Additive Nonparametric Reconstruction of Dynamical Systems from Time Series

We present a nonparametric way to retrieve a system of differential equations in embedding space from a single time series. These equations can be treated with dynamical systems theory and allow for long term predictions. We demonstrate the potential of our approach for a modified chaotic Chua oscillator.Comment: accepted for Phys. Rev. E, Rapid Com

Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems

We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $\omega$-parametric analytic family better than order of $(\omega\ln\omega)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $\omega^{-m}$.Comment: 40 page

Self-attraction effect and correction on three absolute gravimeters

The perturbations of the gravitational field due to the mass distribution of an absolute gravimeter have been studied. The so called Self Attraction Effect (SAE) is crucial for the measurement accuracy, especially for the International Comparisons, and for the uncertainty budget evaluation. Three instruments have been analysed: MPG-2, FG5-238 and IMPG-02. The SAE has been calculated using a numerical method based on FEM simulation. The observed effect has been treated as an additional vertical gravity gradient. The correction (SAC) to be applied to the computed g value has been associated with the specific height level, where the measurement result is typically reported. The magnitude of the obtained corrections is of order 1E-8 m/s2.Comment: 14 pages, 8 figures, submitted to Metrologi

REMOVABLE SETS FOR LIPSCHITZ HARMONIC FUNCTIONS ON CARNOT GROUPS

Abstract. Let G be a Carnot group with homogeneous dimension Q ≥ 3 and let L be a sub-Laplacian on G. We prove that the critical dimension for removable sets of Lipschitz L-harmonic functions is (Q − 1). Moreover we construct self-similar sets with positive and finite H Q−1 measure which are removable. 1

On the Bounds of Function Approximations

Within machine learning, the subfield of Neural Architecture Search (NAS) has recently garnered research attention due to its ability to improve upon human-designed models. However, the computational requirements for finding an exact solution to this problem are often intractable, and the design of the search space still requires manual intervention. In this paper we attempt to establish a formalized framework from which we can better understand the computational bounds of NAS in relation to its search space. For this, we first reformulate the function approximation problem in terms of sequences of functions, and we call it the Function Approximation (FA) problem; then we show that it is computationally infeasible to devise a procedure that solves FA for all functions to zero error, regardless of the search space. We show also that such error will be minimal if a specific class of functions is present in the search space. Subsequently, we show that machine learning as a mathematical problem is a solution strategy for FA, albeit not an effective one, and further describe a stronger version of this approach: the Approximate Architectural Search Problem (a-ASP), which is the mathematical equivalent of NAS. We leverage the framework from this paper and results from the literature to describe the conditions under which a-ASP can potentially solve FA as well as an exhaustive search, but in polynomial time.Comment: Accepted as a full paper at ICANN 2019. The final, authenticated publication will be available at https://doi.org/10.1007/978-3-030-30487-4_3