Seminars from until

This is the third in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.

Symbolic data analysis (SDA) has been developed as an extension of the data analysis to handle more complex data structures. In this general framework the pair observation/variable is characterized by more than one value: from two (e.g., interval-value data defined by minimum and maximum values) to multiple-valued variables (e.g., frequencies or proportions).

This research discusses the clustering of multiple-valued symbolic data. First, we discuss an extension of heuristic clustering based on the symmetric Kullback-Leibler distance combined with a complete-linkage rule within the hierarchical clustering framework. Then, we propose a new model-based clustering framework. These new family of models based on the Dirichlet distribution includes mixture of regression/expert models. Results are illustrated with synthetic and demographic (population pyramids) data.

Some recent results dealing with optimal design problems for energies which describe composite materials, mixed materials and Ogden ones will be presented.

We define a new large $N$ limit for tensor models, based on an enhanced large $N$ scaling of the coupling constants. The resulting large $N$ expansion is organized in terms of a half-integer associated with Feynman graphs that we call the index. This index has a natural interpretation in terms of the many matrix models embedded in the tensor model.

A $2n$-dimension Ornstein-Uhlenbeck (OU) process for which the diffusion matrix is singular is considered. This process is used as a model for the dynamic behavior of vibrating engineering structures such as bridges, buildings, dams, among others. We study the problem of estimating the vibration frequencies of the structure or, equivalently, the parameters of the stochastic differential equation (SDE) that governs the OU process.

Firstly, it is considered the case where the OU process is perturbed by an independent wiener process. The maximum likelihood estimator of the drift matrix is obtained and the properties of the estimator are established. The local asymptotic normality of the estimator is analyzed in detail. Since general regularity conditions do not hold in this case (the diffusion matrix is singular), theoretical results from the classic literature on the subject do not immediately apply and an alternative approach based on the Laplace transform is used.

Secondly, it is considered the case where the OU process is perturbed by two independent fractional brownian motions. Models involving fractional noises have not been widely used in engineering. However, many problems in engineering involve processes exhibiting long memory. For this reason, the estimation of the parameters of multidimensional state space linear models, described by SDEs and disturbed by fractional Brownian motion, has a potential application in different areas of engineering. We analyze the problem of estimating the drift parameters of a $2$- dimension linear stochastic differential equation perturbed by two independent fractional Brownian motions with the same Hurst parameter belonging to $(1/2,1)$. The maximum likelihood estimator of the drift parameters is obtained after a transformation of the original model and making use of the so called fundamental martingale.

In both cases, a simulation study is presented in the context of a real world situation that illustrates the asymptotic behavior of the maximum likelihood estimator of the drift matrix.

Although the study of the Cauchy problem in General Relativity started in the decade of 1950 with the work of Foures-Bruhat, addressing the problem of global non-linear stability of solutions to the Einstein field equations is in general a hard problem. The first non-linear global non-linear stability result in General Relativity was obtained for the de Sitter spacetime by H. Friedrich in the decade of 1980. In this talk the main tool used in the above result is introduced: a conformal (regular) representation of the Einstein field equations — the so-called conformal Einstein field equations (CEFE). Then, the conformal structure of the Schwarzschild-de Sitter spacetime is analysed using the extended conformal Einstein field equations (XCEFE). To this end, initial data for an asymptotic initial value problem for the Schwarzschild-de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild-de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild-de Sitter spacetime in the asymptotic region. Finally, we'll show that small enough perturbations of asymptotic initial data for the Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild-de Sitter spacetime.

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying $L_{\infty}$ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory.