seminar

• IMAG Functional Analysis Seminar
• Title: Lineability, spaceability and Orlicz-Lorentz spaces.
• By Juan Seoane (Universidad Complutense de Madrid)
• Abstract: Vladimir Gurariy showed (1966) that the set of Weierstrass' monsters (classical continuous nowhere differentiable functions) contains (up to the zero function) infinite dimensional linear spaces. On top of that, in 1999, he (jointly with V. Fonf and M. Kadets) showed that, when working within C[0,1], the above infinite dimensional linear space can be chosen to be closed in C[0,1]. These results led Gurariy (2005) to coin the terms “lineability” and “spaceability” (MSC2020: 15A03 and 46B87). The idea behind it is, in a nutshell, to answer the following questions: How common are “bad” properties? And… what do we mean by “common” in the previous question? And by “bad”?Lately, quite a few works have been focusing on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of several researchers within Real and Complex Analysis, Operator Theory, Summability Theory, Polynomials in Banach spaces, Axiomatic Set Theory, Probability Theory, and Functional Analysis.

  Throughout this talk we shall present a brief account on some classical results in this topic as well as some recent joint works on nonlinear subsets of Orlicz-Lorentz spaces with Luis Bernal-González, Daniel L. Rodríguez-Vidanes, and Hyung-Joon Tag.

Título: Bounded \(\lambda\)-harmonic functions on Hadamard manifolds.

Conferenciante: Diego Alfonso Marín (Universidad de Granada)

Resumen:

The study of eigenfunctions of the Laplacian on Riemannian manifolds is a classical topic in differential geometry, closely related to the Uniformization Theorem. This theorem classifies simply connected Riemannian surfaces into three conformal types: elliptic (sphere), parabolic (plane), and hyperbolic (unit disk). Since harmonic functions are conformally invariant in dimension two, this classification determines whether a non-compact surface admits a rich set of bounded harmonic functions or only trivial ones, as in the Euclidean plane. Motivated by extending this idea to higher-dimensional manifolds, recent research has focused on the study of harmonic functions and solutions to the eigenvalue problem \(\Delta u + \lambda u = 0\) (called \(\lambda\)-harmonic functions).

In this talk, we introduce the theory of bounded harmonic functions on Hadamard manifolds (simply connected with negative curvature) and explore the existence of $\lambda$-harmonic functions with zero Dirichlet data on a domain \(\Omega \subset M\). As we will see, this existence is strongly linked to the presence of certain convex sets with specific geometric properties.

This talk is based on an ongoing work with José Espinar and Marcos Petrucio.

Toda la información puede encontrarse en:
Seminario de Geometría.

Título: Bifurcation in a Quasilinear Schrödinger-Type Equation with Two
Parameters: Theory and Applications to Multiplicity of Solutions.

Conferenciante: Miguel Martínez Teruel (Universidad de Granada) (mmteruel@ugr.es).

Resumen:

In this talk we deal with the following family of equations,

\begin{equation}
\begin{cases}
-\Delta u-\lambda m(x) u \Delta(u^2)=f(\mu, x, u)\quad \text{in}\quad \Omega,\\
u=0\qquad \text{on}\qquad \Omega.
\end{cases}
\tag{\(P(\lambda,\, \mu)\)}
\end{equation}

where \(\lambda,\, \mu\in \mathbb{R}\), \(\Omega\) is an open and bounded subset of \(\mathbb{R}^N\) with smooth boundary, \(m(x)\) is a continuous function with \(0\leq m(x)\leq M\) and \(f\, \colon\, \mathbb{R}\times\Omega\times\mathbb{R}^+ \to \mathbb{R}\) is a \(C^1\) function such that \(f(\mu,\, x,\, 0)=0\) for all \(x\in\Omega\) and \(\mu\in\mathbb{R}\) and satisfies:

For every \(\Gamma\) bounded subset of \(\mathbb{R} - \{0\}\) and \(\mu \in \Gamma\),

\[
\lim_{s \to 0^+} \frac{f(\mu, x, s)}{s} = \mu f'_+(x,0), \quad \text{uniformly in } (\mu, x) \in \Gamma \times \Omega,
\]

with \(0 \leq f'_+(x,0) \in L^\infty(\Omega)\) not identically zero.

With these conditions we prove the existence of a continuum of solutions, we study the laterality of the continuum and we give explicit examples in which the number of solutions changes depending on the parameter \(\lambda\).

Descargar resumen aquí.

References:

1. [1] Ambrosetti, A., Arcoya, D., An introduction to nonlinear functional analysis and elliptic problems, in: Progress in Nonlinear Differential Equations and their Applications, Birkäuser, 2011.
2. [2] Arcoya, D., Carmona, J., Pellacci, B., Bifurcation for some quasi-linear operators, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 733–765.
3. [3] Cintra, W., Medeiros, E., Severo, U. On positive solutions for a class of quasilinear elliptic equations. Z. Angew. Math. Phys. 70, 79 (2019).
4. [4] Figueiredo, G.M., Santos J´unior, J.R., Su´arez, A. Structure of the set of positive solutions of a non-linear Schr¨odinger equation. Isr. J. Math. 227, 485–505 (2018).

Toda la información puede encontrarse en:
Seminario de Ecuaciones Diferenciales.

Título: An Extended Symbolic Dynamics.

Conferenciante: Pouya Mehdipour (Universidade Federal de Viçosa, Brasil).

Resumen:

We will present an extended symbolic dynamics, called Zip Shift, and provide an overview of some of its applications across different subareas of dynamical systems.

Toda la información puede encontrarse en:
Seminario de Ecuaciones Diferenciales.

Título: Mean-field limit of particle systems over hypergraphs.

Conferenciante: Nastassia Pouradier Duteil (Sorbonne Université, Paris).

Resumen:

We present a generalization of non-exchangeable particle systems with higher-order interactions, by removing two important assumptions that are usually made in particle systems: that of exchangeability of particles, and that of superposition of binary interactions. In the general model that we consider, individuals instead interact by groups, so that a full group jointly generates a non-linear force on any individual. This interaction is modeled by an underlying hypergraph. We derive the mean-field limit of the particle system (i.e. its limit as the number of particles tends to infinity), and show that it is determined by a Vlasov-type equation, where the limit of the hypergraph is given by a so-called _unbounded-rank hypergraphon_, and the mean-field force admits infinitely-many orders of interactions. This work is in collaboration with N. Ayi (Sorbonne University) and D. Poyato (Universidad de Granada).

Toda la información puede encontrarse en:
Seminario de Ecuaciones Diferenciales.

Título: Stability in Gagliardo-Nirenberg and (Log) Sobolev inequalities.

Conferenciante: Nikita Simonov (Sorbonne Université, Paris).

Resumen:

In certain functional inequalities, the best constants and minimizers are known. The next natural question concerns stability: if a function “almost attains equality,” in what sense is it close to one of the minimizers? We will discuss a recent result on quantitative stability for a certain family of Gagliardo-Nirenberg, Sobolev, and Logarithmic Sobolev inequalities. Our approach is based on the carré du champ method for a family of suitable diffusion equations, enabling us to derive fully constructive estimates. This talk is based on a series of results obtained in collaboration with Matteo Bonforte (UAM), Giovanni Brigati (ISTA), Jean Dolbeault (CEREMADE–Dauphine), and Bruno Nazaret (Paris 1, Panthéon-Sorbonne).

Toda la información puede encontrarse en: Seminario de Ecuaciones Diferenciales.

Title: A minimal road to geometry.

Speaker: Alberto Cerezo Cid (UGR).

Abstract: In this talk, we present the origins and motivations behind the study of minimal and constant mean curvature (CMC) surfaces. We will explore their connections to classical branches of mathematics, including complex analysis, partial differential equations and measure theory. Finally, we will highlight recent advances in free boundary problems, a topic that has gained significant attention over the past decade.

Place: Seminar 1, IMAG. The course will also be streamed at https://meet.google.com/uoh-dyyr-jss.

Title: An introduction to the geometry in the Alhambra.

Person in charge: Miguel Ortega Titos.

Title: Fractional hyperbolic diffusions on sphere with random data.

Speaker: Nikolai Leonenko
(Cardiff School of Mathematics, Cardiff University).

Schedule

Abstracts:

Part I. Introduction to statistical analysis of Gaussian isotropic spherical random fields.

Spherical random fields are very useful for modelling some phenomena in areas such as earth sciences (like, for example, in geophysics and climatology and cosmology). In fact, the application of statistical methods in cosmology has become increasingly important due to the many experimental data obtained in recent years, and spherical random fields are of particular interest regarding the analysis of Cosmic Microwave Background (CMB) radiation discovered by the astronomers Arno Penzias and Robert Wilson in 1964.

As well-known, the CMB is a spatially isotropic radiation field spread throughout the visible universe, originated around 14 billion years ago, and it is the main source of information we have about the evolution of the universe. The CMB radiation can be mathematically modelled as an isotropic spherical random field for which there is a spectral representation by means of spherical harmonics.

We consider statistical analysis of Gaussian isotropic spherical random fields, when only one observation of the field is available. The characteristic function of these empirical covariances is given in terms of cumulants. It turns out that our estimator follows a Rosenblatt type distribution [1,2,3]. Our goal is to estimate the covariance function of the random field, given a single observation at each point of the discretized sphere. We present a methodology for handling the problem of cosmic variance in this framework.

This is joint results with M.Taqqu (Boston University, USA) and G.Terdik (Debrecen University, Hungary). Some results used in the talk are obtained jointly with M.D.Ruiz-Medina (Granada University, Spain).

Part II. Fractional hyperbolic diffusions on sphere with random data.

Our next objective is to study the fundamental solutions to fractional hyperbolic diffusion equation in the time variable using the Caputo derivative, and its properties. The exact solutions of the fractional hyperbolic diffusion equation with random data are derived in terms of series expansions of isotropic in space random fields on the unit sphere. The results are illustrated through motivating numerical examples of interest in several applied fields.

This is joint results with A. Olenko (La Trope University, Melbourne, Australia) and J.Vaz (UNICAMP, Brazil) [4,5].

References:

1. [1] Leonenko, N.N., Taqqu, M.S. and Terdik, Gy (2018) Estimation of the covariance function of Gaussian isotropic random fields, related Rosenblatt-type distributions and the cosmic variance problem, Electronic Journal of Statistics, Vol. 12 (2018) 3114–3146.
2. [2] Leonenko, N.N., Ruiz-Medina,M.D. and Taqqu, M.S. (2017). Rosenblatt distribution subordinated to gaussian random fields with long-range dependence. Stoch. Anal. Appl., 35(1):144–177, 2017.
3. [3] Leonenko, N., Ruiz-Medina, M.D. and Taqqu,M.S. (2017) Non-central limit theorems for random fields subordinated to gamma-correlated random fields, Bernoulli, 2017, Vol. 23, No. 4B, 3469-3507.
4. [4] Leonenko, N. and Vaz, J. (2020) Spectral analysis of fractional hyperbolic diffusion equations with random data, Journal of Statistical Physics, 179,155-175.
5. [5]Leonenko, N., Olenko, A. and Vaz, J. (2024) On fractional spherically restricted hyperbolic diffusion random field, Communications in Nonlinear Science and Numerical Simulation, 131,107866.

For more information, visit:

Red Temática de Procesos Estocásticos y sus Aplicaciones.

Title: Stochastic processes in software reliability.

Speaker: Tadashi Dohi (Department of Engineering, Hiroshima University).

Dr. Tadashi Dohi has served as a Full Professor at Hiroshima University, Japan, since 2002. He is currently appointed as Dean of School of Informatics and Data Science and Associate Dean of Graduate School of Advanced Science and Engineering, Hiroshima University. He received a Doctor of Engineering degree from Hiroshima University in 1995. His research interests include Software Reliability, Dependable Computing, Performance Evaluation, and Operations Research. To date, his research has led to 280 journal papers, 340 peer-reviewed conference papers, 25 book editions, and 47 book chapters in the above research fields. Dr. Dohi is a Regular Member of IEICE, IPSJ, REAJ, a Fellow Member of ORSJ, and a Senior Member of IEEE (Computer Society and Reliability Society). He was acting President of REAJ in 2018 and 2019. He has served as the General Chair of 15 international conferences, including ISSRE 2011, ATC 2012, DASC 2019, and ICECCS 2022. Of note, he was a founding member of the Internationa Symposium on Advanced Reliability and Maintenance Modeling (APARM) and International Workshop on Software Aging and Rejuvenation (WoSAR). He has been a steering committee member in AIWARM/APARM, ISSRE, DASC, DSA. He has also worked as a program committee member in several premier international conferences such as DSN, ISSRE, COMPSAC, SRDS, QRS, EDCC, PRDC, HASE, SAC, ICPE, among numerous others. He is an Associate Editor/Editorial Board Member of over 20 international journals, including IEEE Transactions on Reliability.

Schedule: 9:30-11:00h & 11:30-13:00h.

Abstract:

Software reliability engineering plays a central role to quantify software product reliability in the verification/validation phase before the release. During the last five decades, in fact, over two hundred software reliability models have been proposed in the literature. First, we quickly overview the software reliability modeling based on the software fault count data observed in the testing phase, where homogeneous Markov processes (HMPs) and non-homogeneous Poisson processes (NHPPs) have gained much popularity to assess quantitative software reliability. Next, we provide generalization frameworks on software reliability models, by introducing non-homogeneous Markov processes (NHMPs). Finally, we give some examples on non-Markovian process modeling, including geometric processes, trend renewal processes (TRPs) and Hawkes processes. We show how these generalized software reliability models can work better than the classical ones in terms of goodness-of-fit and predictive performances, through several empirical studies with actual software fault count data.

For more information, visit:

Red Temática de Procesos Estocásticos y sus Aplicaciones.