Search

In this paper we study the local boundary controllability for a non linear system of two degenerate parabolic equations with a control acting on only one equation. We analyze boundary null controllability properties for the linear system via the moment method by Fattorini and Russell, together with some results on biorthogonal families. Moreover, we provide an estimate on the null-control cost. This estimate let us prove a local exact boundary controllability result to zero of the nonlinear system following the iterative method from Lebeau and Robbiano as in \cite{Burgos_2020, Liu_2012}., Comment: arXiv admin note: text overlap with arXiv:2104.00610; substantial text overlap with arXiv:2401.17067 by other authors

In this paper, we are interested in solvable complete Lie algebras, over the field $\K=\R$ or $\mathbb{C}$, which admit a symplectic structure. Specifically, important classes are studied, and a description of complete Lie Algebra with the dimension of nilradical less or equal than six, which supported symplectic structure is given., Comment: 14 pages

The goal of this paper is to analyze the pointwise controllability properties of a one-dimensional degenerate/singular equation. We prove the conditions that characterize approximate and null controllability. Besides, a numerical simulation based on B-splines will be provided, in which the state $u$ and the control function $h$ are represented in terms of B-spline basis functions. The numerical results obtained match the theoretical ones.

We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem., Comment: arXiv admin note: text overlap with arXiv:2212.05264

We study the null controllability for a degenerate/singular wave equation with drift in non divergence form. In particular, considering a control localized on the non degenerate boundary point, we provide some conditions for the boundary controllability via energy methods and boundary observability., Comment: arXiv admin note: text overlap with arXiv:2109.12531

We propose a rate optimal estimator for the linear regression model on network data with interacted (unobservable) individual effects. The estimator achieves a faster rate of convergence $N$ compared to the standard estimators' $\sqrt{N}$ rate and is efficient in cases that we discuss. We observe that the individual effects alter the eigenvalue distribution of the data's matrix representation in significant and distinctive ways. We subsequently offer a correction for the \textit{ordinary least squares}' objective function to attenuate the statistical noise that arises due to the individual effects, and in some cases, completely eliminate it. The new estimator is asymptotically normal and we provide a valid estimator for its asymptotic covariance matrix. While this paper only considers models accounting for first-order interactions between individual effects, our estimation procedure is naturally extendable to higher-order interactions and more general specifications of the error terms.

In this paper we are going to prove existence for positive solutions of the following Schr\"odinger-Maxwell system of singular elliptic equations: begin{equation} \left\{\begin{array}{l} u \in W_{0}^{1,2}(\Omega):-\operatorname{div}\left(a(x) \nabla u\right)+\psi|u|^{r-2} u=\frac{f(x)}{u^{\theta}}, \psi \in W_{0}^{1,2}(\Omega):-\operatorname{div}(M(x) \nabla \psi)=|u|^{r} \end{array}\right. \end{equation} where $\Omega$ is a bounded open set of $\mathbb{R}^{N}, N>2,$ $r>,1,$ $u>0,$ $\psi>0,$ $0 < \theta<1$ and $f$ belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by demonstrating how the structure of the system gives rise to a regularizing effect on the summability of the solutions.

In this article, we investigate the existence and uniqueness of a positive solution for a class of singular nonlinear elliptic problem with boundary condition. Our result holds in fractional Orlicz-Sobolev spaces., Comment: The article is inaccurate

In this paper we study the boundary controllability for a system of two coupled degenerate/singular parabolic equations with a control acting on only one equation. We analyze both approximate and null boundary controllability properties. Besides, we provide an estimate on the null-control cost. The proofs are based on the use of the moment method by Fattorini and Russell together with some results on biorthogonal families., Comment: 26 pages

In this paper we study the existence and regularity of solutions to the following singular problem \begin{equation} \left\{ \begin{array}{lll} &-\displaystyle\mbox{div} \big(a(x,u)|\nabla u|^{p-2}|\nabla u|\big) + |u|^{s-1}u =\frac{f}{u^{\gamma}} &\mbox{ in } \Omega \\ &u>0 &\mbox{ in }\Omega \\ &u=0 &\mbox{ on } \delta\Omega \end{array} \right. \end{equation} proving that the lower order term $u|u|^{s-1}$ has some regularizing effects on the solutions in the case of an elliptic operator with degenerate coercivity.

The quality and generality of deep image features is crucially determined by the data they have been trained on, but little is known about this often overlooked effect. In this paper, we systematically study the effect of variations in the training data by evaluating deep features trained on different image sets in a few-shot classification setting. The experimental protocol we define allows to explore key practical questions. What is the influence of the similarity between base and test classes? Given a fixed annotation budget, what is the optimal trade-off between the number of images per class and the number of classes? Given a fixed dataset, can features be improved by splitting or combining different classes? Should simple or diverse classes be annotated? In a wide range of experiments, we provide clear answers to these questions on the miniImageNet, ImageNet and CUB-200 benchmarks. We also show how the base dataset design can improve performance in few-shot classification more drastically than replacing a simple baseline by an advanced state of the art algorithm., Comment: 23 pages, 11 figures, to appear in ECCV 2020

In this paper, we prove existence and regularity results for solutions of some nonlinear Dirichlet problems for an elliptic equation defined by a degenerate coercive operator and a singular right hand side. \begin{equation}\label{01} \left\{ \begin{array}{lll} -\displaystyle\mbox{div}( a(x,u,\nabla u))&=\displaystyle\frac{f}{u^{\gamma}} & \mbox{ in } \Omega \\ u&>0 &\mbox{ in }\Omega \\ u&=0 &\mbox{ on } \delta\Omega \end{array} \right. \end{equation} where $\Omega $ is bounded open subset of $I\!\!R^{N}(N\geq2),$ $\gamma>0$ and $ f$ is a nonnegative function that belongs to some Lebesgue space.

Iterative solvers preconditioned with algebraic multigrid have been devised as an optimal technology to speed up the response of large sparse linear systems. In this work, this technique was implemented in the framework of the dual delineation approach. This involves a single groundwater flow solve and a pure advective transport solve with different right-hand sides. The new solver was compared with traditional preconditioned iterative methods and direct sparse solvers on several two- and three-dimensional benchmark problems spanning homogeneous and heterogeneous formations. For the groundwater flow problems, using the algebraic multigrid preconditioning speeds up the numerical solution by one to two orders of magnitude. Contrarily, a sparse direct solver was the most efficient for the pure advective transport processes such as the forward travel time simulations. Hence, the best sparse solver for the more general advection-dispersion transport equation is likely to be P\'eclet number dependent. When equipped with the best solvers, processing multimillion grid blocks by the dual delineation approach is a matter of seconds. This paves the way for routine time-consuming tasks such as sensitivity analysis. The paper gives practical hints on the strategies and conditions under which algebraic multigrid preconditioning for the class of nonlinear and/or transient problems would remain competitive., Comment: 27 pages, 5 figures