On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.Comment: Final revisio

Machine Learning for Fluid Mechanics

The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from field measurements, experiments and large-scale simulations at multiple spatiotemporal scales. Machine learning offers a wealth of techniques to extract information from data that could be translated into knowledge about the underlying fluid mechanics. Moreover, machine learning algorithms can augment domain knowledge and automate tasks related to flow control and optimization. This article presents an overview of past history, current developments, and emerging opportunities of machine learning for fluid mechanics. It outlines fundamental machine learning methodologies and discusses their uses for understanding, modeling, optimizing, and controlling fluid flows. The strengths and limitations of these methods are addressed from the perspective of scientific inquiry that considers data as an inherent part of modeling, experimentation, and simulation. Machine learning provides a powerful information processing framework that can enrich, and possibly even transform, current lines of fluid mechanics research and industrial applications.Comment: To appear in the Annual Reviews of Fluid Mechanics, 202

A BPS Skyrme model and baryons at large Nc

Within the class of field theories with the field contents of the Skyrme model, one submodel can be found which consists of the square of the baryon current and a potential term only. For this submodel, a Bogomolny bound exists and the static soliton solutions saturate this bound. Further, already on the classical level, this BPS Skyrme model reproduces some features of the liquid drop model of nuclei. Here, we investigate the model in more detail and, besides, we perform the rigid rotor quantization of the simplest Skyrmion (the nucleon). In addition, we discuss indications that the viability of the model as a low energy effective field theory for QCD is further improved in the limit of a large number of colors N_c.Comment: latex, 23 pages, 1 figure, a numerical error in section 3.2 corrected; matches published versio

An anthology of non-local QFT and QFT on noncommutative spacetime

Ever since the appearance of renormalization theory there have been several differently motivated attempts at non-localized (in the sense of not generated by point-like fields) relativistic particle theories, the most recent one being at QFT on non-commutative Minkowski spacetime. The often conceptually uncritical and historically forgetful contemporary approach to these problems calls for a critical review the light of previous results on this subject.Comment: 33 pages tci-latex, improvements of formulations, shortening of sentences, addition of some reference

Application of the Kalman Filter in Functional Magnetic Resonance Image Data

The Kalman-Bucy filter was applied on the preprocessing of the functional magnetic resonance image-fMRI. Numerical simulations of hemodynamic response added Gaussian noise were performed to evaluate the performance of the filter. After the proceeding was applied in auditory real data. The Kohonen’s self-organized map was employed as tools to compare the performance of the Kalman’s filter with another type of pre-processing. The results of the application of Kalman-Bucy filter for simulated data and real auditory data showed that it can be used as a tool in the temporal filtering step in fMRI data

Expectation values of twist fields and universal entanglement saturation of the free massive boson

The evaluation of vacuum expectation values (VEVs) in massive integrable quantum field theory (QFT) is a nontrivial renormalization-group "connection problem" -- relating large and short distance asymptotics -- and is in general unsolved. This is particularly relevant in the context of entanglement entropy, where VEVs of branch-point twist fields give universal saturation predictions. We propose a new method to compute VEVs of twist fields associated to continuous symmetries in QFT. The method is based on a differential equation in the continuous symmetry parameter, and gives VEVs as infinite form-factor series which truncate at two-particle level in free QFT. We verify the method by studying U(1) twist fields in free models, which are simply related to the branch-point twist fields. We provide the first exact formulae for the VEVs of such fields in the massive uncompactified free boson model, checking against an independent calculation based on angular quantization. We show that logarithmic terms, overlooked in the original work of Callan and Wilczek [Phys. Lett. B333 (1994)], appear both in the massless and in the massive situations. This implies that, in agreement with numerical form-factor observations by Bianchini and Castro-Alvaredo [Nucl. Phys. B913 (2016)], the standard power-law short-distance behavior is corrected by a logarithmic factor. We discuss how this gives universal formulae for the saturation of entanglement entropy of a single interval in near-critical harmonic chains, including log log corrections.Comment: V2: 37 pages, explications and references adde