(j\)'' of the process, the residuals will be a soften version of the original signal having less amount of high frequency signals in contrast with level ``\(j+1\)'', and half number of data. These are the main points by which, in the present work, we have used a signal isolation method based on orthogonal wavelets, in order to analyze the remaining signal with minimum modification of the associated dynamics. Using this filtering method, system noise was virtually eliminated allowing a more precise interpretation of the significance of cooling curves. +Rapid and ultra rapid cooling methods, in which liquid nitrogen (\(LN_2\)) is used as cooling agent, require special equipment and methods in order to record and analyze the thermal history of the sample to be cryopreserved. The temperature profile during the process of cooling is an important parameter in the design and description of any protocol for cryopreservation. We present a home made device, used to record the temperature at high rate during the cooling in \(LN_2\). System noise at room temperature is about \(\pm ~ 2.9 ^{\circ}\)C and at \(LN_2\) is about \(\pm ~ 6.9 ^{\circ}\)C, when in the liquid nitrogen it is (\(-196^{\circ}\)C). This noise introduced when measuring the temperature may, in some cases, difficult the analysis and interpretation of the signal. A related use of Wavelet transform is for smoothing and or de noising data based on wavelet coefficients, by means of removing the undesired frequency components. With the intention of eliminate high frequencies noise, we apply a filter in such a way that in the level ``\(j\)'' of the process, the residuals will be a soften version of the original signal having less amount of high frequency signals in contrast with level ``\(j+1\)'', and half number of data. These are the main points by which, in the present work, we have used a signal isolation method based on orthogonal wavelets, in order to analyze the remaining signal with minimum modification of the associated dynamics. Using this filtering method, system noise was virtually eliminated allowing a more precise interpretation of the significance of cooling curves.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Ana Korol (Universidad Nacional de Rosario, Argentina) - titulo: Aplicación de Transformada Wavelet Continua a señales de Emisión Acústica de fractura de materiales frágiles abstract: | - La Emisión Acústica es un ensayo no destructivo que nos permite monitorear la integridad de un material frágil sometido a esfuerzos. Si, por ejemplo, un cilindro de roca es comprimido hasta su fractura se generan ondas elásticas en el material que a través de un sensor se convierten en ondas eléctricas, denominadas señales de Emisión Acústica. Estas señales nos dan información sobre los procesos complejos que se producen en el material tales como nucleación, crecimiento y coalescencia de microfacturas hasta que finalmente se producen macrofracturas que llevan a la ruptura del material. Las señales explosivas de EA que se generan durante el proceso involucran un amplio rango de tiempos y escalas por lo que a través del análisis con TWC nos permite identificar y evaluar los diferentes procesos. +La Emisión Acústica es un ensayo no destructivo que nos permite monitorear la integridad de un material frágil sometido a esfuerzos. Si, por ejemplo, un cilindro de roca es comprimido hasta su fractura se generan ondas elásticas en el material que a través de un sensor se convierten en ondas eléctricas, denominadas señales de Emisión Acústica. Estas señales nos dan información sobre los procesos complejos que se producen en el material tales como nucleación, crecimiento y coalescencia de microfacturas hasta que finalmente se producen macrofracturas que llevan a la ruptura del material. Las señales explosivas de EA que se generan durante el proceso involucran un amplio rango de tiempos y escalas por lo que a través del análisis con TWC nos permite identificar y evaluar los diferentes procesos.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Rosa Piotrkowski (Universidad de Buenos Aires, Argentina), joint with Miguel Eduardo Zitto - titulo: Uso de la Transformada Wavelet en el análisis de datos de la evolución del Covid-19 abstract: | - Desde el comienzo de la pandemia originada por el SARS-COV 2, se han utilizado distintas herramientas para analizar los datos disponibles, a fin de comprender su dinámica. En gran medida, en el análisis, se hizo uso de modelos matemáticos diseñados para describir la evolución de la pandemia. Se aplicaron, además, técnicas de análisis de series temporales, las cuales tienen la ventaja de acumular información, al contar con un mayor volumen de datos con el paso del tiempo. El número de infectados y muertos diarios, entre otros datos, corresponden a series no estacionarias, dado que sus propiedades estadísticas varían con el tiempo. A modo de ejemplo, la cantidad de personas a infectarse tiende a reducirse a medida que más personas se infectan. Además, si el número de infectados aumenta, suelen tomarse diferentes medidas, tales como las que restringen la movilidad, al mismo tiempo que el avance de la vacunación tiende a producir una disminución de los casos. Todos estos factores, tomados de conjunto, le otorgan una dinámica compleja a las series a estudiar: en ellas conviven múltiples fenómenos localizados tanto en el tiempo como en frecuencia, que se superponen bajo complejas estructuras. La Transformada Wavelet se presenta como una herramienta apropiada para analizar series de este tipo, dado que puede ser aplicada a series no estacionarias, y permite estudiar las relaciones subyacentes entre sus datos en el tiempo y en frecuencia, así como encontrar patrones de comportamiento. En este trabajo aplicamos estas herramientas a las series de datos de infectados, fallecidos y de movilidad de la provincia de Buenos Aires y mostramos, apelando a distintos cuantificadores, tales como la coherencia wavelet, el análisis tiempo-frecuencia de las relaciones entre distintas series. De esta forma la Transformada Wavelet se coloca como una herramienta de análisis relevante a tener en cuenta en el proceso de toma de decisiones. +Desde el comienzo de la pandemia originada por el SARS-COV 2, se han utilizado distintas herramientas para analizar los datos disponibles, a fin de comprender su dinámica. En gran medida, en el análisis, se hizo uso de modelos matemáticos diseñados para describir la evolución de la pandemia. Se aplicaron, además, técnicas de análisis de series temporales, las cuales tienen la ventaja de acumular información, al contar con un mayor volumen de datos con el paso del tiempo. El número de infectados y muertos diarios, entre otros datos, corresponden a series no estacionarias, dado que sus propiedades estadísticas varían con el tiempo. A modo de ejemplo, la cantidad de personas a infectarse tiende a reducirse a medida que más personas se infectan. Además, si el número de infectados aumenta, suelen tomarse diferentes medidas, tales como las que restringen la movilidad, al mismo tiempo que el avance de la vacunación tiende a producir una disminución de los casos. Todos estos factores, tomados de conjunto, le otorgan una dinámica compleja a las series a estudiar: en ellas conviven múltiples fenómenos localizados tanto en el tiempo como en frecuencia, que se superponen bajo complejas estructuras. La Transformada Wavelet se presenta como una herramienta apropiada para analizar series de este tipo, dado que puede ser aplicada a series no estacionarias, y permite estudiar las relaciones subyacentes entre sus datos en el tiempo y en frecuencia, así como encontrar patrones de comportamiento. En este trabajo aplicamos estas herramientas a las series de datos de infectados, fallecidos y de movilidad de la provincia de Buenos Aires y mostramos, apelando a distintos cuantificadores, tales como la coherencia wavelet, el análisis tiempo-frecuencia de las relaciones entre distintas series. De esta forma la Transformada Wavelet se coloca como una herramienta de análisis relevante a tener en cuenta en el proceso de toma de decisiones.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Victoria Vampa (Universidad Nacional de La Plata, Argentina), joint with Federico Holik @@ -2136,49 +2136,50 @@ charlas: - titulo: Small-time global exact controllability to the trajectories of the Boussinesq system abstract: | - In this talk, we consider the global exact controllability problem to the trajectories of the Boussinesq system. We show that it is possible to drive the solution to the prescribed trajectory in small time by acting on the system through the velocity and the temperature on an arbitrary small part of the boundary. The proof relies on three main arguments. First, we transform the problem into a distributed controllability problem by using a domain extension procedure. Then, we prove a global approximate controllability result by following a strategy of Coron and collaborators, which deals with the Navier-Stokes equations. This part relies on the controllability of the inviscid Boussinesq system and asymptotic boundary layer expansions. Finally, we conclude with a local controllability result that we establish with the help of a linearization argument and appropriate Carleman estimates. +In this talk, we consider the global exact controllability problem to the trajectories of the Boussinesq system. We show that it is possible to drive the solution to the prescribed trajectory in small time by acting on the system through the velocity and the temperature on an arbitrary small part of the boundary. The proof relies on three main arguments. First, we transform the problem into a distributed controllability problem by using a domain extension procedure. Then, we prove a global approximate controllability result by following a strategy of Coron and collaborators, which deals with the Navier-Stokes equations. This part relies on the controllability of the inviscid Boussinesq system and asymptotic boundary layer expansions. Finally, we conclude with a local controllability result that we establish with the help of a linearization argument and appropriate Carleman estimates.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Felipe Chaves (Universidade Federal de Paraiba, Brasil) - titulo: 'Identification of a boundary obstacle in a Stokes fluid with Navier–slip boundary conditions: an exterior approach' abstract: | - The problem of identifying an obstruction into a fluid duct has several major applications, for example in medicine the presence of a stenosis in a coronary vessels is a life threaten disease. In this talk we formulate a continuous setting and study from a numerical perspective the inverse problem of identifying an obstruction contained in a 2D elastic duct where a Stokes flow becomes turbulent after hitting the boundary (Navier--slip boundary conditions), generating an acoustic waves. To be precise, by using acoustic wave measurements at certain points at the exterior to the duct, we are able to identify the location; extension and height of the obstruction. Thus, our framework constitutes an external approach for solving this obstacle inverse problem. Synthetic examples are used in order to verify the effectiveness of the proposed numerical formulation. - In collaboration with: L. Breton, J. López Estrada and C. Montoya +The problem of identifying an obstruction into a fluid duct has several major applications, for example in medicine the presence of a stenosis in a coronary vessels is a life threaten disease. In this talk we formulate a continuous setting and study from a numerical perspective the inverse problem of identifying an obstruction contained in a 2D elastic duct where a Stokes flow becomes turbulent after hitting the boundary (Navier--slip boundary conditions), generating an acoustic waves. To be precise, by using acoustic wave measurements at certain points at the exterior to the duct, we are able to identify the location; extension and height of the obstruction. Thus, our framework constitutes an external approach for solving this obstacle inverse problem. Synthetic examples are used in order to verify the effectiveness of the proposed numerical formulation.
+In collaboration with: L. Breton, J. López Estrada and C. Montoya
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Pedro González Casanova (Universidad Nacional Autónoma de México, México) - titulo: Controllability from the exterior of fractional heat equation abstract: | - In this talk, we consider the controllability problem from the exterior for the one dimensional heat equation on the interval \((-1,1)\) associated with the fractional Laplace operator \((-\partial_x^2)^s\), where \(0\lt s\lt1\). - In the first part, we will show that there is a control function which is localized in a nonempty open set \(\mathcal O\subset \left(\mathbb{R}\setminus(-1,1)\right)\), that is, at the exterior of the interval \((-1,1)\), such that the system is null controllable at any time \(T \gt 0\) if and only if \(\frac 12 \lt s \lt1\). - Then, in the second part, we study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator on the interval \((-1,1)\). Our control function is localized in an open set \(\mathcal O\) in the exterior of \((-1,1)\). We will show that there exists a minimal (strictly positive) time \(T_{\rm min}\) such that the fractional heat dynamics can be controlled from any initial datum in \(L^2(-1,1)\) to a positive trajectory through the action of an exterior positive control, if and only if \(\frac 12 \lt s \lt 1\). In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results. +In this talk, we consider the controllability problem from the exterior for the one dimensional heat equation on the interval \((-1,1)\) associated with the fractional Laplace operator \((-\partial_x^2)^s\), where \(0\lt s\lt1\).
+In the first part, we will show that there is a control function which is localized in a nonempty open set \(\mathcal O\subset \left(\mathbb{R}\setminus(-1,1)\right)\), that is, at the exterior of the interval \((-1,1)\), such that the system is null controllable at any time \(T \gt 0\) if and only if \(\frac 12 \lt s \lt1\).
+Then, in the second part, we study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator on the interval \((-1,1)\). Our control function is localized in an open set \(\mathcal O\) in the exterior of \((-1,1)\). We will show that there exists a minimal (strictly positive) time \(T_{\rm min}\) such that the fractional heat dynamics can be controlled from any initial datum in \(L^2(-1,1)\) to a positive trajectory through the action of an exterior positive control, if and only if \(\frac 12 \lt s \lt 1\). In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Sebastián Zamorano Aliaga (Universidad de Santaigo de Chile, Chile) - titulo: On the controllability of a model system for long waves in nonlinear dispersive media abstract: | - Considered here is a higher order generalization of the classical Boussinesq system which models the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal or the propagation of long-crested waves on large lakes and oceans. Our aim is to investigate the controllability properties of this nonlinear model in terms of the values of the parameters involved in the system. We give general conditions which ensure both the well-posedness and the local exact controllability of the nonlinear problem in some well chosen Sobolev spaces. +Considered here is a higher order generalization of the classical Boussinesq system which models the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal or the propagation of long-crested waves on large lakes and oceans. Our aim is to investigate the controllability properties of this nonlinear model in terms of the values of the parameters involved in the system. We give general conditions which ensure both the well-posedness and the local exact controllability of the nonlinear problem in some well chosen Sobolev spaces.
+Joint work with G. J. Bautista Sanchez (Universidad Privada del Norte, Peru) and S. Micu (University of Craiova, Romania).
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Ademir Pazoto (Universidade Federal do Rio de Janeiro, Brasil) - titulo: 'Nematic liquid crystals: well posedness, optical solitons and control' abstract: | - In this talk we present results on well-posedness, decay, soliton solutions and control of the coupled nonlinear Schr\"odinger (NLS) equation +In this talk we present results on well-posedness, decay, soliton solutions and control of the coupled nonlinear Schr\"odinger (NLS) equation \[ \begin{align*} & \partial_{z}u= \frac{1}{2} \mathrm{i} \nabla^{2} u+ \mathrm{i} \gamma (\sin^2(\psi+\theta_0)-\sin^2(\theta_0)) u,\\ &\nu \nabla^{2} \psi= \frac{1}{2}E_0^2\sin(2\theta_0)-\frac{1}{2}(E_0^2+|u|^{2})\sin(2(\psi+\theta_0)), \end{align*} \] - where \(u\) and \(\psi\) depend on the ``optical axis'' coordinate \(z \in \mathbb{R}\), and the ``transverse coordinates'' \((x, y) \in \mathbb{R}^2\). Also \(\nabla^{2} = \partial_x^2 + \partial_y^2\) is the Laplacian in the transverse directions, \(E_0\), \(\nu\) and \(\gamma \) are positive constants, and \(\theta_0\) is a constant satisfying \(\theta_0 \in (\pi/4, \pi/2)\). The model arises in the study of optical beam propagation in nematic liquid crystals, and in particular a set of experiments by Assanto and collaborators since 2000. The complex field \(u\) represents the electric field amplitude of a linearly polarized laser beam that propagates through a nematic liquid crystal along the optical axis \(z\). The elliptic equation describes the effects of the beam electric field on the local orientation(director field) of the nematic liquid crystal and has an important regularizing effect, seen experimentally and understood theoretically in related models. The ``director field'' \(\psi + \theta_0\) is a field of angles that describes the macroscopic orientation of the nematic liquid crystal molecules. The laser beam causes an additional deviation \(\psi\) in the orientation of the liquid crystal molecules. - In this talk we will show well posedness of the coupled system, existence of stationary solutions and a ``saturation'' effect consistent with a bound \( \psi+\theta_0 < \pi/2 \) on the total angle. Finally, we will discuss some recent results concerning an optimal control problem where the external electric field varying in time is the control. - Work in collaboration with: Juan Pablo Borgna, Panayotis Panayotaros, Diego Rial. + where \(u\) and \(\psi\) depend on the ``optical axis'' coordinate \(z \in \mathbb{R}\), and the ``transverse coordinates'' \((x, y) \in \mathbb{R}^2\). Also \(\nabla^{2} = \partial_x^2 + \partial_y^2\) is the Laplacian in the transverse directions, \(E_0\), \(\nu\) and \(\gamma \) are positive constants, and \(\theta_0\) is a constant satisfying \(\theta_0 \in (\pi/4, \pi/2)\). The model arises in the study of optical beam propagation in nematic liquid crystals, and in particular a set of experiments by Assanto and collaborators since 2000. The complex field \(u\) represents the electric field amplitude of a linearly polarized laser beam that propagates through a nematic liquid crystal along the optical axis \(z\). The elliptic equation describes the effects of the beam electric field on the local orientation(director field) of the nematic liquid crystal and has an important regularizing effect, seen experimentally and understood theoretically in related models. The ``director field'' \(\psi + \theta_0\) is a field of angles that describes the macroscopic orientation of the nematic liquid crystal molecules. The laser beam causes an additional deviation \(\psi\) in the orientation of the liquid crystal molecules.
+In this talk we will show well posedness of the coupled system, existence of stationary solutions and a ``saturation'' effect consistent with a bound \( \psi+\theta_0 < \pi/2 \) on the total angle. Finally, we will discuss some recent results concerning an optimal control problem where the external electric field varying in time is the control.
+Work in collaboration with: Juan Pablo Borgna, Panayotis Panayotaros, Diego Rial.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Constanza Sánchez de la Vega (Universidad de Buenos Aires, Argentina) - titulo: Stabilization Aspects of the Boussinesq System on a Bounded Interval abstract: | - In this talk, we present some recent results related to the rapid boundary stabilization for the Boussinesq System of the KdV-KdV Type on a Bounded interval introduced by J. Bona, M. Chen and J.-C. Saut: +In this talk, we present some recent results related to the rapid boundary stabilization for the Boussinesq System of the KdV-KdV Type on a Bounded interval introduced by J. Bona, M. Chen and J.-C. Saut: $$ \left\{ \begin{array} @@ -2187,14 +2188,14 @@ v_{t}+\eta_{x}+vv_{x}+c\eta_{xxx}-dv_{xxt}=0. \end{array} \right. - $$ - This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Firstly, we build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the nonlinear closed-loop system decays exponentially to zero in the \(L^2(0,L)\)--norm and the decay rate can be tuned to be as large as desired if the initial data is small enough under the effects of two boundary feedback. Moreover, by using a Gramian-based method introduced by Urquiza to design our feedback control, we show that the solutions of the linearized system decay uniformly to zero when the feedback control is applied. The decay rate can be chosen as large as we want. The main novelty of our work is that we can exponentially stabilize this system of two coupled equations using only one scalar input. + $$
+This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Firstly, we build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the nonlinear closed-loop system decays exponentially to zero in the \(L^2(0,L)\)--norm and the decay rate can be tuned to be as large as desired if the initial data is small enough under the effects of two boundary feedback. Moreover, by using a Gramian-based method introduced by Urquiza to design our feedback control, we show that the solutions of the linearized system decay uniformly to zero when the feedback control is applied. The decay rate can be chosen as large as we want. The main novelty of our work is that we can exponentially stabilize this system of two coupled equations using only one scalar input.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Fernando A. Gallego (Universidad Nacional de Colombia, Colombia) - titulo: A unified strategy for observability of waves with several boundary conditions abstract: | - We study the controllability of the wave equation acting on an annulus \(\Omega \subset \mathbb{R}^2\), i.e. \( \Omega = B_{R_1} \setminus B_{R_0}\), where \( 0 \lt R_0 \lt R_1\) and \(B_R\) denotes the ball in \(\mathbb{R}^2\) with center at the origin and radius \(R >0\). We are interested in the case of a single control \(h\) acting on the exterior part of the boundary, with a given boundary condition imposed on the interior boundary: +We study the controllability of the wave equation acting on an annulus \(\Omega \subset \mathbb{R}^2\), i.e. \( \Omega = B_{R_1} \setminus B_{R_0}\), where \( 0 \lt R_0 \lt R_1\) and \(B_R\) denotes the ball in \(\mathbb{R}^2\) with center at the origin and radius \(R >0\). We are interested in the case of a single control \(h\) acting on the exterior part of the boundary, with a given boundary condition imposed on the interior boundary: $$ \left\{ \begin{array}{ll} @@ -2204,15 +2205,15 @@ y(t,x) = {h(t,x)}, & \hbox{ on } (0,T) \times \partial B_{R_1}. \end{array} \right. - $$ - We provide a robust strategy to prove the observability of this system with any of several boundary conditions on the internal boundary \((0,T) \times \partial B_{R_0}\), as Fourier conditions, dynamic conditions, conditions with the fractional Laplacian or fluid-structure models. - The main tool we use is given by resolvent characterizations, and the use of adequate estimates. We prove observability estimates with observations performed on the whole external boundary, which are valid for all the mentioned cases for boundary conditions at \(\partial B_{R_0}\). We will mention also recent related results concerning wave and Schrodinger equations in more general domains with the same topological structure. + $$
+We provide a robust strategy to prove the observability of this system with any of several boundary conditions on the internal boundary \((0,T) \times \partial B_{R_0}\), as Fourier conditions, dynamic conditions, conditions with the fractional Laplacian or fluid-structure models.
+The main tool we use is given by resolvent characterizations, and the use of adequate estimates. We prove observability estimates with observations performed on the whole external boundary, which are valid for all the mentioned cases for boundary conditions at \(\partial B_{R_0}\). We will mention also recent related results concerning wave and Schrodinger equations in more general domains with the same topological structure.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Alberto Mercado (Universidad Técnica Federico Santa María, Chile) - titulo: Internal controllability of the Kadomtsev-Petviashvili II equatio abstract: | - In this talk, we present the internal control problem for the Kadomstev-Petviashvili II equation, better known as KP-II. The problem is studied first when the equation is set in a vertical strip proving by the Hilbert Unique Method and semiclassical techniques proving the internal controllability and second, in a horizontal strip where the controllability in \(L^2(\T)\) cannot be reached. +In this talk, we present the internal control problem for the Kadomstev-Petviashvili II equation, better known as KP-II. The problem is studied first when the equation is set in a vertical strip proving by the Hilbert Unique Method and semiclassical techniques proving the internal controllability and second, in a horizontal strip where the controllability in \(L^2(\mathbb{T})\) cannot be reached.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Ivonne Rivas (Universidad del Valle, Colombia), joint with Chenmin Sun @@ -2285,71 +2286,71 @@ charlas: - titulo: Círculos Matemáticos, por el fomento al razonamiento propio... abstract: | - Desde los inicios de nuestra educación, la básica y en el hogar, se tiene una marcada tendencia a sancionar el error; además, producto de nuestros tiempos, se tiene la creencia de que la velocidad es un atributo per se. Esta combinación, en la que se sanciona el error y se induce a aprender las cosas a velocidad, lleva a los infantes a recurrir a la memorización indiscriminada; ello paulatinamente alimenta la inseguridad, la fragilidad y a la larga, la inmovilidad. - Es importante entender que cada individuo requiere de tiempos distintos para hacer suya una idea y desarrollarla, y entender también que errar es parte sustancial del aprendizaje. Comprender las cosas a profundidad no sólo favorece la seguridad del ser humano, sino que fomenta la existencia y desarrollo de perspectivas distintas. - Ante esta situación, el objetivo principal del proyecto de Círculos Matemáticos del Instituto de Matemáticas es despertar e infundir en los jóvenes la confianza en su propio razonamiento, así como el respeto de sus propios tiempos. - Desplazar la noción de éxito basada en la velocidad y las actitudes competitivas, y comprender que el error es algo inherente al proceso de aprendizaje, hace que los estudiantes recuperen la confianza en sí mismos y se atrevan a proponer y explorar caminos. Esto concierne no sólo a las matemáticas sino a la vida misma. +Desde los inicios de nuestra educación, la básica y en el hogar, se tiene una marcada tendencia a sancionar el error; además, producto de nuestros tiempos, se tiene la creencia de que la velocidad es un atributo per se. Esta combinación, en la que se sanciona el error y se induce a aprender las cosas a velocidad, lleva a los infantes a recurrir a la memorización indiscriminada; ello paulatinamente alimenta la inseguridad, la fragilidad y a la larga, la inmovilidad.
+Es importante entender que cada individuo requiere de tiempos distintos para hacer suya una idea y desarrollarla, y entender también que errar es parte sustancial del aprendizaje. Comprender las cosas a profundidad no sólo favorece la seguridad del ser humano, sino que fomenta la existencia y desarrollo de perspectivas distintas.
+Ante esta situación, el objetivo principal del proyecto de Círculos Matemáticos del Instituto de Matemáticas es despertar e infundir en los jóvenes la confianza en su propio razonamiento, así como el respeto de sus propios tiempos.
+Desplazar la noción de éxito basada en la velocidad y las actitudes competitivas, y comprender que el error es algo inherente al proceso de aprendizaje, hace que los estudiantes recuperen la confianza en sí mismos y se atrevan a proponer y explorar caminos. Esto concierne no sólo a las matemáticas sino a la vida misma.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Laura Ortíz (Universidad Nacional de México, México) - titulo: El abrazo del escutoide abstract: | - Sin duda una de las razones más poderosas para insistir en la comunicación de las matemáticas la público general es, al menos para mí, la urgente necesidad de popularización de estas para, sobre todo, acabar con los prejuicios que, tradicionalmente, se han tenido sobre ellas y que conducen, inexorablemente, a la ansiedad de los estudiantes desde las primeras etapas de la educación. Pero, dejando a un lado esta labor del investigador para la sociedad, la divulgación de las matemáticas también pone el trabajo de los investigadores en un escaparate para investigadores de otros ámbitos y puede conducir a la colaboración interdisciplinar. En esta charla les hablaré de cómo la divulgación de la geometría me llevó, nos llevó, a descubrir una nueva forma geométrica mirando a las glándulas salivares de la mosca de la fruta. +Sin duda una de las razones más poderosas para insistir en la comunicación de las matemáticas la público general es, al menos para mí, la urgente necesidad de popularización de estas para, sobre todo, acabar con los prejuicios que, tradicionalmente, se han tenido sobre ellas y que conducen, inexorablemente, a la ansiedad de los estudiantes desde las primeras etapas de la educación. Pero, dejando a un lado esta labor del investigador para la sociedad, la divulgación de las matemáticas también pone el trabajo de los investigadores en un escaparate para investigadores de otros ámbitos y puede conducir a la colaboración interdisciplinar. En esta charla les hablaré de cómo la divulgación de la geometría me llevó, nos llevó, a descubrir una nueva forma geométrica mirando a las glándulas salivares de la mosca de la fruta.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Clara I. Grima (Universidad de Sevilla, España) - titulo: 'Festival de Matemáticas: la experiencia chilena' abstract: | - El Festival de Matemáticas de Chile nació a fines del 2016 como un evento satélite al "Primer Encuentro Conjunto de Matemáticas en Chile y Argentina". Es administrado directamente por la Sociedad de Matemática de Chile y se ha presentado en nueve localidades diferentes del país. En 2020 y 2021 tuvo dos versiones en línea debido a la pandemia COVID19. Se entregará un reporte de la organización de estas actividades, del proyecto de elaboración de material didáctico para las escuelas, y de los planes a futuro de la actividad. +El Festival de Matemáticas de Chile nació a fines del 2016 como un evento satélite al "Primer Encuentro Conjunto de Matemáticas en Chile y Argentina". Es administrado directamente por la Sociedad de Matemática de Chile y se ha presentado en nueve localidades diferentes del país. En 2020 y 2021 tuvo dos versiones en línea debido a la pandemia COVID19. Se entregará un reporte de la organización de estas actividades, del proyecto de elaboración de material didáctico para las escuelas, y de los planes a futuro de la actividad.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Andrés Navas (Universidad de Santiago de Chile, Chile) - titulo: La Comisión de Visibilidad de la Unión Matemática Argentina abstract: | - A fines del 2015 se creo la Comisión de Visibilidad de la Unión Matemática Argentina, conmigo a cargo hasta fines del 2019. En esta charla les contare lo que esta comisión intento/intenta realizar y mi lectura de los resultados obtenidos. Se tratará más de un intercambio de ideas que de una charla en sí. +A fines del 2015 se creo la Comisión de Visibilidad de la Unión Matemática Argentina, conmigo a cargo hasta fines del 2019. En esta charla les contare lo que esta comisión intento/intenta realizar y mi lectura de los resultados obtenidos. Se tratará más de un intercambio de ideas que de una charla en sí.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Teresa Krick (Universidad de Buenos Aires, Argentina) - titulo: La divulgación matemática como un instrumento de integración social abstract: | - Todas las sociedades están formadas por tradiciones, cultura y sobre todo fiestas que crean comunidad y más aún fortalecen el tejido social. De muchas maneras, pareciera que las matemáticas han quedado al margen de estas tradiciones y de la cultura de la sociedades, incluso en muchos casos se han estigmatizado, generando un rechazo per se. - En esta charla platicaré de la importancia de integrar las matemáticas en las tradiciones, en la cultura, y de algunas estrategias que se han desarrollado en el estado de Oaxaca con este fin, a través de las Olimpiadas de matemáticas y el Programa Oaxaqueño de Fortalecimiento a la Educación (PROFE). +Todas las sociedades están formadas por tradiciones, cultura y sobre todo fiestas que crean comunidad y más aún fortalecen el tejido social. De muchas maneras, pareciera que las matemáticas han quedado al margen de estas tradiciones y de la cultura de la sociedades, incluso en muchos casos se han estigmatizado, generando un rechazo per se.
+En esta charla platicaré de la importancia de integrar las matemáticas en las tradiciones, en la cultura, y de algunas estrategias que se han desarrollado en el estado de Oaxaca con este fin, a través de las Olimpiadas de matemáticas y el Programa Oaxaqueño de Fortalecimiento a la Educación (PROFE).
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Bruno Aaron Cisneros de la Cruz (Universidad Nacional Autónoma de México, México) - titulo: Matemáticas (no solo) en las redes abstract: | - En los últimos tiempos hemos sido testigos de nuevas formas de divulgación de las matemáticas. Nuevas en cuanto a los formatos y (tal vez) en cuanto a los públicos. En esta charla hablaré sobre una experiencia particular en este ámbito, tanto en España como en Latinoamérica. +En los últimos tiempos hemos sido testigos de nuevas formas de divulgación de las matemáticas. Nuevas en cuanto a los formatos y (tal vez) en cuanto a los públicos. En esta charla hablaré sobre una experiencia particular en este ámbito, tanto en España como en Latinoamérica.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Eduardo Sáenz De Cabezón Irigaray (Universidad de la Rioja, España) - titulo: Haciendo comunidad en tiempos de confinamiento abstract: | - En medio de la incertidumbre de la pandemia de COVID-19 y en el boom de la virtualidad, se han abierto puertas de comunicación entre divulgadores de matemáticas hispanoparlantes en América Latina y en España. En este camino se han organizado encuentros directamente centrados en crear puentes sólidos de colaboración entre distintos grupos. - Durante esta conferencia, presentaré el trabajo que hemos hecho en este sentido Ágata Timón (Instituto de Ciencias Matemáticas), Andrés Navas (Universidad de Chile), Aubin Arroyo (UNAM), Beatriz Vargas (UNAM) y yo. También me gustaría plantear y oír propuestas de caminos para fortalecer la colaboración entre quienes trabajamos divulgando las matemáticas en español. +En medio de la incertidumbre de la pandemia de COVID-19 y en el boom de la virtualidad, se han abierto puertas de comunicación entre divulgadores de matemáticas hispanoparlantes en América Latina y en España. En este camino se han organizado encuentros directamente centrados en crear puentes sólidos de colaboración entre distintos grupos.
+Durante esta conferencia, presentaré el trabajo que hemos hecho en este sentido Ágata Timón (Instituto de Ciencias Matemáticas), Andrés Navas (Universidad de Chile), Aubin Arroyo (UNAM), Beatriz Vargas (UNAM) y yo. También me gustaría plantear y oír propuestas de caminos para fortalecer la colaboración entre quienes trabajamos divulgando las matemáticas en español.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Lucía López de Medrano (Universidad Nacional Autónoma de México, México) - titulo: Narrativa de no-ficción con temas matemáticos abstract: | - En esta charla trataré de abrir la discusión sobre qué significa la narrativa de ficción dentro de la divulgación de las matemáticas. - En este tipo de narrativa se desarrolla una historia en la que se incluye contenido matemático. Surgen varias preguntas: ¿la historia tiene más importancia que el contenido matemático?, ¿se debe de ser muy riguroso con el contenido matemático? - Estas preguntas están relacionadas con la postura de Humberto Eco, quien en un Postcript a su novela "El nombre de la rosa" criticó fuertemente lo que se conoce como Salgarismo: suspender una narración dentro de la trama para realizar una (extensa) explicación científica. - No se busca dar una respuesta a estas preguntas sino plantear algunas posturas y preguntas que permitan avanzar en una discusión sobre este tema. +En esta charla trataré de abrir la discusión sobre qué significa la narrativa de ficción dentro de la divulgación de las matemáticas.
+En este tipo de narrativa se desarrolla una historia en la que se incluye contenido matemático. Surgen varias preguntas: ¿la historia tiene más importancia que el contenido matemático?, ¿se debe de ser muy riguroso con el contenido matemático?
+Estas preguntas están relacionadas con la postura de Humberto Eco, quien en un Postcript a su novela "El nombre de la rosa" criticó fuertemente lo que se conoce como Salgarismo: suspender una narración dentro de la trama para realizar una (extensa) explicación científica.
+No se busca dar una respuesta a estas preguntas sino plantear algunas posturas y preguntas que permitan avanzar en una discusión sobre este tema.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Javier Elizondo (Universidad Nacional Autónoma de México, México) - titulo: La divulgación de las matemáticas en Costa Rica abstract: | - Se compartirá la experiencia generada por un grupo de personas que nos hemos dedicado a la divulgación de las matemáticas, con el apoyo de organizaciones no gubernamentales, empresa privada y universidades estatales. En este esfuerzo se han realizado diversas actividades que han tenido un impacto sostenido e interesante. En primer lugar, la producción de “Matemática por un minuto”, una serie de programas para la radio con duración de un minuto cada uno que se transmitieron a nivel nacional dos veces al día y su evolución inesperada hacia videos y posteriormente hacia un libro titulado “Las matemáticas de lo cotidiano”, en sus versiones en español e inglés, así como su edición física y e-book. - Además, el diseño, construcción, experiencias, evolución y clonación del “Museo viajante de ciencias y matemática (MUCYM)” y su impacto en la divulgación de las matemáticas a lo largo y ancho de nuestro país, sobre todo a sectores menos favorecidos de su población. - Por último, la organización de congresos académicos, como el evento bianual “Festival Internacional de Matemáticas” en que se inició en 1998 En las doce ediciones que se han realizado, se amalgamaron mágicamente propuestas didácticas, talleres, ponencias y conferencias con juegos, poesía, museos y teatro entre otros que motivan una participación de profesores, estudiantes y varios grupos sociales que propician el aprendizaje activo de esta bella ciencia, las matemáticas. +Se compartirá la experiencia generada por un grupo de personas que nos hemos dedicado a la divulgación de las matemáticas, con el apoyo de organizaciones no gubernamentales, empresa privada y universidades estatales. En este esfuerzo se han realizado diversas actividades que han tenido un impacto sostenido e interesante. En primer lugar, la producción de “Matemática por un minuto”, una serie de programas para la radio con duración de un minuto cada uno que se transmitieron a nivel nacional dos veces al día y su evolución inesperada hacia videos y posteriormente hacia un libro titulado “Las matemáticas de lo cotidiano”, en sus versiones en español e inglés, así como su edición física y e-book.
+Además, el diseño, construcción, experiencias, evolución y clonación del “Museo viajante de ciencias y matemática (MUCYM)” y su impacto en la divulgación de las matemáticas a lo largo y ancho de nuestro país, sobre todo a sectores menos favorecidos de su población.
+Por último, la organización de congresos académicos, como el evento bianual “Festival Internacional de Matemáticas” en que se inició en 1998 En las doce ediciones que se han realizado, se amalgamaron mágicamente propuestas didácticas, talleres, ponencias y conferencias con juegos, poesía, museos y teatro entre otros que motivan una participación de profesores, estudiantes y varios grupos sociales que propician el aprendizaje activo de esta bella ciencia, las matemáticas.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Manuel Murillo Tsijli (Instituto Tecnológico de Costa Rica, Costa Rica) - titulo: 'FUNDAPROMAT: Lecciones y Logros' abstract: | - La Fundación Panameña para la Promoción de las Matemáticas (FUNDAPROMAT) nació el 6 de diciembre de 2019 con el propósito de promover el estudio de las matemáticas en la República de Panamá. Por la pandemia mundial, tuvimos que volvernos creativos para seguir cumpliendo con nuestra misión y comenzamos a organizar eventos virtuales, que son gratis y abiertos a todo público. A la fecha esta Fundación privada sin fines de lucro ha realizado más de 300 eventos virtuales con más de 35,000 participantes, incluyendo panameños y extranjeros de todas partes del mundo. Niños, jóvenes y adultos de todas las edades participan en nuestras actividades. En esta charla, presentaré los diferentes tipos de eventos virtuales que organizamos y compartiré las lecciones aprendidas y los logros alcanzados por la Fundación. +La Fundación Panameña para la Promoción de las Matemáticas (FUNDAPROMAT) nació el 6 de diciembre de 2019 con el propósito de promover el estudio de las matemáticas en la República de Panamá. Por la pandemia mundial, tuvimos que volvernos creativos para seguir cumpliendo con nuestra misión y comenzamos a organizar eventos virtuales, que son gratis y abiertos a todo público. A la fecha esta Fundación privada sin fines de lucro ha realizado más de 300 eventos virtuales con más de 35,000 participantes, incluyendo panameños y extranjeros de todas partes del mundo. Niños, jóvenes y adultos de todas las edades participan en nuestras actividades. En esta charla, presentaré los diferentes tipos de eventos virtuales que organizamos y compartiré las lecciones aprendidas y los logros alcanzados por la Fundación.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Jeanette Shakalli (Fundación Panameña para la Promoción de las Matemáticas, Panamá)